IL KUOVO CIMENI~O V,)L. X L I I B, N. 1 11 Marzo 1966

Some Formal Properties of the Ground-State Wave Function for Even-Even Nuclei.

V. FOlCMISANO (*) ~nd G . VIOLI.~I

Istituto di Fis iea dell'Universit~ - Roma

E. SALVS~I (**) Istituto N'azionale di Fis ica ~Vucleare - Sezione di Roma

Laboratoire de Physique Nueldaire - Orsay (S.-et-O.)

(ricevuto il 6 Luglio 1965)

S u m m a r y . - - Some formal propert ies of the a-cluster wave function and of the n particles component of the quasi-particle vacuum are examined. Exact methods of calculation are also proposed. As a test the Sn isotopes are studied.

I n r e c e n t yea r s , some p a r t i c u l a r fo rms of t h e w a v e func t i ons for t h e g r o u n d

s t a t e of t h e e v e n - e v e n nuc le i h a v e been s u g g e s t e d b y ~ n u m b e r of i n v e s t i g a t i o n s .

In t h e r e g i o n of t h e l i g h t e r nucle i , c o n s i d e r a t i o n of t h e p re sence of n e u t r o n s

a n d p r o t o n s in t h e s a m e shells has s t r e s sed t h e i m p o r t a n c e of n e u t r o n - p r o t o n

c o r r e l a t i o n s (~.~). Thus , a r e a s o n a b l e g r o u n d - s t a t e w~ve f u n c t i o n seems to be

(1) ( y ~ . ~ * *b~b*~" " ia~ j ~: I O/ ,

(*) Present address: M.I.T., Cambridge, Massachussets. (**) Present address: Courant Inst i tute , New York University, New York. (1) B. BR]~MOND and J. *V-ALATIN: ~?~ue~. Phys. , 41, 640 (1963); B. BR]~MOND:

Four-partiele~ correlations in light nuclei (pre-print); P. CAdiZ, A. COV~LLO and M. JEAN : NUOVO Cimeuto, 36, 663 {1963) ; M. BARANGER: Theory of Fini te Nuclei , Carg~se Lectures in Theoretical Physics (New York, 1963).

(3) M. BAI~ANGER: Phys. Rev., 139, 3, 1244 (1963).

72 V. FORMISANO, G. ¥ I O L I N I and E. SALUSTI

where the c~/s are free numers to be de te rmined by a var ia t ional me thod (*). The d /s (b~'s) are the p ro ton (neutron) creat ion operators and i ~ {n~, l~, j~, m~} are the usual shell model q u a n t u m numbers . Clearly, this wave funct ion is character ized b y the vanishing of bo th the to ta l angular m o m e n t u m , and the p ro ton and neu t ron seniorities. Moreover, the analysis of the shell s t ruc ture of the nuclei for which the wave funct ion (1) seems applicable (roughly spe~k- ing, 16 < A < 7"0) shows t h a t the physical ly interest ing cases correspond to v < 3 .

I n the region of heavier nuclei, the Har t ree -Bogol iubov t r ans fo rmat ion and Bla t t ' s approach (2) have suggested wave functions of a different form. I t seems oppor tune to introduce general nuclear pair ing correlations, as in the following wave funct ion:

Clearly, to have a well-defined charge, i t is necessary, t ha t (a} and (fl} are ei ther

protons or neutrons. A par t icu lar case, the B a y m a n (~) wave function, has been extensively

used for the speric~l even-even nuclei. I t is to be pointed out t h a t the impor tance of functions (1) and (2) is not

confined to the even-even nuclei. For instance, the knowledge of the wave funct ion of a sperical even-even nucleus gives essential in format ion abou t t he spect ra of the even-odd or odd-odd neighbouring nuclei (4).

I n the same w~y, the use of funct ion (1) allows us to s tudy nuclei wi th a neu t ron n u m b e r different f rom the p ro ton n u m b e r (**).

Clearly, the difficulty of these approaches consists in the algebraic calcula- t ions of mean values of physical ly interest ing quanti t ies, such as the energy or the part icle number . Cue can easily see t h a t the effective difficulty consists in the evaluat ion of the normalizat ions, because one can express the above- ment ioned mean values as a l inear combinat ion of the normalizat ions of sui- tab le wave funct ions of the k ind of funct ions (]) and (2). The purpose of this note is to describe a common ma thema t i ca l aspect of these wave functions, which makes simpler the calculation of these normalizat ions.

First , let us examine the funct ion

a + ~ A+B

I,=i h=l

/') In fact, once one is able to calculate the energy as a functional in the space of the variables c's~ i t is possible to minimize this functional, using an electronic computer.

(") I t seems possible also to apply this method in the theory of the ~ decay. (3) B. BA~MAN: Nucl. Phys., 15, 33 (1960). (4) E. SALT~STI: NUOVO Cimento, 37, 199 (1965); D. PnOSPERI and E. SALUSTI:

NUOVO Cimento, 36, 1372 (1965).

SOME FOtCMAL P l l O P ~ : R T I E S O F TH:E GI%OUND-ST,&T.E E T C . 73

where k~{ tk , n~¢, l~, j , , m~.}. ]n this way it is unnecessary to indicate expli- -I- 1) t*+m~ a ~ citly the protons and the neutrons. The nota t ion a~ means (-- w',~,-,,,,"

The ma t r ix (ch,) is of course symmet r ic and its elements all vanish in two dia- gonal blocks:

A A + I

A + B

A A + I A + B

0

0

A = n u m b e r of p ro ton

configurations~

B = number of neu t ron

configurations.

The m a t r i x (e~¢) can be easily diagonalized b y means of an un i t a ry t ransfor- ma t ion U, such tha t , defining

A+B

(5) dl, : Z U~a*kal,, k=l

we have

= ~,.~a,) ]o) ft

The quanti~ies k u have

and U, can be de te rmined once one has fixed the eh~:; as we

U ~ i.e., the k, ' s are the eigenvalues of the m a t r i x (Chk) and ( ~ ) is the un i t a ry ma t r i x which diagonalized (Chk).

I t is to be no ted tha t , a l though the t r ans fo rmat ion (5) does not conserve the charge, nevertheless ]yJ} has the same charge also in representa t ion (6). This means t h a t the equal i ty

(7) ~ k u~ v~ = o tt

holds if Tk = ~ . The normal izat ion gives

(s) ~10~. ..... % &-.-~

Because of the shell s t ructure of the nuclei to which the funct ion (3) is appli- cable, we have explici t ly calculated (8) only in the cases v ~ 1, 2, 3. These

74 v. F O R M I , S & N O ~ G. VIOLINI a n d E. SALUSTI

calculations are performed using eqs. (6), (7) and (8). The more general case may be calculated using the same equations.

Thus, we have

(9)

v = l :

- - O .

2

V ~ 3 :

tt #*~ I~ e~ a l t

<vI~> =48{(5 ~) . + 6 ~ ~ ~ + 8 5 ~} + z~s 5 [5 (v; k=)q " - /x pry t t t t o~

° k~ 2}; a# fl a#fl i.~ a

+ f t Once one knows <~pl~o), all quantities like <~vla %iy~ > and Qfla~aa~a;~t7~> are easily obtainable, as we have

(~o)

Thus, we get:

C ~ ~ t a~l~p; v > = 2 v ~ ~,a a~,a, 7 ~ ; v - - t > .

( 1 1 )

y ~ ] . :

r = 3 :

• "t ~r <~f,l I % a 3 a ÷ % [ y , ; 1 > = 4 Z %~,G~ = 4 ~ V 7°'U~, V; ,

oc ¢e

= a~%a~%]~f; v> ;

/ 9 ¢ f

- - .ao, aa)a~a ~ j~p; 1/ = oc

- - i ~ " k - - J ' ~ L k q- afl k # i d

+ 16 2 V~k~ V~ V~: V~,<V, 11... IV,, 1> i o ~ " y

e, f l v

8 0 M E F O R M A L P R O P E R T I E S OF T f t E (}:ROUND-STATE ~ T C . 75

Quantit ies such as

t t

may be calculated using the normalization <V; v IV;v) under the conditions

c~ ~ c~ ---- 0 (a = 1 . . .A + B ) , (k=i, j). A repeated use of the formulae (10)

and (11) can give the numerical value of the quantities of physical interest.

A similar si tuation holds for the wave function (2), tha t is

(~ c~,#a~a~) ]0 .... ( 1 2 ) ]~> = ~ *~ ' " ~fJ

In a recent paper, BARANGER (~) has stressed its connection with the Hartree-

Bogoliubov and Bla t t ' s approaches. He also presents and discusses an approx-

imate method of calculating the mean values of the physica]ly interesting

quantit ies. These approximations are bet ter if v is large.

There is also a simple way of evaluating the mean values of physically

interesting operators between functions as (12). This method, since it is exact,

can be used for every value of v.

As pointed out in (2), by a suitable uni ta ry t ransformation we can reduce

funct ion (12) to the form

(13)

representing I q~> in a new basis defined by U, which is different from the usual

shell model basis.

The quantities kz are real numbers and we can define the operators

(14) d ¢ ~ t It ~- ~ Vl~ao~ •

t The matr ix U is unitary, so the operators d~ are Fermi operators. The matr ix U

is determined by equations

One can define the state fi in the following way: for a fixed #, there is only

one state fi different from zero, c~---- k,. I n this basis the quantities we want

76 V. F O R M I S A N O , G. ¥ I O L I N I and E. S A L U S T I

to c~lculnte are

,u a

It

~uclao

Ira

IT1' lTfi TT~ Ug (eft t ¢ d

So the problem is reduced to t reat wave functions which are analogous to the B a y m a n wave function, tha t is the (( n ,~ particle component of the quasi

particle vacuum. Although there i s an exact method (s) of t reat ing these wave functions,

one can use a simple approach which is analogous to eq. (9) (*). I n this ease,

i t is possible to give the general form of <~1~> for every v~lue of

$2--~+1 -Q--v+2 .0

'6¢ " ' " '

a= l f l=a+l ~ ¢ + 1

where [2 is the number of the available co~figurations.

I n the same way as for eqs. (10), (11) we have

~a (18) <~[a~ ~j~0} = <~19~}--<9o] \

I g"~=o means tha t we pu t k 0 in calculating (.~Ig~};

1 aC a¢ = - - (T; ~']a~a~,lcf; ~} • (19) <cp;v ~ a ~ ; v - - 2 ) vc~

Using these relations other physical quantities may be evaluated. We have applied the preceeding point of view to a simple phenomenological

problem. I n the Sn region, there is a discrepancy between the number of paired

nucleons, calculated by KISSLINGER aIld SDRENSEN (~) and those measured by COHEn- and PrinCE (~). I t is possible tha t this discrepancy is due to the choice

(*) :Recently CI~nS~AN has presented ~ method which seems ~o be substantially equivalen~ ~o this (~).

(5) K, DIETI~ICI{, H. ~¢[ANG ~nd J. I°I~ADAL: Phys. Rev., 135, B 22 (1964). (~) R. C}~ASM),N: Phys. Rev., 138, B 2, 326 (1965). (v) L. KlSSLING~m ~nd R. Son~xs]~': K. Danske Vid. Selsl~., 32, 9 (1960). (~) B. Co~{~N and R. PRecis: Phys. Rev., 121, 1441 (1961).

S O M E F O R M A L , P R O P E R T I E S O F T H E G R O U N D - S T A T E E T C . 77

TABLE I. -- In this table our valeq~s o/energy ( Ea) ave compared with those calculated by (~) ( E.).

Also t h e va lues of t h e p r o b a b i l i t y of f ind ing a n e u t r o n in a s ing le -par t i c le s t a t e are c o m p a r e d : V~ is t h e e x p e r i m e n t a l va lu6 (CoH~N a n d PRrcE (s)), V~ is t h e ~heore t ica l va lue ca l cu la t ed b y KISSLIN(~E~ a n d SO~]~-S~N, V~ is t h e va lue o b t a i n e d b y our m e t h o d .

."°<?Z f-,, <"°YL

112

114

116

118

120

122

124

- - 2 . 2 1

- - 0 . 6 1

2.21

5.88

9.81

14.5

19.7

- - 1 . 3 1

0.00

3.13

6.50

10.3

15.2

23.8

5/2 7/2 172 3/2

1112

5/2 7/2 1/2 3/2

1112

512 7f2 1t2 312

11t2

5/2 7[2 1/2 3/2

11/~

512 7/2 1/2 3/2

11/2

5/2 7/2 lf2 3/2

11/2

512 7/2 1/2 3/2

11/2

0.79 0.78 0.42 0.25 0.27

0.80 0.86 0.50 0.33 0,33

0.87 0.89 0.61 0.55 0.35

0.86 0.92 0.69 0.59 0.47

0.93 0.95 0.74 0.68 0.55

. . . . . .

0.81 0.80 0.86 0.74 0,11 0.11 0.08 0.08 0.05 0.05

0.90 0.90 0.87 0.89 0.20 O.2O 0.13 0.11 0.06 0.04

0.93 0.90 0.91 0.91 0.37 0.42 0.25 0.27 0.11 0.11

0.94 0.96 0.93 0.97 0.53 0,44 0.39 0.35 0.19 0.18

0.95 0.95 0.94 0.95 0.65 0.66

0.53 I 0.54 0.27 ~ 0.26

0.96 0.94 0.95 0.95 0.75 0.79 0.65 0.60 0.38 0.39

0.97 0.97 0.96 0.98 0.88 0.79 0.75 0.71 0.59 0.48

78 V. FO:RMISANO~ G. V l O L I N I and >.'. S A L U $ T I

of the nuclear in terac t ion . Bu t i t is also possible t h a t this effect is due to the

a p p r o x i m a t e min imiza t ion of the pa i r ing Hami l ton ian . Therefore we have

assumed the same pa i r ing H a m i l t o n i a n of Kissl inger and Sorensen and we

have ca lcula ted the same quant i t ies .

The resul ts are given in Table I .

They are not ve ry different f rom those ca lcula ted b y KISSLINGER and

SORENSnN; therefore the above men t ioned effect seems ma in ly due to the

s ingle-par t ic le p a r t of in terac t ion .

Some calcula t ion in order to enlarge the va l id i ty range of the wave func-

t ion (1) and numer ica l calculat ions are being p resen t ly e labora ted .

We mus t t h a n k Profs. W. GROSS, D. PROSPERI, ~ . StIERMAN for m a n y

helpful discussions.

R I A S S UNTO

Si esaminano aleune propriet£ formali della funzione d'onda a cluster di ~ e della componente di n particelle del vuoto di quasi particelle. Sono anche proposti metodi esatti di calcolo. Come prova sono studiati gli isotopi dello Sn.