Correction to intrinsic magnetic moments by central and tensor forces

--~

Nuclear Physics 35 (1962) 49--60; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

C O R R E C T I O N T O I N T R I N S I C M A G N E T I C M O M E N T S BY C E N T R A L AND TENSOR FORCES B. B R E M O N D

t

Laboratoire de Physique Thdorique et Hautes Energies, Facultd des Sciences, Orsay, France Received 27 July 1961 Abstract: We study the magnetic m o m e n t correction by central and tensor interactions between a

nucleon of m o m e n t u m k and spin a and the Fermi sea. The intrinsic magnetic m o m e n t operator takes the form /a = A a + B ( k • a/k~)k. The correction value is calculated by a second order perturbation method. The results are in qualitative agreement with the doubly-magic-plus-orminus-one nuclei magnetic moments.

I. Introduction The reasons for the discrepancies between the observed nuclear magnetic moments and those calculated in the shell model with a pure jj coupling scheme are usually taken to lie in collective effects or in configuration mixing effects. But, as emphasized especially by Blin-Stoyle z), a small mixing of the configurations may affect largely the moment; and, vice versa. These effects can also be studied from the point of view of 'nuclear superconductivity'. It seems clear nevertheless that an appreciable effect is due to the modification of the magnetic moments of the nucleons when they are embedded in nuclear matter. This effect appears explicitly from the study of doubly-magic-plus-or-minus-one nuclei; if some moments (O L7) are in very close agreement with the Schmidt value (1.89 n.m. instead of 1.91), some others ( K 39) a r e clearly different (0.39 instead of 0.12). Furthermore, iris pointed out by De Shalit 2) in a study of K 39 and K 4° that one can define an 'effective moment operator' from which one obtains the moments of different nuclei belonging to the same open shell. If the effects due to the nuclear density gradient are assumed negligible, one can evaluate the modification of the magnetic moment of a nucleon embedded in infinite nuclear matter. In this case, Miyazawa and Drell and Walecka 4) have studied the effect of the exclusion principle on the meson cloud. In the present paper, without any explicit reference to the meson field, our point of view is to investigate the modification of the 'effective' intrinsic magnetic moment of a nucleon added to a Fermi sea of nuclei, or of a hole in this sea, when the nucleons interact by central and tensor forces. * Work supported in part by the United States Air Force through the European Office, Air Research and Development Command. Present adress: Department of Mathematical Physics, University of Birmingham, England. 49

50

B. BREMOND

The central force effects are essentially due to a kinetic m o m e n t u m transfer from proton to neutron; the tensor forces add to this a transfer from spin angular momentum to orbital angular momentum. The nucleon polarizes the medium by transferring charge and kinetic m o m e n t u m to it. We consider this modified m o m e n t as an intrinsic effective magnetic moment of a nucleon in a medium of a fixed density. In a finite nucleus we shall add to this intrinsic moment taken for an average density the orbital magnetic moment, the modification of which has been studied by Bell 5) from a point of view similar to ours. This modification depends on the nucleon-nucleon potential, but seems rather small for a reasonable potential. We first give a formal expression of the magnetic moment, then we compute the modified m o m e n t by the perturbation theory with a fixed potential at variable density, and, finally, we use this result to obtain an effective magnetic m o m e n t of one nucleon in various configurations.

2. Formal Expression of the Magnetic Moment We consider a Fermi sea of Fermi m o m e n t u m f a n d an extra nucleon of spin ~ and m o m e n t u m k. Assuming that each nucleon keeps a magnetic m o m e n t equal to that of a free nucleon, the magnetic m o m e n t & t h e system, for an "initial" extra nucleon with a~ = + 1, is with Z~ = ~ ¢ z etc . . . . sum of each nuclear operator, (Pz) = 0.44(27~) +2.35(2~ T~) +0.5(L~) +0.5(L~ T~).

(1)

In absence of interactions, the Fermi sea contribution is such that (Zz> = ( ~ L >

= 0

(2)

(Lz> = ( L ~ L > = 0,

(3)

and the total contribution is such that

since the wave function of the extra nucleon is a plane wave. In presence of interaction the expectation values of all these kinetic m o m e n t operators are modified. Since we have only two vectors at our disposal, k and a, the spatial and time reversal invariances require that the kinetic m o m e n t u m operators have the form:

0 = A(k2,f2)a+B(k2,f 2) k .trk k2

.

(4)

Since the tensor force involves a transfer of m o m e n t u m from spin to orbital momentum, then there must exist a correlation operator. In infinite nuclear matter, the only operator related to orbital m o m e n t u m is k, the only possible correlation operator is indeed (k. a/k2)k (see the a p p e n d i x ) t . i. we recall that with 'effective operators' use must be made of the unperturbed wave functions to calculate the expectation values.

CORRECTION TO INTRINSIC MAGNETIC MOMENTS

51

3. Effective Calculation of the Correction For more convenience, we shall perform the calculation in two cases: a) k and a are parallel, each value ( O ) is parallel to ~r and (Oz),, = A + B ;

(6)

b) k and a are perpendicular, (O> is also parallel to tr and ± = a .

(7)

As a reasonable choice of tensor force, the calculation is carried out with the G a m m e l and Thaler potential 6 ) , . We use a second order perturbation method starting from plane waves with the relative wave function cut to zero at the core radius r e . This approximation is justified because the G T potential has a hard core independent of spin and isospin; the hard core interaction commutes with the magnetic moment and does not modify it. On the other hand, by an exclusion principle effect, the exact relative wave functions, which are zero inside the core, are quickly "healed" for r > rc and join the regular attractive potential wave function which may be obtained by perturbation theory. We use as unperturbed eigenfunction the one particle functions ft. The non-antisymmetrized wave function of a pair made from the extra nucleon initially in a state of m o m e n t u m k and one particle initially in a state ! inside the Fermi sea (l<), these particles being excited to states m and n outside the Fermi sea (m>, n>) is

I~ki> - - 1 0 > + l l >

= Ikl)+

~ m•n>


(8)

Emn -- Ekl

The expectation value of an operator O = ~ o , the sum being extended to the whole system, when we take = Ore, is

=

<01Ol0>+<11o11> <010>+<111>

=



+

+(lll>

(a) + I < l ~2> n >

(mn[ V[kl> (o m'~-° n - Ok--Ol) (Era. -- Ekl) 2

(b)

+ I < ~2> n >

( mni Vlkl>< kll V[nm > (o m+ on- Ok-- Ot) (Emn -- Eki) 2

(c)

(mn[ VJkl>(kl[ Vlmn) (-o,. - o n + o k "~Ol)

(d)

+ l > mZ< n < +

~ i .....

(9)

(Emn-Ekl) 2

(--°m--°'+°k+°')

1 OlO>+

(e)

(Emn -- Ekl) 2

* The values used here are taken from Brueckner and Gammel 6). ,t Our fundamental states are degenerated but in their subspace the first order energy matrix is diagonal.

52

B. BREMOND

where Ek, is the sum of the kinetic energy and an average potential energy of the two nucleons kl. The five terms (a) to (e) are represented by the following diagrams (the circles indicate the different positions of insertion of the o operator):

t. t. t k

Fig. 1

The Gammel and Thaler potential is V - - E UsT(r)+ E L" SVT(r)+ ~, S , 2 ( t r , tre, r)WT(r ). ST

T

(10)

T

In the case of an infinite model, the spin orbit part gives no contribution. The tensor terms contain

s,2(~,, ~2, ,) = E (-1)qr~(0s;~(~, ~2), q

where 47~

s~ -- i5 (*'' v)(.2, v ) ; Yg(r). After having done the spin and isospin summations in (9), we find that in the case of the b graph ( 1[1),( 1[~zl 1) and ( 11~.zTzll ) become linear combinations of the terms CTST'S' =

Z

,.....

(kllVTs(r)lmn)(mniU~,s,(r)lkt) (E~.--Ekl) 2

T~T' ----- E ,.....

(Era. -- Ekt) 2

ITST" =

(Em.-Ek,) 2

,

X .....

(10 (12) (13)

For the other graphs, they become linear combinations of analogous terms. For plane wave one has = = O. But this is not exact of wave functions perturbed by tensor forces. If we remark that lk + 1, = LCM+/r©lativ©and that we still have = O, (14)

CORRECTION TO INTRINSIC MAGNETIC MOMENTS

5a

the expectation values (11Lzl 1) and (11LzT.I 1) become linear combination of terms ~, (mnlWT'(r)Y2q(r)lkl) = qT~T,. z.. (Era, - - E k t ) 2

Taking P = k + l , Q = re+n, p = ½ ( k - l ) , q = ½ ( m - n ) , and quantizing the waves in a volume I2, we obtain

s = q-p,

(kll V(r)Ytq(r)lmn) = 1 6p. 0 Y~(s)V~(s), where

(15) t = q+p,

(16)

¢o

Vl(s) = 4x(i)'

rc

V(r)Sl(sr)r2 dr,

which integrals were computed numerically. Taking as the particle energy an effective mass approximation E a = kZ/2M * we obtain for the graphs b and d: M *~ (

U~(S)UT's'(S)

CTST'S" -- ( ~ ) 6 J D d 3 S d 3 t

( $ " t) 2

( - 1)'U .2 (

WT(S)WT,(S)[YI(S)[ 2 ,

3o d3sd3t

(2x) 6

'

(17)

(s " t) 2

M*2 ( UTs(S)WT'(S)Y°(s) /TST" -- (27t:)6 !~Dd3Sd3t ($" t) 2 ' and for the graphs c and e M*2

CTST, $, --

T~T,

/*

g~-~_~6| dasd3t (2~z) d o

VTs(S) ~ - t-~-

'

M .2 /" 72Z~_x6| d3sd3t WT(s)WT'(t)Y~(s)Y2q(t)

_

ITST'

__ M .2 f

'

WT'(S)UTs(t)y20($)

(S " 02

(2~)6 3 Dd3sd3t

where the integration domains D are defined by for b and c: for d and e:

Is+k[ > f , Is+kl < f ,

[t-kl >j,

Is-t+k[
It-k]
Is-t+kl

>f.

The T and I terms can be expressed as linear combinations of

tiT" .,

-

m * 2 /* WT($)-WT'-(S) COS n (S), (-~x)6JDdasdat (s" 0 2

M *z (

WT,(S)UTs(S) COS"(s).

(28)

~4

B. BREMOND

For the different expectation values, only n = 0 and 2 are useful; the n = 4 terms disappear because S~2 is a scalar only bilinear in the spins. We take as the spin quantization axis the spin axis of the extra particle, the previous calculations correspond to a coherent choice of axis; the spatial polar axis being the spin quantization axis. But it is useful for integration to take as polar axis the k vector which is axis of symmetry for the D domain. These axis are the coherent ones for calculation of (O~)~. When k and a are perpendicular, one can take the new system with the modifications: M .2 r

sin s (S)WTWT,

-t°,,

t2 - TX',,.

(19)

For an extra nucleon of moment [k I = f, taking _

Jo

S2 f 1 (2n) 2 dod3tdI2s ( s - t ) 2 '

(20) J2

_

S

f

( 2 g ) 2 ~Ddatdf2s

cos (s) (s" t) z

where S = s/fi We obtain as the difference between the graphs b and d, this difference being their only useful combination:

Job-a = S2(1og 2 - ½ ) when S < 2, Job-d -----½S(S+2)log ( S + 2 ) - S 2 log S + ½ S ( S - 2 ) l o g ( S - 2 ) - 2

when S > 2,

J2b-d = $2(~ $ 2 - 1)log 2 + ½ S 2 ( ] ~ $ 2 - 1 ) when S < 2,

(21)

dab-d = S(~J-i$ 3 - ½ S - ½ ) log (S + 2 ) - $2(~S 2 - 1 ) log S s when S >= 2, + S ( ~ S 3 - ½S + ½) log ( S - 2) + ½Sa - 3~ and M,2 ~.~o CTST'S' -- (2n~* do Urs(fS) VT's'(fS)J° dS,

t~-T,

(2~r~4) o WT(fS)WT,(fS)JndS.

(22)

The same method cannot be used for the graphs c and e. If the relative wave function is not made zero inside the core, for a Yukawa central potential V(r) = Voe-Z~/Itr we obtain

V(s) - 4nVo ]2

1 ,U2 "~ S2 '

with p and q as variables. The D domain for the graphs b and c is defined by

Iq!&pl ~J~

Ik-2pl ~f.

(23)

CORRECTION TO INTRINSIC MAGNETIC MOMENTS

55

A p p r o x i m a t i n g this d o m a i n by q > f, [ k - 2 p [ < f , one can do the calculation o f the central terms o f the graphs b and c. This gives approximately for any p value: C,

--

~ ½.

(24)

Cb We assume that this relation is approximately exact for all terms o f the graphs b and c and o f the graphs d and e respectively, and we take for the value o f each c and e term half the value o f the corresponding b and d term. 3.1. EFFECTIVE MASS We take as approximate energy o f a particle o f m o m e n t u m k a quadratic f o r m tangent to the exact Ek: k2 e k = -- + V(k), 2M

(25)

V ( k ) = ~ ( k l l V I k l ) - (kl[ V Ilk).

(26)

where I<

By a calculation similar to the Bell calculation s), we obtain for different Fermi m o m e n t u m values f, for the G T potential for f = 1.3(fm)-X:

M*/M

= 0.86,

for f =

M*/M

= 0.99.

0.7 ( f m ) - l :

3.2. SPIN AND ISOSPIN SUMMATIONS Proceeding along the same lines (the extra particle spin being quantization axis), the spin and isospin summations give: (I[1) = (~c,,

+¼(Cto+Col)+¼Coo+

s o 8tlo +xto}b_d

- {¼c~

(llZ:ll)

- i(c,

o + C o , ) + ¼Coo + s t ° _ 8 to~

= {24t 2 - 24t ° + st g - 8t°}b-d -- {24t2 -- 24t° -- 8t2 + 8t°}¢-e, 40f2

152f0

(llS:T~II> = {-½(Clx+Cxo+Co~+Coo)+C~oo+Gool+ T,x--r-u s 2 s o+2(i2t1+i02ol .2 .2 2 .o .0 .0 + i ° o l - Iloo--/OlO)}b-d --Jo--vto -- t l O O - I o ~ o ) - ~ z 1 1 1 - { - ½(G,

- c ~ o - Co~ + Coo) + c lI o o - c , oo, + - ~ t ~

+2(i2tt_i2o,+i2to

- , ~ .2 oo)-

- ,529 .v, .~_8,2 - o~ . o' °~T,

~2( , i.o i i - , O.oO l + i ° l o - i ° o o ) } o -

.o

o,

( l l L = l l ) = -½(112:~11>, (llL~%I1) = {--8t2+8to}b-d--{--8tz+8to}¢-e.

(27)

•~6

B. BREMOND

3.3. RESULTS

The kinetic m o m e n t u m were computed at two different nuclear densities. At normal density, we assume that the energy at the top of the Fermi sea is 35 MeV. This corresponds to a Fermi m o m e n t u m of 1.3 (fro)-1. We get k.0.

( ~ ) = 2.050.-2.29 ~

k'0.

k,

(12T~) = + 0 . 7 2 0 + 0 . 4 4 _-vw- k, k~

k'0.

( L ) = -0.530.+1.15 --k5- k,

(LT~) = - 0.020. + 0.04

k.0. k2

k.

and by (1) we get k'0.

~Up~oto. = 2.320.+0.60 ~

k,

/~neutron = - 1.020.- 1.50 ~ -

k.

(28)

At lower density with an energy at the top of 10 MeV corresponding to a Fermi m o m e n t u m of 0.7(fro) -1, we get

k.a

<'E) = 1.890.-- 1.90 ~

k'0.

= -0.440.+0.95 ~ -

k.0.

k,

(I~T~) = 0.950.+0.05 ~ - 2 - k,

k,

( L T , ) = - 0.020. + 0.04 V

k.

2.870.-0.25 k "~ k, k2

,Unoutror, = - 1.61o"-0.49 V

k.

And by (I) we have /.lproton

:

(29)

4. Application to Finite Nuclei Let us now consider the magnetic m o m e n t calculated in nuclear matter as an operator. We take the expectation value of this operator in a one nucleon configuration lsJM (with M = J) corresponding to the jj model, and we add to this value the uncorrected orbital magnetic m o m e n t of one nucleon in this configuration to obtain an effective magnetic m o m e n t k-tr (IsJI #effl lsJ) = (1sJJ]A0. + B ~ - k + Cl] lsJJ), where C = 1 for proton, 0 for neutron. A short calculation shows that k.0. (lsJJI ~ k l l s J J ) = (-- 1) t+ 1+s-M (2l+ 1)(2J+ 1)x/2 k" x

( 10101 ~cFO)(/OlOl~'O)

s 1

(JJJ--JIlO),

(30)

CORRECTION TO INTRINSIC MAGNETIC MOMENTS

~7

and

( 0 ±2 2±2 ± I

11

k'a

~ -

k[0½½1)

k ' a k t l ~ _ ~ _ ~ _ , 2~ k2

/9

.-2

2

1 3-32 2 ,I- zk k l•2o"

1 3 ~

-

,

=7, =

1 13-3-1

1

5 (2-~1 5~-~l

i

1 7 (3 ~-~-1 k ' a

kl3 1 7 7\

i

(4½991 k'a

kl 4 1 9

/a ±-s -st k ' a " k13 ~

-~-~,

k~g~ kll ½ 3 3 ) _ k

i

k14 ~ - ~ ) -

~

2

2

21

k-o"

1

7" ~-,

155

k \/ ~~ ±2_ 92 29,1 ~ -

k15

2 2 2/

--

71.

The table 1 compares the effective momentum for two nuclear densities using formulae (28) and (30) or (29) and (30) to the Schmidt value. TABLE 1 C o m p a r i s o n o f t h e effective m o m e n t u m

f o r t w o n u c l e a r densities w i t h t h e S c h m i d t v a l u e

Proton

Neutron

Configuration f = s½ p½ P~z d~ d~ f~ f½ g½ g~h~-

1.3 f m -1 2.52 0.10 3.43 0.53 4.41 1.30 5.25 2.15 6.39 3.06

f :

0.7 f m -x 2.79 --0.37 3.82 0.03 4.83 0.76 5.84 1.61 6.84 2.52

Schmidt 2.79 --0.26 3.79 0.124 4.79 0.86 5.79 1.72 6.79 2.62

f =

1.3 f m -1 - - 1.52 --0.16 - - 1.32 0.31 - - 1.23 0.48 - - 1.20 0.61 --1.16 0.69

f = 0.7 f m -x

Schmidt

- - 1.77 0.37 - - 1.71 0.87 - - 1.68 1.07 - - 1.66 1.11 --1.54 1.18

- - 1.91 0.638 - - 1.91 1.15 - - 1.91 1.37 - - 1.91 1.49 --1.91 1.74

5. Conclusion Our method of investigating the effect of the tensor force on the magnetic moment is just an other way of taking into account the configuration mixing introduced by this tensor force. An inspection of the successive steps involved in the calculation shows that the result is very sensitive to the exchange dependence form of the tensor force. In the case of the Gammel and Thaler potential, the results (28) and (29) are very sensitive to the nuclear density.

58

B. BREMOND

The main result o f this paper is that the intrinsic m o m e n t u m correction increases with I. This is due to the decrease of the expectation value of (k • tr/k2)k with the increase of l. Comparing the theoretical and experimental values of the magnetic m o m e n t given in table 2, we see that: TABLE 2 The magnetic m o m e n t s of doubly magic-plus-or-minus one nucleon nuclei Configu- ExperiNucleus

ration

Theoretical

mental Schmidt

II

I H3 He 3 N ~s O 17 K a8 Sr 87 Pb 2°~ Bi 2°8

s½ s½ p½ d{ d~ g~p½ h~

2.978 --2.127 --0.283 --1.893 0.391 --1.09 0.59 4.08

2.79 --1.91 --0.26 --1.91 0.12 --1.91 0.64 2.62

2.25 --1.52 0.10 --1.23 0.53 --1.16 --0.16 3.06

to to to to to to to to

2.79 --1.77 --0.37 --1.68 0.03 --1.54 0.37 2.52

2.89 to

Phenome-

nological IV

III 2.96

0.27 to --0.20

0.25 to --0.23

0.74 to

0.24

0.99 to

0.49

3.30 to

2.74

3.36 to

2.86

2.59 --1.809 0.138 --1.70 0.547 --1.67 0.35 3.06

I. C o l u m n I contains the corrections due to the present paper and taken from table 1 (the two n u m b e r s correspond to the densities o f Fermi sea o f Fermi m o m e n t a : 1.3 fm -1 and 0.7 fro-l). II C o l u m n II adds to this value the corrections obtained by Jensen and Mayer 8) (in the n e u t r o n cases there are no additional corrections). I I I C o l u m n III adds to the values of column I the corrections obtained by Marty 9). IV The phenomenological values of this column are the result of a least square fit o f the experimental data using formula (4). The phenomenological values o f the parameters are: for a p r o t o n Ap = 2.341, Bp = 0.756; for a neutron Aa = --1.622, B n = --0.563.

- - for H 3 and He 3 the Fermi sea model is especially inaccurate, and hence the experimental values are near the Schmidt lines; - - for N is and O tv the correction is overestimated; for 017 we must notice that the extra nucleon (d~) is embedded in a very weak nuclear density; -f o r K 39 the correction is satisfactory, the hole being embedded in a relatively strong nuclear density; - - for Sr s7 the correction is rather weak; for P b 2°7 it is too strong; - - for Bi 209 the correction is too weak; we remark that, especially for this nucleus, use of an 'effective charge' smaller than the real charge will increase the experimental theoretical discrepancies. Generally, the correction is too strong for I = 1(N 15, pb 207) it is good for l = 2(K 39, 017 (allowing for the weak nuclear density)), and it is too weak for g(Sr 87) and h(Bi2°9). We have used formula (4)

p = Aa+ B k~'-ak from a 'phenomenological point of view' and have adjusted by a least square fit the

CORRECTION TO INTRINSIC MAGNETIC MOMENTS

~

parameters A and B. The results (31) are, especially in the proton case, very near the theoretical value. The phenomenological moments are not much better than the theoretical ones: the main explanation to this seems to lie in our assuming the same peripheral density for the different nuclei. We have neglected the density gradient effects; these can be taken care of by introducing a nucleon-nucleus L. S coupling. This has been done by Jensen and Mayer, and also by Marty, and it gives a contribution only when the extra particle (or hole) is a proton; when we take the results of the above mentioned authors into account (in the two columns preceding the last one in table 2), the general agreement is not better. The experimental results for a given l, J do not seem to discriminate between the neutron case and the proton case (N ~ and Pb 2°7 or H 3 and Hea), and this is in disagreement with an I. S effect. Furthermore, as the 1. S coupling can arise from a two body tensor force, it is not evident that such an effect is entirely independent of the tensor force effect. We are greatly indebted to Dr. B. Jancovici for suggesting the subject of this paper and for m a n y helpful discussions and criticisms.

Appendix A second order perturbation calculation, using a tensor force, gives a term of form

(4). Indeed, the total wave function of a particle 1, the wave function of which is q~l = oq e ik'x, interacting with a particle 2, the wave function of which is ~b2 = ot2 e n'x, these particles being excited to states ~bl = flxei"x~k2 = f l 2 e i U ' x ( o ~ l ~ 2 f l l f l 2 spin wave functions), is given by:

0~2

Emn -- Ek!

and for instance the spin expectation value of particle 1 contains

( ~ ck21S~2(,Tx~2r)V(r)l~xO2>(OtO21S~2(~2r)V(r)l~l ~2) (A2) ,h~,1,2

(Emn -- Eke)2

Taking p = k - l , q = m - n , s = q - p , t = q + p , and by carrying the spatial integration in the matrix element (q~l~b21S12 F(r)l~kl~b2), the energies being scalar functions of s and t, we obtain:

(hz) =

~

st~2#1#2

(~q ¢t2]$12(0 ~02 s)f(st)fl~ fl2)(fl~l'rll/3x>(/~/~21Sxz(Ox o2 s)f(st)lcq ~2)

= E (~tl ~t2lSl2(al a2s)tr t S12(ax 02 s)F(st)l~ ot2). St~t 2

This expression contains a sum of the form:

(A3)

60

B. B R E M O N D

E (cq cql ( a ' ' s)(a2 "2s s) al (trl" s)(a2 "2s s) F(st)l~t 0£2) ~;tGt2

= ~, (a~lal°~(st)+

a, . s sG(st)[~l).

(A4)

The integration domain having a symmetry axis along k, we get: (A4) = ( ~ , l a l A ( k 2 ) + ~tr 1- • k k B ( k 2 ) l ~ l )

'

(A5)

w h i c h is o f t h e f o r m (4).

References 1) R. J. Blin-Stoyle, Rev. Mod. Phys. 28 (1956) 75 2) A. De-Shalit, Proc. Intern. Conf. on Nuclear Structure, Kingston, 1960 (North-Holland Publ. Co., Amsterdam, 1960) p. 90 3) H. Miyazawa, Prog. Theor. Phys. 6 (1951) 801 4) S. D. Drell and J. D. Walecka, Phys. Rev. 120 (1960) 1069 5) J. S. Bell, Nuclear Physics 4 0957) 295 6) J. L. Gammel and R. M. Thaler, Phys. Rev. 107 (1957) 1337 K. A. Brueckner and J. L. Gammel, Phys. Rev. 109 (1958) 1023 7) A. Klinkenberg, Rev. Mod. Phys. 24 (1952) 63 8) J. H. P. Jensen and M. G. Mayer, Phys. Rev. 85 (1952) 1040 9) C. Marty, J. Phys. Rad. 15 (1954) 783