to my family · dicted and experimentally observed states of the nucleus. they do, however, give...
TRANSCRIPT
MICROSCOPIC BASES OF COLLECTIVITY IN LIGHT NUCLEI
Marvin I. Friedman 1970
A Dissertation Presented to the Faculty of the Graduate School of Yale University
in Candidacy for the Degree of Doctor of Philosophy
T O M Y F A M I L Y
v
ACKNOWLEDGMENTS
I t i s a g r e a t p le a s u re to acknow ledge my a p p r e c ia t io n
to th e p e o p le who have made t h is d is s e r t a t io n p o s s ib le .
The s t a f f o f th e A tom ic Energy Commission Computer
C e n te r a t th e New Y o rk U n iv e r s i t y C ouran t I n s t i t u t e o f
M ath e m a tics was v e ry h e lp f u l In th e ru n n in g o f com puter
codes .
I w ould l i k e to exp ress my g r a t i t u d e f o r th e h o s p i t a l i t y
shown to me a t T e l - A v iv U n iv e r s i t y , I s r a e l . The s t a f f th e r e
was o f g r e a t a s s is ta n c e in th e p e r fo rm in g o f a n a ly s is c a lc u
la t i o n s . My v i s i t was a most memorable e x p e r ie n c e .
My a s s o c ia t io n w ith th e W rig h t N u c le a r S t r u c tu r e L ab o ra
to r y is g r a t e f u l l y ackn ow led ged . I am in d e b te d to my f e l lo w
g ra d u a te s tu d e n ts and c o lle a g u e s th e r e f o r in t e r e s t in g d is
cu ss io n s d e a lin g w ith a l l a s p e c ts o f n u c le a r p h y s ic s . The
use o f th e com puting f a c i l i t i e s o f th e la b o r a to r y was o f g r e a t
im p o rta n c e to th e c o m p le tio n o f t h is w o rk .
My a s s o c ia t io n w ith P ro fe s s o r I t z h a k K e ls o n , b o th d u r in g
th e tim e he was a t Y a le U n iv e r s i t y , and a t T e l -A v iv U n iv e r s i t y ,
p ro v id e d s e v e r a l u n fo rg e ta b le e x p e r ie n c e s . H is g u id an ce
th ro u g h o u t my g ra d u a te c a re e r is d e e p ly a p p r e c ia te d .
I t i s w ith trem endous p le a s u re t h a t I e x p res s my g r a t i t u d e
to P ro fe s s o r D . A l la n B ro m ley , th e d i r e c t o r o f t h is d is s e r t a
t i o n , f o r h is p a t ie n t g u id an ce and a s s is ta n c e in th e a n a ly s is
p e rfo rm e d h e r e in . Numerous in v a lu a b le d is c u s s io n s c o n c e rn in g
t h is w ork w i l l be lo n g rem em bered.
ABSTRACTIn an a tte m p t to e lu c id a te th e m ic ro s c o p ic bases o f
c o l l e c t i v i t y in l i g h t n u c le i , i n d i v i d u a l - p a r t i c l e model c a lc u la t io n s have been p e rfo rm ed u s in g b o th r e a l i s t i c and n o n - r e a l i s t i c n u c le a r models w ith th e a id o f a n g u la r momentum p r o je c t io n te c h n iq u e s . M a t r ix methods f o r p r o je c t io n fro m s e v e r a l " i n t r i n s i c " s ta te s o f th e n u c leu s a re d isc u s s e d and d e v e lo p e d . In a d d i t io n , a p o ly n o m ia l p r o je c t io n method w hich is e x tre m e ly u s e fu l in c o n s id e r in g system s o f n e u tro n s and p ro to n s is in tro d u c e d . I t a llo w s f o r t r e a tm e n t o f th e H i lb e r t spaces o f th e two n u c leo n s ta te s as s e p a ra te e n t i t i e s , w hich m ay, a t c o n c lu s io n , be combined in t o th e H i lb e r t space o f th e e n t i r e n u c le u s .
The p h en o m en o lo g ica l p a ir in g -p lu s -q u a d r u p o le n u c le o n - n u c le o n in t e r a c t io n is s tu d ie d in ( j ) n n e u tro n c o n f ig u r a t io n s , 7 /2 $ /« 1 5 /2 . The n-body H a m ilto n ia n o p e ra to r is p a ra m e te r iz e d in term s o f i t s tw o-body m a tr ix e le m e n ts , e n a b lin g f u r t h e r c a lc u la t io n s , f o r any g iv e n s c a la r o p e r a to r , to be p e rfo rm e d s im p ly w ith th e s p e c i f ic a t io n o f i t s m a tr ix e le m e n ts . C o m p le te , e x a c t s p e c tra a re o b ta in e d f o r th e ( j ) n c o n f ig u r a t io n s th ro u g h !th e a p p l ic a t io n o f th e m a tr ix p r o je c t io n te c h n iq u e s . E ig e n fu n c t io n s a re a n a ly z e d in s e n io r i t y space w ith p a r t i c u l a r re g a rd to w ard th e e x h ib i t io n o f c o l l e c t i v e p r o p e r t ie s . V a r io u s e x p e r im e n ta l ly ob served phenomena a re a t t r i b u t e d to th e n a tu re o f th e s im p le , b u t b a s ic , phen o m e n o lo g ic a l fo r c e s .
O ne- and t w o - p a r t ic le /h o le e x c i t a t io n s a re c o n s id e re d as a mechanism f o r " s c a t te r in g " n u cleo n s ac ro ss th e f i n i t e en erg y gap a p p e a r in g betw een o ccu p ie d and unoccup ied H a r tr e e -F o c k deform ed o r b i t a l s o f e v e n -e v e n , N=Z n u c le i in th e 24 -Id. s h e l l . A dm ix tu res o f th e s e c o n f ig u r a t io n s to th e H a r tre e -F o c k s t a te a re [found to be m in im a l and in a d e q u a te to acco u n t f o r th e d is c re p a n c ie s betw een t h e o r e t i c a l l y p r e d ic te d and e x p e r im e n ta l ly o b served s ta te s o f th e n u c le u s .They d o , h o w ever, g iv e q u a l i t a t i v e e x p la n a t io n to th e lo w - en erg y e x c i te d n u c le a r c o n f ig u r a t io n s found to l i e w i t h in o r above th e q u a s i - r o t a t io n a l H a r tre e -F o c k s p ec tru m . S hape- m ix in g c a lc u la t io n s a re d isc u s s e d as a means o f c o r r e c t ly p r e d ic t in g p r o p e r t ie s o f e ig e n s ta te s In th e lo w -e n e rg y re g io n o f e x c i t a t i o n . I t is sugg ested t h a t th e o b je c t io n a b le use o f a s in g le dt^oAmzd n u c le a r c o n f ig u r a t io n as in p u t to th e c a lc u la t io n s , r a t h e r th a n as o u tp u t , may be e l im in a te d by v a r i a t i o n o f th e wave fu n c t io n s a f t e r p r o je c t io n , r a th e r th a n b e fo r e .
TABLE.OF CONTENTS
1 . G e n e ra l C o n s id e ra t io n s ................................................................... 1
2 . The D evelopm ent o f N u c le a r M odels
a . D i r e c t io n s ....................................................................................... 4
b . In d iv id u a l P a r t i c le M o d e ls ............................................ 4
c . C o l le c t iv e M o d e ls ....................................................................... 12
d . U n if ie d M o d e ls .............................................................................
3 . A n g u la r Momentum P r o je c t io n ........................................................ 15
4 . M o t iv a t io n f o r S tu d ie s .....................................................................16
5 . Scope o f P re s e n t I n v e s t i g a t i o n s . * ......................................... 18
C h a p te r I I - P r o je c t io n Techn iques
1 . P r e l im in a r y D is c u s s io n .................................................................... 22
2 . A n g u la r Momentum P r o je c t io n by F i n i t e R o ta t io n s . . 27
3 . I n f i n i t e s i m a l R o ta t io n O p e ra to r P r o je c t io n
P rocesses
a . N o r m a l iz a t io n s .............................................................................. 30
b . T e n s o r O p e r a to r s ......................................................................... 34
c . S c a la r O p e r a to rs ..........................................................................37
4 . A N e u tro n -P ro to n P o ly n o m ia l P r o je c t io n T ech n iq u e
a . In t r o d u c t io n ....................................................................................38
b . Is o to p ic S p in C o n s id e r a t io n s ............................................40
c . P r o je c t io n Method - N o r m a l iz a t io n s ............................. 41
d . P r o je c t io n Method - E n e r g ie s ...........................................43
Chapter I - Introduction
1 . In t r o d u c t io n ................................................ 47
2 . P a ir in g + Q uadrupo le M odel
a . In t r o d u c t io n ...................................................................................... 50
b . The Q u ad ru p o le -Q u a d ru p o le F o r c e ......................................... 53
c . The S e n io r i t y Quantum Number and th e P a ir in g
I n t e r a c t i o n ........................................................................................ 57
3 . The H a m ilto n ia n o f th e In d i v i d u a l - P a r t i c l e M o d e l . . . 63
4 . S e n io r i t y C o m p o s itio n o f Wave F u n c t io n s .............................. 65
5 . R e s u lts o f C a lc u la t io n s - Even Number o f P a r t ic le s
a . D is t o r t io n o f R o ta t io n a l Bands a t H igh A n g u la r
Momentum V a lu e s ................................ 66
b . S e n io r i t y A n a ly s is and V ib r a t io n s A r is in g from
th e Long Range I n t e r a c t i o n ................................................... 69
6 . R e s u lts o f C a lc u la t io n s - Odd Number o f P a r t i c l e s . . 73
7 . Summary.............................................................................................................. 77
C h a p te r IV - H a r tre e -F o c k S tu d ie s
1 . In t r o d u c t io n ................................................................................................. 79
2 . H a r tre e -F o c k E q u a tio n s ........................................................................ 85
3 . Sym m etries o f th e H a r tre e -F o c k S o lu t io n s ........................... 90
4 . H a r tre e -F o c k C a lc u la t io n s ................................................................ 93
5 . P a r t ic le - H o le A d m ix tu res to H a r tre e -F o c k
a . P r e l im in a r y D is c u s s io n .............................................................. 102
b . Is o to p ic S p in C o n s id e ra t io n s ...............................................104
c . The Use o f R e fe re n c e N u c le i and H o le S t a t e s . . . . 108
d . P r o je c t io n E q u a tio n s and Wave F u n c tio n
A n a ly s is ................................................................................................. 112
Chapter III - Studies in Single / Configurations
6 . R e s u lts o f C a lc u la t io n s
a . G e n e ra l R em arks............................................................................ 115
b . Neon 2 0 .................................................................................................H g
c . Magnesium 2 4 ....................................................................................119
d . S i l ic o n 2 8 ......................................................................................... 122
e . S u l f u r 3 2 ........................................................................................... 126
f. Argon 36......................................... 1287 . Summary........................................................................................................... 129
C h a p te r V - Summary and C o n c lu s io n s ............................................................ 132
References..................................................... 135
A ppendices
A - I M a th e m a tic a l F o o tn o te s
1 . The Group U3 ....................................................................................141
2 . The M a t r ix E lem ents <jm\y^\jtn> .......................................142
A - I I C o l le c t iv e N ucleon M o tio n s
1 . The N u c le a r S u r fa c e ............................................................. . . 1 4 5
2 . C o u p lin g to th e N u c le a r S u r fa c e ....................................... 148
B - I ( / ) n Com puter Codes
1 . P r o je c t io n C ode.............................................................................153
2 . A n a ly s is C ode..................................................................................I 65
B - I I P a r t i c le H o le Com puter Code
1 . In p u t C ode......................................................................................... 169
2 . PHEXCIT .................................................................................................175
B - I I I G e n e ra l Purpose Computer C odes................................................ 200
1
The te rm " n u c le a r s t r u c tu r e " encompasses a l l m ic ro s c o p ic
a n d /o r m acroscop ic a s p e c ts o f th e m o tio n o f n u c leo n s bound
to g e th e r to fo rm a n u c le u s , e . g . , s p a t ia l p a th s , momentum
d i s t r ib u t io n s , a n g u la r momenta, n u c leo n c o r r e la t io n s , b in d in g
e n e r g ie s , n u c le a r sh ap es , e t c . T h e o r e t ic a l ly a l l o f t h is
In fo r m a t io n is em bodied w i t h in th e t r u e com ple te n u c le a r wave
f u n c t io n s , and may be o b ta in e d by a p p l ic a t io n o f th e a p p r o p r i
a te quantum m e c h a n ic a l o p e r a to r s . Wave fu n c t io n s f o r th e
n u c le a r m any-body system may be w r i t t e n in a v a r ie t y o f w ays;
th e most u s e fu l o f th e s e em phasizes th e p h y s ic a l ly s ig n i f i c a n t
a s p e c ts o f th e p a r t i c u l a r n u c le a r system u n d e rg o in g in v e s t ig a
t i o n . F o r e xam p le , th e p resen ce o f c o r r e la te d m otio ns in
some n u c le i may make i t advantageous to choose a d e s c r ip t io n
w hich a c c e n tu a te s e i t h e r c o l l e c t i v e o r p a ir e d -n u c le o n degrees
o f free d o m ; in o th e r cases th e m o tio n o f c e r t a in in d iv id u a l
n u c leo n s may be em phasized because many o f th e o b s e rv a b le
phenomena a p p e a r to be a t t r i b u t a b l e to o n ly a few r e l a t i v e l y
a c t iv e p a r t i c l e s .
A t p re s e n t I t is im p o s s ib le to d e s c r ib e th e b e h a v io r
o f a l l o f th e degrees o f freedom o f th e in d iv id u a l nu c leo ns
c o m p ris in g a dynam ic n u c le u s . T h is is t r u e p a r t l y because
no c lo s e d t h e o r e t ic a l e x p re s s io n has y e t been found f o r th e
n u c le o n -n u c le o n fo r c e . Even I f such an e x p re s s io n w ere known
CHAPTER I
INTRODUCTION
1. General Considerations
f o r e x t r a - n u c le a r p a r t i c l e s , h o w ever, i t w ould have to be
m o d if ie d in s id e th e n u c leu s where th e p resen ce o f o th e r
p a r t ic le s s e v e r e ly l i m i t s th e p o s s ib le f i n a l s ta te s to w h ich
a n u c leo n may s c a t t e r as a r e s u l t o f i t s in t e r a c t io n s .
T h is p ro b lem has been bypassed somewhat by th e em ploy
ment c f e f f e c t i v e in t e r a c t io n s . We assume t h a t we a re g iv e n
a H a m ilto n ia n H = H + V , w itho
h0 ■ i h0(i> - 1 [T(1) + v i ) ]° 1=1 ° i = l °
V = I U ( i , j ) - I V ( i ) . i < j i = l 0
T ( J ) i s th e k in e t i c en erg y o f th e p a r t i c l e ; U ( i , j ) th e
tw o -b ody in t e r a c t io n betw een th e i th and p a r t i c l e s ; V Q( j )
some c o n v e n ie n t one-body o p e ra to r w h ich s im p l i f ie s th e p ro b lem .
The H a m ilto n ia n H is e x a c t and c o n s is ts o f two p a r ts : Hq , a
one-body o p e r a to r , and V , th e " e f f e c t iv e in t e r a c t io n " . I t i s
hoped t h a t by p ro p er, c h o ic e o f VQ( J ) , th e e f f e c t i v e in t e r a c
t io n may be t r e a t e d by p e r t u r b a t iv e te c h n iq u e s , i . e . , i t s
e f f e c t i s s m a ll com pared to t h a t o f Hq . The e f f e c t i v e i n t e r
a c t io n may have l i t t l e to do w ith th e a c tu a l in t e r a c t io n
betw een n u c leo n s w i t h in th e n u c le u s . P a s t e f f o r t s have been
a lo n g fo u r d i s t i n c t l i n e s : ( i ) The e f f e c t i v e in t e r a c t io n is
assumed to ta k e some s im p le and re a s o n a b le fo rm . S tr e n g th s ,
ran g es and m ix tu re c o n s ta n ts a re c o n s id e re d p a ra m e te rs w h ich
a re o b ta in e d by f i t s to e x p e r im e n ta l d a t a . 1 ( i i ) The i n t e r
a c t io n m a tr ix e le m en ts th em selves a re c o n s id e re d d i r e c t l y as
3
a d ju s ta b le p a ra m e te rs w ith o u t s p e c ify in g t h e i r a lg e b r a ic
fo rm s . 2 ( i i i ) M a t r ix e lem en ts a re d e te rm in e d d i r e c t l y from
th e e x p e r im e n ta l s p e c tra o f p a r t i c u l a r l y s im p le n u c le i .
( i v ) The e f f e c t i v e in t e r a c t io n is d e r iv e d from f r e e n u c le o n -
n u c le o n s c a t t e r in g p o te n t ia ls a d ju s te d to f i t d e u te ro n
p r o p e r t ie s and s c a t t e r in g d a ta up to a p p ro x im a te ly 350 MeV.
E f f e c t iv e m a tr ix e lem en ts a re c a lc u la te d u s in g re a s o n a b le
assum ptions a p p l ic a b le to th e n u c le a r m any-body p ro b le m . 3""6
I n a c t u a l i t y , th e fo rm a l s o lu t io n o f th e p rob lem w ould
n o t c o n t r ib u te much to w ard th e u n d e rs ta n d in g o f n u c le a r
s t r u c t u r e ; th e com ple te n u c le a r wave fu n c t io n s a re to o com
p l ic a t e d to p ro v id e any s im p le p h y s ic a l p ic t u r e o f th e n u c le u s .
A d e s c r ip t io n in term s o f fe w e r p a ra m e te rs must be found so
t h a t , h o p e f u l ly , n u c le i can be u n d ers to o d by th e human m ind
as w e l l as by e le c t r o n ic co m p u ters . T h is le a d s to th e concept
o f th e n u c le a r m o d e l, w h e re in c e r t a in a s p e c ts o f th e a c tu a l
pro b lem a re em phasized to th e e x c lu s io n o f o th e r s . T h is is
done to p ro v id e a m a th e m a tic a lly t r a c t a b le re p ro d u c t io n o f
c e r t a in o b s e rv a b le phenom ena, n o t s i g n i f i c a n t l y dependent
upon th e ig n o re d p a ra m e te rs o r n u c le a r degrees o f freed o m .
In e s s e n c e , th e n u c le a r m odel re p re s e n ts a c h a r ic a tu r e o f th e
com ple te n u c le a r p ro b le m . On th e b a s is o f an a c c e p ta b le
n u c le a r m odel i t sh o u ld be p o s s ib le to p r e d ic t most o f th e
im p o rta n t p r o p e r t ie s o f s p e c i f ic n u c le i u n d er s tu d y . In d e e d ,
th e u l t im a t e v a l i d i t y o f th e m odel is m easured by th e e x te n t
to w h ich i t s p r e d ic t io n s a g re e w ith subsequent o b s e rv a b le s
found by e x p e r im e n t.
2 . The D evelopm ent o f N u c le a r M odels
a . D ir e c t io n s
The deve lopm ent o f n u c le a r m odels has been in two
seem in g ly opposing d i r e c t io n s , w ith a p p a re n t ly c o n t r a d ic to r y
a ssu m p tio n s . C o l le c t iv e m odels ( e . g . , th e l i q u i d drop m odel
f i r s t sug g ested by Bohr and W L e e le r7 ) p ic t u r e th e n u c leu s as
a c o n tin u o u s drop o f in c o m p re s s ib le n u c le a r m a t te r , th e shape8
o f w h ich may d e p a r t a p p r e c ia b ly from s p h e r i c i t y . . F o r a
s p h e r ic a l n u c le u s c o l l e c t i v e m otions co rresp o n d to s u r fa c e
v ib r a t io n s abou t th e e q u i l ib r iu m shape; th e m o tio n o f a
s p h e r o id a l n u c le u s , on th e o th e r h an d , can be c o n s id e re d as
a c o m b in a tio n o f r o t a t io n s o f a s t a t i c a l l y deform ed n u c le a r
co re and s u r fa c e v ib r a t io n s abou t an e q u i l ib r iu m d e fo rm a tio n .
A t th e o th e r ex trem e a re th e in d iv id u a l p a r t i c l e m o d e ls ,
a c c o rd in g to w h ich th e m o tio n o f any one g iv e n n u c leo n is
in f lu e n c e d d i r e c t l y by a l l o th e r n u c leo n s c o m p ris in g th e
n u c le u s . These m odels a re n o t e n t i r e l y In d e p e n d e n t; th e y
com plem ent each o th e r . In d iv id u a l p a r t i c l e m odels may be
co u p led to m odels o f c o l l e c t i v e m o tio n to produce th e s o -
c a l le d u n -t^ e d m o d e ls , 1 w h ich have le d to an e x p la n a t io n o f
a v a s t amount o f n u c le a r d a ta . In th e f i n a l a n a ly s is , a l l
o f th e s e re p re s e n t d i f f e r e n t a p p ro x im a tio n s to a th e o ry w hich
does n o t y e t e x i s t . The s tu d y o f th e r e l a t i o n betw een th e
m odels is t h e r e fo r e e s p e c ia l ly im p o r ta n t .
b . In d iv id u a l P a r t i c le M odels
In th e 1930s I t was d is c o v e re d t h a t n u c le i w ith c e r t a in
4
p ro to n numbers Z , o r n e u tro n numbers N , known as "m agic
nu m b ers", e x h ib i t enhanced s t a b i l i t y , w h ich may be in t e r p r e t e d
as a r is in g fro m th e c lo s in g o f n u c le a r s h e l l s . In p a r t i c u l a r ,
th e f i r s t p ro to n o r n e u tro n beyond such a m agic number is le s s
bound th a n th e p re c e d in g nu cleo ns by 1 -2 MeV; th e c r o s s -s e c t io n
f o r c o rre s p o n d in g n u c leo n c a p tu re is r a th e r low f o r th e m agic
n u c le i , in d ic a t in g an anomalous d eg ree o f s t a b i l i t y ; th e e le c
t r i c q u ad ru p o le moment is v e ry s m a ll and in c re a s e s in m agni
tu d e w ith th e a d d i t io n o f a n o th e r n u c le o n , in d ic a t in g th e •
em ergence o f s p h e r ic a l symmetry f o r th e m agic n u c le i . C le a r ly ,
s in c e c o l l e c t i v e m odels c o n ta in v e ry l i t t l e re fe r e n c e to th e
number o f n u c leo n s p r e s e n t , th e e x p la n a t io n o f th e s e phenomena
must be sought w i t h in th e fram ew ork o f in d iv id u a l p a r t i c l e
m o d e ls .
The e x p e r im e n ta l ly o b served d is c o n t in u i t ie s n o te d above
le d to th e in t r o d u c t io n o f th e s h e l l m odel o f n u c le a r s t r u c
t u r e , In d e p e n d e n t ly , by M ayer9 and by H a x e l t t a l . xo The
assu m p tio n o f an u n d e r ly in g n u c le a r s h e l l s t r u c tu r e makes i t
o f i n t e r e s t to exam ine th e o r ie s o f n u c le a r p h y s ic s w h ich a re
p a r a l l e l to th o s e o f a to m ic p h y s ic s . I t w ould be te m p tin g
to d e s c r ib e th e e f f e c t i v e fo rc e a c t in g on each n u c leo n by
means o f a c e n t r a l p o t e n t i a l V ( / l ) , an a lag o u s to th e in t e r a c
t io n o f a to m ic e le c t r o n s w ith th e c e n t r a l e l e c t r i c f i e l d
g e n e ra te d by th e n u c leu s and in n e r e le c t r o n s . H ow ever, a p a r t
fro m th e above m e n tio n ed f a c t t h a t th e n u c le a r fo rc e is un
known, a f u r t h e r p ro b lem im m e d ia te ly a r is e s — th e re is no
n a t u r a l c e n t e r - o f - f o r c e f o r th e n u c le o n s . In l i e u o f t h is
5
6
c o n c e p t, i t i s assumed t h a t th e n u c le a r p o t e n t i a l re p re s e n ts
an a v era g e o v e r n u c leo n m o tio n s . A c c o rd in g ly , th e c e n t e r - o f -
fo r c e does n o t have any s p e c ia l p h y s ic a l s ig n i f ic a n c e , b u t
m e re ly c o in c id e s w ith th e c e n te r -o f-m a s s o f th e n u c le u s .
T h u s , as a f i r s t a p p ro x im a tio n , th e fo rc e b in d in g each
n u c leo n to th e c e n te r -o f-m a s s o f th e n u c leu s is assumed cen
t r a l . I t seems p la u s ib le to assume t h a t th e a v e ra g e fo r c e
a c t in g on a n u c leo n lo c a te d a t th e c e n te r o f th e n u c leu s must
v a n is h . T h is im p lie s t h a t th e p o t e n t ia l I s f l a t in t h is
r e g io n . A p o t e n t i a l w h ich s a t i s f i e s th e s e re q u ire m e n ts and
i s m a th e m a t ic a lly s im p le to w ork w ith is t h a t o f th e is o
t r o p ic harm on ic o s c i l l a t o r
V (; t ) = -V + A Mw2* 2 ,O 2
where VQ is th e p o t e n t i a l d e p th , M th e mass o f th e o s c i l l a t i n g
p a r t i c l e , w th e c la s s ic a l a n g u la r fre q u e n c y o f o s c i l l a t i o n ,
and A th e d is ta n c e o f th e p a r t i c l e from th e c e n te r o f o s c i l l a - +
t i o n . I f th e harm onic o s c i l l a t o r p o t e n t i a l is to be o f u s e ,
i t must in some way be a b le to rep ro d u ce th e m agic num bers.
U n fo r tu n a te ly , h o w ever, I t g e n e ra te s a s i n g l e - p a r t i c l e spectru m
w ith en erg y gaps a t p a r t i c l e numbers w hich a g ree w ith o n ly th e
f i r s t th r e e m agic num bers.
^ I t must be borne in m in d , h o w ever, t h a t th e harm onic o s c i l l a t o r p o t e n t i a l cannot be r e a l i s t i c f o r o th e r th a n r a th e r s t r o n g ly bound s ta te s inasm uch as i t cannot rep ro d u ce e i t h e r an unbound s ta te o r th e b e h a v io r o f n u c le a r wave fu n c t io n s o f s l i g h t l y bound s t a t e s , p a r t i c u l a r l y in th e re g io n o f th e n u c le a r s u r fa c e .
7
The o b s e rv a t io n o f s p in - o r b i t d o u b le ts , p a r t i c u l a r l y5 5 17 17th o s e in He and L i , and l a t e r in 0 and F , sugg ested
th e s tro n g c o u p lin g o f th e i n t r i n s i c s p in a n g u la r momentum
s o f th e n u c leo n to th e o th e rw is e f ix e d o r b i t a l a n g u la r
momentum Z o f th e n u c leo n v ia th e s p in - o r b i t p o t e n t ia l
V = -V U ) L s , w here Is a c e n t r a l s c a la r p o t e n t ia lo U S O S O
d epend ing on th e d is ta n c e fi o f th e n u c leo n from th e c e n t e r -
o f-m ass o f th e n u c le u s . T h is in tro d u c e s an a d d i t io n a l quantum
number 7 > th e t o t a l a n g u la r momentum o f th e n u c le o n , g iv e n by
7 “ Z + s , w h ich has th e c o n s ta n t m agnitude / / ( / + ) ) . The
s p in - o r b i t p o t e n t i a l does n o t m ix s ta te s o f d i f f e r e n t I s in c e
Z2 commutes w ith Z ' S ; th e p r o je c t io n o f Z o n to th e a x is o f
q u a n t iz a t io n , h o w ever, is no lo n g e r a c o n s ta n t o f th e m o tio n ,
and o n ly 7 has a f ix e d component on th e q u a n t iz a t io n a x is .
S in ce th e i n t r i n s i c s p in o f a n u c leo n is 1 /2 , th e r e a re
o n ly two p o s s ib le v a lu e s o f th e t o t a l a n g u la r momentum / f o r
a g iv e n l - - j s I *—. The en erg y o f a n u c leo n h a v in g a n g u la r2momentum j w i l l depend on w hich o f th e s e two o r ie n t a t io n s o f
/ th e n u c le o n assum es. S in ce th e e ig e n v a lu e s o f 2 *s in s ta te s
o f / » l + — and j - L - — a re £ / 2 and - ( £ + / ) / 2 , r e s p e c t iv e ly , i t 2 2
may im m e d ia te ly be seen t h a t th e en erg y d i f fe r e n c e betw een
th e s e s ta te s is p r o p o r t io n a l to 2 Z + I; w ith V ( * ) p o s i t i v e ,s o
th e s ta te w ith h ig h e r / l i e s lo w e r In e n e rg y .
I f th e c e n t r a l s c a la r p o t e n t ia l b in d in g n u cleo n s to th e
c e n te r -o f-m a s s is chosen to be t h a t o f a harm onic o s c i l l a t o r ,
w h ic h , as n o te d ab o ve , is f r e q u e n t ly th e case , th e quantum
* The shape o f th e a c t u a l p o t e n t i a l w e l l l i e s somewhere betw een
8
numbers f o r a s in g le p a r t i c l e o r b i t a re th e s e t ( n , £ , / ) , n
b e in g th e o s c i l l a t o r quantum num ber. The group o f s ta te s
c o rre s p o n d in g to th e quantum number N + l (N = 2n+1) i s b ro ken
up by th e s p in - o r b i t i n t e r a c t io n , and th e s t a te w ith th e
h ig h e s t s in g le p a r t i c l e a n g u la r momentum is lo w e re d to w ard
th e group o f o s c i l l a t o r s ta te s w ith quantum number N . Some
s in g l e - p a r t i c l e o r b i t s fro m t h is group a re r a is e d as a con
sequence o f th e s p in - o r b i t in t e r a c t io n , r e s u l t in g In a re g io n
c o n ta in in g s ta te s from th e groups w ith quantum numbers N and
N + l. S in g le - p a r t i c le o r b i t s w h ich l i e c lo s e in en erg y com prise
a s in g le m a jo r s h e l l ; when a l l s in g l e - p a r t i c l e o r b i t s in a
m a jo r s h e l l a re f i l l e d , we speak o f a c lo s e d s h e l l . The s p in -
o r b i t fo rc e produces en erg y gaps betw een c lo s e d s h e l ls w h ich
c o in c id e w ith th e o b served m agic num bers.
S in ce th e r e is o n ly one t o t a l l y a n tis y m m e tr ic quantum
m e c h a n ic a l s t a te f o r a c lo s e d s h e l l , m agic n u c le i a re n eces
s a r i l y e x c i te d by m oving p a r t ic le s in t o h ig h e r s h e l l s . How
e v e r , c o n s id e ra b le amounts o f energy a re needed to s c a t t e r
ac ro s s th e s in g le p a r t i c l e en erg y g a p s , whence th e e x t r a
s t a b i l i t y a s s o c ia te d w ith m agic num bers. The mean f r e e p a th ,
i . e . , th e d is ta n c e a n u c leo n can t r a v e l in s id e th e n u c leu s
w ith o u t u n d erg o in g c o l l is io n s w ith o th e r n u c le o n s , i s p r e d ic te d
to be h ig h because o f th e P a u l i p r in c ip le w h ich in h i b i t s c o l -
an o s c i l l a t o r p o t e n t ia l and a square w e l l w ith a t a i l . Harm onic o s c i l l a t o r s ta te s a re used m e re ly as a m a tte r o f m a th e m a tic a l c o n v e n ie n c e . C o n s id e ra b le work has been acco m p lish ed u s in g o th e r form s f o r th e c e n t r a l p o t e n t i a l , e . g . , th e Woods-Saxon p o t e n t i a l , e s p e c ia l ly in th e a n a ly s is o f th e m u lt ip o le and t r a n s i t i o n p r o p e r t ie s o f e x c ite d s t a t e s , w h ich a re more sens i t i v e to th e d e t a i l s o f th e p o t e n t ia l th a n th e en erg y spectrum i s .
9
l i s io n s i f th e quantum s ta te s to w h ich th e c o l l id i n g fe rm io n s
can s c a t t e r a re a lr e a d y f i l l e d . T h is is o n ly t r u e because
n u c le a r e n e rg ie s a re e x tre m e ly lo w . T y p ic a l n u c leo n e n e rg ie s
i n bound s ta te s a re to o low to p e rm it c o l l is io n s w h e re in th e
p ro d u c ts a re prom oted to p r e v io u s ly open o r b i t s .
One o f th e e a r l i e s t a tte m p ts to em ploy th e s h e l l m odel
to u n d e rs ta n d th e s y s te m a tic b e h a v io r o f n u c le i in v o lv e d th e
E xtrem e S in g le P a r t i c le M odel (ESPM ), a c c o rd in g to w hich
in d iv id u a l n u c leo n s a re c o n s id e re d to move in s ta t io n a r y s h e l l
m odel o r b i t s w ith f ix e d a n g u la r momentum / . I t I s assumed
t h a t n e u tro n and p ro to n s ta te s f i l l in o r d e r , in d e p e n d e n t ly ,
i . e . , th e s t a te to be f i l l e d by an a d d i t io n a l n e u tro n Is
in d e p e n d e n t o f th e number o f p ro to n s , and v ic e v e rs a . The
n u c leo n s a re th e n p a ir e d o f f in such a way t h a t th e v a lu e s
o f many n u c le a r p a ra m e te rs a re d e te rm in e d s o le ly by any
rem nant s in g le u n p a ire d n u c le o n . T h u s , a c c o rd in g to t h is
m o d e l, a l l e v e n -e v e n n u c le i have z e ro ground s ta te s p in ,
w hich a g re es w ith e x p e r im e n t; odd-A n u c le i have th e s p in o f
th e l a s t u n p a ire d p a r t i c l e , w h ich is g e n e r a l ly t r u e ; odd-odd
s p in s can n o t be p r e d ic te d d i r e c t l y s in c e th e re is n o th in g to
in d ic a t e w h ich r e s u l t a n t c o u p lin g o f th e u n p a ire d n e u tro n and
p ro to n a n g u la r momenta w i l l have th e lo w e s t e n e rg y .^ C o l le c
t i v e m otio ns o f many n u cleo n s a r e , o f c o u rs e , n o t p o s s ib le
w it h in th e m odel fram ew o rk .
*f* The "N ordheim r u le s " a re g e n e r a l ly s u c c e s s fu l In p r e d ic t io n s o f s p in s o f odd-odd n u c le i . The Nordheim number is d e f in e d as N = j - I + j - I , where p and n r e f e r to th e odd p ro to n and n e u t r 8n , pr e § p e c t iv e ly . A cco rd in g to th e r u l e , i f N =0, J = l / n -J p l> and i f N = ± l, J i s e i t h e r T / n- J p l o r Jn+ Jp *
10
The m ag n e tic moment o f odd-A n u c le i is a t t r i b u t e d to b o th
th e o r b i t a l and s p in a n g u la r momenta o f th e la s t u n p a ire d
n u c le o n . T h is g iv e s r i s e to two v a lu e s f o r th e m ag n e tic moment
o f th e n u c le u s , one c o rre s p o n d in g to s ta te s in w h ich th e s p in
and o r b i t a l a n g u la r momenta a re p a r a l l e l (/= •£+—) , and th e2
o th e r to s ta te s in w h ich th e y a re a n t i p a r a l l e l ( / = £ - —) . P lo ts2
g iv in g th e m ag n e tic moment as a fu n c t io n o f / a re known as
Schm idt l i n e s 1 1 . E x p e r im e n ta l ly , I t is o b served t h a t a lth o u g h
th e m easured m ag n e tic moments do n o t show such s im p le s y s te
m a t ic s , th o s e f o r odd mass n u c le i f a l l ro u g h ly in t o two
g ro u p s , one n e a r each o f th e Schm idt l i n e s ; e s s e n t ia l ly a l l
odd mass n u c le i have m ag n e tic moments betw een th e two Schm idt
l i m i t s .
The m odel p r e d ic ts e x c i te d s ta te s w h ich d u p l ic a te th e
s i n g l e - p a r t i c l e s p e c tru m , i . e . , th e u n p a ire d n u c le o n , in s te a d
o f o ccu p y in g th e lo w e s t s in g l e - p a r t i c l e s t a te a v a i l a b le , may
be e x c i te d to a h ig h e r o r b i t w i t h in th e m a jo r s h e l l . T h is
i s o b served e x p e r im e n ta l ly in th e lo w - ly in g s p e c tra o f n e a r
c lo s e d s h e l l n u c le i . The s p in s and p a r i t i e s o f th e s e e x p e r i
m e n ta lly o b served le v e ls a g re e v e ry w e l l w ith th e s h e l l m odel
a s s ig n m e n ts .
A c c o rd in g to th e P a u l i p r i n c i p l e , th e wave fu n c t io n o f
i d e n t i c a l n u c leo n s o u ts id e o f a c lo s e d s h e l l must be t o t a l l y
a n t is y m m e tr ic . O b v io u s ly , any more q u a n t i t a t iv e n u c le a r
m odel must ta k e in t o acco u n t n o t o n ly th e u n p a ire d n u c le o n s ,
as does th e ESPM, b u t r a th e r th e s e v e r a l n u c leo n s o u ts id e o f
th e c lo s e d s h e l l . T h u s , th e S in g le P a r t i c le M odel (SPM)
11
a t t r i b u t e s th e lo w -e n e rg y p r o p e r t ie s o f n u c le i to a few
" a c t iv e " p a r t ic le s o u ts id e o f th e c lo s e d , t i g h t l y bound,
" in e r t " co re o f th e s h e l l m o d e l. I t is th e SPM w hich is
commonly r e f e r r e d to in th e l i t e r a t u r e as th e s h e l l m ode l.
T h a t th e n u c leu s cannot be a c c u ra te ly d e s c r ib e d s o le ly
in te_-ms o f In d e p e n d e n t p a r t ic le s m oving in s h e l l m odel
e ig e n - o r b i t s seems r a t h e r c le a r . As d is c u s s e d e a r l i e r , th e
a c tu a l n u c le o n -n u c le o n in t e r a c t io n w i t h in th e n u c leu s is
c o n s id e re d to c o n s is t o f two p a r t s . The f i r s t g iv e s r i s e to
th e s h e l l m odel and re p re s e n ts th e a v erag e e f f e c t on a s in g le
n u c leo n produced by a l l o th e r p a r t ic le s in th e n u c le u s . The
seco n d , known as th e " e f f e c t iv e in t e r a c t io n " is re s p o n s ib le
f o r th e rem o va l o f th e h ig h o r ie n t a t io n a l degeneracy o f
nu cleo n s w ith th e same s e t o f quantum numbers ( n ,< £ , / ) . I t
i s a p p l ie d o n ly betw een p a ir s o f a c t iv e p a r t i c l e s ; i t must
be s tro n g enough to l i f t th e degeneracy and y e t n o t so s tro n g
t h a t j ceases to be a good quantum num ber.
V a r io u s a lg e b r a ic form s have been used f o r th e c e n t r a l
p a r t o f th e e f f e c t i v e in t e r a c t io n in m odel c a lc u la t io n s .
Some o f th e s e in c lu d e :
V - { - Vo K < K n (s q u a re w e l l )0 K>fin
V = -V c e x p ( - * 2A 2 ) (G a u s s ia n )
V = _ V 0 e x p ( - * A n )
* A „(Y u k a w a ).
The 6- i n t e r a c t i o n , s u r fa c e d e l t a i n t e r a c t io n , p a i r in g and
q u ad ru p o le fo rc e s have a ls o been e x te n s iv e ly em ployed in more
s o p h is t ic a te d s tu d ie s . These re p re s e n t a v a r ie t y o f shapes
and ra n g e s , and y e t g iv e q u ite s im i la r r e s u l t s . T h is len d s
s u p p o rt to th e b e l i e f t h a t a t r u e u n d e rs ta n d in g o f n u c le a r
s t r u c tu r e does n o t depend on th e know ledge o f th e e x a c t
f u n c t io n a l fo rm o f th e In t e r a c t io n , b u t r a th e r on i t s g e n e ra l
p r o p e r t ie s o n ly .
c . C o l le c t iv e Models
The fo r m u la t io n o f th e s h e l l m odel d isc u s s e d up to t h is
p o in t assumes a s p a t i a l l y is o t r o p ic p o t e n t i a l . H ow ever, i t
has been found t h a t n u c le i in th e mass re g io n s A^25> 150<A <190,
and A<222 have la r g e s t a t i c d e fo rm a tio n s — th e y e x h ib i t la r g e
e l e c t r i c q u ad ru p o le moments and n o n -s p h e r ic a l shapes . I f th e
s im p le s h e l l m odel wave fu n c t io n s a re used to c a lc u la te m a tr ix
e le m en ts o f th e q u ad ru p o le moment, m ag n e tic moment, t r a n s i t io n
p r o b a b i l i t i e s , e t c . in th e s e r e g io n s , th e r e s u l t s d is a g re e
w ith e x p e r im e n t, o f te n by a f a c t o r o f as much as te n o r m o re . 12
A n a ly s is o f th e s p e c tra o f th e s e n u c le i in d ic a te s t h a t
th e shape o f th e n u c leu s may g e n e r a l ly be assumed to be n o n -
s p h e r ic a l , b u t a x i a l l y s y m m e tric . These n o n -s p h e r ic a l shapes
may be e n v is io n e d as a r is in g as a r e s u l t o f th e opposing te n
d e n c ie s o f th e e x t r a -c o r e nu cleo ns to p o la r iz e th e n u c le a r
co re ( t h is is due to th e lo n g range p a r t o f th e n u c le o n -n u c le o n
i n t e r a c t i o n ) , and o f th e co re i t s e l f to r e s i s t th e s e fo rc e s
and to m a in ta in s p h e r ic i t y (due to th e s h o r t range p a r t ) . 8
12
The q u ad ru p o le moment and t r a n s i t io n a m p litu d e s a r e , th e r e
f o r e , th e r e s u l t o f c o n t r ib u t io n s n o t o n ly from a few p a r t i c l e s ,
b u t r a th e r fro m th e e n t i r e n u c le u s . The d e t a i ls o f t h is m odel
w ere f i r s t d eve lo p ed by Bohr and M o tte ls o n 13 in th e 19 50 s .
The n u c leu s in t h is m odel is c o n s id e re d as a co re and e x t r a
co re n u c le o n s . The c o r e , f o r n u c le i f a r from m agic num bers,
i s n o t t h a t o f th e s h e l l m odel— i t c o n ta in s many more n u c le o n s .
The co re is t r e a t e d m a c ro s c o p ic a lly as a deform ed drop o f
n u c le a r m a tte r w h ich in t e r a c t s w ith th e e x t r a -c o r e n u cleo n s
on a m ic ro s c o p ic b a s is ( v . A ppend ix A - I I ) .
d . U n i f ie d Models
T h ere is no a pn.i.onL reaso n why th e e f f e c t i v e i n t e r
a c t io n sh o u ld n o t m ix s ta te s w ith d i f f e r e n t v a lu e s ( n , £ , / ) .
In d e e d , much p ro g re s s has been made to w ard u n d e rs ta n d in g
n u c le a r s t r u c tu r e by r e la x in g t h is assu m p tio n . The n u c le a r
p o t e n t i a l i s no lo n g e r assumed s p h e r ic a l , and th e d is t o r t io n
w h ich y ie ld s minimum en erg y is ta k e n as th e e q u i l ib r iu m de
fo r m a t io n . Two s p e c i f ic m odels f a l l i n g in t o t h is c a te g o ry
a re th e N ils s o n and H a r tre e -F o c k m odels .
S .G . N i ls s o n 14 c a lc u la te d s in g le p a r t i c l e s ta te s o f an
a x i a l l y sym m etric s p h e ro id a l harm onic o s c i l l a t o r p o t e n t i a l ,
c o n s id e r in g a d i s t o r t i o n o f th e p o t e n t ia l up to th e q u ad ru
p o le te rm o n ly , and in c lu d e d a f l a t t e n i n g e f f e c t f o r h ig h
v a lu e s o f o r b i t a l a n g u la r momentum. The wave fu n c t io n s w ere\expanded on th e s e t o f Is o t r o p ic o s c i l l a t o r fu n c t io n s . T h is
le d to th e s u c c e s s fu l c o r r e la t io n o f a la r g e f r a c t i o n o f th e
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n u c le a r d a ta known a t t h a t t im e . 15 N ew ton16 ex ten d ed t h is
w ork to a c o n s id e ra t io n o f n o n - a x ia l ly sym m etric n u c le i , and
more r e c e n t ly B. N i ls s o n 17 in c lu d e d h exad ecu p o le d is t o r t io n s
w hich a re in d ic a te d e m p ir ic a l ly fro m a n a ly s is o f in e l a s t i c
a s c a t t e r in g d a t a 1 8 * 1 9 .
In th e H a r tre e -F o c k m o d e l, f i r s t a p p l ie d to n u c le a r
c a lc u la t io n s by K e ls o n 2 0 , each n u c le o n is assumed to move in
a deform ed p o t e n t i a l w e l l a r is in g s e l f - c o n s is t e n t ly fro m i t s
tw o -b ody in t e r a c t io n s w ith a l l o th e r n u c le o n s , w ith th e
r e s t r i c t i o n t h a t th e r e e x is t some " m a th e m a tic a l" s t a te ( th e
H a r tre e -F o c k s t a t e ) f o r w h ich th e H a m ilto n ia n o f th e system
i s a minimum. A deform ed s i n g le - p a r t i c le b a s is , w h ich may
be expanded on th e u s u a l s h e l l m odel b a s is , is g e n e ra te d .
The m ain d i f f e r e n c e betw een th e s e two m odels Is th e em ploy
ment o f n o n - lo c a l p o t e n t ia ls in th e H a r tre e -F o c k m o d e l. T h is
a llo w s f o r th e tre a tm e n t o f th e n u c leu s as a system o f non
in t e r a c t in g p a r t i c le s in th e deform ed b a s is , w ith th e a t te n d
a n t s im p le S la t e r d e te rm in a n ta l wave fu n c t io n s .
N u c le i in th e f i r s t h a l f o f th e 2 6 - 1 d s h e l l ap p e a r to
d is p la y r o t a t i o n a l p r o p e r t ie s c lo s e ly r e la t e d to deform ed
shapes o f th e n u c le u s . H a r tre e -F o c k c a lc u la t io n s f o r th e
ev en -e v e n N=Z n u c le i in t h is s h e l l g e n e ra te q u a s i - r o t a t io n a l
band s t r u c tu r e s whose sp ac in g s a re s y s te m a t ic a l ly s m a lle r
th a n th o se o f th e e x p e r im e n ta l s p e c t r a . As a consequence o f
u s in g n o n - lo c a l p o t e n t i a ls , an energy gap a r is e s in th e
deform ed H a r tre e -F o c k p a r t i c l e s p ec tru m . I f t h is gap is
la r g e H a r tre e -F o c k th e o ry sh o u ld re p re s e n t a good ap p ro x im a
t io n to th e t r u e s i t u a t io n . I f , on th e o th e r h an d , t h is gap
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i s s m a l l , i t is e x p e c te d t h a t p a r t ic le s w ould be a b le to
s c a t t e r ac ro ss i t , and th e r e fo r e t h a t n - p a r t ic le / n - h o l e s ta te s
w ould m ix a p p re c ia b ly w ith th e q u a s i - r o t a t io n a l band p r o je c te d
fro m th e H a r tre e -F o c k s t a t e . T y p ic a l en erg y gaps f o r th e
26 - 1 d s h e l l n u c le i a re c a lc u la te d to be o f th e o rd e r 6 - 8 MeV,
as compared to gaps o f th e o rd e r 18 MeV in th e 1p s h e l l .
P a r t ic le - h o l e s ta te s sh o u ld th e r e fo r e re p re s e n t v a l id adm ix
tu r e s to th e H a r tre e -F o c k s ta te in th e 2 6 - 1 d s h e l l .
3 . A n g u la r Momentum P r o je c t io n
The o r ig in a l p rob lem fa c in g n u c le a r s t r u c tu r e th e o r is t s
was to e x p la in th e p r o p e r t ie s o f n u c le i as a consequence o f
th e p resen ce o f A n u c le o n s . T h is fo rm id a b le p rob lem le d to
th e developm ent o f th e m odels d isc u s s e d a b o v e , w ith th e r e s u l t
t h a t i t i s no lo n g e r n e c e ss a ry to work in th e space o f A
n u c le o n s , b u t r a t h e r o n ly in th e space o f th e a c t iv e n u cleo n s
o u ts id e o f an i n e r t c o r e .+ But even t h is becomes to o p r o d i
g io u s a p ro b lem f o r more th a n a few a c t iv e nu cleo ns s in c e th e
number o f m u l t i - p a r t i c l e s ta te s in c re a s e s v e ry r a p id ly w ith
e i t h e r th e number o f p a r t ic le s o r o s c i l l a t o r quantum num ber.
C o n v e n t io n a lly , th e a n g u la r momentum s t r u c tu r e o f a
com posite system is s tu d ie d by v e c to r c o u p lin g th e a n g u la r
momenta o f th e in d iv id u a l members. F o r m u l t i - p a r t i c l e system s
th e c o n s tr u c t io n o f e ig e n fu n c t io n s o f th e a n g u la r momentum
o p e ra to r is r a t h e r d i f f i c u l t . In d e e d , th e method becomes
+I t som etim es becomes n e cessary to c o n s id e r v a r io u s e x c i
t a t io n modes o f th e co re a n d /o r in t e r a c t io n s o f th e core w ith th e e x t r a -c o r e p a r t i c l e s .
15
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c o m p lic a te d even f o r th re e p a r t i c l e s , s in c e th e r e is no un ique
way o f c a r r y in g o u t th e c o u p lin g . T h is p rob lem has been in v e s
t ig a t e d by s e v e r a l a u th o rs ; p a r t i c u l a r l y e x te n s iv e work has
been c a r r ie d o u t by W ig n e r21 u s in g group t h e o r e t ic a l te c h n iq u e s .
Lo w din 22 d eve lo p ed an a n a ly t ic method f o r a to m ic p h y s ic s
"w hich c o n s id e rs th e com posite system as an e n t i t y to w n ich
th e v a r io u s components c o n tr ib u te in an e q u iv a le n t ly and n o t
n e c e s s a r i ly in an o rd e re d w ay"2 3 . The e s s e n t ia l p rem ise is
t h a t an a r b i t r a r y fu n c t io n f o r th e t o t a l system must be r e
s o lv a b le in a u n ique way in t o o rth o g o n a l components o f sharp
a n g u la r momentum. Each o f th e s e may be fo u n d , in p r i n c i p l e ,
by means o f a p r o je c t io n o p e ra to r w h ich a n n ih i la t e s a l l b u t
th e s e le c te d com ponent, w h ich rem ain s unchanged. The em
p loym ent o f p r o je c t io n o p e ra to rs c o m p le te ly o b v ia te s th e
need to c o n s tru c t s ta te s o f good a n g u la r momentum, w h ich is
e x a c t ly th e s im p l i f i c a t io n n e c ess ary in d e a lin g w ith m u l t i
p a r t i c l e system s.
4 . M o t iv a t io n f o r S tu d ie s
M ic ro s c o p ic and m acroscop ic m odels re p re s e n t o p p o s ite
avenues o f approach to th e problem s o f n u c le a r s t r u c t u r e .
Each g iv e s r e fe r e n c e to e s s e n t ia l c h a r a c t e r is t ic s ob served
e x p e r im e n ta l ly . L o g ic a l ly , any good m odel must In c lu d e
a s p e c ts o f b o th s im u lta n e o u s ly . C o l le c t iv e m odels t y p i c a l l y
d isp en se w ith th e m ic ro s c o p ic p ic t u r e o f th e n u c le u s . The
d eg rees o f freed om o f th e in d iv id u a l nu c leo ns a re re p la c e d
by a few c o l l e c t i v e c o o rd in a te s w h ich a r e , in th e o r y , d e r iv a b le
from th e p a r t i c l e c o o r d in a te s . A b a s ic assu m p tion is t h a t
o n ly a few p a r t ic le - c o n f ig u r a t io n s a re im p o rta n t in th e m ic ro
s c o p ic t r e a tm e n t .
The m o t iv a t io n f o r th e work re p o r te d h e r e in is th e d e s ir e
to u n d e rs ta n d th e r e l a t i o n betw een th e two c o n c e p tu a l m o d e ls ,
i . e . , th e deve lopm ent o f c o l l e c t i v i t y , in p a r t i c u l a r r o t a
t io n a l m o tio n s , w i t h in th e fram ew ork o f an i n d i v i d u a l - p a r t i c l e
m o d el, and i n d i v i d u a l - p a r t i c l e a s p e c ts w i t h in th e fram ew ork o f
th e c o l l e c t i v e m ode l. In p a r t i c u l a r , we w ould l i k e to answ er
th e fo l lo w in g q u e s tio n s : ( i ) In w hat way a re c o l l e c t i v e
m otio ns d e s c r ib a b le in term s o f th e s im p le g ross fe a tu re s o f
th e e f f e c t i v e n u c le o n -n u c le o n in t e r a c t io n ? ( i i . ) How a re we
to r ig o r o u s ly u n d e rs ta n d th e shape t r a n s i t io n s w i t h in a s h e l l
as a consequence o f th e c o m p e tit io n betw een th e s h o r t and
lo n g ran g e components o f th e e f f e c t i v e in t e r a c t io n ? ( i i i ) Does
th e lo n g ran g e p a r t o f th e e f f e c t i v e In t e r a c t io n b r in g ab o u t
tem p o ra ry s p h e r ic i t y a s s o c ia te d w ith th e change in s ig n o f
th e q u ad ru p o le moment in th e m id d le o f th e s h e l l , o r is t h is
caused by th e s h o r t range in t e r a c t io n ? ( i v ) Can c o l l e c t i v e
m odels make any p r e d ic t io n s abou t in d iv id u a l p a r t i c l e m otions?
(v ) Can we g iv e an e x p la n a t io n w i t h in th e fram ew ork o f an
i n d i v i d u a l - p a r t i c l e m odel f o r th e ob served d is t o r t io n s o f
r o t a t io n a l bands a t t h e i r h ig h e r a n g u la r momentum s ta te s ?
( v i ) What is th e e x te n t o f th e p a r t i c le - h o l e a d m ix tu re s to
th e H a r tre e -F o c k q u a s i - r o t a t io n a l bands in th e 2 4 - Id. s h e l l
n u c le i? ( v i i ) A re th e a d m ix tu re s s u f f i c i e n t to e x p la in th e
d is c re p a n c ie s betw een c a lc u la te d and e x p e r im e n ta l s p e c tra ?
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( v i i i ) Are p a r t i c le - h o l e s ta te s c a p a b le o f ,e x p la in in g th e
e x c i te d s ta te s above th e q u a s i - r o t a t io n a l band p r o je c te d
fro m th e H a r tre e -F o c k s ta te ? ( i x ) I s H a r tre e -F o c k a good
a p p ro x im a tio n in t h is mass re g io n ? A l l o f th e s e q u e s tio n s
c o n c e rn in g th e in t e r r e la t io n s h ip betw een m acroscop ic and
m ic ro s c o p ic m odels o f n u c le a r s t r u c tu r e may be answ ered
u s in g th e p r o je c t io n fo rm a lis m d is c u s s e d e a r l i e r to d e r iv e
th e p r o p e r t ie s o f e ig e n s ta te s o f th e n u c le a r sys tem , and
th u s to exam ine th e em ergence o f c o l l e c t i v e b e h a v io r from
an i n d i v i d u a l - p a r t i c l e d e s c r ip t io n o f th e n u c le u s .
5. Scope of Present InvestigationsThe purpose o f p r o je c t io n c a lc u la t io n s is to o b ta in
p r o p e r t ie s o f e ig e n s ta te s o f p a r t i c u l a r symmetry o p e ra to rs
w ith o u t e x p l i c i t know ledge o f th e s t r u c tu r e o f th o s e s t a t e s .
K e ls o n 21* has o u t l in e d t h is ty p e o f p r o je c t io n fo rm a lis m as
i t may be a p p l ie d to n u c le a r s t r u c tu r e c a lc u la t io n s and has
su g g ested v a r io u s ty p e s o f prob lem s to w h ich i t may be a p p l ie d .
The fo rm a lis m o f p r o je c t io n te c h n iq u e s w i l l be d e ve lo p ed in
C h a p te r I I . I t w i l l be shown how p r o je c t io n may be accom
p l is h e d u s in g e i t h e r f i n i t e r o t a t io n o r i n f i n i t e s i m a l r o t a t io n
o p e r a to r m ethods. F i n i t e r o ta t io n s w i l l be d isc u s s e d b r i e f l y
i n th e f i r s t c h a p te r , and from th e r e on o n ly I n f i n i t e s i m a l
r o t a t io n o p e ra to r p r o je c t io n s w i l l be t r e a t e d . Methods w i l l
be d e v e lo p ed f o r h a n d lin g p r o je c t io n s from any number o f s t a t e s ,
r a t h e r th a n from a s in g le s t a te as is u s u a lly th e c a se . I t
th e n becomes most c o n v e n ie n t to p e rfo rm th e c a lc u la t io n s In
19
bases h a v in g n o n -u n it m e t r ic s . R e fe re n c e is f i n a l l y g iv e n
to o rth o n o rm a l bases by th e in t r o d u c t io n o f th e Schm idt
o r th o n o r m a liz a t io n p ro c e d u re .
I n C h a p te r I I I th e p r o je c t io n te c h n iq u e s d isc u s s e d
and d e v e lo p ed in th e p re v io u s c h a p te r a re a p p lie d to system s
o f n e u tro n s r e s t r i c t e d to a s in g le a n g u la r momentum s h e l l .
The p a r t i c l e a n g u la r momentum spans th e range 5 /2 < /< 1 5 /2 ,
and c o n s id e ra t io n is g iv e n to n u c leo n numbers fro m two to
m id - s h e l l , th e f i l l i n g o f th e l a t t e r h a l f o f th e s h e l l b e in g
sym m etric w ith t h a t o f th e f i r s t h a l f f o r a s i n g l e - / s h e l l . ^
V a ry in g p a r t i c l e a n g u la r momentum and n u c leo n num ber, a s tu d y
i s made o f th e developm ent o f c o l l e c t i v i t y a r is in g from a
s im p le p h en o m en o lo g ic a l H a m ilto n ia n c o n ta in in g s h o r t and lo n g
range com ponents. A p u re p a i r in g in t e r a c t io n is f i r s t
em ployed in o rd e r to o b ta in th e s e n io r i t y co m p o s itio n s o f
th e e ig e n s ta te s o f th e above H a m ilto n ia n .
I f th e n u c le a r system is c o n s tru c te d such t h a t M=£ m .,i 1
where th e a re p r o je c t io n s o f in d iv id u a l n u c leo n a n g u la r
momenta o n to th e a x is o f sym m etry, and M is th e p r o je c t io n
o f th e t o t a l a n g u la r momentum, th e n th e p r o je c t io n m ethod f o r
th e system ta k e s on a p a r t i c u l a r l y e le g a n t form w h ich u t i l i z e s
th e e x p a n s io n o f c e r t a in o p e ra to rs in a p o ly n o m ia l e x p re s s io n .
In p a r t i c u l a r , f o r th e s p e c ia l case o f M = ^n+^p* where is
th e p r o je c t io n f o r th e n e u tro n s and Mp t h a t f o r th e p ro to n s ,
u t i l i z i n g th e b in o m ia l theorem o b ta in s a p r o je c t io n o p e ra to r
ex p an s io n w h ich is e q u iv a le n t in th e two c o n s t i tu e n ts . T h is
^ We n o te t h a t th e s p in - o r b i t fo rc e in tro d u c e s an assym etry w ith re s p e c t to th e m id d le o f th e s h e l l ( c f . s e c t io n I V . 4) .
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te c h n iq u e p ro ves to be e x tre m e ly u s e fu l .
In C h a p te r IV th e s e p o ly n o m ia l p r o je c t io n te c h n iq u e s
a re a p p l ie d to th e c a lc u la t io n o f p a r t i c le - h o l e e x c i t a t io n s
o f th e H a r tre e -F o c k s ta te s in e v en -e v e n N=Z n u c le i c a lc u
la t e d e a r l i e r by R ip k a 2 5 , in an a tte m p t to e x p la in th e sys
te m a t ic d is c re p a n c ie s d isc u s s e d e a r l i e r betw een th e m odel
c a lc u la t io n s and th e e x p e r im e n ta l e x c i t a t io n s p e c tr a . I t
i s a ls o e x p e c te d t h a t some o f th e h ig h e r e x c i t a t io n s ta te s
w i l l be found to be p re d o m in a n tly based on p a r t ic le - h o l e
c o n f ig u r a t io n s . The need to p r o je c t sharp is o s p in s ta te s
s im u lta n e o u s ly w ith a n g u la r momentum s ta te s w i l l be o b v ia te d
by th e em ploym ent o f s im p le c o m b in a tio n s o f S la t e r d e t e r -
m in a n ta l s ta te s w hich a lr e a d y have sharp is o s p in .
As d is c u s s e d a b o ve , th e aim o f t h is w ork is to c l a r i f y
th e com plem entary r e la t io n s h ip betw een m acroscop ic and m ic ro
s c o p ic m odels o f n u c le a r s t r u c t u r e . By th e p erfo rm an ce o f
d e t a i le d m ic ro s c o p ic c a lc u la t io n s , we w i l l d em o n stra te and
e lu c id a t e how a system o f nu c leo ns may undergo b e h a v io r w h ich
i s t y p i c a l l y d e s c r ib e d by c o l l e c t i v e p a ra m e te rs . We w i l l
th u s show t h a t th e o r ig in s o f c o l l e c t i v e m otio ns in n u c le i
may be u n d e rs to o d from fu n d a m e n ta l m ic ro s c o p ic many-body
th e o r y .
The bases f o r p r o je c t io n c a lc u la t io n s from th e p a r t i c l e -
h o le e x c i t a t io n s o f H a r tre e -F o c k s ta te s a re r a th e r s m a ll ,
c o n s is t in g o f o n ly a few s t a t e s , in c o n tr a s t to th e ( / ) w
c a lc u la t io n s o f C h a p te r I I I in w h ich th e f u l l H i lb e r t s p a c e ,
d e r iv e d from th e c o u p lin g o f n p a r t i c l e s , i s u sed . The te c h -
21
n iq u e s f o r h a n d lin g th e s e v a r ia b le s iz e bases a re d e ve lo p ed
in th e fo l lo w in g c h a p te r .
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CHAPTER I I
PROJECTION TECHNIQUES
1 . P r e l im in a r y D is c u s s io n
A n u c le a r i n t r i n s i c s ta te <f>, w i t h in th e c o n te x t o f t h is
p r e s e n ta t io n , i s an i n d i v i d u a l - p a r t i c l e m odel s ta te w h ich is
s im p le and easy to w ork w i t h , and y e t c o n ta in s a la r g e amount
o f in fo r m a t io n ab o u t p r o p e r t ie s o f th e n u c leu s in th e lo w -
en erg y re g io n o f e x c i t a t i o n . I t i s n o t an e ig e n s ta te o f th e
H a m ilto n ia n , b u t r a th e r a m a th e m a tic a l c o n s tr u c t , o u t o f
w hich th e p h y s ic a l s ta te s o f th e system may be e x t r a c te d .
I t may e i t h e r be guessed a t , i . e . , th e b a s ic assu m p tion o f a
m o d e l, o r be a d e r iv e d f u n c t io n , e . g . , th e H a r tre e -F o c k
s t a te ( v . C h a p te r I V ) . P r o je c t io n o p e ra to r te c h n iq u e s a re
used to e x t r a c t th e d e s ir e d in fo r m a t io n from th e s e i n t r i n s i c
s t a t e s .
A b r i e f re v ie w o f th e b a s ic p r o p e r t ie s o f a n g u la r
momentum o f im p o rt to th e developm ent o f p r o je c t io n o p e ra to rs +
f o l lo w s .
A g e n e ra l a n g u la r momentum if = m easured inx y z
u n its o f H may be d e f in e d by th e com m utation r e l a t i o n ifxif = i i f ,
o r c y c l ic p e rm u ta tio n s o f The assu m p tion o fx y y x »
th e is o t r o p y o f space r e q u ir e s t h a t any H a m ilto n ia n v a l id in
n u c le a r p h y s ic s be in v a r ia n t under s p a t ia l r o t a t io n s . I f
deno tes th e g e n e ra to r o f r o t a t io n s abou t th e x ^ - a x is , t h is
^ A l l a n g u la r o p e ra to rs d isc u s s e d fo l lo w th e d e f in i t io n s and phase c o n v e n tio n s o f Rose2 6 .
23
- i j , e i j . e i j . e i j . ee He = H o r e H - He = 0 .
E xpand ing th e e x p o n e n tia ls in power s e r ie s and s e t t in g each
power o f 0 I d e n t i c a l l y e q u a l to ze ro shows t h a t H commutes
w ith any power o f A s im p le c a lc u la t io n d em o n s tra tes t h a t
J 2 a ls o commutes w ith any o f th e components o f J . From th e
d e f i n i t i o n o f a n g u la r momentum a b o v e , i t is c le a r t h a t no two
components commute. Hence ? 2 and any component o f ? may be
chosen as commuting o b s e rv a b le s . J is u s u a lly s e le c te d asz
t h is second o p e r a to r . The o th e r two com ponents, J and J ,x y
may be r e p la c e d by t h e i r h e r m it ia n c o m b in a tio n s , g e n e r a l ly
r e f e r r e d to as r a is in g (+ ) and lo w e r in g ( - ) o p e ra to rs :
J . = J ± iJ . I t i s e a s i ly v e r i f i e d t h a t J „ J + = J . ( J ± 1 ) , z x y z - - z
whence t h e i r names. D i l i g e n t a p p l ic a t io n o f th e s e o p e ra to rs
to e ig e n fu n c t io n s o f a n g u la r momentum j and z - p r o je c t io n m,
w i l l v e r i f y th e e ig e n v a lu e r e la t io n s
- mv *
where j may have in t e g e r o r h a l f in t e g e r v a lu e s g r e a t e r th a n
o r e q u a l to z e r o ; m is in t e g e r o r h a l f in t e g e r as is / , and
ranges
The p r o je c t io n approach may be u n d ers to o d by c o n s id e r in g
a m any-body H a m ilto n ia n H w h ich commutes w ith a l l o f th e
g e n e ra to rs o f a group G .21* The e ig e n s ta te s o f H may be c la s
s i f i e d by th e r e p r e s e n ta t io n s o f G; th e y may be grouped in t o
invariance may be written mathematically as
r e p r e s e n ta t io n s c h a r a c te r iz e d by th e e ig e n v a lu e s o f th e
C a s im ir o p e ra to rs o f G. W ith in each r e p r e s e n ta t io n , a com
p le t e s e t o f o p e ra to rs is d ia g o n a liz e d and th e e ig e n v a lu e s
a re used to c h a r a c te r iz e th e c o rre s p o n d in g s t a t e s . F o r
ex am p le , when th e group is S I^ , th e C a s im ir o p e ra to r is J 2 ,
and th e o p e r a to r w h ich is d ia g o n a liz e d w i t h in each re p re s e n
t a t io n is J z .
The i n t r i n s i c s ta te s c o n s id e re d h e re a re assumed to be
sym m etric ab o u t th e q u a n t iz a t io n a x is , i . e . , e ig e n s ta te s o f
J _ . They a r e , h o w ever, g e n e r a l ly n o t e ig e n s ta te s o f. th e
a n g u la r momentum o p e ra to r J 2 b u t r a th e r a l i n e a r s u p e r p o s it io n
o f s ta te s w ith d i f f e r e n t a n g u la r momenta. A p r o je c t io n o p e ra
t o r P7 , w h ich e x t r a c ts th e component P7 <J> h a v in g a n g u la r
momentum J , i s fo r m a l ly in tro d u c e d . The e x p l i c i t fo rm o f
PJ<J> i s , in g e n e r a l , v e ry c o m p lic a te d in com parison to th e
i n t r i n s i c s t a te i t s e l f . The u s e fu ln e s s o f th e p r o je c t io n
o p e ra to r te c h n iq u e l i e s in th e f a c t t h a t th e e x p l i c i t form
i s n e v e r n eed ed , s in c e a l l m a tr ix e lem en ts o f p h y s ic a l ob
s e rv a b le s may be c a lc u la te d u s in g th e i m p l i c i t fo rm P <f».
T h e r e fo r e , i t i s n o t a t a l l n e c e s s a ry to a c t u a l ly c o n s tru c t
wave fu n c t io n s o f good J .
A p a r t i c u l a r l y s im p le form f o r th e p r o je c t io n o p e ra to r
may be o b ta in e d by o b s e rv in g t h a t th e e ig e n v a lu e r e l a t i o n fo r
th e o p e r a to r J 2 may be w r i t t e n in th e form
[ J 2 - J ( J + 7 ) ] | J , M , a > = 0 ,
where | J , M , a > is a n o rm a liz e d m u l t i - p a r t i c l e e ig e n s ta te o f
24
b o th ? 2 and J , w ith e ig e n v a lu e s J ( J + 7 ) and M, r e s p e c t iv e ly . z
T h is e q u a t io n im p lie s t h a t th e s im u lta n e o u s e ig e n s ta te | J , M , a >
p o s s ib le to e x t r a c t a s p e c i f ic a n g u la r momentum component from
an i n t r i n s i c s ta te by a n n ih i l a t in g a l l o th e r com ponents. T h is
may be acco m p lish ed u s in g th e p ro d u c t o p e ra to r
The n u m e ra to r is a p ro d u c t o f th e a n n ih i la t io n o p e ra to rs
d e f in e d a b o v e , o v e r a l l v a lu e s o f a n g u la r momentum e x c e p t
t h a t o f th e d e s ire d com ponent. The d en o m in a to r i s s im p ly
a n o r m a liz a t io n te rm w h ich g iv e s th e v a lu e u n ity when
a p p l ie d to th e d e s ire d com ponent.
U t i l i z i n g an e x p l i c i t fo rm o f th e p r o je c t io n o p e ra to r
f o r b o th th e t o t a l a n g u la r momentum and i t s p r o je c t io n on to
th e a x is o f q u a n t iz a t io n :
where | J , M , a > is a co m p le te o rth o n o rm a l s e t o f e ig e n s ta te s
Is a n n ih i la t e d by th e o p e ra to r [ J 2 - J ( J + 7 ) ] . I t I s th e r e fo r e
pJ = n ? 2 - J * ( J ' + l ). J ’ t J J {J+ 7 ) - J ' ( J ' + 7)
( I I - l )
a
o f J 2 and J , a d e n o tin g a l l o th e r quantum num bers, th e w e l l z
known p r o p e r t ie s o f p r o je c t io n o p e ra to rs a re e a s i ly demon
s t r a t e d :
26
In g e n e r a l , i t may be n e c e ss a ry to c o n s id e r s e v e r a l
d i f f e r e n t i n t r i n s i c s t a t e s . These w i l l be la b e le d by s u p e r
s c r i p t s . The i n t r i n s i c s ta te s used in th e fo l lo w in g c a lc u
la t io n s a re S la t e r d e te rm in a n ts o f s in g l e - p a r t i c l e s h e l l
m odel s ta te s in th e / - / c o u p lin g schem e, i . e . , | n , £ , / , m > ,
o r l i n e a r c o m b in a tio n s o f them . The l i n e a r c o m b in a tio n s ta te s
a re n o t n e c e s s a r i ly e ig e n fu n c t io n s o f th e s i n g le - p a r t i c le
a n g u la r momentum o p e ra to r J 2 , and may, f o r exam p le , be
H a r tre e -F o c k o r N ils s o n s in g l e - p a r t i c l e wave fu n c t io n s .
The i n t r i n s i c b a s is {<}>} is a com plete s e t o f e ig e n
v e c to r s , a l l h a v in g th e same e ig e n v a lu e o f J . F o r an evenz
number o f p a r t i c l e s , s ta te s o f a re chosen; f o r
an odd number o f p a r t ic le s s ta te s o f M^n^ = l / 2 . W ith t h is
c h o ic e one is c e r t a in th a t a l l a n g u la r momentum components
p re s e n t can be o b ta in e d v ia th e p r o je c t io n p r o c e s s .
F o r m a lly , th e b a s is s ta te s may be decomposed in t o com
po nents w hich a re s im u lta n e o u s e ig e n s ta te s o f J 2 and J byz
th e p r o je c t io n o p e ra to r P-7:
* ( i ) ’ j X f * ( 1 ) = 5 * a ) = I aJ i ) | J ’ M* « * ’ ( 1 ) > ( I I " 2>
w here J is some maximum v a lu e o f th e t o t a l a n g u la r t n d x
+momentum a llo w e d by th e P a u l i p r in c i p le . The a m p litu d e o f
th e n o rm a liz e d s ta te | ( i ) > in th e o r ig in a l s t a t e
is g iv e n by • I f th e s t a te is n o rm a liz e d , th e s e
+In p r i n c i p l e , th e r e need n o t be an upper l i m i t to th e angu
l a r momentum o f th e system . T h is w i l l depend on th e p a r t i c u la r m odel b e in g em ployed . I f a s h e l l m odel is used th e r e is an upp e r l i m i t , w h ile a hydrodynam ic m odel g e n e r a l ly does n o t have one
27
a m p litu d e s a re d e f in e d to w i t h in a p h a s e , and th e sum o f
th e squares o f t h e i r a b s o lu te v a lu e s is u n i t y . I t is
im p o r ta n t to n o te t h a t th e s ta te | ( i ) > Is g e n e r a l ly
n o t o r th o g o n a l to th e s ta te | 3 ( J ) > .
P r o je c t io n may be acco m p lish ed fro m any number o f
i n t r i n s i c s t a t e s . F o r exam p le , C h a p te r I I I d e a ls w ith
p r o je c t io n s fro m a com ple te b a s is , r e s u l t in g in e x a c t ( / ) w
s p e c t r a ; C h a p te r IV d e a ls w ith p r o je c t io n s from th e H a r t r e e -
Fock s ta te and I t s p a r t i c le - h o l e e x c i t a t io n s . H ow ever, even
i f th e e x a c t p ro b lem is n o t s o lv e d , th e com plete b a s is m u st,
i n g e n e r a l , be c o n s id e re d .
2 . A n g u la r Momentum P r o je c t io n by F i n i t e R o ta t io n s
The f i r s t c a lc u la t io n a l method o f p r o je c t in g a n g u la r
momentum makes use o f th e H i l l - W h e e le r 2 7 i n t e g r a l . The
m ain co n cern is w ith s ta te s w hich fo rm a r e p r e s e n ta t io n o f
a group w h ich is th e d i r e c t p ro d u c t o f a number o f SU2
g ro u p s . H e re , th e most im p o rta n t o f th e s e a re th e SU2
a s s o c ia te d w ith a n g u la r momentum ? and th e one a s s o c ia te d
w ith is o s p in The d is c u s s io n w h ich fo l lo w s may be g e n e r
a l i z e d e a s i ly to a s im u lta n e o u s c o n s id e ra t io n o f th e s e two
g ro u p s , o r in d e e d to m ore.
A fu n c t io n o f r o t a t io n a n g le s is o b ta in e d f o r th e
o p e r a to r un der c o n s id e r a t io n . T h is fu n c t io n must be a n a ly z e d
as a sum o f Legendre p o ly n o m ia ls ( o r , more g e n e r a l ly , o f
r o t a t io n m a tr ix e le m e n ts ) ; th e c o e f f ic ie n t s o f th e p o ly
n o m ia ls a re th e m a tr ix e lem en ts o f th e o p e r a to r . T h is
a n a ly s is is u s u a lly p e rfo rm e d by in t e g r a t io n . The o p e ra to r
f o r f i n i t e r o ta t io n s is g iv e n b y 28
-iocJ -1 8 J - i y j R (ft) = e e y e
The r o t a t io n R is c a r r ie d o u t by th r e e s u c c e s s iv e E u le r
r o t a t io n s . The f i r s t is a r o t a t io n a a lo n g th e z - a x i s , th e
second a r o t a t io n 8 ab o u t th e y ' - a x i s , and th e t h i r d a r o t a
t io n y ab o u t th e z " - a x is . We n o te t h a t th e axes o f r o t a t io n
a re c o o rd in a te axes in d i f f e r e n t c o o rd in a te system s, nam ely
th e c o o rd in a te system o b ta in e d by th e p re v io u s r o t a t io n .
The r o t a t io n s may a c t u a l ly be c a r r ie d o u t in th e same c o o r
d in a te system i f th e o rd e r o f r o t a t io n s is in v e r t e d .
The r o t a t io n m a tr ix is d e f in e d as th e m a tr ix
e le m en t o f R (ft) in th e |JMA> b a s is o f th e p re v io u s s e c t io n :
< J M A | R ( f t ) | J ' M * X * > = 5 j j » 6xA’ PMM'
o r , in v e r s e ly ,
R (ft) = 7 | , ( f t ) <JM1A| .JMM'A
S in ce th e r o t a t io n m a tr ic e s obey th e o r th o n o rm a lity
r e l a t i o n
/ dnPM1M l ( f i ) PM22M i ( n ) = ^ 7 j " 6J 1J 25MiM26M{Mi ,
7 1m u lt ip ly in g by V £ ^ (ft) and in t e g r a t in g o v e r th e E u le rJLnt Z n t
a n g le s ft o b ta in s an e x p l i c i t r e p r e s e n ta t io n o f th e p r o je c t io n
o p e r a to r :
JLYlt JLYlt 2J ' + l A
28
29
? J =2J+J
Sir2
The q u a n t i t ie s o f in t e r e s t a re th e m a tr ix e lem en ts
«J>( i ) |PJ |4>( J ) > = aJ ( i ) aJ ( J ) < J , ( i ) | J , ( j ) >
and (II-4)Si j = < ^ ( 1 ) |SPJ U ( J ) > «* a J ( 1 ) a J ( J ) < J , ( i ) | s | J , ( j ) >
w here S is a s c a la r o p e r a to r . I f , f o r exam p le , S is th ei 7
H a m ilto n ia n , th e n S J is d ia g o n a liz e d a f t e r t ra n s fo rm in g to
an o rth o n o rm a l b a s is to y ie ld en erg y e ig e n v a lu e s and e ig e n
s ta te s o f th e system .
U s in g th e e x p l i c i t form o f th e p r o je c t io n o p e ra to r ( I I - 3 )
o b ta in s th e d e s ire d r e s u l t s :
<<j) i ) |PJ |<J> > = f d n V t u <<j>(1 ) |R(fi) |<|)( ^ >* tt2
( I I - H a )
|SPJ |<|)( J ) > = fdnV*.J u <<|>( i ) |SR(f l ) |<t>( ; } ) >Sir2 i.n t A.nt
The i n t e g r a l s , known as H i l l - W h e e le r i n t e g r a l s , a re c a lc u la te d
n u m e r ic a l ly . The n o rm a liz e d wave fu n c t io n is e a s i ly seen to
be .,, (n)R(f l ) |<l>( i ) >, ( H - 5 a )
" i n t 8tt2Nj ( i )
w here N j ( i ) is a n o r m a liz a t io n c o n s ta n t g iv e n by th e d ia g o n a l
fo rm o f th e f i r s t o f e q u a tio n s ( I I - * J a ) :
Ma ( 1 ) - tJ+1 M . «l>( i ) | R ( f i ) k ( 1 ) > • ( I I - 5 b )
A s i m i l a r , a l t e r n a t iv e approach is p o s s ib le i f th e r e is
30
a n a t u r a l l i m i t to th e a n g u la r momentum to be p r o j e c t e d ,
i ‘ e *» “ * T h en » f o r each Pa i r ( i » J ) o f i n t r i n s i c
s t a t e s , th e o v e r la p o f th e i bb i n t r i n s i c s t a t e w ith a r o ta te d
i n t r i n s i c s t a t e may be c a lc u la te d f o r as many d i f f e r e n t
r o t a t io n s ftk as th e r e a re p o s s ib le d i f f e r e n t v a lu e s o f
a n g u la r momentum. These th e n fo rm th e inhomogeneous p a r t
o f th e s e t o f - l i n e a r e q u a tio n s
«J>( i ) | R( f lk ) U ( J ) > = [ p j » (f ik )<d>( i ) | PJ |4>( J ) > ( I I - 6a )J I n t i n t
w ith V m a tr ix e lem en ts as c o e f f i c ie n t s . S im i l a r l y , one
o b ta in s
«|>(i) |SR(ftk )|<J>(J)> = I PjJ M (ftk )«|.(i)|SPJ |<|>(J)>. (II-6b)
E q u a tio n s ( I I - 6a , b ) a re b o th o f th e fo rm X ® V%. The column
v e c to rs b e in g e i t h e r < < j> ^ |P "7 1 <j> o r <<|>^^ | SP"7 | <J>
f o r a l l p o s s ib le v a lu e s o f a n g u la r momentum, a re e a s i ly
o b ta in e d by in v e r t in g th e m a tr ix V:
t = P " 7Jt . (II-6c)
3 . A n g u la r Momentum P r o je c t io n w ith I n f i n i t e s i m a l
R o ta t io n O p e ra to rs
a . O r th o n o r m a liz a t io n o f M u l t i - P a r t i c l e Wave F u n c tio n s
A second p r o je c t io n p ro c e d u re was f i r s t in tro d u c e d by
Lo w din 22 f o r th e s tu d y o f a to m ic s p in o r b i t a l s . I t le a d s to
a s e t o f l i n e a r e q u a t io n s , n o t u n l ik e e q u a tio n s ( I I - 6 ) , th e
s o lu t io n s o f w h ich have been found f o r b o th s c a la r and te n s o r
31
operators 2 3The a p p l ic a t io n o f th e r a is in g o p e r a to r J + to a s im u l
taneous e ig e n s ta te o f J 2 and J has th e e f f e c t1 /2
J + |J ,M> = [ J (J + / ) - M ( M + I ) ] |J ,M+ 7> ,
i . e . , th e m a g n e tic quantum number is in c re a s e d by one u n i t ,
and th e s ta te is m u l t ip l i e d by th e c o e f f i c ie n t [ J (J + 1 ) - M ( M + I ) ]
S u c c e s s ive a p p l ic a t io n o f th e r a is in g o p e r a to r y tim e s y ie ld s
J +y |J ,M> = Bl / 2 ( J , M , y ) | J ,M+y> ,
where th e c o e f f ic ie n t s B ( J , M , y ) a re d e f in e d by:
B ( J , M , 0) = 1
M+y- 7B ( J , M , y ) = n [ J ( J + I ) - v ( v + 7 ) ] y+0 ( I I - 7 )
v=M /
o rb ( j , m , i i ) = U t W . U J - J H J .
I J - M - u l ! 1J+M) I
The r e s u l t o f a p p ly in g th e r a is in g o p e ra to r y tim e s to th e
s t a te <|>^^ is
i/
a T1 ) j + y |J,M-cw*C,( i )>
J.max- i
1 /2
w hich is an e ig e n s ta te o f J z w ith e ig e n v a lu e T h is
s e t o f e q u a t io n s , f o r a l l p o s s ib le v a lu e s o f y ^ , may be w r i t t e n
(1 )f o r each o f th e i n t r i n s i c s t a t e s <f> I f th e m a tr ix o f o v e r -
^ S in ce th e sum o v e r J must have a l i m i t Jmax f o r t h is m ethod to be a p p l ic a b le , y can range 0 4Vl4-7max”^ n^*
laps of these states (II-8) is formed, it Is immediately seen that the raising operator must operate the same number of times y in the bra as in the ket because of orthogonality of states of different angular momentum projection:
states, the coefficients B being given by equation (II-7). The metric of the subspace of states of total angular momen turn J in the space spanned by the intrinsic basis vectors
the inhomogeneous parts of equation (II-9), labeled by increasing values of y, beginning with y=0. B is the coefficient matrix whose columns and rows are labeled by J and y, respectively; is a column vector whose components are the angular momentum components of the (i,j) metric element.
< J + y | J +y <j>( J ) >
( I I - 9 )
The result is a set of linear equations for the unknown products |P^<|> for each pair (i,j) of intrinsic
{4*} is denoted byMIj = <PJ^ (i)|PJ«l>(j)> = aji)ajJ)<J,(i)|J,(j)> . (11-10)
Equation (II-9) may now be rewritten:
J- B& for each (i,j)
where J+<j> represents a column vector whose components are
( I I - l l )
The inhomogeneous parts of equations (11-11) must be calculated by actually performing the J+y operations on the intrinsic determinantal states and computing the scalar products, after which the metric is obtained using matrix algebra. The coefficients B form a triangular matrix in the upper triangle (the number of terms in the summation decreases when y increases), so that the (i,j) element of the metric may be obtained for successively smaller values of angular momentum, beginning with J=Jmax> by successive elimination. For M^n^=0, V~JmCLX» there is only
<J+JBa ^ ( l ) | J+JmaS (j)> . B (Jmax,0,JmaJC) M ^ ax ,
which gives directly the value M . ^ a x . Then the equationsJ 13-Jfor V=Jmax~1 give the values MjJax » etc. This is the essential part of the projection process, in which the properties of eigenstates of total angular momentum are retrieved from eigenstates of angular momentum projection onto the axis of quantization.
Having obtained the metric of the subspace of a particular eigenvalue of the total angular momentum, it remains only to apply an orthonormalization procedure to obtain transformation matrices U"7 which, when applied to the Intrinsic basis, will yield an orthonormal basis for each eigenvalueof J? For example, a Schmidt orthonormalization process may
(k)be used, in which case the orthonormal function ¥j are given (for a real metric) by
33
(k) _
34
(II-12)
( < * J ( k ) i # J ( k ) > - Y < ^ { i ) \ ^ M > z ) l / 21=1
Expanding the above,k
t
•>< (k) _ J ^ *J W)
- *j(k>S t i DW ) J i ° ^ * J« )
(11-13){ < * / * > i * _ / * > > I ( i y l m „ ^ j ) i
2, l / 2k-1 i• 1 ( 1 1=1 m=l
The transformation matrix elements are recursively obtained be equating the coefficients of the intrinsic basis. They are:
U
U
U
k,k+n
Jk,k
Jk,k-n
= 0
{ V k ) i * J ( k ) > - T ( i < M ] 2 ) }1=1 m=l
- 1 / 2)(II-14)
n k-m . T , 7
The solution is unique up to a unitary transformation.
b. Tensor OperatorsThe tensor operator is denoted by T , k being the rank
of the tensor, and q the magnetic component. The quantities of interest are the reduced matrix elements
35
Applying the operator Tq to the states (IIr-8) obtains
Forming scalar products again with the states (II-8) yieldsi i r ,, m i J m a x J m a x i / 2
<J+% (1)|T^|J+I,4.(j)> = 6x>y+q ^ ^ B
Mi n t + V ** U i n t lv
1
The dependence on orientation can be removed by Invoking the Wigner-Eckhart Theorem21,
<J1,M1|Tk |J2,M2> = C(J2,k,J1;M2,q,Mi)<Jl||Tk ||J2> , (11-16)
where <Ji||Tk ||J2> is the reduced matrix element of Tk and is independent of orientation. All of the geometrical dependence Is included in the Clebsch-Gordan (CG) coefficient C.Thus,
y r*\ , /,\ ^max ^max i/2<J+ * J > = { X ,y + q I I B
V J 2 =U i n t * X U M * V
<J,,(l)||Tk ||J2,(j)> .
This set of linear equations, for » andis solved to obtain the products a l ^ a l ^ c J x ,i| |Tk | | J2,j>
1 2
36
The inhomogeneous part poses some technical difficulty, as it is easier to solve for products of the form <J+p+q<}>^ | J+pT^<f>^ . This is accomplished by successively invoking the commutation rules of irreducible tensor operators with angular momentum operators26,
CJ+,Tq] = [k(k+l)-q(q+l)]l/2 T*+1 .
One easily obtains
<J+X*(i)|Tq|J+M4>(3)> - «x>u+qJ o <-l>m (t;) B ,/2(k>q,m)
<J+X<f>(1) |J+y-m Tq+m<J)(J)>, (11-18)
where (p) is the binomial coefficient defined by
© = HJ— . (11-19)m!(y-m)!
We note that m has a further restriction m+q<k, arising from the tensor operator in equation (11-18). Thus, to
Ifobtain these scalar products for T , it is also necessaryto work with the higher magnetic components.
Equation (II-18) is used to find the inhomogeneous partsof equations (11-17), which are then solved for the productsal^a l ^ c J , ,i | |Tk | | J,, j>. The transformation matrices (11-14)
J l J 2 2
are applied to these to obtain the reduced tensor matrix elements between orthonormal states. If, for example, T^ is the quadrupole operator, the diagonal matrix elements in the orthonormal basis are the quadrupole moments of the states, and the non-diagonal elements are related to the BE(2)s.
Tensors of rank k=0— scalars— are treated as a special case of the previous section. The scalar operator is denoted by S. The matrices of interest, S ’ ^ were defined by equation (II-4). The elements of these matrices may be obtained as follcws:
37
c. Scalar Operators
S * (J> ■ I S3 k * (k) ' £ Sj k j ■
where Sjk is the matrix element of S in the intrinsic basis. The raising operator applied successively to yields
II M l _ ' > /2 ( HJ + s * ■ I s J k B '
Mint*vForming scalar products with the states (II-8), as in the previous section, utilizing the fact that the scalar operator S commutes with the raising operator J+ , obtains
t (a \ / * \ ' RKIX<J+P ♦(1)|s|j»;JJ)> " t w , l SJk I
If M • y.+ U ‘tnt (11-20)
a j ^ a j ^ J ^ ^ + y , (i) | J,M^nJt+y, (k)>
or, using equation (II-4),<J+y,4,(1)|s|J+y<|»U)> = 6yp, I B(J,M^,p)SiJ
J (11-21)J , £ = B$' for each (i,J)T •
These equations have the identical structure as equations (11-11). Here represents a column vector whose components are the inhomogeneous parts of equations (11-20), labeled by
increasing values of y beginning with y=0, and is a column vector whose components are the angular momentum components of the (i,J) matrix elements of the operator S in the intrinsic basis. The solutions of equations (11-21) are obtained In a manner similar to those of euqtions (11-11).
We note that instead of calculating the expectation values of S between the many pairs of states of a complicated form, generated by the Hamiltonian, operation is required only on those states which are intrinsic and of simpleform. The raising operator is, of course, a one-body operator. The only place in which a two-body operator (the Hamiltonian) may enter is the initial set of simple intrinsic states.
Applying the transformation matrix U"7, (equation (11-1*1)) to the matrix
SJ = S ’J UJgives the matrix in the orthonormal basis {U^$}. This matrix is usually diagonalized to yield eigenvalues and eigenvectors. The eigenvectors are expressed as linear combinations of the wave functions projected from the intrinsic states. The explicit form of the projected wave functions is unknown.
M. A Neutron-Proton Polynomial Projection Techniquea. IntroductionThe projection techniques discussed In the previous
sections are certainly applicable to systems of neutrons and protons if a suitable isotopic spin formalism is introduced. Present-day calculations are still limited to near
38
closed shell nuclei because the number of possible multiparticle states increases extremely rapidly with increasing number of neutrons and/or protons outside of a closed core.^For this reason, following a suggestion by Lowdin23, a method has been developed which greatly decreases both the number and complexity of the intrinsic states necessary for the calculation.
The part of the projection calculation which involves the determinantal multi-particle states is the calculation of the inhomogeneous parts of equations (11-11) and (11-21), i.e., scalar products of states having various angular momentum projections onto the axis of quantization, which have been generated by successive application of the raising operator to the intrinsic states. Once this has been accomplished, It is necessary only to solve the sets (11-11) and (11-21) of linear equations by successive elimination to arrive at the desired projection results. In order to make the task practical, the basis {<(>} of the Hamiltonian operator is written as the outer product of the neutron space {a} and the proton space {&), coupled to total angular momentum projection M=0,
W = [ ( « } 8 { 8 > ] m=0- ( H - 2 2 )
Since the separate neutron and proton spaces have zero overlap, the antisymmetrization between them need not be considered, as would be necessary if t_ were not a good quantum number for the calculations. Therefore, the intrinsic states need not be
For Neon 20 there are 2498 possible determinantal states with M^O.
39
totally antisymmetrized, rather the neutron and proton parts may be antisymmetrized separately.
The need to utilize a large number of states if the polynomial technique is not employed may be understood by realizing that for each neutron state it would be necessary to consider a large portion of the proton space, and vice versa. With the technique developed below, projection results may be calculated in the separate spaces and then
4*combined to yield the results for the total system.
b. Isotopic Spin ConsiderationsThe procedure is demonstrated for nuclei in which protons
and neutrons move in the same single-particle eigen-orbits.They need not, however, have the same one-body energy eigenvalues. This becomes important in nuclei above mass 40, where the Coulomb potential raises the proton single particle levels significantly above the neutron levels. If this type of calculation is to be performed for nuclei in which the protons and neutrons do not move in the same single-particle eigen- orbits, eigenstates of isotopic spin may be obtained by successive application of the raising operator T+ in isospin space, the mathematics of which is analagous to equations (II-8) through (11-11), but which is practically very difficult to perform.
* These separate bases contain a total of only 40 possible neutron states and 40 proton states for Neon 20.
*10
It is assumed that projection is to be accomplishedfrom a single determinantal state iji, such as results froma Hartree-Fock calculation.- Since this state is generallya pure T = |T | state, (where T(T+1) and Tw are the eigen- z z
values of the isotopic spin operator 'f2 and the projection operator in isospin space T ) isospin projections need not be of concern at this time. The method may be extended to projection from many states, such as partlcle-hole excitations of the Hartree-Fock state. This will be done in a later chapter. It is further assumed that the nucleus is even-even, although this is not necessary for the concepts which follow.
It is more convenient to write the Hamiltonian basis (11-22) as
■ {<*>} = (v> 8 {*) ® I { v B-m'} » (H-23)m'+O
where the neutron states {v} having Mn=0, and the proton states {tt} having Mp=0, are the bases for the separate problems of projection from neutron states and projection from proton states. The intrinsic state considered here contains only combinations of states from these separate bases.
c. Projection Method - NormalizationsThe intrinsic state ip may be written as a product of an
intrinsic neutron and intrinsic proton wave function. These are denoted by |v> and |ir>, respectively:
41
| iJ/> = | v> | ir> .The raising operator is now applied successively to this combined intrinsic state as in section II.3a:
J+ | 9> = | «J + V> | TT> + |v>|j+7T>
J+2 |<|»> = |J+ 2v >|tt> + 2 1 J+v> | J+ir> + |v >|J+2tt>
and p . .J / I * > = 1 ( X) I J + V > I J + R> > (1 1 -2 4 )
X=0 *
where (y) is the binomial coefficient defined by equation(11-19). Forming the scalar product of the states (11-24)with themselves obtains
<J+% I J+V = I (5J)2<J+Xv|J+Xv><J+y_XTr|j+y"XTr>. (11-25) X—0
Each of the overlap expressions appearing on the right hand side may be decomposed into its separate angular momentumcomponents, analagous to equation (11-11):
Jn wax
42
<J+ V IJ+ v> = I B ( J j , 0 , X ) M 1
where
Jj-XJp
<J+y_XTr|j+y"XTT> = j B( J2 ,0 ,p-X)MP 2J 2 = li-X
Mn J * = a j aV <PJ i v | P J iv>J i J i
MpJz = a* a l <PJ 2t t |PJ 2tt> .J 2 J 2The usefulness of the polynomial technique now becomes
apparent. Rather than actually performing the J+ operations on the left hand sides of equations (11-25), these operations are carried out on the separate simpler neutron and proton
states |v> and |ir>. The metrics Mn" 1 and Mp"L are calculated using equations (11-11), and are then substituted into the equations for the combined system:
< J + P < H j + y 'J» = ^ (V) 2 I B ( J 1 ,0 , X ) B ( J , , 0 , y - X ) M n J iMp J *. (11-26)A= 0 J j =A
J 2=y-X
Thus, the numberous complicated neutron-proton states are no longer required to obtain the system metrics.
d. Projection Method - EnergiesBecause electric charge is a conserved quantity, the
nuclear Hamiltonian can be divided into three parts— one involving only neutrons, one only protons, and one both neutrons and protons:
H = H + H + Hnn pp np
H|*> - I Hnn(i)|v(1)>|lt> + Hpp(i)|v>|ir(i)>
+ I + H ’ (ij)|a(1)> | 6 (3)>(II-27)i j
where, for example,
H„ n (1 ) " < v ( 1 ) lHn n lv>
Hnp(iJ) “ <»(1>’'(3)|Hnp|vi.> .
The bases (v)and {it} must contain all states of angular momentum projection Mn=0 and Mp=0. The states (a) (neutrons) and (B) (protons) do not have this property, but the combinationsincluded in H' have total angular momentum M=M +M =0. We np n p
43
note that the states | a ^ > and | 3 ^ > differ from |v> and |ir>, respectively, by only one particle each.
Applying the raising operator successively to the states (11-27) is easily seen to be a generalization of equations (11-24):
44
- ! (> ) { Hn n I J+ X v > I J+ u_Xtt> + Hp p |J + xv > | j + » - \ >a=o
+ Hn D I V v > | j +'1- V + H- I j / v > ! ! / - % > ) .y-A,np np
Defining
HpJ2 = a 7 a 7 <PJ2ir|H |PJ2tt> J2 J2 pp,(i)
= ay a® <PJ iv|PJ ia(i)>
NpJ2 = * a ^ < P J 2ir|PJ 2 3 ^ ) >J J 2 J 2
? ( 1 1 - 2 8 )
the expectation value of the Hamiltonian in the state |J+y^> becomes
<J yi H H | J . V = I {(J)2 jBCJj.O.AWJj.O.y-A)A J , J 2
[MnJiHpJ2 + HnJlMpJ2 + I Hnp(ij)MjJlMpj2]
+(5) l (^ li j
Bi/2(J2,0,y-A)B1/2(J2,-mi ,y-A+mi)NjJ »WpJ2}
(11-29)
where m^ is the magnetic quantum number of the state |c/^>.The metrics M-7 have already been calculated in separate neutron and proton spaces for equations (11-26); the Hamiltonian matrices defined for separate bases by equation (II-4), are calculated as solutions to equations (11-21).The matrices N"7 are similar to the metrics M-7, except that the kets are not the bases {v} or {tt} but rather {a} or(3); they are obtained as solutions of the equations
Jmax<J+Xv|j+X”mka (k)> = ^ XBi/2(J,0,X)BV2(J,mk ,X-mk )NjJJ
1 1 m ( o \ J L v ( 1 1 - 3 0 )✓ t A | t A—mo n \ J v widx w+ ' + B = l B1/ 2(J ,0 ,X )B 1/ 2(J,m X-m,)wPJ
J=X
Part of the coupling of the two spaces is achieved merely by calculating two-body matrix elements Hnp and between the bases. The only part for which further calculations must be performed is the solutions for the matrices N-7. These again, involve only separate basis calculations.The inhomogeneous parts of equations (11-21) for the total system are calculated using equation (11-29), and the solution by successive elimination follows immediately.
The utility of the above polynomial approach cannot be20overemphasized. The Hilbert space for Ne , for example, is
reduced from a required consideration of 2498 neutron-protonM=0 states, to one of 40 M =0 and 40 M =0 states. For the * n peven-even, N=Z nuclei, as will be treated later, consideration need not even be given to the proton states, since the matrices
45
developed in that space are identical to those developed in the neutron space. These factors cut typical computational times by a factor of the order ten or more. This allows calculations to be performed in the middle of the 2 i - 1 d shell which were not possible with previous techniques.
Having developed the theory and formalism of angular momentum projection techniques, we turn now to an application of these techniques to the study of the connection between microscopic and macroscopic models of nuclear structure. In particular, in the following chapter consideration is given to simple microscopic configurations interacting through phenomenological forces which have the main characteristics of the nucleon-nucleon force within the nucleus. A study is made of the development of collective phenomena within the framework of this microscopic model.
46
CHAPTER III STUDIES IN SINGLE j CONFIGURATIONS
1. IntroductionThe number of different states of a multi-particle
system increases with an increase in the possible angular momentum values of the individual particles comprising that system. Eigenstates of the Hamiltonian will, in general, be linear combinations of these states. As the number of particles, and therefore the number of degrees of freedom, is also increased, it is expected that some of the particle coordinates will find a natural grouping in collective coordinates, resulting in the possibility of exhibition of collectivity in the energy spectrum, transition, and multipole properties of the nucleus. For example, a two particle system interacting via a central potential has a wave function which may be considered as the product of a rotation function and an intrinsic function. The spectrum of such a system does not, in general, appear collective because these two modes of excitation are strongly coupled, and the energies associated with them are of the same order of magnitude.
Throughout the nuclear mass table, many phenomena are observed— ground state spins, magnetic moments, excited states, magic numbers, etc.— which have natural explanation in an individual-particle model. At the same time, among the particle-like spectra are found phenomena of an undeniably collective nature, such as rotational and vibrational excita
47
tion spectra, which find natural explanation in a collective model.
As has been previously mentioned, exact shell model calculations become prohibitive in both time and required computer memory storage demands, even when considering only a few nucleons outside of a closed, inert core. For this reason, a suitable truncation of the very large shell model Hilbert space— a model space— must be chosen. It is desirable to have a relatively simple means of control of both the particle angular momentum and the number of nucleons in the system, while maintaining the proper quantum mechanical properties of the nucleons. A model space which satisfies these requirements is that of ( j ) n- - n particles, degenerate in their single particle energies, restricted to one value of the single-particle total angular momentum /. For simplicity only neutrons will be treated.
Utilizing the projection techniques developed in the previous chapter, the emergence of collective behavior from an individual particle description of the nucleus will be examined. Studies along these lines have been carried out mainly for spinless particles.29 This represents an unjustified oversimplification of the situation. For example, the inclusion of spin and the spin-orbit force splits the
and *d3/2 -'-eve -s by approximately 5 MeV, which would otherwise be treated as degenerate. Bargmann30 introduced spin in the form of a spin-orbit force, which however, tends to destroy the L-S coupling scheme used in his calculations.
48
49
It is desirable to consider the intrinsic spin of the nucleon explicitly. This is accomplished in the calculations reported here by the employment of the j - j coupling scheme.The deformations produced in this particular model have been investigated by Baranger31, but eigenstates of the system were not found.
The complete (/)n space is considered. Projection techniques are applied to generate all of the exact eigenstates of the ( j ) n system interacting through a phenomenological Hamiltonian containing varying mixtures of short and long range components. These will be shown to be in agreement with the general predictions of a collective model in which individual particles are coupled to a nuclear core having a quadrupole distortion (v. Appendix A-II). Nomura32 has used the seniority scheme to obtain approximations to the low-lying excitation spectra in this model space. This procedure will be shown to be invalid by generating an exact seniority analysis of the eigenstates of the Hamiltonian.This analysis can also be used to determine the degree of pairing and therefore of sphericity of the wave functions.This point will be discussed in detail later in the chapter.
Considering the shell model, (/)n configurations may exist in the regions of very low excitation of nuclei with one closed shell and a few nucleons in or 1g9^configurations. However, past calculations have shown that such a configuration represents a poor approximation to the true situation. Therefore, no attempt will be made to match
the results of the calculations reported here to experimental
data in these regions. Rather we shall concentrate, as sug
gested above, on an attempt to elucidate the emergence of
general collectivity from this microscopic basis.
2. Pairing + Quadrupole Model
a. Introduction
Elliot33 has shown that collective static deformations
and attendant rotational excitation spectra are obtained for
nuclei far from a closed shell if the nucleons, moving in a
mixed configuration space produced by an harmonic oscillator
potential, Interact via a two-body force of angular dependence
?2 (cos 0). This force also gives rise to quadrupole vibrational
excitation spectra for nuclei with or near closed shells.
Dayman31* showed that this exhibition of collective properties
is a general characteristic of the /-/ coupling scheme arising
from interactions having slow angular dependence, such as
P2 (cos 0); it is independent of the use of harmonic oscillator
states and of the particular radial dependence of the interac
tion; it is dependent upon the adiabatic condition that intrin
sic excitations are much greater than rotational ones. The
low multipoles of the force, which are associated with a long
range, represent that part of the interaction where the
effects of many nucleons upon a single given nucleon are ex
pected to contribute coherently, those contributions from
higher multipoles having a tendency to cancel because of their
dependence on the rapidly varying spherical harmonics.
50
There are, however, additional interaction effects which
cannot be included in a nuclear deformed field. They may be
inferred from the following observations: (IL) Surface deforma
tions submerge with an approach to closed shell regions.
(ILL) Even-even nuclei invariably display J=0 ground state
spins. (-ii-L) The low-energy excitation spectrum for e/en-
even nuclei is particularly simple. There is an energy gap,
corresponding to the energy required to break a J=0 pair,
below which only collective states appear. (-tv) The last
nucleon is less strongly bound in an odd-mass nucleus than
in the neighboring even-even nucleus, where it can form a
pair. These effects arise from the relatively short range
part of the two-body interaction.
In view of the fact that many nuclear phenomena can be
explained by considering the nucleon-nucleon interaction to
consist of a combination of these two simple forces— a short
rang pairing force and a long range field producing force—
a particularly simple model has evolved. Since the nucleon-
nucleon interaction is a complicated entity, a model in
which the interaction is represented by these two phenomeno
logical components cannot be expected to yield quantitative
properties of nuclei in precise agreement with experimental
data. The model does, however, predict the main qualitative
features of the observed spectra35. In particular, it accounts
for the generally observed, gradual transitions from closed
shell spherical region to regions of static deformation, and
51
it seems to work well in the Samarium region, where abrupt
shape transformations are observed.
Three types of nuclei are distinguishable by their
resistance to deformations of the quadrupole type. Starting
with a doubly closed nucleus exhibiting spherical symmetry,
the addition of extra nucleons causes the onset of deformations,
first of the (-c-c) dynamic type, and then finally of the (xXt)
static type. The rate of this transition from spherical sym
metry to permanent distortion depends on the region of the
nuclear periodic table being considered, but the three steps
are broadly characterized by: (Z) The near-closed shell
nuclei exhibit low-lying nuclear structure (i.e., energy
levels, transition rates, moments, stripping and pickup
cross sections, etc.,) due primarily to the valence nucleon(s).
(iJ.) Adding more nucleons produces a stronger net force on
the core, resulting in a dynamic deformation and shape oscil
lations interpretable through a vibrational excitation spectrum.
(ill) The addition of still more particles produces a perma
nent static deformation of the core, which may exhibit rota
tional motion. A further increase in the number of nucleons,
as the shell becomes filled, causes the situation to revert
back to (-Ac) and to {<.) above, and finally to sphericity for
the next closed shell.
Near ciosed shells the short range pairing interaction
dominates over the long range, field producing forces, resul
ting in spherically symmetric systems. While each pair of
52
53
particles is on the average spherically symmetric, the pair
will, at all times, be undergoing fluctuations in which it
is in a non-spherical configuration. If the long range
force is not too important, these fluctuations occur inco
herently and independently, resulting in a spherical overall
system. Adding more particles to the system increases the
importance of the long range forces^, so that the fluctua
tions of one pair will result in a time dependent non-
spherical field, inducing fluctuations in other pairs which
will be coherent with those of the first. The system thus
undergoes shape oscillations.36 With the addition of still
more particles, until the middle of the shell, the long
range force becomes still more important and its effeict
overrides that of the pairing interaction, giving rise to
statically deformed nuclear shapes.
b. The Quadrupole-Quadrupole Interaction
The central part of any two-body interaction potential
between nucleons may take the form
v l Vk (r1>rj)Pk (coS
where Pk is the Legendre polynomial and is the angle be
tween the vectors r^ and r ^ . Expanding V k as a power series
in gives
^ The number of long range correlations in an N particle system is N(W-71/2, while the number of pairs is W/2.
v = I i I vmn r i +m I y * y ( 01 »<<>1 ) r j +n ^ <e. ,4 * . ) ( I I I - l )i< J k mn y y i i j y J j
k lrwhere are expansion coefficients, y* are spherical har
monics, and (r^,0£,$^) are the polar coordinates of the 1th
particle. At first glance such an expression appears objec
tionable because it diverges as the radial distances grow
large. This is certainly so. However the wave functions go
to zero much more rapidly at distances large compared to a
nuclear diameter, so the divergence of the expansion is of
little consequence. It is also to be noted that negative
powers of r^ and/or r^ are not permissible because the poten
tial would then represent an infinitely hard core, which is
not generally accepted as an effective interaction.
The low multipoles of the force are associated with the
relatively long range part of the interaction. Kugler37 has
shown that in the long-range limit any reasonable potential
contains a considerable amount of the above quadrupole-quad-
rupole interaction. If only initial and final states of the
same parity are considered, the dipole-dipole interaction
k=l term vanishes. This leaves the quadrupole-quadrupole
Interaction term k=2 as the dominant component having long
range characteristics. Elliot33 has obtained the exact solu
tion for this term, as well as a classification scheme. It
is, however, limited to spinless particles, the coupling
scheme being associated with the group of three dimensional
unitary transformations (v. Appendix A-I ). The coupling
scheme groups together states whose L-values are just those
of rotational bands cut off at some upper value of L.
That these states do, in fact, belong to a rotational band
was shown by exhibiting them as Hill-Wheeler integrals (v.
section II.2) over a single intrinsic function having a good
quantum number K, the component of angular momentum along
the intrinsic z-axis. With the above considerations, the
quadrupole-quadrupole force was chosen to represent the long
range part of the Interaction:
V a d = VQ r2 C y2( ° x ) ^ 2 ( ° 2) 30°
= VQ *i r 2 I ( - D y *J(nx) ( n i - 2 )
where V q is a strength parameter; y2(ft1) operates on particle
number 1 and y2(ft2) operates on particle number 2, these
coupled to zero angular momentum. The matrix elements of
interest are
<(n/)SJ,M.«|Vquad|(n/)2,J,M»(l> -
V Q I X ( - l ) u m m 1 y
<ym , |y2 | /m></-m»|/2y |y-m > . (III-3)
The radial integral of r2 between the shell model wave func-+
tions, being constant for a single shell , has been absorbed
into the strength parameter V q . The dependence on orientation
may be removed from equation (III-3) with the help of the
tA shell is defined here as having only one value of the
single-particle angular momentum j.
55
Wigner-Eckhart theorem (11-16). Since the CG coefficient
C (J,0,J;0,0) is equal to unity, writing V^uad for the re
duced matrix element < (n/)2,J| |VqUaciI I
J i 2Vquad = VQ ^ C(/f/,J;m,-m)C(/,/,J;m',-m')
m=-/y=-2
C(/,2,/;m,y)C(/,2,/;-m,-y)</|\ y z \ | / > 2 where m'=m+y. Invoking symmetry relations of the CG coeffi
cients2 6
Vquad= VQ c (/»2,J>,P)C(j,j>J;m,,-ffl')
C(2,/,/;-y,-m*)C(/f/,J;m,-m)</||V2 ||/
The summations over the (artificial) indices m and y of the
CG coefficients may be replaced by a Racah coefficient
W (a,b,c,d; e., by means of the identity26
[ ( 2 e + n ( 2 r f + m l/2W(a,b,c,d;e,*) =
C(a,b,e;a,3)C(e,d,c;a+6,6)C(b,d,j(;8,6)C(a,)J,c;a,B+6) .
Thus TV a d - J ) V q w c y , z , j , / ; y , / ) < y 1 | y 2 j | y > 2 . . .
Noting that for identical particles, only even values of J
are allowed by the Pauli principle,
V a d - ( * / ♦ > ivQ w ( y , i , y , / ; 2 , j ) < y | | y 2 lii> 2 .
With the reduced matrix element (A-I-5)
V quad = - W r . W J + U W * ) . VQ W(/,/ ,/ ,/; 2,J)
j l j + ’ l
56
(III
where again the constants have been absorbed into the strength
parameter V q , and explicit use has been made of the fact that
the single-particle states all have the same parity, given by £
(-1) , where the value of Z would depend on the oscillator
quantum number n.
57
c . The Seniority Quantum Number and the Pairing Inter
action
Shell model wave functions | m a y be used to
characterize the states of a nucleon in a central field. The
states of a system of two non-interacting nucleons in a cen
tral field may be characterized by the product wave functions
\nl , l l , j1, m l> \ n2, l z ,j2,m 2>. If the particles are allowed to
interact through a Hamiltonian which mixes configurations,
the individual angular momenta 7i and 7 2 are no longer con
stants of the motion, and their projections m x and m 2 onto
the axis of quantization are, of course, no longer good
quantum numbers. Instead, the constant total angular momen
tum of the system 7 = 7i+7z and its projection M remain as
the only good quantum numbers. If the interaction is not
too strong, the individual orbital (Z) and total (/) angular
momenta may still be approximate quantum numbers. The con
figuration-mixing interaction will have no matrix elements
between states of different parity, i.e., (Zl+Z2 ). =
(£i+£2)i n i t i a l .
Carrying the development of the system one step further,
for a system of three interacting nucleons, the quantum num
58
bers j ly iiy S 3> J and M (suppressing now the quantum numbers
n and I) are no longer, in general, sufficient to fully de
scribe the states. The angular momentum of, say, the first
two particles J 12 = 7 i+?2 may also be used to describe the
system. If, however, the particles are equivalent, i.e.,
/ = j V j V / 3, there is no longer meaning to choosing a particu
lar pair of particles, and thus ? 12i°ses its physical signi
ficance. There are then not enough quantum numbers to describe
the interacting system completely.
One possible solution to this problem was given by
Racah38. He introduced the operator Q which measures the
number of particles coupled to zero angular momentum, and
classified the states of the system such that this operator
is diagonal. Jahn39 demonstrated that for short range
forces, the energy eigenstates are nearly eigenstates of Q,
and in the limiting case of 6-forces the states of the
system are simultaneous eigenstates of the Hamiltonian,
angular momentum, and the operator Q. This is understandable
since, for a short range attractive interaction, contribu
tions to the matrix element of the force will be greatest
when the particles are correlated in pairs with maximum
spatial symmetry and consequently antiparallel spins.
For spheroidal nuclei exhibiting the phenomenon of
"dynamic pairing" these "saturated" pairs coupled to zero
angular momentum contribute very simply to many character
istic properties of a state; a number of such pairs may be
added to a given system of particles without changing its
properties very much. It is, therefore, advantageous to
consider the number u of non-saturated particles, which
determine most of the properties of the state, plus a number
of saturated pairs. This number of non-saturated particles
u is called the "seniority number" because it gives the
smallest number of particles required for forming a state
with given properties, and thus specifies the simplest con
figuration which contains such a state in cases where dynamic
pairing is applicable. There are, however, regions where
dynamic pairing is not applicable. For example, in the region
O s ^ ^ - 0 s ^ Z the addition of a "static" pair of nucleons
alters the properties of the nucleus considerably, so that
while Os*®^ is highly deformed, 0 s ^ Z is spherical.1*0
The concept of dynamic pairing is employed in these
calculations, not so much because consideration is being
given to nuclei where this type of interaction predominates,
but rather because it gives rise to the concept of seniority,
which will be used to determine the degree of sphericity in
the wave functions arising from the deformed field part of
the interaction. The association of pairing with sphericity
may be understood from a consideration of two particles
moving in the time-reversed orbits |/,m> and 1,1 In
such cases the particles are (classically) in close proximity
to each other two times during each period of revolution
about the nuclear core (which gives rise to the shell model
states). If these particles Interact through a strong short
range force, they are frequently scattered into different,
but still time-reversed orbits. In this way the particles,
59
6o
The notation of second quantization is used here to
obtain the matrix elements of a particular short range
pairing interaction in the seniority scheme.'*2 The closed-
shell or vacuum state is denoted by |0>. The creation
operator a* is defined as that operator which produces a
particle of spin / with z-projection m when operating on the
vacuum state:
a* |0 > - |j,m>.
The corresponding annihilation operator am , of course, gives
zero when applied to the vacuum state. These fermion creation
and annihilation operators obey the anticommutation rules
{a* a.,) = <S , { A V . ) * - 0. (III-5)m m mm m m m m
The operators which create a pair of particles with spin J
and z-projection M are defined by
a jm * • (III=6a)
Since the particles being considered are identical, only
even values of angular momentum J are allowed. The associated
annihilation operator is defined as
The commutation rules for these operators may be calculated
by invoking the definitions of the operators, with the result
within only a few cycles, cover the entire angular range,and the average mass density distribution is spherical.
61
+ C(/,/,J;M-y,y)y
C C / ^ J ' j y . M ' - y J a
We note that because the second term ends with an annihila
tion operator, when the commutator is applied to the vacuum
state- this term gives no contribution. Using equation (III-7)
obtains
which gives the number of particles in the state, denoted by
N. Defining the quantity ft = /+!/2, the above commutator
becomes
Experimental data from spheroidal nuclei typically show a
depressed J=0 ground state. Other observations indicating
the presence in the true nucleon-nucleon interaction of a
pairing contribution are discussed in the introduction to
this section. We therefore chose a Hamiltonian which will
depress two particle states of zero angular momentum but
leave other states unaffected. Such an interaction is given
by
where G is a strength parameter.
In order to find the two particle spectrum, the pairing
Hamiltonian is applied to a state of spin J and projection M:
The operator well known fermion number operator
(III-8)
62
Vpair4JMI0> = - G( ^ ~ ~ ) A00A00AJmI0>‘4
Since A^^|0> = 0, the term 10> may be added with no
effect. Thus,
VpairAJM10> * .
For J£0 the commutator applied to the vacuum state vanishes,
leaving two-particle levels of J*0 unaffected. For J=0
VpairA00|o> " -0!MSol°> (111-10)
since the number operator W applied to the vacuum state gives
zero. Thus A ^ | 0 > is an eigenfunction of Vpalr with eigen
value -Gft. The spectrum of vpair is degenerate for all
values of J*0, and the J=0 level is depressed for G>0, by
E=-Gft.
The generalization of the above results has been carried
out1*3 with the result that the energy for an N particle sys
tem of seniority o is given by
E(N,u) = -|(N-u)(2fl-N-u+2) . (III-ll)
Inspection of this equation yields the following conclusions:
(.t) For N<<ft the levels of lowest seniority lie lowest, with
energy increasing with u and level spacings M5. For every
pair which couples to zero, we gain Gft in energy. The ex
clusion principle, however, prevents the process from continu
ing indefinitely. (-cc) The level density is predicted to be
very low near the ground state, although the total number of
levels increases with N. (jLIa,) The energy gain for the ground
state (u=0) is, for N<<ft proportional to N, in contrast to
the energy of a quadrupole interaction, which Is proportional
to N 2.
Casting equation (III-ll) in a slightly different form:
E(N,u) = - | [N(2ft+2-N) -i) (2ft+2-u)] (III-lla)
shows the very similar dependence on the number of particles
and on the seniority more explicitly. Clearly, this expression
for constant N is minimized for u=0, which can only occur for
the J=0 ground state, since all pairs will then be coupled
to ■Jpaij,''0 * Figure 1 shows the parabolic dependence of the
energy E(N,u) on both the number of particles N and the senior
ity u. The graphs for both N and u are symmetric and peaked,
not about the middle of the shell (ft=/+7/2), but rather about
ft+1. The seniority (upper) curve, however, does not pass
beyond ft or ft-1, depending on whether ft is even or odd, re+
spectively. The number (lower) curve, of course, does not
pass beyond 2ft=2/+J, which is the maximum number of particles
allowed in the shell. Keeping seniority fixed, the energy
(III-lla) decreases with increasing number of particles until
N=ft+1, after which it increases, but does not return to its
original value for N=2ft. Instead, there is a residual pairing
energy given by -Gft, which, coincidently, is the same as the
lowering of the J=0 level in the two-particle spectrum.
63
3. The Hamiltonian of the Individual-Particle Model
The Hamiltonian used in these calculations is a linear
combination of the pairing (III-9) and quadrupole-quadrupole
(III-2) interactions discussed in the previous section. We
Figure 1. Variation of pairing force energy in a (j)n configuration with the seniority and the number of particles N. The seniority curve terminates at ft-1 for /+I/2 odd, and at ft for y+7/2 even. The number curve terminates at 2ft, leaving a residual pairing energy associated with a closed shell of -Gft.
PAIRING
FORCE
ENERGY
-G £2
o o*
a = - ^ r ( 2 i l + 2 - N ) + - f - U U 1 ) 2 4 4
b = --r-(2G + 2 - N )
64
h(x) = LilLZlL(i_x)vpalr + (x) vquad. ox<x<i (111-1 2)Fnorm
The normalization factor (/+//2)/E , yields, for G=1 MeV,J norm' J * *a separation energy of 1 MeV between the highest and lowest
seniority states. E is E(N,0) or E(N,1) for even or oddnorm ’
N, respectively.
The single-particle states used are shell model
states. Since the particles are restricted to a single j
shell, the only quantum number needed to describe the individual
particle states is the magnetic quantum number. As discussed
in section II.1, the intrinsic states all have M. .=0 or* -cn t
h n t * 1 / 2 -
The projection techniques discussed in section II.3
have been applied to the above problem for 5/2</<15/2, to
yield the complete exact spectra of the Hamiltonian (111-12).
The energy matrices were calculated parametrically in the
strengths of the two-body angular momentum components of the
matrix elements of the Hamiltonian, so that the N-body
matrices of any other two-body force may, at this point, be
easily obtained from its two-body matrix elements. The com
puter codes used in these calculations are presented in
Appendix B-I.
define a mixing parameter X by the following:
65
4. Seniority Composition of Wave Functions
For X=0— for the pairing interaction (III-9)— seniority
is a good quantum number. If the overlap of the eigenvectors
of the Hamiltonian (111-12) for X#0 with those for X=0 is
calculated, the exact seniority composition of the eigenstates
may be obtained for the entire range of the mixing parameter
X. This may be seen as follows.
The time independent Schrodinger equation may be written
for the system for any value of the mixing parameter X:
H(X)^(X) = E^(X)^(X) (111-13)
where H(X) is the Hamiltonian (111-12). The wave functions
of total angular momentum J are further labeled by the sub-
script k, and are continuous functions of X . In particular,
for X-0, the Hamiltonian is a pure pairing interaction, so
that the eigenfunctions may also be labeled by the seniority
u:
H ( 0 ) ^ » U (0) = E^(0)i|^’u (0) . (III-14)
The wave functions of (111-13) may be expanded in terms of
the seniority eigenfunctions of (111-14) at X=0:
^ ( X ) = I ( i n - 1 5)^ p \)
The amplitudes may be easily obtained by taking the over
lap of (111-15) with the seniority functions:
< ^ ' U (0)|i^(X)> = I fa ^ ( X ) < ^ ,U(0)|^'°'(0)>.& p V)
^ The continuity of the eigenfunctions is guaranteed if the operator H is holomorphic in the parameter X.1*5
Since the seniority functions form an orthonormal basis,
this reduces to
< ^ ' U( 0 ) | ^ ( X ) > = a ^ i ( X ) *
The desired numbers are the intensities of the seniority mix
tures for the wave functions of (111-12):
Xk ’V (X) = £ (ak ; i ( X ) ) 2 = I C < ^ f V (0 ) l ^ ( X ) > J 2 . (1 1 1 - 1 6 )
In this manner, the exact seniority mixtures have been
obtained for all states throughout the range of the mixing
parameter X.
5. Results of Calculations - Even Number of Particles
a. Distortion of Rotational Bands at High Angular
Momentum Values
The calculated spectra of the Hamiltonian (111-12)
(Figures 2-8) may be analyzed with respect to a variation of
the two parameters / and n. Since the main concern is with
a comparison of these projection calculation results- with the
predictions of a collective model (v. Appendix A-II), consid
eration is given only to the quadrupole-quadrupole Interaction,
which Is known to give rise to collectivity because of its
relatively long range.
The association of a long range interaction with non-
spherical equilibrium shapes of the nucleus leads to the
expectation that the spectrum of the quadrupole-quadrupole
interaction in (/)n configurations would be highly rotational,
i.e., it would follow the J(J+I) rule for rigid rotators. If
66
67
we consider comparing the calculated energy levels to a
J(J+7) spectrum, with the moment of inertia parameter obtained
from the excitation energy of the first J=2 state relative
to the ground state, we find that the calculated excited
states are all depressed relative to those of the rigid rota
tor. Figure 9 shows the ratios = (E^-Eq )/(E2-Eq ) and
R62 = ^E6_E0^‘ E2'‘Eo L where Eq , E2 etc. are the energies
of the first J=0, 2, etc. levels, respectively. Also shown
are the pure rotational limits R ll2=10/3 and Rg2=7* We see
that increasing the particle angular momentum / has a pro
nounced effect on the deviation of the calculated spectra
from the rotational limits. We can make the observation that
the nuclear configuration becomes more rigidly deformed with
n 4an increase in /. However, even for (/) = (15/2) (which
might occur in the low-energy regions of excitation only of
heavy mass nuclei such as are found in the lead region), the
deviation from the rotational J(J+J) energy dependence is
significant at and above J=8. This type of distortion of
high angular momentum states of rotational bands is a rather
common experimental observation. It has also been interpreted
as a cut-off rather than distortion at high angular momentum.
In order to find the reason for the deviations - from a
pure rotational spectrum noted above, the exact form of the
quadrupole-quadrupole force should be examined. The only
angular momentum dependent factor in the two-body energy
expression (III-4) is the Racah coefficient W(y///;2J). The
closed expression for this coefficient is1*1*
W(///j;2J) = - 6 flliilLi] {A(A+1) - ik/(/*I)3*L (III-17)M 2 / + 3 ) " 3
whereA = = | </!.J|?-?l/2.J>.
It is easily seen that the coefficient has the form
W = a(/)[J(J+n]2 + 7 (J+7) C 7-4/(/+ 7) ] + b(/) (III-1 8)
where a (/) and b (/) tend to constants with increasing j. Thus
the pure rotational J {J+ 7)energy dependence is not even
indicated in the two-particle spectrum except for j » J . We
can therefore expect the energy spectrum to follow the J(J+/)
rule only for the first few values of J. The dependence for
higher J-values is given by
W 'v, J(J+H - c (/)J2 (J+ 7) 2
which is reminiscent of a nonrigid rotator for which the
equatorial diameters increase while the polar diameter shortens,
thus causing a slight decrease in the energy. This is just
the expression describing the rotational states of an axially
symmetric nucleus including rotation-vibration interaction.
Of course, for j becoming very large, equation (111-18)
effectively predicts the rotational J(J+I) energy dependence
exactly.
The Racah coefficients, for various values of the par
ticle angular momentum j are presented in Figure 10. Except
for multiplicative factors, which change the scales for dif
ferent values of j, these are the two-particle spectra. Here
the deviations from pure rotational spectra for higher J-values
is very evident. Indeed, the rotation-like bands may be con-
68
sidered to be cut off, not at the maximum value of angular
momentum permitted by vector coupling, but rather at some
lower value. For example, the highest rotational state for
(7/2) is the J=4 level, rather than the maximum J=6 .
Figure 11 is a convenient way of presenting results in
the form of a comparison with those of a pure rotator. In
this diagram Rg2 versus R^2 is plotted (cf. Figure 9) for the
2-, 4-, and 6-particle systems for the range of j considered.
We note that Rg2 for (7/2 )** is not shown because it is believed
the rotational band in this problem should be cut off at J=4,
as discussed above. A perfect rotor would lie at the point
(7,10/3).
We have thus shown that the experimentally observed
distortions of rotational band structures, which have previously
found explanation in rotation-vibration competitions, may be
considered to arise solely as a result of the long range
part of the effective interaction. The vibrational modes of
excitation, which are generally associated with pairing, and
consequently with the short range interactions, will be shown
below to arise from the long range part of the effective
interaction.
b. Seniority Analysis and Vibrations Arising from the
Long Range Interaction
Beginning with an empty shell in regions of static pairing,
adding a few particles generally polarizes the core, thus intro
ducing an effective deformation, and adding still more parti-
69
cles (up to the middle of the shell) generally increases
this deformation and makes the nucleus more rigid. From
Figures 4, 7 and 5, 8 , it may be seen that for configurations
of particles described by a single angular momentum, inter
acting through a pure quadrupole force, increasing the number
of particles from four to six (closely approaching the middle
of the shell) increases the distortion of the rotational
spectrum until it appears quite vibrational. We are forced
to conclude that the long range part of the nuclear force
causes nuclei approaching the middle of a shell to have
vibrational modes of excitation in competition with rotational
modes as they undergo transition from prolate to oblate defor
mations. An explanation of this phenomenon follows from a
seniority analysis of the eigenfunctions of these configura
tions. 1
The low-lying levels are not good seniority states (v.
Tables I-VII), but seniority does give a fair classification
of the low-energy region of excitation. The states do have
a major component of seniority u=J. The intensities of these
components may be as low as 0 .6 , so that producing, say, the
first J=4 level by a quadrupole excitation of the first J=2
level considered as a pure seniority 2 state is not at all
valid. It is seen that the seniority of this J=4 level is
usually very pure (>95%), while the seniority mixtures of
the J=0 and 2 levels increase with single particle angular
momentum /. For />ll/2, although the low-lying states de
velop continuously from pure pairing u=2 states, they are
The number of pure seniority states increases suddenly
for n=(2/+7)/2, the middle of the shell (v. Tables I and VI).
This is not a characteristic of the quadrupole-quadrupole
force alone, but rather, as shown below, of any two-pa.’t i d e
scalar operator which may be given as a sum of scalar products
of tensor operators:
no longer u=2 states at X=l, I.e., for a pure quadrupole-quadrupole interaction.
The particular Interaction discussed here, of course, has only
k=2. A simple expression may be obtained for matrix elements
It may be observed that these vanish for n=(2/+7)/2, for which
u<(2/+7)/2; they also vanish for n=u={2/+J)/2. As a conse
quence, many matrix elements which are non-diagonal in senior
ity are identically zero, resulting in pure seniority states
in the middle of the shell. This may be considered to re
present an increase in the amplitude of the J=0 part of the
long range quadrupole force, and an increasing tendency toward
sphericity in the middle of the shell (cf. section III.2c).
This trend toward sphericity is accompanied by a decrease in
V 1 2 = E fK ( D - r K (2).
of (/)W in the seniority scheme in terms of the (/)u matrix
elements;4 6
i<h Tih
2j +1 - 2n i n - v + Z ) (2y+3-n-u) < -u
2/+J-2u ^ 2(2y+3-2u)<y u ,u,a,J|Ei<hV i h |yu , u - 2 ,a'
(111-18)
J
the frequency of shape oscillations, and consequently a
lowering of vibrational modes of excitation. This may be
understood from a consideration of equation (A-II-la),
which shows that the effective equilibrium radius R q of the
nuclear surface is increased with the approach, in the middle
of the shell, toward sphericity. Equation (A-II-4) then
shows that an increase in the equilibrium radius is accompa
nied by a lowering in the frequency of shape oscillations,
and therefore a lowering of the energy associated with vibra
tions of the nuclear surface about some equilibrium distortion.
Experimentally, no such trend toward sphericity is
observed. For example, nuclei in the first half of the 26-1dp Q
shell have prolate equilibrium shapes. At Si there is an
abrupt change to oblate shapes, which continue throughout
the latter half of the shell. The fact that the associated
vibrational spectra are not observed In mid-shell regions
reinforces the hypothesis that nuclei do not contain low-
energy (/)n configurations (cf. section III.l).
72
Figure 2. Pairing + Quadrupole spectrum of (7/2)4 configuration for complete range of mixing parameter X. Pure pairing states are labeled by.seniority u. All states are labeled by spin J.
MIXING
PARAMETER
ENERGY (MeV)
Figure 3. Pairing + Quadrupole spectrum of (9/2) configuration for complete range of mixing parameter X. Pure pairing states are labeled by seniority u. All states are labeled by spin J.
ij
ENER
GY
(MeV
)
XMIXING PARAMETER
Figure 4. Pairing + Quadrupole spectrum of (11/2)^ configuration for complete range of mixing parameter X. Pure pairing states are labeled by seniority u. All states are labeled by spin J.
XMIXING PARAMETER
Figure 5. Pairing + Quadrupole spectrum of (13/2)1* configuration for complete range of mixing parameter X. Pure pairing states are labeled by seniority u. All states are labeled by spin J.
XMIXING PARAMETER
Figure 6. Pairing + Quadrupole spectrum of (15/2)** configuration for complete range of mixing parameter X. Pure pairing states are labeled by seniority u. All states are labeled by spin J.
XMIXING PARAMETER
Figure 7. Pairing + Quadrupole spectrum of (ll/2)b configuration for complete range of mixing parameter X. Pure pairing states are labeled by seniority u. All states are labeled by spin J.
XMIXING PARAMETER
Figure 8. Pairing + Quadrupole spectrum of (13/2)^ configuration for complete range of mixing parameter X. Pure pairing states are labeled by seniority u. All states are labeled by spin J.
° V# G V r -
Figure 9. Comparison of Projection Calculations to Rigid
Rotator Data. The curves are the ratios R^^ = E 4“E2 ^ (E 2—E0 and Rb2=(Eg-E2 )/(E2-E0 ), where EQ , E2 , etc., are the excita
tion energies of the first J=0, 2, etc. levels.
PARTICLE
NUMBER
8
<0it 4 c
0 L
8
•• 4C
0 L-
8
CJii 4c
I
R 62
R 4 2
R 6 2
R 4 2
R 6 2
1 1 J5 / 2 7/2 9/2 11/2 13/2 15/2
]S I N G L E - P A R T I C L E A N G U L A R M O M E N T U M
Figure 10. Two-particle Quadrupole-Quadrupole Spectra for 5/2</<J5/2. The energy spectrum is simply proportional to the Racah coefficient W(////;2J) (v. equation (III-4)).
0.08 -
0 . 0 6
0 . 0 4
0 . 0 2
0 . 0 0
4,6
8
8
10
108
612
•1012
8
6,14
_ - 0 . 0 2
OJ • ^
- 0 . 0 4
- 0 . 0 6
2
0
- 0 . 0 8
-0. I 0
■0. I 2
-0. 14
-0. 16
J i i i I I I I I I 1-----15 / 2 7/2 9/2 11/2 13/2 15/2
J S I N G L E - P A R T I C L E A N G U L A R M O M E N T U M
Figure 11. Comparison of Projection Calculations to Rigid Rotator Data. EQ , E2 , etc., are the excitation energies of the first J=0, 2, etc. levels. 2P, 4P, and 6P represent 2-, 4-, and 6-particle configurations'.
0UJ1CM
UI\0
UJ1<0
UJ
E 4 - E q / E 2 - E o
Quadrupole-Quadrup ole Int e ra c t ion
4Table I - Seniority Analysis of (7/2) Configuration
J i)-0 u=2 u=4
0 1 - -
2 - 1 -
4 - - 1
4 - 1
6 - 1 -
2 - - 1
8 - - 1
4Table II - Seniority Analysis of (9/2) ConfigurationQuadrupole-Quadrupole Interaction
u S0
0
2
46
6
48
2
10
8
46
0
1 2
,9390
, 0 6 1 0
u® 2
.9313
. 9 1 6 6
.8462
. 9 2 0 0
.0687
. 0 8 0 0
.1538
.0834
. 0 6 1 0
.06871
1
.0834
.1538
.0800
.93131
. 9 2 0 0
.8462
. 9 1 6 6
.93901
Table III - Seniority Analysis of (11/2)** ConfigurationQuadrupole-Quadrupole Interaction
J u -0 ua40 .8368 - .1632
2 - .8437 .1563
4 - .0149 .9851
6 - .1696 .8304
8 - .2140 .7860
8 - .5742 .4258
2 - .1291 .8709
4 - .8372 .1628
10 - .9101 .0899
6 - .6832 .3168
12 - - 1
10 - .0433 .9567
4 - .1080 .8920
8 - .0025 .9975
6 - .0083 .9917
0 - .1632 .8368
14 — - 1
6 - .1390 .8610
16 ' - - 1
8 — .2092 .7908
4 - .0399 .9601
2 - .0273 .9727
10 - .0466 .9534
12 — — 1
v
Table IV - Seniority Analysis of (13/2)^ ConfigurationQuadrupole-Quadrupole Interaction
J u®0 u = 2 v=4
0 .6698 .3302
2 - .7486 .2514
4 - .0445 .9555
6 - .0850 .9150
10 - .1874 .8126
8 - .2517 .7483
2 - .1452 .8548
10 - .4400 .5600
4 - .7489 .2511
12 - .8478 .1522
8 - .4461 .5539
6 - .7246 .2754
12 - .0008 .9992
14 - - 1
4 — .0380 .9620
0 — .3302 .6698
6 - .1431 .8569
16 — - 1
18 ■ - - 1
14 - - 1
8 - .2399 .7601
20 - - 1
16 — — 1
Table V
J u=0 u*2 u*4
4- Seniority Analysis of (15/2) ConfigurationQuadrupole-Quadrupole Interaction
0 .5683 - .43172 - .6902 .3098
4 - .0795 .92056 - .0392 .9608
12 - .1720 .8280
8 - .2077 .7923
10 - .2511 .74892 - .1036 .89644 - .5986 .4014
14 - .6781 .3219
10 - .3097 .69030 .4284 - .5716
16 1
12 - .0251 .974924 - 1
22 1
1 8 1
14 .0066 .9934
20 1
Table VI - Seniority Analysis of (11/2)^ ConfigurationQuadrupole-Quadrupole Interaction
0
2l«
6
2
6
48
1 0
6
8
0
12
410
6
8
41410
6
12
1 2
16
46
140
8
10
418
u®0
.8755
u g 2
.9663
. 0 0 3 6
.4512
.6639
.6853
.6748
. 2 8 3 6
.3672
.0411
.1245
.0416
.0931
. 1 8 1 6
u - 4
,1245
1
1
1
1
1
1
1
1
1
1
1
1
1
,8755
u g 6
.0337
.9964
.5488
.3361
.3147
.3252
1
.7164
. 6 3 2 8
1
.9589
.9584
.9069
.8184
1
Table VII - Seniority Analysis of (13/2)b ConfigurationQuadrupole-Quadrupole Interaction
J u 30 u =2 o=4 u -60 . 6 8 3 6 - . 2 6 2 2 .0542
2 - .7236 .1741 . 1 0 2 2l\ - .0086 . 8 0 8 0 .1834
6 - .0347 .0459 .9194
8 - • .0227 .1596 .8177
2 - . 1 6 6 1 .6103 . 0 2 3 6
8 - .0249 . 7 8 1 0 .1941
4 - .3338 .1695 .4967
10 - . 0 1 7 8 .7420 .2402
10 - .4686 .0367 .4947
12 - .4880 .0644 .4476
6 - .4386 .1291 .4323
0 . 1 0 3 2 - .0123 .8845
4 - .0369 .5440 .4190
14 - - .8404 .1596
8 - .0354 .1667 .7979
1 2 - .0487 .6511 . 3 0 0 2
1 0 - .0517 .4039 .5444
6 - .0004 . 6 3 6 6 .3630
4 — . 2 7 1 6 . 1 1 2 0 .6164
2 - .0223 .1736 .8041
16 — - .1299 .8701
1 8 - - .1349 .8651
20 - - .7929 .2071
0 .1679 - .4617 .3704
22 - - - 1
20 — - .2071 .7929
24 _ — 1
6. Results of Calculations- Odd Number of Particles
The complete quadrupole-quadrupole spectra for /*9/2
and /*M/2, and the low-lying spectra for /*/3/2 and /* 7 5/2
are shown for w=3 and 5 particles in Figures 12 and 13.
Of special Interest is the approach, with increasing particle
angular momentum /, to an evenly spaced band of J*K, K+l,
K+2,.... In the shell model these bands would begin at K=/
because of pairing effects of the short range part of the
nuclear interaction. However, here the absence of this
short range part of the force is conspicuous by the bandsC
beginning with K=/-2 for (/) configurations, and with K=/-/O
for (/) configurations. This is true in the cases shown
except for /=9/2, which reflects the fact that n=5 marks
the middle of the 9/2 shell. It appears that the farther
from the middle of the shell the cleaner this band will be.
A spectrum beginning with K=/ can certainly be obtained by
choosing values other than unity for the mixing parameter X,
i.e., by including short range interactions. It is to be
noted that these results are in agreement with a collective
model (cf. Appendix A-II) for oblate deformations.113
At first glance, it might appear that the surface-
coupled model cannot be applied to the (j)n calculations
since the system contains too few particles, and thus the
core is conspicuously absent. There are two ways out of
this predicament. First, the "core" may be replaced by the
average field it produces which, in Its turn, produces the
good quantum numbers j of the individual particles. Particle
73
Figure 12. Complete quadrupole spectra of (9/2)3, (11/2)
(13/2)3, and (15/2)3 configurations.
ENERGY
(MeV)
27/2
1.0
0.5
-0.5
-1.0
1.5
15/2 21/215/2
■ 2 7 / 2
•15/2
•21/2
21/29 / 2
•17/2
3 / 2
5 / 2
13/2,11/2
• 9 / 2 • 2 3/2
• 2 7/2
•3/2
•17/2• 11/2• 19/2
5 / 2■7/2
•9/2,23/2- 2 9 / 2" 1 7 / 2
.,3/2" 2 5 / 2
■5/2' 3 3 / 2
11/2
• 13/2
■ 21/2
19/2■ 9 / 2• 7 / 2
15/2
13/2
9 / 2
■17/2
15/2
7 / 211/2
9 / 21 3 / 2
11/2
I___L
•21/2
• 3 3 / 2
•15/2
■ 2 3 / 2 , 2 9 / 2
9 / 2.17/2-31/2^ 3 5 / 2 , 3 / 2
11/2-5/2,19/2 :2 7 / 2 , 2 5 / 2 7 / 2
• 3 9 / 2 rl 3 / 2 15/2
2 3 / 2
• 11/2 r 9 / 2
21 /2
19/2
17/2
15/2
1 3 / 2
9 / 2 11/2 J 13/2 15/2
S I N G L E - P A R T I C L E A N G U L A R M O M E N T U M
Figure 13. Quadrupole spectra of (9/2)^, (11/2)^, (13/2)^ (15/2)^ configurations.
ENERGY
(MeV)
2 .0
1.5
.0
0.5
-0.5
- 1.0
-1.5
-2 . 0
-2.5 L
■ 25/2
• 9/2 ,13/2 19/2
■ 3/2 "7/2
17/2
•21/2,9/215/2
11/2"15/2
1/2
17/2
11/2
13/25/2
9/25/27/2
I__ L9/2
23/2y^f-17/2,15/2"25/2,9/2,7/2
'11/2 N5/2.I7/2
21/2'23/2,13/2
✓17/2-11/2-3/2"l9/2
13/2
11/2
■9/2■7/2
-15/2
,25/2-11/2
5/213/221/25/2,9/2,19/27/2
■ 17/2
■15/2
■ 13/2
- 11/2
-9/2
21/2,29/2
27/2
25/2
.11/2 ✓9/2 "23/2 7/2 -21/2
19/2
17/2
15/2
13/2
11/2J___I
11/2 13/2 15/2J
S I N G L E - P A R T I C L E A N G U L A R M O M E N T U M
74
excitations of this core are negligible compared to valence
nucleon excitations since the low energy excitations of
nucleons below the Fermi level are primarily collective,
neighboring particle states being fully occupied (cf. sec
tion 1.2b).
A different viewpoint may be assumed; the "core" may
be considered-made up, self-consistently, of the "extra-
core" particles (cf. Chapter IV). To appreciate this model,
we consider a system of two nucleons interacting via a
central force. The Hamiltonian for this system is given by
= - ^ ( V2 + V| ) + V(A), (111-19)H2M
where M is the mass of the nucleon. Separating the Hamil
tonian into relative and center-of-mass coordinates:
Hrel = - — V 2 + V U )M
H ' = - — V2cm ijjvj R
The time-independent Schrodinger equation for the relative
motion of the nucleons Is
V 2i|> + -2 [E-V(*)]iJ/ = 0. (III-20)
As is well known, this equation is separable in spherical
coordinates, and has the solution
75\
£The spherical harmonic may be regarded as a rotational
function and the radial function as intrinsic. then
has the same form as the eigenfunctions (A-II-15). We consider,
for example, two nucleons Interacting via the harmonic oscil
lator potential. The energy levels of this system may be
arranged in bands, each characterized by an Intrinsic, or
radial quantum number (n = number of oscillator quanta). The
rotational motion undeniably exists, although the spectrum
does not follow the 1(1+7) energy rule of a rotor. This is
because the rotational energy Is of the same order as the
intrinsic excitations and the centrifugal force distorts the
intrinsic structure. The two degrees of freedom are strongly
coupled, as evidenced by the dependence of the radial function
on the quantum number Z.
The example above demonstrates how a collective inter
pretation may be given to a system comprised of only two
particles, but does hot clearly indicate the applicability
of the surface-coupled model discussed in Appendix A-II.
To determine the extent of its validity for the calculations
reported in this chapter, consider a system of particles
interacting via the quadrupole-quadrupole force (III- 2)
The average field experience by particle i, expressed as a
function of the coordinates of the intrinsic frame is
with(1 1 1 - 2 2 )
where an averaging over the coordinates of the particle
has been performed. This clearly demonstrates that each
individual particle undergoing quadrupole interactions with
all of its neighbors behaves as if, on the average, it were
interacting with a quadrupole deformed surface.
Equation (A-II-16) indicates that for oblate deformations
the ground state of a system of odd n would be I=K=ft=/. Since
this is not borne out by the calculations performed, (as seen
above, the ground state is given by K=/-J for n=3, and by
K=j-2 for n=5) it can only be inferred that the nucleon
averaging discussed above is not applicable since there are
too few particles present. Consideration must therefore be
given to a surface-coupled model involving several external
particles.
The competition between particle forces and surface
interactions will determine the nuclear coupling scheme most
appropriate. If the forces are weak compared to the coupling
of the individual particles to the nuclear surface, the
particle angular momenta remain good quantum numbers and the
coupling scheme may be depicted as in Figure 14a. The effect
of the particle forces is then to contribute a small energy
shift, which depends on the ft quantum numbers. Such effects
may be significant if there are near-lying states of different
ft, such as in odd-odd nuclei.
With increasing strength the particle forces introduce
non-diagonal terms in the ft^ For very strong forces, the
particle structure is coupled to a resultant angular momentum
76
Figure 14. Coupling schemes for many-particle configurations.
In many-particle configurations, the coupling scheme results from a competition between surface coupling and particle forces. Two extreme cases are shown: (a) Surface coupling dominates over particle forces. The particles move indepen
dently of each other in the deformed nucleus, each having a constant component ft of angular momentum along the symmetry axis. The total ft equals ^ ft^ and the nuclear ground state has I=K=ft. The figure illustrates the coupling scheme for a (j) configuration. The three lowest particle states have
and leading to I=ft=y~7. (b) Particle forces
dominate over surface coupling. The particles are coupled to a resultant 5, which is then coupled to the surface as a single particle. The figure refers to a (y)^ configuration, where the particle forces, in general, favor the state J=y. The resultant ground state has I=ft=J=y.13
z1
(b)
J . T h is i s th e n coup led to th e s u r fa c e in th e same manner
as a s in g le p a r t i c l e . The ground s ta te s p in I= J is d e te rm ine d
by th e p a r t ic le fo rc e s (v . F ig u re 1 4 b ).
I t i s im m e d ia te ly seen th a t weak c o u p lin g p redom ina tes
in th e c a lc u la t io n s re p o r te d s in c e th e ground s ta te s p in s f o r
odd n a re g iv e n by ( n - 1 ) / 2 .
7 . Summary
P r o je c t io n c a lc u la t io n s have been c a r r ie d ou t in th e
com plete ( j ) n space 5 /2 $ /$ 1 5 /2 , th e nuc leons In te r a c t in g th ro u g h
a s im p le phenom eno log ica l fo rc e w h ich accoun ts f o r th e main
c h a r a c te r is t ic s o f r e a l n u c le i . An e xa c t s e n io r i t y a n a ly s is
was p e rfo rm ed on th e above e ig e n s ta te s . The fo l lo w in g r e s u l ts
have been o b ta in e d :
( JL) D e v ia t io n s from an expected t ru e r o ta t io n a l spectrum
have been a t t r ib u te d to th e tw o - p a r t ic le quadrupo le in te r a c t io n
i t s e l f , w h ic h , as has been shown, i s capab le o f e x h ib i t in g
r o t a t io n a l s p e c tra o n ly f o r low a n g u la r momentum v a lu e s . F o r
h ig h e r va lu e s th e two p a r t i c le spectrum is th a t o f an a x ia l l y
sym m etric n u c le u s u nd e rg o in g r o ta t io n - v ib r a t io n in t e r a c t io n .
( U ) The pure quad ru p o le in t e r a c t io n was shown to g iv e
r is e to a v a r ia t io n in th e shape o f th e "n u c le u s " . The wave
fu n c t io n s , w ith an in c re a s in g number o f p a r t ic le s , a t f i r s t
approach those o f a t ru e r o t o r , b u t th e n , w ith the approach
to th e m id -s h e l l r e g io n , become more s p h e r ic a l in unde rgo in g
t r a n s i t io n from p r o la te to o b la te d e fo rm a tio n s .
{JU.JL) S p e c tra in th e m id - s h e l l re g io n s lo o k rem a rkab ly
77
78
l i k e pure v ib r a t io n a l e x c i t a t io n s p e c tra . T h is i s a t t r i b
u ta b le to a marked in c re a s e in th e number o f pure s e n io r i t y
s ta te s in t h is re g io n . I t has been shown th a t t h is is a
g e n e ra l fe a tu re o f any tw o - p a r t ic le in te r a c t io n w h ich may
be g iv e n as a sum o f s c a la r p ro d u c ts o f te n s o r o p e ra to rs .
{ i . v ) The lo w - ly in g e x c i ta t io n s p e c tra was found to
have a m a jo r component o f s e n io r i t y o=J.
(v ) Odd mass c o n f ig u ra t io n s were found to d is p la y
r o t a t io n a l band s t ru c tu re s b e g in n in g w ith K = / - ( n - J ) /2 .
r a th e r th a n th e expected K = /. T h is was shown to be e x p l ic a
b le in te rm s o f th e weak c o u p lin g m ode l, and to agree w ith
p r e d ic t io n s o f th e c o l le c t iv e model in w h ich th e n p a r t ic le s
a re c o n s id e re d coup led to a deform ed n u c le a r f i e l d .
As p re v io u s ly m e n tion e d , e x p e rim e n ta l d a ta does n o t
u pho ld th e assum ption o f pu re ( j ) n c o n f ig u ra t io n s w i t h in th e
n u c le u s . In an a tte m p t to f u r t h e r s tu d y th e r e la t io n between
m acroscop ic and m ic ro s c o p ic models o f n u c le a r s t r u c tu r e , we
tu r n now to a d is c u s s io n o f th e more r e a l i s t i c H a rtre e -F o c k
c a lc u la t io n s . H ere , in d iv id u a l p a r t ic le s a re assumed to move
in an average (m a cro sco p ic ) n u c le a r p o t e n t ia l , gene ra te d s e l f -
c o n s is te n t ly by a l l o f th e a c t iv e p a r t ic le s . E x c i ta t io n s o f
th e H a rtre e -F o c k s ta te a re c o n s id e re d in an a tte m p t to f in d
c o r re c t io n s to th e q u a s i- r o ta t io n a l H a rtre e -F o c k band.
79
CHAPTER IV
HARTREE-FOCK STUDIES
R o ta t io n a l s t r u c tu r e has lo n g been th o u g h t to e x is t
in th e n u c le i o f th e 24- I d s h e l l (16<A<40). The main
c h a r a c te r is t ic s o f these n u c le i w h ich le a d to an in te r p r e
t a t io n as r o t a t io n a l s t ru c tu re s a re th e s im i la r i t y o f th e
lo w -e n e rg y e x c i t a t io n s p e c tra to th a t o f a r o t o r , and the
e x h ib i t io n o f s tro n g q uad rupo le t r a n s i t io n and s t a t i c m u l t i
p o le moments. The appearance o f r o t a t io n - l i k e s t r u c tu r e
in d ic a te s a degree o f asymmetry in th e average p o te n t ia l
f e l t be each n u c le o n ; a s p h e r ic a l ly sym m etric system cannot
undergo quantum m echan ica l r o ta t io n s . T h e re fo re , in c re a t in g
an in d iv id u a l - p a r t i c le model to in t e r p r e t th e e x p e rim e n ta l
d a ta f o r these n u c le i , s p e c i f ic accoun t must be ta ke n o f the
d e p a rtu re o f th e average n u c le a r p o te n t ia l f i e l d from sp h e r
i c i t y . The use o f th e s p h e r ic a l j - J c o u p lin g scheme does
n o t seem s u ita b le f o r such cases ; a deform ed b a s is w ould be
m ore .a p p ro p r ia te . A s p h e r ic a l b a s is may, how ever, be u t i l i z e d
in c o n f ig u r a t io n in te r a c t io n c a lc u la t io n s , in w h ich an a tte m p t
is made to in c lu d e asymmetry e f fe c ts by m ix in g d i f f e r e n t
s p h e r ic a l s o lu t io n s w h ich l i e c lo s e in e n e rg y . U n fo r tu n a te ly ,
these c a lc u la t io n s a re p r o h ib i t i v e ly lo n g and c o m p lic a te d f o r
a l l b u t th e s im p le s t o f n u c le i .
An i n t r i n s i c deform ed n u c le a r s ta te re p re s e n ts a system
o f n o n - in te r a c t in g p a r t ic le s In a deform ed p a r t i c le b a s is
g e n e ra te d by some " s e l f - c o n s is te n t " t re a tm e n t o f e i t h e r a l l
1. Introduction
o f th e nuc leons co m p ris in g th e n u c le u s , o r m ere ly those
n uc leons co n s id e re d to be " a c t iv e " . I t i s n o t an a c tu a l
s ta te o f th e n u c le u s , b u t r a th e r a c ts as a b a s is from w h ich
th e p h y s ic a l s ta te s may be e x t ra c te d . R e d lic h 1' 6 has demon
s t r a te d th a t th e r e s u l ts o f s h e l l model c o n f ig u ra t io n i n t e r
a c t io n c a lc u la t io n s may be app rox im a ted by p r o je c t in g o u t
th e e ig e n s ta te s o f a n g u la r momentum from t h is i n t r i n s i c
deform ed s ta te o f th e n u c le u s . The p ro je c te d q u a s i- r o ta t io n a l
band co rresponds to th e a c tu a l s ta te s o f th e n u c le us i f most
o f th e n u c le o n -n u c le o n in te r a c t io n is absorbed in p ro d u c in g
th e deform ed s ta te , i . e . , c o n ta in e d in th e s e l f - c o n s is te n t
one-body p o t e n t ia l , so th a t th e r e s id u a l in te r a c t io n s a re
s m a ll.
The deform ed p a r t i c le b a s is may be o b ta in e d by a s e l f -
c o n s is te n t H a rtre e -F o c k (h e re a f te r HF) c a lc u la t io n . H a r tre e -
Fock th e o ry p ro v id e s a c o n n e c tio n between th e m acroscop ic
d e s c r ip t io n o f n u c le a r d e fo rm a tio n s g iv e n by an average
n u c le a r p o t e n t ia l , and th e m ic ro s c o p ic d e s c r ip t io n in term s
o f in d iv id u a l p a r t ic le s . I t i s assumed th a t each n uc leo n
moves in d e p e n d e n tly in a deform ed p o te n t ia l w e l l a r is in g
s e l f - c o n s is t e n t ly from i t s one- and tw o-body in te r a c t io n s
w ith a l l n u c le o n s , averaged ove r th e wave fu n c t io n s o f th e
o th e r n u c le o n s .
The e f f e c t o f s e lf - c o n s is te n c y on r a d ia l m o tion s may
be im p o r ta n t f o r an e x p la n a t io n o f th e s a tu ra t io n p ro p e r t ie s
o f th e n u c le u s , i . e . , b in d in g e n e rg ie s and e q u i l ib r iu m r a d i i ,
b u t n o t f o r th e lo w -e n e rg y e x c i t a t io n s p e c tra . T h e re fo re ,
80
81
two typ e s o f HF c a lc u la t io n s a re g e n e ra lly p e rfo rm e d —
r a d ia l c a lc u la t io n s f o r c lo s e d - and n e a r - c lo s e d -s h e ll
n u c le i , and deform ed c a lc u la t io n s f o r n o n -s p h e r ic a l n u c le i .
R a d ia l wave fu n c t io n s a re v a r ie d when i t i s th o u g h t th a t
m a jo r s h e l l m ix in g sh o u ld be ta ke n In to a cco u n t. In th e
. l a t t e r , th e r a d ia l p a r t o f th e p a r t ic le wave fu n c t io n i s
f ix e d , u s u a l ly as th e harm onic o s c i l l a t o r wave fu n c t io n
(as a m a tte r o f c a lc u la t io n a l c o n v e n ie n c e ); s e lf - c o n s is te n c y
is Imposed o n ly on th e o r b i t a l and s p in p a r ts o f th e wave
fu n c t io n . I t i s th u s assumed th a t th e ,la c t I v e ,, p a r t ic le s
may be c o n fin e d to a h ig h ly t ru n c a te d H i lb e r t space. I t is
t h is typ e o f c a lc u la t io n to w h ich re fe re n c e i s made in t h is
p re s e n ta t io n .
S ince an e xa c t s o lu t io n o f th e n u c le a r many-body
p rob lem is im p o s s ib le ( c f . C hap te r I ) , th e ch o ice o f an
in d iv id u a l p a r t i c le re p re s e n ta t io n cannot be made on th e
b a s is o f m a th e m a tica l conven ience a lo n e ; th e v a l i d i t y o f
a p p ro x im a tiv e te ch n iq u e s depends upon th e p a r t i c le re p re
s e n ta t io n b e in g used. Warke and Gunye1*7 have a rgued ,
a lth o u g h n o t r ig o r o u s ly , th a t th e deform ed s ta te o u t o f
w h ich th e lo w - ly in g q u a s i- r o ta t io n a l band i s p ro je c te d ,
sh o u ld be th e HF s ta te r a th e r th a n any o th e r . There i s a
b a s ic d if fe re n c e between the HF re p re s e n ta t io n and o th e r
bases o b ta in e d m e re ly by th e a p p l ic a t io n o f a u n i ta r y t r a n s
fo rm a tio n to th e s p h e r ic a l s h e l l model r e p re s e n ta t io n : th e
HF p a r t i c le wave fu n c t io n s c o n ta in in fo rm a t io n about the
one- and tw o-body in te r a c t io n s in th e system . C a lc u la t io n s
to t e s t th e p ro je c te d wave fu n c t io n s in th e 2 6 - 1 d s h e l l have
re s u lte d * *8 in good agreem ent w ith s h e l l model r e s u l t s * 9 .
E a r ly successes o f th e p r o je c t io n m ethod20>50-55 in
o b ta in in g deform ed p a r t i c le wave fu n c t io n s w h ich e x p la in e d
p ro p e r t ie s o f n u c le i in th e 2 6 - 1 d s h e l l im p lie d th e e x is te n c e
o f an u n d e r ly in g independen t p a r t i c le b e h a v io r in th e t ru e
wave fu n c t io n s o f these n u c le i . The wave fu n c t io n i s expec
te d to f a c to r in t o in d iv id u a l p a r t ic le and c o l le c t iv e term s
when p a r t i c le e x c i t a t io n e n e rg ie s a re la r g e r th a n th e energy
a s s o c ia te d w ith c o l le c t iv e e x c i ta t io n s . These c o n d it io n s
a re met by th e even-even N=Z n u c le i o f th e 2 6 - 1 d s h e l l , where
p a r t i c l e e x c i ta t io n s a re in h ib i t e d by a HF energy gap o f
5-8 MeV (v . s e c t io n I V . 4 ) , w h ile th e f i r s t 2+ le v e l o f th e
ground s ta te q u a s i- r o ta t io n a l s e r ie s occu rs a t a p p ro x im a te ly
1 -2 MeV. The 2 6 - I d s h e l l i s a p a r t i c u la r ly c o n ve n ie n t re g io n
o f th e n u c le a r mass ta b le in w h ich to examine th e in t e r r e la
t io n s h ip s between models o f n u c le a r s t r u c tu r e . T h is is th e
f i r s t s h e l l in w h ich m ix in g o f d i f f e r e n t £ -s ta te s o c c u rs .
I t has th e fe a tu re o f h a v in g enough degrees o f freedom to make
th e p rob lem n o n t r i v i a l , and n o t so many degrees o f freedom
th a t th e s o lu t io n becomes to o d i f f i c u l t ; th e re a re few enough
p a r t ic le s so th a t in te rm e d ia te c o u p lin g c a lc u la t io n s can be
p e rfo rm e d , and y e t enough p a r t ic le s so th a t c o l le c t iv e models
can be te s te d . I f m acroscop ic and m ic ro s c o p ic th e o r ie s o f
n u c le a r s t r u c tu r e a re to be u n i f ie d , as is th e aim o f t h is
w o rk , no o th e r re g io n appears more f e r t i l e th a n th e 2 6 - 1 d
s h e l l .
82
83
The p ro p e r t ie s o f n u c le i in th e f i r s t h a l f o f th e 2 6 - Id.
s h e l l a re s t ro n g ly s u g g e s tiv e o f lo w -e n e rg y e q u i l ib r iu m shapes
w h ich a re de form ed. E x c i ta t io n s p e c tra in t h is mass re g io n ,
o b ta in e d by p r o je c t in g q u a s i- r o ta t io n a l bands from HF s ta te s ,
e x h ib i t spac ings w h ich a re s y s te m a t ic a l ly s m a lle " th a n those
o f e x p e r im e n ta lly observed e x c ite d s ta te s by a fa c to r o f20a p p ro x im a te ly .1 /2 . A n o ta b le e x c e p tio n is Ne , th e p ro je c te d
spectrum o f w h ich agrees f a i r l y w e l l w ith e x p e rim e n ta l o b se r
v a t io n s ( v . s e c t io n I V . 4 ) . T h is r e f le c t s a more com plete con-20s id e r a t io n , in th e HF c a lc u la t io n f o r Ne , o f th e degrees o f
freedom o f th e a c t iv e p a r t i c le s , th a n in th e c a lc u la t io n s f o r
any o th e r n u c le u s in t h is re g io n .
The use o f n o n - lo c a l one-body p o te n t ia ls ( in th e fo rm o f
a s p a t ia l exchange te rm ) w i l l be shown (v . s e c t io n I V . 4) to
r e s u l t in an energy gap in th e deform ed p a r t ic le spec trum .
HF th e o ry is expected to re p re s e n t a good a p p ro x im a tio n to
th e t ru e s i t u a t io n i f t h is gap is la r g e . I f , on th e o th e r
hand, th e energy gap i s s m a ll, p a r t ic le s sh o u ld be a b le to
s c a t te r across i t , g iv in g r is e to p a r 't ic le - h o le s ta te s w h ich
m ix w ith th e HF s ta te . C a lc u la t io n s in d ic a te th a t t y p ic a l
energy gaps f o r 2 6 - I d s h e l l n u c le i a re o f th e o rd e r 5-8 MeV,
as compared to those o f th e I p s h e l l , w h ich a re a p p ro x im a te ly
18 MeV. I t i s th e re fo re expected th a t p a r t ic le - h o le s ta te s
m ig h t re p re s e n t v a l id a d m ix tu re s to th e lo w -e n e rg y e x c i ta t io n
s p e c tra o f n u c le i in th e re g io n - Ca**^. I t i s th e purpose
o f th e c a lc u la t io n s re p o r te d h e re in to f in d these a d m ix tu re s .
84
By i t s v e ry n a tu re , how ever, th e HF s ta te i s s ta b le
a g a in s t 1 - p a r t i c le / l - h o le e x c i t a t io n s . T h is r e f le c t s th e
f a c t th a t th e se e x c ita t io n s have a lre a d y been in c lu d e d in
th e 1-body s e l f - c o n s is te n t p o t e n t ia l . T h is f a c t has been
m is ta k e n ly used by P a l and Stamp56 to ig n o re these e x c i ta
t io n s and to c o n c e n tra te on 2 - p a r t ic le /2 - h o le a d m ix tu re s
th ro u g h a r e s id u a l p a ir in g in t e r a c t io n . The a d m ix tu re s w h ich
a re o f im p o rta n c e , how ever, a re n o t those to th e HF s ta te
i t s e l f , b u t r a th e r those to th e in d iv id u a l s ta te s o f th e
q u a s i- r o ta t io n a l band p ro je c te d o u t o f th e HF s ta te . T e w a r i57
has re c o g n iz e d t h is f a c t in a p p ly in g a Tamm-Dancoff a p p ro x i
m a tio n to o b ta in m ix tu re s to th e HF band and h ig h e r e x c i t a -20t io n s p e c tra in Ne , w ith q u ite rem arkab le r e s u l ts ( v . s e c t io n
V .6 b ) . T h is , how ever, i s an a p p ro x im a tiv e te ch n iq u e in w h ich
a s in g le " p a r t ic le - h o le " s ta te is o b ta in e d as a co m b in a tio n
o f 1 - p a r t i c le / l - h o le e x c ita t io n s o f th e HF s ta te , t h is com
b in a t io n d e te rm in e d by th e re q u ire m e n t th a t th e f lu c tu a t io n s
o f th e H a m ilto n ia n v a n is h . U n lik e T e w a r i's w o rk , th e c a l
c u la t io n s re p o r te d here a re e x a c t. A l l 1 - p a r t i c le / l - h o le
e x c i ta t io n s w ith K=0, and th o se 2 - p a r t ic le /2 - h o le c o n f ig u ra
t io n s w h ich have maximum o v e r la p w ith these and th e HF s ta te
a re c o n s id e re d . The H a m ilto n ia n m a tr ix is c a lc u la te d and
d ia g o n a liz e d in th e f u l l space o f these e x c i ta t io n s . The
e xa c t p a r t ic le - h o le m ix tu re s a re th u s o b ta in e d . We may, from
t h i s , conc lude how s ta b le th e HF s o lu t io n s a re w i th in th e
26 - 1 d s h e l l , what th e c o r re c t io n s to th e lo w -o rd e r q u a s i-
r o ta t io n a l band sh o u ld be , and w he the r o r n o t p a r t ic le - h o le
85
e x c i ta t io n s accoun t f o r th e e x c ite d s ta te s above th e HF
q u a s i- r o ta t io n a l band.
2. H a rtre e -F o c k E q u a tio n s
The n u c le a r H a m ilto n ia n may be w r i t t e n , in th e n o ta t io n
o f second q u a n t iz a t io n , In term s o f any com plete o r th o n o rm a l
s in g le - p a r t ic le b a s is {<J>}, such as th e s h e l l model wave fu n c
t io n s | i n j - j c o u p lin g :
H ‘ + I (IV'1)k#H
where <<J> 1114>j > a re m a tr ix e lem ents o f th e one-body o p e ra to r
w h ich i s a sum o f th e k in e t ic e n e rg y , th e harm onic o s c i l l a t o r
energy and I 2 and ! • s fo rc e s ; |V | a re a n tis y m m e tr iz e d
m a tr ix e lem ents o f th e e f f e c t iv e tw o-body in t e r a c t io n , i s
th e fe rm io n c re a t io n o p e ra to r a s s o c ia te d w ith th e s in g le
p a r t i c le s ta te <f> ; i t obeys th e a n tic o m m u ta tio n ru le s
t a i ,a j-*+ = 6i j » [ > i> ajJ + = 0 •
E s s e n t ia l ly HF th e o ry seeks a p a r t ic u la r u n i ta r y t r a n s
fo rm a tio n from th e b a s is { } to a n o th e r com plete o rth o n o rm a l
b a s is { X>, in w h ich th e HF wave fu n c t io n 4 ^ is d e f in e d as
a S la te r d e te rm in a n t o f A o f th e fu n c t io n s X , known as th e
"o c c u p ie d " o r b i t s , where A i s th e number o f nuc leons b e in g
c o n s id e re d . The t ra n s fo rm a t io n to th e new b a s is i s g iv e n by^
XX = ^ c I 4.± . ( IV -2 )
L a t in in d ic e s a re used f o r th e b a s is {$ } and Greek in d ic e s f o r th e b a s is { X} .
I f re p re s e n ts th e fe rm io n c re a t io n o p e ra to r a s s o c ia te d
w ith th e p a r t i c l e s ta te x ^ , th e n I t may im m e d ia te ly be seen
th a t
- I « i * I
and
''HP = bt bT • • - bA l 0>* ( IV - 3)
where |0> i s th e vacuum s ta te . The o r th o n o rm a lity o f th e
b a s is { X> im p lie s
<XA I Xy > = ]• ° i c i 6Xy »
where e x p l i c i t use has been made o f th e o r th o n o rm a lity o f th e
b a s is {<|>}. The b a s is { X} i s chosen so as to g iv e a s ta t io n a ry
and minimum f o r th e HF energy ERF o f th e system d e s c r ib e d by
th e wave fu n c t io n
86
The p rim e s in d ic a te th a t o n ly th e "o c c u p ie d " o r b i t s a re i n
c lu d e d in th e sum m ations. Ejjp may be expressed in term s o f
th e c o e f f ic ie n t s c ^ :
The s ta t io n a r y s o lu t io n s o f e q u a tio n ( IV - 5 ) , f o r a r b i t r a r yX*i n f i n i t e s im a l v a r ia t io n s o f th e c o e f f ic ie n t s ci , may be found
n o t in g th e o r th o n o rm a lity o f th e b a s is (x > , by s o lv in g th e
e q u a tio n
TTv* CeHF - V I cj * cJ " 1 ) ] = °*OC J
where ev is in tro d u c e d as a Lagrange m u l t i p l i e r . P e rfo rm in g
th e d i f f e r e n t ia t io n o b ta in s a s e t o f homogeneous e q u a tio n s
f o r th e Cj c o e f f ic ie n t s :
IC«J>i | t|<j>J > + r < * i X x | V | V x > ] c ” = e v c ^ # ( I V - 6 )j X *
where th e m a tr ix e lem ents l v l a re d e fin e d by
<*1XX|V |* J XX> - « *1* k |T |+ J * 1» 0A . (XV-7)
D e f in in g th e HF H a m ilto n ia n h by i t s m a tr ix e le m e n ts :
= <<l>i | t |<|>j> + I <<J>i x x IV | Xx> , ( IV -8 )X
e q u a tio n ( IV - 6 ) becomes
- ev c i
o r e q u iv a le n t ly
* l x v> * ev lxv> • ( iv -9)Thus, th e HF o r b i t a ls a re e ig e n fu n c t io n s o f th e HF H a m ilto n ia n
The H a m ilto n ia n ft, w h ich is re p re s e n ta t iv e o f th e deform ed
f i e l d , may be e i t h e r phenom eno log ica l o r c a lc u la te d e x p l i c i t l y
as in e q u a tio n ( IV - 8 ) . The cho ice o f a deform ed harm onic
o s c i l l a t o r w ith a s p in - o r b i t and £ 2 c o r re c t io n
ft e V2 + [ w 2 x 2 w 2t /2+ w 2 z 2 ] + a t « s + Dt 2 ( I V - 1 0 )2m 2m A y z
87
88
has o f te n been made. The s p in - o r b i t and Z z c o n t r ib u t io n s
a r is e q u ite n a tu r a l ly from a c o n s id e ra t io n o f th e te rm 7 2 in
a c o l le c t iv e tre a tm e n t o f a n u c le a r co re ( v . e q u a tio n ( A - I I - 1 0 ) ) .
Expand ing 7 2=( ^ +s ) 2 y ie ld s an Z2 te rm , an Z-s te rm , and an s 2 ^ 2
te rm , s b e in g a c o n s ta n t, can be ig n o re d . Assuming a x ia l
sym m etry, a>=wx =Uy, and a llo w in g f o r se p a ra te pa ram ete rs a and
D in th e H a m ilto n ia n ( IV -1 0 ) , r a th e r th an th e f ix e d r a t i o 2 ,
o b ta in s th e N ils s o n 11* o r b i t s ( IV - 9 ) . Newton16 re la x e d th e
assum p tion o f a x ia l sym m etry, b u t m a in ta in e d a and D as se p a ra te
p a ra m e te rs .
The expans ion ( IV -2 ) o f th e o r b i t a l s , in g e n e ra l, c o n ta in s
v e ry many te rm s . F o r d e fo rm a tio n s w h ich a re n o t to o la r g e ,
th e b a s is {<(>} may be chosen as s h e l l model wave fu n c t io n s
b e lo n g in g to one m a jo r s h e l l ; any e x t r a - s h e l l e f fe c ts th e re
m ig h t be a re e i t h e r n e g le c te d o r assumed to be p a r t i a l l y i n
c lu d e d in th e one-body fo r c e . In h e re n t in t h is ch o ice a re
th e assum ptions th a t th e in n e r core rem ains i n e r t , and th a t
p a r t i c l e e x c i ta t io n s to h ig h e r s h e l ls a re r e la t i v e l y un im por
t a n t . I t i s s ig n i f i c a n t th a t th e HF energy ( IV -4 ) is th e
o n ly q u a n t i ty w h ich i s s ta t io n a r y w ith re s p e c t to s m a ll v a r i
a t io n s in th e o r b i t a ls ( x ) . T r a n s it io n and m u lt ip o le moments
a re v e ry s e n s it iv e to s l ig h t a d m ix tu re s to th e wave fu n c t io n s
fro m e i t h e r p o la r iz a t io n s o f th e core o r p a r t i c le e x c ita t io n s
in t o h ig h e r s h e l ls . The r e s t r i c t i o n o f th e p a r t i c le b a s is
to a s in g le m a jo r s h e l l may, th e re fo r e , p re c lu d e a f u l l un d e r
s ta n d in g o f these phenomena.
89
E q u a tio n s ( IV -8 ,9 ) a re g e n e ra lly r e fe r r e d to as th e
H a rtre e -F o c k e q u a tio n s . They a re so lv e d by th e fo l lo w in g
i t e r a t i o n p ro c e s s : An i n i t i a l guess i s made f o r th e HF
o r b i t a ls ( IV - 2 ) . These o r b i t s a re used in e q u a tio n ( IV -8 )
d ia g o n a l iz a t io n , gene ra tes a new s e t o f HF o r b i t s . Thr>
H a m ilto n ia n i s th e n re c a lc u la te d w ith th e new o r b i t s , and
so on , u n t i l su cce ss ive d ia g o n a liz a t io n s y ie ld th e same b a s is .
The method does n o t y ie ld a un ique s o lu t io n because o f th e
n o n - l in e a r i t y o f e q u a tio n s ( IV - 8 ,9 ) , and th e re may be s e v e ra l
lo c a l m inim a on th e energy s u r fa c e ( IV - 4 ) . D i f f e r e n t lo c a l
m in im a may be reached by v a r io u s i n i t i a l guesses o f th e
o r b i t a ls ( IV - 2 ) . T h is amounts to b e g in n in g a t d i f f e r e n t p o in ts
on th e m u lt i-d im e n s io n a l energy s u r fa c e . The ch o ice o f t h is
s t a r t in g p o in t i s th u s c r u c ia l . One g e n e ra lly has to search
f o r th e s o lu t io n w h ich g iv e s an a b s o lu te minimum f o r th e HF
e n e rg y .
Once th e b a s is ( x ) i s d e te rm in e d , th e HF energy may be
c a lc u la te d by expand ing th e HF s in g le - p a r t ic le energy e ig e n
va lu e s e ^ :
The m a tr ix e lem ents o f th e HF H a m ilto n ia n h a re g iv e n in
e q u a tio n ( IV - 8 ) ; th u s
to c a lc u la te th e H a m ilto n ia n m a tr ix <<1 1 h I w h ich th ro u g h
eX
The HF energy i s e a s i ly seen to become
( I V - 12)
90
A t each s te p o f th e i t e r a t i o n p ro c e s s , i t must be
d ec ide d w h ich o r b i t s a re to be c la s s i f ie d as "o c c u p ie d " .
T h is i s g e n e ra lly done by e v a lu a t in g e q u a tio n ( IV -1 1 ) f o r
a l l a llo w e d va lu e s o f X. The A o r b i t s h a v in g th e lo w e s t
e ig e n va lu e s e^ a re chosen to be o ccu p ie d .
I t i s to be n o te d th a t unoccup ied o r b i t a ls and t h e i r
e n e rg ie s a re a ls o o b ta in e d by b r in g in g A to a d ia g o n a l fo rm .
I t sh o u ld n o t be conc luded th a t th e nu c leus o f odd mass A + l
can be t re a te d by c o n s id e r in g th e la s t n u c le on in th e v a r io u s
unoccup ied p a r t i c le o r b i t a ls g e n e ra te d s e l f - c o n s is t e n t ly by
th e f i r s t A n u c le o n s . C le a r ly , such a model w ould n e g le c t
a l l rea rrangem en t o f th e i n t r i n s i c co re s t r u c tu r e produced
by th e tw o-body in te r a c t io n s between th e co re nuc leons and
th e odd one. M oreove r, th e s t re n g th o f t h is p o la r iz a t io n
depends on w h ich le v e l th e la s t n u c le on o c c u p ie s .
3. Sym m etries o f th e H a rtre e -F o c k S o lu t io n s
The s o lu t io n o f th e HF e q u a tio n s ( IV -8 ,9 ) o b ta in s an
o rth o n o rm a l b a s is o f e ig e n s ta te s (x ) o f th e H a m ilto n ia n h .
Not a l l o f th e se s ta te s a re occup ied by n u c le o n s— o n ly those
s ta te s a p p e a rin g in th e d e f in i t i o n ( IV -2 ) o f Any
p a r t ic u la r s o lu t io n o f th e HF e q u a tio n s depends upon th e
ch o ice o f w h ich s ta te s a re to be o cc u p ie d , b o th i n i t i a l l y
and d u r in g th e i t e r a t i o n p ro c e s s . A l te r n a t iv e s o lu t io n s ,
n o t n e c e s s a r i ly o r th o g o n a l, may be g e n e ra te d by occupy ing
d i f f e r e n t s ta te s a t any p o in t in th e i t e r a t i o n p ro ce ss .
The HF H a m ilto n ia n does n o t a p t iio t u . e x h ib i t th e same
symmetry p ro p e r t ie s as the n u c le a r H a m ilto n ia n ( IV —1 ) ; n o r
i s a g iv e n symmetry n e c e s s a r ily p re se rve d in su cce ss ive
i t e r a t io n s . N e v e rth e le s s h may share some symmetry p ro p e r t ie s
w ith th e n u c le a r H a m ilto n ia n , depending upon th e p a r t ic u la r
ch o ice o f o ccu p ie d o r b i t s . Tq u n de rs ta nd t h is phenomenon, we
c o n s id e r an o p e ra to r ft w h ich commutes w ith th e n u c le a r H a m il
to n ia n H
[H , f t ] = 0 ,
and l e t {A } be th e s e t o f o ccup ied HF o r b i t s , i . e . , th e s e t
(A ) be longs to th e s e t o f HF o r b i t s ( x K I f ft leave s th e
s e t {A } in v a r ia n t
ft{A ) = {A } ( IV -1 3 )
th e n th e HF H a m ilto n ia n h w i l l commute w ith ft
[A , f t ] = 0 .
F o r exam ple, i f ft i s a one-body o p e ra to r , e q u a tio n ( IV -1 3 )
im p lie s th a t th e s ta te o b ta in e d by o p e ra t in g w ith ft on an
o ccup ie d o r b i t may be expressed as a l in e a r co m b in a tio n o f
o ccup ie d o r b i t s o n ly :
ftA = y w A ,P { yv v ’
where a re th e m a tr ix e lem ents
" i i v = •
The m a tr ix e lem ents <Xv lnl^p> va n is h id e n t i c a l l y i f xv does
n o t b e lo n g to th e s e t {A } . I f ft o p e ra te s on th e HF s ta te ,
i t w i l l have no e f f e c t o th e r th a n m u l t ip l i c a t io n by a c o n s ta n t,
i . e . , i s an e ig e n s ta te o f ft:
91
Thus, f o r exam ple, th e o r b i t s ( IV -2 ) a re e ig e n s ta te s o f J_ .zT h e re fo re J leave s th e s e t o f occup ied o r b i t s in v a r ia n t , z
The o p e ra to r R ( tt) = e x p ( - i ir J ) p e rfo rm s a r o ta t io n o fV J
tt ra d ia n s abou t th e y - a x is . W ith each o r b i t x Is a s s o c ia te d
th e o r b i t X
|X> = Rv ( tt) | X>_y ( IV -1 4 )
Ry( it) |X> = - |x> •
When b o th o r b i t s x and x ar® o ccu p ie d , R ( tt) le a ve s th eys e t (X ) o f occu p ie d o r b i t s in v a r ia n t . T h e re fo re k commutes
w ith R ( t t ) , and th e HF s o lu t io n f o r even-even n u c le i mayv
possess th e symmetry
Ry ^ I'I'h f> = I^HF* *
N e x t, we l e t th e HF o r b i t s be e ig e n s ta te s o f J 2 and
^ z * i , e * » , .X = I
J 2X = j l j + U x
j zX = mx •
Then h w i l l commute w ith J o n ly i f 3” le a ve s th e s e t {X}
in v a r ia n t . S ince j and j m ix s ta te s o f d i f f e r e n t m, th ex ys e t w i l l rem a in in v a r ia n t o n ly i f a l l p o s s ib le m -s ta te s f o r
a g iv e n j a re o ccu p ie d . Such a s ta te is a c lo s e d s h e l l .
F in a l l y , we c o n s id e r n e u tro n -p ro to n exchange o p e ra to rs .
I f a n u c le u s has an e q u a l number o f p ro to n s and n e u tro n s , a
HF s o lu t io n e x is ts f o r w h ich n e u tro n s and p ro to n s a re in th e
same o r b i t s . The th re e components o f th e t o t a l is o s p in ope ra
t o r w i l l th e n le a ve th e s e t {X } o f o ccu p ie d o r b i t s in v a r ia n t .
N e g le c tin g th e Coulomb in t e r a c t io n , th e HF f i e l d i s an is o
s p in s c a la r , n e u tro n and p ro to n o r b i t s a re d e g e n e ra te , and
th e HF s ta te has ze ro is o s p in .
I t s h o u ld , how ever, be s tre s s e d , th a t a lth o u g h a s o lu
t io n o f th e HF e q u a tio n s ( IV -8 ,9 ) may e x h ib i t a g iv e n sym
m e try , t h is need n o t be th e lo w e s t s o lu t io n . There may be
a n o th e r s o lu t io n w ith lo w e r energy w ith o u t t h is p a r t ic u la r 28sym m etry. S i , f o r exam ple, has a s p h e r ic a l s o lu t io n c o r
re sp o n d in g to th e c lo s u re o f th e ^ 5 / 2 ( ° f * s e c t io n28IV .6 d ) . Yet deform ed s o lu t io n s e x is t f o r S i w ith lo w e r
e n e rg y . In g e n e ra l, th e r e la x a t io n o f a symmetry c o n d it io n ,
in fa v o r o f a le s s r e s t r i c t i v e c o n d it io n , low e rs th e HF
e n e rg y , th e symmetry energy no lo n g e r b e in g in c lu d e d .
4 . H a rtre e -F o c k C a lc u la t io n s
The b a s is { <t>} on w h ich th e HF o r b i t a ls ( x ) a re expanded
(e q u a tio n ( IV - 2 ) ) i s , in th e o ry , a r b i t r a r y . However,- in
r e a l i t y , such im p o rta n t fa c to r s as th e convergence o f the
s o lu t io n s , th e conven ience o f c a lc u la t in g m a tr ix e lem ents o f
th e tw o-body in t e r a c t io n , and th e sym m etries o f th e HF f i e l d ,
impose r e s t r i c t io n s upon th e ch o ice o f b a s is fu n c t io n s . The
convergence p rob lem has been s tu d ie d . f o r s p h e r ic a l c lo s e d -
s h e l l n u c le i , and harm onic o s c i l l a t o r o r b i t s were found to
g iv e e x c e l le n t co n ve rg e n ce .58 Thus, harm onic o s c i l l a t o r wave
fu n c t io n s w i l l be used f o r th e r a d ia l p a r t o f th e b a s is {<J>}.
S h e ll model s ta te s \ n t j m r > a re used f o r th e a n g u la r - s p in - is o -
s p in p a r ts because o f th e conven ience o f o b ta in in g m a tr ix
93
S in g le - p a r t ic le e n e rg ie s o b ta in e d 59 from p icku p e x p e r i- 40ments on Ca a re s l i g h t l y ' d i f f e r e n t from those o b ta in e d
17from th e 0 spec trum . I t sh o u ld be rea so n a b le to e x tra p o la te
f o r in te rm e d ia te n u c le i . These e n e rg ie s , to g e th e r w ith th e
o s c i l l a t o r c o n s ta n ts d e r iv e d from e le c t ro n s c a t te r in g 60 a re
p re se n te d in T ab le V I I I . T h is method has th e advantage o f
g iv in g and Ca**^ e q u a l ro le s as re fe re n c e n u c le i .
The tw o-body in t e r a c t io n used is th e Gaussian p o te n t ia l
w ith a R o s e n fe ld 61 exchange m ix tu re :
2 . 2 "*■V = V e " r / y IjlaL z . [ o .3 + 0 .7<*i *<*2 ] . ( IV -1 5 )
0 3
T h is i s a p a r t i c u la r l y u s e fu l in t e r a c t io n because o f th e
ease w ith w h ich i t s m a tr ix e lem ents a re c a lc u la te d . I t
w i l l be shown th a t th e space exchange component o f th e i n
te r a c t io n g iv e s r is e to energy gaps in th e HF s in g le - p a r t ic le
s p e c tra , w h ich a re necessa ry f o r H a rtre e -F o c k to be a v a l id
a p p ro x im a tiv e te c h n iq u e .
R ip k a 25 has s o lv e d th e HF p rob lem as d e s c rib e d above w ith
th e assum ptions o f t im e r e v e r s a l, is o to p ic s p in , and a x ia l
sym m e tries . The r e s u l t in g s in g le - p a r t ic le s p e c tra and e ig e n
fu n c t io n s a re p re se n te d in F ig u re 15 and T ab le IX , r e s p e c t iv e ly .24When th e r e s t r i c t i o n to a x ia l symmetry was re la x e d , Mg and
S32 were found to have t r i a x i a l s o lu t io n s 3-** MeV (b e fo re
p r o je c t io n ) lo w e r th a n th e a x ia l l y sym m etric s o lu t io n s . The
n a tu re o f these s o lu t io n s w i l l be d iscu sse d la t e r in t h is sec
t io n (see a ls o s e c t io n s IV .6 c ,e ) .
94
elements of the two-body Interaction.
T ab le V I I I V a lues o f o s c i l l a t o r c o n s ta n t a and o f th e s in g le - p a r t ic le e n e rg ie s e . uped in s in g le m a jo r s h e l l HF c a lc u la t io n s w ith a R o se n fe ld exchange c h a r a c te r is t ic
.Nucleus a ( fm“ 1) 5/2(MeV)
* * 1 / 2(MeV)
3/2(MeV)
VO i—io 0 .568 -4 .1 4 -3 .2 7 0 .93
He20 0.559 -4 .3 8 -3 .2 6 0 .79
Mg2*1 0.547 -4 .7 1 -3 .2 6 0 . 6 1
COCM•HCO 0 .548 -4 .6 8 -3 .2 6 0 .62
S32 0.531 -5 .1 4 -3 .2 5 0 .36
A r^ 6 0.496 -6 .0 9 -3 .2 2 -0 .1 9
Ca4° 0.492 -6 .1 9 -3 .2 2 -0 .2 5
\
F ig u re 15. S in g le - p a r t ic le e n e rg ie s ev o b ta in e d from s in g le m a jo r s h e l l HF c a lc u la t io n s w ith a R o se n fe ld exchange ch a ra c t e r i s t i c f o r even-even N=Z n u c le i in th e 26 - 1 d s h e l l . The s p e c tra shown a re f o r th e lo w e s t a x ia l l y sym m etric s o lu t io n s in each case . Shown to th e r ig h t o f each s in g le p a r t i c le le v e l i s th e a n g u la r momentum p r o je c t io n o n to th e a x is o f sym m etry, and th e p a r i t y . The I p s h e l l o r b i t s have n e g a tiv e p a r i t y , and th e 2 6 - I d o r b i t s have p o s i t iv e p a r i t y . The Ferm i s u r fa c e and i p s h e l l c lo s u re s a re in d ic a te d . A lso shown a re th e Ferm i le v e ls f o r th e lo w e s t t r i a x i a l s o lu t io n s o f Mg24 and S32 .
SIN
GL
E-P
AR
TIC
LE
EN
ERG
Y (M
eV)
-5
-10
-15 -
-2 0
-25 -
-30
-35 -
-40
DEFORMED PARTICLE E IGENVALUES
3/2 ' +
1/2"+
.5 /2 + l / 2 ' +
•3 /2 +
•3/2'+
■1/2"+
•5 /2+
•1/2’ + :
■1/2" +
. 1/2' + ’3/2'+ .1/2"+
'3 /2 ‘+ 1/2"+ELLIPSOIDAL
-3 /2 + FERMI LEVELS
\ 5/2+
3 /2 + y W l /2 ‘+ / / / ' |/2' +
AXIALLY SYMMETRIC FERMI LEVEL
•P SHELL CLOSURE
Ne 2 0 Mg24 Si 28 "32 Ar 36
T ab le IX - A x ia l ly sym m etric HF s o lu t io n s w ith a Gaussian tw o-body in t e r a c t io n and R o se n fe ld exchange c h a r a c te r is t ic . The u n d e r lin e d numbers a re th e e n e rg ie s ev and z - p ro je c t io n s o f p a r t i c le a n g u la r momentum o n to th e symmetry a x is o f th e HF o r b i t s . The energy o f each o r b i t i s fo llo w e d by expans ion c o e f f ic ie n t s c^ f o r th a t o r b i t ( v . e q u a tio n ( IV - 2 ) ) . The s ix s ta te s a re p o s i t iv e p a r i t y 26 - 1 d s h e l l o r b i t s . The
k = l /2 p o s i t iv e p a r i t y o r b i t s a re fo llo w e d by t h e i r components
on th e 6 1 /2 * and d l / 2 s ta te s > in o rd e r . Thek= 3 /2 p o s i t iv e p a r i t y o r b i t s a re fo llo w e d by t h e i r components
on th e ^ 3 / 2 and d3 /2 s ta te s * The k =5 /2 p o s i t iv e p a r i t y o r b i t s have no in d ic a te d com ponents, and th e y a re pure s ta te s . The la s t l in e g iv e s th e HF energy E^p in MeV. A s te r is k s in d ic a te w h ich o r b i t s a re o ccu p ie d .
Table IX - Axially Symmetric HF Solutions
Ne20 Mg24 S i 2 8 ( p r o l ) S i 2 8 ( o b l ) S3 2 ( o b l ) S3 2 ( p r o l ) Ar36- 1 4 . 5 8 * ’ - 1 6 . 2 5 * - 1 9 . 3 7 * - 1 8 . 5 3 * - 1 7 . 9 5 * - 1 9 . 7 4 * - 2 0 . 9 2 *( k = l / 2 ) ( k = l / 2 ) ( k = l / 2 ) ( k = 5 /2 ) ( k = l / 2 ) ( k = l / 2 ) ( k = 5 / 2 )- 0 . 7 5 7 6 - 0 . 7 9 0 4 - 0 . 7 7 6 3 - 0 . 7 5 3 6 - 0 . 8 8 5 6
0 . 5 2 7 3 0 .5 45 2 0 .6 1 8 8 - 1 7 . 9 8 * - 0 . 5 9 0 4 0 .4 6 4 1 - 1 9 . 7 8 *0 .3 8 4 7 • 0 .2 7 9 4 0 . 1 2 0 3 ( k = l / 2 ) 0 . 2 8 9 1 0 .0 20 3 ( k = 3 / 2 )
- 6 . 5 8 - 1 1 . 9 4 * - 1 7 . 5 3 * - 0 . 5 7 8 3- 0 . 7 5 9 6
0 : 2 9 7 7
- 1 7 . 6 1 * - 1 9 . 9 1 * 0 .9 9 4 1( k = 3 / 2 ) ( k = 3 / 2 ) ( k = 3 /2 ) ( k = 5 / 2 ) ( k = 3 / 2 ) 0 .1 0 8 4
- 0 . 9 9 3 2 - 0 . 9 7 0 6 - 0 . 9 5 3 1 . - 0 . 9 7 5 3 - 1 8 . 8 0 *0 . 1 1 6 7 0 .2 4 0 6 0 .3 0 2 7 - 1 4 . 9 9 * - 1 6 . 4 8 * 0 .2 20 9 ( k = l / 2 )
0 . 8 0 5 40 . 5 8 7 30 . 0 8 0 1
( k = 3 /2 ) ( k = l / 2 1)- 5 . 1 9 - 9 . 8 1 - 1 4 . 6 7 * 0 .6 9 3 5 - 0 . 5 6 5 5
0 .8 0 6 4- 1 5 . 5 1 *
( k = l / 2 » ) ( k = l / 2 *) ( k = l / 2 1) 0 .7 20 4 ( k = l / 2 ' )- 0 . 6 3 5 7 - 0 . 5 4 3 6 - 0 . 4 5 7 2 - 8 . 3 2
( k = 3 / 2 ' )0 . 1 7 2 9 - 0 . 2 9 8 5
- 0 . 7 2 9 8 - 0 . 4 1 3 9 - 0 . 4 2 1 2 - 0 . 5 3 5 0 - 1 5 . 8 3 *- 0 . 2 5 1 6 - 0.7302 - 0 . 7 8 3 3 - 1 5 . 0 7 * - 0 . 7 9 0 3 ( k = l / 2 1)
0 .7 20 4 ( k = 3 / 2 )- 0 . 4 3 9 3
0 .6 8 2 2- 0 . 5 8 4 4
- 5 . 1 4 ( k = 5 / 2 )
- 7 . 5 0 ( k = 5 /2 )
- 9 . 2 6 ( k = 5 /2 )
- 0 . 6 9 3 5
- 8 . 1 10 . 9 0 8 10 .4 1 8 7
- 1 4 . 3 3 * ( k = 5 / 2 )
- 2 . 3 5 - 4 . 89 - 6 . 5 0 ( k = l / 2 ' )- 1 1 . 3 7 - 1 0 . 2 3 - 1 4 . 7 4 *
( k = l / 2 " ) ( k = l / 2 " ) ( k = l / 2 " ) 0 .8 04 4 ( k = 3 / 2 ’ ) ( k = 3 / 2 1) ( k = 3 / 2 » )- 0 . 1 4 8 1 0 .2 82 5 0 .4 3 4 0 - 0 . 4 7 0 2
0 .3 6 3 0 0 .4 1 8 7 0 .2 20 9 0 .1 0 8 4- 0 . 9 9 4 10 . 4 3 5 1 0 .7 29 0 0.6630 - 0 . 9 0 8 1 0 .9 7 5 3
- 0 . 8 8 8 1
- 0 . 2 5
- 0 . 6 2 3 5
- 2 . 9 9
- 0 . 6 0 9 9
- 4 . 7 0- 4 . 1 0 ( k = l / 2 " ) - 1 0 . 0 3
( k = l / 2 " )- 1 0 . 0 3( k = l / 2 " )
- 1 0 . 1 4 ( k = l / 2 " )
( k = 3 / 2 ’ ) ( k = 3 / 2 ' ) ( k = 3 / 2 ’ ) 0 . 1 358 0 .3 3 5 2 0 . 355 9 0 .3 9 7 90 .1 16 7 0 .2 40 6 0 .3 0 2 7 - 0 . 4 4 9 4 0 .0 3 3 2 0 .7 06 0 - 0 . 4 3 5 50 .9 9 3 2 0 .9 706 0 .9 5 3 1 - 0 . 8 8 3 0 0 .9 41 6 - 0 . 6 1 2 3 - 0 . 8 0 7 5
p - 3 5 . 7 8 - 7 2 . 9 1 - 1 2 2 . 0 1 - 123.00 - 1 6 8 . 4 4 - 168.80 - 2 2 0 . 4 8
The a p p l ic a t io n o f p r o je c t io n o p e ra to rs to th e HF
s ta te ( IV -3 ) gen e ra te s a q u a s i- r o ta t io n a l s e r ie s in deform ed
n u c le i . The n u c le a r s ta te s w ith d e f in i t e a n g u la r momentum
p re d ic te d by th e HF model may be d e s c rib e d by a p ro d u c t wave
fu n c t io n in w h ich r o t a t io n a l m o tion is d is t in g u is h e d from
i n t r i n s i c m o tio n ( c f . Appendix A - I I ) :
95
2 J + 1 V 216 ir ‘
where M i s th e p r o je c t io n o f J in th e la b o ra to ry fra m e , t h is
s ta te b e lo n g in g to a r o t a t io n a l band ge n e ra te d by a deform ed
in t r i n s i c s ta te |i|> > , w h ich has a x ia l symmetry and p r o je c t io n- K
K o f J o n to th e z -a x is o f th e i n t r i n s i c fra m e , and i|> =
e x p ( - iu J )i|>K.«/
An a l t e r n a t iv e way o f d e s c r ib in g t h is s ta te i s by th e
p r o je c t io n method in tro d u c e d in s e c t io n I I . 2 . U t i l i z i n g
e q u a tio n s ( I I - 5 ) . ,
" ~ T ~ ~ ,
where
n j k c f d Q V K K Cn)<ipK | r ( n ) k K> .S IT2
Because o f th e assum ption o f a x ia l sym m etry, th e HF s ta te
JJKi s an e ig e n s ta te o f J . The n o rm a liz a t io n in t e g r a l N?
reduces to
NJK ■ /d B s in ( B ) d ^ 6 l< > ( > K | e “ i p J y |<eK> , ( I V - 1 6 )
where
dMK{01 = <JM|e”16Jy | JK> . (IV-17)
96
S im i la r ly , th e e n e rg ie s o f th e q u a s i- r o ta t io n a l band, genera
te d by th e HF s ta te , a re g iv e n by
T IS “ i8«J IS
/ y .B in e d ^ ( e i < 4 l e ■ yH | * S _ ( I V _ l 8 )
hJ = --------------------------------------------------- :---------T V - i8 J y
/ d8 s inB d ^ { Q ) < ] p |e y |i|» >
P a r t ia l p ro je c te d s p e c tra o b ta in e d by R ip k a 25 in t h is way f o r20 28 -jfi
Ne , S i , and A r a re p re se n te d in F ig u re 16, to g e th e r
w ith th e lo w -e n e rg y p o s i t iv e p a r i t y e x p e rim e n ta l s p e c tra o f
th e se n u c le i .20As can be r e a d i ly seen, th e s o lu t io n is f a i r f o r Ne ,
b u t p o o r f o r th e o th e r n u c le i shown. T h is may be p a r t ly
u n d e rs to o d fro m an e xa m in a tio n o f th e s in g le p a r t ic le s p e c tra
(F ig u re 15) (see a ls o s e c t io n I V . 6 ) . In s p ite o f th e fa c t
th a t th e wave fu n c t io n o f th e Ferm i le v e l i s unde rg o ing con
t in u a l change from n u c leu s to n u c le u s , th e e x c i ta t io n energy+
o f th e Ferm i le v e l rem a ins a p p ro x im a te ly c o n s ta n t. The
energy gap between occup ied and unoccup ied o r b i t s , how ever,20f a l l s o f f r a p id ly a f t e r Ne ( v . s e c t io n IV .5 a ) .
To u n d e rs ta n d th e o r ig in o f these phenomena, we c o n s id e r
th e expans ion o f th e s p a t ia l p a r t o f a tw o-body p o t e n t ia l in
te rm s c o rre s p o n d in g to d e c re a s in g ra n g e 52. T h is has a lre a d y
been d iscu sse d in s e c t io n I I I . 3 , in c o n n e c tio n w ith th e phe-
+ 24The n o ta b le e x c e p tio n is Mg , w h ich has a Ferm i le v e la p p ro x im a te ly 2 .5 MeV above th e o th e rs . T h is i s due to th e f a c t th a t th e a x ia l s o lu t io n p re se n te d here is n o t th e lo w e s t s o lu t io n ; th e lo w e s t energy s o lu t io n s o f Mg24 and S3 2 a re t r i a x i a l . The Ferm i le v e ls f o r these s o lu t io n s a re a ls o i n c lu d e d in F ig u re 15, and dem onstra te th e c o n s ta n t e x c i t a t io nenergy o f th e le v e ls q u ite w e l l .
F ig u re 16. P ro je c te d H a rtre e -F o c k s p e c tra o f Ne^9 , S i^ 3 , and A r3A The number to th e r ig h t o f each le v e l i s th e s p in o f th e s ta te . The lo w -e n e rg y p o s i t iv e p a r i t y e x p e r im e n ta l spectrum is drawn to th e l e f t in each case. Is o s p in a s s ig n ments a re a l l z e ro , excep t when in d ic a te d as J ( T ) .
EX
CIT
ATI
ON
EN
ERG
Y (M
eV
)
[10
Ne20 S i28 A r 36
12 1 — 8
-6^2'0
8 ---------8
^2(1)"3(1)
46 ,3
‘68 | - 2 -2 8
•4■0•3
0>0 2
■ — — I
EXPT PROJ EXPT PROJ EXPT PROJ
n o m e n o lo g ica l in te r a c t io n used in th e ( / ) w c a lc u la t io n s .
The le a d in g te rm , o f zero o rd e r , i s s im p ly a c o n s ta n t. The
n e x t te rm o f im p o rtan ce is th e q u a d ru p o le -q u a d ru p o le fo rc e
o f s e c t io n I I I . 3 , w h ich is re p re s e n ta t iv e o f the lo n g range
p a r t o f th e n u c le a r fo r c e . The f i r s t te rm , VQ, a c o n s ta n t
p o te n t ia l w ith no s p a t ia l dependence, has o n ly an exchange
c h a r a c t e r is t ic , w h ich may g e n e ra lly be w r i t t e n as
VQ = W + BPa + HPt + MPX ( IV -1 9 )
where Pa , P^, and Px a re th e s p in , is o s p in , and p o s i t io n
exchange o p e ra to rs :
P = 1 + Q i • < * ,
0 " 2, ->■ -+ p - T ,
T 2
P = -P P x a x
and W (W ig n e r) , B ( B a r t l e t t ) , H (H e is e n b e rg ), and M (M a jo rana)
a re s t re n g th c o e f f ic ie n t s . F o r s im p l ic i t y , we l e t th e 2 6 - 1 d
s h e l l be degenera te in i t s s in g le p a r t ic le e n e rg ie s , e=ey.
The HF s o lu t io n s ( IV -2 ) may th e n be fa c to re d in t o se p a ra te
space , s p in , and is o s p in components:
IX * = I V I ° X> ITX> • ( IV -2 0 )
S ince VQ commutes w ith a l l s p a t ia l m o tio n s , th e HF H a m ilto n ia n
i s , o f c o u rs e , d ia g o n a l In any o rth o n o rm a l re p re s e n ta t io n o f
th e s p a t ia l wave fu n c t io n s . The m a tr ix e lem ents o f h (e q u a tio n
( IV - 8 ) ) become
e a =<V V rJ ' l K aa V
=<<Y V a I K ° a V + <ax V a : * x V XlV I V ’ a V V x V
97
H ere , a g a in , p rim es in d ic a te summation o ve r occup ie d o r b i t s
o n ly .
The two term s in each b ra c k e t a re th e d i r e c t and exchange
c o n t r ib u t io n s . Depending upon th e ch o ice o f fu n c t io n s |X >,Ae i t h e r a x ia l l y sym m etric o r n o n - a x ia l ly sym m etric s o lu t io n s
/may be c o n s tru c te d . The o n ly c h a r a c te r is t ic o f im p o rta n ce
here is th e o v e r la p in te g r a ls «x |X > , w h ich a re e i t h e r zeroA Ao r u n i t y ; th e se term s cannot d e te rm in e th e s p a t ia l dependence
o f th e HF o r b i t a l s . F i l l i n g each o ccu p ied o r b i t |X > w ithAtwo n e u tro n s and two p ro to n s o f o p p o s ite s p in s , each o f th e
o ccu p ie d HF e n e rg ie s ( IV -2 1 ) ( f o r even-even N=Z n u c le i)
e. = e + ^ -G ( IV -2 2 a )X 2j
and th e unoccup ied o r b i t s
e = e + — ( IV -2 2 b )a ij
where A is th e number o f nuc leons b e in g c o n s id e re d , and
S = 4W + 2B + 2H - M( IV -2 2 c )
G = W + 2B + 2H -4M .
The degenera te o ccup ied o r b i t a ls a re th e re fo re depressed
r e la t i v e to th e degenera te unoccup ied o r b i t a ls by an energy
gap G, independen t o f th e number o f p a r t ic le s A. F o r th e
R o se n fe ld exchange m ix tu re ( IV -1 5 ) , S=0, im p ly in g a c o n s ta n t
Ferm i le v e l . The energy gap is thus d e te rm in ed by th e cornbin-
a t io n G o f th e exchange m ix tu re , w h ich is dom inated by th e
M ajorana component o f th e in t e r a c t io n . T h e re fo re , a s tro n g
a t t r a c t iv e M ajorana fo r c e , fa v o r in g maximum s p a t ia l sym m etry,
w i l l p roduce a la rg e energy gap, and th e a tte n d a n t s ta b le HF
s o lu t io n s ( v . s e c t io n IV .5 a ) .
F o r a fo rc e o f a r b i t r a r y r a d ia l dependence, f i l l i n g each
o r b i t |X > w ith fo u r n u c le o n s , th e m a tr ix e lem ents o f th e HFA
H a m ilto n ia n ( IV -8 ) become
n n< a |h |B > = < a |t|& > + s £ < a X |V |3 X>n-GJ<aX|V|X$>n ( IV -2 3 )
X=1 v X=1 v
where n=A/4 re p re s e n ts th e number o f space o r b i t a ls |X >.AThe s u b s c r ip t D in d ic a te s th a t th e m a tr ix e lem ent i s th e
d i r e c t te rm
<aX | V | 8X>D = / d r 1d r 2<j)a* ( r 1)«J)A* ( r 2 ) V ( r 1 , r 2) ( r 2 )<j>x ( r 2) ,
( IV -2 4 )
T ra n s fo rm in g th e HF H a m ilto n ia n ( IV -2 3 ) to c o n f ig u ra t io n
space may be accom p lished o n ly a f t e r th e r e la t io n between
momentum-and c o n f ig u ra t io n -s p a c e wave fu n c t io n s is d e f in e d :
|a> = fdr<r|<{> ( r ) > | r > . ( IV -2 5 )vX
Then th e H a m ilto n ia n ( IV -2 3 ) may be w r i t t e n
<£ i l M ? 2> = - 6 ( r x- r 2) — + S 6 ( r 1- r 2 )/dr^|< j>x ( r ) | 2V ( r x , r )2 2m X=1 A
- ) V ( r x , r 2) . ( IV -2 6 )X
S i s th e in t e n s i t y o f th e lo c a l te rm o f th e HF f i e l d , and G
is th e in t e n s i t y o f th e n o n - lo c a l te rm . The m a n ife s ta t io n
99
o f an energy gap in th e HF spectrum is th e re fo re a d i r e c t
consequence o f th e n o n - lo c a l i t y 6 f th e HF f i e l d .
A G aussian p o te n t ia l w ith R o se n fe ld exchange cha ra c
t e r i s t i c s and v a n is h in g s p in - o r b i t s p l i t t i n g y ie ld s th e HFA l|
r e s u l t s 25 shown in F ig u re 17. In th e Mg a x ia l s o lu t io n ,
a n e u tro n and p ro to n a re p la c e d In th e d ^ f and o r b i t s .
In th e e l l i p s o id a l s o lu t io n , fo u r nuc leons f i l l th e same
s p a t ia l o r b i t . The e l l i p s o id a l s o lu t io n , w h ich has
lo w e r energy and a la r g e r gap th a n th e a x ia l s o lu t io n , has
maximum s p a t ia l sym m etry. T h e re fo re , in o rd e r o f p re fe r re d
sym m e trie s , a R o se n fe ld fo rc e p re fe rs HF s o lu t io n s w ith
maximum s p a t ia l symmetry to s o lu t io n s w ith a x ia l sym m etry.20 24Both a re co m p a tib le in Ne , b u t n o t in Mg . In th e p ro la te
p Os o lu t io n o f S i b o th th e (d ^ -d ^ ) and (d^+d ^ ) o r b i t s a re
f i l l e d . In t h is case , a x ia l symmetry is re s to re d because th e
o p e ra to r J s im p ly tra n s fo rm s one occu p ied o r b i t in t o a n o th e r zand le a ve s th e s e t o f occup ied o r b i t s in v a r ia n t ( c f . s e c t io n
I V . 3 ) . E s s e n t ia l ly , th e o b la te s o lu t io n may be o b ta in e d by
exchang ing empty and f i l l e d o r b i t s o f th e p r o la te s o lu t io n .
The two a re n e a r ly o r th o g o n a l and degenera te in th e HF
e n e rg ie s ( c f . s e c t io n IV .6 d ) .
The e x is te n c e o f o r th o g o n a l s o lu t io n s d e s c r ib in g d i f f e r -p O
e n t e q u i l ib r iu m shapes in S i is co n firm e d by th e re a c t io n
A l27(He3 ,p ) S i2 '*. M e y e r-S c h u tz m e is te r62 has found th a t th e
r e a c t io n c ro ss s e c t io n f o r a s ta te near 4 .7 MeV e x c i ta t io n29 29energy in S i i s v e ry much la r g e r th a n th a t f o r th e S i
ground s ta te . A ve ry l i k e l y e x p la n a t io n o f t h is phenomenon
1 00
i s th a t th e 4 .7 MeV s ta te in S i2^ is a r e s u l t o f a n e u tro n
occu p y ing th e lo w e s t HF o r b i t a l a v a i la b le 1” in th e S i2® p r o la te27c o n f ig u r a t io n . S ince A l has a p r o la te e q u i l ib r iu m deform a
t io n in th e ground s ta te 63, th e re a c t io n c ro ss s e c t io n f o r
the 4 .7 MeV s ta te s h o u ld , th e r e fo r e , be la r g e r th a n th a t f o r
th e ground s ta te , th e fo rm e r r e a c t io n b e in g shape c o n se rv in g
( c f . s e c t io n , IV .6 d ) .
Im p o rta n t e f fe c ts on th e HF wave fu n c t io n s a re i n t r o
duced by th e in c lu s io n o f an t*s s p in - o r b i t te rm in th e HF
H a m ilto n ia n . I t m ixes s ta te s w ith sp in s o f o p p o s ite s ig n s ,
so th a t th e o r b i t s cannot be fa c to r iz e d in t o space and s p in
p a r ts as in e q u a tio n ( IV -2 0 ) . I f th e s t re n g th o f th e s p in -
o r b i t te rm were s m a ll, i t w ould cause a s m a ll p e r tu r b a t io n
in even-even N=Z n u c le i (w h ich have c lo se d s p in - is o s p in20s h e l ls ) . F o r Ne , F ig u re 14 shows a gap o f 8 MeV. N u c le i
such as S32 and A r3® have c a lc u la te d gaps o f le s s th a n 5 MeV,
w h ich i s o f th e o rd e r o f th e s p in - o r b i t s p l i t t i n g . In th e
absence o f s p in - o r b i t s p l i t t i n g , th e wave fu n c t io n s o f th e28
2 i - 1 d s h e l l n u c le i w ou ld be sym m etric w ith re s p e c t to S i
The s p in - o r b i t s p l i t t i n g , how ever, in tro d u c e s an asymmetry
between p ro p e r t ie s o f th e n u c le i b e lo n g in g to th e f i r s t and•la *ta
second h a lv e s o f th e s h e l l ( c f . s e c t io n I V . 6 ) . F o r n e g l i -32g ib le s p in - o r b i t s p l i t t i n g , th e e l l i p s o id a l s o lu t io n f o r S
^ We speak here o f f i l l i n g th e lo w e s t a v a i la b le o r b i t a l in a schem atic way o n ly . As p re v io u s ly n o te d , th e a d d it io n o f a s in g le n u c le o n to an even-even n u c leus a l t e r s th e deform ed o r b i t a l s o lu t io n s . I t is n o t expected how ever, th a t t h is w ould change th e n a tu re o f th e d e fo rm a tio n .
++ A s p in - o r b i t te rm is in c lu d e d in b o th th e n u c le a r H a m ilto n ia n ( IV -1 ) and th e HF H a m ilto n ia n ( IV -8 ) by th e use o f e x p e r im e n ta l s in g le - p a r t ic le e n e rg ie s .
1 0 1
20 24F ig u re 17. The spectrum o f HF o r b i t s o f Ne and Mg inth e case o f v a n is h in g s p in - o r b i t in t e r a c t io n . The o r b i t sa re la b e le d by t h e i r wave fu n c t io n in c o n f ig u ra t io n spaceand th e o r ie n ta t io n + and + o f th e s p in . N eu trons and p roto n s have s im i la r o r b i t s . O ccupied o r b i t s a re marked by ad o t . The n o ta t io n means £=2 ,m = l. The o r b i t s o f th ee l l i p s o id a l s o lu t io n a re la b e le d by th e s ta te ( n ,n , n )x y zto w h ich th e y a p p ro x im a te c lo s e ly .
DEFORMED ORBIT COMPOSITIONS (Vso=0)
‘M W0.58d0+0.8l s0
-0.8ld0+0.58s0
N e 2 0 ( A X IA L )
E h f = - 3 4 . 2 9
A \ ’ \ ( 2 0 0 ) 4 t W r
W W f f i < " ° > W0.63do+0.77s„ (020) 4jA|r
t l 0 l ) i 4 f j r
- f t - —
^ t d| d- ' T H "
10111
(0 0 2 )-0.77 d0+ 0.63s0
M g 2 4 (A X IA L ) M g 2 4 (E L L IP S O ID A L )
E ^ p 2 —6 5 .9 6 E ^ p = - 7 3 . 6 3
1 02
i s a p p ro x im a te ly 15 MeV be low an a x ia l l y sym m etric s o lu t io n .
W ith in c re a s in g s p in - o r b i t s t r e n g th , th e HF e n e rg ie s o f th e
e l l i p s o id a l and a x ia l l y sym m etric s o lu t io n s approach each
o th e r , and f i n a l l y merge w ith th a t o f th e s p h e r ic a l e q u i l ib r iu m
shape. Recent e x p e r im e n ta l r e s u l t s 64 in d ic a te th a t S32 is
a lm o s t s p h e r ic a l . I t i s th e re fo re expected th a t th e re shou ld
be n o n -n e g l ig ib le c o n f ig u ra t io n m ix in g , w h ich is n o t ta ke n
in t o accoun t by th e HF wave fu n c t io n s . These w ould ten d to
d e s tro y th e s t a b i l i t y o f th e HF deform ed s ta te a g a in s t B and
Y v ib r a t io n s ( c f . s e c t io n s IV .6 e and I V . 7 ) .
5 . P a r t ic le - H o le A dm ix tu res to H a rtre e -F o c k
a . P re lim in a ry D is c u s s io n
I m p l i c i t i n H a rtre e -F o c k th e o ry is th e assum ption th a t
o n ly th e lo w e s t energy s in g le - p a r t ic le o r b i t s a re o ccu p ie d .
From e q u a tio n ( IV -2 6 ) i t i s im m e d ia te ly seen th a t i f th e
e f f e c t iv e n u c le o n -n u c le o n in t e r a c t io n c o n ta in s an exchange
te rm , h w i l l n o t be a lo c a l o p e ra to r . A p h y s ic a l ly s i g n i f i
c a n t consequence o f t h is n o n - lo c a l i t y i s th e p e rs is te n c e o f
an energy gap between o ccup ied and n o n -o ccu p ie d le v e ls . I f
t h is gap is s m a ll, a p a r t ic le - h o le expans ion around th e HF
i n t r i n s i c s ta te may indeed p ro v id e a su cce ss ive a p p ro x im a tio n
to lo w - ly in g n u c le a r s ta te s . F o r v e ry l i g h t n u c le i , such asO 1 p
Be and C , th e energy gap i s o f th e o rd e r 16-18 MeV, and
th u s th e assum ption o f a sharp Ferm i s u r fa c e i s v a l id . On
th e o th e r hand, f o r n u c le i in th e 26 - I d . s h e l l , th e energy
103
gap is s ig n i f i c a n t l y s m a lle r (^5 -8 MeV), th e la r g e r gaps
a p p e a rin g in th e b e g in n in g o f th e s h e l l . The e f f e c t o f th e
decrease in gap s iz e i s e v id e n t from F ig u re 15. The Ne20
s o lu t io n , w h ich e x h ib i t s a la rg e gap (^8 MeV) in com parison
to th e o th e r n u c le i s tu d ie d , compares f a i r l y w e l l w ith th ep o p/r
e x p e r im e n ta l sp e c tru m ; S i and A r , how ever, w ith gaps
v5 MeV, compare p o o r ly w ith e xp e rim e n t.
A s s o c ia te d w ith th e decrease in s iz e o f th e energy gap
is an in c re a s e in th e p r o b a b i l i t y th a t p a r t ic le s in th e HF
s ta te w i l l s c a t te r ac ross th e gap to unoccup ied le v e ls . The
r e s u l t i s th e in t r o d u c t io n o f a degree o f d if fu s e n e s s to the
p re v io u s ly sharp Ferm i s u r fa c e , i . e . , th e upper-m ost "o c c u p ie d "
le v e ls in th e Ferm i sea may have le s s th a n u n i t p r o b a b i l i t y
o f b e in g o ccu p ie d , w h ile th e "u n o ccu p ie d " le v e ls may have
f i n i t e p r o b a b i l i t y o f b e in g o ccu p ie d .
I f i t i s s t i l l d e s ira b le to speak o f an i n t r i n s i c n u c le a r
wave fu n c t io n , th e n , based on th e above d is c u s s io n , i t sho u ld
be a m ix tu re o f th e H a rtre e -F o c k d e te rm in a n ta l s o lu t io n
and n - p a r t ic le /n - h o le d e te rm in a n ts . The p a r t ic le - h o le d e te r
m inan ts a re d e fin e d as h a v in g n "o c c u p ie d " s in g le - p a r t ic le
s ta te s empty (h o le s ) and n "u n o ccu p ie d " s ta te s f i l l e d (p a r
t i c l e s ) . The number n o f p a r t ic le s o r h o le s in a s ta te
|np-nh> th a t can s ig n i f i c a n t l y m ix w ith ¥^F shou ld be s m ^ ll
because these s ta te s w i l l have an u n p e rtu rb e d energy o f fh e
o rd e r nS above the HF e n e rg y , where <5 i s th e s iz e o f the
energy gap between o ccu p ie d and unoccup ied o r b i t s .
The one-body H a rtre e -F o c k s e l f - c o n s is te n t p o t e n t ia l i s
de te rm ine d such th a t tw o-body r e s id u a l in te r a c t io n s canncft
cause lp - lh e x c i ta t io n s . I t has been argued th a t " th e HF
p o te n t ia l b e in g a one-body p o te n t ia l sh ou ld i t s e l f be a b le
to cause l p - l h typ e e x c i t a t i o n s ; . . . " th e H a m ilto n ia n i s thus
s ta b le a g a in s t 1 - p a r t i c le / l - h o le e x c i t a t io n s , and th e re fo re
th e lo w e s t-o rd e r c o r re c t io n to is th ro u g h a d m ix tu re o f
2p-2h typ e d e te rm in a n ts .56 W h ile i t i s t ru e th a t i s
s ta b le a g a in s t l p - l h e x c i t a t io n s , i t i s n o t th e t o t a l ad
m ix tu re s w h ich a re im p o r ta n t , b u t r a th e r a d m ix tu re s to
s p e c i f ic t o t a l a n g u la r momentum components o f th e HF s ta te .
The sum o f these a d m ix tu re components v a n is h e s , b u t s in c e th e
t o t a l a n g u la r momentum c h a ra c te r iz e s th e e ig e n s ta te s o f th e
system , i t i s th e in d iv id u a l term s w h ich a re s ig n i f i c a n t .
Thus, o n e -p a r t ic le /o n e -h o le e x c i ta t io n s sh o u ld y ie ld th e
lo w e s t-o rd e r c o r re c t io n s to th e a n g u la r momentum components
P^ H F of> th e ^ s t a t e *
b . I s o to p ic S p in C o n s id e ra tio n s
P a r t ic le - h o le e x c ita t io n s o f th e HF s ta te r e a d i ly le n d
them se lves to th e p o ly n o m ia l p r o je c t io n te ch n iq u e s deve loped
in C hap te r I I i f th e is o s p in c o m p o s itio n o f these s ta te s is
s im p le enough. The one- and tw o - p a r t ic le - h o le c o n f ig u ra t io n s
co n s id e re d range in is o s p in from T = |T _ | to T= |T 1+2. Inz zth a t w h ich fo l lo w s , (-c) w i l l denote m u l t i - p a r t i c le s ta te s o f
th e v a r io u s typ e s d e s c rib e d be lo w . (1 ) s ig n i f ie s th e HF s ta te .
There a re two b a s ic typ e s (assum ing N=Z) o f 1 - p a r t i c le /
o n e -h o le e x c i ta i to n s to be c o n s id e re d :
(2 ) one e x c ite d n e u tro n , le a v in g a n e u tro n h o le In th e
104
105
(3 ) one e x c ite d p ro to n , le a v in g a p ro to n h o le in th e HF
s ta te (F ig u re 1 8 b ). These s ta te s a re n o t a n n ih i la te d by the
is o s p in r a is in g o p e ra to r T+ . They a re , how ever, a n n ih i la te d
by T 2 , in d ic a t in g th a t th e y may c o n ta in b o th T=0 and T=1
is o s p in com ponents. These s ta te s may be combined in a p a r
t i c u l a r l y s im p le m anner, to produce pu re is o s p in s ta te s .
D e no tin g by ( 2 ) ' and ( 3 ) ' s ta te s o f th e above typ e s w ith the
same p a r t i c le and h o le s ta te s occup ied ( th e is o s p in p r o je c
t io n s a re exchanged), th e two n o rm a liz e d co m b in a tio n s a re
/ - | [ ( 2 ) , + ( 3 ) 1] and/ | [ ( 2 ) ' - ( 3 ) ' ] , w ith T=0 and T = l, re sp e c
t i v e l y . We n o te th a t th e fo rm e r , w h ich w ould u s u a lly be
c a lle d sym m etric because o f th e p lu s s ig n , i s a c tu a l ly a n t i
sym m etric in is o s p in space.
Fou r typ e s o f tw o - p a r t ic le / tw o - h o le s ta te s a re co n s id e re d
f o r even-even N=Z n u c le i :
(4 ) an e x c ite d p ro to n p a i r 1 (F ig u re 1 8 c ),
(5 ) an e x c ite d n e u tro n p a i r (F ig u re l8 d ) ,
(6 ) an e x c ite d n e u tro n -p ro to n p a i r w ith p o s i t iv e n e u tro n
z - p r o je c t io n (F ig u re l8 e ) ,
(7 ) an e x c ite d n e u tro n -p ro to n p a i r w ith n e g a tiv e n e u tro n
z - p r o je c t io n (F ig u re l 8 f ) . O the r n o n -p a ir tw o - p a r t ic le / tw o -
h o le c o n f ig u ra t io n s are n o t c o n s id e re d u s in g these methods
because t h e i r is o s p in co m p o s itin n s a re to o c o m p lic a te d . F o r
these i t w ou ld be necessary to in tro d u c e th e is o s p in r a is in g
t W ith a x ia l symmetry b e in g assumed in these c a lc u la t io n s , a p a i r i s d e f in e d as those degenera te s in g le - p a r t ic le HF e ig e n fu n c t io n s w ith o p p o s ite z - p r o je c t io n s .
HF state (Figure 18a),
F ig u re 18. 1 - and 2- p a r t ic le - h o le c o n f ig u ra t io n s . N eutronsa re shown to th e l e f t and p ro to n s to th e r i g h t . P a r t ic le s a re denoted by " x " and h o le s by " o " . (a ) and (b ) a re 1 -p a r -t i c l e / l - h o l e typ e e x c i t a t io n s , w h ile (c ) th ro u g h ( f ) a re 2 - p a r t ic le /2 - h o le typ e e x c i ta t io n s .
I - AND 2 - P A R T IC L E H O LE C O N F IG U R A T IO N S
X XX o X X X X X o
n p n p
(a) (b)
X X X X X X X X
X X o o o o X X O X X o X o o x
n p n p n p n p
(c) (d) (e ) ( f )
106
o p e ra to r d iscu sse d in s e c t io n I I . 4.
The HF s ta te o f even-even N=Z n u c le i c o n s is ts o f p a ire d
s in g le - p a r t ic le s ta te s w h ich are t im e -re v e rs e d o f each o th e r
and degenera te in e n e rg y . The HF s ta te is th e r e fo r e , by
d e s ig n , in v a r ia n t under tim e re v e rs a l ( c f s e c t io n I V . 3 ) .
T h is in v a r ia n c e p ro p e r ty sho u ld be re ta in e d by th e c o r re c te d
i n t r i n s i c s ta te c o n s is t in g o f th e o r ig in a l HF s ta te and p a r
t i c le - h o le a d m ix tu re s . T h e re fo re , those 2 - p a r t ic le /2 - h o le
s ta te s w h ich m ix most w ith th e o r ig in a l HF s ta te w i l l be
those w h ich have b o th t im e -re v e rs e d h o le s and t im e -re v e rs e d
p a r t ic le s in p a i r s ta te s ( ty p e s (4 ) and ( 5 ) , a b o ve ). The
same re a s o n in g a p p lie d to is o s p in in v a r ia n c e p re d ic ts th a t
those s ta te s w h ich m ix most w ith th e HF s ta te w i l l have h o le s
o r p a r t ic le s w h ich a re e i t h e r b o th p ro to n s o r b o th n e u tro n s .
S ta te s o f th e typ e s (6 ) and (7 ) a re in c lu d e d because these
have la rg e o v e r la p w ith th e 1 - p a r t i c le / l - h o le s ta te s . They
a re a ls o re q u ire d f o r th e s im p le c o n s tru c t io n o f T=0 s ta te s .
We have , th e r e fo r e , in c lu d e d those 2 - p a r t ic le /2 - h o le s ta te s
w h ich have maximum o v e r la p w ith th e HF s ta te and i t s 1 -p a r
t i c l e / l - h o l e e x c i ta t io n s . The use o f an in co m p le te s e t o f
2 - p a r t ic le /2 - h o le s ta te s sh o u ld n o t , th e r e fo r e , be to o d e t r i
m en ta l to th e r e s u l t s . The above ch o ice s a re s u f f i c ie n t to
c o n s tru c t s im p le is o s p in s ta te s , as shown be low .
As in th e case o f th e 1 - p a r t i c le / l - h o le c o n f ig u ra t io n s ,
pure is o s p in s ta te s may be o b ta in e d from th e above fo u r type s
(w ith p rim es a g a in s ig n i f y in g s ta te s w ith id e n t ic a l p a r t i c l e -
h o le s t r u c tu r e and exchanged is o s p in p r o je c t io n s ) by fo rm in g
107
the following simple combinations:
T=0 i [ {(4)■ + (5 ) ' } - 1(6) ' + ( 7 ) ' ) ]
T=1 k C t*1) ' “ ( 5 ) ’ ]
| C ( 6 ) ' - ( 7 ) ' ]
T=2 | [ { (4 )> + ( 5 ) ’ } + { ( 6 ) ' + ( 7 ) ' ! ]
To f a c i l i t a t e th e w r i t in g o f n o rm a lize d i n t r i n s i c s ta te s o f
good is o s p in , th e fo l lo w in g n o rm a liz a t io n s and phases a re
in tro d u c e d :
U s ing th e se s im p le co m b ina tio n s ( IV -2 8 ) as i n t r i n s i c
s ta te s fro m w h ich p r o je c t io n i s to be accom p lished c o m p le te ly
o b v ia te s th e need to s im u lta n e o u s ly p r o je c t is o s p in and angu
l a r momentum. T h is w ou ld re p re s e n t a fo rm id a b le p rob lem s in ce
an N -n e u tro n Z -p ro to n p rob lem w ould in v o lv e an N+Z n e u tro n
re p re s e n ta t io n and th e employment o f th e is o s p in r a is in g o r
lo w e r in g o p e ra to r , in a manner ana lagous to th e use o f th e
a n g u la r momentum r a is in g o p e ra to r d iscu sse d in C hap te r I I .
I n f a c t , w ith th e above c o m b in a tio n s , is o s p in need h a rd ly be
a c o n s id e ra t io n when p e rfo rm in g th e c a lc u la t io n s .
( IV -2 7 )
N o rm a lized s ta te s may now be w r i t t e n
( IV -2 8 )
1 08
^ab^ = N1 | v ( a ) ir (b ) + p1v (b ^Ti( a ) > . ( IV -2 9 )
These a re th e n combined to th e s ta te s ( IV -2 8 ) . (We no te
th a t th e se s ta te s ( IV -2 9 ) w i l l have sharp is o s p in o n ly f o r
th e HP s ta te and i t s 1 - p a r t i c le / l - h o le e x c i t a t io n s . )
c . The Use o f R eference N u c le i and Hole S ta te s
The use o f re fe re n c e c lo s e d -s h e l l n u c le i i s o f g re a t
a s s is ta n c e in p e rfo rm in g s h e l l model c a lc u la t io n s , b u t as
a lre a d y m e n tion e d , th e assum ption o f an in e r t co re m igh t
p re c lu d e a f u l l u n d e rs ta n d in g o f th e e le c t r i c and m agne tic
p ro p e r t ie s o f th e n u c le u s . S ince th e main in t e r e s t In
these c a lc u la t io n s i s in th e energy s p e c tra and wave fu n c
t io n s , w i l l be used as th e re fe re n c e n u c leu s f o r th e
lo w e r h a l f o f th e 2 6 - 1 d s h e l l (Ne20 , Mg2*1 and S i2® ), and
Ca**° f o r th e upper h a l f o f th e s h e l l (S32 and A r3® ). The
f i l l i n g o f o r b i t s in th e 2 6 - 1 d s h e l l n u c le i may be re fe re n c e d
to th e f i l l i n g in 0 1® o r Ca**° as fo l lo w s :
A M N1 = 1 + I + 1
A=1 l e r e f y = lv = l
The M " p a r t i c le o r b i t s " y a re f i l l e d in th e n u c le u s under
c o n s id e ra t io n , b u t empty in th e re fe re n c e n u c le u s ; th e N
"h o le o r b i t s " v a re empty in th e n uc leu s b e in g s tu d ie d ,
b u t f i l l e d in th e re fe re n c e n u c le u s . Thus, f o r n u c le i below 28S i th e f i r s t summation on I i s o ve r th e 16 and 1p s h e l ls .
Projections are actually performed using states of the
form
109
The p a r t ic le o r b i t s y may b e lo n g to th e 2 6 - 1 d s h e l l o r
h ig h e r s h e l ls , and th e h o le o r b i t s v to th e 16 and 1p s h e l ls? Rf o r n u c le i above S i , th e o r b i t s I may b e lo n g to th e 74,
1p and 2 6 - 1 d s h e l ls , th e p a r t ic le o r b i t s y to the 2 p - 1 £
s h e l l o r h ig h e r s h e l ls , and th e h o le o r b i t s v to th e 2 6 - 1 d
s h e l l o r lo w e r s h e l ls .
The HF H a m ilto n ia n ( IV -8 ) now becomes
<4>i |7i|<j)j> = <<J»i 111 <f>j> + ^ I f <<l,i Xj l |v |< j)jX )l>
M N+MI < * i xv M - J < q x v iv| V v > .
The f i r s t l in e above is ju s t th e HF H a m ilto n ia n o f th e r e fe r
ence c lo s e d -s h e l l n u c le u s , w h ich i s d ia g o n a l and has s p h e r i
c a l s o lu t io n :
M N- e+ i«± j + I ^ i X y |V |4 'JXU> ~v I< * iX v (VI * jX v>
where o n ly p a r t i c le and h o le o r b i t s appear and the are
s in g le - p a r t ic le o r h o le e n e rg ie s i n th e 0 "^ o r C a ^ f i e l d ,
r e s p e c t iv e ly .lin
The use o f Ca f o r th e upper s h e l l n u c le i im p lie s th a t
th e HF s ta te s o f S32 and A r3^ must now be e n v is io n e d as con
s is t in g o f h o le s r a th e r th a n p a r t ic le s . That w h ich was p re
v io u s ly c o n s id e re d n - p a r t ic le /n - h o le e x c ita t io n s o f th e HF
s ta te must how be co n s id e re d n - h o le /n - p a r t ic le e x c i t a t io n s .
T e c h n ic a l ly , t h is poses o n ly a s m a ll p rob lem w hich is e a s i ly
d e a lt w i th . I t w i l l be shown below th a t an a lm ost c lo se d
s h e l l I s e q u iv a le n t , in a sense, to an a lm os t empty s h e l l .
The t o t a l number m o f p a r t ic le s (n e u tro n s o r p ro to n s )
w h ich can be p la ce d in a g iv e n s h e l l is
m = I (2 /+ 1 )S
where th e summation extends o ve r a l l va lu e s o f p a r t ic le
a n g u la r momentum a llo w e d in th e p a r t ic u la r s h e l l under s tu d y .
Thus, f o r th e 26 - 1 d s h e l l , m=2+4+6=12. F o r an n - p a r t ic le
s ta te , w ith n>m/2, i t i s more co n ve n ie n t to s p e c ify a s ta te
o f th e system by th e s e t o f m-n uncocup ied o r h o le s ta te s .
A o n e -to -o n e correspondence may be e s ta b lis h e d between each
n - p a r t ic le s ta te and a co n ju g a te n -h o le s ta te . The number
o f these c o n ju g a te h o le s ta te s i s , o f n e c e s s ity , e q u a l to th e
number o f p a r t ic le s ta te s .
We c o n s id e r now th e tw o-body p a r t V o f th e H a m ilto n ia n .
D en o ting th e s e t o f a l l quantum numbers necessary to s p e c ify
a g iv e n p a r t i c l e s ta te by y , each row and column in th e m a tr ix
I v i k may be c h a ra c te r iz e d by a s e t o f n numbers y^> y2> • • >Vln *i r i ' ' '
S ince is a tw o - p a r t ic le o p e ra to r , < 1 ^ 2 • • • V * v i k l pi p2 - - * V
d i f f e r s fro m ze ro o n ly i f th e s e t y^ d i f f e r s from th e s e t y |
by a t most two members. F o r th e d ia g o n a l e lem ents o f th e
n - p a r t ic le c o n f ig u ra t io n
<yl y 2* ’ ^ Vi k l ul y 2 ’-“ y n> = I < ^ ’ |v 1 2 | y y ’ > . ( iv - 3 0 )i , k = l y > y *
y , y ' = y 1y 2 * • *Un S im i la r ly , f o r th e n o n -d ia g o n a l e lem ents
where i t is assumed th a t th e s e t y.£ d i f f e r s from th e s e t y^
in a t most th e f i r s t two e le m e n ts . S ince th e m u l t i - p a r t i c le
s ta te s c o n s id e re d a re S la te r d e te rm in a n ts , th e y can a lways
be re o rd e re d so th e d i f f e r i n g s ta te s appear f i r s t .
L a b e lin g th e s ta te s o f th e n -h o le c o n f ig u ra t io n by th e
unoccup ied v a lu e s o f y r a th e r th a n th e o ccup ied va lu e s
<y - 11j J 1 . . . u * 1 | i f V . J u - R i - L - . y - L = I< v y ' |V |p y ’ > ( IV -3 2 )l , k = t t + l
y ,y» = * “ y n*
Here th e summation extends o ve r a l l va lu e s o f y d i f f e r e n t
from y ^ , y 2 , • • , y n » i . e . , o ve r th e m-n occu p ied y - s ta te s . Now,
Ill
I <yy ' |V|yy'> = \ \ <yy•|V|yy * >y>U * d . .
y , y - ^ y 1 . . . y n
= \ { I < y y ’ |v |y y '> - I I < y , y ' | V | y , y ’ >2 yy* i = l y 1
- I I< y y jL|v |y y 1>} + | I < yy1 |V |y y '> . ( IV -3 3 )1 -1 y 2y y '= y 1 - * y n
That i s , th e f i r s t summation in b ra c k e ts has no r e s t r i c t io n s
on e i t h e r y o r y ' , th e second summation has no r e s t r i c t i o n on\
y ' , and th e t h i r d no r e s t r i c t i o n on y . The e x p re s s io n above
in b ra c k e ts , denoted by EQ, is independen t o f th e s p e c i f ic
va lu e s o f y ^ y 2 ’ ’ *y n* c o n t r i b u t i ° ns to th e second and t h i r d
te rm s a re each independen t o f y ^ . Comparing e q u a tio n s ( IV -3 1 )
and ( IV -3 3 ) , f o r d ia g o n a l e lem ents
1 12
> = E + oi,k = w + l
<V lV 2 - - - V n l | Z Vi , k = l
A s im i la r c a lc u la t io n shows th a t f o r th e n o n d ia g o n a l e lem ents
C o m p le te ly ana lagous r e s u l ts can be shown to be t ru e f o r the
one-body p a r t o f th e H a m ilto n ia n . T h e re fo re , th e e n t i r e
H a m ilto n ia n m a tr ix in a h o le c o n f ig u ra to n i s e qua l to a con
s ta n t Eq p lu s th e tra n sp o se d m a tr ix o f th e co n ju g a te p a r t ic le
c o n f ig u r a t io n . The s p e c tra f o r th e two c o n f ig u ra t io n s a re
th e same, b u t s h i f t e d by th e amount Eq . T h is p roves p a r t ic u
l a r l y u s e fu l in t r e a t in g S32 and A r3^ as 8 - and 4 -h o le40system s, r e s p e c t iv e ly , w ith Ca as re fe re n c e n u c le u s , r a th e r
th a n as 16- and 2 0 - p a r t ic le system s, r e s p e c t iv e ly , w ith
as re fe re n c e n u c le u s .
d . P ro je c t io n E q u a tion s and Wave F u n c tio n A n a ly s is
C a lc u la t io n s have been c a r r ie d o u t u s in g i n t r i n s i c s ta te s
o f th e fo rm ( IV -2 9 ) . Each s ta te is a s u p e rp o s it io n o f
s ta te s w ith v a r io u s a n g u la r momentum components ( c f . s e c t io n
i , k = n + l
i , k = l
II.2):
( IV -3 6 )
The p o ly n o m ia l p r o je c t io n te ch n iq u e s o f s e c t io n I I . 4 c a r ry
o ve r q u ite s im p ly . The ana logue o f e q u a tio n (1 1 -2 4 ) may be
deve loped as
<J+W*a b >l J +P* c d )> , 0 , y ) a jba j d< J ,y , (a b ) | J , y , (cd )> ( IV -3 7 )
= (2 -6 1 J )N1NJ <J+»1Cv( a ) ir ( b ) +p1v ( b ) ir ( a ) ] | J +*lv ( c ) i r ( d ) >
= (2 -6 )N N J W 2 I I B (J ,0 ,X )B (J 2 ,0 ,y -X ) 1 J =0 l AJ j 1=x J 2=y-X 1
[ Ma c 'Mb d 2+ P iMSo‘ Mad2;l ( IV ' 38)
S im i la r ly , ana lagous to e q u a tio n (1 1 -2 7 ) ,
<J+P*a b ) l H l J +P,,,c d )> = I B (J»0»Vi)a5ba5d< J ,y , ( a b ) |H |J ,y , ( c d ) >J ( IV -3 9 )
= (2 -6 , )N N I { M 2[ I I B (J , ,0 ,X )B (J 2 ,0 ,y -X )1 Jx=0 W J j J 2
i K l ' Hb f 2 + Ha e X f 2) +pi K e lH d f 2+HSe1Md f 2) }
+ I Mb s 2 + pi Hnp^rs , e f Mc r X Md s 2 ) ]r s
+ M I [ x !!m | I / B ( J 1>0 ,X ) /B ( J 1,mr ,X-mr ) /B ( J 2 ,0 ,y -X ) * rs ' -
113
• 'B (J2 ,-m r ,u-X+mr ) [H ^ p ( r s ,e f ) M ^ > N ^ 2
+ P iHn p ( l ’s - e f)N c r 1 Md f 2 ]>
( IV -4 0 )
114
Once a g a in , the m e tr ic a p p e a rin g in e q u a tio n ( IV -3 7 )
is o b ta in e d by su cce ss ive e l im in a t io n o f a n g u la r momentum
components b e g in n in g w ith Jmax, and th e o r th o n o rm a liz in g
m a tr ix (1 1 -1 2 ) is o b ta in e d v ia th e Schm idt o r th o n o rm a liz a t io n
p roce du re ( v . s e c t io n I I . 3 ) . The H a m ilto n ia n m a tr ix a p p e a rin g
in e q u a tio n ( IV -3 9 ) is th e n c a lc u la te d , and th e m a tr ix (1 1 -1 2 )
is a p p lie d t o . y ie ld th e H a m ilto n ia n m a tr ix in an o rth o n o rm a l
subspace f o r each v a lu e o f a n g u la r momentum and is o s p in .
These a re th e n d ia g o n a liz e d , r e s u l t in g in energy s p e c tra and
wave fu n c t io n s f o r th e e n t i r e system .
Wave fu n c t io n s o f th e H a m ilto n ia n may be expanded
as l in e a r co m b in a tio n s o f th e J -T components o f th e i n t r i n s i c
n e u tro n -p ro to n s ta te s :
* j £ > - I (PJT'l>np>> » ( I V - i l l )k
i kwhere a .j^ a re a m p litu d e c o e f f ic ie n t s . These wave fu n c t io n s
a re o b ta in e d v ia th e d ia g o n a liz a t io n o f th e H a m ilto n ia n
m a tr ix in a n o th e r o r th o n o rm a l b a s is ^
'1JT ) ° I < I V - 1 , 2 )m
w h ic h , in t u r n , is d e te rm in e d from th e i n t r i n s i c b a s is by the
Schm idt o r th o n o rm a liz a t io n p ro ce d u re :
♦ j " ’ = I BJ ? (pJT '*'nD>) • ( I V - Ws
The a m p litu d e s al k may th us be expanded in term s o f th e o r th o
n o rm a liz a t io n and e ig e n v e c to r m a tr ic e s. ik _ j Aim Dmk
mi k _ r Aim nmkJT “ L J T J T ( IV -4 4 )
JT (\c)We n o te , how ever, th a t th e e x p l i c i t fo rm P d> is n o tnpo b ta in e d by t h is p r o je c t io n p ro ce ss .
6 . R e s u lts o f C a lc u la t io n s
a . G enera l Remarks
The m e tr ic o f th e J = 0 subspace o f T=0 s ta te s used in 20th e Ne p a r t ic le - h o le m ix in g c a lc u la t io n is p re se n te d in
T ab le X. The s ta te s la b e le d l p - l h (k ) and 2p-2h (k ) a re
th e n o rm a liz e d co m b in a tio n s o f p a r t ic le - h o le e x c i ta t io n s
d iscu sse d in s e c t io n IV .5 b , w ith (k ) s ig n i f y in g th e a n g u la r
momentum p r o je c t io n on to th e a x is o f symmetry o f th e deform ed
p a r t i c le s ta te ( v . T ab le IX ) to w h ich th e p a r t ic le ( p a i r ) is
e x c ite d . O f p a r t ic u la r im po rtan ce I s th e e x is te n c e o f non
ze ro 1 - p a r t i c le / l - h o le m e tr ic com ponents, and th e la c k o f
o r th o g o n a l i ty o f these to th e HF component. T y p ic a l o v e r
la p s o f th e HF and p a r t ic le - h o le J=0 , T=0 components a re o f
th e o rd e r 1 /1 0 , based on n o rm a liz e d a m p litu d e s . The o v e rla p s
o f th e 1 - and 2 - p a r t ic le - h o le components o f t h is m e tr ic a re
o f th e same o rd e r o f m agn itude . We n o te th a t th e 2p-2h s ta te s
have a r a th e r la rg e J=0 com ponent, and th a t th e y th u s re p re s e n t
an in t r o d u c t io n o f p a i r in g c o r r e la t io n s .
The s p e c tra o b ta in e d from th e p a r t ic le - h o le m ix in g c a l
c u la t io n s f o r N e ^ , M g ^ , S i ^ , S ^ , and A r^ b a re p re se n te d
in F ig u re s 19-23 f o r p r o je c t io n s fro m : (a ) th e HF s ta te ,
(b ) th e HF s ta te and 1 - p a r t i c le / l - h o le e x c i t a t io n s , (c ) th e
HF s ta te and 2 - p a r t ic le /2 - h o le e x c i t a t io n s , and (d ) th e HF
s ta te and 1 - and 2 - p a r t ic le - h o le e x c i t a t io n s . P a rts (e ) show
115
TABLE X
M e tr ic o f J=0, T=0 Subspace o f S ta te s Used in Neon 20 M ix in g C a lc u la t io n s
HFlp - l h( 1 /2 '
HF .1088 .0196lp - l h ( 1 / 2 ' ) .0196 .1451lp - l h ( 1 /2 " ) -.0 3 3 5 -.0 1 7 02p-2h ( 1 / 2 ' ) .0466 .06602p-2h ( 1 /2 " ) .0439 .0325
lp - l h 2p-2h 2p-2h( 1 /2 " ) ( 1 /2 1) ( 1 /2 " )
- .0 3 3 5 .0466 .0439-.0 1 7 0 . 0660 .0325
.1172 -.0 2 8 3 -.0 1 7 4
-.0 2 8 3 .7332 .0773-.0 1 7 4 .0773 .6462
116
th e lo w -e n e rg y p o s i t iv e p a r i t y e x p e rim e n ta l s p e c tra . The
HF deform ed p a r t ic le bases used a re those l i s t e d in T ab le IX .
A p re s e n ta t io n o f e ig e n fu n c t io n s o f th e H a m ilto n ia n ,
expanded on th e n a tu r a l bases o f a n g u la r m om entum -isospin
subspaces o f th e H a rtre e -F o c k , 1 - p a r t i c le / l - h o le , and 2 -p a r -
t i c le / 2 - h o le s ta te s , ( e . g . , those o f th e J=0 , T=0 m e t i ic o f 20Ne d iscu sse d above) w ould be e x tre m e ly d i f f i c u l t to com
prehend because o f th e la c k o f o r th o n o rm a lity w i t h in these
subspaces. In s te a d , th e e ig e n fu n c t io n s a re expanded on
o rth o n o rm a l bases o b ta in e d by a p p l ic a t io n o f th e Schm idt
p roce du re to th e subspace m e tr ic s . The Schm idt te ch n iq u e
i s e x tre m e ly u s e fu l in t h is a p p l ic a t io n s in c e th e f i r s t T=0
fu n c t io n in th e o rth n o rm a l b a s is may be chosen as th e no rm a l
iz e d H a rtre e -F o c k component a lo n e ; th e f i r s t T=1 fu n c t io n a
n o rm a liz e d 1 - p a r t i c le / l - h o le component; th e f i r s t T=2 fu n c t io n
a n o rm a liz e d 2 - p a r t ic le /2 - h o le com ponent. S uccess ive o r th o
no rm a l fu n c t io n s a re o b ta in e d by ta k in g those p a r ts o f th e
a p p ro p r ia te p a r t ic le - h o le components o rth o n o rm a l to the
fu n c t io n s p re v io u s ly chosen in each case , e tc .
b . Neon 20
F ig u re 19 is a p re s e n ta t io n o f th e r e s u l ts o f th e m ix in g20c a lc u la t io n s pe rfo rm e d on Ne . The s p e c tra a re n o rm a liz e d
to ze ro ground s ta te e x c i t a t io n e n e rg y . The " T o ta l11 spectrum
(F ig u re 1 9 d ) , w h ich c o n ta in s b o th 1 - and 2 - p a r t ic le - h o le
c o n f ig u ra t io n s in a d d it io n to th e HF s ta te , l i e s 220 KeV
be low th e pu re HF spectrum (F ig u re 1 9 a ).
117
As can be seen from F ig u re 16, H a rtre e -F o c k th e o ry suc
c e s s fu l ly p r e d ic ts th e e x c i t a t io n e n e rg ie s o f th e ground K=020r o t a t io n a l band o f Ne . The e x p la n a t io n o f th e e x c ite d J=0
le v e l a t 6 .72 MeV e x c i t a t io n energy has been a tta c k e d by
v a r io u s a u th o rs , n o ta b ly by B a s s ic h is and K e lso n 55 by the
method o f tim e -d e p e n d e n t H a rtre e -F o c k th e o ry , and by B a r-T o u v61*
as a one-phonon ^ - v ib ra t io n ( c f . Appendix A - I I ) o f th e HF
ground s ta te . Both a u th o rs have o b ta in e d e n e rg ie s f o r t h is
s ta te in rough agreem ent w ith th e e x p e rim e n ta l v a lu e .
Based on these w o rks , s im p le p a r t ic le - h o le e x c i ta t io n s
o f th e HF s ta te appear to be a p a r t i c u la r l y a p p e a lin g way
to e x p la in th e appearance o f th e 6 .72 MeV s ta te . In l in e
w ith t h is b e l i e f , T e w a r i57 has been q u ite s u c c e s s fu l in h is
a p p l ic a t io n o f a Tamm-Dancoff a p p ro x im a tio n to th e HF s ta te .
H is r e s u l ts a re shown in F ig u re 1 9 f. In th e p re s e n t c a lc u
la t io n s , no such r e s t r i c t i o n has been a p p lie d . As can be
seen fro m F ig u re 19, th e 1 - p a r t i c le / l - h o le e x c i t a t io n to
th e deform ed k = i / 2 ' le v e l ( v . T ab le IX ) p r e d ic ts a r o t a t io n a l
band w i t h in th e v i c i n i t y o f th e e x p e r im e n ta lly observed band
a t 6 .72 MeV. Tab le X I shows th a t th e s ta te s o f t h is band have
some m ix tu re from th e h ig h e r p a r t ic le - h o le e x c i t a t io n to th e
k = l /2 " s ta te (v . Tab le IX ) . Because o f th e la rg e energy gap
(^8 MeV) between occup ied and unoccup ied o r b i t a ls in th e HF 20s ta te o f Ne (v . F ig u re 1 5 ), 2 - p a r t ic le /2 - h o le c o n f ig u ra t io n s
a re expected to be s ig n i f i c a n t l y h ig h e r in e x c i t a t io n energy
th a n th e 1 - p a r t i c le / l - h o le c o n f ig u r a t io n s , and th e r e fo r e , n o t
to m ix w ith t h is e x c ite d K=0 band. T h is is th e case , as shown
1 1 8
in F ig u re 19. Tab le X I shows th a t th e re a c tu a l ly Is a v e ry
s l i g h t m ix in g among these c o n f ig u ra t io n s .
The e x p e r im e n ta l J=0 le v e l a t 7 .2 MeV e x c ita io n energy
i s n o t accounted f o r by e i t h e r 1 - p a r t i c le / l - h o le o r 2 - p a r t i c le /
2 -h o le e x c i t a t io n s . T h is s ta te may be th e r e s u l t o f m u l t ip le -
p a r t ic le - h o le e x c i ta t io n s from th e 1p s h e l l . C o r ro b o ra tio n
i s g iv e n to t h is s u p p o s it io n by th e appearance o f a 4p-4h
J=0 s ta te a t a p p ro x im a te ly th e same e x c i t a t io n energy (6 .0 6
MeV) in Ol 6 ®3. I t i s , o f c o u rs e , p o s s ib le th a t such I p s h e l l
e x c i ta t io n s w ou ld m ix w ith th e p a r t ic le - h o le e x c i ta t io n s
c o n s id e re d h e re , th u s y ie ld in g b e t te r agreem ent between th e
c a lc u la te d e x c ite d band and th e e x p e r im e n ta l v a lu e s . A one-
to -o n e correspondence between th e t h e o r e t ic a l and e x p e rim e n ta l
band s t ru c tu r e s is co m p lic a te d by an appa re n t r o ta t io n a l
spectrum b u i l t on th e 7 .2 MeV J=0 s ta te .
F ig u re 19. C a lc u la te d and e x p e r im e n ta l s p e c tra o f Neon 20.The spectrum la b e le d (a)HF is th e energy le v e ls p ro je c te d fro m th e H a rtre e -F o c k s ta te . The spectrum la b e le d (b ) l p - l h in c lu d e s a d ia g o n a liz a t io n o f a l l 1 - p a r t i c le / l - h o le s ta te s c o n s id e re d w ith th e H a rtre e -F o c k s ta te ; th a t la b e le d (c )2p-2h in c lu d e s a d ia g o n a liz a t io n o f a l l 2 - p a r t ic le /2 - h o le s ta te s c o n s id e re d w ith th e H a rtre e -F o c k s ta te . The spectrum la b e le d (d ) T o ta l in c lu d e s an e xa c t d ia g o n a liz a t io n o f b o th l p - l h and 2p-2h e x c i ta t io n s c o n s id e re d w ith th e HF s ta te .The e x p e r im e n ta l p o s i t iv e p a r i t y spectrum is la b e le d (e )E x p t . I n d iv id u a l le v e ls a re la b e le d by a n g u la r momentum and s p in ( J ,T ) . The spectrum o b ta in e d by T e w a r i57 is la b e le d ( f ) .
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Tab le X I
Wave F u n c tio n C om pos ition o f L o w - ly in g Neon 20 S ta te s
J (T ) HF lp - lh( 1 /2 * )
l p - l h( 1 /2 " )
2p-2h( 1 / 2 ' )
2p-2h( 1 /2 " )
E x c itEnergy
0 (0 ) .9902 -.0 4 9 9 -1259 .0342 • 0065 0 .002 (0 ) .9939 -.0 4 3 9 .1010 .0023 .0047 1 .324 (0 ) .9754 -.1 7 7 4 .1274 .0256 .0287 3.750 (0 ) .0960 .9129 -.3 9 5 5 .0130 -.0 2 7 7 5.286 (0 ) .9899 -.1 3 7 4 .0012 .0013 .0348 7.802 (0 ) .0812 .9109 -.4 0 2 7 -.0 3 7 3 -.0 0 8 2 8.244 (0 ) .2170 .8330 -.5 0 3 8 -.0 7 1 0 .0159 10.338 (0 ) .9407 -.3 3 8 2 - - -.0 2 4 5 10.773 (1 ) - .9949 -.0 9 6 4 .0207 -.0 2 0 0 12.148 (2 ) - - - .8626 .5059 12.330 (1 ) - .9962 .0749 .0433 .0072 12.46
1 (1 ) - .9786 -.2 0 5 3 -.0 0 7 5 -.0 0 8 2 12.76
2 (1 ) - .9982 .0388 -.0 0 3 4 -.0 4 5 5 12.88
3 (0 ) - .9984 -.0 5 7 4 - - 12.990 (2 ) - - - .9871 .1600 13.451 (0 ) - - .0 6 8 6 .9888 -.1 3 2 7 -.0 0 0 3 13.50
5 (1 ) — ;9968 -.0 6 2 1 -.0 3 7 4 .0331 13.68
Mg a re p re se n te d in F ig u re 20 and T ab le X I I . The s p e c tra
a re n o rm a liz e d to ze ro ground s ta te e x c i t a t io n e n e rg y . The
" T o ta l" (F ig u re 20d) sp e c trum , c o n ta in in g b o th 1 - p a r t i c le /
l - h o la and 2 - p a r t ic le /2 - h o le c o n f ig u ra t io n s in a d d it io n to
th e HF s ta te , l i e s 230 KeV lo w e r in energy th a n th e pure HF
spectrum (F ig u re 2 0 a ).s A 25P a r ik h has made a n a ly s is o f th e wave fu n c t io n s o f Mg
24based on th e assum ption o f a n e u tro n in th e Mg HF f i e l d ,
f o r b o th t r i a x i a l and a x ia l l y sym m etric HF s o lu t io n s . The
a x ia l l y sym m etric s o lu t io n was found to g iv e b e t te r ag re e
ment w ith s p e c tro s c o p ic fa c to r s and d e c o u p lin g pa ram ete rsnji OC
o b ta in e d 67 fro m th e s t r ip p in g r e a c t io n Mg (d ,p )M g , th an
does th e t r i a x i a l s o lu t io n . T h is occu rs in s p i te o f the
f a c t th a t th e t r i a x i a l s o lu t io n is lo w e r in energy ( c f . F ig u re
15 and Tab le IX ) th a n th e a x ia l s o lu t io n . W ith t h is in m ind,
i t seems o n ly a p p ro p r ia te to p e rfo rm m ix in g c a lc u la t io n s based
on th e a x ia l l y sym m etric s o lu t io n .
The appearance o f a lo w - ly in g K=2 r o ta t io n a l band a t
4 .27 MeV e x c i t a t io n energy in th e e x p e r im e n ta l spectrum
(F ig u re 20e) cannot be e x p la in e d on th e b a s is o f th e p a r t i c l e -
h o le c o n f ig u ra t io n s co n s id e re d h e re . T h is is because these
were r e s t r ic t e d to c o n f ig u ra t io n s w h ich w ou ld m ix w ith th e
HF s ta te , i . e . , K=0 c o n f ig u ra t io n s . I t s h o u ld , how ever, be
n o te d th a t the se c o n f ig u ra t io n s w ith K=2 can be o b ta in e d from
an a x ia l tre a tm e n t in th e fo l lo w in g m anner. We can c o n s id e r
119
c. Magnesium 24
The results of particle-hole mixing calculations for24
120
*2 “ ^ AaXba 6XVHF a x
where th e s e t {X } a re o ccup ied o r b i t s and (a ) a re unoccup ied
o r b i t s , a re a m p litu d e c o e f f ic ie n t s , and ka- k x=2. T h is
s ta te i s a l in e a r co m b in a tio n o f 1 - p a r t i c le / l - h o le e x c i t a t io n s ,
a lb e i t n o t o f . th o s e c o n s id e re d in t h is c a lc u la t io n . We must
be c a r e fu l , how ever, n o t to in c lu d e p a r t ic le - h o le c o n f ig u ra
t io n s w h ich may u l t im a te ly be d e s c rib e d as These
w i l l , o f c o u rs e , be o r th o g o n a l to th e HF s ta te , b u t w i l l r e
p re s e n t s p u r io u s s o lu t io n s s in c e a n g u la r momentum p r o je c t io n s
fro m th e two w i l l be id e n t ic a l . A n o th e r tre a tm e n t is th e
r o t a t io n a l d e s c r ip t io n o f fe re d by B ar-Touv and K e lso n 68, in
w h ich th e y c o n s id e re d a s in g le asym m etric d e te rm in a n ta l
i n t r i n s i c s ta te
' 4> = n bx |0> ,
X AK=0 K=2o u t o f w h ich th e y p r o je c t P ” <f> and P ~ <j>. They found th e
K=pva lu e 0 .94 f o r th e o v e r la p <<f>2 l p ” w h ich dem onstra tes
th e coa lescence o f th e v ib r a t io n a l and r o t a t io n a lv —p
(P <J>) d e s c r ip t io n s o f these s ta te s .
An in t e r e s t in g fe a tu re ( v . F ig u re 20d) w h ich a r is e s is
th e appearance o f a n o th e r K=0 r o ta t io n a l band a t 6 .50 MeV
e x c i t a t io n e n e rg y , w h ich agrees re m a rka b ly w ith th e p o s i t io n
o f a lo w - ly in g J=0 le v e l in th e e x p e rim e n ta l spectrum (6 .4 7
MeV). The e x p e r im e n ta l spectrum (F ig u re 20e) shows a J=2
le v e l a p p ro x im a te ly 300 KeV above th a t p re d ic te d by the
m ix in g c a lc u la t io n . A n a ly s is o f th e wave fu n c t io n s (v . Tab le
another intrinsic state of a vibrational nature:
X I I ) o f t h is band shows th a t I t a r is e s from a m ix tu re o f
two k = l /2 1 - p a r t i c le / l - h o le c o n f ig u ra t io n s , and extends a l l
th e way to J=10 a t 9 .7 MeV e x c i ta t io n e n e rg y .
A no th e r r o ta t io n a l band, e x te n d in g in a n g u la r momentum
to J=12, appears a t 11.33 MeV e x c i ta t io n e ne rg y . U n lik e th e
lo w e r band, i t has an is o s p in assignm ent T=2. The band is
somewhat d is to r te d a t h ig h a n g u la r momentum va lu e s ( th e J=8
and J=10 le v e ls a re w i t h in 50 KeV o f each o th e r ) . T h is is
p ro b a b ly due to th e la c k o f c o n s id e ra t io n o f o th e r p a r t i c l e -
h o le (b o th 2 - and h ig h e r ) c o n f ig u ra t io n s w h ich w ould m ix
s ig n i f i c a n t l y a t th e e x c i t a t io n e n e rg ie s co n s id e re d h e re .
I t sh o u ld be remembered th a t a d d i t io n a l 2p-2h c o n f ig u ra t io n s
were n o t t re a te d because (+.) t h e i r is o s p in a n a lyse s a re
e x tre m e ly c o m p lic a te d , and (-Li.) th e y do n o t m ix s ig n i f i c a n t l y
w ith th e HF s ta te ( v . T ab le X I I ) . An in c lu s io n o f th e t o t a l
2 - p a r t ic le /2 - h o le space w i l l un d o u b te d ly a l t e r th e p o s i t io n
o f th e le v e ls in t h is band.
121
F ig u re 20. C a lc u la te d and e x p e rim e n ta l s p e c tra o f Magnesium 24. The spectrum la b e le d (a ) HF i s th e energy le v e ls p ro je c te d from th e H a rtre e -F o c k s ta te . The spectrum la b e le d (b ) l p - l h in c lu d e s a d ia g o n a liz a t io n o f a l l 1 - p a r t i c le / l - h o le s ta te s co n s id e re d w ith th e H a rtre e -F o c k s ta te ; th a t la b e le d (c )2p-2h In c lu d e s a d ia g o n a liz a t io n o f a l l 2 - p a r t ic le /2 - h o le s ta te s co n s id e re d w ith th e H a rtre e -F o c k s ta te . The spectrum la b e le d (d ) T o ta l in c lu d e s an e xa c t d ia g o n a l iz a t io n . o f b o th l p - l h and 2p-2h e x c i ta t io n s c o n s id e re d w ith th e HF s ta te .The e x p e r im e n ta l p o s i t iv e p a r i t y spectrum is la b e le d (e )E x p t. L e ve ls a re la b e le d by a n g u la r momentum and is o s p in (J , T ) .
lp-lh
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T a b l e XI I
W a v e F u n c t i o n C o m p o s i t i o n of L o w - l y i n g
M a g n e s i u m 24 S t a t e s
J( T) H F l p - p h(1/2')
l p - l h(1/2")
l p - l h (3/2')
2 p - 2 h(1/2')
2 p - 2 h(1/2")
2p -2h(3/2')
E x c i tE n e r g
0(0) .9945 .0437 .0002 .0099 .0685 .0604 .0239 0 . 00
2(0) .9955 .0568 - .0071 .0090 .0553 .0451 .0217 0 . 6 3
4(0) .9940 .0910 - .0214 .0218 .0365 .0348 .0153 1 . 8 7
6(0) .9938 .0992 - .0071 .0451 .0179 .0100 .0044 3 . 55
8(0) .9852 .0547 - .1285 .0779 .0412 .0455 - . 0 0 8 1 5 . 5 2
0(0) - . 0 4 0 1 .8333 - .5444 - .0684 .0117 .0532 - . 0 0 8 6 6 . 4 9
2(0) - . 0 5 0 9 •8539 - .5143 - .0376 - .0416 . 0 2 3 0 .0032 7 . 0 8
4(0) - . 0 8 4 9 .7872 - .6033 .0508 - .0707 .0386 - . 0 0 0 6 7 . 5 8
4(1) - .9111 .0812 .2227 - •3360 - .0270 - . 0 0 4 5 8 . 24
2(1) - .9526 .0268 .1713 - .2486 - .0003 .0254 8 . 4 1
3(1) - .9453 - .0740 .3083 - .0760 .0119 .0031 8 . 50
6(0) .0703 - . 6 5 0 7 .7456 - .0451 .1165 - .0098 .0100 8 . 5 3
8(0) .1346 - . 2 9 4 7 .9376 .0284 .1079 - .0585 .0057 8 . 5 4
0(1) - .9868 .0304 .0852 - .1191 .0322 .0529 8 . 7 2
5(0) - .9537 - . 0 2 1 8 .2999 - - - 8 . 74
5(1) - .9411 - .0484 .3266 - .0618 .0370 . 0 10 1 8 . 78
1(1) - .9885 - .0507 .1309 - .0515 - . 0 2 1 6 - . 0 6 1 6 8 . 8 9
3(0) - .9879 .0163 .1540 - - - 9 . 0 1
10(0) .9766 - - .0081 - .1961 .0872 - . 0 0 9 0 9 . 0 1
6(1) - .9085 .1444 .2050 - .3302 - .0509 - . 0 0 6 7 9 . 0 3
1(0) - .9692 .0906 - ,2290 - - - 9 . 2 2
7(0) - .9694 - .0933 .2270 - - - 9 . 2 7
4(0) - . 0 3 1 3 .5602 .7732 .2276 - .1833 - .0444 - . 0 0 0 3 9 . 3 9
6(0) - . 0 7 2 0 .6775 .6459 .2795 - .2002 - .0117 .0147 9 . 5 2
10(0) - . 1 6 0 6 - - .5083 - .8275 - .1228 - . 1 2 6 4 9 . 69
0(0) - . 0 0 5 6 .4773 .7736 - .3947 - .1260 - .0407 - . 0 1 7 8 9 . 90
12(0) 1 — - - - - - 1 1 . 3 3
0(2) — — - - .9819 .1767 .0683 1 1 . 4 1
2(2) — - - - .9982 .0572 .0196 1 2 . 4 4
4(2) — — - - .9948 .0938 1 • 0 u> CO 1 2 . 9 7
6(2) — — - - .9992 .0208
COC\lon01 1 3 . 1 4
8(2) — - - - .9643 .2611 .0439 13 . 4 2
10(2) - - - - .8174 - .5607 .1316 1 3 . 4 7
12(2) _ — - .8008 - .5990 - 1 4 . 0 6
122
s c h e m e h a s b e e n p e r f o r m e d b y B e r n i e r a n d H a r v e y 69 i n an
28a t t e m p t to d e s c r i b e a l l of th e l o w - l y i n g s t a t e s of Si
SU^, h o w e v e r , is u n a b l e t o e x p l a i n the o b s e r v e d e x p e r i m e n t a l
s p e c t r u m . In fact , f o r a p u r e q u a d r u p o l e - q u a d r u p o l e f o rc e ,
it l e a d s to a p a i r o f o r t h o g o n a l d e g e n e r a t e r e p r e s e n t a t i o n s
at l o w e n e r g y (cf. s e c t i o n I I I . 4 ) , a s i t u a t i o n w h i c h is n o t
e x p e r i m e n t a l l y o b s e r v e d . T h e tw o d e g e n e r a t e r e p r e s e n t a t i o n s
a r e a s s o c i a t e d w i t h q u a d r u p o l e m o m e n t s o f o p p o s i t e s i g n
(cf. b e l o w ) . T h e m o d e l d o e s , h o w e v e r , p r e d i c t t h e e x i s t e n c e
o f l o w - l y i n g J = 3 + l e v e l s , w h i c h ar e o b s e r v e d e x p e r i m e n t a l l y
(v. F i g u r e 21e). B a r - T o u v a n d K e l s o n 68 c o n s i d e r e d t h e g r o u n d
? fts t a t e o f Si to be d e f o r m e d , t a k i n g th e a b s o l u t e m i n i m u m
7 0(o bl a t e ) s o l u t i o n f r o m H F c a l c u l a t i o n s a n d a p p l y i n g th e I n g l i s
c r a n k i n g f o r m u l a f o r a d e t e r m i n a t i o n of th e m o m e n t o f i n e r t i a
o f t h e g r o u n d s t a t e b a nd . T h i s m e t h o d e f f e c t e d th e c o r r e c t
p r e d i c t i o n o f t h e f i r s t e x c i t e d J= 2 s t a t e at 1 . 77 M e V e x c i t a
t i o n e n e r g y . Da s G u p t a a n d H a r v e y 71 e x t e n d e d th e w o r k o f
B a r - T o u v a n d K e l s o n i n c o n s i d e r i n g t h e n e a r b y ('vl M e V h i g h e r )
p r o l a t e " l o c a l m i n i m u m " , a n d a l s o a T a m m - D a n c o f f a p p r o x i m a t i o n
a p p l i e d t o t h e o b l a t e s o l u t i o n . T h e y , h o w e v e r , f o u n d it t o o
d i f f i c u l t t o p e r f o r m t h e a c t u a l p r o j e c t i o n s , a n d i n s t e a d a s s u m e d
a r o t a t i o n a l c h a r a c t e r ~ E j = A + B J ( J + 1 ) f o r the s p e c t r u m . T h e i r
r e s u l t s , p a r a m e t e r i z e d b y th e s t r e n g t h o f th e s p i n - o r b i t fo rc e,
w e r e in f a i r a g r e e m e n t w i t h the e x p e r i m e n t a l s p e c t r u m w h e n
m u l t i p l i e d b y a n i n e x p l i c a b l e f a c t o r o f 2.
d . S i l i c o n 28
An e x t e n s i v e c a l c u l a t i o n u s in g th e SU^ c l a s s i f i c a t i o n
123
R e s u l t s of p a r t i c l e - h o l e m i x i n g c a l c u l a t i o n s p e r f o r m e d
28f o r Si a r e r e p o r t e d in F i g u r e 21 a n d T a b l e XI II . T h e g r o u n d
s t a t e q u a s i - r o t a t i o n a l b a n d ( F i g u r e 21a) s h o w s th e sa m e f a c
t o r o f 2 d i s c r e p a n c y w i t h the e x p e r i m e n t a l s p e c t r u m ( F i g u r e 21e)
as n o t e d a b o v e . T h e e x c i t e d s t r u c t u r e s , h o w e v e r , u n l i k e
t h o s e o f Das G u p t a a n d H a r v e y , do not.
V a r i o u s e x p l a n a t i o n s h a v e b e e n o f f e r e d fo r t h e a p p e a r a n c e
o f th e l o w - l y i n g J= 0 l e v e l o b s e r v e d at 4. 97 M e V e x c i t a t i o n
e n e r g y . A l l o f t h e s e a s s o c i a t e th e l e v e l w i t h a n e q u i l i b r i u m
s h a p e o f the n u c l e u s , b u t t h e y d i f f e r as t o w h a t e q u i l i b r i u m
s h a p e is r e p r e s e n t e d . T h e p o s s i b l e s h a p e s , a l l a x i a l l y s y m
m e t r i c , c o v e r t h e f u l l r a n g e , i . e. , s p h e r i c a l 7 2 , o b l a t e 7 1 ,
a n d a m i x t u r e o f p r o l a t e a n d o b l a t e 7 3 .
B a r - T o u v a n d G o s w a m i 72 a t t r i b u t e t h i s l e v e l t o " i n v e r t e d
c o e x i s t e n c e " , i. e . , th e e x i s t e n c e o f a s p h e r i c a l e q u i l i b r i u m
s h a p e * , h i g h e r i n e n e r g y t h a n the d e f o r m e d g r o u n d st ate.
B a s e d o n an e l e m e n t a r y t h e o r y of s h a p e m i x i n g b e t w e e n the
s p h e r i c a l J= 0 l e v e l a t 4 . 97 M e V , a n d the o b l a t e g r o u n d st a t e ,
t h e y p r e d i c t e d a l o w e r i n g o f the c a l c u l a t e d (by p r o j e c t i o n )
g r o u n d s t a t e , a n d t h e r e b y an i n c r e a s e in th e e x c i t a t i o n o f the
J = 2 le ve l , W h e n t h i s is do ne, the g r o u n d s t a t e fa l l s r i g h t
i n t o p o s i t i o n i n a J( J+ 7) s p e c t r u m (v. F i g u r e 21f).
E x p e r i m e n t s 7 ** h a v e s h o w n an e n h a n c e m e n t (8 W e i s s k o p f
U n i t s ) f o r th e B ( E 2 ) t r a n s i t i o n f r o m th e J= 0 l e v e l at 4. 9 7 M e V
* W e n o t e t h a t th is s p h e r i c a l s t a t e is n o t the s p h e r i c a l HF s o l u t i o n o b t a i n e d b y f i l l i n g th e I d c / v s u b s h e l l , w h i c h liess o m e 30 M e V a b o v e th e l o w e s t o b l a t e s o l u t i o n . T h e s t a t e c o n s i d e r e d h e r e c o n t a i n s a d d i t i o n a l p a i r i n g i n t e r a c t i o n s . 4
e x c i t a t i o n e n e r g y to th e J = 2 l e v e l at 1 . 77 M e V e x c i t a t i o n
e n e r g y (v. F i g u r e 21e). Da s G u p t a a n d H a r v e y h a v e i n f e r r e d
f r o m t h i s t h a t th e 4 . 97 M e V l e v e l m u s t b e o b l a t e , as is th e
g r o u n d s t a t e , s i n c e th e E 2 t r a n s i t i o n o p e r a t o r is a s i n g l e
p a r t i c l e o p e r a t o r a n d t h us c a n n o t c o n n e c t s t a t e s w h i c h a r e
h i g h l y d i f f e r e n t i n t h e i r s t r u c t u r e , e . g . , s p h e r i c a l or p r o
l a t e vs. o b l a t e . T h e J=0 l e v e l at 6 . 6 8 M e V e x c i t a t i o n e n e r g y
d o e s h a v e a n i n h i b i t e d E2 t r a n s i t i o n to th e J = 2 l e v e l at
1 . 7 7 M e V 7 \ so th i s , t h e y a r g u e , is th e s t a t e w i t h w h i c h to
a s s o c i a t e t h e p r o l a t e e q u i l i b r i u m s h a p e (cf. s e c t i o n I V . 4).
I n l i n e w i t h t h i s a s s o c i a t i o n o f t h e J= 0 l e v e l at 6 . 6 8
M e V w i t h th e p r o l a t e e q u i l i b r i u m s h a p e , C a s t e l a n d S v e n n e 73
a t t r i b u t e t h e 4 . 97 M e V l e v e l w i t h a m i x t u r e of t h e p r o l a t e
a n d o b l a t e e q u i l i b r i u m s h a p e s . T h e i r m o d e l d e s c r i b e s t w o
v i b r a t i o n s , on e a b o u t e a c h of th e d e f o r m e d H F m i n i m a , c o u p l e d
t h r o u g h t h e i r q u a d r u p o l e f i e l d s . T h e b a s i s s p a c e fo r the
c a l c u l a t i o n s c o n s i s t e d of th e 1-6, Ip, 2 6 Id, a n d * 1(7 / 2 o s c il ~
l a t o r s h e l l s . As a t w o - b o d y i n t e r a c t i o n , th e e f f e c t i v e p o
t e n t i a l o b t a i n e d f r o m th e K u o - B r o w n 75 G - m a t r i x c a l c u l a t e d
w i t h t h e H a m a d a - J o h n s t o n p o t e n t i a l 5 w a s u s e d , r a t h e r t h a n the
m o r e e l e m e n t a r y p h e n o m e n o l o g i c a l i n t e r a c t i o n u s e d i n t h e s e
c a l c u l a t i o n s . T h e i r r e s u l t s for t h e i n t e r m e d i a t e s t r u c t u r e
P 8o f Si a r e p r e s e n t e d i n F i g u r e 21f. As s h o w n , t h e y a g r e e
a m a z i n g l y w e l l w i t h t h e o b s e r v e d s p e c t r u m .
T h e c a l c u l a t i o n s r e p o r t e d h e r e p r e d i c t the a p p e a r a n c e
o f t h e l o w e s t J= 0 l e v e l at a p p r o x i m a t e l y 7*5 M e V e x c i t a t i o n
e n e r g y . In fa c t , w e p r e d i c t t w o q u a s i - r o t a t i o n a l b a n d s w h i c h
124
a r e n e a r l y d e g e n e r a t e i n t h e i r b a n d h e a d e n e r g i e s , ( w i t h i n
40 K e V) b u t w h i c h h a v e m o m e n t s o f i n e r t i a w h i c h d i f f e r b y
a f a c t o r o f a p p r o x i m a t e l y 2 ( b a s e d on th e e n e r g y o f the
e x c i t e d J= 2 l e v e l s r e l a t i v e to t h e b a n d h e a d e n e r g i e s ) .
A n a l y s e s of w a v e f u n c t i o n s o f t h e s e b a n d s (v. T a b l e X I I I )
s h o w s t h a t t h e y b o t h a r i s e f r o m 1 - p a r t i c l e / l - h o l e e x c i t a t i o n s
o f t h e o b l a t e s o l u t i o n , a n d n o t, as m i g h t b e s u s p e c t e d f r o m
t h e a b o v e d i s c u s s i o n s , on e e a c h f r o m the o b l a t e a n d p r o l a t e
s o l u t i o n s . If th i s w e r e the ca se, w e w o u l d e x p e c t to see
s i g n i f i c a n t a d m i x t u r e s f r o m th e 2 - p a r t i c l e / 2 - h o l e c o n f i g u r a
t i o n s f o r the p r o l a t e b a n d . T h i s is no t the case. O n e b a n d
(w it h t h e s m a l l e r v a l u e o f the m o m e n t of i n e r t i a ) h a s as a
m a j o r c o m p o n e n t the 1 - p a r t i c l e / l - h o l e c o n f i g u r a t i o n k = l / 2 f
(v. T a b l e IX), th e o t h e r th e 1 - p a r t i c l e / l - h o l e c o n f i g u r a t i o n
k = 3 / 2 ' , t h e s e at l o w e n e r g y r e l a t i v e to the b a n d h e a d s . At
t h e h i g h a n g u l a r m o m e n t u m s t a t e s o f the b a n d s , t h i s s e p a r a t i o n
is n o l o n g e r m e a n i n g f u l , h o w e v e r , a n d th e b a n d s ar e h i g h l y
m i x e d .
T h e e x p e r i m e n t a l J = 3 + l e v e l at 6 . 2 8 M e V e x c i t a t i o n e n e r g y
is b e l i e v e d to be a s s o c i a t e d w i t h a K = 3 r o t a t i o n a l b a nd . As
d i s c u s s e d i n s e c t i o n I V . 6 c , t h i s s t r u c t u r e c a n n o t b e r e a c h e d
b y th e c o n f i g u r a t i o n s c o n s i d e r e d h e r e , b u t th e g e n e r a l m e t h o d s
m a y b e a p p l i e d t o m a k e p r e d i c t i o n s c o n c e r n i n g t h is ba nd.
E x a m i n a t i o n o f F i g u r e 2 1 d a l s o r e v e a l s the p r e s e n c e of a
T=2 q u a s i - r o t a t i o n a l b a n d at a p p r o x i m a t e l y 1 3 .9 M e V e x c i t a t i o n
e n e r g y . T h i s o f c o u r s e , a r i s e s as a r e s u l t of 2 - p a r t i c l e / 2 -
h o l e c o n f i g u r a t i o n s , a n d is e x p e c t e d to u n d e r g o e n e r g y s h i f t s
w h e n th e c o m p l e t e s p a c e o f 2 p - 2 h c o n f i g u a t i o n s is c o n s i d e r e d .
125
F i g u r e 21. C a l c u l a t e d a n d e x p e r i m e n t a l s p e c t r a o f S i l i c o n 28.
T h e s p e c t r u m l a b e l e d (a) HF is th e e n e r g y l e v e l s p r o j e c t e d
f r o m t h e H a r t r e e - F o c k s t a t e . T h e s p e c t r u m l a b e l e d (b) l p - l h
i n c l u d e s a d i a g o n a l i z a t i o n o f a l l 1 - p a r t i c l e / l - h o l e s t a t e s
c o n s i d e r e d w i t h th e H F s t a t e ; t h a t l a b e l e d (c) 2 p - 2 h i n c l u d e s
a d i a g o n a l i z a t i o n of a l l 2 p - 2 h s t a t e s c o n s i d e r e d w i t h the HF
s t a t e . T h e s p e c t r u m l a b e l e d (d) T o t a l i n c l u d e s a n e x a c t
d i a g o n a l i z a t i o n of b o t h l p - l h a n d 2 p - 2 h e x c i t a t i o n s c o n
s i d e r e d w i t h th e HF s t a t e . T h e e x p e r i m e n t a l p o s i t i v e p a r i t y
s p e c t r u m is l a b e l e d (e) E x p t . L e v e l s a r e l a b e l e d b y a n g u l a r
m o m e n t u m a n d i s o s p i n (J,T). T h e r e s u l t s o f tw o s h a p e - m i x i n g
c a l c u l a t i o n s a r e r e p o r t e d i n (f). T h e s e a r e b y B a r - T o u v
a n d G o s w a m i 72 a n d C a s t e l a n d S v e n n e 7 3 .
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W a v e F u n c t i o n C o m p o s i t i o n of L o w - l y i n g
S i l i c o n 28 S t a t e s
J ( T) HF l p - l h(3/2'
l p - l h ) (1/2')
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0(0) .9947 .0546 .0115 .0767 .0317<0 20 1—10• . 0 163 0 . 00
2(0) .9953 .0530 .0129 .0712 .0288 .0154 .0176 0.70
4(0) .9954 .0414 .0296 .0691 .0266 .0186 .0259 2.29
6(0) .9956 .0275 .0450 .0606 .0218 .0263 .0349 4.76
0(0) - . 0 6 1 3 .9766 .1877 .0812 - . 0 2 4 8 .0010 .0043 7.48
0(0) - . 0 3 6 0 - . 2 0 5 6 .8646 .4481 .0145 . 0 7 7 0 .0450 7 . 5 2
8(0) .9970 .0082 .0414 .0433 .0161 .0286 .0368 7. 99
2(0) - . 0 6 9 2 .8071 .5175 .2717 - . 0 2 6 7 .0323 .0194 8 . 0 3
1(0) - .9939 - . 1 0 8 8 .0161 - - - 8 . 1 4
0(1) - .9874 .1373 - . 0 5 3 5 - . 0 5 5 8 .0159 .0032 8 . 4 4
2(0) - . 0 0 7 4 - . 5 8 7 4 .7062 .3922 .0095 .0379 .0295 8 . 4 6
1(1) - .9825 - . 1 8 2 3 .0332 .0135 - .0103 -
^T<NOO• 8 . 7 2
2(1) - .9868 .1330 - . 0 4 8 8 - . 0 7 7 8 .0135 .0004 8 . 9 7
3(0) - .9950 - . 0 9 4 0 .0334 - - - 9 . 1 4
4(0) - . 0 7 5 4 .5526 . 6 9 8 9 .4454 - . 0 2 1 2 . 0 3 2 2 .0257 9 . 1 6
3(1) - .9818 - . 1 8 2 0 .0523 . 0 1 1 8 - .0122 .0007 9 . 7 3
4(0) .0024 .8315 - . 4 5 6 7 - . 3 1 3 7 - . 0 2 8 7 - .0153 - .0212 9 . 90
4(1) - .9852 .1295
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126
e. S u l f u r 32
T h e r e s u l t s o f p a r t i c l e - h o l e m i x i n g c a l c u l a t i o n s f o r S 32
a r e r e p o r t e d i n F i g u r e 22 a n d T a b l e XIV. F i g u r e 2 2 f r e p r e s e n t s
t h e w o r k o f F a r r i s a n d E i s e n b e r g 7 6 . T h e y c o n s i d e r the g r o u n d
32s t a t e c o n f i g u r a t i o n of S to be th e c l o s e d ^ ^ 5 / 2 _7,&7/2 s u b ~
s h e l l s , a n d p e r f o r m e d p a r t i c l e - h o l e 1 c a l c u l a t i o n s o n t h is
s p h e r i c a l c o n f i g u r a t i o n . E x a m i n a t i o n of t h e i r r e s u l t s s h o w s
t h a t t h e y h a v e a c c o u n t e d f o r m o s t of th e e i g e n s t a t e s , b u t
c a n n o t p r o d u c e n u m e r i c a l a g r e e m e n t w i t h the o b s e r v e d e x c i t a
t i o n e n e r g i e s (v. F i g u r e 22e). A p a r t i c u l a r l y w e a k p o i n t in
t h e i r r e s u l t s is t h e i n a b i l i t y to a c c o u n t fo r t h e 2- p h o n o n
v i b r a t i o n s t r u c t u r e at a p p r o x i m a t e l y 4-5 M e V e x c i t a t i o n e n e r g y
(v. F i g u r e 22e).
32T a b l e I X s h o w s t h a t the p r o l a t e HF s o l u t i o n f o r S J is
o n l y s l i g h t l y (by 3 6 0 KeV) f a v o r e d o v e r th e o b l a t e s o l u t i o n .
A l t h o u g h t h e a x i a l l y a s y m m e t r i c s o l u t i o n is s o me 3 M e V b e l o w
th e a x i a l s o l u t i o n s , it is n o w k n o w n (v. s e c t i o n I V . 6c) t h a t
t h e l o w e s t H F s o l u t i o n is n o t n e c e s s a r i l y r e p r e s e n t a t i v e of
th e b e s t d e s c r i p t i o n o f the n u c l e u s i n t e r m s o f a n i n t r i n s i c
st a t e . In v i e w of t h i s , a n d c o n s i d e r i n g th e cl o s e s i m i l a r i t y
?4 32o f th e b e h a v i o r o f t h e H F e n e r g y s u r f a c e s 68 o f M g a n d S ,
it w a s d e c i d e d to p e r f o r m p a r t i c l e - h o l e m i x i n g c a l c u l a t i o n s
32o n th e l o w e s t a x i a l l y s y m m e t r i c p r o l a t e s o l u t i o n s o f S
T h e b a r e H F s o l u t i o n , b e i n g r e p r e s e n t a t i v e o f a p e r m a
n e n t l y d e f o r m e d p r o l a t e s h a p e , y i e l d s a r o t a t i o n a l s p e c t r u m
( F i g u r e 22a). S 32 is n o t, h o w e v e r , d e s c r i b e d as a r o t a t i o n a l
n u c l e u s , b u t r a t h e r as v i b r a t i o n a l , e x h i b i t i n g th e t y p i c a l
J = 0 , 2, 0 - 2 - 4 e x c i t a t i o n s p e c t r u m ® . T h e t r a n s f o r m a t i o n f r o m
a r o t a t i o n a l d e s c r i p t i o n to a v i b r a t i o n a l one is e f f e c t e d b y
the 1 - p a r t i c l e / l - h o l e a d m i x t u r e s t o th e H F s t a t e (v. F i g u r e 22b),
a l t h o u g h t h e r e is s o m e m i x i n g w i t h 2 - p a r t i c l e / 2 - h o l e c o n f i g u
r a t i o n s (v. T a b l e XIV) . T h e l p - l h e x c i t a t i o n s c a u s e a n i n
c r e a s e in th e e x c i t a t i o n e n e r g y o f the l o w e s t J = 2 le vel, as
it s d e s c r i p t i o n c h a n g e s f r o m b e i n g p a r t of a r o t a t i o n a l b a n d
t o t h a t o f a 1 - p h o n o n e x c i t a t i o n . In a d d i t i o n , a J= 0 a n d
J = 2 l e v e l a p p e a r in t h e v i c i n i t y of th e l o w e s t J = 4 s t a t e
(the e x c i t a t i o n e n e r g y o f w h i c h Is r a i s e d o v e r t h e p u r e HF
v a l u e ) , p r o d u c i n g th e J = 0 - 2 - 4 t r i p l e t o f the 2 - p h o n o n e x c i t a
tion .
127
T h e J= 2 l e v e l a c t u a l l y lies s o m e 2.5 M e V h i g h e r t h a n the J = 0 a n d J= 4 l e v e ls . T a b l e X I V s h o w s t h a t the w a v e f u n c t i o n of t h i s l e v e l ha s s o m e a d m i x t u r e f r o m the 2 p - 2 h s t a t e s c o n s i d e r e d .It is b e l i e v e d t h a t w h e n th e e n t i r e s p a c e of 2 p - 2 h c o n f i g u r a t i o n s is c o n s i d e r e d , t h i s l e v e l w i l l be l o w e r e d i n t o th e v i c i n i t y of th e t r i p l e t .
F i g u r e 22. C a l c u l a t e d a n d e x p e r i m e n t a l l o w - e n e r g y s p e c t r a
o f S u l f u r 32. T h e s p e c t r u m l a b e l e d (a) HF is th e e n e r g y
l e v e l s p r o j e c t e d f r o m the H a r t r e e - F o c k s t a t e . T h e s p e c t r u m
l a b e l e d (b) l p - l h i n c l u d e s a d i a g o n a l i z a t i o n o f a l l 1 - p a r t i c l e /
1 - h o l e s t a t e s c o n s i d e r e d w i t h th e H a r t r e e - F o c k s t a t e ; t h at
l a b e l e d (c) 2 p - 2 h i n c l u d e s a d i a g o n a l i z a t i o n of a l l 2 - p a r t i c l e /
2 - h o l e s t a t e s c o n s i d e r e d w i t h th e H a r t r e e - F o c k s t a t e . T h e
s p e c t r u m l a b e l e d ( d ) T o t a l i n c l u d e s a n e x a c t d i a g o n a l i z a t i o n of
b o t h l p - l h a n d 2 p - 2 h e x c i t a t i o n s c o n s i d e r e d w i t h the H F st a t e .
T h e e x p e r i m e n t a l p o s i t i v e p a r i t y l o w - e n e r g y s p e c t r u m is
l a b e l e d (e) E x p t . L e v e l s a r e l a b e l e d b y a n g u l a r m o m e n t u m a n d
i s o s p i n (J , T ) .
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0 (0 ) .8651 .3107 .1137 - . 3 5 7 4 .0902 .0138 .0774 0 . 0 0
2(0) .9874 .1269 .0862 - . 0 0 7 7 . 0 3 2 2 .0191 .0043 1 . 95
0(0) .4638 - . 3 2 8 4 .0199 .8189 - . 0 5 8 3 .0167 - . 0 4 8 0 2 . 9 1
4(0) .9187 .2854 .0704 - . 2 4 8 3 .0629 .0228 .0586 3.49
2(0) .0608 - . 4 0 5 7 -.008t| .9112 - . 0 3 2 0 .0067 .0193 6 . 37
6(0) .9485 . 2 3 2 1 .0454 - . 2 0 3 9 .0086 .0155 .0496 6 . 8 4
4(0) .3837 - . 5 6 1 0 - . 0 4 7 0 .7192 - . 0 5 3 5 .0097 - . 1 2 5 2 6 . 9 5
1(1) - - . 0 8 9 5 .0494 .9730 - . 0 3 9 0 - . 0 0 6 2 - . 2 0 3 0 7 . 5 0
0(2) - - - - .1687 .0562 .9841 7 . 50
1(0) - .0112 .0213 . 9 9 9 1 - - - 8 . 1 0
0(1) - .0383 - . 2 3 3 3 .9683 - . 0 4 9 8 .0331 .0544 8 . 1 3
2(1) - .0777 - . 1 7 4 4 .9398 - . 1 1 7 2 .0002 - . 2 5 8 1 8 . 3 0
8(0) .8242 .2225 - . 3 2 8 5 - . 3 0 5 7 .2607 .0423 - 8 . 5 2
3(1) - .0755 . 2 3 0 1 .9658 .0185 - . 0 0 0 5 - . 0 9 0 7 9 . 4 9
3(0) - .0502 .1775 .9828 - - - 9 . 8 4 .
4(1) - .5243 .0738 .7223 - . 0 0 1 4 - . 0 2 1 1 - . 4 4 4 4 1 0 . 1 1
0(0) - . 0 9 7 2 .8219 - . 3 5 6 9 .4059 .1440 - . 0 4 0 0 .0248 1 0 . 8 7
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f . A r g o n 36
T h e H F s p a c e u s e d in th e p a r t i c l e - h o l e m i x i n g c a l c u l a t i o n s
f o r A r -5 is t h a t l i s t e d in T a b l e IX, w h i c h Is a s s o c i a t e d w i t h
a n o b l a t e d e f o r m a t i o n . M u t h u k r i s h n a n 8 3 h a s o b t a i n e d a p r o l a t e
s o l u t i o n u s i n g a n o n - l o c a l p o t e n t i a l w h i c h a c ts o n l y i n r e l a
t i v e s - s t a t e s . T h i s s o l u t i o n lies s o m e 6.5 M e V a b o v e th e o b l a t e
s o l u t i o n o b t a i n e d f o r t h e s a m e i n t e r a c t i o n .
T h e r e s u l t s o f m i x i n g c a l c u l a t i o n s a r e r e p o r t e d in F i g u r e
23. A s d e p i c t e d , the 1 - p a r t i c l e / l - h o l e e x c i t a t i o n s m i x w i t h
t h e H F s o l u t i o n to y i e l d b e t t e r a g r e e m e n t o f th e g r o u n d s t a t e
q u a s i - r o t a t i o n a l b a n d w i t h th e l o w - l y i n g e x c i t a t i o n s p e c t r a
o b s e r v e d e x p e r i m e n t a l l y . I n a d d i t i o n , a l s o a r i s i n g f r o m the
l p - l h c o n f i g u r a t i o n s is a l o w - l y i n g J = 0 l e v e l at a p p r o x i m a t e l y
4.9 M e V. A r o t a t i o n a l b a n d s t r u c t u r e is b u i l t on th is le ve l.
T h e r e is c o n s i d e r a b l e e v i d e n c e t h a t t h e e q u i l i b r i u m
s h a p e o f th e g r o u n d s t a t e o f A r 3^ is s p h e r i c a l , r a t h e r t h a n
d e f o r m e d , as s u g g e s t e d b y H F r e s u l t s . T h e p a r t i c l e - h o l e
m i x i n g c a l c u l a t i o n s p e r f o r m e d h e r e c l e a r l y s h o w t h e e x i s t e n c e
o f t h e 2 - p h o n o n t r i p l e t , w h i c h e x p e r i m e n t a l l y lies at ^ 4 . 4
M e V e x c i t a t i o n e n e r g y . Its t h e o r e t i c a l d e s c r i p t i o n as a
r o t a t i o n a l s t r u c t u r e is, h o w e v e r , i n e s c a p a b l e . C l e a r l y , HF
t h e o r y f a i l s t o d e s c r i b e t h i s n u c l e u s .
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F i g u r e 23. C a l c u l a t e d a n d e x p e r i m e n t a l l o w - e n e r g y s p e c t r a
o f A r g o n 3 6 . T h e s p e c t r u m l a b e l e d (a)HF is th e e n e r g y
l e v e l s p r o j e c t e d f r o m th e H a r t r e e - F o c k s t a t e . T h e s p e c t r u m
l a b e l e d (b) l p - l h i n c l u d e s a d i a g o n a l i z a t i o n o f a l l 1 - p a r t i c l e /
1 - h o l e s t a t e s c o n s i d e r e d w i t h the H a r t r e e - F o c k s t a t e ; th a t
l a b e l e d (c) 2 p - 2 h i n c l u d e s a d i a g o n a l i z a t i o n o f a l l 2 - p a r t i c l e /
2 - h o l e s t a t e s c o n s i d e r e d w i t h th e H a r t r e e - F o c k s t a t e . T h e
s p e c t r u m l a b e l e d (d) T o t a l i n c l u d e s a n e x a c t d i a g o n a l i z a t i o n of
b o t h l p - l h a n d 2 p - 2 h e x c i t a t i o n s c o n s i d e r e d w i t h th e H F st ate.
T h e e x p e r i m e n t a l p o s i t i v e p a r i t y l o w - e n e r g y s p e c t r u m is
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H a r t r e e - F o c k s o l u t i o n s o f th e e v e n - e v e n N = Z n u c l e i in
the 2 4 - / d s h e l l h a v e b e e n s t u d i e d . T h e f o l l o w i n g g e n e r a l
c o n c l u s i o n s h a v e b e e n d i s c u s s e d :
( i . ) T h e m a n i f e s t a t i o n o f a n e n e r g y g a p b e t w e e n o c c u p i e d
a n d u n o c c u p i e d HF o r b i t a l s is du e to a p a r t i c u l a r c o m b i n a t i o n
(v. e q u a t i o n (I V - 2 2 ) ) o f e x c h a n g e p o t e n t i a l s , d o m i n a t e d b y
th e s p a t i a l e x c h a n g e ( M a j o r a n a ) term. It t h us r e f l e c t s th e
e m p l o y m e n t of a n o n - l o c a l s e l f - c o n s i s t e n t p o t e n t i a l .
24( l - L ) T h e a s y m m e t r y o f th e l o w e s t HF s o l u t i o n s of M g
32a n d r e f l e c t a p r e f e r e n c e o f th e H F f i e l d fo r s p a t i a l ,
r a t h e r t h a n a x i a l s y m m e t r y . I n t h e o t h e r e v e n - e v e n N=Z
n u c l e i o f t h e 2 4 - I d s h e l l , t h e t w o s y m m e t r i e s ar e c o m p a t i b l e .
( U l ) P a r t i c l e - h o l e c o n f i g u r a t i o n s h a v e b e e n s h o w n t o
r e p r e s e n t a p o s s i b l e a d m i x t u r e t o t h e l o w - l y i n g H F s o l u t i o n s .
P r o j e c t i o n c a l c u l a t i o n s h a v e b e e n c a r r i e d ou t f o r th e
1- a n d 2 - p a r t i c l e - h o l e e x c i t a t i o n s of th e H F s t a t e s o f e v e n -
e v e n N = Z n u c l e i in t h e 2 6 - 1 d s h e l l to d e t e r m i n e t h e e x t e n t
o f t h e s e a d m i x t u r e s . T h e p o l y n o m i a l p r o j e c t i o n t e c h n i q u e s
d e v e l o p e d i n C h a p t e r II w e r e o f g r e a t a s s i s t a n c e i n p e r f o r m i n g
t h e s e c a l c u l a t i o n s , as w a s th e m e t h o d of p e r f o r m i n g c a l c u l a
t i o n s i n n o n - o r t h o n o r m a l H i l b e r t s p a c e s . T h e n e c e s s i t y to
s i m u l t a n e o u s l y p r o j e c t i s o s p i n a n d a n g u l a r m o m e n t u m w a s o b v i a t e d
b y th e c o n s t r u c t i o n of s i m p l e c o m b i n a t i o n s o f S l a t e r d e t e r m i n -
a n t a l s t a t e s w h i c h a l r e a d y h a d g o o d i s o s p i n . T h e f o l l o w i n g
r e s u l t s h a v e b e e n o b t a i n e d f r o m t h e s e c a l c u l a t i o n s :
129
7 . Summary
130
(-t) A n g u l a r m o m e n t u m c o m p o n e n t s o f th e HF a n d p u r e
p a r t i c l e - h o l e c o n f i g u r a t i o n s h a v e c o n s i d e r a b l e o v e r l a p (v.
T a b l e X), a l t h o u g h H a m i l t o n i a n m i x i n g b e t w e e n th e H F c o m
p o n e n t s a n d a n o r t h o n o r m a l p a r t i c l e - h o l e b a s i s g e n e r a t e d
b y a S c h m i d t p r o c e d u r e a r e n e g l i g i b l e in th e f i r s t h a l f of
th e s h e l l , w h e r e r o t a t i o n a l p r o p e r t i e s a r e m o r e p r o m i n e n t
t h a n i n the l a t t e r h a l f .
{ aJ , ) T h e i n s t a b i l i t y of th e H F s o l u t i o n s f o r S 3 2 a n d
A r m a n i f e s t t h e m s e l v e s i n t h e a p p e a r a n c e o f a l o w - l y i n g
K = 0 q u a s i - r o t a t i o n a l b a n d (at ^ 3 a n d 5 M e V e x c i t a t i o n
e n e r g y , r e s p e c t i v e l y ) , w h i c h c a u s e s i g n i f i c a n t c h a n g e s i n
t h e l o w e n e r g y s p e c t r a w h e n m i x i n g is t a k e n i n t o a c c o u n t .
(-Let) P a r t i c l e - h o l e c o n f i g u r a t i o n s a c c o u n t f o r e x c i t e d
s t a t e s o f l o w a n g u l a r m o m e n t u m o b s e r v e d in a l l o f th e n u c l e i
u n d e r c o n s i d e r a t i o n , b u t n u m e r i c a l a g r e e m e n t wit-h t h e e x p e r
i m e n t a l e i g e n v a l u e s is n o t a c h i e v e d . T h i s is p r o b a b l y
b e c a u s e o f t h e l i m i t e d s h a p e m i x i n g w h i c h is i n t r o d u c e d b y
t h e p a r t i c l e - h o l e c o n f i g u r a t i o n s . It is c l e a r (cf. d i s c u s s i o n
P 8o f Si ) t h a t a n e x p l a n a t i o n o f th e l o w - e n e r g y p r o p e r t i e s of
t h e s e n u c l e i c a n o n l y b e a c h i e v e d w h e n the i n t e r a c t i o n a m o n g
th e v a r i o u s e q u i l i b r i u m s h a p e s is t a k e n i n t o a c c o u n t .
B a s e d o n the a b o v e , it c a n o n l y b e c o n c l u d e d t h a t w h i l e
H F is an e x t r e m e l y u s e f u l t o o l f o r t h e s t u d y of l o w - e n e r g y
n u c l e a r s t r u c t u r e , c o n s i d e r a b l e i m p r o v e m e n t is n e e d e d I f it
is t o y i e l d , w i t h o u t e x t e n s i v e c o m p l i c a t i o n s , q u a n t i t a t i v e
a g r e e m e n t w i t h e x p e r i m e n t a l data . T h e r e Is a p a r t i c u l a r i m
p r o v e m e n t w h i c h m a y p r o v e f r u i t f u l t o t h is end. W e n o t e t h at
a v a r i a t i o n a l c a l c u l a t i o n is p e r f o r m e d w i t h i n s t e a d of
w i t h A v a r i a t i o n w i t h s h o u l d i n v o l v e th e a p p l i c a t i o n
o f a n i n t r i n s i c H a m i l t o n i a n r a t h e r t h a n the a c t u a l one.
T h e o n l y i n t r i n s i c H a m i l t o n i a n e m p l o y e d o f t h is n a t u r e is
H - I J 2 , w h e r e I is r e l a t e d to the m o m e n t o f i n e r t i a p a r a m e t e r .7 7
It h a s b e e n p o i n t e d o u t b y V i i l a r s t h a t t h e s t a t e s o b t a i n e d
b y p r o j e c t i n g out of ^ h a v e a J(J + / ) d e p e n d e n c e i n th e f i r s t
o r de r . T h e l o w - e n e r g y e x c i t a t i o n s p e c t r a is t h u s f o r c e d to
h a v e a r o t o r - l i k e s t r u c t u r e . As h a s b e e n s h o w n , t h i s is n o t
a l w a y s a p p r o p r i a t e , a n d w h e n it is, w e w o u l d p r e f e r to o b t a i n
it a s o u t p u t o f a c a l c u l a t i o n , r a t h e r t h a n as in pu t.
T h e s e o b j e c t i o n s a r e r e m o v e d , in p r i n c i p l e , i f the
v a r i a t i o n is p e r f o r m e d b y s o l v i n g t h e f o l l o w i n g e q u a t i o n for
e a c h J va l u e :
6[ < P J '1'K |H|PJ 4'K > / < P J 1'K |PJ 4'K > ] = 0,
i . e . , t h e c o e f f i c i e n t s c^ a r e v a r i e d a f t e r p r o j e c t i o n , r a t h e r
t h a n b e f o r e . C a l c u l a t i o n s to c o m p a r e w a v e f u n c t i o n s o f this
t y p e w i t h t h o s e p r o j e c t e d f r o m a n i n t r i n s i c HF s t a t e a f t e r
m i n i m i z a t i o n h a v e b e e n p e r f o r m e d b y S a t p a t h y a n d N a i r 7 8 . ..
T h e y h a v e f o u n d e s s e n t i a l l y n o d i f f e r e n c e i n th e w a v e f u n c -
20 20 t i o n s f o r Ne . H o w e v e r , as h a s b e e n n o t e d , Ne is a
r a t h e r u n i q u e n u c l e u s in th e 2 6 - I d s h e l l , i . e. , H F c a l c u l a t i o n s
20a g r e e r e m a r k a b l y w e l l w i t h e x p e r i m e n t a l d a t a ; Ne a l s o e x h i b i t s
m a r k e d r o t a t i o n a l p r o p e r t i e s , m u c h m o r e so t h a n th e n u c l e i
o f t h e l a t t e r h a l f of the s h e l l (v. s e c t i o n I V . 6). It is
s u g g e s t e d t h a t s u c h c a l c u l a t i o n s b e p e r f o r m e d in t h i s re g i o n .
W e e x p e c t t h a t the c o m p l i c a t i o n o f s h a p e - m i x i n g c a l c u l a t i o n s
f o r t h e s e n u c l e i w i l l be p r e c l u d e d b y m i n i m i z i n g a f t e r p r o j e c t i o n .
131
132
C H A P T E R V
S U M M A R Y A N D C O N C L U S I O N S
T h e w o r k r e p o r t e d h e r e i n is a n a t t e m p t to e l u c i d a t e
the c o m p l e m e n t a r y r e l a t i o n s h i p b e t w e e n m a c r o s c o p i c a n d
m i c r o s c o p i c m o d e l s o f n u c l e a r s t r u c t u r e . T o t h i s e n d, a
s t u d y o f t h e d e v e l o p m e n t o f c o l l e c t i v i t y w i t h i n the f r a m e
w o r k o f a n i n d i v i d u a l - p a r t i c l e m o d e l , a n d o f i n d i v i d u a l -
p a r t i c l e m o t i o n s w i t h i n th e f r a m e w o r k -of a c o l l e c t i v e m o d e l
h a s b e e n u n d e r t a k e n . W e h a v e c o m e t o th e f o l l o w i n g c o n c l u
si on s:
(+) T h e q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n g i v e s r i s e
to a n a t u r a l g r o u p i n g o f p a r t i c l e d e g r e e s o f f r e e d o m in
c o l l e c t i v e c o o r d i n a t e s . T h e s e m a n i f e s t t h e m s e l v e s in
e x c i t a t i o n s p e c t r a w h i c h m a y b e d e s c r i b e d b y a f e w c o l l e c t i v e
p a r a m e t e r s .
( i t ) T h e o b s e r v e d s y s t e m a t i c s h a p e t r a n s i t i o n s ( f r o m
s p h e r i c a l t o p r o l a t e , to o b l a t e , a n d f i n a l l y b a c k to s p h e r i c a l )
w h i c h o c c u r w i t h t h e a d d i t i o n of p a r t i c l e s to a n e m p t y sh e l l ,
a r e a c c o u n t e d f o r e x c l u s i v e l y by t h e l o n g r a n g e c o m p o n e n t of
t h e n u c l e o n - n u c l e o n i n t e r a c t i o n . T h i s h a s b e e n d e m o n s t r a t e d
i n ( / ) w c o n f i g u r a t i o n s ; t h e r e is s o m e e v i d e n c e t h a t t h i s is
n o t g e n e r a l l y th e c a s e i n th e m o r e r e a l i s t i c s h e l l m o d e l .
U U ) F o r ( / ) w c o n f i g u r a t i o n s , th e l o n g r a n g e p a r t of
t h e e f f e c t i v e i n t e r a c t i o n b r i n g s a b o u t a t e m p o r a r y s p h e r i c i t y
o f t h e n u c l e a r s u r f a c e a s s o c i a t e d w i t h the c h a n g e i n s i g n o f
th e q u a d r u p o l e m o m e n t In th e m i d d l e o f th e sh e l l .
(-tv) T h e p a r t i c l e - t o - s u r f a c e c o u p l i n g m o d e l is o f
v a l u e i n d e s c r i b i n g t h e g r o u n d s t a t e o f m u l t i - p a r t i c l e
s y s t e m s o n th e b a s i s o f a c o l l e c t i v e d e s c r i p t i o n .
(v) T h e e x p e r i m e n t a l l y o b s e r v e d d i s t o r t i o n s o f r o t a
t i o n a l b a n d s at h i g h a n g u l a r m o m e n t u m Is d i r e c t l y a t t r i b u t e d
t o t h e n a t u r e o f th e l o n g r a n g e fo r c e . T h i s d i s t o r t i o n is
s m a l l f o r s m a l l v a l u e s of t o t a l a n g u l a r m o m e n t u m J, a n d f o r
l a r g e v a l u e s o f s i n g l e - p a r t i c l e a n g u l a r m o m e n t u m /.
(v-c) P a r t i c l e - h o l e a d m i x t u r e s t o th e H a r t r e e - F o c k s t a t e s
o f t h e e v e n - e v e n N = Z n u c l e i o f the 26 - 1 d s h e l l a r e m i n i m a l .
(v-U) P a r t i c l e - h o l e a d m i x t u r e s a r e i n a d e q u a t e t o e x p l a i n
th e s y s t e m a t i c d i s c r e p a n c i e s b e t w e e n c a l c u l a t e d a n d e x p e r i
m e n t a l s p e c t r a .
(v-Lt-i) P a r t i c l e - h o l e s t a t e s a r e o f us e i n e x p l a i n i n g the
n a t u r e o f e x c i t e d s t a t e s o f th e e v e n - e v e n N = Z n u c l e i o f the
26 - 1 d s h e l l , b u t n u m e r i c a l a g r e e m e n t w i t h e x p e r i m e n t a l l y
o b s e r v e d e i g e n v a l u e s o f th e H a m i l t o n i a n ar e n o t o b t a i n e d .
(-ox) I n d i c a t i o n s ar e t h a t H a r t r e e - F o c k d o e s r e p r e s e n t
a g o o d a p p r o x i m a t i o n to th e m o r e r e a l i s t i c m a n y - b o d y p r o b l e m
i n t h e r e g i o n 0 " ^ - C a ^ . H o w e v e r , th e e x p l a n a t i o n o f l o w -
l y i n g s p e c t r a h a s b e e n s e e n to lie, n o t w i t h a s i n g l e d e f o r m e d
s t a t e (e.g., th e HF s t a t e ) , b u t r a t h e r w i t h a n i n t e r a c t i o n
a m o n g the f e w l o c a l m i n i m a o f th e c o m p l i c a t e d H F e n e r g y s u r
f a c e .
(x) It is s u g g e s t e d t h a t th e m e t h o d o f p r o j e c t i n g a n g u
l a r m o m e n t u m s t a t e s b e f o r e th e HF v a r i a t i o n o f w a v e f u n c t i o n s
b e e m p l o y e d , r a t h e r t h a n t h a t of p r o j e c t i o n a f t e r v a r i a t i o n .
133
T h i s s h o u l d p r o v e u s e f u l i n o b v i a t i n g i n t e r a c t i o n c a l c u
l a t i o n s a m o n g the l o w - l y i n g d e f o r m e d H F s o l u t i o n s d i s c u s s e d
a b o v e .
M i c r o s c o p i c c a l c u l a t i o n s h a v e s h o w n t h a t s y s t e m s o f
n u c l e o n s m a y u n d e r g o b e h a v i o r u s u a l l y d e s c r i b e d b y c o l l e c t i v e
p a r a m e t e r s . T h e p a r t i c l e d e g r e e s of f r e e d o m t h e n f i n d n a t u r a l
g r o u p i n g s in c o l l e c t i v e c o o r d i n a t e s . T h u s , w e h a v e a c c o m
p l i s h e d th e a i m of t h is w o r k i n s h o w i n g t h a t the d e t a i l e d
o r i g i n of c o l l e c t i v e b e h a v i o r i n n u c l e i m a y be u n d e r s t o o d
f r o m f u n d a m e n t a l m i c r o s c o p i c m a n y - b o d y t h e o r y .
134
135
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A P P E N D I X A - I
M A T H E M A T I C A L F O O T N O T E S
T h e a v e r a g e s h a p e of the s h e l l m o d e l p o t e n t i a l is
g e n e r a l l y c o n s i d e r e d to lie b e t w e e n t h a t of a n h a r m o n i c
o s c i l l a t o r p o t e n t i a l a n d a s q u a r e w e l l w i t h a tail. T h e
u s e o f t h e m a t h e m a t i c a l l y c o n v e n i e n t h a r m o n i c o s c i l l a t o r
l e a d s to a set of p a r t i a l l y d e g e n e r a t e s i n g l e - p a r t i c l e e n e r g y
l e v e l s . T h e m i x i n g of t h e s e l e v e l s is e x p e c t e d to p l a y a
p h y s i c a l l y i m p o r t a n t r o l e i n t h e p r o p e r d e s c r i p t i o n o f li g h t
n u c l e i . T h e g r o u p o f t h r e e d i m e n s i o n a l u n i t a r y t r a n s
f o r m a t i o n s m a y b e u s e d to c l a s s i f y th e s t a t e s of p a r t i c l e s
i n an y of t h e s e d e g e n e r a t e o s c i l l a t o r c o n f i g u r a t i o n s . 33 T h i s
m a y b e s e e n to be a c o n s e q u e n c e of th e s y m m e t r y o f th e h a r
m o n i c o s c i l l a t o r H a m i l t o n i a n
H q = n. 2 + b V (A - I - l )
w h i c h is i n v a r i a n t , no t o n l y w i t h r e s p e c t to r o t a t i o n s , b u t
a l s o w i t h r e s p e c t to t h e g r o u p d e s c r i b e d b y th e n i n e o p e r
a t o r s
H Q = f i 2 + b “ p 2
L y = ( t x p ) y U - I - 2 )
Q y = / W ^ [ / L ?y'2 (e/t, ^ ) + b 4p 2yJ(0p ,4>p )]/b2J
T h e L y ar e the t h r e e i n f i n i t e s i m a l r o t a t i o n o p e r a t o r s i n c o o r
d i n a t e s p a c e , a n d the Q y a r e th e f i v e c o m p o n e n t s o f a s e c o n d
r a n k t e n s o r o p e r a t o r r e l a t e d t o the o p e r a t o r s of i n f i n i t e s i m a l
141
1 . The Group
142
q u a d r u p o l e d i s t o r t i o n s . T h e i n v a r i a n c e of H q w i t h r e s p e c t
to t h e g r o u p is e q u i v a l e n t to t h e c o m m u t i n g of H q w i t h
t h e o p e r a t o r s (A-I -2 ) of t h e ' g r o u p . 79 T h e d e g e n e r a t e l e v e l s
b e l o n g to th e s a m e i r r e d u c i b l e r e p r e s e n t a t i o n of U^, i.e.,
the o p e r a t o r s (A-I-2) a r e th e n i n e p r o d u c t s of th e t h r e e
o s c i l l a t o r c r e a t i o n o p e r a t o r s ( t - i b 2p) a n d th e t h r e e d e
s t r u c t i o n o p e r a t o r s (£ + i b 2p). C l e a r l y , t h e s e p r o d u c t
o p e r a t o r s m u s t l e a v e th e e n e r g y u n c h a n g e d .
22. C a l c u l a t i o n o f < j m |Y q |j m >
T h e d i a g o n a l m a t r i x e l e m e n t s of th e s p h e r i c a l h a r m o n i c2
V q a r e c a l c u l a t e d in th e s h e l l m o d e l b a s i s |jm>. T h e b a s i s
m u s t b e s e p a r a t e d i n t o its o r b i t a l a n g u l a r m o m e n t u m | a n d
1 / 2s p i n |xm > c o m p o n e n t s s i n c e th e s p h e r i c a l h a r m o n i c a f f e c t s £
o n l y t h e o r b i t a l p a r t of th e w a v e f u n c t i o n :
9 j + 1 / 2 i x i 2< j m \ V 0 \ j m > s I I I C ( l 1 , 1 / 2 , j ; m 1 , m - m 1 ) C ( l z , 1 / 2 , j ; m 2 , m - m z )
h l z s mi- m z - j - 1 / 2 -I x - l z
( A -I -3)
2T h e r e d u c e d m a t r i x e l e m e n t s o f V a r e w e l l k n o w n , b e i n g g i v e n
b y 261/2
<l^ \ \Yl \ 11 ^ > = C [lv l , 1 ^ 0 , 0 )4-n ( 2 1 ^ + 1 )
In o r d e r t h a t th e CG c o e f f i c i e n t C U ,2,£ ;0,0) n o t v a n i s h ,
L l + L ^ + 2 m u s t b e even. W i t h t h e r a n g e o f s u m m a t i o n s d e f i n e d in
e q u a t i o n (A-I*3 ) a b o v e , o n l y t h e t e r m s h a v i n g ^ j = -^2 a n d m j ~ m 2
c o n t r i b u t e to th e sum:
< j m \ v U j m > I C ( l , 2 , l ; 0 , 0 ) I C ( 1 , 1 / 2 , j ; m ' , m - m ’)4 tt I = m ' = - £
y-i/2
C [ l , 1 / 2 , j ; m ’ , m - m ’ )C ( £ , 2 , £ ;m 1 , 0 ) .
I n t e r c h a n g i n g s o m e of th e i n d i c e s b y i n v o k i n g s y m m e t r y r e l a t i o n s
of t h e CG c o e f f i c i e n t s 26
143
C { Z t Z , 2; m' , - m' ) .
S u m m a t i o n o v e r th e ( a r t i f i c i a l ) i n d e x m' of th e CG c o e f f i c i e n t s
m a y b e r e p l a c e d by th e p r o d u c t of a R a c a h c o e f f i c i e n t a n d
a n o t h e r CG c o e f f i c i e n t by the i d e n t i t y 26
[ (2e+7) (Z£+1) ] i / *W (abed; etfJC (a, rf,c;a,e+<5)
= ][C{a,b,e.;a,B)C(e.,d,c;a+B,6)C(b,df6 ; 8 , 6 ) ,3
w h i c h g i v e s t h e d e s i r e d r e s u l t :
2 t - s t ]t 2 £ + y - * / 2......< y m | V J y m > = /— C [ j , 2 , j ; m , 0 ) / 2 j + 1 £ (-7) / 2 Z + T
U 4 l - j - 1 / 2
C U , 2 , l j 0 , 0 ) \ H j l j l ; 1 / 2 , 2 ) . (A-I -4)
U s i n g a r e l a t i o n g i v e n b y B r i n k a n d S a t c h l e r 80 f o r t h e s p e c i a l
3/ c o e f f i c i e n t
( - 1 / 2 7/2 fl) = -C(2a+/)(2b+I)]i/2W(abcd;e7/2)C(abe;O0)
a + b + e = e v e n
< J m I I j m > - /— C ( j , 2 , j ; m , 0) C (/, 2, /; - 1 / 2 , 0 ) , u 4
w h e r e it ha s b e e n a s s u m e d t h a t o n l y on e £ - v a l u e is a l l o w e d in
144
t h e s u m m a t i o n . U t i l i z i n g t h e c l o s e d e x p r e s s i o n g i v e n b y
A b r a m o w i t z 81 f o r the s e c o n d CG c o e f f i c i e n t a b o v e , o b t a i n s1 / 2
<y 11v2 i|j> 5 [ 2/r 7) ( 2 j ' + 3 )
6 4 k j l j + 1 )(A-I-5)
A P P E N D I X A - I I C O L L E C T I V E N U C L E O N M O T I O N S
T h e t h e o r y of n u c l e o n s m o v i n g i n a d e f o r m e d f i e l d w a s
d e v e l o p e d 1 3 " 15 in the 19 50 s f o l l o w i n g s u g g e s t i o n s b y
R a i n w a t e r 8 . T h e s u m m a r y b e l o w c l o s e l y f o l l o w s t h e s e w o r k s ,
a n d i n n o w a y d o w e i n t e n d it t o b e c o m p l e t e .
A n o n s p h e r i c a l s u r f a c e R m a y b e d e s c r i b e d b y its
e x p a n s i o n i n s p h e r i c a l h a r m o n i c s
1 . The N u c le a r S u r fa c e
w h e r e R q is th e e q u i l i b r i u m s p h e r i c a l r a d i u s ; the e x p a n s i o n
d e f o r m a t i o n o f t h e n u c l e a r s u r f a c e . If t h e d e v i a t i o n s
f r o m s p h e r i c i t y ar e to c o n s e r v e the n u c l e a r v o l u m e , t h e n R Q
m u s t b e r e p l a c e d b y t h e e x p r e s s i o n
o s c i l l a t i o n s p r e d i c t s t h e p o t e n t i a l e n e r g y o f d e f o r m a t i o n to
b e o f t h e f o r m
R(6,4>) = R q (1+X , y
c o e f f i c i e n t s are g e n e r a l i z e d c o o r d i n a t e s d e s c r i b i n g th e
* x-1/3 ( A - I I - l a )
A s s u m i n g th e c o e f f i c i e n t s a r e s m a l l , th e t h e o r y o f s m a l l
V d e f = 2 ^ C x J a x J 2( A - I I - 2 )
a n d th e a s s o c i a t e d k i n e t i c e n e r g y
w h e r e a n d m a y b e d e v e l o p e d f r o m a s s u m p t i o n s c o n c e r n i n g
t h e n a t u r e of n u c l e a r m a t t e r . B r i e f l y , f o r i r r o t a t i o n a l f l o w
o f a c o n s t a n t d e n s i t y f l u i d u n d e r s u r f a c e t e n s i o n a l o n e
146
C x = S R 0 2 ( X - l ) U + 2 ) ,
w h e r e p is t h e m a s s d e n s i t y a n d S th e s u r f a c e t e n s i o n .
T h e L a g r a n g i a n L = T - V is t h u s a s u m o f s e p a r a t e t e r m s
f o r e a c h of th e g e n e r a l i z e d c o o r d i n a t e s a X y> a n d e a c h o f t h e s e
is t h e L a g r a n g i a n o f a s i m p l e h a r m o n i c o s c i l l a t o r . T h e o s c i l
l a t o r f r e q u e n c y a s s o c i a t e d w i t h th e v a r i a b l e is
B,
or, w i t h th e a b o v e r e s u l t s
U A =S A (A - l ) ( A + 2 )
PR,
1/2( A - I I - 4 )
S i n c e d e f o r m e d n u c l e a r s u r f a c e s a r e k n o w n to h a v e d o m i n a n t +
A = 2 c o m p o n e n t s , h i g h e r o r d e r d e f o r m a t i o n s w i l l n o t b e c o n s i d e r e d
i n t h i s p r e s e n t a t i o n .
W e c o n s i d e r the t r a n s f o r m a t i o n f r o m a f i x e d f r a m e of
r e f e r e n c e to th e b o d y c o o r d i n a t e s y s t e m w h i c h c o i n c i d e s w i t h
t h e p r i n c i p a l a x e s of th e e l l i p s o i d a l s u r f a c e . U n d e r this
t r a n s f o r m a t i o n th e g e n e r a l i z e d c o o r d i n a t e s a 2p d e s c r i b i n g the
d e f o r m a t i o n a r e g i v e n i n the b o d y f r a m e b y a v , w h e r e
fT y p i c a l r a t i o s of h e x a d e c u p o l e / q u a d r u p o l e a m p l i t u d e s m a y
b e o f th e o r d e r 2 0 % 1 9 .
147
( A - I I - 5 )
s e c t i o n I I . 2.
T h e m o d e l e m p l o y e d h e r e w i l l a s s u m e a x i a l s y m m e t r y .
T h e r e ha s b e e n s o m e e v i d e n c e as o f l a te t h a t c o n s i d e r a t i o n
s h o u l d be g i v e n to t r i a x i a l s h a p e s . T h e s e h o w e v e r , a r e
g e n e r a l l y o b t a i n e d b y f i t t i n g e x p e r i m e n t a l d a t a w i t h p a r a m
eters,. T h e t r i a x i a l f i t s , h a v i n g a n o t h e r f r e e p a r a m e t e r ,
a r e e x p e c t e d to g i v e b e t t e r a g r e e m e n t w i t h t h e da ta . T h is
d o e s n o t, t h e r e f o r e , i n d i c a t e t h e v a l i d i t y o f t h e a s s u m p t i o n
I n a n y c a s e , th e i n t e n t h e r e is to s t a y w i t h i n th e f r a m e w o r k
of s i m p l e m o d e l s .
F o r a n e l l i p s o i d of r e f o l u t i o n a 2= a _ 2 , a n d a ^ = a _ ^ = 0 .
D e f i n i n g t h e n e w p a r a m e t e r s g a n d y by
3, t h e r e f o r e , p a r a m e t e r i z e s th e t o t a l d e f o r m a t i o n of th e
n u c l e u s . T h e c o o r d i n a t e y is a s h a p e p a r a m e t e r w h i c h
d e s c r i b e s th e d e v i a t i o n f r o m r o t a t i o n a l s y m m e t r y . G e n e r a l l y
t h e i n c r e m e n t s o f t h e t h r e e a x e s o f the e l l i p s o i d f o r a
d e f o r m a t i o n (3,y) a r e g i v e n by
a Q = 3c o s y( A - I I - 6 )
a ~ = a o = — 3s i n Y 2 - 2 / 2
it m a y e a s i l y be v e r i f i e d t h a t
6R, = / 5 gR c o s C y - ^ ) 1 ETT ° J i = l , 2 , 3 .
T h e r e f o r e , f o r a n e l l i p s o i d o f r e v o l u t i o n , Y = 0 a n d Y=ff
c o r r e s p o n d to p r o l a t e a n d o b l a t e d e f o r m a t i o n s , r e s p e c t i v e l y .
T h e p o t e n t i a l e n e r g y of d e f o r m a t i o n ( A -I I-2) in
th e b o d y s y s t e m n o w t a k e s the f o r m
V d e f = | c ® 2 • ( A - I I - 2 a )
T r a n s f o r m i n g the k i n e t i c e n e r g y t o th e b o d y f r a m e of
r e f e r e n c e l e a d s to a n a t u r a l s e p a r a t i o n i n t o a v i b r a t i o n a l
t e r m — b y w h i c h th e e l l i p s o i d a l s h a p e v a r i e s — a n d a r o t a t i o n a l
t e r m — b y w h i c h th e s h a p e is m a i n t a i n e d w h i l e th e e l l i p s o i d
r o t a t e s . T h e v i b r a t i o n a l e n e r g y m a y b e e x p r e s s e d as
T v i b = | b I I I * - | ( « 2 + • (A -I I- 7)
T h e r o t a t i o n a l t e r m m a y be e x p r e s s e d i n a m a n n e r c o m p l e t e l y
a n a l o g o u s t o t h a t o f a r o t a t i n g e l l i p s o i d i n c l a s s i c a l
m e c h a n i c s :
T r o t " 7 I < L (A- I I ‘ 8)
w h e r e a r e the c o m p o n e n t s o f a n g u l a r v e l o c i t y of the
e l l i p s o i d a l o n g its p r i n c i p a l ax e s . I a r e th e m o m e n t '
o f i n e r t i a c o m p o n e n t s w h i c h ar e p r o p o r t i o n a l to the s q u a r e
o f th e d e f o r m a t i o n p a r a m e t e r 8:
I0 = 4 B 8 2s i n 2 (Y - ^ £ ) . (A -I I - 9 )
2. C o u p l i n g to the N u c l e a r S u r f a c e
S o m e n u c l e i m a y b e c o n s i d e r e d to b e h a v e as if t h e r e
148
w e r e i n d i v i d u a l n u c l e o n s o u t s i d e of a r e l a t i v e l y i n e r t
d e f o r m e d co re , s u c h as d e s c r i b e d a b o v e. If t h e r e w e r e
n o c o u p l i n g at a l l b e t w e e n th e c o r e a n d e x t e r n a l n u c l e o n s ,
t h e c o r e w o u l d u n d e r g o r o t a t i o n s a n d s u r f a c e o s c i l l a t i o n s
i n d e p e n d e n t of p a r t i c l e e x c i t a t i o n s o f th e o u t e r n u c l e o n s .
A d e g r e e o f c o u p l i n g is a l w a y s i n t r o d u c e d h o w e v e r , b y the
c o n s e r v a t i o n of t o t a l a n g u l a r m o m e n t u m of the s y s t e m .
A d d i t i o n a l c o u p l i n g m a y b e i n t r o d u c e d b y c o n s i d e r i n g th e
i n t e r a c t i o n o f th e e x t e r n a l p a r t i c l e w i t h the q u a d r u p o l e
f i e l d o f th e core.
T h e d i s c u s s i o n w h i c h f o l l o w s w i l l b e l i m i t e d t o a
c o n s i d e r a t i o n of a s i n g l e n u c l e o n o u t s i d e of a core. It
m a y , a n d l a t e r w i l l , b e g e n e r a l i z e d to a c o n s i d e r a t i o n of
s e v e r a l e x t e r n a l n u c l e o n s .
T h e t o t a l a n g u l a r m o m e n t u m it of t h e n u c l e u s m a y b e
s e p a r a t e d i n t o c o n t r i b u t i o n s f r o m th e a n g u l a r m o m e n t u m of
t h e c o r e (§, a n d f r o m the e x t e r n a l n u c l e o n -J (cf. F i g u r e 24).
T h e p r o j e c t i o n s of J a n d f a l o n g the b o d y z - a x i s (z') ar e
d e n o t e d b y ft a n d K, r e s p e c t i v e l y , a n d o f it a l o n g th e s p a c e
f i x e d z - a x i s by M. T h e c o m m u t a t i o n r u l e s f o r th e c o m p o n e n t s
o f 1 a n d J a l o n g th e a x e s of t h e n u c l e u s are
V b - V a ■ - 1 I a * b
* V b ” " V a " ^ a x b
V b - V a = °-T h e r o t a t i o n a l e n e r g y (A - I I - 8 ) m a y n o w b e w r i t t e n as
Trot ’ I J 7 (W * •° 2 I o
F o r a n a x i a l l y s y m m e t r i c c o r e Q c a n n o t h a v e a c o m -
p o n e n t a l o n g the b o d y z - a x i s ; r o t a t i o n s ar e I n d i s t i n g u i s h
a b l e i n th e p l a n e p e r p i n d i c u l a r to th e s y m m e t r y a x i ~ . In
t h i s c a s e K=ft a n d the d i r e c t i o n o f i n F i g u r e 24 is p e r
p i n d i c u l a r to the z ' - a x i s . F o r a n a x i a l l y s y m m e t r i c s u r
f a c e d e f o r m a t i o n I j = I 2=I T h e r o t a t i o n a l e n e r g y m a y n o w
b e e x p a n d e d as
150
E l i m i n a t i n g Q o b t a in s
w h i c h , a f t e r s o me m a n i p u l a t i o n b e c o m e s
T ro t = — 2 [ I ( I + l ) + j ( j + l ) - K 2-ft2 ] + — 2 (K-ft)2 r0t 21 2 1 3
- — ( I . J + 1 1 ) . ( A -I I- 1 0 )21 “ +
T h e l a s t t e r m , w h i c h c o u p l e s the t o t a l a n g u l a r m o m e n t u m w i t h
w i t h t h a t of the e x t e r n a l n u l c e o n is g e n e r a l l y r e f e r r e d to
as th e R P C t e r m 8 2 . T h i s h o w e v e r , is a m i s n o m e r , s i n c e
t h e c o u p l i n g I n t r o d u c e d b y th e i n c l u s i o n of t h i s t e r m is
b e t w e e n t h e t o t a l a n g u l a r m o m e n t u m a n d t h a t o f th e s i n g l e
p a r t i c l e , a n d not, as t h e n a m e i m p l i e s , b e t w e e n th e r o t a
t i o n a l m o t i o n of th e c o r e a n d th e s i n g l e - p a r t i c l e a n g u l a r
m o m e n t u m . B e c a u s e of t h e a p p e a r a n c e of th e r a i s i n g a n d
l o w e r i n g o p e r a t o r s , it g i v e s r i s e to b a n d m i x i n g f o r AK =± 1.
If t h i s m i x i n g is s m a l l , as f o r s t r o n g l y d e f o r m e d n u c l e i ,
K m a y b e r e t a i n e d as a n a p p r o x i m a t e q u a n t u m n u m b e r .
F i g u r e 24. T h e t o t a l a n g u l a r m o m e n t u m 1 of a n o n - s p h e r i c a l
n u c l e u s Is t h e r e s u l t a n t o f th e p a r t i c l e a n g u l a r m o m e n t u m J
a n d the c o r e a n g u l a r m o m e n t u m (§. T h e p r o j e c t i o n s o f J a n d f
a l o n g th e z - a x i s ( z 1 ) o f th e n u c l e u s , a r e ft a n d K, r e s p e c -
t i g v e l y . T h e p r o j e c t i o n of 1 o n t o the s p a c e - f i x e d z - a x i s
is M. If t h e n u c l e a r s u r f a c e is a x i a l l y s y m m e t r i c , $
c a n n o t h a v e a c o m p o n e n t a l o n g t h a t a x i s , s i n c e q u a n t u m
m e c h a n i c a l r o t a t i o n s a b o u t a s y m m e t r y a x i s a r e u n o b s e r v a b l e .
I n t h a t c a s e w i l l be p e r p i n d i c u l a r to th e z ' - a x i s .
G
K
/
151
T h e t o t a l H a m i l t o n i a n of th e s y s t e m m a y n o w b e
w r i t t e n as the s u m o f th e n u c l e a r s u r f a c e H a m i l t o n i a n , the
p a r t i c l e H a m i l t o n i a n , a n d a n i n t e r a c t i o n term:
H ■ " s u r f + Hp a r t + H ln t • U - U - l l )
T h e s u r f a c e t e r m Is g i v e n b y e q u a t i o n s ( A - I I - 2 a , 7 , 1 0 ) :
H s u r f = T vi b + T r o t + V d e f . ( A -I I - 1 2 )
T h e p a r t i c l e ( s h e l l m o d e l ) H a m i l t o n i a n is s e p a r a b l e i n t o
a k i n e t i c e n e r g y te rm , a c e n t r a l p o t e n t i a l e n e r g y te rm ,
a n d a s p i n - o r b i t term:
H p a r t ■ T p a r t + V p a r t (r) + V 5oJ 'a ( A - I I - 1 3 )
w h e r e V gQ is a s t r e n g t h p a r a m e t e r . R j_n b r e p r e s e n t s th e
i n t e r a c t i o n o f t h e p a r t i c l e w i t h th e n u c l e a r q u a d r u p o l e
s u r f a c e d e f o r m a t i o n , a n d m a y b e w r i t t e n i n the f o r m
H m t = (a - i i - i H)
w h e r e k ( r ) is a d e n s i t y d e p e n d e n t s t r e n g t h f u n c t i o n . T h e
e x p e c t a t i o n v a l u e s of H i n t d e p e n d o n the p a r t i c l e s t a t e i n
q u e s t i o n , a n d its o r i e n t a t i o n r e l a t i v e to the d e f o r m e d s u r
f a c e i n t h e l a b o r a t o r y f r a m e . T h e c o o r d i n a t e s a ^ y i n the
l a b o r a t o r y f r a m e a r e r e l a t e d to t h o s e i n th e b o d y f i x e d
f r a m e b y t h e t r a n s f o r m a t i o n e q u a t i o n ( A - I I - 5 ) , y i e l d i n g
Hint ’ ®5vav ^ < e **)Sy v ^
the t r a n s f o r m a t i o n of th e s p h e r i c a l h a r m o n i c to th e b o d y -
f i x e d c o o r d i n a t e f r a m e is g i v e n b y
y M
T h e r e f o r e , the I n t e r a c t i o n H a m i l t o n i a n i n the b o d y s y s t e m
is g i v e n b y
H i n t = - k ( r ) J a / J C e ' , * ' ) . ( A - I I - l 4 a )
T h e w a v e f u n c t i o n f o r t h e s y s t e m c a n b e w r i t t e n as th e
p r o d u c t of a f u n c t i o n d e s c r i b i n g t h e r o t a t i o n a l m o t i o n a n d
a p a r t i c l e f u n c t i o n '
152
t M K «21+1
1/2 v m * l . (A-it-15)8tt2
F o r m u l a t i n g the e x p e c t a t i o n v a l u e of H i n t :
. / • 2 T Bc0SYk(r) BOj-Ji l t l l - (A -II-1 6 ) 64ir j ( J + l )
F o r a n o b l a t e d e f o r m a t i o n ( y = ir ), th e l o w e s t s t a t e
is s e e n to h a v e Q=j . F u r t h e r d i s c u s s i o n c o n c e r n i n g t h e
p r e d i c t i o n s o f t h i s m o d e l ar e m a d e i n C h a p t e r III.
A P P E N D I X B-I
( j ) n C O M P U T E R C O D E
T h i s c o d e w a s d e s i g n e d f o r u s e o n th e C o n t r o l D a t a C o r
p o r a t i o n 6 6 00 at th e A t o m i c E n e r g y C o m m i s s i o n C o m p u t e r C e n t e r ,
N e w Y o r k U n i v e r s i t y C o u r a n t I n s t i t u t e o f M a t h e m a t i c s . T h e
r e q u i r e d i n p u t is in th e f o l l o w i n g form:
(a) J T W , N N 1 , N N 2 - t w i c e the p a r t i c l e a n g u l a r m o m e n t u m
(JTW), the n u m b e r of p a r t i c l e s f o r the f i r s t p r o b l e m (NN1),
a n d th e n u m b e r o f p a r t i c l e s f o r th e l a s t p r o b l e m (NN2), a p p e a r i n g
i n (313) f o r m a t .
(b) N J - th e m u l t i p l i c i t y o f J - s t a t e s f o r the f i r s t ( / ) W
c o n f i g u r a t i o n .
(c) N J - th e m u l t i p l i c i t y o f J - s t a t e s f o r the s e c o n d
( / ) n c o n f i g u r a t i o n .
(d) etc.
P r o c e s s i n g p r o c e e d s as f o l l o w s :
(a) D a t a is r e a d in.
(b) P R E L I M is ca l l e d . It c a l c u l a t e s a f e w a r r a y s , i n
c l u d i n g th e t r i a n g u l a r a r r a y u s e d f o r c o n v e r t i n g M - a m p l i t u d e s
to J - a m p l i t u d e s .
(c) P R E P N D g e n e r a t e s the l i s t of a l l S l a t e r d e t e r m i n a n t
s t a t e s in th e ( / ) n c o n f i g u r a t i o n w i t h M^O.
(d) HOP, th e H a m i l t o n i a n o p e r a t o r , s c a t t e r s e a c h of the
i n t r i n s i c s t a t e s , th e m a t r i x e l e m e n t s b e i n g e v a l u e a t e d in
F U N C T I O N V.
153
1 . P r o j e c t i o n Code
(e) J P L U S o p e r a t e s o n e a c h of the i n t r i n s i c s t a t e s , a n d
a l s o o n t h o s e s t a t e s s c a t t e r e d to b y th e H a m i l t o n i a n .
(f) S C H M I T f o r m s a l l r e q u i r e d o v e r l a p s , s o l v e s f o r th e
m e t r i c s f o r e a c h v a l u e of th e t o t a l a n g u l a r m o m e n t u m , a n d
f i n d s th e m a t r i c e s w h i c h w i l l o r t h o n o r m a l i z e t h e s e m e t r i c s .
(g) S T P T W O f o r m s a l l H a m i l t o n i a n o v e r l a p s a n d a p p l i e s
t h e o r t h o n o r m a l i z a t i o n m a t r i c e s f o r e a c h a n g u l a r m o m e n t u m v a l u e .
T h e s e a r e t h e n o u t p u t f o r . t h e a n a l y s i s code.
(h) S t e p s (d), (f), a n d (g) a r e r e p e a t e d f o r e a c h v a l u e
o f th e t w o - b o d y a n g u l a r m o m e n t u m a l l o w e d by v e c t o r c o u p l i n g
a n d the P a u l i p r i n c i p l e . In t h is w a y the H a m i l t o n i a n is
c o m p l e t e l y p a r a m e r t e r i z e d by its t w o - b o d y i n t e r a c t i o n s . It
m a y t h e n b e s u p p l i e d i n the a n a l y s i s c o d e w h i c h f o l l o w s .
154
D I M E N S I O N N C M A X ( 3 5 ) . C O ( 6 7 0 0 1 , C K 6 7 0 0 1 » S Q M ( 2 0 1 . C O E D ( 3 5 . 3 5 ) , N U ( 3 5 ) . 1 K U K ( 8 ) , N O O ( 6 7 0 0 > , I N D ( 7 0 > , M S < 1 0 ) , C O V I 3 5 ) , S ( 1 0 ) , N J ( 3 5 )
0 1 M E N S I ON N T R N S C I S 3 0 ) , N T R E F L ( S 3 0 >D I M E N S I O N 7 ( 1 0 1COMMON N N N , J F C R C E , N U M B E RC C M M . O N / D L K I / T J P O , J T W , N U M J , L N G T 0 T , N N N M 1 , N S P S » N S P S P 1 . J M A X V , J M A X M O ,
1 L E N G T H , L N C i n , J M L T M X , N N N P l , J T W 1 2 C O M M O N / B L K 2 / C O . C l , S O M . C O E D , C O V , N C M A X , N U . K U K . N O O , I N O . H S ,
1 S . N JC O M M O N / B L K 3 / F j l , F N N C O M M O N / B L K 6 / K E R R S ( 7 ) . N O O R U HD O U B L E P R E C I S I O N C O , C 1 , C , S Q M , C O E D , C O V , S O O , F K G . T E M A , T E M E • T J P OD O U B L E P R E C I S I O N F J I , TV, J . C O U N T D O U B L E P R E C I S I O N F NN D O U B L E P R E C I S I O N B I C O O O U B L E P R E C I S I O N R l , R L M I , R N S T , R N S T P l D O U B L E P R E C I S I O N B C u N T
J T W I S T W I C E T HE J OF E A C H P A R T I C L E ------- NNN I S T H E N U M B E R O FP A R T I C L E S J M A X V I S T H E N U M B E R O F D I F F E R E N T T O T A L ‘ J * V A L U E SP O S S I B L E - - - N J I S T H E N U M B E R O F S T A T E S W I T H A G I V E N ' J * V A L U E -------L E N G T H I S T HE T O T A L N U M B E R OF S T A T E S F OR T H E NNN P A R T I C L E S ( W I T H NONN E G A T I V E T O T A L ' M M N U MJ I S T H E N U M B E R O F S T A T E S W I T H A T O T A LV A L U E OF Z E R O , I F T H E R E A R E AN E V E N N U M B E R O F P A R T I C L E S , OR A T O T A L ' H *V A L U E OF 1 / 2 I F T H E R E A R E AN ODD N U M B E R OF P A R T I C L E S ------ J M L T M X I S T H EM A X I M U M J - M U L T I P L I C I T Y O F S T A T E S
CO 1 1 - 1 , 7 1 K E R R S t 1 1 = 0
R E A D I S , 1 0 0 0 ) J T W , N N N 1 , N N N 2 1 0 0 0 F OR MA T ( 3 I 3 )
N S P S - J T W * 1 N S P S P l = N S P S * l J T W I 2 - N S P S / 2 T J P O = N S P S
DO SOOO N N N - N N N l , N N N 2 W R I T E ( 6 , 4 0 0 ) N N N , J T W
4 0 0 F 0 R M A T ( l H l , I 2 t I X . 2 3 H P A R T I C L E S R E S T R I C T E D T 0 . I 3 . 8 H / 2 S H E L L )N N N P 1 - N N N * 1 N N N H l - N N N - l F N N - N N NJ M A X V - ( N N N * J T W - N N N * N N N M l * 2 ) / 2 J M A X M O - J M A X V —IR E A D ( 5 , 7 0 4 ) ( N J I I ) , I » 1 , J M A X V )
7 0 4 F OR MA T ( 2 4 1 3 )W R I T E 1 6 , 4 0 1 ) ( N J I I ) , 1 = 1 , J M A X V )
4 0 1 F O R M A T ( 6 H N J 1 I ) • , 3 X , 1 7 1 4 / 1 9 X , 1 7 1 4 / ) )
4 0 3 L E N G T H = 0N U M J - 0 . ‘NCMAX ( J M A X V * 1 ) - 0j m l t m x - o
0 0 1 0 1 = 1 . J M A X V I L - J M A X V - I * 1N C M A X I L L ) - N C M A X ( 1 1 * 1 ) * N J I L L )L E N G T H - l E N G T H + N C H A X ( L L )J M L T M X - M A X O I J M L T M X , N J I I ) )
1 0 N U M J - N U M J . N J ( I )L 0 N G - 2 * l E MGT H
noon
N U M B E R - J M L T M X 4 J M L T M X / 2 L N G T O T - B I C O I T J F 0 » F N N ) - . 9 L N G T T 1 - L N G T 0 T T W J - J T W F J 1 - T W J / 2 . D 0
C A L L P R E L I M
C A L L P R E P N D
M N - 10 0 l i I - l . N U M J N T R N S C ( I ) = 0
1 1 N T R E F L ( I ) - 00 0 8 1 N S T - l . N U M J
C T H E S T A T E S A R E T E S T E D F O R T H E C R I T E R I A D I S C U S S E D B E L O W
1 2 4 C A L L C 0 D 2 L ( K U K , N O O ( N S T ) , N N N )
0 0 4 3 I - l . N N N R K U K - K U K ( I )
4 3 S < l l - R K U K - l . - F J l M M M - N N N / 21 F I M N . E Q . D G 0 T O 1 0 0
C T H E S T A T E I S T E S T E D T O S E E I F I T I S A R E F L E C T I O N O F A N Y O F T H E .C P R E V I O U S I N T R I N S I C S T A T E S . I F I T I S I T I S P L A C E D I N T H E A R R A Y N T R E F l
N M - M N - 1 DO 5 6 1 = 1 , NM K - N T R N S C I I )C A L L C 0 D 2 L ( M S , N O O ( K ) , N N N )0 0 5 7 J - l . N N N R M S - M S I J )
5 7 T < J I - R M S - l . - F J l DO 5 8 J - l . M M M L L - N N N P 1 - JI F ( S ( J ) * T ( L L ) » G T . 0 . 1 ) G 0 T O 5 6 ( F ( T ( J ) * S ( L L ) . G T . O . D G O T O 5 6
5 8 C O N T I N U E1 F I MOD 1 N N N , 2 ) . E Q . 1 • A N D . A B S ( S I M M H 4 1 ) —T ( H H M * 1 ) ) . G T . 0 . 1 ) GO T O 5 6N T R E F L I I ) - N S T GO TO 8 1
5 6 C O N T I N U E
1 0 0 I F ( M O O ( N N N , 2 ) . E O . O ) G O T O 4 9
I F T H E R E A R E AN 0 0 0 N U M B E R OF P A R T I C L E S T H E S T A T E I S T E S T E D F O R I N C L U S I O N OF T H E V A L U E 1 / 2 . I F I T I S I N C L U D E O T H E S T A T E I S F U R T H E R T E S T E D F O R S Y M M E T R Y A B O U T T H E V A L U E 1 / 2 . I F T H E R E A R E AN E V E N N U M B E R OF P A R T I C L E S T H E S T A T E I S T E S T E O F O R S Y M M E T R Y A B O U T 0
DO 4 6 I - l . N N N L J - I
155
I F ( A 8 S ( S M ) - . 5 ) . L T . 0 . l ) G 0 T O 4 74 6 C O N T I N U E
GO TO 4 54 7 P P » N N N - 1
I F ( L J . E O . N N N ) G O T O 4 8 0 0 5 0 K = L J , P H
5 0 S I K I = S t K * 1 )GO TO 4 8
4 9 P P = N N N4 8 0 0 4 4 1 = 1 , HP H
L H = P M - I ♦ 1I F ( A B S ( S < I ) » S ( L M > > . G T . O . l > G O T O 4 5
4 4 C O N T I N U E GO TO 8 1
C t h e S T A T E I S GOOO I F J - P L U S C A N O P E R A T E H O R E T H A N O N C E
4 5 0 0 5 2 I - l , N N N M l L K « I » lI F I ( K U K ( I ) - K U K I L K ) ) . G T . 1 ) G 0 T O 5 3
5 2 C O N T I N U E GO TO 8 1
5 3 I F ( L K . E O . N N N ) G O T O 9 4 0 0 5 4 I = L K , N N N M 1I F I ( K U K ( I ) - K U K I I + 1 ) ) . G T . 1 > G 0 T O 5 5
5 4 C O N T I N U E9 4 I F I N S P S P 1 - K U K ( 1 ) . G T . 1 ) G 0 T O 5 5
GO TO 8 1
C I F A L L C R I T E R I A A R E M E T , T H E S T A T E I S P L A C E D I N T H E A R R A Y N T R N S C
5 5 N T R N S C ( H N ) - N S T M N * M N * l
8 1 C O N T I N U E M N = M N - 1
C P R O C E S S I N G B E G I N S H E R E
J F O R C E - O I N D E X - 0 N 0 0 R U P = 7 5
5 0 5 1 8 0 U N T - 1 . 0 0 C Q U N T - 1 . 0 0
5 0 5 0 C A L L K Y I 0 I 4 , 1 , 0 , 0 )R E W I N D 1 R E W I N D 2I N T E G E = J T W + 1 0 0 * N N N + 1 0 * J F O R C E !C A L L 0 1 S P L A ( 1 0 H P R Q G R E S S , I N T E G E )I C O U N T = 0I F I I N U E X . E 0 . 1 I G 0 T O 9 9 8 DO 2 5 0 5 . 1 - 1 , N U M J
2 5 0 5 C 0 ( I ) = 0 . 0 0
C A C O M B I N A T I O N OF ONE I N T R I N S I C ANO ONE N O N - I N T R I N S I C * N O N - R E F t E C T E D “C I N T R I N S I C S T A T E I S C H O S E N A S T H E S T A T E W I T H W H I C H TO W O R K . F R OM H E R E
u o u u
C ON T H E C O M B I N A T I O N I S R E F E R R E D T O A S T H E I N T R I N S I C S T A T E
J - l5 9 L - l6 0 I F ( L . G T . N U M J ) C O U N T - C O U N T * ! . 5 0 0
I F I L . G T . N U H J I L - IK = LI F ( H N . E Q . O ) GO T O 4 4 4 5 DO 1 0 0 6 M P N = l . M NI F I K . E O . N T R N S C ! M H N ) . O R . K . E Q . N T R E F L I M M N 1 1 G O T O 1 0 0 5
1 0 0 6 C O N T I N U E L = L + 1GO TO 4 4 4 4 *
1 0 0 5 l = L + lGO TO 6 0
4 4 4 4 N S T = N T R N S C ( J )R N S T = NS T R N S T P 1 = N S T * 1C 0 I N S T I = 8 0 U N T * R N S T P I / R N S T
4 4 4 5 R L = K R L M I = K + 1C O ( K ) = C 0 U N T * R L H 1 / R L
9 9 8 I C O U N T = I C O U N T + 1 .I F ! I N O E X . E O . O I G O T O 4 4 4 6 R E A D ) 1 ) ( C 0 ( I ) , I - I , L E N G T H )
4 0 0 0 C A L L HOP GO TO 7 7 7
J - P L U S O P E R A T E S ON T H E I N T R I N S I C S T A T E . T H E R E S U L T I S W R I T T E N O U T ONT A P E I
4 4 4 6 C A L L J P L U S ( C O )
I F ( I C O U N T . G E . N O O R U H ) G O T O 6 6 46 6 5 C A L L H Y 1 0 ( 4 , 2 , C O , L O N G )
I F I K E R R S i n . E Q . O I G G T O 6 6 6 0 0 2 1 = 1 , 7I F ( K E R R S ( I ) . G T . 2 ) G 0 T O 2 2 2 6
2 C O N T I N U EN O D R U H = M I N O ( I C O U N T , N O O R U H )
6 6 4 W R I T E I 2 ) ( C O ( l ) , 1 = 1 , L E N G T H )6 6 6 W R I T E ( l ) ( C O ( I ) , 1 = 1 , L E N G T H )
0 0 6 6 7 1 = 1 , N U MJ6 6 7 C 0 ( l l = 0 . 0 0
C O TO 4 4 4 7
J - P L U S O P E R A T E S ON T H E R E S U L T O F T H E H A M I L T O N I A N O P E R A T I O N ON T H E I N T R I N S I C S T A T E . T H E R E S U L T I S W R I T T E N O N T O T A P E 2
7 7 7 C A L L J P L U S ( C l )
I F I I C O U N T . L T . N O D R U M I G O TO 6 6 8 ,6 6 9 W R I T E 1 2 ) ( C l 1 1 ) . 1 - 1 . L E N G T H ) (
GO TO 4 4 4 7 j6 6 8 C A L L P Y I 0 ( 4 , 2 , C 1 , L 0 N G )
I F ( K E R R S t 1 ) . E O . O I G O T O 4 4 4 7 0 0 3 1 = 1 , 7I F ( K E R R S U ) . G T . 2 ) G 0 TO 2 2 2 6
3 C O N T I N U EN O D R U M = M I NO I I C O U N T , N O D R U M )GO TU 6 6 9
4 4 4 7 I F < I C O U N T . E Q . N U M B E R I G O TO 3 7
I F ( I N D E X . E Q . 1 ( G O T C 9 9 8 I F U . E Q . M N I J « 0 I F I J . E O . Q I B O U N T * 8 0 U N T / 1 . 4 0 0 J * J * 1 C O TO 6 0
3 7 I F ( I N D E X . E Q . I ) G 0 TO 3 8 C A L L S C H H I TI NOE X * I J F O R C E * I C O TO 5 0 5 0
3 8 C A L L S T PT WO
1 0 0 2 [ F ( J F 0 R C E . E 0 . J T H 1 2 ) G 0 T O 5 0 0 0 J F O R C E * J F O R C E + 1 C O TO 5 0 5 0
5 0 0 0 W R I T E < 6 , 4 0 1 M N J ( I ) , I » l , J H A X V )
2 2 2 6 W R I T E . ! 6 , T O M K E R R S . NODRUH S T O P E N D
oo non
non
D I M E N S I O N N C M A X ( 3 5 ) , C O < 6 7 0 0 ) , C 1 ( 6 7 0 0 ) , S Q M ( 2 0 ) , C O E D ! 3 5 , 3 5 ) , N U ( 3 5 ) , I K U K ( 8 ) , N 0 0 ( 6 7 0 0 ) , I N 0 ( 7 0 ) , M S ( 1 0 ) , C O V ( 3 5 ) , S ( 1 0 ) , N J ( 3 S )
COMMON N N N . J F C R C E , N U M B E RC O M M O N / B L K 1 / T J P O , J T W , N U M J , L N G T 0 T , N N N M 1 , N S P S , N S P S P 1 . J M A X V , J M A X M O ,
1 L E N G T H , L N G T T I , J M L T M X , N N N P 1 , J T W 1 2 C O M M O N / B L K 2 / C 0 . C 1 , S Q M , C O E O , C O V , N C M A X , N U , K U K , N O O , I N O , M S ,
I S . N JD O U B L E P R E C I S I O N C O , C 1 , C , S Q M , C O E D , C O V , S O D , F K G , T E M A , T E M E , T J P O O O U B L E P R E C I S I O N F I
SUBROUTINE PRELIM
SQM I S T HE A R R A Y C F C O E F F I C I E N T S W H I C H R E S U L T F R OM O P E R A T I O N W I T H J - P L U S - N O T E T H A T WE MUST A D J U S T T H E V A L U E S O F * J * A N D » M ' W H I C H WOULD N O R M A L L Y O C C U R B E C A U S E OF OUR WAY OF C O U N T I N G
0 0 2 0 1 * 1 , N S P S F 1 * 1 .S O D = F I * I T J P O - F I I
2 0 S Q M ( I J = 0 S Q R T ( S 0 0 )
C O E D I S T H E A R R A Y OF P R O D U C T S OF T E R H S L I K E T H O S E W H I C H O C C U R I N S Q M . T H I S A R R A Y I S T H E T R I A N G U L A R C O E F F I C I E N T M A T R I X W H I C H I S U S E O TO S O L V E F O R T H E T R A N S F O R M A T I O N M A T R I X I N S T E P T W O
. N O N E * N N N - I N N N / 2 ) * 2 0 0 2 1 J * 1 , J M A X V
2 1 C O E O 1 1 , J ) * 1 . 0 0 0 0 2 2 1 * 2 , J M A X V 0 0 2 2 J * 1 . J M A X V 1 F I J . L T . D G 0 TO 2 3 F K G = ( J - I + l ) * ( J * I ~ 2 ♦ N O N E ) C O E D ! I , J ) = C O E O ( I - 1 , J ) * F K G GO 1 0 2 2
2 3 C O E O ( I , J ) = 0 . 0 02 2 C O N T I N U E
NU I S T H E L E N G T H O F T H E S U 8 V E C T 0 R U P T O ( B U T N O T I N C L U D I N G ) T H E *M» V A L U E C O R R E S P O N D I N G T O T H E I N D E X O F NU
N U 1 1 ) ° 0DO 2 0 0 0 1 * 2 , J M A X V
2 0 0 0 N U ( I ) * N U ( I - l ) + N C M A X ( t - l )
R E T U R NE N O 1
57
uuoo
S U B R O U T I N E P R E P N ODIMENSION NCMAX135),C016700) »C 1167001, SQM1201»C0E0(35,35)»NU(351, 1KUKI8I,NOOI6700>. INDI70),MSU0),C0V(35),S<10),NJ(35)COMMON N N N . J F O R C E , N U M B E RC O M M O N / R L K 1 / T J P O , J T W , N U M J , L N G T 0 T , N N N H 1 , N S P S , N S P S P 1 » J M A X V , J M A X H 0 ,
l L E N G T H . L N G T T l , J M L T M X . N N N P l , J T W 1 2 C O M M . O N / D I K 2 / C 0 . C 1 , S O M , C O E D , C O V , N C M A X , N U , K U K , N O O . I N D . M S ,
1 S , N JD O U B L E P R E C I S I O N C O , C I . C , S Q M . C O E O , C O V , S O O , F K G , T E M A , T E M E , T J P O
THE LIST OF ALL POSSIBLE STAIES OF NNN PARTICLES WITH NON-NEGATIVE TOTAL M IS GENERATED IN NOD. THE STATE IS CHARACTERIZED BY THE H VALUES OF THE INDIVIDUAL PARTICLES, THE HIGHEST BEING FIRST. THE LIST INCREASES IN TOTAL 'H1 VALUEN J l l = ( N N N * ( J T W * 2 l - l ) / 2
DO 1 3 1 * 1 , L E N G T H 1 3 N O D I I ) * 0
DO 11 1=1,JMAXV1 1 I N O I I ) 3 I
0 0 1 0 1 * 1 , NNN 1 0 M S I I ) * I
M S I N N N P U - N S P S P 1 '
0 0 8 K * 1 , L N G T T 10 0 1 I * t , NNN1 F I M . S I I I ♦ l - M S U + l 1 0 2 , 1 . 6 6 6 6
6 6 6 6 S T O P 6 6 6 62 M S i n * M S ( t m 11=1-1
I F ) 1 1 - 1 1 3 1 , 3 2 , 3 2 3 2 0 0 3 J = 1 , I 1
3 M S ( J ) = J 3 1 GO 1 0 A
1 C O N T I N U E
A N S 1 G * 00 0 5 1 = 1 , NNN
5 N S I G = N S I G » M S ( I )N S I G = N S 1 G - N J 1 1 I F ( n S I G ) 8 , 8 , 9
9 DO 1 2 1 = 1 , NNN N P 1 M I s N N N P l - I
1 2 K U M I I = M S ( N P I M I » I X X = I N O ( N S I G ) * N U ( N S I G )
C A L L L 2 C 0 D I K U K , N 0 0 1 1 X X 1 , N N N !
I N O ( N S l G l » l N O ( N S l G ) * l B C O N T I N U E
R E T U R NEND
DIMENSION NCMAXI351,CO167001,C116700),SQMI201»COED135,351,NU(35), 1KUKI8), NOD(6700),I NO I 70),MS I 10),COV{35)tS(10),NJ(35)COMMON NNN,JFORCE,NUMBERCOMMON/BLR I/TJPO,JTW,NUMJ,LNGTOT.NNNHI,NSPS.NSPSP1,JMAXV,JHAXMO, lLENGTH.LNGTTl,JMLTMX,NNNPL,JTH12COMMON/HLK 2/CO,C1,SUM,COED,COV,NCMAX,NU.KUK.NOO.IND,US,1S.NJCOMMON/BLK 3/FJl.FNNDOUBLE PRECISION CO,C1,C, SQM,COED,COV, SOO,FKG,TEMA, TEME,TJPO DOUBLE PRECISION V DUUOLC PRECISION CHECK DOUBLE PRECISION FJI.FNN
SUBROUTINE HOP
IF THE COEFFICIENT OF THE STATE, CO, IS NON-ZERO. THE STATE IS OE- CODEI). THE TWO/BODY HAMILTONIAN THEN OPERATES ON THE STATE. TWO PARTICLES AT A TIME, SO AS TO CONSERVE THE VALUE OF THE TWO PARTICLES, REFERRED TO AS MPAIR THE STATE IS THEN ARRANGED, COOEO ANO SEARCHED FOR, BEING FOUND IN POSITION NP. THE MATRIX ELEMENT IS THEN EVALUATEO IN FUNCTV ANO THE COEFFICIENT OF THE NEW STATE IS AOJUSTEODO 1 0 2 ! = 1 , N U H J
1 0 2 C 1 ( I ) = 0 . D 0
DO 1 0 0 1 * 1 . N U M J I F ( C O I I ) 1 1 0 1 , 1 0 0 , 1 0 1
1 0 1 C A L L C 0 D 2 L ( K U K , N 0 0 1 1 ) , N N N )
* 0 0 0 0 0 1 I P I * 1 , NN N M 1 I P 1 P 1 = I P 1 + 1 0 0 1 ! P 2 = I P I P 1 , N N N M P A I R = K U K ( I P I ) + K U K ( ! P 2 )0 0 1 I S 1 * t , N S P S 0 0 * I S 2 = l , I S l I F 1 1 S 2 - 1 S 1 ) 5 , 4 , 4
5 I F ( I S I * I S 2 - M P A 1 R ) 4 , 6 , 4
6 0 0 7 I K = I , NNN7 MS 1 1 K ) * K U K ( I K )
MS 1 1 P 1 ) * I S I M S ( I P 2 ) = I S 2
C A L L A R R A N G I M S , I C H E C K )
I F ( I C H E C K ) 8 , 4 , 8
8 C A L L L 2 C 0 D I M S , N E M P , N N N )
C A L L S E A R C H I N E H P . l . N P ) H*U1C H E C K * I C H E C K 0 2C l ( N P ) = C 1 ( N P M C 0 C I ) * C H E C K * V I K U K I I P 1 ) , K U K ( I P 2 ) , I S 1 , I S 2 )
4 C O N T I N U E 1 C O N T I N U E
1 0 0 C O N T I N U E
RETURNENO
D O U B L E P R E C I S I O N F U N C T I O N V I N l . H 2 . N 3 . M A ) COMMON N N N . J F O R C E . N U M B E R C O M M O N / B L K 3 / F J 1 . F N ND O U B L E P R E C I S I O N F M l . F M 2 . F M 3 . F N A . F M . F J . C O F C CO C U B L E P R E C I S I O N F J 1 . F N N
T H E M A T R I X E L E M E N T OF T H E H A M I L T O N I A N I S G I V E N B Y T H E P R O O U C T OF T H E C I E B S C H - G O R D A N C O E F F I C I E N T S F O R T H E P A I R OF P A R T I C L E S B E F O R E AND A F T E R O P E R A T I O N . T H E 2 . DO I S I N C L U D E D I N V SO T H A T T H E N O R M A L I Z A T I O N M I L L A G R E E W I T H T H A T OF OE S H A L I T AND T A L M I WHEN V O . V 2 , E T C . , A R E T A K E N E Q U A L TO U N I T Y , S U C C E S S I V E L Y
V » 0 . 0 0I F ( H I . M 2 - M 3 - M A ! 1 , 2 , 1
2 F H 1 = M 1 F N 2 = M 2 F M 3 = M 3 F M A = H AF M l = F M I - F J l - l . D OF M 2 « F M 2 - F J 1 - 1 . 0 0 -F H 3 = F M 3 - F j l - l . D OF M A = F M A - F J 1 - 1 . 0 0F M = F H 1 + F M 2
F J « 2 * I J F O R C E - 1 )V = V * C O F C G ( F J l , F J 1 » F J , F M I » F M 2 , F M ) P C O F C G I F J 1 , F J 1 , F J , F M 3 . F M A . F M ) * 2 . 0 0
I R E T U R N END
O I M E N S I O N N C H A X ( 3 5 ) » C 0 1 6 7 0 0 ) , C 1 ( 6 7 0 0 ) , S Q M ( 2 0 ) . C O E D ( 3 5 , 3 5 ) , N U ( 3 5 ) , 1 K U M 8 ) . N O D ( 6 7 0 0 ) , I N D ( 7 0 ) , M S I 1 0 ) , C O V ( 3 5 ) , S t 1 0 ) . N J ( 3 5 )
O I M E N S I O N C ( 6 7 0 0 )COMMON N N N . J F O R C E , N U M B E RC O M M O N / B L K 1 / T J P O , J T W , N U M J , L N G T O T , N N N M 1 , N S P S . N S P S P l , J M A X V , J M A X M O ,
I L E N G T H . L N G T T l , J M L T M X , N N N P I , J T W 1 2 C O M M O N / B L K 2 / C O , C 1 , S C M , C O E O , C O V , N C H A X , N U . K U K , N 0 0 , I N O , M S ,
I S , N JO O U B L E P R E C I S I O N C O . C I . C . S Q M , C O E O , C O V » S O D , F K G , T E M A , T E H E , T J P O
SUBROUTINE JPLUS(C)
I F T H E C O E F F I C I E N T C I S N O N - Z E R O , T H E S T A T E I S O E C O O E O ANO E A C H S U C C E S S I V E TERM I S I N C R E A S E D . T H E NEW WORD I S T H E N A R R A N G E O ,S E A R C H E D F O R ANO F OU ND I N P O S I T I O N L E X . T H E C O E F F I C I E N T OF T H I S S T A T E I S 1 H E N A D J U S T E D A C O I T I V E L Y W I T H T HAT O F T H E ONE J U S T O P E R A T E D O N , ANO U S I N G T H E A R R A Y SCM W H I C H WAS P R E P A R E D I N P R E L I H
N U M J P 1 = N U M J * 1 0 0 6 K = N U M J P 1 . L E N G T H
6 . C ( K I * 0 . DO
M S I l ) = N S P S P 1 DO b M = I . J M A X M O L M A X = N C M A X I H )
DO 5 L = 1 » L MAX L N X = L * N U ( MI I F ( C I L N X ) ) 4 , S , 4
4 C A L L C O O 2 L I K U K , N O 0 I L N X ) , N N N ) 0 0 7 I e 1 , NNN
7 M S ( I ♦ I ) = K U K ( I )
DO 1 1 = 2 . N N N P I I F ( M S I 1 1 - M S I 1 - 1 ) 4 1 ) 2 , 1 , 6 6 6 7
6 6 6 7 S T O P 6 6 6 7 2 H S ( I ) = M S ( I ) . l
DO 8 J » 1 , N N N 6 K U K t J ) = M S t J * I )
C A L L L 2 C 0 D ( K U K . N E M P . N N N )
C A L L S E A R C H ( N E H P , M * l , L E Q )
L E X » L E Q . N U ( H * 1 I I X X - M S I I ) - IC I L E X ) = C I L E X ) * C I L N X ) * S Q M I I X X ) M S ( I ) = H S ( I » - 1
1 C O N T I N U E5 C O N T I N U E
R E T U R NENO
159
noon
SUBROUTINE SCHMIT01 MENS I ON NCMAX(35).CO(6700) .Cl(67001•S Q M(20).COED(35.35)»NU(35) , 1K U K 1 8 ) . NOO(6700).IND(70).MS( 10).COV(35)« S( 10),NJ(35)01 MENS I ON YI2500),AST<70,70)COMMON NNN.JFORCE,NUMBERCOMKON/BLK l/TJPO.JTWfNUMJ.LNCTOT.NNNMlfNSPS.NSPSP1.JHAXVtJMAXMO. 1 LENGTH,LNGTT1,JMLTMX.NNNPI,JTW12 COMMON/BLK 2/CO.Cl.SQMfCOCO.COV.NCMAX »NU»KUK.NOD,INO.MS.1S.NJ C 0 M M 0 N / B L K 5 / Y . A S T C 0 M M 0 N / 6 L K 6 / K E R R S I 7 ) . N O O R U HD O U B L E P R E C I S I O N C O , C I , C . S Q M . C O E O . C O V , S O D . F K G , T E M A , T E M E . 7 J P O D O U B L E P R E C I S I O N Y , A S 7 , 7 E M P , S I
C D U E T O T HE L A C K OF A D E Q U A T E C O R E S T O R A G E . A O V A N T A C E I S T A K E N OF T H EC S Y M M E T R I C N A T U R E OF T H E N O N - O R T H O N O R M A L I Z E O A M P L I T U D E S OF S T A T E S ANOC H A M I L T O N I A N M A T R I X E L E M E N T S . I N S T E A D O F C R E A T I N G AN A R R A YC D I M E N S I O N E D AT ( J M L T M X , J M L T M X , J M A X V ) , ANO H A V I N G T H O S E E L E M E N T S B E L O WC T H E D I A G O N A L ! I N T H E F I R S T TWO S U B S C R I P T S ) . AN A L G O R I T H M ( I N O E X ) I SC U S E D F O R L O C A T I N G THE P O S I T I O N I N T H E F I C T I T I O U S A R R A Y A N D T HEC S M A L L E R A R R A Y Y I S U S E D I N S T E A O
I N D E X ( 1 1 , 1 2 ) = N U M B E R * ( M I N 0 ( 1 1 . 1 2 ) - 1 1 + M A X O I 1 1 . 1 2 > - I M [ N 0 ( 1 1 , 1 2 ) • 1 ( M I N O I 1 1 , I 2 ) - l ) 1 / 2
MAX I N D = I N D E X ( N U M B E R , N U M B E R )L 0 N G - 2 * L E N G T H
C A L L I N V E R T ( C O E D , J M A X V . 3 5 . O E T E R M )W R I T E ( 6 , 1 0 0 1 I C E T E R M _
R E W I N D 2 R E W I N D 3C A L L M Y t O t n , 1 , 0 , 0 1 0 0 1 1 1 = 1 , N U M B E R I F I I l . L T . N O O R U M I G C TO 4 + 3
R E A D ! 2 ) I CO I I I . 1 = 1 , L E N G T H )R E W I N D 2 GO TO 4 4 4
4 4 3 C A L L MY I 0 I 4 , 3 , C O , L O N G )I F I K E R R S I D . E Q . O I G C T O 4 4 4 W R I T £ ( 6 , 1 0 0 2 ) K E R R SS T O P
4 4 4 C A L L M Y I 0 I 4 , 1 , 0 , 0 1 00 1 1 2 = 1 , 1 1I F I I 2 . L T . N C O R U M I G O T O 4 4 5 R E A O I 2 ) ( C l 1 I ) . 1 = 1 . L E N G T H )GO TO 4 4 6
4 4 5 C A L L M Y I 0 I 4 . 3 . C 1 . L 0 N G I I F I K E R R S I I ) . E O . O > G O T O 4 4 6 W R I I E 1 6 . L 0 0 2 1 K E R R SS T O P
T H E S C A L A R P R O D U C T S OF A L L T H E S T A T E S A R E F O R M E D F O R A G I V E N H V A L U E . N OT E T H A T T H E R E I S NO M - M I X I N G ANO W I T H I N A G I V E N M S U B V E C T O R T H E P K C O U C T S A R E 1 - 1 , 2 - 2 , 3 - 3 , E T C . , B E C A U S E O F T H E O R T H O G O N A L I T Y OF T H ED I F F E R E N T M ' S
4 4 6 0 0 2 M = I , J M A X V
r% n o
C 0 V I H I - 0 . 0 0 N C M - N C M A X ( M )DO 2 1 = 1 , NCM N U M X = N U ( M ) ♦ I
2 C O V I M ) = C O V I M ) + C 0 ( N U M X I * C 1 ( N U M X )1 W R I T E I 3 ) ( C O V t M ) , M = 1 , J M A X V )
C O V F O R M S T H E I N H O M O G E N E O U S P A R T O F T H E L I N E A R E O U A T I O N S T O B E S O L V E O , R E S U L T I N G I N T H E A M P L I T U D E S , Y , A S D I S C U S S E D A B O V E . N O T E T H E T R I A N G U L A R N A T U R E O F C O E O , W H I C H WAS P R E P A R E D I N P R E L I H
R E W I N D 8 R E W I N O 9N U M B = J M L T M X .0 0 2 S 1 M = l , J M A X V N X = N J ( M )I F ( N X . E Q . O ) G O T O 2 5 1
R E W I N O 3 S4 4 9 0 0 5 l T l = l , N U M B E R
I N O I I T 1 ) = 0 0 0 5 I T Z = I , N U M B E R I F I I T 2 . G T . I T l ) GO T O 5 R E A D ! 3 ) I C O V ( J ) , J = 1 , J M A X V )K = I N D E X ( I T I . I T 2 )YIK I=0.00 OU 3 1 = 1 , J M A X V
3 Y ( K ) « Y ( K ) + C 0 E D ( M , 1 ) * C O V I 1 1 5 A S I I I T 1 , 1 7 2 1 = 0 . 0 0
C A S C H M I D T O R T H O N O R M ! L I Z A T I O N I S D ONE ON T H E S T A T E S
1 = 14 K » I N O E X I I , I )
I F ( Y ( K ) . G T . 1 . 0 - 4 ) G 0 T O 6 1 = 1 + 1I F I I . G T . N U M B E R ) G O TO 6
GO TO 4 6 A S T ( I . I ) = 1 . 0 0 / D S Q R T ( Y ( K ) )
I P l = I * l INOII 1 = 1I F ( N X . L T . 2 ) G 0 TO 7 8 11 = 10 0 1 1 1 = I P I , N U M B E R1 M l - I - 1i m = m iA S T I I , I ) = 1 . 0 00 0 7 J = 1 , I M 1I F I I N D l J I . E 0 . 0 1 G 0 TO 7T E M P = 0 . 0 0N X 1 = 00 0 8 0 K 2 = 1 , 1 1
8 1 K 4 = N X 1 + K 2 I F ( 1 N 0 ( K 4 ) . N E . 0 ) G 0 T O 8 2 N X l = N X mGO TO B I
8 2 K = I N O E X ( I , K 4 )L = M A X 0 ( K 4 , J )N X 2 = 0DO 6 0 K l = l , 1 1
8 3 K 3 = N X 2 * K L
091
IF(INO(K3) . NE.01GO TO 84 NX2=NX2*1 GO TO 83 84 IFIK3.LT.L)GO TO 80T E M P = T E M P * A S T < K 3 , K 4 ) * A S T { K 3 , J ) * Y ( K )
8 0 C O N T I N U EA S T ( I , J 1 = - T E M P
7 C O N T I N U E
T E M P = 0 . 0 0 0 0 9 2 J 1 = 1 , II F ( A S T I J 1 , J 1 ) . E Q . 0 . D 0 ) G 0 T O 9 20 0 9 1 J 2 = I , II F I A S T I J 2 , J 2 ) . E Q . O . O O ) G O T O 9 1 J * I H D E X < J 1 . J 2 I
9 T E M P = T E M P . A $ T ( I , J 1 ) * A S T ( I , J 2 ) * Y ( J )9 1 C O N T I N U E9 2 C O N T I N U E
1 F H E M P . G T . 1 . E - 0 4 ) GO TO 1 0 0S 1 * 0 . 0 0GO TO 1 0 1
1 0 0 S I “ I . O O / O S Q R T I T E M P I I N D ( l l = l1 1 » 1 1 P I
1 0 1 0 0 1 0 J = l , l1 0 A S T I I , J ) * A S T 1 1 , J ) * S 1
1 F 1 1 1 • E Q . N X ) G 0 TO 7 81 1 C O N T I N U E
W R I T E 1 6 , 5 0 0 0 ) 1 1 . NX 5 0 0 0 F O R M A T ( 2 1 5 )8 W R I T E ( 6 , 1 0 0 1 ) t (A S T ( I , J ) , J « I,N U M B E R ) , 1 = 1 . N U M B E R )
W R I T E ( 6 . 1 0 0 1 ) ( Y I K ) , K « 1 , M A X I N 0 )1 0 0 1 F 0 R M A K B D 1 5 . T )
N J ( M l = 0N X * 0GO TO 2 5 1
7 8 N U M 8 = M A X 0 ( N U M B , I )w R I I E I B ) ( I A S T ( I , J ) , J = 1 , N U M B E R ) , 1 = 1 . N U M B E R )W R I T E 1 6 , 1 0 0 2 ) ( I N D ( J ) , J = 1 . N U M B E R )
1 0 0 2 F O R M A T I 1 X . 1 0 0 I 1 )W R I T E ( 9 ) I I N D l J ) , J » 1 , N U M B E R )
2 5 1 C O N T I N U E
N U H B E R o NUMBR E T U R NE N D
no no
SUBROUTINE STPTWOD I M E N S I O N N C M A X 1 3 5 ) , C 0 ( 6 7 0 0 ) , C l ( 6 7 0 0 > , S O M I 2 0 ) , C O E O I 3 5 1 3 5 ) , N U I 3 5 ) «
1 K U K ( 8 ) , N O O ( 6 7 0 0 ) , I NO 1 7 0 ) , MS I 1 0 ) , C O V ( 3 5 ) , S 1 1 0 ) , N J I 3 5 )0 1 M E N S I O N Y ( 2 5 0 0 ) . A S T I 7 0 , 7 0 ) , H { 7 0 , 7 0 )COMMON N N N . J F O R C E , N U M B E R
C O M M O N / B L K l / T J P O , J T W , N U M J , L N G T O T , N N N M l , N S P S , N S P S P 1 , J M A X V , J M A X M O , 1 L E N G T H . L N G T T 1 , J M L T M X , N N N P 1 , J T W 1 2
C O M M O N / B L K 2 / C 0 . C 1 , S O M , C O E O , C O V , N C M A X , N U , K U K , N O O , I N O , M S ,1 S . N J
C 0 M M 0 N / B L K 5 / Y , A S T C O M M O N / 0 L K 6 / K E R R S 1 7 > , NODRUMO O U O L E P R E C I S I O N C O , C 1 , C , S O M . C O E D . C O V , S O D , F K G , T E H A , T E M E , T J P O O O U B L E P R E C I S I O N Y , A S T , T E M P , S I
-I N O E X 1 1 1 , 1 2 1 = NUMB E R * ( M I NO ( I I , I 2 ) - 1 ) » M A X 0 ( I 1 , I 2 ) - I M I N 0 ( I 1 , 1 2 ) * l ( M INO(11,121-111/2
H A X I N U = I N O E X ( N U M B E R , N U M B E R I L 0 N G = 2 * L E N G T H N U M B = J M L T M X . J M L T M X / 2
R E W I N D 1 R E W I N D J0 0 1 0 1 1 1 * 1 , N U MB E R R E A D I I H C H I I , 1 * 1 , L E N G T H )
1 0 0 C A L L M Y I 0 I 4 , 1 , 0 , 0 )R E W I N D 2 00 101 12*1,11 I F I I 2 . L T . N C D R U M ) GO T O 1 4 4 7 R E A D 1 2 ) ( C 0 ( l ) , l = l , L E N G T H )GO TO 1 4 4 8
1 4 4 7 C A L L M Y I O ( 4 . 3 . C O , L O N G 1I F ( K E R R S U ) . E Q . 0 ) G 0 T O 1 4 4 8 W R I 1 E ( 6 . 5 0 0 0 1 K E R R S
5 0 0 0 F OR MA T I 7 I 5 1 S T O P
T H E N Q N - O R T H O N Q R M A L l Z E O H A M I L T O N I A N P R O D U C T S A R E F O R M E D I N C O V , MlT H T H E SAME T Y P E OF M - S E L E C T I O N R U L E S A S F O R C O V
1 4 4 8 0 0 1 0 2 M = l , J M A X V C O V ( M 1 * 0 . 0 0 N C M = N C M A X ( M IDO 1 0 2 1 = 1 , NCM N U M X = N U ( M 1 ♦ I
1 0 2 C 0 V ( M ) * C 0 V ( M ) + C 1 ( N U M X ) * C 0 I N U M X )1 0 1 MR I T E ( 3 ) ( C O V ( M ) , H * 1 , J M A X V )
A G A I N T H E T R I A N G U L A R E Q U A T I O N S A R E S O L V E D , T H I S T I M E F O R T H E N O N - U R T H O N O R M A L I Z E O M A T R I X E L E M E N T S , V
R E W I N D 6 R E W I N D 9DO 2 5 1 M = l , J M A X V N X = N J ( M lI F ( N X . E O . 0 ) GO TO 2 5 1 R E AO ( 9 1 ( I N D U l , J = 1 . N U M B E R )
1 4 4 9 R E A D ( 8 ) ( ( A S T I I , J ) , J = 1 . N U M B ) , 1 = 1 , N U M B )R E W I N D 3
191
o o
0 0 1 5 I T l » l . N U M B E R 0 0 1 5 I T 2 » 1 , N U M B E R I F ( I T 2 . G T . I T 1 ) G 0 T C 1 5 R E A D ! 3 1 ( C O V I J ) , J = 1 .JMAXV) K M N D E X I I T 1 , I T 2 )Y!K)=O.UO 00 103 I = 1» JMAXV 103 YIK)=YIK)+C0E01M,1)*C0VII) 15 H1IT1,IT2)=0.T H E T R A N S F O R M A T I O N M A T R I X A S T I S T H E N A P P L I E D T O T H E H A M I L T O N I A N H A T R I X TO Y I E L D T H E H A T R I X B E T W E E N O R T H O N O R M A L I Z E O S T A T E S
DO 1 6 2 1 1 = 1 , N U M B E RI F U N U U l > . E Q . O I G O T O 1 6 2DO 1 6 1 1 2 = 1 , 1 1I F I I N O I 1 2 ) . E Q . O I G O T O 1 6 10 0 1 7 2 I S 1 = I , N U M B E RI F ( I N D I I S I I . E C . O I G O T O 1 7 20 0 1 7 1 I S 2 = 1 , N U M B E R I F U N D t I S 2 > . E C . O I G O T O 1 7 11 * I N O E X ( I S I . I S 2 )
1 7 H i u , ! 2 1 = H ( I L , 1 2 ) + A S T ( I I , I S 1 1 * A S T U 2 , I S 2 ) * Y ( I )1 7 1 C O N T I N U E1 7 2 C O N T I N U E
1 6 H I 1 2 , 1 1 ) = H I I I , 1 2 )1 6 1 C O N T I N U E1 6 2 C O N T I N U E
K F C R C f c = 2 * l J F O R C E - l )1 = 00 0 3 I 1 = 1 , N U M B E R .I F 11 MO t I L > . E Q . O I G O T O 31 = 1 + 1 J = 00 0 1 J l = l , N U M B E R I F ( l N O U l ) . E Q . O I G O T O 1 J = J + lH I I , J ) = H < 1 1 , J 1 1 I F I J . E Q . N X I GO T O 2
1 C O N T I N U E2 I F ( I . E O . N X I GO TO 43 C O N T I N U E4 W R I T E 1 1 0 , 1 0 0 0 1 J T W , N N N , K F O R C E , M , N X , I ( H I I . J ) , J - l . N X > , ! • 1 , N X >
W R I T E ! 6 , 1 0 0 0 1 J T W , N N N , K F 0 R C E , M , N X , « ( H ( I , J ) , J » 1 . N X ) , I - 1 , N X )1 0 0 0 F O R M A T U X , 1 2 , 1 1 , 3 1 2 , 5 E 1 4 . 7 / I 1 0 X , 5 E 1 4 . 7 ) )
2 5 1 C O N U N U E
R E T U R NEND
I
OU
U
S U B R O U T I N E C 0 D 2 L ( K U K , N E M P , N N N ) O I M E N S I O N K U K I B l L E H P - N E M P DO 1 1 = 1 , NNN J = 6 * t + 6
1 K U K I I ) = I S H I F T U S H I F T ( L E H P , J ) , - 5 4 )R E T U R NE N D
S U B R O U T I N E L 2 C 0 D I K U K , N E M P , N N N )D I M E N S I O N K U K ( B )I F I N N N . E Q . 8 1 G 0 TO 2 N N N P 1 = N N N + 1 0 0 1 I = N N N P 1 , 8
1 K U K I I 1 = 02 NEMP=KUKI8) + ISHIFT(KUK(7),61♦ISHIFT(KUK(6)»t2)+ISHIFT(KUK(51,18)4 1ISHIF T(KUK(4I,24) + ISHIFT 1KUK(31,30)♦!SHIFT(KUKt 2),36)+ISHIFTIKUKt 21) ,42)R E T U R NENO
SUBROUTINE ARRANGIMS, IC)OIMENSION NCHAX1351,CO 16700),C116700),SOMI20),COEOI35,35>.NU<35). 1KUKI8), NOD 16700),INDI70),MSI 101, COVI35>,SI10),NJI35)COMMON NNN,JFCRCE,NUMBERCOMMON/BLK 2/CO,C1,SOM,COEO,COV,NCHAX,NU,KUK,NOD,I NO,MS,1S.NJDOUBLE PRECISICN CO,Cl,C,SQM,COEO,COV,SOD,FKG,TEMA,TEME,T J P O
T H E A R R A Y MS I S P L A C E O I N O R D E R S O T H A T T H E L A R G E S T T E R M I S H S f l ) . IF. A N Y TWO T E R M S A R E E Q U A L I C I S G I V E N T H E V A L U E Z E R O . O T H E R W I S E 1 C I S - 1 F O R AN ODD P E R M U T A T I O N , AND + 1 F O R AN E V E N P E R M U T A T I O N
NNNMl=NNN-lIC-17 1 = 12 IF(MStt)-MSII+l))6,4,l1 IF 1 1—NNNM1) 3 , 5 , 53 1=1+1 GO TO 26 N=HSI 11MSII I=MS11+1)
MS 1 1 + I ) = N IC=-IC GO TO 74 IC-05 R E T U R N ENO 162
non
S U B R O U T I N E S E A R C H I N E M P . M . L E Q )01 PENSION NCMAX(35)<C0(6700),C1(6700),SQM(20),C0ED(35»35),NU(35), 1KUX(8).NOOI6700), I NO(70).MS 110),COV(35),S( 10),NJ(35>COMMON N N N . J F C R C E . N U M B E RC O M M O N / B L K 1 / T J P O . J T W . N U M J , L N G T O T . N N N M 1 , N S P S . N S P S P 1 » J M A X V . J M A X M O .
1 L E N G T H . L N G T T 1 , J M L T M X . N N N P l , J T W 1 2 C O M M O N / B L K 2 / C O , C 1 , SQM., C O E D , C O V , N C M A X , N U . K U K , N O D , I N O , M S ,
1 S . N JD O U B L E P R E C I S I O N C O , C 1 , C , S Q M , C O E O . C O V , S O O , F K G , T E M A , T E M E , T J P O
T H E WORD N E MP I S S E A R C H E D F O R I N T H E L I S T N O O . I F T H E WORD I S L A R G E R T H A N T H A T I N T H E L I S T T H E S E A R C H S K I P S 1 0 P O S I T I O N S , I F S M A L L E R , T H E S E A R C H I S C O N T I N U E O B A C K W A R D , U N T I L T H E WORD I S F O U N D I N P O S I T I O N L E Q
M E M P = N E M P I B = N U ( M ) * 1 I E = l B * N C H A X ( M ) - l I E Q = 1
6 I F ( N E M P - N 0 0 t I B ) ) l , 2 , 3
3 I B = I B ♦ 1 0 L E C = L E Q * 1 0 I F ( I B - I E ) 6 , 6 , 1
1 DO 5 1 = 1 , 9 L = I1 E N = I B - II F ( NE M P - N O O ( I E N ) ) 5 , 7 , 5
5 C O N T I N U E
W R I T E ( 6 , 1 0 ) M , L E Q , L 1 0 F 0 R M A T ( 3 ! 6 , 2 2 H E Q U A L I T Y - S E A R C H - F A I L E O )
W R I T E ( 6 , 2 1 2 1 1 NE MP 2 1 2 1 F 0 R M A T I I 1 7 )
0 0 2 2 1 = 1 , L E N G T H 2 2 W K I T E I 6 , 2 2 2 4 ) 1 , N O O ( I )
2 2 2 4 F O R M A T ( 1 5 , 1 1 7 )S T O P
7 L E Q = L E Q - L2 R E T U R N
E N D
S U B R O U T I N E M Y I O I I T A P E , M A N A G E , A R R A Y , N U M B E R )D I M E N S I O N A R R A Y ( N U M B E R )C 0 M M 0 N / B L K 6 / K E R R S ( 7 ) . N O O R U M
GO TO ( 1 , 2 , 3 ) . M A N A G E1 C A L L D R U M I 0 ( 5 L T A P E 4 , 6 L R E W I N 0 , 0 . 0 , 0 , I R E S P , K E R R S )
R E T U R N
2 C A L L 0 R U M I 0 ( 5 L T A P E 4 , 5 L U R I T E , 0 , N U M B E R , A R R A Y . I R E S P , K E R R S )R E T U R N
3 C A L L 0 R U M 1 0 1 5 L T A P E 4 , 4 L R E A 0 . 0 , N U M B E R , A R R A Y , I R E S P , K E R R S ) R E T U R NE N D
ONO J
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m c s n o o o o o o o o o nZ c * C O O O O O O O O O OO V 1 2 Z 2 Z 2 Z Z Z Z Z Z
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O LWrg »— w * - o t—S3) O is; O fs> o rg O X' o o o o o o o o o o o o o ^ *+ * oCC CD
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491o o o o o o o o o o o o o o o o o o o o o o o o o o o o oX X X X X X X X D O X X X T J X X X X X X X X X X X X X X X X X T r T t T t t X I C T ^ C ^ J t - C T - C - C C I - C T C ^ T C t o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * 3 0 0 0 0 < A < o x a a a ( c a o c t a c c - g * g - g - g - g * * w * g -+i'~j'~4OO0-oO0‘0' p'OOftiNO'XXwMw'C'JCLgOu'XuMM'O'XfflgO'ViXwQ O C Q Q O n Q o o r * ) r . a n n n n r » n r » f t r t r v r t n r t / > « i ^
T h i s c o de w a s d e s i g n e d f o r u s e on t h e C o n t r o l D a t a
C o r p o r a t i o n 3 40 0 at T e l - A v i v U n i v e r s i t y , Is r a e l . T h e r e q u i r e d
i n p u t is as f o l l o w s :
(a) A t i t l e c a r d to be p r i n t e d o n the o u t p u t (80 c o l u m n s ) .
(b) J T W , N N N - T w i c e th e p a r t i c l e a n g u l a r m o m e n t u m (JTW),
a n d th e n u m b e r of p a r t i c l e s (NNN) i n (213) f o r m a t .n
(c) NJ - T h e m u l t i p l i c i t y o f J - s t a t e s in th e (/) c o n
f i g u r a t i o n i n (2413) f o r m a t .
(d) A N G L E - T h e a r r a y o f l i n e a r m i x i n g p a r a m e t e r s f o r
t h e p a i r i n g a n d q u a d r u p o l e f o r c e s . A N G L E = 0 c o r r e s p o n d s to p u r e
p a i r i n g a n d A N G L E = 1 to a p u r e q u a d r u p o l e f o r c e .
(e) H - O u t p u t f r o m the p r o j e c t i o n c o de f o r e a c h v a l u e
o f th e t w o - b o d y a n g u l a r m o m e n t u m .
(f) T I T L E S - 20 c o l u m n t i t l e s a p p e a r i n g on t h e e n e r g y
l e v e l d i a g r a m s f o r e a c h v a l u e of th e m i x i n g p a r a m e t e r s u p p l i e d
t o A N G L E .
P r o c e s s i n g p r o c e e d s as f o l l o w s:
(a) T h e f i r s t t h r e e d a t a c a r d s a r e r e a d in.
(b) R a c a h c o e f f i c i e n t s a r e c a l c u l a t e d f o r the q u a d r u p o l e
f o r c e .
(c) A N G L E is r e a d in.
(d) T h e p a r a m e t e r i z e d H a m i l t o n i a n is r e a d in, a n d th e
p a i r i n g a n d q u a d r u p o l e f o r c e s a r e c o m b i n e d .
(e) T I T L E S a r e r e a d in.
(f) T h e H a m i l t o n i a n f o r e a c h v a l u e o f t o t a l a n g u l a r m o
m e n t u m is d i a g o n a l i z e d b y the g e n e r a l p u r p o s e s u b r o u t i n e H D I A G
165
2 . A n a ly s is Code
(v. A p p e n d i x B - I I I ) . If A N G L E = 0 th e e i g e n v e c t o r s a r e s t o r e d
f o r u s e i n s e n i o r i t y a n a l y s e s . If A N G L E = 0 , th e o v e r l a p s w i t h
t h o s e f o r e A N G L E = 0 ar e c a l c u l a t e d , r e s u l t i n g in t h e s e n i o r i t y
c o m p o s i t i o n s o f th e e i g e n s t a t e s of th e H a m i l t o n i a n .
(g) E i g e n v a l u e s ar e s t o r e d f o r t r a n s m i s s i o n to the
p l o t t i n g r o u t i n e .
(h) D I A G R M is c a l l e d , w h i c h g e n e r a t e s o u t p u t i n th e
f o r m o f e n e r g y l e v e l d i a g r a m s (v. A p p e n d i x B - I I I ) .
166
PROGRAM SPECTRUM 01 MENS I ON V(13.13 >,U(13,13),PIVOT<13>,IP IV0TI13), T ( 201 01 MENS I ON H(13,13).ENERGY(28,100),NJ(28),W(10)•SENIOR<1 3 , 5 )0 1 ME NS I ON P L O T ( 1 5 0 0 ) , J C O U N T I 1 5 0 0 )D I M E N S I O N P R 0 8 L M ( 2 0 ) » T I T L E S ( 5 , 1 0 )0 1 MEN S I ON A N G L E ( 9 ) , N O S T A T I 2 8 , 5 ) , N U M S R I 2 B )D I M E N S I O N O C C U P Y ( 1 3 )L O G I C A L F I R S T , O C C U P Y
CC G ( F L ) = - F L * I F L + 1 . ) / S O R T I C 2 . * F l - l . ) * F L * ( F L * l . » ♦ ( 2 . * F L * 3 . »>
CF I R S T * . T R U E .
1 0 0 0 R E A 0 ( S , l 0 0 9 ) l P R 0 B L M ( n » I = l , 2 0 )1 C 0 9 F O R M A T ( 2 0 A 4 )
I F ( E O F , 5 1 7 3 , 4 0 0 4 0 0 W R I T E ( 6 . 1 0 0 2 1 P R O 0 L M
1 0 0 2 F O R M A T ! 1 H 1 . 2 0 X . 2 0 A 4 I R E A 0 1 5 . 1 0 0 1 ) J T W . N N N
1 0 0 1 F O R M A T I 2 4 I 3 )S P J = F L 0 A T ( J T W 1 / 2 .J T W 1 2 = ( J T W . D / 2 T J P t = J T w M F J T W 1 2 = J T W 1 2 M O O U L = M O O ( N N N , 2 ) ■0 1 V I D E " F L O A T ( ( N N N - M O O U L I * ! J T W - N N N - M 0 0 U L + 3 ) > / I 4 . * F J T W 1 2 )
CJ M A X V " ! N N N * ( J T W - N N N + 1 ) * 2 ) / 2 R E A D I 5 , 1 0 0 1 ) I N J ( J ) , J « l . J M A X V )j m l t m . x = o NUMBE R = 0 0 0 3 9 J = l , J M A X V N X = N J ( J )J M L T M X " M A X O t J H L T M X . N X )
3 9 N U M B E R = N U M B E R * N X
c0 0 I I J * 1 . J T W 1 2 T 0 I J = 2 * ( J - l )
1 1 W ( J ) * R A C A H ( S P J , S P J . S P J , S P J , 2 . , T 0 T J )WI 7 ) " 0 .R E 0 Y 2 = 0 .L " J T W1 20 0 1 2 L L “ 1 , 2 F L » LR E D Y 2 " R E 0 Y 2 * ( - 1 . ) * * L * S Q R T C 2 . * F L * 1 . ) * R A C A H C S P J , F L , S P J , F L . . 5 , 2 .
C G ( F L )1 2 L » L - 1
R E D Y 2 = ( R E 0 Y 2 * T J P 1 ) * * 2
CI C 0 U N T " 0 N UMP L T " 11 U N I T * 5R E AD I 5 . 1 0 2 5 ) ( A N G L E I J ) . J - 1 , 4 )
1 0 2 5 F O R M A T ( 4 F 1 0 . 4 )DO 6 6 I A N G L E * 1 . 4 R E W I N D 3G P A I R * I A N G L E < I A N G L E ) - l . ) / D I V I D E G Q U A D * A N G L E 1 1 A N G L E ) * R E 0 Y 2
CC C O M B I N E T H E M A T R I C E S F O R D I F F E R E N T J - C O U P I I N G
0 0 5 J F O R C E " I , J T W 1 2 T E M P 2 = 0 .W C O E F = W ( J F O R C E )
oo
D O 5 J - l , J M A X V N X " N J ( J )I F ( N X . E Q . O ) G O T O 5
C
R E A D U U N I T , 1 0 0 3 X ( H ( 1 1 , 1 2 ) , I l - l . N X ) , 1 2 - l . N X )1 0 0 3 F 0 R M A T ( 1 0 X , 5 E 1 4 . 7 )
2 I F ( I U N I T . E Q . 5 ) W R I T E ( 3 , 1 0 0 3 ) ( ( H ( I 1 , I 2 ) , I 1 " 1 , N X ) , I 2 " 1 , N X )
0 0 4 1 1 = 1 , NX K - l l - N X DO 4 1 2 = 1 , 1 1 K = K + N X —I 2 * 1 I E M P 1 = H ( I 1 , 1 2 )
I F ( J F O R C E . N E . I ) G 0 TO 3 E N E R G Y ! J , K ) = 0 .
. I E M P 2 = T E M P 1 * G P A I R3 T E M P I = T E M P 1 * G C U A 0 * W C 0 E F * T E M P 24 E N E R G Y ! J , K ) = T E M P I * E N E R G Y t J , X >5 C O N T I N U E
I F ( F I R S T ) R E A D ( 5 , 1 0 0 9 ) ( T I T L E S ( ! , N U M P L T ) , 1 " 1 , 5 )W R I T E ( 6 , 1 0 0 4 ) ( T I T L E S ! I . N U M P L T I , 1 = 1 , 5 )
1 0 0 4 F O R M A T ! 1 H 0 . 5 A 4 )c
0 0 6 J = l , J M A X V N X = N J ( J )I F ( N X . E O . O ) G O TO 6 MOMEN T = J * 1 1 * M 0 0 U L ) - 1
C A L C U A L T E E I G E N V A L U E S 4 3 M A X 1 N 0 = ( N X * ( N X + 1 ) ) / 2
I F I N X . N E . D G O TO 3 3 3 H I 1 , 1 ) = E N E R G Y ( J , 1 )GO T O 4 4
3 3 3 K = 0 0 0 7 1 1 = 1 , NX 0 0 7 1 2 = 1 1 , NX K = K * 1
7 H I I I , I 2 ) = E N E R G Y ( J , K )1 0 2 0 F O R M A T ( 1 X . 9 E 1 3 . S )
C A L L HO I A G ( H , N X , 0 , U , N R , P I V O T , I P I V O T « 1 3 )
3 3 4 I F ( I A N G L E . NE • 1 1 GO TO 4 5
W R I I E ( 3 ) ( ( U ( I 1 , I 2 ) , I 2 = 1 , N X ) , I I = 1 , N X )0 0 9 4 1 1 = 1 , NX
9 4 O C C U P Y I l l l = . F A L S E .KK = 00 0 9 5 1 1 = 1 , NX I F ( O C C U P Y I 1 1 ) ) G 0 TO 9 5 K 0 U N T R = 1 T E S T = H ( 1 1 , 1 1 )I 1 P I = 1 1 * 10 0 9 6 1 2 = 1 1 P I , N XI F I O C C U P Y I 1 2 ) I G O T O 9 6! F I A B S ( T E S T - H ( I 2 , I 2 ) ) . G T • l . E - 3 I G O T O 9 6K 0 U N T R = K 0 U N T R * 10 C C U P Y ( I 2 ) * . T R U E .
9 6 C O N T I N U E KK = K K ♦ 1N O S T A T ( J , K K ) = K O U N T R
9 5 C O N T I N U E N U M S R I J ) = KK 0 0 8 0 U > 1 , N X
o o
DO 8 0 1 2 = 1 1 , NX S U M - 0 .0 0 8 1 K » 1 » NX
8 1 S U M - S U M * U ( K , l l ) * U ( K , 1 2 )V ( I I , 1 2 1 - S U M
8 0 V ! I 2 , m = S U M DO 8 2 1 1 “ 1 , NX82 W R I T E 16, 1020)(VI11 ,12 ) , 12-1, NX)GO TO 4 4C
4 5 R E A O ( 3 ) I ( V I 1 1 , 1 2 ) , 1 2 = 1 , N X ) , 1 1 = 1 , N X )0 0 9 7 1 1 - 1 , NX K K - N U M S R I J l 12-00 0 9 7 1 = 1 , KK K K K - N O S T A T ( J , I )S U K 2 - 0 .0 0 4 6 1 = 1 , K K K 12=12*1 S U M 1 - 0 .0 0 4 7 K - l . N X
4 7 S U M 1 = S U M 1 » U ( K , I I ) * V ( K , I 2 )4 6 S U M2 = S U M . 2 * S U M 1 * S U M I 9 7 S E N I 0 R 1 I 1 . 1 I - S U M 2
S T O R E E I G E N V A L U E S I N T H E A R R A Y P L O T 4 4 DO 8 1 = 1 , N X
I C O U N T - I C O U N T ♦ I P L O T ! I C O U N T ) * H I I , I )
8 J C O U N T I I C O U N T I - M O M E N TI F I I A N G L E . E C . l . O R . N X . E Q . D G O T O 6 W R I T E 1 6 , 1 O S O ) MOMENT
1 0 5 0 F O R M A T 1 / 4 H * J = , 1 3 )I F I M O G U L . E C . 1 ) W R I T £ 1 6 , 1 0 5 1 )
1 0 5 1 F O R M A T t l H * , 6 X , 2 h / 2 / )I F ( M O G U L . EC*. 0 ) W R I T E ! 6 , 1 0 5 2 )
1 0 5 2 F O R M A T ( 1 H * / 1 DO 4 8 1 1 = 1 , NX
4 8 W R I T E ( 6 , 1 0 1 8 ) 1 1,I S E N I O R I 1 1,12) ,1 2 = 1 , K K )1 0 1 8 F 0 ? . M . A T 1 1 X , I 2 , 1 X , 1 3 F 9 . 5 )
6 C O N T I N U EC
K = ( N U M P L T - 1 > * N U M B E R + 1 I F I H O O U L . E C . D G O TO 5 0 W R I T E 1 6 , 1 0 0 7 )
1 0 0 7 F O R M A T 1 1 H 0 , 6 1 3 X . 6 H E N E R G Y , 7 X , 1 H J , 3 X ) )WRITE<6,1006)I PLOT 1J),JCOUNT( J ) , J = K , ICOUNT)1 0 0 6 F 0 R M A T I 6 I E 1 5 . 7 , 1 3 , 2 X 1 1
GO TO 5 15 0 W R I T E I 6 . 1 0 1 7 )
1 0 1 7 F O R M A T 1 1 H O , 6 I 3 X , 6 H E N E R G Y , 8 X , I H J , 2 X ) )W R I T E 1 6 , 1 0 1 6 1 I P L O T ( J 1 , J C O U N T ( J ) , J - K , I C O U N T )
1 0 1 6 F O R M A T ( 6 1 E 1 5 . 7 , I 3 , 2 H / 2 ) )5 1 I U N I T - 36 5 N U M P L T = N U M P L T « 16 6 C O N T I N U E 1C
N U M P A C - N U M B E R / 1 0 * 15 2 C A L L 0 1 A G R M I P R O B L M . N U M P A G , T I T L E S , N U M P L T - 1 , N U M B E R , P L O T , J C O U N T ,
M C O U L • 3 )CF I R S T = . F A L S E .
GO TO 1000 73 STOP ENO
H1cnCD
A P P E N D I X B - I I
P A R T I C L E - H O L E C O M P U T E R C O D E
1. I n p u t C o d e
T h i s c o d e w a s d e s i g n e d f o r u s e o n th e I B M 709*1/70*10
D i r e c t C o u p l e d S y s t e m at Y a l e U n i v e r s i t y . It g e n e r a t e s
i n p u t m a t r i x e l e m e n t s of th e o n e - a n d t w o - b o d y p a r t s of
th e H a m i l t o n i a n a n d th e J + o p e r a t o r , i n th e H a r t r e e - F o c k
s c h e m e , i. e . , it t r a n s f o r m s t h e s e o p e r a t o r s f r o m a s h e l l
m o d e l b a s i s to th e H a r t r e e - F o c k b a s i s .
P r o c e s s i n g p r o c e e d s as f o l l o w s:
(a) J N U M , N S P S a r e r e a d in. J N U M is th e n u m b e r of
d i f f e r e n t s i n g l e - p a r t i c l e a n g u l a r m o m e n t a in the s h e l l m o d e l
b a s i s ; N S P S t h e n u m b e r of s i n g l e - p a r t i c l e s t a t e s fo r n e u t r o n s
a n d / o r p r o t o n s .
(b) A l i s t of C l e b s c h - G o r d a n c o e f f i c i e n t s is g e n e r a t e d .
(c) T w o - b o d y m a t r i x e l e m e n t s i n th e s h e l l m o d e l b a s i s
a r e r e a d i n e i t h e r f r o m c a r d s o r a t a p e , c o n t r o l l e d b y a
c o n t r o l ca rd , th e f i r s t f o u r c o l u m n s of w h i c h sa y C A R D o r TA PE.
(d) P R O B L M - a n 80 c o l u m n t i t l e to b e p r i n t e d o n th e
o u t p u t , is r e a d in.
(e) M C O D E - a n a r r a y g i v i n g th e M - v a l u e s o f th e i n d i v i d u a l
p a r t i c l e s t a t e s , is r e a d in.
(f) T h e H a r t r e e - F o c k b a s i s is c h e c k e d f o r p r t h o n o r m a l i t y
to th e a c c u r a c y o f t h i s m a c h i n e . T h e S c h m i d t p r o c e s s is e m p l o y e d .
(g) O n e - b o d y e n e r g i e s a r e r e a d in.
169
(h) T h e t r a n s f o r m a t i o n to th e H a r t r e e - F o c k b a s i s is
p e r f o r m e d .
(i) O n e - a n d t w o - b o d y m a t r i x e l e m e n t s a n d t h o s e o f J + ,
a l l i n th e H a r t r e e - F o c k b a s i s a r e o u t p u t to b e r e a d i n b y
P H E X C I T , w h i c h f o l l o w s .
170
D I ME NS 1 3 N R 0 0 T ( 4 , 8 ) , I M U L T ( 2 3 ) , M V A L U ( 8 , 6 3 )D I M E N S I ON C I 4 , 2 3 > , M C 3 D E ( 2 O ) , H F J P L S ! 2 0 , 2 3 ) . H F l B 0 V C 2 0 , 2 0 ) D I M E N S I O N D ! 4 , 2 3 ) , Y ( 2 1 0 > , A S T I 4 0 0 ) , P R 0 B I M ( 2 0 )0 1 ME N S I DY M V A L U E ! 8 , 5 3 , 2 > . N J ( 8 ) , V 3 ! 4 , 4 , 4 , 4 , 8 ) , V I ( 4 , 4 , 4 , 4 , 8 > , T ( 4 ) 0 I M E N S I 3 N B 0 Y 2 1 ! 1 8 3 3 ) , B 0 Y 2 2 ! 1 5 9 6 ) , B D Y 2 3 ( 1 0 8 1 ) , B D Y 2 4 ! 6 1 1 ) ,
8 0 Y 2 5 I 2 1 0 ) , B D Y 2 6 < 5 5 ) , 3 D Y 2 7 ( L O ) , B 0 Y 2 8 ( 1 )COMMON C C < 1 3 , 8 , 5 4 )03J3L E PRECISION C,0,Y,AST,SCAL4R,SJMI,SUM2,TEMP DOJ3LE PRECISION FJ,FM1 ,FM2,FFM,HFJPLS 03J3LE PRECISION ROOTO A T A C A R 0 , T A P E / 4 H C A R D , 4 H T A P E /R E W I N D 4R E A 0 ( 5 , 1 3 3 1 ) I S K I PI F ! I S X I P . G T . 3 ) C A L L R E A D E R ! I S K I P )R E A D ! 5 , 1 3 3 1 1 J N U M , N S P S
1 3 3 1 F 3 R M A M 2 4 I 3 )S 3 R T 2 = S 3 R T ( 2 . ) j n j m s u = j n u m « j n u m
j m A X M = 2 * J N U Mm a d j s t = j m a x m u
ML I M I T = J M A X M - 1 C A L L C C C O E F ( J N U M )R E A 3 I 5 , 1 3 3 3 )HOW I F I H 3 W . E 3 . C A R D I RE AD C 5 , 1 3 3 4 )
1 ( ( 11 I Y D ( J l , J 2 , J 3 , J 4 , J ) , V 1 ( J 1 , J 2 , J 3 , J 4 , J ) , J » l , J M A X M ) , 2 J 4 = I , J N J M ) , J 3 = 1 , J N U M ) , J 2 = l , J N U M ) , J l ° l , J N U M )
1 3 3 4 F D R M A M 5 E 1 6 . 8 )I F ( M 3 w . E 3 . T A P E ) R E A O ( 3 ) V 3 , V l R E A O I 5 , 1 3 0 2 ) S T R N T H 0 0 7 3 J 1 = 1 , J N U M 0 0 7 3 J 2 = 1 , J NUM
7 1 T E M P I = S T R ' I T HI F ! J 1 . E 3 . J 2 I T E M P 1 = 2 . ♦ S T R N T H 0 0 7 3 J 3 = 1 » J NUM 0 3 7 3 J 4 = l , J N U M T E M ? = T E M P lI F ! J 3 . E 3 . J 4 ) T E M P = T E M P 1 * 2 .DO 7 3 J = 1 , J M A X MV 3 I J l , J 2 , J 3 , J 4 , J ) = V 3 1 J l , J 2 , J 3 , J 4 , J ) * T E M P Y 1 ( J 1 , J 2 , J 3 , J 4 , J ) = V 1 ( J 1 , J 2 , J 3 , J 4 , J ) * T E H P
7 3 V 3 ( J 1 , J 2 , J 3 , J 4 , J ) = ( V 0 ( J 1 , J 2 , J 3 , J 4 , J ) * V 1 ( J 1 , J 2 , J 3 , J 4 , J ) ) * . 5CN S P S P = N S P S * N S P S N S P S ? 1 = N S P S P * 1
5 3 0 3 R E A 3 ( 5 , I 3 3 0 1 P R O B L M 1 0 3 3 F ORMA T ( 2 D A 4 )
w R I I E I 6 , 1 3 5 0 ) P R O B L M 1 3 5 3 F O R M A T ! I . H 1 , 2 3 X , 2 0 A 4 )
R E A D ! 5 , 1 3 3 1 I ( M C 3 0 E ( N ) , N » 1 , N S P S )N I C E = 30 3 5 5 N = 1 , N S P S N I C E = N I C E » N S P S P 1
6 5 I M J L T ( N ) = N I C E00 54 J = I,JNUM ,F J = J * J0 0 5 4 M = I , J MAXMF F M = M - J N U MT E M P s F J - F F M * F F MI F ( I F M P . L E . 3 . ) G 3 TO 6 3R O O M J , M ) = D S 3 R T I T E M P IGO T 3 6 4
o o
63 ROOT(J,M)«0.64 CONTINUEC LOWl“1 LI Ml = NSPS 1X1 = 3 L0W2=1 LI M2 = NSPS 1X2 = 3WRITE(6,1010)1313 FORMAT!47H3HARTREE-F3CX (OEFORKEO) SINGLE PARTICLE STATES)00 139 ISPlN-1,3C REMOVE NEXT CARO IF PROTON 1-B03Y .NE. NEUTRON 1-BODY ENERGIESC IF!ISPIN.E3.21G3 TO 102IF! ISPIN.E0.21G3 TO 102GO TO (801,802,99),ISP1N 831 WRITE( 6, 1320)(J,J = l,ML I MIT,2)1323 FORMAT!I3H3STATE NO.,2X,4!11X,2HJ«,11,2H/2))00 I N=I,NSPS I REAOI5,1002)(0!J,N),J°1,JNUM)1002 FORMATI4F10.4)ORTHONORMALI2E THE SINGLE PARTICLE SPACE 00 435 N=1,MLIMIT,2 INDE X =3 KOUNT = 300 432 Nl=l,NSPS H1=MC3DE(.N1)IF(Ml.E3.N1KOUNT=KOJNT*l DO 432 N2=N1,NSPS 1N0EX=IN0EX*I SCALAR»3.IF(M1.NE.N.0R.MC30E(N2).NE.N)G0 TO 402 00 431 J = I,JNUM401 SCALAR=SCALAR*OIJ,Nl)*D(J,N2)402 Y{ IN3EX)=SCALARCALI SCHMITtXOUNT,NSPS,Y,AST)00 405 N1=1,NSPS,2 IF(MC0DE(N1).NE.N)G3 TO 405 00 433 J = I,JNUM SJM1=3.SJM2=0.NJM3ER=Nl-NSPS 00 434 N2=1,NSPS NUM3ER=NUMBER»NSPS IFIMC00EIN2).NE.N)GO TO 404 TEMiP = AST( NUMBER)SJM1 =SdMUTEMP*D( J,N2)SJM2 = SJM2HEMP*0( J,N2*1I404 CONTINUEC<J,NI)=SUM1403 C(J,Nl«l)=SUM2405 CONTINUE00 436 N=1,NSPS 4 36 WRITE(6, 1005 IN,MCODE(N),IC(J,N),J>1,JNUM)1035 FORMAT!16,3X.3H X =,12,2H/2,4E16.7)
C SET JP I HE M VALUE OF THE 2-BODY STATES99 00 103 M«l,JMAXM 103 NJ!MI= 3
03 105 91=1,9SPS M1»MC0DE(N1)MVALJI=IHULTTN1)03 135 92=L3W2,LIH2 IFI 91.EQ.92)GO 13 105 992=92-1<2M=(Ml♦m; 0DE(N92))/2*l IFtM.LT.llS3 TO 105 9'JM3ER = 9J(M)»1 9 J I M ) = 9 J M B E R MV41JE< M,9UMBER,1)=91 MVALJEIM,9UMBER,2)=92 MVAL J( M, 9UM8ER) =MVALU1*92 105 C39TI9UEIF(ISPI9.EQ.31G3 TO 85 832 READI5,1D32>ITU>,3 = 1,390*1)I F t ISPM.£3.2)33 TO 87 85 HR I (El 5,1006)IT(J) , J = l,J9UM)1335 FORMAT 141H09EUTR09 SI95LE PARTI OLE E9ER3IES - J=1/2,F8.4,5X.7H .3/2,F8.4,5x,7H J«5/2,F8.4,5X,7H J=7/2,F8. 4 , 5X, 5H(MEV)),<R ITE ( 5,1001 19UM3ER 30 T3 85 87 „R I TE(6, 1308)(T(J) , J«I,J9UM)1303 FORMAT( 41H0PR3T09 SI93LE PARTICLE ENERGIES - J« 1/2• F8.4,5X,•7H J=3/2,F8.4»5X,7H J=5/2,F8.4,5X, 7H J=7/2,FB.4,5X, 5H(MEV)»CC BES19 CALCJLAU9G VARIOUS MATRIX ELEMENTS85 03 2 91=L0W1,LIM1 N91=9I-I<1 mi=m; ooein91i M MI =(Ml.MAOJST)/2 MMM1=( IABSMl) + l )/2 9PRIM1=IMULT(91)IMP1=JMAXM*(MMl-l)03 2 92=L0rf2,LlM2 992=92-1<2 H2=MG00E(992)M 92 = tM2.9A0JSTI/2 M 9 92 = ( IAHSIM2)»l)/2 I9?2=I MR I»MM2CC FORM Jf MATRIX ELEMENTSGO T 3 (833,834,32),ISPI9803 TEM»=0.■ IF(92-Ml.9E.2)G3 TO 3 DO 4 J=liJNUM 4 IE9P = IE9P»C( J,91)*C( J,92)*R00TU,MM1)3- HFJPLSI 92, 91 ) = TE9PCC FORM 1-33DY PART OF HAMILTONIAN804 IF(91.Gr.92)GO 13 33 TEMPO.IF19).9E.921G0 13 31 DO 33 J = I, J9UM 30 TEMP=TEMPvttJ,N91)*ClJ, 992 >*T(J). 31 HF130T(99I,N92)=TEM?HF130YI992,991>=TEMP IFIISPIM.E0.21G0 TO 2CC FORM 2-330Y PARI OF HAMILTONIAN32 1FI91.E3.92JG0 TO 233 IF(ISPI9.EQ.2)G3 TO 2
M=(Ml+M2)/2+l IFlM.LT.llGO TO 2 NPRIME = 9PRIMl«-N2 MAX I 9 = 90(M)00 43 NJMB=1,MAXIM 43 IF(MVALJ<M,9UMB).EQ.NPRIME)G0 T3 42 WRITE(6,1234)1234 FORMAT(1 OH STOP 1234)STOP 42 L=9JMR-MAXIM00 23 NJMBER=1, NUMB TEMP3=0.93=MVALJE(M,NUMBER,1)993=93M3=MC00E(9N3)MM3=(M3AMADJSTI/2 . MMM3 =(IAHS(M3)*l)/2 94=MVALJE(M,NUMBER,2)N-9 4 = 94-1 <2 M4 = m;0DE(N.94)MM4=(M4.MA0JST)/2 MMM4=( IABS(M4Ul)/2 IMP4=JMAXM*(MM3-1)+HM4J1CG= 000 53 J1=1,J9UM IEMP1=C(JI.N91I DO 53 J2 = 1,J9UMjicg=j i : g»iIFIJl.Lr.MMMl.0R.J2.LT.MNN2)C0 TO 33 TEMP2=TEM?1*C(J2.NN2)J1PRVM = J102 J 3PR YM =1ABS(J1-J2)J2CGO00 51 J3=l,J9UM TEMP3=TEMP2*C(J3.NN3)DO 51 J 4=1» J9UM J2CG=J2CGtl1F1J3.LT.MMM3.0R.J4.LT.MMM41G0 TO 51TEMP4=TEMP3*C(J4.NN4)J2PRTM0304JMI 9=MAX 01J3PRYM,IABSIJ3-J4))*1 JMAX=MI901J1PRYM,J2PRYM)IF(JMI9.GT.JMAX1G0 TO 51 T EMP = 0.03 6 J=JMIN,JMAX CG11=CG1J1CG,J.1MP2)Ca21=C,IJ2CG.J,IMP6)10 IFMSPI9.E0.3IG3 TO 862TEMP=TEMP*CG11*V1(Jl,J2,J3,J4,J)*CG2l GO TO 6862 TEMP=TEM?.CG11*V0IJl,J2,J3,J4,J)*CG21 6 CONTINUETEMP3=TEMP0*TEMP*TEMP4 51 CONTINUE 53 CONTINUE
25 L = L»MAXIM-9UM8E.R*1GO TD (501,502,503,504,505,506,507,538), 501 8DY21(L)=TEMP0 GO T 3 20
502 80Y22(L)“TEMPO30 TO 20503 BDY2 3 IL)-TEMPO30 TO 20504 30Y241LI * TEMPO30 TO 20505 BDY25IL)8TEMPO30 TO 20505 B0Y25IL)-TEMPO30 TO 20507 80Y27(LI=TEMPOGO TO 20508 B0Y28IL1-TEMPO20 CONTINUE2 CONTINJE30 ro (21.23.82),I SPIN 21 WRITE(6,2002)2002 FORMAT<27H0NEUTRDN 1-BODY HAMILTONIAN)30 TO 2423 nR IIE I 5,2003)2003 FORMAT(26H0PR0T0N 1-BODY HAMILTONIAN)24 WRITEIS,2001)1 (HFIBOYII,J),J*1,NSPS),1*1,NSPS)2001 FORMArI IX,12F10.6)WRITE(4) (IMF1B0Y(I,J),J=1,NSPS),I=1,NSPS)1F(!SPIN.E9.2)G0 TO 102 BI wRirE(6,10ll)(NJ(M),M=l,JM4XM)1011 FORMAT!13HINEUTRON-NEUTRON M-MULTI PL ICITIES,1114)30 TO 8482 WRITE(5,1012)(NJ(HI,M=1,JMAXM)1012 F.ORMAT(33H1NEJTRON->ROTON M-MULTI PL ICITIES,111 4)84 DO 106 M=1,JMAXMMAXIM-NJTM)IFIMAXIM.E0.0130 TO 106wRirE(S, 2005 )MM\/ALJE<M,NUM3ER,I>,I = l , 2 ) tNUM3ER*l,MAXIM) 2005 FORMAT!17(213,IH,))106 CON!INUElFtISPIN.EO.3130 TO 2725 WRITE(6,2004)2004 FORMAT!35H0NEUTR0N-NEUTR0N 2-BODY HAMILTONIAN)30 TO 2927 wRITE(f>.3232!3232 FORMAT!34HONEUTRON-PROTON 2-B00Y HAMILTONIAN)29 WRITEI4) (NJ(M),H=1,JM4XM)00 SI M-l.JMAXM MAXIM-NJIM)IFIMAXIM.EQ.3IG0 TO 61 • MM , M - 1wRIT EI 5, 1007)MM 1007 FORMAT! HO,3HM =,12,17H (UPPER TRIAN3LEI)WRITE!4) I MmU(M,NJM3ER), NUMBER* I, MAXIM)1 0W= 1LI MM MAXIM*I MAXIM* 11 1/2600 30 TO (SOI,602,503,504,605,606,607,608),M601 WRI TEI 6, 200111 B0Y2K < ) ,<«L3W,LIN)701 WRITEI4) (3DY2l(<),<=L0W,LIM) ,30 TO 61602 WRIIE(6,2001)(B0Y22I<),K*LOW,LIM)702 WRITEI4) <BDY22(<),<-L3W,LIM)30 TO 61603 »RITE(6,2O01)(B0Y23(<),<=lOW,LIH)703 WRITEI 4)’ (80Y23(<),<*L0W,LIM)
GO 70 61604 MR I TE(6,2001)(B0Y24I<I, <«LOH,LIM)704 MR ITE14) <8DY24(<),<=L0W,LIM)GO TO 61605 WRITE(6,2001)(B0Y25(<),X*L0W,LIM)705 WRITE(4) (8DY25IO,<=LOW,LIM)GO TO 61606 WRITE(6,20011(B0Y26IR), K-LOW, LIMI706 WRIT c(4) (BOY2 6 ( 0 ,<=LDW,LIM»30 TO 61607 WRITE(6,2001)(B0Y27(<),<*L0W,LIM)WRITEI4) (BDY27<<),<=L3W,LIM)GO TO 61608 WRITE(6,2001)(B0Y28(<),K*L0H,LIM)WRITE14) (BDY28I<),<=LOW,LIM)61 CONTINUE. IF(ISPIN.EQ.3)G0 TO 109 901 HRITE(6,2000)2000 F0RM4TISH1JPLUS)903 WRITEI6,2001)11NFJPLS(I«J),J=liNSPS),1*1,NSPS) WRITE!4) ( ( HFJPLSfI,J),J*1,NSPS),1*1,NSPS)101 LOWl=NSPS*1limi- nspspIKl-NSPS LUW2-L0WI L I M2 = LI M1 IX2=I XI 888 30 (0 109102 LOWl8 1 LIM1-NSPS
U\=0109 CONTINUE GO TO 5000 STOP tND
SJBR3JTINE SCHM t T (NX*NUM3ER,Y»AST) DIMENSION Y(1) ,ASTI NUMBER,11*1NDI20) D3J3LE PRECISION Y,AST,TEHP, SI LOGICAL IND03 5 -I T1 = 1, NUMBER INDIIT1>=.FALSE.DO 5 IT2 = 1,NUMBER5 ASTI ITl,IT2)=0.1 = 111 = 34 <=NJM8ER+U-ll-U*M-3) )/2 IFIYI<).ST.0.0031)C3 TO 6 1 = 1 + 1IF! I.LE.NUMBER)S3 T3 4 WRirEI6,l(1 F ORM A T I 7rl STOP I)STOP
6 ASTI I,I) = 1.D0/DS3RT(Y(K))1P1 = I ♦ 1INDII>=.TRUE.1FINX.E3.1)S3 T3 78 11 = 1IFI IPl.LE.NUMBERIGO TO 3 wRirE(6 , 2 )2 F QRMA I ( 7rl STOP 2)STOP3 03 11 I = I P1,NUMBER IM1=I-111PI» 11 + I ASTI 1 , I 1 = 1.DO 7 J=1. IHl IFI.N3T.IN0IJ1IG3 T3 T 1EM?=3.NX 1 = 303 93 <2=1,II81 K4=NXl+<2I F ( I stDC <4 >) C3 73 82NXI=NX1+IGO TO 8182 K=LINfcAR(NUMBER,I,K4) l = MAOU4,J)NX 2 = 300 93 <1=1,1183 K3=NX2+<1 IFUND(<3))G0 73 84 NX2 = NX2U• GO T3 8384 IFU3.Lr.LIGn 73 80 TEMP=rEMP+AST(K3,K4l*ASTIK3,J)*Y(K)83'C3NTINJEASTI I , J)»-TEHP7 C3NTINJE1EMP=3.03 42 31=1,1IFIASTIJI,J1).E3.0.)G0 TO 9203 31 J2 = 1,1IFIASTIJ2.J2I.E3.0.IG0 TO 91 J=LINEA1(NUMBER,JI.J2)TEM?=rEMP+AST(I,J1)»ASTII,J2)*YIJ)
91 CONTINUE92 C O N T I N U E I F t r E M P . G T . l . E - 3 4 ) G 3 T O 10 0 Sl = 3.GO TO 131C 100 Sl=l.DO/OSQRTITEMP)INOII I = .TRUE.11 = 11 PI101 03 13 J => 1 , I10 AS r CI, J)=ASTI I , J)+s1 IFI11.E3.NXIRETJRN11 CONTINUE 78 RETURNENO
FUNCTION LINEARINUM8ER,11,12)C N1=MIND(II,I2I-1 N2«MAX0<11,12)LINEAR=NJMBER*N1+N2-INI*1N1+1))/2C RETURNEND
SUBROUTINE CGCOEFIJNUM)COMMON CG(10,8,&4)03J3LE PRECISION FJ1,FJ2,FJ, FM1, FM2, FM,NINHAFC JMAX=2+JNUMNINHAF=-JNUMNINMAF=NINHAF-.5D003 1 J=I, JMAXFJ=J-1FJl=-.5C JPR!ME=3 00 1 J1=1,JNUM FJ1=FJ1+1.00 FJ2=-.5DO 1 J2=1,JNUMFJ2=FJ2+1.00JPRIME=JPR1ME+1C FMI=NINHAF M = 0on 1 Mt=l,JMAX FMUFMl + l.DO FM2=NINMAF 00 1 M2= L * JMAX F M2 = F M2 + I. DO FM=FM1+FM2 H = M + 1c 1 CALL C3FCG(FJI,FJ2,FJ,FM1,FM2,FM,CGUPRIME,J,M1 1C RETJRNENO
2. PHEXCIT
PHEXCIT is a multi-phase Fortran IV program designed for
use on the IBM 360/44 of the Wright Nuclear Structure Laboratory,
Yale University, its purpose is the calculation of the mixture
of one- and two-particle/hole excitations to the Hartree-Fock
state with an analysis of the individual contributions.
The overlay structure is shown in Figure 25. Phases,
at any given level, are loaded into core (temporally) in the
direction left to right.
(a) R00T2 is the root phase, which is always resident in
core storage. The following subroutines and main program make
up this phase:
PH2 is the main program. Its function is to load all
phases into core at the correct times.
TWOBDY locates two-body matrix elements in the input
data, taking account of the antisymmetry of two-body states,
positive and negative angular momentum z-projections, and
isospin components.
LOCATE finds the correct position of determinantal multi-
particle states in a list of all positive M-states using a bi
nomial coefficient algorithm.
DECODE accepts a coded one-word description of the multi
particle state and populates the individual single-particle
states which make it up.
TRIANGLE is a triangular flow output routine for symmetric
or antisymmetric matrices.
S0LVER2 extracts angular momentum components from over-
175
lap functions with good z-projections by solving a triangu
lar matrix equation.
BICO produces binomial coefficients.
PLUSJ applies the raising operator successively to any
combination of multi-particle states with the same z-projection
of angular momentum.
(b) READR finds the correct set of input data on the
input tape.
(c) BEGINR controls all sub-phases having to do with
neutrons or protons separately.
(d) PRPARE
PRPARE2 prepares the list of all possible neutron or
proton states with z-angular momentum projection greater than
or equal to zero, and tests these states for their one- or
two-particle nature. It decides which of these states are to
be treated as intrinsic.
CODER accepts individual particle states, generating
a single coded word to identify the multi-particle determinantal
state.
(e) HAMLTN applies both the one- and two-body operators
to the intrinsic states. If the problem is to be solved as a
hole calculation with the shell closed, then the proper addi
tive adjustments are made for the energy spectra.
(f) FINEE controls the extraction of angular momentum
components from M-overlaps, and the calculation of metrics for.
1-particle/l-hole non-zero M-states.
176
(g) COMBIN
KOMBINE2 extracts angular momentum components from
M-overlaps for both the neutron (proton) metrics and energy
matrices, generating separate contributions of each state to
the Hamiltonian matrix.
BLEND2 calculates metrics for one-particle/one-hole,
non-zero (positive and negative) M-states.
MINUSJ applies the angular momentum raising operator
to the negative M-states generated by BLEND2.
(h) WORK begins the formation of isospin states, gen
erating the coupled neutron-proton solutions in angular
momentum subspaces. The neutron-proton Hamiltonian interac
tion is calculated and added to the energy solutions.
(i) COUPLE completes the mapping of angular momentum
subspaces onto isospin subspaces for both the metric and the
Hamiltonian. The final analysis programs are then loaded to
produce the separate contributions of 1- and 2-particle/hole
excitations to the final solutions.
(j) SCHMIT
SCHMITS2 applies the Schmidt orthonormalization pro
cedure to the metric angular-momentum-isospin subspaces, gen
erating transformation matrices to be applied to the Hamiltonian
matrices.
0RTH02 operates with the transformation matrices from
SCHMITS2, resulting in final energy matrices between orthonormal
states. Eigenfunction compositions and occupation number
expectations are also calculated.
177
1 7 8
S Y M E I G d i a g o n a l i z e s t h e f i n a l e n e r g y m a t r i c e s a n d g e n
e r a t e s o r t h o n o r m a l e i g e n f u n c t i o n s .
( k ) P L O T g e n e r a t e s c o m p a r a t i v e e n e r g y l e v e l d i a g r a m s .
Figure 25. PHEXCIT Phase Overaly Structure. Phases, at
any given level, are loaded into core (temporally) in the
direction left to right.
P H E X C I T P H A S E OVERLAY S T R U C T U R E
DIMENSION NCMAXI8,l>.NUC8.lt.MCODE(12>,ISKATR(142).KRSPNDCIO),NJC15), STATE! 10),<OUNTR(8. 10,1), ITYPE! 142,1) .ADJUST (2) DIMENSION PRD3LMI20), OCCNUM!12,10)COMMON/LINE/INDEX!10,10,1)CDMMDN/S<ALAR/CDED(15,15),COVI2,55,8),COOI2,142,8),YC2,15,55).BINO.MI 15,15)CDMMON/NTRAXT/HFIBDYI12,12), B0Y21(405),B0Y22(325),30Y23!171).B3Y24I55), 8DY25 110) ,BDY26II) , B0Y2711),B0Y28( I ) CDMMDN/INTEOR/NJTRON«PRDrDN,N'JCLON,MDOUL»JMAXM« JMAXP, JHAX , NSPS,1 JID,LENTHN,LENTHP,LENGTH,NUMJN,NUMJP,NUMJ,MAXIN,2 MAXIP,NUMJHF,K0NTRL(5), MATRON, NSPSPl, 1 OF I NOlogical*i iskatrINTEGER PROTON,OOUBLDDJ3LE PRECISION COED,BI MOM, COVINTEGER*2 NJ.NCMAX.NU,MCODE,I NOE X,XRSPNO, STATE, I TYPEDOES THIS HAVE TO BE REDEFINED******--------- ♦»*+♦♦♦++*DEFINE FILE 2 2100,360,E.IOFIND
NOTE - BECAJSE OF THE TREATMENT OF (POINT) FOR STORAGE. THE PROGRAM WILL NOT WORK FOR KONTRLI1)“2** THIS PROGRAM ONLY WORKS ON EVEN-EVEN, N«Z NUCLEI* SINGLE PARTICLE STATES MUST BE ARRANGED WITH POSITIVE M BEFORE* CORRESPONDING NEGATIVE M STATES4♦KDNTRLII) EXPLANATION* 1 1-SOLVE ONLY ONE PRD3LEH 2-SOLVE 2 PROBLEMS* 2 O-PROJECT FROM STATES ONLY 1-PART ICLE-HOLE EXCITATIONS4 2-BOTH OF THE ABOVE» 3- O-SAME 1 -BODY OPERATOR FOR N ANO P 1-01FFERENT 1-BODY OPERATORS* 4 8-THIS IS A PARTICLE PROBLEM 1-HOLE PROBLEM* 5 O-THIS IS A NEUTRON/PROTON PROBLEM 1-NEUTRON PROBLEM♦DATA INPJT ON UNIT 5 UNLESS OTHERWISE SPECIFIED* IS<IP - NJM3ER OF CASES TO BE SKIPPED ON INPUT TAPE* PR03LM* I XONTRLII) ONTRL(2) <DNTRL(3)' ONTRL(4) X0NTRLC5)* 2 NEJT RONS, PROTONS, JMAXN, JMAXP, NSPS* 3 MCODE* 4 NJ - NE J T RON MULTIPLICITIES* 5 HF130Y - NEUTRONS* 6 NM* 7 M/ALJE, B0Y21,BDY22,...« .8 HFJPLS* --------- --------------- KONTRL11) =2--------* 9 NJ - PR3T0N MULTIPLICITIES•10 HF133Y - PROTONS (IF 01FFERENT FROM NEUTRONS)•11 NM FOR INTERACTION*12 MVALJE,30Y21,BDY22,...F0R INTERACTION* N'JMP AG•
UNIT 4 UNIT 4 UNIT 4 UNIT 4UNIT 4UNI T 4 UNIT 4
DO 5000 IJKL"1,10 REWIND 4CALL LOAD READR RE AD(5,1001)1 SKIP CALL READER!ISKIP) READ(5,1000)PR03LH 1000 FORMAM20A4)
WRI TE(6,1002)PROBLM 1002 FORMAT!1H1.20X.20A4)READ(5,1001KONTRL 1001 FORMAT(BI3)RE AO I 5,1001)NUT RON,PROTON,JMAXN,JMAXP,NSPS,JTONSPS?l=NSPS+NSPS+lAOJJS1(1)=0.4DJJST(2)=0.CALL LOAO REG I NRCALL BEGIN!NJ.NCMAX.NU.NCODE,ISXATR. XOUNTR, AOJUST,ITYPE, DOUBLI IF(<0NTRL(51.NE.1)G0 TO ISL = 000 1 L1=1,NUMJHF 00 I L2=L1,NUMJHF L*L*lDO 1 J-l.JMAX I Y(l,J,L)=COV(l,L,J). CALL LOAO WORKCALL SCHMITIY.AST.NJ.NUMJHF,NUM3ER,IND.INOEXl.l.JMAX)MAXIN0=I NUMBER*!NUMBER*!))/2 00 10 L=1,MAXIND00 10 J=1,JMAX10 Y(1,J,L)=C0V<2,L,J»1 ISPIN = 1 GO TO 18
C 15 NUCLON»NUTRON*PROTON M0DJL=M30(NIJCL0N,2) .JMAX=JMAXN+JMAXP-1c II SPIN"I(K0NTRLC21+11/21*2*1 IFCONTRLC 3) .EO.O)ADJUST! 1)»2.*A0JUSI(1)CALL LOAD WORKCALL WORKER(NCMAX,NJ.MCOOE.NJ,11SPIN,KOJNTR,ADJUST!2 ) , I TYPE, KRSPNO.SIATE.OOUBL,OCCNUM)C IB CONTINUEADJJS TI 1)"ADJUST I1)*ADJJST(2)CALL LOAO COUPLECALL COJPLEI11S>IN,Y,DOJ3L.XRSPNO,STATE,INDEX,NJ,ADJUST,PROBLM, OCCNUM)PAUSE PAUSE SOOO CONTINUEC STOPEND
179
SJBROJTINE HFZBOYCM.SIGN)CQMM3v/JPLUSR/HFJPLS(12.12),M03JLOI12) SO SHELLCOMM3N/3LOC</HV41UE(28>,NM16),IPR0D(24),MVALJI6,12,12)C0MM3N/3L0C<2/INI»M1,IN2.M2»IN3»M3»IN4,N4,MM»LCOMMON/INTEGR/NJTRON.PR3T3N,NUCLON,M30UL, JMAXV, JMAXP,JMAX,NSPS,1 • JT3,LSNTHN»LENTHP,LE NGTH «NUMJN» NJMJP»NUMJ*HAXIN*2 HAXIP»NUMJHF»<0NTRL(5)«MATR0N»NSPSP1»10FIND 03J3LE PRECISION HFJPLSINTEGER*2 MVALUE.N.M, IPR30.MVALU,MODULOH = MM N I = I N 1 N2=IN2 N3=IN3 N4=IN4IFIM.SE.1IG3 TO 3 M = 2 - M<< = ( IABSlMl)MABSIM2)fIABSIM3IMABSCH4l>/2 IF(M33KK,2).EQ.l )SIGN*-S!GNN L * MOOJL 0 ( N1)N2=M0DJl0(N2>N3*M30JLO(N3)N4 = M30JL 31N4)3 NUM3U = MVALU(M,Nl,N2)NJM3R2=MVALU(M,N3,N4)MAXI M = NM ( M)L1= MIN3< NUMBRl,NUMBR2)-l L2=MAX3(NUHBRl,NUMBR2)L=M4XIM*LI*L2-(L1*(L1*1))/2REIJRN End
SJBROUTINE S3LVEIC0T)DIMENSION C3TII)COMMON/SKAL AR/C3E0115.15),C3V12,55,B),COO(2.142,8),YI2,15,55),8 INOMt15,15)COMM3N/INTESR/NJTRON,PROTON, NUCLON,M3DUL,JMAXV,JMAXP,JMAX,NSPS,1 JTOtLENTHN*LENTMP»LENSrH,NUMJN*NJMJP»NUMJ«MAXlN*2 K4XIP,NUMJHF,<0NTRL(5).MATR0N,NSPSP1,I0F!ND D3J3LE PRECISION COED,8 IN3M, COT, TEMP, COVJ*JMAXU DO 4 1*1,JMAX LAHB3A*J J = J-lTEMP.COriJ)IF(I. EO.1)GO TO 403 3 JPRIME*LAMBDA,JMAX3 TEMP* TEMP-C3E0IJ,JPRIME)*COTCJPR1ME)4 COT!J)=TEMP/COEO(J*J)RETURNEND
SJBR3JTINE LOCATE I SIGN,XUK.LE3,NUCLON,MATRON) DIMENSION <U<(1),HU<(10)COMM3N/LDKAT/IBICOI12,6),LOXATE I 925,I ) INTESER*2 LOXATES13N* 1.DO 23 N*1,NUCLON
2 3 MU<(N)*KU<(N)1 = 132 IFIMJXIIl-MUXI1*1)133,34,3633 1=1+1IFI I-NUCL0N)32,21,21 36 N = M’J< 11)MUXII)=MUX(1+1)MUX(I+1)*N SISN*-SISN ■ IFt I.EO.DGO TO 33 1 = 1-1 GO TO 3234 SISN*0.RETURN
21 IFIMATRON.EQ.OIRETURN 1 = 1D3 1 N=1, NUCLON L-MJXIN)1 I=I+IBICO(L,N)LE3*L0.<ATE(I, MATRON)RETURNENDSJBR3UTINE C302L(XU<, I, J, NUCLON) DIMENSION KUKII) COMMON/STATES/NOOI 533,1) C0MM0N/C0DE/X0DERI6*12)NN=NJCL3N NEMP* NOOII,J)NNN=NNDO 1 N=l,NNH = NEHP/OOER(NNN,1)KUK!NNN)=MNEMP = NE MP-XOOERINNN, M)1 NNN= NNN-1RETURNENDSJBR0JT1NE TRIOJT(A.N)DIMENSION All)1 = 1DO 1 11=1,N LOW* ILIMIT=L0W*I1-1 I-LIMITM 1 WRITE!4 ,1000)(A(J),J=LOW,LIMIT) 3 FORMAT!IX,10F10.5)RETURNENO
)
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181
SJ8R3JTINE BEG!N(NJ,NCMAX,MU»HC3DE, ISKATR,KOJNTR,ADJUST, ITYPE.DOUBLE)DIMENSION NJ(I),MCOOEII),ISXATRI1),A0JUST(2)01 MENS 13N NCMAX(8,l),NU(B,l),H(142,10),ITYPEC142,1>,VECT0RC533>,X3UNTRI8,10,1)C3MM3N/LINEZ INDEX I 13,10,1)C0MMON/C3DE/<QDER(6,12) SO SHELLCOMMON/JPLUSR/HFJPLS!12,12),HODULO(12> ■ SO SHELLC3MM3N/ I NTEGR/N JT RON , ? R 0 T ON , NUC LO.N, M3DUL , JMAXN, JHAXP, JHAX.NSPS,1 J r0, LEnthn. lenthp,LENGTH,numjn, njmjp, numj, max in,2 MAXIP,NDMJHF,<0NTRL<5),MArR3N,NSPSPl,I0FIN0 LOGICAL*! ISRATRINTEGER PROTON.OOUDLE E3JIVAl = NC£INUM3ER.ONTRL<l) )D0J3LE PRECISION HFJPLS, VECTOR, HINIEGER*2 NJ.NCMAX.NJ.MCOOE,INOEX, M3DUL3, 1 TYPEI3FInD=1 NJCLON=NjrRON JMAX = JMAXN D3J31t =3 REWIND 8DO 50 MATR0N*1»NUMBER 1F<MATR3N.£3.2)33 TO 10MCOOE IS TWICE THE ANGULAR MOMENTUM PROJECTION OF EACH OF THE SINGLE****** PARTICLE STATES *REAOI5,10011 IMC30EII) , I*l,NSPS) *MOOJL = MOD(NUCL ON,2)ODER IS JSED FOR COOING AN3 DECODING STATES (SEE SUBROUTINESL2COO ANO CDD2L)XODE = 1NN=MAX3INJTR3N,PROTON)- NN=MIN0<NN,61 00 2 N=l,NN DO 3 1=1,NSPS 3 K00ERIN,M)=<3DE*H 2 <OOE=<OOER(N,NSPS)*<ODEMOOJL0 ARE FUNCTIONS OF THE SINGLE PARTICLE STATE NUMBERS,JSEO TO CONVERT NEGATIVE M STATES T3 POSITIVE M STATES ANO VICE VERSA (SEE SUBROUTINES HF233Y ANO JMlNUS) no 9 N=1,NSPS,2 M33JL3!Nl=N*1 9 MOOJLO(NH)*NNJ IS THE NUMBER OF STATES WITH A GIVEN TOTAL ANGULAR MOMENTUM 13 REA3(5,1331MNJ(J),J = l,JMAX)1301 F 3.RM A T ( 2 4 I 3 )NCMAX IS THE NUMBER 3F DIFFERENT STATES WITH A GIVEN TOTAL ANGULARM3MENTDM PR3JEC11 ON <=NJ(JMAX)NCMAX(JMAX,MATRON)**L E N31H= X JPRIME=JMAX-1 D3 I J=2,JMAXK=NCMAX!JPR I ME*I,MATRON)*NJ(JPRIME)NCMAXt JPRIME,MATRON)*K length=length*x I JPRlME=JPRIME-lNJMJ=NCMAX<1,MATRON)CALL LOAD PRPARECALL PR5PARINCMAX( I,MATRON),NU(1,MATRON),MC33E.X0UNIR,
ITYPEI1,MATRON),D3U3LE)1*000 4 Il*l,NUMJHF DO 5 12*11,NUMJ.MF 1*1*1INOEXTII,I2,MATR3NI*I 6 INDEX(I2,I1,MATR0N)=I IF(NJMBER.E0.2)G3 T3 U WRITE(6,1032)NUMJHF1332 FORMAT!16H3PR0JECTI3N FR3MI2,24M NEUTRON (PROTON) STATES)GO TO 711 IF(MATR0N.EQ.2IG3 TO 12 WRITE(6,1033INUMJHF1333 FORMAT!16H0PR0JECTION FR3MI2.15M NEUTRON STATES)GO TO 712 WRITE 16, 1035JNUMJHF^1335 FORMAT! 15H0PROJECTION FR3M,I2,14H PROTON STATES)
T CONTINUECALL LOAD HAMLTNCALL HAMLrN(H,MC30E,ITYPE(l,MATR3N),I SKATR,A3JUST)• REWIND 1IFIMAIRON.EO.DRE AD I4)((HFJPLS(I,J),J»1,NSPS),I-l, NSPS)1 PR I ME* 100 4 1=1,NDMJIF(I TYPE I I.MATRON).EQ.0)30 TO 4 00 8 K=l,LENGTH B VECTOR!<)=0.D0 VECTOR! 11 = 1.00CALL JPLUSIVECTOR,NU!I,MATRON),MCOOE,2)WRITE!1)(VECTORIX),X=l,LENGTH)IFIIPRIME.EO.NUMJHFIGO TO 45 IPRIME=IPRIHE*1 4 CONTINUE
C 45 IFIMATR3N.E0.2IG3 T3 40 MAX IN = NJMJHF NUMJN=NJMJlenmn*length43 CONTINJECALL LOAD FINEECALL FINISHIH,NCMAXTl,MATRON),NJ(1,MATRON),MC33E,!TYPE(1,MATRON), ISKATR)IFI<3NTRL(5).E3.1)RETURN NUCL ON = P R3 TON 50 JMAX*JMAXP MAXIP=NJMJHF NJMJP=NJMJ LENTHP=LENGTH RETURN ENO
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SJBR3JTINE PREP4RINCMAX,NU,MC3DE,K0U.NTR,ITYP6,30UBLE)DIMENSION NCM4X<1),NU(1),MC30E<1),K0U,NT(21,ITYPE(1>,MUX(6),XUX(&) DIMENSION POINT 15333,2),X3UNTR<3.10,1)COMM ON/S TATES/NOD I 533,1)C0MN0N/i.3<AT/ IB I GO! 12,6),LOXATE(925,1)C3MM3V/JPLUSR/HFJPLSI12,12).MODULO!12)C3MMON/INTE3R/NjrRON,PROT3N»NUCLON,M3DUL»JM4XN»JMAXP»JMAX»NSPS»1 UI3,LENTHN,LENTHP,LEN3TH,NUMJM,NUMJP,NUMJ,MAXIN,2 MAXIP.NUMJHF.XONTRLIS), MATRON,NSPSPl, I OF I NO D3J3L E PRECISION HFJPLS EQU!YAL = NCEH3LES,K3NTRL(2))INTEGER HOLE,AUORES,PR3I3N,DOUBLE,HOLES INTEGER*2 NCMAX,NU,MG3DS,P3INT, L0X4TE, MODULO,I TYPEMINl*MlN3< NUTRON,PRO ION)N P I 3 M I N I UMAXI3MAXO(NUTRON,PROTON)MATRIN32«MATR0N-2 IFIMATR3N.E3.2IGO TO 2IBIG3 ARE BINOMIAL COEFFICIENTS USED TO LOCATE STATES IN THE LISTNOD (SEE SUBROUTINE LOCATE)03 20 I-l.NSPS <3I-1 L*XIBIC3(I,1)*L 00 23 J=2,MAXI < = <-1IF !J.GT.1-1IG0 TO 22 L*(LM)/J IBI CO I I * U)*L GO 10 23
22 -!8!C3(I,J)*023 GONTINJEc 2 NJMJHF°l N J I I I O00 133 M»2,JMAX MM 3 M“ INJIM)=NJ(MM)♦NCMAX(MM)103 NCMAX(MM)=3NCM4XI JMAXIOC 00 1 N31,NUGLON 1 XJ<(N).»N KU<(1)-0 IIY=*EI l ) - lC GREAT E A LIST (NOD) OF STATES HAVING ANGULAR M3MENTUH PROJECTIONC ■ GREATER THAN OR EQUAL TO ZERO. STORE THEM 3Y M-OROER. THE ARRAYC L0<ATE GIVES THE CORRES»ONOENCE BETWEEN H-OROERIN3 AND THE ORDERC OF PRODUCTION
LX*300 10 <31,LENGTH3 00 9 N3I. NUGLON [TEST3XU<(N»l)-l IF<N.E3.NUCL3N)ITEST*NSPS 1FKJ<(N).E3.ITEST)S0 TO 9 XU<(N)=<J<(N)Hl IFIN.EO.1IGO TO 5M=N-1DO 3 Nl*l,M B XU<IN1)3 NI
C 5 NSI3*0DO 4 Nl*1,NUCLON M*XU<(N1)4 NSIG*NSIG»HCODE(M)LX=LX*1IFINSIG.LT.MDDUDGO TO 3 NS IG* NS IG/2MC IX»NGMAX(NSIG)+1 NGMAXINSIG)*IX MG = NU(NSIG) MX
GALL L2G00IXJX, NOD I MG,MATRON) , NUCLON)LOXATEILX.MATRONloMG IFINSIG.NE.1.0R.MG.E3.D30 TO 10 IFHOLES.EQ.OIGO TO 205CC _ FIND THE PARTICLE-HOLE STATESC IT IS ASSUMED THAT IF NUTR3N .NE. PROTON THEN THERE IS A NEUTRON EXCESSHOLE 3 0 NNM= 1DO 231 NM*I, NUCL3N iF(<JX(NNM).EQ.NM)G3 TO 202 IFIHJLE.E0.2IG0 TO 205 MISING«NM HOLE3 HOL EM GO T3 231202 NNM=NNMH 201 CONTINUEC GO TO (208,209),HOLE C TYPE THE IP-1H STATES208 NJ.MJHF = NUMJHFM IFIMISING.GT.MINDGO TO 203 IF(XJX(N'JCLON).3T.NJTRON)30 TO 204203 I TYPE I MG)*2 GO TO 13234 ITYPE(MG)»3 GO TO 13CC TYPE THE 2P-2H STATES209 NEMRl-XJXINUCLON-l)■NEMP2*XU< I NUGLON)IF(MG0r)c(NEMP2).NE.MC00E(HISING))S0 TO 205 IF(NSMP2.NE.NEMPl+l.DR.M0DCNEMP2,2).NE.0)6O TO 205 NUMJHF3NUMJHF+1 11YPEI M3)=4IF(M4rRDN.E0.l)D0UBLE=00UBLE*l GO 10 13235 ITYPE(MG)*0 GO TO 139 CONTINUE 10 CONTINUEC FINO THE PARTICLE-HOLE STATES HAVING ANY ANGULAR MOMENTUM PROJECTIONC ANO XEEP TRACX OF WHICH STATES THEY GOME FROM. THESE ARE REQUIAEO-iC IN SUBROUTINE BLENO COC AODRES 3 (LENGTH,NUMJHF) UJXX20 ADORE S* 3 ICOUNIO03 336 <2*1 ,NUMJ
!FUTYPE(<2).EQ.0)G0 TO 306 K<2«XX2*lCALL C002LIHJK, K2, MATRON,NUCLON)<1 = 003 304 M=1,JMAX <0JNT(1)=3 XDJNT (21=0 NCM=NCMAXt M)03 300 1=1,NCM X I » < 1 » IADORES-ADDRESHCALL CD021IXU<,<1,MATRON,NUCLON)00 300 ITRIY=1,2 IIRY=ITRIY»MATRIN lFIITRIY.EO.DGD TO 307 IFtM.EO.llGO TO 300 00 3DB N=l,NUCLON NM=<J<(N)303 <J<(N)=m30UL0(NM)307 HOLE = 0DO 301 N= 1,NUCLON 00' 302 NM=1.NUCLON IFKJ<(N).EQ.MUX<NM)IGO TO 301 302 CONTINUEIFHOLE.EQ.11GO TO 305HOLE = 1 ^301 CONTINJE<*<OJNT(ITRIYDI XOJNTtIT RIYI=K POINTIAOORES.ITRYHR GO TO 300 335 POINT(ADORES,I T R Y )» 0 300 ICOJNT=MAXO(ICOUNT,<1304 XOJNTRIM,<<2,MATRDN)=HAXO(<OUNT(1),K0UNTI2))306 CONTINUE
c WRITEIS,1000)ICOUNT 1000 F0RMATI23M0THERE ARE A MAXIMUM 3FI3.38H PARTICLE-HOLE STATES HAVIN.G A GIVEN M)C IRITHDMATRIN IRIT2=IRlTl»l 00 310 M = 1,J MAX NCM*N3MAX(M)N'JM = NU I M)DO 310 .1 = 1, NUM J HF LOW- I I-D•LENGTH*NUMFl LIMIT=L0H*NCM-1 310 WRITEI8X (POINT(J,K),J = LOM,LIMIT),K=tRITl,IRIT2) 'RETURNEND
SUBROUTINE L2C0D(KU<,K0DE,NUCLON) DIMENSION KUX( 1 ) , KODE(2) COMMON/COOE/XOOER(6,12)
c NEMP»0 .NN=NJCLON DO 1 N=1,NN M=<J<(N)NEMP=NEMP*KODER(N,M)1 CONTINUE KDDEt1) = NEMP
cRETURNEND
481
no
n
SJBR3JTINE HAMLTN(H,MCODE»ITYPE»ISKATR»AOJUST)DIMENSION H(1)»MC3DE(1)»ITYPE!1)*ISX4TRI1)«ADJUST(2)»KUK(6) ,IBUFR(28)CDMM3N/NTRAXT/HF1BDY!12,12), B0Y2I(405),BDY22I325),• 30Y2 3!171),B3Y24!55), BDY25(101,B0Y26<I ) , B0Y27(1 ) ,BDY28!1) COMM3N/3lOC</MVALUE( 28) ,NM(6), I PROD! 24) ,MVALUti,12.12) CDMM3n/ 3LOC<2/NM1,MM1,NN2,MM2,M1SAVE.M1,M2S4VE,M2,M,LL C3MM3N/INTE3R/N'JTR0N,PR3TDN, NUCL3N t M3DUL»JM4XN»JMAXP»JMAX»NSP$»1 JI3,LENTHN,LENTHP,LEN3TH,NUMJN,NJMJP,NJMJ,MAXIN,2 MAXIP,NUMJHF,<ONTRl(5),M4TRON,NSPSPl,IDFl.ND INTE3ER AODRES.HDLEL0G!CAL*l ISXATR E3UIVALENCEMDLE,XONTRL(4) )DOJ3LE PRECISION H , T E.MP , TBME INTS3ER*2 MVALJE,NM,[PR33,MVAIU,M:03E.ITYPEc IF tMATR3N.E3.1.0R.X3NTRL( 3).EQ.l)READ!4)((HF1 BOY 11, J) , J = 1, NSPS),1=1,NSPS)IFIMATR3N.E0.2I33 TO 99 NSPSP'NSPSPL-1 II=NSPS?1 03 98 I = L, NSPSP IPR33( I) = 1 I 98 11=11*NSPSP1CC INM) IS THE NUM3ER 3F DIFFERENT 2-B30Y STATES FOR A 31 YEN 2-PARTICLEC M valueREAD!4 ) ( IBUFR(H),M=l,JT3)33 52 M = 1, J T 3 62 NMIM)=I3JFRIM)00 133 M=1,JTOMAXIM=NM(M)IFIMAXIM.E3.3I33 TO 103REAOI4)I!BUFR(NJMBER)«NJNBER“1«MAXIM)33 64 NJMBER=l,MAXIM 64 MVALJE(NJMBER)=IBUFRINUMBER)03 53 NJMBER=1,MAXIM I= MV A LU E(NUMBER)1 l = l/NSPSPl12 = 1-1PR30II 1)53 MVALJIM,I 1 , I2) = NUMB£RL=IMAXIM*1 MAX IM♦1))/2GO 73 I 11, 12,13,14,15,16,17,18),M11 REA3!4)(bOY21(X),K=1,L)33 13 13312 REA3!4)130Y22IX),X=1,L)33 T 3 13313 RE A3(4)(B0Y23IX),K = 1,L)33 T 3 13314 RE A3(4)I30Y24IX),X«1,L)30 TO 13315 REA3!4)(BDY25IX),K=1,L)33 T 3 13015 RE A3(4)(3DY26IX),X = 1,L)33 T3 13317 READ!4)1RDY27IXI,X=1,L)33 T3 133
IB READ! 4)(3DY28(X),K = 1,L)133 CONTINUENOTE THAT (ISXATR) IS DEFINED INVERSELY, I.E., HOSE S T A T E S W H I C H ARE SCATTERED TO HAVE ELEMENTS (.FALSE.)
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99 00 131 I-1 ,NUMJ 131 ISXATRII)■•TRUE.OPERATE ON THE HARTREE-FOCK STATE ANO, IF OESIRED, ON THE PARTICLE- HOLE STATES WITH ANGULAR MOMENTUM PROJECTION ZERO 11 = 1ADDRES=3 03 22 1=1,NUMJ IF( t T YPE(1 ) .EQ.D)GO TO 22 <«ADDRES DO I J = 1,NUMJ K= <♦ I 1 H(X)=3.30CALL C002LIXUX,I, HATRON,NUCLON)CHOOSE THE FIRST PARTICLE AND SCATTER IT 03 21 Nl=l,NUCLON ‘ N1P1»NI*LM1SAVE=<J<(N1I ML=MCODc(MLSAVE)DO 90 NN1=1,NSPS mM1=MC0JE(NNI)XU<(N1)=NN1IFIN1.E3.NUCLON.OR.NNi. EO.NSPS)30 T3 24 NN1P1=NN1*1CHOOSE THE SECOND PARTICLE AND SCATTER IT CF3R THE 2-BODY OPERATOR) DO 23 N2=NIPI,NUCLON"M2S A V E = X'JX I N2 )M2=MC03E(M2SAVE)HPRIME=M1+M2 M=M»RIME/2*1 DO 13 NN2=NNLP1,NSPS MM2 = MC03E(NN2)IF(M?RIME.NE.MM1*HM2)G0 TO 10 XU<(N2)=NN2LOCATE THE NEW STATE IN THE LIST, 03TAIN ITS 2-300Y MATRIX ELEMENT, AND ADO ITS CONTRIBUTION TO THE HAMILTONIAN CALL LOCATE!SIGN, XUX, LEX, NUCLON, MATRON)IF(SIGN.E3.3.)30 TO 10 CALL HF2B0Y!MRETRN.SIGN)GO TO ( 31,32,33,34,35,36,37,38),MRETRN31 TBME=BDY21(LL)GO 73 4332 TRME = B0Y22!LL)GO T3 4333 TBME=BOY23(LL)GO TO 4334 TBMC=BDY24(LL)GO T 3 4335 TBME=BDY25(LL)30 TO 4336 TBML=B0Y26(LL)GO TO 43 |_j37 TBME=B0Y27(LL) COGO T3 43 UR3B THME=B0Y28(ll>40 X=ADDRES*LEXIF(SIGN.EQ.-1.)TBME*-TBME ISX4TR(LEXI=.FALSE.HI XI =H( < ) ♦ TBME
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SJBROJTINE COMBIN(H»NCMAX,NU»MCODE»ITYPE»IS<ArR)DIMENSION VECTR1(5331.VECTR2I533) ,C0T(8) ,COT TI 3 > ,RITE UDO) DIMENSION HI 1) .NCMAXIII,NJ<1) , MCDDE(1>,I TYPE(I ) , ISKATRI1> COMMON/SXALAR/COED!15,15),COV!2,55,8) ,COD(2,142,81 ,Y(2,15,55) ,B I NOMI15,15)COMMON/INTE3R/NUTR0N.PROT ON.NUCLON,M3DUL.JMAXN,JMAXP, JMAX,NSPS,1 JT3,LENTHN.LENTHP,LENGTH,NUMJN.NUMJP,NUMJ,MAXIN,2 MAX IP,NUMJHF,<ONTRL(SI(MATRON,NSPSP1,I0FINDinteger adoreslogical*i isxatrINTEGER*2 NCMAX,NU,MC3DE,irrPED0J9LE PRECISION COED,3 INOM, COT, COV,SCALAR, VECTRl, VECTRZ ,HC IFIXONTRLI ll.EQ.21GO TO 40 WRITE(6,1031)1001 FORMATI27HONEUTRON AND PROTON METRICS)GO 10 4240 IFIMATR0N.E0.2)G0 TO 41 WRITE I 5, 1002)1002 FORMAT! 15H0NEUTR0N. METRICS)GO 1.0 4 241 „RITE(6,1004)1034 FORMAT!15H3PROT0N METRICS)COVIN,-,j) CONTAINS THE AMPLITUDES OF THE NORMALIZATION AND ENERGY MATRICES FOR BOTH NEUTRON AND PROTON STATES WITH ANGULAR MOMENTUM PROJECTION ZERO FOR EACH VALUE OF TOTAL ANGULAR MOMENTUM J ( I )-NEJT RON NORMALIZATION, 121-NEUTRON ENERGY,(31-PR0I0N NORMALIZATION, (41-PROTON ENERGY42 RE «I NO 1 MANAGE*MATR3N*MATR0N LI MI I 1*1MAXI NO*(NUMJHFM NUHJHF + 11)/2 DO 5 IJ X* 1, 2DO 1 <=1,NUMJHF ll*<-numjhf IFllJR.EO.l)GO TO 47 REWIND I LIMIT1 = <47 DO 45 <1=1,LIMIT145 REAO!l)(VECTR2(L),L*l,LENGTH)IF(IJK.E0.1)llMIT2«<
' REWIND 1 00 11 I»I,LIMIT2 IF!IJX.E3.11G0 TO 16 . IF!IS<A!R(I) ICO TO 11 IF( I TYPE I I ) .EO.OIGO TO 20 16 REA0(1)(VECTR1(L),L-1,LENGTH)GO TO 45 20 00 13 L=l,LENGTH ID VECTR1Il)*0.D0 VECTRl!11*1.DOCALL JPLJS(VECTR1,NU,MC30E,2) i46 NJMX«0 00 32 M=I, JMAX NCMX = NCMAX IM1 SCALAR*3.00 03 2 1*1,NCMX NJMX = N'JMX»1
2 S C A L A R » S C A L A R + V E C T R 1 I N U M X 1 * V E C T R 2 ( N U M X )32 : 0 T ( M ) « S C A L A RC
C A L L S O L V E (C O T )I F U J X . E 3 . 1 I G 0 TO 13
C C O D ! L , J ) C O N T A I N S THE S E P A R A T E C O N T R I 3 U T I O N S OF E A C H T E R MC H A M I L T O N A l A N TO THE E N E R G Y M A T R I C E S OF C O VC II) - P R O D U C T S W I T H C O N T R I B U T I O N S F R O M S T A T E S W I T H M ■ 0
D O 35 J* l . J M A X 35 C O D ! 1 , I , J ) = C O r < J )
CA D O R E S * I - N U M J LL * X - N U M J H F D O 31 L * 1 , X L L = L L * N J M J H F - L + I A D O R E S * A O O R E S + N U M J TEMPO*!!! A D O R E S )- D O 33 J * I , J M A X
33 C O V ! M A N A G E i L L , J ) * C O V ( M A N A G E , L L , J ) » C O T ( J ) * T E M P O 31 C O N T I N U E
G O TO 11C13 L L * L L + N U M J H F - I + 1
D O 34 J * 1 , J M A X3 4 C O V ! M A N A G E , L L , J ) » C O T I J )11 C O N T I N U EI IF IJX.E3.2 WRITE 2 IOFIND,1000 COO LM1, LM2,LM3 ,LM1 1,2
1 , N U M J ,LM3 l.JMAX 1030 FORMAT!VDA4)C4 9 D O 43 J * 1 , J M A X
J P R I M E * J - lL * 0D O 44 L l = l , N U M J H F D O 44 L 2* 1 , L 1 L = L + lL M * ( L 2 - 1 ) * N U M J H F + L I - I L 2 * ( L 2 - l ) ) / 2
44 R I T E ( L ) * C O V ( M A N A G E , L M , J )5 0 W R i r E ( 6 , 1 0 0 3 )JPRIME
1 0 3 3 F O R M A T ! 3 H 3 J = I 2 )43 C A L L r R I O U T I R I T E , N U M J H F )
IF! I J < . E 3 . 2 ) R E T U R NCW R I !E ( 6 ,1005)
1 3 3 5 F O R M A T ! 1 H 1 , 5 X , 8 H E N E R G I E S )■MANAGE = M A N A G E + 1 0 0 1 0 0 l=1,MAXIND D O 1 0 3 J * 1 , J M A X
1 3 0 C O V ( M A N A G E , L , J ) » O . D OL I M I T 2 * N U M J
5 C O N T I N U E R E T U R N E N D
no
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SJBROUT I NE BLEND ( NCMAX,NU« MCODE )DIMENSION NCMAX(l),NU(l),MCODEIl)DIMENSION VECTR1(533),VECTR2(533),VECTR3(533),C3SI8)«COT 18),POINT!142,10,2)COMMON/SXALAR/COEDI15,15),CDVI2,55,8),C0D(2,142,B), VI2,15,55),BINOMI15,15)COMMON/INTEGR/NJTRON,PRO TON,NUCLON,MJDUL, JMAXN,JMAXP,JMAX.NSPS.1 JT0,LENTHN,LENTHP,LEN3TH,NUMJN,NUMJPfNUMJ,MAXIN,2 MAXIP»NUMJHF,<0NTRU5), MATRON,NSPSP1 , 1 OF I NO INTEGER ADORESLOGICAL MANAGEINTE3ER*2 NCMAX,NU.M30DE,POINT00J3LE PRECISION COED,31NOM, COT, COS, SCALRl, SC4LR2, XI, X2, COV, VECTR1.VECTR2.V6CTR3SELECT PARTICLE-HOLE STATES NOT HAVING ANGULAR MOMENTUM PROJECTION ZEROANO FIND THEIR CON T RIBU11 ONS COO(I,-,J) TO HE HAMILTONIAN MATRICES(1) - PROOJCTS WITH POSITIVE M 12) PROOUCTS WITH NEGATIVE MX<=2«MATR0N-1AKP t = <<♦1REWIND B00 52 L M2 - I»NUMJHF52 READ< 3)T( POINT(LM1,LM2,LM3I,LMl»l,NUMJI,LM3-1,2)00 70 M-2.JMAXMP1=M*1 MM1 = M-l NC.M- NCMAX I M)REWIND 100 53 LM2=1.NUMJHF53 RE AO IB)I IP0INT{LM1,LH2,LM3),LM1=1,NCM),LM3=1,2)00 70 <-1,NUMJHF• 90 54 1 = 1,NUMJ 00 54 J-l.JMAX COOI 1 , I , J >=0.54 COOI2 , I, J > = 0.»62 REAOt1){VECTR1(LK),LK"1,LENGTH)NJMX = NIJ(M)00 51 1 = 1 , NCM NJ.M< = NJMX»1DO SO L<=1,NUMJHF 50 I F I »0 INTI I ,LK,K<)+POINT< I , LK ,KKP1) .3T.0) GO TO 60 30 TO bL 50 00 53 L4-1,LENGTH VECTR2(l<)=0.0063 VECTR3.lLO-0.00 VECTR2INJMXI-1.00 VECTR3INJMX)=1.00CALL JPLUSIVECTR2,NU,MC00E,MP1I call JMINUSIVECTR3,NU,MC00E,NUMX,MM1|
71 M'J-JMAX J-MJ00 54 JJ-l.JMAX IFIJ.LT.M)GO TO 68 SCALRl-3.00SCALR2-0.00 . |SCALR2-3.NJM3=NJIMJ*l)*lIF < MJ.E3.JMAX)NJMB=LENGTH*1NCMX-NCMAXIMU)00 55 L-l.NCMX
SC AL R l-SCALRl+VECTRK NUMB )*VECTR2( NUMB)65 SCALR2-SCALR2FVECTR1(NUMB)*VECTR3(NUMB)IF( JJ.E3.D30 TO 69 LAMBOA-Jfl00 55 JPRIME-LAMBOA,JMAX Xl-COEOI .MU,JPRI ME)X2-0S0RT(C0ED(M,JPRIME))SCALRl-SCALRl-CDTIJPRIME)*XI/X266 SCALR2- SCALR2-COS(JPRIME I*Xl*X2 69 X1-C0ED(MU,J)X2=DS0RT(C0ED(M,J))COT!JI=SCALRl*X2/Xl COS!JI=SCALR2/(X2*X1)GO TO 67 , 6B COTIJl-O.OO C OS IJ >- 3 .DO67 COOI1 , I,J)=30T(J)C00(2,I,J)-C0SIJ)MJ-MJ-I64 J-J-lC 61 CONTINUE73 WRITE 2 I OF I NO,1000 COD LM1.LM2.LM3 ,LM1 1,2 ,LM2 l.NUMJ ,LM3. 1, J M AX1000 FORMA TI90A4)RETURNEND
NJMB-NUMB-1188
SJBR3JTINE JHINUS(VECT3R«NU,MC03E,I,MLOM)DIMENSION VECTOR!1),NUI11. MCODE<U , KUK110),MUX(13)C3MM3N/J9IUSR/HFJPLS(12?12)«M30UL0(12) SD SHELLCOMMON/INTE3R/NJTR0N,PR3T3N,NUCLON,M0DUL,JM4XN,JMAXP,JMAX,NSPS»1 • JT3,L5NrHN»LENTHP»LENjTH,NU.MJN,NUMJP,NUMJ,MAXIN,2 MAX IP.N'JMJHF, KONTRL (5), HA IRON, NSPSP1 , I OF I NO INTE3ER+2 NJ,MC33E,M3DUL003J3LE PRECISION HFJPLS, VECTOR, X, TEMPMPl-MLOW+103 II <<=l,MLOWIFKK.NE.IIGO T3 4L = I + 1NCM= 103 T 3 5
4 L = NJ(MP1+1I+l NC M = L-NJ(MP1I-I5 03 13 K=1,NCM L = l-1< = VEC TOR(L)• IF(X.E0.3.130 TO 10 CALL C032L(MUK,L,MATRON,NUCLON)03 3 N= 1,NUCLON M = M J < 1 N 1 3 KUON)=M3DUL0!M)CALL LOCATE!SIGNS. KUK, LEX,NUCLON,0)D3 1 N=l,NUCLON M = K JO N I HPRI-E-MUKIN)• MM=MC3DEtMl+2 03 2 NN=1,NSPS IF1MC3DEINN1.NE.NHI30 TD 2 KJK(N1=NN MJKINI = M3DUL01NN1 PH4S E = 1.CALL LOCATE!SIGN,XUX,LEX,NUCLON,MATRON)IF(S!GN.E3.0.)G3 TO 2 IF1S13N.NE.SIGNS)PHASE--1.IF ( U.LT.MLOWICALL LOCATEISIGN• MUX,LEX,NUCLON,MATRON)
3 TEM?=HFjPLS(NN,M)oxIF!9M4SE.E3.-l.>TEM>=-TEMP VECT3R(LEX 1=VECTOR!LEX)+TEMP 2 CONTINUEMJUNUMPRIME 1 <J<( MUM10 CONTINUE11 MPUMPl-1MANAGE-NJMJ+l 03 13 K=M4NAGE,LENGTH 43 VECTOR! 0=0.00CALL JPLJS(VECT3R,NU,HC0DE,2)RETURNENO ,
no
n
SJBROJTINE WORKER!NCMAX,NU»MCOOE»NJ,11SPIN.XOJNTR,ADJUST,I TYPE, KRSPNO.STATE,30U8LE,OCCNUM)DIMENSION MC0DE(1),XRSPND(1)»XUX(12)»MUX(12)»4ST(1),NJ!1)»ST ATE!I)DIMENSION NCMAX(8, I),NUI 8 ,1).POINT!142,10,2),COT(15 ) . IBUFR!28),1 XOJNTRI 8, 13, I ). ITY>E ( 142,1) ,0CCNUM( 12,10) ,C0F (2,142, B) ,
2 corn IS) .COUPLE 110,10) ,C0UPEL(13,13)COMMON/LINE/I NOE XI10,10,1)COMMON/JPLUSR/HFJPLS!12,12),MOOJLO(12) S D S H E L LC3MM0N/310CX2/XXI,M1,XX2,M2,KX3,M3,KX4,M4,MSTAR,LLLCOMMON/3LOCX/MVALUE(28) ,NM( 6 ) , IPROD(24),MVALJ!5,12,12) S D S H E L LC0MM3N/NTRAXT/HF18DY(12,12), BOY2I(436),B3Y22(325),83X23(171),80Y24(55),BOY25I1D),BOY26II), B0YZ7(1 ) ,BDY2B!1) COMM3N/S<ALAR/COED(l5t l5),COV(2,55,U),COO(2,142,3),Y(2,I5,5S),81N0M115,15)COMMON/INTESR/NJTRON,PROION,NUCLON,MODUL,JMAXN,JMAXP, JMAX, NSPS,1 J10,LENrHN,LENTHP,LENGTH,NUMJN,NJMJP,NUMJ,MAXIN,2 MAX IP,NUMJMF,X0NTRL(5), MATRON,NSPSPI, I OF I NO INTEGER ADORES,PRO TON,81 OCXS,DOUBLEINTEGER‘2 NJ,NCM4X,NJ,MCOOE,POINT,XRSPNO,INOEX, I TYPE,. • MOOULO.MVALUE.NM,IPR30.MVALU,STATEOOJ3L E PRECISION HFJ>LS, C3E0,3 INOM,COT, COTT, X, XX1,XI, XX2, I , 1 1,1 YPRIMl,YPRIM2»XPRIMl,XPR[M2,TEMP,GIANT,ClANT1»BIN0,CUPL»CUPLI2 ,COEOS,SIGMA,SIGMAl,RATIO,YY1,YV2,;0UPEL,COUPLE,COV DATA LEFT/IH)/,l1NE/IH /.RIGHT/1H /.PLUS/1H /REWIND 3 MANAGE =0IF(XONTRL(5).EQ.3)AOJUST=AOJUST*2.JJJMAX=1LU0T=JM4XN*MAX1Nblo: xs=numjn*jmaxn*2IAOJOIFIMODI3LOCXS,90).NE.O)1400*1BL0:XS=3L0CXS/93»IADDMAXInD=<NUMJHFM NUMJrtFM ) )/2NSPSP=NSPSP1-1wRITE(6,1010)1010 FORMAT!17H1INTRINSIC STATES)GO TO 893(NM) IS T H E N U M B E R .OF 0 1 F F E R E N T 2 - B O D Y S T A T E S F O R A G I V E N 2 - P A R T I C L E * * * * * * M V A L U E . *
9 1 3 R E A O l 4 ) ( I B U F R I M J ) , M J * i , JTO)0 3 54 H J * I , j r O 54 NM(MJ)='IBUFR(MU)03 130 MU*I, J TO MAXI M*NM( .MU)IF(MAXIM.EQ.0)G3 TO 100READ! 4) ( I BUFR (XOUNT) , KOJNT*I, MAXIM)DO 53 XOJNT*!,MAXIM 53 MVALJE!XOUNT)=I3UFR(XOUNr>DO 200 <3UNT*I,MAXIM I=MVALUE(X3UNT)U*I/NSPSP1 12 = 1-1 PROD!11l-NSPS 203 MVALJIHj, I 1 , I2)=X0UNT LMMAXlM«tM4XlM»l>)/2 GO TO (11.12,13,14,15,16,17,18),MU11 REAOI4) (BDY21(<),X»1,L)GO TO 10012 REAOI4) (BDY22(K),K"1,L)
G O T O 1 0 013 R E A D ! 4) ( B D Y 2 3 I K ) ,X *1
G O TO 1 3 314 R E A D ! 4) (B 0 Y 2 4 ! < 1 , X * 1
G O TO 1 3 015 R E A D ! 4) ( B 0 Y 2 5 (X 1 , X * 1
G O 10 13 016 R E A D ! 41 (B 0 Y 2 6 ! X ) ,X *1
G O TO 10 017 R E A D ! 4) ( B D Y 2 7 ( K ) , K * 1
G O TO 10318 R E A O I 4) (B D Y 2 8 (X ) , X * 1
1 0 0 C O N T I N U EJ.MAXM = MI NO I J M A X N , J M A X P )J J J M A X = J M A X M J J M A X = J M A X H * J M A X M - 1
: ' h e r e t h e a d j u s t m e n t f o r h o l e s t a t e s i s c a l c u l a t e dI F ( X 0 N T R L ( 4 ) . E Q . 3 ) G 0 TO 8 9 0 DO 951 X X 1 * 1 , N S P S M l = M C O D E ( K K l >XX 3 = X X 1 H3 = MI0 0 951 K K 2 * 1 > N S P S M 2 = M C 0 D E ( K K 2 )X X 4 = X X 2 M 4 ° M 2M S T A R = I M 1 * M 2 ) / 2 * 1 S I C N * 1•C A L L H F 2 B D Y ( M R E T R N , S I G N )G O TO ( 9 5 1 , 9 6 2 , 9 6 3 , 9 6 4 , 9 6 5 , 9 6 6 , 9 6 7 , 9 6 8 ) , M R E T R N
96 1 T E M P = B D Y 2 l ( L L L )G O TO 9 5 5
9 6 2 T E M P = B 0 Y 2 2 ( L L L )G O TO 955
9 6 3 T E M ? = B D Y 2 3 ( L L L )G O TO 9 5 5
9 6 4 T E M P = B D Y 2 4 ( L L L )GO TO 95 5
9 6 5 T E M P * B 0 Y 2 5 ( L L L )G O TO 9 5 5
9 6 6 T E M P = B O Y 2 6 1 L L L )GO TO 95 5
9 6 7 T E M P * B D Y 2 7 I L L L )GO TO 9 5 5
9 6 8 T E M P = B 0 Y 2 8 ( L L L )9 5 5 I F ( S I G N . E Q . - 1 . ) T E M P * - T E M P
A O J J S T * A O J U S T * T E M PI F ( < X l . L E . N U T R O N ) A D J U S T * A D J U S T - T E M P I F ( X X 2 . L E . N U T R 0 N ) A 0 J U S T * A 0 J U S T - T E M P
951 C O N T I N U EW R I T E ( 6 , 1 0 0 0 )
1 0 0 0 F O R M A T (1 HI, 1X )
8 9 0 X K ° X O N T R L ( 1)KKKK=2*KK X * X X X X - 1 X X X = N U T R 0 N * 1
S E L E C T I N T R I N S I C N E J T R O N - P R O T O N C O M B I N A T I O N S F O R S C A L A R P R O D U C T S (RMS)X R 2 P N D = 2 * D O U 8 L E * 2 LI *3
190
113*303 27 13*1,NUMJN 131*1 TYPE(13,1)IF(I3I.E0.0)G3 13 27 I I 3*I I 3+ 1IF(I3I.E3.4)G0 T3 27 CALL C302L(KUX,I3,1,NUTRON)NEMPl=XJX(NUTRON)C J J 3 = 0DO 25 J 3* 1,NUM JP J3J=ITYPE(J3.XX)IF(J3J.E3.0IG0 TO 26 JJ3*JJ3*lIF113I.E3.3.AND.J3J.NE.3)G0 TO 26 CALL C032LIXUXIXXX),J3,XX, PROTON)NEM?2=<JXINUCLON)IF( I3I+J3J.E3.6.AND.INEMP2.NE.NEHPl*l.OR,HOD(NEMP2 *21• NE.O))CO TO 25 IF(MANA35.EQ.O)<RSPND(LD1)=0 IF( I 3 I. E3.3)GO TO 872 GO TO (373,873,874,875),J3J872 ST'4TE(L1»1)=4 GO TO 933873 SrRTE(Ll»l>*l G3 T 3 933874 lFIM3DtNEM?2,2).EQ.3)G0 TO 26 STATEILIM>=2GO TO 933875 STATEILl+l1*3 IFIMANACE.E3.0)XRSPND(LDl)»KR2PND <RZPN3=XRZPND*1933 L 1 *L1♦1L = L1-'IJMJHFIF(MANAGE.Ea.l)33 T3 62 00 931 LM=1,NSPS 931 0CCNJM(LM,L1)*3.DO 932 LM = 1,NUCLON L MI = XU<(LMI(F(LM.Gr.NurRON)GO TO 935 3CCNJKILM1.L1)=3CCNUM(LM1,L1)♦!.935 IF(l1.E3.1.JR.LM.LE.NUTR3N)30 TO 902 904 DCCNJM(lM1,L1)*0CCNUM(LHI,L1)»1.902 CONTINUE
C WRITE I 6, 1002)LI,IXUX(LM), LM*1,NUCLON)1002 FORMAT!HO,13,5X,1213)!F(J3J.N5.1)WRITEI6,1003)PLJS,PLUS,<XUXILM) , LM*KKX,NUCLON), (XJ<(LM),LM*l,NUrRON)1003 F3RMAT(A1,50X,A1,12I3)GO TO 93CC CALCJLATE THE NEJTR3N-PR3T3N INTERACTIONS62 REWIND 1 REWIND 3WRITE(9,33S3)LINE,LI,RIGHT 3353 FORMAT!IX,2HH , Al, 12 , IX,A1)DO 55 ISJB*1,JJMAX iI*.{ ISJ3»2)/2 . 'IF(MOO(ISJB,2).E0.1.AND.ISJB.NE.1)GO TO 912 NCHM= NCMAXI 1,1)00 913 L M2 * 1» MAX IN
913 REA0I8)((P0INT(LMl,LM2,LM3),LMl*l,NCMM),LH3-t,2) 912 LIMIT*MAXO(XOUNTRI1,113,1),XOUNTRII,JJ3,KK)1 DO 906 MN=1,LIMIT DO 906 MP=1,LIMIT 906 COUPLE(MN,MP)=O.DO DO 39 KI*l,NUTRON <<1*<JXI<1)M1=MC00E1XK1)DO 33 KX1=1,NSPSM3 = MC0DEI XX 3 IXJ<( <D=<<3MDIFDM3-M1M*IIABS(MDIFII1/2*1IFIM0IF1.GE.0IG0 TO 65IF( ISJB.NE.-M0IF1IGD TO 3800 153 NN*I« NUT RON NE MP *XUXINNJ163 MU<(NN)=MODULO(NEMP)■ CALL LOCATE!SIGN1,XUX,MM,NUTR0N,0)IFISIGNl.EQ.O.130 TO 38CALL LOCATE!DUMMY,MUX, MN,NUTRON,1)1 T RY = 2 GO TO 6765 IF! ISUB.NE.MOIFDDGO TO 3BCALL L0CATE(SIGN1,KUX,MN,NUTRON,1) IFISIGN1.E0.0IG0 TO 38 IT RY * 1C 67 MN*MN-NJ(1,1)MMN = POINT IMN,I 13,ITRY)03 60 X2=XXX,NUCLON XX2*<UX(<2)M2 = MC ODE IXX2)MPRIME=M1*M2 MSTAR=MPRIME/2*1 DO 51 XX4=1,NSPS MA = MC ODE IXX4)IFIM3+M4.NE.MPRIME)SO TO 61 XJXlX2)*XX4IFIMDIF1.GT.0IG0 TO 182CALL LOCATE ISIGN2.KUX(XXX),MP,PROTON,XXI IFISIGN2.E0.0.IG0 TO 61 ITRY*X GO TO 191182 00 133 NN=XXX,NUCLON NEMP=XU<( NN)183 MUXINN)=M3DUL0(NEMP)CALL LOCATE!SIGN2.XUX(XXX), MP,PROTON,0)IFISIGN2.EQ.0.130 TO 61CALL LOCATEI DUMMY,MUXIXXX),MP,PROTON,XX)ITRY*XXXXc 181 CALL HF2B0YIMRETRN.SIGN2)GO TO ( A1,A2,43,44,45,46,47,48),MRETRN41 TEM?=B0Y21(LLLI GO TO 4042 TEMP»BOY22(ILL)GO TO 4043 TEMP=B0Y23(LLL)GO TO 4044 TEMP=B0Y24(LLL)GO TO 43
191
192
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IF(MMPl.LE.O)GO TO 22X=C0UPLEIMMNl, MMP1)Z3C30U,MN,.Xl)*CDFU,HP,,JK21 Z 13ZIF CI3UBLL.E3.OI30 T3 208TEM9 = C3f t 1,MN, J.<1)*C30(1,MP, JK2)z=z+tempZ1=Z1-TEMP238 cj?l=cj?l+x*z: JPL1=cJPL1+X*ZI22 C3NMNJE23 CONTIN JEC 87 00 34 MJ=1,JMAX 31 AN T = 3.GIANT 1 = 3.LM33A13MAX0(3,MU-JK2)«-1 LM33»2=MIN0(JK1.MU)IF(LMB341.GT.LM3DA2)G0 73 34C <AP?A3MJ-LMBDA1+1 *00 23 IAMBOA=LM30A1,LMB3A2 bIN0=B!N3M<MJ,LAMBDA)S I CM A =0.SICMAIO.COE3S«C3E0I LAHBDA, J<11*C3ED(KAPPA,JX21*8INO IF(MANA3E.E3.O.3R.H.EQ.l.0R.MU.EQ.l)SO TO 21 I3TA1=LAMB3A+M-1 )OTA2=LAMB3A“M+lIFIMIN3IJ<1,J<2,MAX3ILAM334,KAPPA>).LT.M>30 TO 20 RAT !3 0S3RT(C3E3(M, JXD/COEOIM, JK2II 9I-N31 = 3In3M(MU,I3TA1)+RATIO IFILAM33A.3E.M)3IN02-BIN3MIMU,I3TA2I/RATI0
C NNN=NCM4X(M,1 I 03 52 MN=1,NNN1 031 ? 3 3 I 031 M = 3MMNl = P3INT(MN, 113,1)I F tMMN1.3T.3I 1031P3l MMN3-P3[NT(MN,113,21 1FIMMN3.3T.31 I 031M= 1 IFd30VPd031K.E3.0IG0 TO 52C NNP=NCM4X(M,XX1 DO 51 HP=I, NNP ID33P=3 ID33M=3 • MMP1=P3INT(MP,JJ3,K)IF(MMP1.3T.0)I033P=l MMP3=P0INT(MP,JJ3,K<XK)IFIMMP3.3T.31I033M°1 IFd333Pd033M.E3.0>50 TO 51 I035P=I031P+ID33M I005M=I331M*ID03P IFII305PUD35M.EQ.OIGO TO 51C Xa3.X 1 =3.
1 = 3.11-3.X X13 3 •
IF( ID05P.EQ.01G3 TO 732 XX13CQ'J?EL(MMN1» MHP3) *B IND1 X=COOI1,MN,JX11*COF(2,MP,JX2I XI = XIF(IOUBLL.EQ.OIGO T3 732 TEM?=COF(1,MN,JU 1 *C3D( 2,MP, JK21 X=X+TEMP XUX1-TEMP 732 IFII005M.E0.01G3 TO 849XX23C0J?LElMMN3.MMP1I*BIN02 Z=C33I2,MN,JXI)+C0F(I,MP,JX2)Z1 = ZIFd3UBLL.E0.01G3 T3 849 TEM?=C3F(2,MN,JUUCODII,MP,JK2)Z3ZdEMP ZUZl-TEMP 849 IFILAMB3A.lt.H1G3 TO 8492 SI3mA=SI3MA+Z*xX2 SIGMA1=SIGMA1+Z1*XX2 8492 IF1MJ.LT.I0TA1130 T3 51 SIGma=sigma+x*xxi sigmai=sigmai+xi*xxi51 CONTINUE52 CONTINUEc 21 XO.XlO.IFIM.NE.DG0 TO 215 X=YPRtMl+yPRIM2 X13Y?RIM1-YPRIM2 IFIMANA3E.E3.0I30 TO 215 X-X+YYI+YY2 X1=X1+YY1-YY2215 3I4Nr=GIANT + C0E3S*(3IN0*IX«-CUPL»+SI3MA)3IANri=3IANTl+C3EDS*IBIN0*(Xl+CJPLlI+SI3MA1) 20 <APPA=X4PPA-1COTT ( MUUCOTT (MJI+GIANTl 34 COTIMJ)=C0T(MU)+GIANT 86 CONTINUE 84- CONd NUE334 CONTINUECALL SOLVE I COT I CALL SOLVEICOTT)335 00 2535 J31,JMAX IFILI.NE.1IG3 TO 2534 C3T(JI=C3T(JI*.500 C3TTI JIOOTTI Jl*.5002534 Y( I, J, L1=COT(JI Yt2, J.LUCOTTI J)2535 IF(13JBLL.NE.1)Y(2,J,L)B0.802 IF! JJI.E3.MAXIP1G0 TO 2525 CONTINUE26 1F(JJ3.EQ.MAXIPIG0 TO 2727 IF(II3.E3.MAXINI50 TO 2400C 2403 WRIT E(3 I VIFIMANAGE.EQ.DRETURNMANAGE3!GO 13 910 2010 RETURN ENO
XX2»3.
SJBROJTINE COJPLEIIISPIN.Y.OOUBLE.KRSPNO.STATE.INDEX.NJ.ADJUST, PROBLM.OCCNUH)DIMENSION Y(2,15,55),YY(15,55).INDEXI10,10).OCCNUM(12,10) ,ASTI 133),I NO( ID>,NUMBR(3,41,OCCUPY I12,10,3)DIMENSION <RSPN3< I) , STATEI I1,NJ11) ,PR03LMI20),PLOT 12253), NM3RS(2250),TITLEStS,41 C OMMON/1 NTEGR/NJTRON, PROTON, N'JCLON, MOOUL , JMAXN, JMAXP, JMAX, NSPS, JTO.LENTHN.LENTHP,LENGTH,NUMJN.NUMJP.NUMJ,MAXIN,2 MAXIP,NUMJHF,<0NTRL(5),M4TR0N,NSPSPI»10FINDINTEGER DOUBLEINT£GER*2 KRSPNO,STATE,NJ,INDEX,INODATA TIT LES/4HALL , 4HC3NF, 4H13UR, 4HATI 0 ,4HNS , 4HHART, 4HREE • 4HF3C<,4H STA.4HTE ,4HH = A, 4HN3 2,4HP-2H,4rt STA,2 4HTES , 4 HHF A, 4HNJ 1,4HP-1H,4H STA,4HTES /CALL LOAD SCHMIT I FLAG-3 XOJNTR-OMAXIN0=INUMJHF*(NUMJHF*in/2 REWIND 300 SOD MANAGE-!,?REWIND 8 I3FIN0-1 RE AO(3)Y REWIND 1DO SOO ISPIN-l.IISPIN DO I L-l.MAXIND DO 1 JJ-l.JMAX1 YY(JJ.Ll-O.IF(MANAGE.E0.2IGO TO 3 DO 2 JJ-l.NSPS 00 2 L-l.NJMJHF2 OCCJPYIJJ,L,ISPINJ-D.
3 L - 1DO 450 11-1,4 00 400 J 1 - I 1,4GO TO (1000,2000,3000),ISPIN-: T = 0 STATES1000 GO TO ( HOD, 1203, 1330,450), II 1100 GO TO (1110,1120,1130,450),J11110 DO 1111 JJ-l.JMAX1111 YYIJJ , 1) «YI l ,JJ, l ) IFIMANAGE.E3.21G0 TO 400 DO 1112 JJ-l.NSPS1112 OGGJPYIJJ,1,1)-OGCNJMIJJ,I)GO TO 4031120 DO 1122 <1=2,NUMJHFIF(STATEK1).NE.2)G0 TO 1122 L-LH00 1121 JJ-l.JMAX YY<JJ,L)»Y(1,JJ,K1) 1FIMANAGE.E3.21G0 TO 1122 00 1123 JJ-l, NSPS OCCJPYI JJ,L, ll-OCCN'JMI JJ,<1)*.5 CONTINUE 30 TO 403
1121
11231122
1133 DO 1132 <1-2,NUMJHF IF(STATE(<1).NE.3)G0 TO 1132
K2-XRSPN0IK1)L»L*1DO 1131 JJ-l.JMAX1131 VY(JJ,L)«Y(l,JJ,<n-r(l,JJ,K2>1FIMANA3E.E0.2130 TO 113200 1133 JJ-1,NSPS1133 OCCJPYIJJ,L,1)-(0CCNUM(JJ,<1)*0:CNUM(JJ,K2))*.251132 CONTINUE GO TO 450C1200 GO TO (400,1230,1230,450),J1 C ALSO USEO FOR T » 1 STATES1230 Ml=1 1F(I1.GE.3)HI«M1»D0UBLE1 F( I 1 ,E3.4)M1 = M1*00U3LE LOW-200 1233 <1-2,NUMJHF IF(STATE(<1).NE.I1)G0 TO 1233' MI-MI♦ 1IFIIl.NE.JDGO TO 1234M2-W1-1LOW-<150 TO 12351234 M2 -1IF!Jl.GE.3IM2-M2*DOJBLE1 FIJ1 .E3.4)M2-M2*D0UBLE1235 00 1232 <2-L0W,NUMJHF IF(STATE(<2).NE.J1)G0 TO 1232 M2-M2 *1X4=INUEX(<1,<2)K3-<RSPND(<21 X5=IN0EK(R1,X3I L- INOEX(Ml,M2)00 1231 JJ-1,JMAX YY(JJ,L)-Y( ISPIN,JJ,X4)1231 IFIISPIN.E9.1.AN0.J1.E9.3)YYCJJ, L)■YYIJJ, L) - Y(I , JJ, X5)1232 CONTINUEIF! ISPIN.NE.2.OR.MANAGE.EQ.21G0 TO 1233 00 1236 JJ-l,NS?S1236 0CCJPY(JJ,M1,2)-OCCNUM(JJ,X1)*.S1233 CONTINUE GO TO 400CC ALSO USEO FOR T - 21330 Ml =DOURL E*1 SIGN-l-l.)**(ISPIN/2)00 1333 <1-2,NUMJHF IFISTATEKII.NE.11130 TO 1333 Ml-Ml*1K3=<RSPND(<1)M2-M1-100 1332 <2-Kl,NUMJHF IF(ST4TE<K21.NE.J1>50 TO 1332 M2-M2 * IX5-INDEXI <1*, <21 X8=IN0EX(<3.<2)K4*<RSPND(K2)K7-INDEX(<3,<4)K6=INDEX(<1,<4)L-IN0EX1M1.M2)DO 1331 JJ-l.JMAX1331 YYIJJ.LI-YI 1,JJ,<5)*YIl,JJ,K7)*SISN*CV(l,JJ,X3)fYIl,JJ,Xi))
1332 CONTINUE IF(ISPIN.NE.3.3R.MANAGE.EQ.2)GO TO 1333 DO 1334 J J• I, NSPS1334 OCCUPY!JJ,Ml,3)=(0CCNUM(J J , < 1 > OCCNUMIJJ,K3))*.251333 CONTINUE GO TO 4-50s~200D IFtll.ED.llGO TO 450 GO TO 1230 3003 I F(I 1.E3.3.4ND.J1.69.3)30 TO 1330 400 CONTINUE 450 CONTINUE503 WRITE 2 I OF I NO.4444 YY J,< ,J l.JHAX ,K l.MAXINO 4444 FORM A T(90A4)C XOJNT = 1DO 545 < ASE* I« 3GO TO (526,527,528),<ASE526 L0* = lLlMIT =NJMJHF GO TO 529527 L0W=DDJ3LE+2 LI MlT =NJMJHF GO TO 529528 L3w=1L1 MlT =D0U8LE + lC 529 I OF IN0=100 544 ISPIN*1,11 SPIN IF(<ASE*ISPIN.E3.6130 TO 544READ 2 I OF I NO,44441 Y 1,J,K ,J 1,JMAX ,K l.MAXINOL * 000 530 1=1,NUMJHF DO 530 J = I.NUMJHF L = l + 1IF!I I.NE.l.ANO.II.IT.LOW.OR.I.GT.LIMIT)).OR.(J.NE.l.AN0.1J.LT.LOW.OR.3*GT.LIMIT))IGO TO 534 00 532 JJ=1,JMAX 532 YY(JJ,L)=Y(l,JJ,L)IFIMANAGE.EO.l.OR.J.NE.IIGO TO 530 DO 536 JJ=1,NSPS 536 OCCNJM!JJ,I)=OC:uPY(JJ,ItISPIN)GO TO 530534 00 535 JJ=1,JMAX535 YY1JJ,LI=0.IFIMANAGE.EO.l.OR.J.NE.IIGO TO 530 DO 5 3 T J J = 1,NSPSP■ 637 OCCNJM!JJ,I)=0.530 CONTINJEC GO TO 154D,541),MANAGE540 DO 542 JJ*1,JMAX 542 NJ(JJ)=NJMJHFCALL SCHMU ( YY, AST , N J , NUM JHF .NUM3ER, I NOEX, ISPI N-I , XASE ) NJM3R!I SPIN,RASE)“NUMBER IF(<ASE.EO.l)KOJNTR = OUNTR*NUMBERIF!RASE.EO.l.ANO.NJ!JMAXI.E3.0)IFLAG=1 ,GO TO 544541 CALL ORMDN! YY, AST, NUMJHF. I SPIN-1,PLOT! AOUNTI.NMBRSKOim),< AS E.OJNTR, I NOEX, AD JUST, OCCNUM) <DJNI=<DJNT+NUMSR<ISPIN,<ASE)544 CONTINUE
IFIMANA3E.E0.UG3 TO 545NUMBER*MODIXOUNT.KOUNTR 1-1IF((ASE.E0.1)NUMBER“JMAX/2UIFKASE.EO.l.ANO. I FLAG.E3.1)NUM3ER-NUMBER-lIF!XASE.EO.l )<DUNT = <DUNT + NUMBER<=<0UNTR-NUM3£RIFK.EO.OIGO TO 544DO 546 L* I , <PLOT!OJNT)*PLOTUOJNT-1)NMbRS ! RJ'JNT ) = 7777 546 XDUNT=<3UNT*1 545 CONTINUE 603 CONTINUECALL LOAD PLOT REA0!5,1032)NUMPAG 1332 FORMAT!13)CALL 01 AGRMlPRD3LM,NUMPAG,TITLES,2,KOUNTR,PLOT,NMBRS,O,5,l,60.)X3UnT = <0UNTRUCALL DIAGRMIPROBLM.NUMPAG,TITLES!1,2) ,3 ,K3UNTR,PLOT(KOJNT), NMBRSIXOUNTI,0,5,1,60.)RETURNEND
195
no
n
S JBRO'JTINE SCHMIT(Y,AST,NJ« NUMBER, NUMB, INDEX,ISO*KASE)DIMENSION ASTINJMBERtDtNJIDDIMENSION Y( 15,55),BST(10,10),I4RAN3U0),JARANGU0),!NE<10),1N0(10) ,INDEX!10,10)COMMON/INTEGR/NJTRON.PRDTON,NUCLON,MODUL, JM4XN,JMAXP,JMAX,NSPS,1 ‘ jrO,LENTHN.LENTHP,LENGTH,NUMJN,NUMJP,NUMJ,MAXIN,2 MAX1P,NUMJHF,<0NTRL 5 , MAT RON,NSPSP1, I OF I NO INTEGER PROTONI NTE 3ER*2 NJ, INDEX, I NO.IAR4NG.JARANG,INEA SCHMIDT 0RTM0N3RM4LIZAT10N IS OONE ON THE SCALAR PRODUCTS OF STATES.MAXI ND=(NUMJHF*(NUMJHF*1))/2 NJMB = 3 244 00 251 M=1» J MAX MM = M- 1 N X =N JIM)WRITE(B)(Y(M,X),<«1,MAXIN0)IFINX.EO.OIGO TO 251 BOUND*.01 < OunT 5 O 355 00 5 IT 1*1,NUMBER INDI IT1) = 100 5 IT2=1.NUMBER 5 ASTI IT1, 1T2)=0.
C 1 = 1II = 04 <=NJMdER*<1-1)-(I * CI-3 ) )/2 IF1Y(M,O.GT.I.E-3)GO TO b 1*1*11FIf. GT.NUMBER I 30 TO 3 GO TO 4C 5 ASTI I,I I«l./SORT(VCM,X))IFIOJNIS.EO.l) 1FIN0*IARANGC I)101*1*1I NO I I ) = 0IFIOJNTS.EO.D33 TO 30 IAR4N3I1)*I JARANGII)=1 30 IFINX.EO.DGO TO 78II = 1IF!I?1.3T.NUMBER)G0 TO 300 11 I * IP1,NUMBER1 Ml = I-1I1PL*I1*1CC CALCJLATE NUMERATORASTI I ,11 = 1.DO 7 J*l,IM1IF!INDIJ).£0.1)30 TO 7T E M° = 0.N X 1 = 000 30 <2=1,II81 <4*NX1*<2IF(IND(<4).EO.O)30'TO B2 , jNX1*NX1*1GO TO 3182 < = I 'O E X II,X4)S 1 = Y 1 M , < )L = MA X 0 I<4,11
o o
00 80 Xl*l ,l l83 X3»NX2*<1 IFIINDI<3).EO.OIGO TO 84 NX2=NX2*1 GO TO 8384 !F(<3.ir.L)G3 TO 80 TEM, =TEMP*ASTIK3,X4)*ASTIX3,J)*S180 CONTINUEASTI I, J)=-TEMP 7 CONTINUEFIND NORMALIZATION TEMP=0.00 92 JI*1,IIFIASTI J 1, JD.EO.O.IGO TO 92DO 91 J 2 = 1,JIIF(ASf(J2,J2).EO.O.)GO TO 91 J=INDEX(J1,J2)S1 = AST< I.J1KASTI !,J2)*YCM,J)TEMP*TEMP+S1IF(J1.NE.J2ITEMP*TEMP*S191 CONTINUE92 CONTINUE IFITEMP.GT.BOUNOIGO TO 10093 DO 12 J*l,I 12 ASTI I , J) = 0.GO 10 11
100 Sl=l./S3RT(TEMPI INDII 1 = 0IF(<0UNrS.EO.l)IFINO*IARANS(I)11 = 11 PIIFKOUNTS.EO.DGO TO 31 IARANGI11)°I JARANGII)* 11 31 DO 10 J=1,I10 ASTI I , J)°AST(I,J)*S1101 IFIIl.EO.NXIGO TO 7711 CONTINUE
C REARRANGE SO THAT STATES ALREADY FOUND ARE PROCESSED FIRST 77 IFK3JNTS.E0.DG0 TO 62 XOUNTS=<OUNTS*1 x=o L = U03 63 1 = 1,NUMBERIFC INOII).EO.OIGO TO 59L = LHIARANGIL)=I JARANGI D»L59 00 60 J*I,NUMBER X = <*1ASTI I , JI*Y(M,K)60 ASTIJ, I)=AST I I , J)DO 51 1=1,NUMBER1 11 = IARANGII)DO 51 J=l,NUMBER J1=IARANGIJ)61 BSTI I , JXASrt I II, Jl) x=o
n n
DO 66 1=1,NUMBER 03 66 J»I,NUMBER < = <♦1 66 Y(M,<)=3ST(t,JI B3JN3=« 3338 G3 T} 366RETJRN 13 ORIGINAL 3RDERIN3 6? 33 63 1 = 1,.NUMBER INE CI I * IND(1 I DO 63 J=l,NUMBER63 BSE!I,J)=AST<I, J)DO 64 I-l.NJMBERI I 1 = JARANGII I INDI IU1NEIIII I 03 64 J=l,NUMBER Jl=JARAN3(J)64 AST ( I , J UBSTI 11 1, J 1»3 NJIMUI178 NJMR N'JM3 IIWR I Is I 11 11. < INQI J> ,J»1, NUMBER). MAST ( I . J ) .J*1 . NUMBER) , 1 * 1 , NUMBER! 251 CONTINJERETURNENO
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APPENDIX B-III
GENERAL PURPOSE COMPUTER CODES
The following general purpose subroutines are included
in this section:
(1) Double Precision Function COFCG - generates Clebsch-
Gordan coefficients in double precision.
(2) Double Precision Function BICO - generates binomial
coefficients in double precision - needed for COFCG.
(3) INVERT - Inverts a matrix.
(4) HDIAG - Diagonalizes a real symmetric matrix by the
Jacobi Method.
(5) Function RACAH - calculates Racah coefficients.
(6) SYMEIG - Calculates the eigenvalues and eigenvectors
of a real symmetric matrix.
(7) READER - skips input data cases on input tape.
(8) DIAGRM - plots energy level diagrams with variable
scales.
OOUBLE PRECISION FUNCTION COFCGIA.B.C.X,Y.ZIOOUBLE PRECISION A,B,C,X, Y.Z,X1, VI,Z1,X2,Y2.Z2,Z3.R,S,T.U.GOOUBLE PRECISION BICO OOUBLE PRECISION SUM,TEMP IF(C-A-B-0.1)301,301,302302 COFCG“0.00GO TO 300 0030301 IFIA-B-C-0. 1)303,303,302303 IFtB-A-C-O.1)304,304,302304 IF IABS(Z-X-Y)-0.1)305,302,302305 NMAx»NINHC-A*B*.l,C*Z*.l)IF((ABS(Z)-C).GT.0.1)GO TO 302 NM[N=MAX1( .1 ,B-A*Z*.1)SUM*0.00CCFCG*1. DOX13 A* X 0110Y1 =H ♦ Y 0120Z1=C*Z 0130X2 = A-X 0140Y2 = B-Y 0150Z2=C-Z 0160Z3=2.00*C+1.00 R*A* 8 *C*1,000S=C*B*X 0190T = B*C -A 0200J * Y 1 * • 100 306 l=NMIN,NMAX 0220U» I306 SUM=SUM»(-1.D0>*MI+2>*BIC0IZ1,U>*8IC0IS-U,X1)*81C01X2*U,YIJ SUM*(-1.0001**IJ*2)*SUMIF|ABSIX>-0.1)12.12.112 112 IF(X)11,12,1311 TEMP=BIC0(X2,X1)IFIIEMP.GT.0.5)G0 TO 401G=0.00GU TO 201401 G*I.00/TEMP201 L=-lGO TO 14 029012 G=1.D0L*I 0310GO TO 14 032013 G=BIC0(Xl,X2l 0330L*1 034014 IF(ABSIY)-0.1)16,16,114 114 IFIY) 15,16,1715 TEHP*BIC0(Y2,Y1IIF(TEMP.GT.0.5IG0 TO 402 G*0.00 1GO TO 202402 G*G/TEMP202 l=L-2GO TO 18 041016 L = L»2GO TO 18 038017 G=G*8IC0(Y1,Y2) . ' 0420L=L*2 043018 IFIABS(Z)-0.1)20,20,118 118 IF(Z>19,20,2119 G = G»8IC0U2,Z1) 0450L=L-3 0460GO TO 22 0470
20 L»L*1GO TO 22 osoc21 TEMP*B1COIZl,Z2)IF(TEMP.GT.0.5)C0 TO 425 G*0.00GO TO 225 425 G*G/TEMP 225 L=L*122 L=(8»L)/2 05J0GO T0(23,24,25,26,27,28),L 054023 G=G*BICO(-2.D0*Z,-2.00*X)GO TO 307 056„24 TEMP=8IC0(-2.D0*Y,-2.D0*Z)•IF(TEMP.GT.0.5)C0 TO 403 G*0.00GO TO 203 403 G=G/TEMP 203 GO TO 30725 rEMP=BlCO(-2.00*X,-2.00*Z)IF(TEMP.GT.0.5)G0 TO 407 G*0.00GO TO 307 407. G*G/ TEMPGU TO 307 060026 G=G*81C0I2.D0*X,2.00*Z)GO ro 307 062027 G=G«BICO(2.D0*Y,2.00*Z)GO 10 307 064028 TEMP=8ICO(2.00*Z,2.00*XJ IFITEMP.GT.O.SIGO TO 40B' G°0.00 GO TO 307408 G*G/TEMP •;307 TEMP*BIC0(R-l.00,2.00*O*BIC0«2.00*C,T)*R IF( rEMP.GT.0.5)G0 TO 499 C0FCG=0.DO GO TO 399499 COFCG=CUFCG*SUM*OSQRT(G*Z3/TEMP)399 RETURN 067„ENO 068()
1201202203 1203204
20820520/
DOUBLE PRECISION FUNCTION BIC0(A,8)DOUBLE PRECISION A,8,X,Y,UX-A-B 0010IF(ABS(Xl-0.1)202,202,1 IFIXI201,202,203 BIC0-0.00CO TO 207 0040BICO-l.OOCO TO 207 0060IF(ABSIB)-0.1)202,202,1203 IF(6)201,202,204 Y = DMINMD,X>BICO-A/YJ-Y-.91FIJI207,207,208 CONTINUE 00 205 I>1,J U-IBlCO = BICO*t A-UI/ ( Y-UI 0130RETURN 0140ENO ' 0150
SUBROUTINE INVERT!A,N,KKK,0ETERM)C DOUBLE PRECISION MATRIX INVERSIONDOUBLE PRECISION A,B,WORK,AMAX.T,PIVOTDIMENSION A(KKK,KKK),IPIV0T(50),INDEX!50,2) 0ETERH-1.0 00 1 I"I, Nl iPivorm=o00 108 1=1,NC SEARCH FOR PIVOT ELEMENTAMAX-0.00000 103 J = 1,NIFMPIVUTt JI.EQ.l) GO TO 103 DO 3 K=1,NIFUPIVUT(K).EQ.l) GO TO 3 IF(OABS(AMAX).GE.OABS(AIJ,KI II GO TO 3 IROW = J ICOLUM=K AMAX = A IJ, KI3 CONTINUE 103 CONTINUE1 PIVOT! ICOLUM)*I PIVOT1ICOLUM)*!IFIIPIVUTI ICOLUM).GT.l) GO TO 13C INTERCHANGE ROWS TO PUT PIVOT ON 01AGONALIF IIROW.EQ.ICOLUM) GO TO 5 00 4 J = 1,N T=A(IROW.J)A< !ROW,J)=A(ICOLUM,J!
4 A(ICOLUM,J)=T5 INDEX 11, 1) = IRCW INOEXII,2)=ICOLUM PIVOT-AIICOLUM,ICOLUM)C DIVIDE PIVOT ROW BY PIVOT ELEMENTAlICOLUM,ICOLUM)>1.000 00 6 J=1,N6 A(ICOLUM,J)=A(ICOLUM,JI/PIVOT C REDUCE NON-PIVOT ROWSDO 8 J=l,NIF I J.EQ.ICCLUMI GO TO 8 T-AIJ,ICOLUM)Al J,ICOLUM)=O.ODO 00 7 K=1, N7 A(J,K)=A(J,K)-A(ICOLUM,K) *T8 CONTINUE108 CONTINUEC INVERSION COMPLETE, PUT COLUMNS IN CORRECT ORDER00 109 1*1,N J=N*1-IIF(INOEX(J,11.EQ.INOEXIJ,2)) GO TO 109 IROW=INOEXIJ.l)ICOLUM-INOEX(J,2)DO 9 J=1.N T=A(J.IROW)A(J,IKOW)=A(J,ICOLUM)Al J,ICOLUM)-!9 CONTINUE109 CONTINUE RETURNC SINGULAR MATRIX INDICATION13 DETERM=0.0 RETURN ENO
202
no
n
10
11121415
17
2030
SUBROUTINE HOIAGIH.N,IEGEN,U*NR,X,IQ,NDIM) 'HD1AG - DIAGONALIZATION OF A REAL SYMMETRIC MATRIX.USES THE JACOB I METHOD CALLING SEQUENCE FOR 01 AGONAL IZAT ION‘CALL HOIAG(H,NtIEGEN,U,NR,XtIQ,NDIM)WHERE H IS THE ARRAY TO BE DIAGONALIZED.N IS THE ORDER OF THE MATRIX, H.IEGEN MUST BE SET UNEQUAL TO ZERO IF ONLY EIGENVALUES ARE TO BE COMPUTED.IEGEN MUST BE SET EQUAL TO ZERO IF EIGENVALUES AND EIGENVECTORS ARE TO BE COMPUTEO.U IS THE UNITARY MATRIX USEO FOR FORMATION OF THE EIGENVECTORS.NR IS THE NUMBER OF ROTATIONS.X » WORKING STORAGE IQ = WORKING STORAGENDIM » SIZE OF ARRAYS IN MAIN PROGRAM DIMENSION STATEMENTTHE SUBROUTINE OPERATES ONLY ON THE ELEMENTS OF H THAT ARE TO THERIGHT OF THE MAIN 01 AGONAL. THUS, ONLY A TRIANGULAR SECTION NEED BE STORED IN THE ARRAY H.NOTE THAT THIS PROGRAM PRODUCES ORTHONORMAL EIGENVECTORS FOR DEGENERATE EIGENVALUESDIMENSION HINOIH.NOIMI, UI NO IM,NOIM1,XINDIM), IQ INDIM)SET I NO ICATOR FOR SHUT-OFF.HDTEST=l.0E300RAP=2.0**t-48)IF IIEGEN .NE. 01 G0T015DO 14 1 = 1,N00 14 J=1,NIF( I .EQ. J i l l , 12U(I.J)=1.0GO TO 14U(I,J1=0.CONTINUENR = 0IFIN .LE.•11GOTOIOOOSCAN FOR LARGEST OFF DIAGONAL ELEMENT IN EACH ROMXII) CONTAINS LARGEST ELEMENT IN ITH ROM10(1) HOLOS SECOND SUBSCRIPT OEFINING POSITION OF ELEMENTNMIl=N—I DO 30 1 = 1,NM11 XII) ° 0.IPH=I*1 00 30 J=IPL1,NIF ( X(l) - ABSFI HI I , J11) 20,20,30 'XI I) = ABSFIH tI,J)I IQ(I )=JCONTINUE ^FIND MAXIMUM OF XII) S FOR PIVOT ELEMENT ANO TEST FOR END OF PROBLEM
6HDAG000 6H0AG001 6HOAG002 6HOAG003 6H0AG004 6HDAG00S 6HDAG006 6HDAG007 6HDAG008 6H0AG009 6HUAG010 6UOAGOU 6HCAC012 6HCAG013 6H0AG014 6H0AG015 6HDAG016 6HOAGO17 6H0AG018 6H0AG019 6H0AG020 6H0AG021 6HUAG022 6HDAG02 3 6H0AG0246MDAG025 6HDAG026 6HDAG027 6HUAG028 6H0AC029 6HDAG030 6H0AG031 6H0AG032 6H0AG033 6H0AG034 6HDAG035 6H0AG036 6H0AG037 6HDAG038 6H0AG039 6HDAG040 '6H0AG041 6H0AG042 6H0AG043 6H0AG044 6HUAG045 6H0AG046 6H0AG047 6HDAG048 6HDAG049 6HDAG050 6HDAG051 6H0AC052 6HUAG053 6HDAG-054 6H0AG055 6MDAG056 6H0AG057 6MDAGU58 6HQAG059 6HOAG060
u o
404560
70
BO8590
150
DO 70 I-L.NMllIF( I .LE. 11G0T060IFIXMAX . G E . X(I) 1 G 0 T 0 7 0XMAX-XlllIPIV=IJPI V=I0(11CONTINUEISIFIFIF
100110
MAX. XII) EQUAL TO ZERO*I XMAX) 1000,1000,60 IHDTEST) 90,90,85 IXMAX - HOTEST) 90,90,148 HOIMIN = ABSFI HI 1,1) )00 110 1= 2,NIF (HOIMIN- ABSFI HU,DM 110,110,100 HOIMlN=ABSFIH(1,1 I )CONTINUE
IF LESS THAN HOTEST* REVISE HDTECT
HDTEST=HDIMIN*RAP
148RETURN IF MAX.HI I *JILESS THAN!2**-481ABSF(H(K,K1-MIN) IF (HOTEST- XMAX) 148*1000,1000 NR « NR*1
6HDAG06I6H0AG062 6HDAG062 6H0AG064 6H0AG065 6HCAG066 6H0AG061 6HDAG068 6HOAG069 6HCAG0 70 6HCAG0 7 I 6HDAG0 72 6H0AG0 7 3 6H0AG0 74 6H0AG075 6HDAG076 6HUAG077 6HCAG078 6HDAG079 6HDAG0d0 6HGAG0BI 6H0AG0B2 6HDAG033 6H0AG0B4 6H0AG085 6HCAG0B6COMPUTE TANGENT* SINE ANO COSINEtHII,I) ,H(J*J)TANG=SIGNF(2.0,(H(IPIV,IPIV)-H(JPtV.JPIV)))*H(IPIV,JPIV)/(ABSFIHII6HDAG0B7 IP I V.IPI VI-HI JPIV«JP1VM+SQRTFI IHI IP IV , IP IV >-HI JP1V, JP IV ) )**2*4.0*H6HDAG0BB 2(IPIV,JPIVI**2I) 6HCAG069COSINE=1.0/SQRTF(l.0+TANG**2) 6HDAG090SINE=IANG*COSINE 6HDAG091HII=H(IPIV,IPIV) 6H0AG092HIIPIV,IPIV)=C0SINE**2*(HII*TANG*(2.*H(IPIV,JPIV)*TANG*HIJPIV,JPIV6HUAG0931)1)HIJP1V,JPIV)=C0SINE**2*(HIJPIV.JPIV)- iim HI IPIV,JP)V)=0.
TANG*(2.*H(IPIV,JPIV)-TANG*H
152
153
PSEUDO RANK THE EIGENVALUESAOJUST SINE ANO CCS FOR COMPUTATION OF HI IK) ANO UIIK) IF ( HUPIV.IPIVI - HIJPIV, JPtVl I 152.1S3.153 HTEMP = HIIPIV,IPIV)HIIPIV,IPIV) = HiJPIV,JPIV)HIJPIV,JPIV) = HTEMP RECOMPUTE SINE ANO COS HTEMP = SIGNF (1.0, -SINE) * COSINE COSINE = ABSF (SINE)SINE » HTEMP CONTINUEINSPECT THE IQS BETWEEN 1*1 WHETHER A NEW MAXIMUM VALUE THE PRESENT MAXIMUM IS IN THE I OR
ANO N-L TO DETERMINE SHOULD BE COMPUTEO SINCE J ROW.
200210230240250
00 350 I = 1,NM11IF (I-IP1V)210,350,200IF(I-JPIV)210,350,210IF I IQ I I)-IP IV1230,240,230IF( IQ 11l-JPIV)350,240,350K = IU ( I )HTCMP = H11,K>HI I, K) = 0•
6HCAGQ94 6HDAG095 6HOAG096 6HUAG097 6H0AG09B 6H0AG099 6H0AG100 6H0AGI 01 6H0AG102 6H0AG10 3 6HDAGL04 6H0AG105 6HDAG106 6H0AG1J7 6H0AG108 6H0AG109 6H0AG110 6HUAG111 6H0AG112 6H0AG113 6HCAG114 6H0AGI15 6HDAGI16 6HDAGI17 6hCAG118 6H0AG119 6HDAG120 6HUAG121 6HCAG122
IPLl-I+l 6HDAG123XII) >0. 6HDAG124c 6H0AG125c SEARCH IN DEPLETEO ROW FOR NEW MAXIMUM 6HDAG126c 6H0AG12700 320 JMPLl.N 6HDAG128. IF I XII>- ABSFI KII.JIl 1 300,300,320 6HDAG129300 XIII *> ABSFIHI I,J) I 6HDAG130IQU) = J 6HDAG131320 CONTINUE 6HDAG132H(I,K)=HTEMP 6HDAG L 33350 CONTINUE 6HDAG134C 6HDAGI35XIIPIVI *0. 6H0AG136X(JPIV) *0. 6HDAG131C 6HDAG138C CHANGE the other elements of h 6HDAG139C 6HDAG14000 530 1*1,N 6HDAG141C 6HDAG142IFII-IPIV1370.530.420 6HDAG143370 HTEMP * H(I.IPIV) 6HDAG144Htl.lPIV) * COS INE*HTEMP ♦ SINEYHII,JPIV) 6HDAG145IF ( XII) - ABSFI HIl.IPZV)1 1380,390,390 6HDAG146380 Xll) = ABSFIHU .IPIV)) 6HDAG14710(1) = IPIV 6HDAG14B390 HI I,JPIV) » -SINE*HTEMP ♦ COSINE*HII, JPIV) 6HDAG149IF ( XU) - ABSFI HU,JPIV)) ) 400,530,530 6HDAG150400 XU 1 = ABSFIHI I, JPIV) ) 6IIDAG1 5 IIQtl) = JPIV 6IIDAG15ZGO tO 530 6HDAG153C 6HDAG154420 IF II-JPIV)430,530,480 6HDAG155430 HTEMP * HIIPIV,I) 6HDAGI 56H( IPI V, I ) - COS INE*HTEMP ♦ SINE*HU, JPIV) 6HDAGI57IF ( XIIPIVI - ABSFI HIIPIV,I)) ) 440,450,450 6HDAG158440 XUPI VI = ABSFIHI IPIV, I 1) 6HDAGL 59ICUPIV) - I 6IIDAGI 60450 HI I,JPIV) » -SINE*HTEMP ♦ COSINE*HU,JPIV) 6H0AG161IF ( XII) - ABSFI HU,JPIV)) ) 400,530,530 6HDAG162C 6HDAG163480 HTEMP * HIIPIV.I) 6HDAG164HIIPIV,I) = COSINE*HTEMP ♦ SINE*H<JPIV,I) (SHDAG165IF ( X(IPIV) - ABSFI HIIPIV,!)) ) 490,500,500 6HDAG166490 XIIPIV) = ABSF(HIIPIV,I)1 6H0AG167IOUPIV) = I 6HDAG168500 HIJPIV.I) » -SINE*HTEMP ♦ COSINE*H(JPIV,I) 6HDAG169IF 1 X(JPIV) - ABSFI H(JPIV.I)) I 510,530,530 6H0AG170510 X(JPIV) = ABSF(HIJPIV,I)) 6H0AG171ICIJPIV) * I 6H0AG172530 CONUNUE 6H0AG173C 6HDAG174C TEST FOR COMPUTATION OF EIGENVECTORS 6H0AGI75C 6II0AG176IFUEGEN .EG. 0)540,40 6HDAGI77540 DO 550 1=1,N ■ ! 6HIJAG1 78HTEMP=U(I, IPIV) 6HDAGI 79Ull, IPIV) = COS!NE*HTEMP + SINE*U(I.JPIV) 6HDAG1B0550 U( I , JPIV)=-SINE*HTEMP + COSINE*UI k*JPIV) 6HDAG1UIGO TO 40 6HUAG1R2
1000 RETURN 6HDAGI 83END 6HDAG1U4
FUNCTION RACAH(A,B.Y,X,C,Z)C IFIC-A-B.LE.O.llGO TO 401 402 C0FJU=0.0 RACAH=0.• RETURN 401 IFCA-B-C.GT.O.IICO TO 402 IFIB-C-A.GT.O.l)G0 TO 402 IFIC-X-Y.GT.0.11G0 TO 402 IF (X-Y-C.GT.0.1IGO TO 402 IFIY-X-C.GT.0.1 IGO TO 402 IF(Z-X-B.GT.0.1)G0 TO 402 IFIX-Z-B.GT.O.MGO TO 402 IF(B-X-Z.GT.0.1IGO TO 402 IF(Z-A-Y.GT.0.1ICO TO 402 IF(A-Z-Y.CT.O.IIGO TO 402 IF(Y-Z-A.GT.O.IIGO TO 402413 NMAX* MINI (A*B-C,X+Y-C,A*Y-Z,B4X-Z»NMIN= MAXI (O..A-C+X-Z,B-C+Y-Z)
IFINMIN.LT.OICO TO 402SUM=0.COF JU = l •I X3 = A*B + X +Y+ 1 . 1 1 Y 3 = C + X + Y *■ 1 • 1 12C=C«C+.1 I Z 3= 12C +1 Z3 = IZ 3IX4=A+C+X-Z+t.l IZ5=Z+Z+1.1 Z5=IZ S 122=I 25-1 IY4=IX3-I2Z I Z4 = B + X-Z + 1.1 IR=A+B+C+1.1 R = IR1T=B+C-A*.l I R1=IY3 R1=IR1IR2 = X + 8 + Z + 1• I R2*IR2IR3=A+Y+Z+1.1 R3 = 1R 3 IT1*Y + C-X +.1 IT2=B+Z-X+.l IT3=Y+Z-A+.1SOI 00 414 I=NMIN,NHAX
IA3=IZ3+I IA4=IY4-1 IAS*I+1414 SUM* SUM+I-1.)**I*BICO(1X3—t«IY3l*8IC0(IY3,1A3)*BIC0(IA3,IX4I*BICOIIX4,IA4)*BIC0(IA4,IZ4)*B’CO(IZ4,IA5I*FL0AT(IA51 COFJU=CUFJU*SUM*SORT(Z3*Z5/(BICO(IR-l,I2CI*BtCOII2C,ITI*R*BICO1 <IR1-1, I2C)*BIC0(I2C,ITlI*R1*B!C0(IR2-1,I2ZI*BICO 112Z, IT2>•2 R2*BIC0(IR3-l,!2ZI*BICO(I2Z,IT3)*R3)I 502 INTEGE = 1X3-1506 COFJU*(-1. )**INTEGE*COFJU/SQRT(FLOATI IZ3*IZ51)499 RACAM=CUFJU INTEGE=IX3-1 D=(-ll**INTEGE racah=racah*dRETURNEND
f7 02
SJBRDJTINE SYMEISIA,N,C,JMLTMX.T.KNTRLl,KNTRL2) DIMENSION A(1) .C(1I,T(1) 003EIGENVALUES AND EIGENVEC TORS OF 4 REAL SYMMETRIC MATRIX 004N=ORDER 3F MATRIX A 005A=JP»ER TRIANGLE OF SYMMETRIC MATRIX TO BE 01AGDNALIZED 305A IS A LINEAR ARRAY, UPPER TRIANGLE ST0RE3 ROWWISE 007EIGENVA.JES ARE RETJRNE3 IN THE FIRST N LOCATIONS OF THE ARRAY A C = E I GENVECTDR MATRIX 008JMLTMX IS THE DIMENSIONALITY 3F C IN THE CALLING PROGRAMC<I,J>=I-TH COEFFICIENT OF J-Trt EIGENVECTOR 303T = 90 CHARACTERS 3F JOB TITLE<NTRLl=l FOR CALCULATION OF EIGENVECTORS, «D FOR NO CALCULATION <NTRL2=L FOR OUTPUT, =0 FOR NO OUTPUT
011IN0EXlI,N) = I«N-l l*t l -m/2»l <=(N*(N*1)1/2n oDO 53 1=1,N 03 53 J=l,I 11=11*11l = N*( J - l ) ♦!-( J*( J - l ) 1/2 53 CCI I I =41 IIIIFI<NTRl2.E0*1)CALL TRI3UTCC.N)
C GENERATE IDENTITY MATRIX 012NMIN = N-1 013IFKNTRLl. EO.OIGO TO 61 03 T3 1=1,N <=I-J“LfMX DO 73 J=1,N <=<*JMLTMX IF I I . EG.J)G0 TO 60 tIRIO.GO 10 7360 Cl 0 = 1.70 CONTINUE ^ 02961 IFIN.CT.l)GO TO 71N’JM = 3 GO TO 553 71 TRACEA=3.03 75 1 = 1,N M=IN3EX(I-1,N)7b TRACEA=TRACEA*4IM) 024C 025N J M = 3I ND = 3 AN3RI=0.C COMPJ T E NORMS 030DO 140 I=1,NMIN M= I ♦ 1NPR33=IN9EXII -1,M»-I DO 143 J = M,N NTEMP=NPR3D*J143 AN3RI=AN3RI*2.*A(NTEMPI*A(NTEMP» 036ANORI=SORTIANORII160 ANDR M = AnDRI 038AN3RF=<.5E-D8)*ANORIIFIANORF) 160,631,190 ' 1 040
C 041C COMMUTE THRESHOLD VALUE, SEARCH FOR ELEMENTS LARGER THAN THIS190 AvDRM=AvDRM/FLDAT(N)200 DD 530 < = 2,N
M»X-I NTEMP=X-N DO 530 1 = 1,M NTEM»=NTEMP*N-L*lIFIABSIAINTEMPM.LT.ANORMIGO TO 530I ND= 1C PIVDT ELEMENT FDUND, COMPUTE SINE ANO COSINE OF. ROTATION ANGLEXLAM=-AINTEMP>N?R30=IN0EX(L-1,N)N0JD=INJEX(X-l,N)XMU=.5*< A INPROOI-AIN3UO))3MEGA=XLAM/(SORT{XLAM*XlAM*XMU*XMU))1FIXMU.LT.0.)OME GA=-OMEGASINE>DMEGA/S3RT(2.*(l.*S3RTI1.-OMEGA*DMEGAII)CDSI=S3RT(1.-SINE*SINE)CC ROTATE L-TH AND K-TH COLUMNS AND ROWS WHERE PIVOT IS AINTEMPIDO 423 1=1,N IF I I.E3.LIGD TO 420
I F ( I - < ) 347,420,343 343 NX = N3J0M-KGO TO 348347 N< = INDEXII-1,N)*K-I348 IFII.ST.DGO TO 353 NL=INOEXIl-l,NI*L-I GO TO 354353 NL * NPROD*I-L354 r EMP = A( NLIA(NLI=TEMP*C3SI-A(NXI*SINE At NX I=TEMP*S1NE»A«NX)*C3S1 423 CONTINUE C IFKNTRLl.EO.OIGO TO 425 M4TRDN=(L-1)*JMLTMX MANAGE = K-1>*JMLTMX00 424 1=1,N M4T RDN = MATR0N*1 MAN4GE=MANAGE*1 TEM?=C(MATRON!ClMATRDNI=TEMP*COSI-CIMANAGE)*SINE424 C(MANAGE)=TEMP*SINE«-C(HANAGE»*C3SI425 NJM=NJM*1 SINCDS=SINE*COS!SINS3=SINE*SINECOSS3=CDS1*COS!TEMP = A(NPROUI-AIN3U0)WDR<=2.*A(NrEMP)*SlNC0SSINE=4IN>R0D>AINPRDO)=SINE*COSSO*AIN3UOI*SINSQ-WORX AIN3JD)=S1NE*S1NS3*A(N3JO I*C0SS3*WORX AINrEMP) = TEM?*SINC0S*A(NTEMP)*IC0SS3-SINS0I 530 CDNTINJc C IF!IND.EO.OIGO TO 5401 ND= DGO TO 230 540 IFIANDRM.GT.ANORFIGO TO 190CC MOVE EIGENVALUES TO FIRST ROW OF MATRIX 601 NL = 1DO 6D4 1=2,N NL=NL*N-I*2
205
634 A(I)•AlML)CC WRITE OJT RESULTS653 IF(<NTRL2.EO.O)RETURNWRITE(6,66D)(TII), I»1,20),MUM 650 FORMAT! 1H3, 2DA4,6X, 3DHNUMBER OF ITERATIONS RSOUIREO , I3//15X,UHE llGENYAtJES)03 570 1*1,N 670 W RITE(6,630)I, A( I )680 F0RM4r(2X,I2,2X,E15.7>IFKNTRLI.EO.OIRETURN WRITEI6>1333)1003 F3RMAT(/.15x,12HEIGENVE;TORS/»■33 71 3 I = 1, M MAIR3'l=t 1-1>*JMLTMX Lj WER = MATR3MU LIMIr=MATR3M*N 710 WRITE(5,733)I,(G(J),J=LDWER,LIMIT)733 F3RMATIIX,13,7E15.7/(4X,7E15»7))RETJRNEmD
SJ8R3JTI NE READER! ISXIP)DIMENSION HFJPLSI12,12),HF1BOY 112 ,12) ,NM16 ) , MVALUE<2BI. B0Y211410>D3J3LE PRECISION HFJ»LS IF!IS<I?.E3.0)RETURM 03 1 XASE-l.ISKIP RE AD(4)HF1 BOV 00 3 J= 1,2 'Re AO I 4) MM 00 2 M=1,6 MAXIM=NM(M)IF(MAXIM.E0.3IG3 TO 2READ! 4 1 ( MYALUEI OUNT) ,K3UNT*l,MAXIM)L = (MAXlMMMAXIM*l))/2 READ!4)(C0V21(K), K>1, L)2 COMTIMJEIF(J.EO.1)READ(4)HFJPLS3 GONT!MUc I CONTINUEWRITEI6,2500)!S<IP 2500 FORMAT!25H1TAPE PROPERLY POSITIONEO,13 , 14H FILES SKIPPEO)RETURNEND
104105
10B109112
119
206
SUBROUTINE DIASRM(PRDBLMt NUMPAG,TITLES,NUMPLT,NUMBER.PLOT,JCOUNT,MDDUL,ISCALE,ITRUNK,TRUNK)DIMENSION PR0BLM(l>,TlTLESt5,l),PL0TIl),JCOUNTtI) , EQUAL!2>,LOCATE!2253),IARANGI200).GROUND(25)(PR33LM) IS A 80 LETTER TITLE, CENTERED ON TOP OF THE PAGE, AND (TITLES) ARE 2D LETTER NAMES ASSOCIATED hITH SUCCESSIVE PLOTS.THIS ROJTINE PRINTS (NUMJLT) ENERGY LEVEL D1AGRMS, SIDE BY SIDE,THREE TO A P GE, EACH OF (NJMPAG) PACES LONG. EACH DIAGRAM MUST CONTAIN THE SAME NUMBER (NUMBER) OF LEVELS. THE ENERGIES ARE IN THE ARRAY (PLOT). THE ROUTINE SORTS (PLUT) IN DESCENDING ORDER FOR EACH LEVEL DIAGRAM, ANO RETURNS (PLOT) ANO (UCOUNT) THJS OILREARRANGED. THE ROUTINE SCALES TO (NUMPAG) PAGES, AND PRINTS THE 015ENERGY AND (1) 3JANJJM NJMJtR (JCOJND ASSOCIATED WITH EACH LEVEL. DliIF IMOOJL)= 0 THE OJANTJM NUMBERS ARE INTEGER, ANO 1= UNITY THE 017OJANTJM NJM3ERS ARE HALF INTEGER. IF TWO (OR MORE) LEVELS FALL AT 013THE SAME LOCATION, ALL THE OUAnTUM NJMU-RS ARE LISTED.(jcojnt) should contain twice he ojantjm numbers if they are 1 /2 023INTEGER. IF (ISCALE) - 1,2 THERE ARE SEPARATE SCALES FOR EACH DIAGRAM WITH 2 3EING ADJUSTED TO ZERO. IF (ISCALE) = 3,A,5 THERE IS ONE SCALE FOR ALL DIAGRAMS, WITH A HAVING ONE ADJUSTMENT To ZCRO, AND 5 HAVING SEPARATE' ADJUSTMENTS TO ZERO.SETTING IITRJNK) EUJAL TO JNITY ALLOWS GUTTING OFF THE SPECTRUM ANDAOJJSTING THE SCALE TO A MAaIMuM EXCITATION OF (TRJNK)LEVELS MAY 3E DELETED FROM THE DIAGRAMS BY SPECIFYING IJCOUNT) «7777, 3JT THE ARRAY (PLOT) FOR THESE LEVELS SHOULD BE WITHIN THE RANGE OF THE LEVELS TO 3E PLOTTEDDATA PLJS/1H*/,DVRPLD/H3/,EQUALI1>/1H»/,EQJAL(2)/9H«--*/, .PARENl/H(/,PARcN2/lH)/IT AP E = 3 REWIND ITAPE IC0JNT=NJMBER*NJMPLT MANA3EOSKIP=NJMPAG*66-121SK1P-SKIPM.ICC ARRANGE EIGENVALUES IN PLOT IN DESCENDING ORDER KK- - 1IFINJMBER.GT.DGO TO 300WRITE I6,2009I3VRFLD,PLUS,PLOT!1)RETURN300 DD 3 K-l,ICOUNT,NUMBER D69KK=KK«NJMBER91 N= <93 IF(»LOT(N).GE.PLOT(N*1))3D TO 92x=plotin)PLOT!N)«PLOT(N»l). PLOT!N * 1 ) = XI-JCOUNTIN) / Njcojntini- jcountinm)JCOJNTIN»1)«I IFIN.E3.OGO TO 92 N = N-l GO TO 93C 19892 IFIN.ED.KOGO TO 8N = N*1 ' 200GO TO 9J 8 CONTINUE1FIISCAlE.LT.3130 TO 305 202
ENMAX-PLOTI I)ENMIN-PLOTI NUMBER)DELTA-0.KK-0DO 7 K-l,ICOUNT,NUMBER KK-K <.NUMBERIF!ISCALE.NE.5)30 TO 190 EDELTA-PLDTIKI-PLOTIKK)DELTA-AMAXK EDELTA.OELTA)GO TO 7190 ENMIN = AMINl(ENMIN,PLOTKKI)ENMAX=AMAXUENMAX,PL0T(K) )7 CONTINUEIF!ISCALE.NE.510ELTA-ENMAX-ENMIN DELTAI=DELTAIFIITRJNK.E3.1.AND.DELTA.GT.TRUN.KIDELTA-TRUNKINTERPOLATE TO FIND THE LINE POSITIONS 305 KKODO 9 I-l.NUMPLT X-KKUKK-XX.NJMBERGO TD (191,192,193,199,195),ISCALE191 ADJJST-»LOT(K)OELTA-AOJJST-PLOTIKX)IFIlTRUNK.NE.l.OR.DELTA.LE.TRUNK)GO TO 196DELTA-TRJNKAOJJST=?LDT(K<)*DELTA.GO ro 196192 ENMIN=PLOT(KK)DELTA=PLOT(K)-ENMINIF! I TRUNK.EO.l.ANO.DELTA.GT.TRUNK)DELTA-TRUN<ADJJST-DELTA GROJNDII)= ENMIN GO TO 196193 ADJJST-EN.MAXIF!I TRUNK.ED.1.AND.DELTA1.GT.TRUNK)AOJUST-ENMIN»TRUNK GO TO 196 199 ADJJS T = DEL T A GO TO 195195 AOJJST-DELTA ENMIN-PLOTIKK)GROJNDII)=E NHIN196 DO 9 J = < , K K IFUSCALE.NE.1.AND.ISCALE.NE.3)PL0T(J)-PLOT IJ)-ENH!N ENERGY-AOJUST-PLOTI J)I FI ENERGY.LT.O.)ENERGY-0.IF!JCOJNTIJ) .ED.7777IG0 TO 10 LOCATE! J )=ENER&Y*SKIP/DELTAM.l GO 10 9 13 LOCATEIJI-O.9 CONTINUE
130 I 1 -MANAGE♦1 .I2=MANAGE*NUMBER*1I3=MANAGE*2*NUM3ER»1KONTRL-1WRITE I 6,1031 I(PROBLMIIJ<),IJK-1,201 IDDl FORMAT!HI,12X.2DA9)IF!ISCALE.E3.9I WRITE I 6,2001IE3UALI1),ENMIN 2001 FORMAT!99X,13HGROUNO STATE , Al, IX,E12.5) IF(ISCALE.NE.9)WRITE(6,13D2)
DO 5 J = l ,3IFI IC3JNT.Ea.MANAGE*J*NUN8ER>50 TO 55 CONTINJ:J * 36 I=MAN4SE/NUN8ER*l L = I ♦ J -1WRITE(6,1332)((UTLES(LI,12) ,L1-1,5>. LZ-I,L>1332 FORMAKIX,3(3X,5A4,20X))33 TD <2,3,2,4,3),ISC4LE2 «RITE(6,(002)33 T 3 43 WRITE(6,1333)(E3JALI1),3ROJNDU2>,12=I, L)1333 FORMAT! IX,313X,7HGR3UNO ,41, E12.4,23X))4 WRITE(6,1032)CC PL3T I HE SPECTRAITESTO03 13 J=1,!S<!P53 IF(J.NE.L3CATE(I1))G0 TD 11 <1 = 3
111=11 i rem3 = 1103 54" X3-1,NUMBER IF(!ll.E3.MANAGE*NUM3ER)33 TO 12 111=111*1i f(l3:4te( i i ).E3.i.o: ate( i i i ) )30 ro 52 lFtLOCATEIIlD.EO.OIGO T3 54 53 T 3 12 52 <1 = <1 ♦ 1IARAnG(<1)=JC0UNT(I1)ITEST=111=11154 CONTIN JE
C 11 IFIL3CATEIIl).NE.0)WRITE(6,2003)3VRFL0,PLUS25 ISE r = 327 IF(l3:aTECI1).NE.0)33 T3 20 ISEI * I 11=11*1 53 ID 2728 IFIISET.E3.1IS3 TO 53 11=11-153 ID 152
C 12 IFII IESr.Ea.3IG3 TO 125 IF(<1.5T.8)53 TD 124 IF(M30JL.E3.l)G3 TO 123 >,wRir£I5,2333)3VRFLD,>LUS»PL3T{Il), E3UAL.(IAR4N3(X)»K=1«KI), JC 3UNT(II)2033 F3RMATCAl/4l,F9.4,Al,A4,9I3)53 13 15212 3 MRirE( 5. 2333>3VRFLD.»LUS,PL3T(Il),E3UAL,(IA»AM3«) ,X=1,K1),JCOUNT(ll)2033 FORMAT I 41/Al, F9.4 ,Al,44,IX,9 It 2 , lrt/))50 13 152124 WRITE IS, 3033)3VRFLO,, LJS»PL3T(I1)«E3U4L*P4REN1» KONTRL tPAREN2 3333 F3RM4TI41/41,F9.4,41,44,IX,41,12,41)WRITE!ITA?E)<1,IIARAN5(<),X=1,XI),JC3UNTill)ONI RL=X3NTRL*1 53 TD 152125 WRITE! 5, 2 33413VRFL0 ,3LUS, PL 3T(11).JCOUNTIIt)2334 FORMAT!41/41,F9.4,5M ,13)
097093099
133101102
107
11111112
119
IFM3DUL.EQ.1)NRITE(6,2005)PLUS 2335 FORMAT!1A1.17X,2H/2)152 11-11*1 I TEST = 3153 IFIIC3JNT.LT.2*NUM8ER*MANA3E)33 TO 13 151 IF(J.NE.L0CATE«I2))53 T3 111XI =3 121=1203 154 0=1, NUMBER IFII2I.E3.MANASE*2*NUMBER)53 TO 120 121=121*1IFUDCATEI12).E3.LOCATE!121)ISO T3 155 IFIL3CATEI 121).£3.0)30 T3 154 50 T 3 123 155 Xl=Xl*lIARANSKll-JCOUNTI 12)HESr = l 12=121154 C3NTINUE111 (SET = 3112 IFIL3CATEI 12).NE.0)50 TO 128 I SET - 1. 12=12*1 GO 13 112 128 IF(ISkT.EQ.l)GO TO 151 12=12-1 GO 13 133120 IFIITEST.E0.0IG3 TO 225 IF(<l.Sr.8 )G0 T3 224 IF(K30UL. EO.I)G3 TO 223WRirE(6,2037)?LJS,PL3TI12 ) .EQUAL, IIARAN51X) , X-l, XI) . JCOJNT(12) 2007 F3RMAII1A1,43X.F9.4,AI,A4,9I3)GO T3 133223 WRITEI6,2337)PLJS,PL3T(I 2 ) ,EQUAL, 11ARAN5(X) , K=1, XI) •JCOUNT( 12)2337 FORMAT!IA1.43X.F9.4,A1,A4,IX,9(12,1H/))GO TO 130224 WRITE(S, 3037)PIJS,PLOT!I 2 ) ,EQUAL,PAREN1, K3NTRL,PARENZ 3337 F3RMAI(1A1,43X,F9.4,A1,A4,IX,A1,I2,A1)WRITE! ITAPE)<1,(IARANG(X),X-I,XI), JCOUNT(12)ONTRL = X3NTRL*l GO T3 133225 WRITE(6,2308)PLJS,PL3I(12),JC3UNT(1212338 FORMAT!1A1,43X,F9.4,5H------- ,13)IF IMOOUL.EO.l)WRITE(6,2039)PLUS2339 FORMAT!141,SOX,2H/2I133 12=12*1 HESr = 0453 IF(IC0JNT.LT.3*NJMBER*MANA5E)50 TO 13 451 IFIJ.NE.LOCATE!13))G3 13 411
Xl = 3 131=13 IrEMP=I3DO 454 <3=1,NUMBERIFI I3I.E3.MANAGE*3*NUMBER)G3 TO 420131=131*1IFIL3C4TEI13).EO.LOCATE!1311)G0 TO 455 IF(L3CATE(131I.EO.OIGD 13 454 GO 10 420
208
455 KI«<1*1IAR4N5(<1)=JC0UNTII3>ITEST=1 13=131 454 CONTINUE411 ISET=0412 IFILOCATEt13).NE.0)50 TO 428 ISET=113=13*1 GO TO 412 428 IF( ISET.EO.HGO TO 451 13=13-1 50 TO 430
: 420 1F(ITEST.EO.OJGO TO 425 IF(<1.3T.8)50 TO 424 IFMOOUL.EO.DGO TO 423wRirE(S,2010)PLJS,PLOT(13)•EQUAL,( IARANS!K),K*1,K1)tJCOUNT( 13)2010 FORMAT!1A1,86X,F9.4,A1.A4,913)30 TO 413423 WRITE(6,2310)PLJS,PL0T(I 3 ) ,EQUAL,( IARANOiK),K>1 ,K1) , JCOUNT! 131 2310’F0RN4T(141,B6X,F9.4,41,A4,IX,9(12,14/))30 TO 410424 WRITE I 6,3010)PLJS, PL3T( I 31, E0U41, P4RENl, KONTRL,PAREN2 3010 FORMAT!lAlf86X,F9.4,Al,A4,lX,Al,12,Al)WRirE(irAPE)<l,(IARANa(X),< = l ,Kl) .JCOUNT11 3)<ON!RL = ONTRL*1 30 !0 413425 wRITE(St2011)PLJS«PL01(!3),JCOUNT!! 3)2011 FORMAT!1A1,B6X,F9.4 ,5H ,13)IF(MOOJL.E3.1)WRITE!6,5450)PLUS5450 F0RMATI1A1,103X,2H/2IC 430 13=13*1 IT ES T = 0 13 CONTINUEMAN43E=MAN43E*3*NUM3ERWR1TE(S,1001)IF! ONTRl. £0.1)30 TO 575 REWIND ITAPE<ONT RL = ONTRL-1 00 500 !NDEX = 1, ONTRLREAD!(TAPEIK1,!IARAN3(K)*K»1*X1)»KOUNT IHM0DJL.E3.DG3 TU 550WRITE(b.4000)PARENl, I NOEX,PAREN2, 11ARAN3(X) , K=1,Kl>,KOUNT 4000 FORMAT!IX,Al,12,41,3513/I5X,3513))30 TO 500553 WRI!E(6,4030IPAREN1,INDEX,PAREN2,!IARAN3!K),K*l,Kl), KOUNT 4030 FORMATI1X,41,12,41,23(13,2H/2)/(5X,23!I3,2rl/2)))500 CONTINUE
c '57S IF!ICOJNT.ST.MANAGE)50 TO 100 RETJRN ENO
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