OE'bE.llJ.1HEHHbl~ J.1H CTJ.1TYT ll.llEPHbiX

J.1 CCnED OBAH J.1 ~

.lly6Ha

I • -• • ·• s = =r

! A. • = I • i • .. ~

E4 • 3445

}.Blank, I.N.Kuclitina ·

'MATRIX ELEMENTS

OF THE HAMADA JOHNSTON POTENTIAL . FOR THE HARMONIC OSCILLATOR

WAVE FUNCTIONS

' -

lf67r •,

E4 • 3445

J.Biank, I.N.Kuchtina

MATRIX ELEMENTS

OF THE HAMADA JOHNSTON POTENTIAL

FOR THE HARMONIC OSCILLATOR

WAVE FUNCTIONS

1, Introduction

Recently a new method for s olving the integral equation for the re­

action matrix has been develo ped ( r ef.1). This method is based on the

use of the oscillator s ing le- particle wave functions a nd tra nsforms the

o rig inal integral e quation in a system o f a lgebraic e qua tions . The coef­

ficients of this system contain the matrix elements o f the chosen potential.x)

Thus if we solve our system for a g iven p otential and a given oscillator

frequency we must know a ll the matrix e lements involved,

Calculating them is a very te dious procedure s ince the inte rva l of

integration r a nges to infir.ity, a nd the integrands a re rather complica ted

functi ons a nd consequently a numerical method of integration must be used,

'the method described in r e f.1 h a s been a pplied for the calculation

of the g round state characte ristics o f He 4, o 16 and Ca. 40 ( ref,2 ), the

HJ- potential being u sed ( ref.3). The tables of all matrix elements which

occur in the system are p r esented for five values of oscillato r frequency

distributed around the shell- model values of thes e qua ntity for the conside­red nuc lei,

The tables can be simply modified for the potential of Bressel et a l.

( r ef.4) which has the same lo ng- range part as the HJ- potential and thus

the corresponding correction integrals a re restricted to a small interval

and can be simply calcula ted o r estimated.

X

Herea fter by a matrix e lement of a p o tential w hich contains a hard co­re, only the contribution of the regular ( outside the core) p a rt is meant. The total matrix element is of course infinite.

3

The same refers to the application of the tables when the Scott- Moszkowski

separation method with oscillator single-- particle functions is used ( cf.ref.5).

2, Characteristics of the HJ- Potential

The general form of the HJ- potential is

V•Vc(x)+S 12 VT(x)+(t.S)VLS(x)+ L 12 VLL (x)

where the subscripts, C, T,LS, LL denote respectively central, tensor, spin­

-orbital and quadratic spin- orbital terms, x is the relative coordinate, f

the relative angular momentum and S the total spin of a pair of nucleons.

The operators S12 and L 12 are defined by

s 12- 3

~ .. ... ... (ul'x) (u 2 .x)

2 X

... ... ... ... ...2 ... 2 - ( u 1•u 2 ) , L 12 • [ 8 f l + ( u 1ou 2) ] f - (f. S )

The radial functions V 1 f..x) depend further on S and on the parity of f

Since for s-o only the central part is non- vanish ing x/ we have for a

given parity of f five different functions Y1 : y~) y!s-tly C ' T, ·yLS

and VLL • Hereafter they will be numerated in this order ( i • 1,2, .. 5 ), and

the Coulomb interaction which is taken into account as well is denoted

by V ( x). These functions are given in ref.3. However for odd V LL we 0

use the modification proposed in ref,6: for x ;t ·689 fm VLL remains

unchanged while for "c::; x ::;·689 fm ( x c- • 485 fm is the hard core

radius) VL!s a ·constant ( -37.28 MeV) which is chosen as to make

V LL continuous,

ons by multiplying

numbers by n , f

FUrther it is convenient to pass to dimensionless functi­xx/ '

each V1 by 2/(JJ • Denoting oscillator quantum ... ... ... the total angular momentum by j (i.e. j • f + S

and the total isospin and its projection by T, T • we get in the repre--

sentation defined by these quantum numbers, for a given (JJ

x/ The system of units with h· c• 1 is used.

xx/ The contribution of the V L L potential for s. 0 multiplied by - 2 .£ (t+ 1) ( ct. the. form of L u ) is included in V ~ 8

• 0 >

4

.

...... +Bee· d s j le·s l es j > <i~-e <w) +

( 1)

(I )

Here I •. e. 'e, ( w ) d e n o tes the contribution of the i-- th part of potential

( i• 0 , 1, ••• ,5) a nd just these q u antities are tabulated. They are all diagonal

with r espe c t to e xcept of the tens or force ( i• 3) which gives non- va­

n i s h i ng c ons tributio n s for I e• - e I ·• 2 as w ell. H owever most of these non­

-diagonal terms can be expressed by the diagonal ones as will be shown

in the n e xt s ection.

3. Formulae for the Radial Functions

All the functions involved a re express ed in dependence on a dimen-

s io nless v a rio.b l e which i s related to the relative coordinate x

r-V~ x 2

where m i s the nucleon ma ss. Consequently a = v ~ x 2 . c

x c = .48 5 fm- the h a rd core r a dius) is the lower bound of integration.

Note tha t the integrals depend on w ( or a ) not only by means

by

of the l ow er bound, s ince the potential functions are giver1 in dependence

o n p • 11 x x) ( 11 is the averaged pion mass equal (1.415)- 1fm- 1 ).

H ence p a ssing to the variable we obtain the integrand dependent

e xplicitly on a. Thus we mus t integrate for each value of a. over the

c o mp lete interval ( a , + .. ) .

·x) The Coulomb interactio n form V ( x ) • ___IL_ __!__ •

0 137 p

if expres sed as a function of h a s p

5

.the

Two basic radial functions appear in the method of ref.l. First of

them is R nf ( r) which describes the radial motion of an oscillator with

angular momentum 3

and energy (LJ ( 2 + 2 n + e )

~(-- 2n! ~f+l cf+-{-> 2 R n t r ) • y ~---- 3 r exp ( - r 2 /2 ) L n ( r )

f(n + f+T)

where L is a Laguerre polynomial as defined in ref. 7. The normalization

relation reads

joo 2 [ R o ( r)]

!I n L d r • I .

For the numerical treatment it is convenient to express the Laguerre po­

lynomial by means of the confluent hypergeometric function (ref. 7) which

leads to

"· (.rl- f ;-;; K 0 C 0 r f + 1 exp ( - r 2 / 2 ) F ( - n , f + ...!_, r 2 )

.L n L I I 2

where

2 f + 1

ii ) . I( f (2f+l)!!

c n e· J•l

2j+2f+l

2 j

( 2)

( 3)

The second function is c' f) ( r. r') m which is defined and studied in detail

in ref,l. We recall its most important property

+oo

I 0

Gcf> (r,r')l(r')dr'• I n•O

n fa m

I 4(m-n) o

R nf (r) +oo R nf ( r ') f ( r ') d r

which is valid for any quadratic integrable function f. For applications

connected with the calculation of matrix elements only the special case

r-a, r ' ~ a, m• 0, ~ 1, + 2, ••• is of interest. The corresponding formula

reads

(f) -G m ( a, r' ) • - M _ m, f ( a ) W _ m, f ( r ') ••• m • - I , - 2 , ••

6

2 p -

1-)R f(a)R f(r')+

2p-l m m

m • 0, I, ...

where C is the Euler constant. The functions W and W which enter

the integ rals will be specified later.

Now w e can list all the integrals occuring in the method of ref. 1

( i- 0,1 ••• 5 ):

where

[ 0 l

[ I )

[ 2 l

[ 3 l

[ 4 l

v ( r) I

+oo f Rmf(r)v

1(r)R.f(r)dr

• ·~ J \Vmf(r) v

1(r) Rnf(r)dr

+"'

J •

1Vmf(rlv1

(r lR.f(r)dr

+oo 2

f . [wme<r>l v 1

( r) d r

+oo f

- 2 [wmf(r)] v ( r) d r

• I

is related to V ( x l I

v ( r) I

2 . --by

Note that integ rals of the type [ o ]

m• 'O,l,2, ... n• 0,1,2, ...

mm0,1,2, . .. n•O,l,2, .. (5)

m•l,2,3,, .. D•0,1,2, ...

m • 0,1,2, .. .

Pi~ 1,2,S, ...

are symmetrical with respe-ct to

n, m and hence it is sufficient to calculate them for n ~ m only.

As to the norr- diagonal integrals for the tensor force the following

recursion relation lea ds to a straig hforward expression of the types [ O I, [ I ] , ( 2 j by means of the corresponding diagonal integrals

Rn, f+2• --;:::===::::-[r.Rn-t,f+2 (r)+jn+f++R.e(rl-J;.;:IR,.+4 f (r)J.

/n +f+-5-2 (~.)

Only the integrals

7

+oo +oo f R00 ( r )v

8(r)Wm

2(r)dr , f R

00(r)v

8(r) Wm

2(r) d r

cannot b e evaluated by means of ( 6) and they have been calculated sepa­

rately as well as the n o rr- diagonal integrals of the type [ 3 ] and [ 4 ]

(see tabl e 1).

Now we s h a ll define the functions \Vmf and IV m f

I. ( r 2) - 1

w t•>--• e a e 1 m Vr -m+-;- +4'2+'7

where .,.>. P. i s the Whittaker function (ref. 7) . For r not too l arge , say

r $ b (the value b-2.5 has been chosen for the numerical calculation),

the W::. functio n c a n be calcul<?-ted by means of the hyper geometric functionx)

where

f+l 2 W n ( r ) • _1_ [ r exp ( - r I 2 )

mt. m t

+F~ 2 e +I Ke

C m, -f- 2

exp ( r2 I 2)

C a nd K are d efined by ( 3) .

3 2

1 F I ( m , f + -

2- , 1 ) +

(1)

1 F

1 (1-m,~ -f, - r

2 ) J

For r > b formula ( 7 ) is not convenient for a numerical trea t.mnent,

since W which te nds to zero for r .. .., • i s expressed as a difference

of two diverg ing functions. The following asymptotic for mul a is u sed irr­

si:ead of ( 7 )

where

n• -2m+f+l

W n ( r ) • exp ~r 2 I 2 ) r ( I + L ak(r)) m L ka1

8 k+l(rl---~-2

r

a k( r ) I -- ( m + k. ) ( m .+ k. - e - -- ) k.+l 2

m(m-f-+) a

1 ( r ) .. -

( 8) k. • I , 2, . .•

x) The confluent ' hypergeometi-ic functio n is calcula ted b y means of its Taylor series, which c o nverges very

6rapidly for a ll values o f par ameters,

with an absolute e rror less than 10 •

8

'The upper bound in the sum has been chosen in tlependence on r

a ccording to the following . conditions

a { r ) k +I

----- \ < I a { r )

k

and \a \ > 10-7

k a 1,2, ... , n k

If these con ':iitions are not fulfilled for any k > 0 (this case can occur

only if is near to b ) we take

-2 m + e +I 2 r exp {- r /2 ). w me< f) D ----- - ----:-- -

2 -2 m +e+l exp {- b / 2) b

n. 'The general formula for w me is rather complicated and here

only explicit expressions for • 0,1,2 w ill be given (for the n otatio n see

eq. ( 3)):

( 9) 1 -A lr)-8

1{r) ea1

2 r m+f m+

3 --A

1{rl---8

1{rl-A <r> ... e-2

4 r1. m + 2 r m + m+ 2

The functions A 0

{ r l 8 0

{ r l are defined recursively by mea ns of two

il. . f ti H1 + l { ) i - l { ) h . h t ' al t th H "t aux tary unc ons 0

r , 0

r w tc are p rop Qr to n o e ermt e

polynomials of degree 2p- 2 a nd 2p-1 respectively , multip lied by exp ~r 2 / 2 l.

For calculating them, recursion relations are again the most convenient.

'Thus we have

8 1 ( +l

{rl•(p-- )8 {i)-rA {r)+H {r) 2 p-I P • P

(-) (+) (-) H ( rla-r H (r)+(p-l)H (r)

p p p -1

u1+) { r) P +I

1 (+) • r H (- '< r ) + ( p - - } H { r )

p 2 p

A { r) • p A { r } + r B { r } - H 1 _, { r ) p+l p p p

Finally B 0

{ r ) a nd A 1

{ r ) a r e defined by

p = 1, 2, ...

"I' t 2

B 0

{ r ) • - exp ( - r 2 / 2 ) [ C + 2 In 2 + 2 \[Ti' J e [ <ll ( t ) - 1 I d t I

9

( 10)

A 1

( r) = r B 0

( r) + Vrr exp ( r 2 / 2 ) [ cl> ( r)- I )

w h e r e C i s the E ule r con stant a nd

1' <ll ( r) a J ,;-;;

2 -t

dt-2 r F (_!_ , ~,-r 2 ) .

I 2 2 .;-;;

( 11)

Her e again the tro uble w ith numer ical a c c ura c y a ris e s b y ev alu a ting the

expressi o n

exp ( r 2 / 2) ( ¢J ( r ) -I ) .

F'or small values o f we can u s e directly the fo rmula fo r the 1 F 1 a nd

s imilurly the integr a l in ( 11) can be expressed as a diffe r e n ce o f two

rapidly conve r gent series

1' 2

2 y7;" J e t ( ¢J (I ) -I ) d t a L n :a 0 n +

(2

ra)n+l

(2n+l) !! ( 12 )

r- I 3 2 -2yrr r 1F

1 (-,-. r ) .

2 2

H owever f o r l a r ge r ( in our case r > 2 . 5 ) w e must use the fo l-

lowing a s ympto tic rela tio n ( r e f. 7 )

2 n- I k 2k r

[<l>(rl-1 l- _:_:_ (-I)

e I. + • Z n ( r) ( 13 ) .r;r, ( 2 It - I) ! ! r 2k

kaO V rr

w h e r e ., Z n ( r) \ <

2 --(2n-ll !t 2 n + 1

r

and con s eque ntly f o r r > y 2 we g et a g o od a ccura c y r e ta ining only few

te r ms ( we put n• 6 ). The inte rval of i nteg r a tio n in ( 1 1 ) i s then divided in.

two p a rts : <0, 2 .5 > a nd < 2 .5 , r > The integ r a tio n over the

firs t inte rval i s perfo rme d a c c ording to e q. ( 12 ) w hile for the sec o nd in­

te rva l w e integr a te dire ctly the rig ht- h a nd s ide of ( 13).

The accura c y o f these methods fo r dete r m ining o f W a nd W h a s been

tes ted as d escribed in the n e xt sectio n.

10

4. Arra n gement a nd Accuracy of the Tables

For the numerical integ r a tion the R omberg method h as been a dopted

( r ef.B) . However the determination o f the accuracy has been modified. Let

u s denote the result after the n--th iteration by Then the accuracy

parameter is compared w ith

if

or if n < 5 ( n- 5 means tha t the integra nd i s evalua ted in 32 points of

the basic interval of integration) . In the remammg cases is compa-

1 -4

red with - I n _ 1 • The value t •10 has been taken throughout

the tables.

The integration starts with the "basic interva l" < a, 3 > (note

tha t the values of a r ange from .168 to . 267) which should give the main

contributio n. Let us denote the res ult by t Then integrals j n over· in-

tervals < n, n+ 1 > ( n- 3 , 4 ••• ) are calcula ted a nd a dded to l, and for

each of them the qua ntity t n i s determined by

where

(na { l inl if

li n/J' J

I} J <I

i, + 1 .

This process i s continued until for three s ubs equent 'h the values

are less than . Thus the l owest value fo r the a pproxima ted u pper

bound of integration. i s 6 .

The accuracy o f this appr oximation of the upper bound has been - 6 :...7

tes ted by calculati <ng some integrals with l •10 and t •10 •

It turns out that: ( I) the change in from 10-6

to 10-7

a ffects the

value o f the con s i dered integr a l s less tha n i s the a ccuracy of the compu­

ter ( 9 dig its ) a nd consequently the values so obtained may be consi.­

dered as exact.

( II) if t •10-4

then for integra ls with the absolute value

a t least four dig its behind the decima l point are exact, while for the

other s a t least three digits behind the decimal point a re exact.

11

< 1

The accura cy o f the methods for calcula ting W a nd W desc ribed

in sec.3 h as been tested putting V 1

( r ) a 1 • In th i s case exact f o r-

mulas for the integr a l s of the firs t three types in ( 5) can be derived

since all the functions involved a r e solutio n s of the differential e qua tio n

d 2 [ -~~- 2

d r 2 - r e ( £2+1 l -A j £<2 ( r l = g < f' ( r l

r

for differe nt values of A a n d d iffere nt rig ht- hand s ides ( r ef.1). Compa--

ring them w ith the nume rica l calculations performed w ith

- 7 obtai n an e rro r l ess than 10 •

- 6 • 10 we

In table 1 the non- diagona l integrals for the tensor force ( i a 3)

a r e g ive n , E ach column contains 18 integrals for a g iven i\ i n the fo l­

lowing order ( again only integ r a l s w hich have been u sed in r e f. 2 are

l i s ted) :

+oo f IV (r) v ( r )W (r ldr

mO 3 m -1 ,2 Ill= 1.2

+oo -f w 1,2 ( r)v a ( r ) ~~'oo( rl dr

+oo

f Wm 0 ( r ) v3

( r ) \Vm+l.2

( r ) d r m a 1, 2, . . . , 6

+ oo

f R0 0 ( r ) v 3

( r ) W m 2

( r ) d r m = 0, I

+ oo f R ( r )v ( r) W ( r) d r

0 0 3 m 2 m= l ,2, ... ,7

I n tabl e 2 th e values of integrals diagon a l with respect to r are

g iven. The y are subdivided according to the values of parameter a ( w )

and e • The fo llowing values of have been considered: a •.267,

.246,.221, . 197, .168 which corresponds t o w =2 5 .10, 21.31, 17.20, 13. 67,

9 .94 MeV. For each equals 0 , 1,2 i.e. we h a ve 15 pairs of va-

lues a These values are i ndica t e d at the head of each p age.

Futhe r each page cons i s ts of seven colutnns. The very left one i s a n

" ind ex col umn" a nd contains in e ach row three integer number s : firs t

o f them denotes the g ive n type, the second refers to index m and

the third t o index n of int egr als placed in this row ( fo r the notation

12

cf, eq. (5)). Further the ( i+2)-th column (i•0,1,2, ... 5) from the left

contains integrals (with indices a s given by the "index column") for the

potential v 1 (e.g . the second column from the left contains Coulomb . in,..

· teg rals, the third central- singlet integrals etc)~/ The limitation of indices

e I m n corresponds to the limitations used in ref,2.

An ALGOL- 60 programm for tt-ie calculati on of integ rals is availa ble, I

The authors would like to thank to _:Professor I. Uiehla for permanent I

interes t in this work. The technica l help of M.Plchova is g reatly apprecia-

ted.

R e f e r e n c e s:

1. J.Blank (to be published in Czech.Journal of Physics) I

2, J,Blank, l,Ulehla, preprint (Dubna, 1967)

3. T,Hamada, !,Johnston, Nucl,Phys., ~ ( 1962) 328

4. C ,Bressel, A,Kerman, E,Lomon: Bull.Am.Phys.Soc., .1Q_( 1965) 584

5 . T ,Kuo, G ,B rown, Nucl.Phys., 85 ( 1966 )40

6. T .Hamada et al . Progr.Theor,Phys., _;u_ ( 1965) 769

7. I,Gradstein, I.Ryzhik., 'Tablicy integra lov ... 4-th ed.

( Fizmatgiz-Moscow, 1963).

8 , F ,Bauer, ,CACM !_ ( 1961) 225

CACM .!2_ ( 1962) 169 and 281.

R ecived by Publishing Department

on July 18, 1967.

x/ For e • 0 the last two columns are e mpty, since the spin- orbital and quadratic spin- orbital integrals a re multiplied by zero· in ( 1). On the other hand the tensor integrals for f~ 0 are necessary for the cal­culation of the nondiagonal integra ls e. 0, e -·- 2 ( cf, ( 6)).

13

il

IX)

"'

..

..... :

..

"' "'

..

"' ..,., "'

<f

"' "'

<f

co t"'\ r-- \0 I""'\ .. 0 M I'""\ I""\ 0\ 0\\0N...tOt-t'--­..:t oct ,.... C\J 0\ 0\ r-t

"' "' "' IX) "' 1' co 0 C\J \0 ,.... .. ('IJ C"'J ,....,

..... '"' ..... I 1 1

. .... n M n r- ~ 1""'\ ~ ~ ...t ~ o lt'\ONO\r-t00\0<\ICOt­OOV'\\OIC'\If\11'\C\ItOI""'\a>

• •0\CD rl\0 Q)ll'\t"--U"\0 1 • • • • •••

It\ co Q) .. It'\ ,.... C\1 f""'' t- r-- ("'\ r-t

1 I 1 1

..... '"' C\J 0 0\ \0 r- \0 If\ co co 0 0\ 0 co 0\ ..... \0 0 N It'\ M t- \0 N n t- 0 \0 CO ...t r-t \0 0 N CO 0\

~ ~ ~ ~ ~ ~ ~ ~ g ~ ~ p ~ g ~ ~ ~ 8 0 t"'\ 0 • • • • . • . • . • .•.• M •0\0\0\ 1 r-1\0\0r"\ll\COr-t

0 CO \0 M If\ It'\ C\J

"' 1'

~ g ~ ~ g ~ ~ ~ ~ ~ ~ ~ ~ r- n r- ...t r-t M 00 r-t 0\ 0\ \0 0\ M 0 0\ If\ 0\ 0 N ll\ CO t- "lit r- \0 \0 "" o co n o o • n 0\ r- \0 It'\ II'\ r- •

"' ..... "' I I \0 co U"\ If\ "' . 1'

•••• If\ r-t ..,. C\1 0 0\ If\ .. ,.... • • •

• .. (7\ \0 r-t .....

.... '"' "' ..... 0 ..... 0\ o • 0\ n n co • 0\

~ ~ ~ ~ ~ g ~ s ~ ~ ~ ~ @ ~ ~ N 0 t- N 0 0 t- ...t N t"'\ 0 \0 r-1 r-t 1""'\

0 f""\ t- M C\1 0

N 0\ , N '"' .... . "i

. . .. ..... 0\ 0 t"'\ ... IX) '"'

1

.... . "' t"'\ I"'\ It'\ .. ,....

'"' ..... 1 I I

r- 0\ 0\ M It'\ r- n a> • 0\ o r- \0 0\ ..:t r-t 0\ CO rl 0 0 N I""'\ ..:t IC'\ 1""\ If\ \0 t- 0\ \0 0\ 0\ If\ \0 r-t co • r- o n m "' r- n m o \0 N 0\ M 0 N a> \0 t- 0 0 CO Cl) \0 ll\ \0 CO 0\ M 0

• • • • • MoO..Ot- • • • • C\11""'\<o:t t-r-1 • • •• • eOt'--C\It"'\ I It"'\~~~ I I I\Or"'\7<-t'1

14

&• .267

0 0 0 0 0 0 0 2

0 0 3

0 0 'I 0 0 5 0 0 6 0 0 7

0 0 8 0 0 9 0 0 10

0 0 11 0 0 12 0 0 13

0 1 1

0 1 2 0 1 3 0 1 'I

0 1 5 0 1 6 0 1 7 0 1 8

0 1 9 0 1 10 0 1 11 0 1 12

0 1 13 0 2 2 0 2 3

0 2 'I

0 2 5 0 2 ' 6

0 2 7

0 2 8

0 2 9 0 2 10 0 2 1l 0 2 12

0 2 13 0 3 3

0 3 'I

0 3 5 0 3 6 0 3 7 0 3 8

1·0

.0663

.023 2

.0131

.0083

.005'1

.0035

.0021

.0011

.0003

.1}00'1

.0009

.0013

.0011)

.0019

.0523

.0215

.0128

.0083

.0055

.0035

.0020 .0010

.0001

.0006

.0011

.0016

.0019

.0'1'15

.0195

.0119

.0078

.0051

.0032

.0018

.0008

.0001

.0008 .0013

.0018

.0391

.0177

.0109

.0072 .00'17

.0029

- 3.3276 - 3.3180 - 3.0882

- 2 .8086

- 2 .5 230 - 2.2'!6'1

- 1.98'!5

- 1. 7392

- 1.5108 - 1. 2990

- 1··1029 .9216

.75'12

.5997

- 3.5'176

3.'1027

- 3.1511 - 2.8677

- 2.5800

- 2.2998 - 2.0322 - 1.7796

- 1.5'127 - 1.321'1

- 1.1155 .92'11

.7'167 - 3.3790 - 3.1968

- 2.9527

- 2.6873 - 2.'1189

- 2 .1563

- 1.90'12

- 1.66'16 - 1.'1387 - 1. 2265 - 1.0281

.8'130 - 3.097'1

- 2.9108

- 2.68'!0 - 2.'1'120 - 2.1977

- 1.9579

Table 2

- 1.0381 .9132 .7699 .6'!32

.53'16 .'!'118

.3622

. 293'1

. 2338

.1819

.1365

.0967

.0617

.0309

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- 2 . 952'i

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0 5 13 0 6 6 0 6 7

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0 6 12 0 6 13 0 7 7 0 7 8

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- 2 . 2797 - 2.1077 - 1.9397

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2 .9116 - 2. 7691 - 2 .6138

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2 'i 1 2 'i 2 2 'i 3 2 'i 'i

2 'i 5 2 'i 6 2 'i 7

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2 5 2 2 5 3 2 5 'i

2 5 5 2 5 6 2 5 7 2 5 8 2 6 0

2 6 1 2 6 2 2 6 3

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2 7 0 2 7 1 2 7 2 2 7 3

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1=2 .0062

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1 . 7352 . 8506

129 . 867

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26.796'! 2 .7316

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356. 267 36 . 6580

2 . 1290

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a-.197

0 0 0

0 0 1 0 0 2 0 0 3 0 0 'i 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 10 0 0 11

0 0 12 0 0 13 0 1 1 0 1 2

0 1 3 0 1 'i 0 1 5 0 1 6 0 1 7 0 1 8

0 1 9 0 1 10

0 1 11 0 1 12 0 1 13

0 2 2

0 2 3

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0 2 13 0 3 3

0 3 'i 0 3 5

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- 2.9958 - 2 .9003 - 2.7753 - 2 .6378

- 2.'!957 - 2.3533

- 2.2129 - 2.0758 - 1.9'!28 - 1.81'i'i

- 1.6906

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- 3.3319 - 3.2128 - 3.071'! - 2.9196 - 2.7638 - 2.6078

- 2.'!537 - 2.3029

- 2.1562 - 2.01'!0 - 1.8767

- 3.5093

- 3.'!792 - 3.3838

- 3.2553 - 3.110:0

- 2.9585 - 2. 7997

- 2.6'i27 - 2.'!876 - 2.335'! - 2.1869

- 2.0'i28 - 3.'!931 - 3.'1296 - 3.3227

- 3.1918 - 3.0'179 - 2 .897'1

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- .'1632

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- .2990 - .2$9 - .9978 - .9567

- .8883 - .8151 - .7'!38 - .6767

- .61'13 - .5566

- .5032 - .'i5'i0

- .'!085 - .366'!

- .3275 - . 9628

- .9203 - .8596

- .79'12

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- .5529

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- .3671

- 1-9359 - 1.8680

- 1.71'18 - 1.561'1 - 1.'!210

1.29'18

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- .9858 .9010 .8232

- .7516

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- 1.8103 - 1.6706 - 1.5360 - 1.'1103

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- .9130 - .83'!6

- .7617

- 1.96'i'i

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- t.t975 - 1.1008 - 1.0106

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60

a= .197 0 3 9

0 3 10 0 3 11

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0 'i 5 0 'I 6 0 'i 7

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0 5 12 0 5 13 0 6 6 0 6 7 0 6 8

0 6 9 0 6 10 0 6 11 0 6 12 0 6 13 0 7 7

0 7 8

0 7 9 0 7 10 0 7 11

0 7 12 0 7 13 0 8 8 0 8 9

0 8 10 0 8 11

0 8 12 0 8 13 0 9 9

1=0 .0073 .0058

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2 .8509 - 2 .7'!'13 - 2.628'1 - 2 .5067

2.3818 2 . 2556 2.7755 2 . 68'!6

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2 . 2391 - 2 . 6090

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- .6626

- .6151 - . 5680

. 5222

.'l782

.'1361

.7300

. 6979 - • 6593

- . 6175 . 57'15

. 5316

.'!897 - . 'l'l91

.6766

.6'172

.612'l - . 57'l7 - .5357

.'l966

.'1581

.6270

. 6o01

.5687 - . 5 3 '15

.'l991

.'l63'l

.5812

6 1

- 1.2689 - 1.1708

- 1 . 078'l

- . 9915 - .9097

- 1. 7725

- 1· 70'l3

- 1.61'l9 - 1.5161 - 1.H51 - 1.3155

- 1.2192 - 1.1272 - 1.0398

- . 9570

- 1.6685 - 1.60 '!6

- 1 . 5237

- 1.'13'15 - 1.3'l2'1 - 1.2508

1.1616

1.0756 - . 9933

1.566'1 1.5068

- 1.'l332

- 1.3521 1 . 2679

1 . 1836

- 1.1008 - 1.0205 - 1.'1683

1.'1 127 1.3'15'1

- 1.2713 - 1.19'l2

1.116'! - 1.0396 - 1.3750

1.3233

1. 2615 - 1. 1936

- 1.12 26

- 1.0507 - 1.2869

a= .l97 1 =0

0 9 10

0 9 11 0 9 12 0 9 13 0 10 10

0 10 11

0 10 12 0 10 13

0 11 11

0 11 12 0 11 13 0 12 12 0 12 13

0 13 13 1 0 0

0 1 0 2

1 0 3 1 0 'I

1 0 5

1 0 6

0 7 0 8

0 9 1 0 tO

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1 0 13 1 1 0

1 1 1 1 1 2 1 1 3

1 1 'I

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7

1 8

1 9 1 10

1 11 1 12

1 1 13 1 2 0

2 1 2 2

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.0209

. 01'19

. 011'1

.0090

. 0370

. 0200

. 01'13

. 0110

. 035'1

. 0192

. 0138 .03'10 . 0185

.0328 - .0310 - .0398

.0320

- .0269 .0230

- .0198

.0173

- .015·1 - .0133

- .0117

- .0103 - .0090 - . 0079 - .0069

.0127

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- • 0181

- .0162 - .01'15

.0129

.01 15

. 0103

. 0092 - .0083

.007'1

.0066

. 0059

.01'17

.018 0

.0106

.0099

- 2 .5208

- 2 .'1 232

- 2 . 3 190 2. 2105

- 2 .'1'166

2 .3620 2 . 2693

- 2 .1710 - 2 . 2900

- 2 . 2093 - 2 .1217 - 2 .1'103 - 2 .0638

- 1. 9980 3.'1910 3.9866

'1.1529

'1.18 12 '1.1360 '1.0'17'1

3.9316

3.798'1 3.65'10

3.5027 3 .3 '173

3.1901 3 . 032'1

2 . 875'1 1.6601

1. 9983 2 .t690

2 . <529

2 . 28'11 2. 2809 2 . 2s'H 2 . 2 107

2 .1553 2 .09 1'1 2.0212

1.9 '165 1.8688 1.7890

. 637'1

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1.0736

- .5567

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- .'1972 . '16'19

- . 5392

.5168 - . '1909

.'1626

- .5006

. '180 1 - . '1 585 - .'1651 - .'1'185

- .'1327 .9056

1.0071

1.0186

.9961

.9580

. 9 125

. 8636

.8135

.7635

.7177

.6699

. 6235

.5788

.5358 .3120

.'109'1

. '16'19

.'1918

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.'1980

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. '1735

.'1559

. '1363

.'1155

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.3722

. 350'1

.0172

. 071'1

.127'1

.17'16

62

- 1. 2388

- 1.18 19

- 1.1195 t.05'10

- 1.20'11

- 1.1593 - 1.1068

1.0"'193

- 1.12 6'1

- 1.08 '17 - 1.0363 - 1.0537 - 1.0150

.9859 1.8920 2.1158

2 .1533

2 .1197 2 .0535 1.971'1

1.8815

1.7885 1.69'18

1.6019

1-5107 1.'1218 1.335'1

1.2517 .6830

. 8998 1.02'1'1

1.0870

1.1100 1.108 1 1.0902

1.0.620

1.0272 .9881 .9'16'1

.9031

.8592

.8150

.0915

.2151

.3366

.'1367

a•.l97

1 2 'I

2 5 2 6

1 2 7 1 2 8

1 2 9 2 10 2 11

1 2 12

2 13

2 1 0 2 1 1

2 1 2

2 1 3 2 1 'I

2 1 5 2 1 6

2 1 7

2 1 8 2 1 9 2 1 10

2 1 11

2 1 12 2 2 0

2 2 1

2 2 2 2 2 3 2 2 'I

2 2 5 2 2 6 2 2 7

2 2 8 2 2 9

2 2 10 2 2 11 2 3 0

2 3 1 2 3 2 2 3 3

2 3 'I

2 3 5 2 3 6 2 3 7

2 3 8

2 3 9

2 3 10

1=0

.0111

- . 0107 - .0100

.0093

.0085

- .0078

- . 0071

.0065 - .0059

- ' . 005'1

. 0886

.0739

.0627

.05'11

. 0'172 .0'115 . 0367

. 0325

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. 0256 . 0227

.0201

. 0178

. 0'197

. 0'172

.0'1 28

. 0386

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. 0313

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.0255

.0230 . 0207

.0186

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. 0175

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. 01'10

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.0106

. 0097

.0088

. 0080

1.1'1'1'1

1.1901 1. 2163

1. 227'1 1.2268

1.2169

1.1996

1-176'1 1.1'186

1 . 1171 - 7.6'123 - 8 . 6862 - 9 .0360

- 9 .0968 - 9.0035 - 8.8182 - 8 .57'18

- 8 . 2936 - 7.9880 - 7 . 6667 - 7 .3361

- 7.0006 - 6 . 6635 - 5 .5268

- 6.33 8 '1

- 6.6'131 - 6. 7311 - 6. 700'1 - 6.5966

- 6.'1'152 - 6. 2618 - 6.0565 - 5 .836'1

- 5..606'1 - 5.3702 - 2.23'!0

- 2 .5737

- 2 .7081 - 2.7538

- 2 .7500

- 2. 7 15.6 - 2 .6607 - 2.59 18

- 2 . 5 132

- 2 .'1277 - 2 . 33 75

. 2090

• 2321 . 2'165

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.2568

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. 2331 - 2 .0011 - 2 .188'1 - 2 . 2008

- 2.1'191 - 2 . 0682 - 1.9730 - 1.8712

- t. 7670 - 1.6627 - 1.5599 - 1.'1595

- 1.3619

. 5085

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. 58'12

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'1.60'1 ~

- '1.6571

- '1.5772 - '1.'136'1 - '1.265 2 - '1.0787

- 3.8855 - 3 . 6903 - 3.'!961 - 3.30'17

- 3 . 117'1 - 1.2676 . - 2 . 93'17 - 1.3708 - 2. 9056

- 1.5285 - 3 .2571

- 1.5612 - 3 .3'16'1 - 1.5'1'1'1 - 3 .3315. - 1.5030 - 3 . 26'11 - 1.'1'181 - 3 . 1675 - 1.3858 - 3.05'12 - 1. 3195 - 2 . 9311 - 1 .251'1 - 2 . 8028 - 1.1827 - 2 .6719

- 1.11'1'1 - 2 .5'!06 - 1. 0'171 - 2 .'1099 - .5366 - 1.1'!77 - .60'11 - 1. 2990

- .6220 - 1.3'!53 - .6196 - 1.3'185 - .6068 - 1 . 3291 - .5880 - 1. 2967 - .5656 - 1.2563 - .5'112 - 1.2109

- .515.6 - 1.1626

- . '1895 - 1.1125 - .'1632 - 1.0616

63

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2 't 3 2 't 't

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2 5 1 2 5 2 2 5 8 2 5 't 2 5 5 2 5 6 2 5 7 2 5 8

2 6 0

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- 5.2183

- 1.7'107 - .70'18 -25.0068

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- .0055

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- .006'1 - .0062 - .0060 - 1.1768

- .8638 - .17:5 - 5.5806

- 2 .9718

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- .sass - .8201 - .068't - .0782

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- 2.5729

- .8'108 - .'1005 - 12.2989

- 6.7088 - 1. 1336 - .0927 - .00'15

- .0002

64

a=.l97

0 0 0

0 0 1

0 0 2

0 0 3

0 0 'I 0 0 5

0 0 6

0 0 7

0 0 8

0 0 9 0 0 10 0 0 11

0 0 12 0 0 13

0 1 1 0 1 2

0 1 3 0 1 'I

0 1 5 0 1 6

0 1 7 0 1 8

0 1 9

0 1 10

0 1 11 0 1 12 0 1 13

0 2 2

0 2 3 0 2 'I

0 2 5 0 2 6

0 2 7 0 2 8

0 2 9

0 2 10 0 2 11 0 2 12

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1=1

. 06'13

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2 . 2873 2 .'1'103

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2 .6905 2 . 79 10 2 . 8772

2 . 950'1 3 . 0120 1.7828 2 . 2362

2.6169 2.9513 3.2'180

3.5111 3.7'136 3. 9'180

'1.1267

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2.8616

3.368'1 3.7999

'1.1778

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5 . 2978

5.'1979 5.6713 5.8203

5.9'168 3.99'16 '1.5202 '!.972'1 5,.368'1

5.7179 6.0267

- .01'16 - .0300

- .0386 - ,0'1 26

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- .0620

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- .0091

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- . 27'11 . - . 00'18 - .3'127 - .006'1

- . 3963 - .0080 - . '1386 - .0095

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- .517'1 .532 1

- .5'126

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- • 6871 .6973 .7037 .7069

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- . 0197 - .0216 - .023'1

• • • 197 0 3 9

0 3 10

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0 7 7 0 7 8 0 7 9 0 7 10

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1 ·~ . 0066

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7.'1'137 7.6810

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7 .2081 7.5983 7.9 '115 a . 2'i30

8 .50 68

a. 7361 8 , 9337 7.3 282

7. 8 193 8". 2 '176

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9. 239'1 9 .'1902 9 .7067

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9 .7569 10.1381 10 .'!705

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2.2871

2.6'!08 2.9759 3.29'!0

3.5960 3.8829 'l.155'i

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1.5221 1. 7168 1.8993

2 .0710

2 . 2328 2 .3855 2 .5297 2 .6660

2 .79 '!8 .1899 .3625

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a= .197 1 2 'i

1 2 5 2 6 2 7

2 8 1 2 9 1 2 10

1 2 11

1 2 12 1 2 13

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2 1

2 1 2 2 1 3

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2 1 6 2 1 7

2 1 8 2 1 9

2 1 10 2 1 11

2 1 12

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2 2 'l 2 2 5 2 2 6 2 2 7 2 2 8

2 2 9 2 2 10

2 2 11 2 3 0

2 3 1 2 3 2

2 3 3 2 3 'l 2 3 5 2 3 6

2 3 7 2 3 8

2 3 9

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1=2

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2 .386'1 2 .518 '1

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-18.0199 -19 .9799 -21.87'i3 - 23 .70 11

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- 3. 8006

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- 6 . 8281 - 7 .'!978 - 8 .1336 - 8 .736'1

9 . 3072

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1. 7'l'i0 - 2 .12'1'1 - 2 . 'l90'l - 2 . 8'l l'l

- 3 .1769 - 3 . 4971

- 3 . 8 020

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1. '1 599 1.5769 1. 6863 1 . 7888

1. 8850 1. 9754

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-12 .2830

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1.107.::1 1.9692

2 . 8 153

3 . 6426 4.'1467

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1 .3392 . 07 17 .1288 .1855

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2 .1170

3.10'12 'i. 1097 5.1238

6 . 1386

7 .1'l8 'i 8 .1'189 9. 1365

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11.0629 11.9980

12 .9123

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7.7750 8. 7218 9 . 65'13

10.5706

11.'1690 • 52'!6 .9606

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3 . 7237 '1 .1787 '1 . 6271

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1.3380

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119.'157 36.?187

17.5327

2953.01 2?58.00 6'!6.326

67.'!919

3.9?26

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a= .l68

0 0 0

0 0 1 0 0 2 0 0 3 0 0 'I

0 0 5 0 0 6

0 0 ? 0 0 8

0 0 9

0 0 10

0 0 11 0 0 12

0 0 13

0 1 1

0 1 2 0 1 3

0 1 'i

0 1 5 0 1 6 0 1 ?

0 1 8 0 1 9

0 1 10

0 1 11

0 1 12 0 1 13 0 2 2 0 2 3

0 2 'i 0 2 5 0 2 6

0 2 ?

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0 3 3 0 3 'i

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- 1.8985

- 3.16'!5

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- 3.2012

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- 3.0251 - 2 .9 216

- 2 . 8 13? - 2 .?038

- 2 .5933

- 2 .'!835

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- 3.'!122

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- 3.2'158

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- 3.5001

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75

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1. 7202

1.60?3 - 1.'19?8

1.3958 - 1.3018

1.215'! 1.135?

1.062 1

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1.98 '!9

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1. ?60?

1.65'16

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1. 2002

1.1256

1.0556 .9898

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1.8 ?'!9 - 1.780'!

- 1.6825

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- 1.1665 - 1.0956

- 1.9589

- 1.9119

- 1.8 383 - 1· ?52? - 1.6629

- 1.5?31

a :.169 1=0

0 3 9

0 3 10

0 3 11 0 3 12

0 3 13

0 '1 '1

0 '1 5 0 '1 6 0 '1 7 o '1 a 0 '1 9 0 '1 10

0 '1 11 0 '1 12

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0 5 5 0 5 6 0 5 7

0 5 8 0 5 9

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0 5 11

0 5 12

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0 6 6 0 6 7 0 6 8

0 6 9 0 6 10

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0 6 12 0 6 13

0 7 7

0 7 8 0 7 9 0 7 10

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2 .5986

3.5026

3 . '1692

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3.3227 - 3.2273

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2 . 7 9 30

2 .680 6 3 . '15'18

3 . '107 '1

- 3 . 3375

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3 . 1570

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- 2.9'192

- 2 . 3 '108

- 2 . 7 313

- 3 . 3761 - 3 . 3 20 1

3 . 2 '166 3 . 1609

3.0667

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2.8638

2. 7 585

3 . 27 80

3 . 2169

3 . 1'118 3 . 0566

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1.3394

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1.648 '1

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1 . 3672

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1 . 6779

1 . 6336

1.5787

1.5169

1 . 4513

1.38 '10

1.3165

1.6035

1.5610 - 1.5095

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76

a= .168 1=0

0 9 10

0 9 11

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0 9 13 0 10 10

0 10 11 0 10 12 0 10 13

0 11 11

0 11 12

0 11 13

0 12 12 () 12 13

0 13 13

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- 2 . 9857

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3 . 70 9 1

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1.327 2 1. 2921

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2 .'1 277

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1.13 68 1 .2709

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17.3597

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20.1296 21.36'l'l

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73.7633 '1.2330

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3 . 20 '15 3 . 'l525 3 . 6933 3 .9272

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5.5633 . 3295 .6011

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1. 2768

1 . '1 9'17 - 1. 7 130

1. 9307

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2 . 9950 . 1301 . 2393

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n= . l68

2 'I

2 5 2 6 2 7 2 3

1 2 9

1 2 10

2 1 1

2 12

2 13

2 1 0

2 2 1 2

2 1 3

2 1 'I

2 1 5 2 1 6 2 1 7

2 1 8

2 1 9 2 1 10

2 1 11

2 1 12 2 2 0

2 2 1

2 2 2

2 2 3

2 2 'I 2 2 5

2 2 6 2 2 7

2 2 8

2 2 ,;

2 2 10

2 2 11 2 3 0

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2 3 2 2 3 3

2 3 'I

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1=2

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2 .5756

2 . 7379

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