Two-body interaction currents and nuclear magnetic moments

Nuclear Physics A123 (1969) 449--470; (~) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint oi" microfilmwithout writtenpermissionfrom the publisher

TWO-BODY INTERACTION CURRENTS AND NUCLEAR MAGNETIC MOMENTS MARC CHEMTOB Service de Physique Th~orique, Centre d'Etudes Nucldaires de Saclay, B.P. no 2-91-GlF-sur-YVETTE

Received 7 October 1968

Abstract: The meson-theoretical approach to nuclear interaction currents is considered in the onemeson exchange approximation. Results are presented for two models of charge-symmetric :rN coupling: the75 model and the static model. The effect of interaction currents on the nuclear dipole magnetic moment of odd-mass-nuclei with j j closed-shells plus (or minus) one nucleon is calculated within the framework of the pure spherical shell model. For nuclei with a closed ls core (15N, I70, 39K and 41Ca), first-order perturbation corrections are also taken into account. The contribution of interaction magnetic moments is found to be generally small in magnitude, but to depend strongly on the angular momentum of the nucleus; the net effect is a translation of both Schmidt lines, upwards for odd-proton nuclei and downwards for odd-neutron nuclei. For 2°9Bi, the correction is 0.5 nuclear magneton. The magnetic moment associated with the two-body spin-orbit interaction is also calculated: it accounts for approximately ~ of the contribution from the one-body spinorbit force.

1. Introduction T h e response of nuclei to electromagnetic disturbances is usually described by m e a n s of o n e - b o d y operators. I n this description, each n u c l e o n is given the same charge a n d m a g n e t i z a t i o n d i s t r i b u t i o n that it carries in free space. Such a picture is however still incomplete. It has already been k n o w n for a long time 1) that to a n y n o n - c h a r g e conserving p a r t in the t w o - b o d y N N interaction (e.g., terms i n v o l v i n g ~1 " ~2 or n o n - l o c a l terms) one m u s t associate a n a d d i t i o n a l electromagnetic i n t e r a c t i o n c u r r e n t of t w o - b o d y type, in order to allow for charge c o n s e r v a t i o n i n the t w o - n u c l e o n system. The p h e n o m e n o l o g i c a l a p p r o a c h 2 - 4 ) to i n t e r a c t i o n currents is based o n this observation, or what is essentially the same, o n the r e q u i r e m e n t of gauge i n v a r i a n c e of the nuclear H a m i l t o n i a n . This a p p r o a c h becomes however prohibitive for multipoles of the c u r r e n t higher t h a n the first, since it requires a large n u m b e r of parameters: m a g n i t u d e s a n d radial dependences o f m o s t of the terms are left wholly arbitrary. It was also early recognized 5 - 7 ) that the i n t e r a c t i o n currents come from the mesic degrees of f r e e d o m of the nucleons. This is already clear f r o m the fact that charged n - m e s i c q u a n t a exchanged between n u c l e o n s can interact i n d e p e n d e n t l y with a n electromagnetic field. Such a n a p p r o a c h using the relativistic field theory has the a d v a n t a g e o f r e q u i r i n g a limited n u m b e r of parameters, which are actually k n o w n f r o m other sources: the n - N c o u p l i n g constant, the m e s o n mass, a n d possibly, data related to the n - N scattering amplitude. 449

450

M. CHEMTOB

In fact, the existence of nuclear interaction currents seems to be well established by experiment. They contribute roughly 40 ~o to the electric dipole sum rule in the nuclear photoeffect 8), and explain rather well the ~0.2 n.m. (nuclear magneton) anomaly of the isovector part of the magnetic moments of mirror nuclei 3He and 3H 9). For magnetic scattering of the electrons from deuterons, their contribution improves slightly the agreement with experiment 10). There is also strong evidence for a 10 ~o interaction effect in the radiative n - p capture cross section at thermal neutron energies 11). Little however is known directly about the role of interaction currents in heavy nuclei. Our main purpose in this work will be, in fact, to calculate the interaction magnetic moments of odd-mass-nuclei. Previous works were devoted to this subject. These however restricted to specific aspects of either the interaction currents (exclusion principle effect 12,13), phenomenological approach 14,15)) or the nuclear structure (core of the odd nucleus described by an infinite Fermi-gas model 6,7) or by a shellmodel state saturated in spin and orbital angular momentum 16), i.e., with L -- O, S = 0). In sect. 2 of this work, we derive the two-body interaction currents in the onemeson exchange approximation. The expression for the dipole moment has already been considered by a number of authors s-7), but our results apply equally to arbitrary multipoles of the current. In sects. 3 to 5, we discuss the effect of interaction currents on the static magnetic moments o f odd-nuclei with closed shells in the jj coupling sense + one nucleon; the spherical shell model is used to describe the groundstate wave function. In sect. 6, we calculate for nuclei 15N, 170, 39K and 41Ca, with a closed ls core, the contribution o f first-order perturbation corrections to the nuclear wave function.

2. Mesic theory The electromagnetic properties of nuclei should, in principle, be derivable from the fundamental interactions between nucleons, mesons and the electromagnetic field. The electromagnetic interactions of nucleons and mesons are of course, very well understood. On the other hand, the n N coupling is well described by the relativistic pseudoscalar Yukawa theory. Its non-relativistic description in the ChewLow static model is also justified for the regions of energy of the nucleons inside the nucleus. The remaining problem is to take into account the presence of many nucleons. Consider two nucleons isolated from the nucleus and placed in an electromagnetic field. Each nucleon is clearly the source of an individual current induced by its charge and spin: this gives the ordinary one-body potential. The mechanisms involving exchange of mesons between the nucleons lead to two-body potentials. Two kinds of processes can then take place: (i) the mesonic interaction current process, where the photon interacts with the exchanged meson, as shown ~n fig. 1 ; (ii) the nucleonic

451

NUCLEAR MAGNETIC MOMENTS

interaction current processes, where the photon interacts with the nucleons before or after exchange of a meson. These processes are shown in fig. 2.

Fig. 1. The two-body mesonic interaction current process.

(a)

lb)

1")

(e)

Fig. 2. N u c l e o n bremsstrahlung processes contributing to the two-body nucleonic interaction current.

We now discuss, in the non-relativistic limit, the expressions of the mesonic and nucleonic interaction currents in two models of charge-symmetric 7rN coupling: the 75 Yukawa pseudoscalar model and the static a . V Chew-Low model. Consideration of both models is meant to give better insight into the problem. 2.1. RELATIVISTIC MODEL Our purpose is to find the representation, in configuration space, of the potentials associated with the mesonic and nucleonic interaction current processes. The procedure used is the following. First, we write in m o m e n t u m representation the scattering amplitude, or the S-matrix element, for the processes in figs. 1 and 2. We denote by Pl, P2 and p~, p~ the energy-momentum four-vectors of the initial and final nucleons, and take for convenience a monochromatic plane wave dependence for the external field a.(x, t) = V.(q)d (q°t-''') (1) The m o m e n t u m space representationju(q ) of the current is defined in terms of the S-matrix element by the relation

= -2ircO(El + E2-E'l-E'2-qo)ju(q)V~'(q).

(2)

The effective interaction potent_ial nef f reads then

(P'I P[]HefflPx P2) =" J~,(q)VU(q),

(3)

and its expression in configuration space is finally found by the change of representation

ff

, , = (2rc)6 1 (XlX2[H,,fflXlX2) "'"

dPl dp2 dPl dpz(Pl p2lHefflPl P2) .

.

.

.

x e ~w'~" x'l +v'2. x ' 2 - p , - ~, - v 2 - x 2 )

(4)

452

M. CHEMTOB

Let us first consider the mesonic current. An application of the Feynman rules gives

eG 2 M2 J~(q) = cS(P'l+ P'2 + q - Pl - P2) (2rt) 3 (E 1E2 Etl E'2) ~ A~(pl - P'I) X [Pl -- P'I + P; -- P2]. AF(p'2 -- P2) [W(P'I)r5 ~i w(Pl)] i(6n 5i2 -- 612~jl)

× [ (pl)Ts

jw(p2)].

(5)

We have set h = c = 1, and denoted by e the absolute value of the electronic charge, G the renormalized pseudoscalar rrN coupling constant, M the nucleon m a s s . . . The meson propagator is defined by A~(p) = 1/(pZ-#2+ie). We adopt the conventions of Schweber 17) for the space-time metric (g0o = - g l l = 1) and the Dirac matrices (7o = fl, 7 = fl~, 75 = 7 o 7 1 7 2 7 3 ) • Only the expression ofj~,(q) in the non-relativistic limit: E1 ~ M, e t c . . , is of interest to us. The matrix elements with respect to Dirac spinors are then expressed in terms of Pauli spinors in such a way

g'(P')T5 w(p) ,-, ~

i

Z + [a . (p'-P)]Z-

(6)

The following formula for the product of the meson propagators appearing in eq. (5) is also used

A~(p;-pl)Ab(pl-p2),~

fo

dz[(pl-pt+q(1-z))2+#2-q2z(l-z)-ie]

-2.

(7)

The mesonic interaction energy H ~ is obtained by substituting the current j~,(q) into eq. (3). Its representation in configuration space is then derived with the help of eq. (4)

(x'lx'2lH"lXlX2) = - 5 ( x , - x ' , ) a ( x 2 - x 2 )' ~ e ( G ) (v(1) x *(2))3

X ( a 1 " V l ) ( a 2 • Vz)

(Xl--X2) °

dz

Ixl-x21

a((l-z)xl+zXz)

,

(8)

where the radial range parameter L is given by L = [ / ~ 2 - q 2 z ( l - z ) ] ~t. It reduces to #, the inverse meson Compton wavelength, for a real photon (q2 = 0). Let us recall that the renormalization, due to radiative effects, of the mesonnucleon and meson-photon vertices appearing in the process of fig. 1 has a negligible effect. This is because the maximum kinetic energy of the nucleons inside the nucleus is comparable to the meson mass. On the other hand, the difference between the renormalized and bare meson propagators is given by a weighted integral of the bare propagator over the pion mass spectrum, extending from 3# to infinity. Propagator renormalization affects therefore the short range part of the interaction energy. We shall verify in the numerical applications of sect. 5 that its effect is negligible. For

NUCLEAR MAGNETIC MOMENTS

453

the same reason, contributions from multimeson exchange or from exchange of pion resonances (p . . . . ) should have a minor role too. Let us now turn to the nucleonic interaction energy, associated with the four graphs in fig. 2. The diagrams where the initial nucleons interact with the photon lead to a current which is hermitian conjugate of the one where the interaction with the photon is on the final nucleons. Furthermore the process (b) of fig. 2 is deduced simply from the process (a) through permutation of the coordinates of the initial nucleons and the final nucleons. We therefore consider only the current j(ua)(q) associated to the process (a) eG 2

j ( a ) ( q ) = a ( p , 1 .4_ P2 q- q -- e l -- P 2 )

x I#(p;)(7uF~(q 2) ×

M 2

(2~z)3 (El E2 Ei El) ~

Av(pl -

P2)

K ty~vc[Fa(q2)) S'~(p; +q)ys"qw(pi) 1

[#(pl)y,

(9)

where the nucleon propagator is defined as S~(p)= (Y "P-M+i~) -1. We take for the nucleon form factors the values Fz = 0 and F t = (1 + % ) / 2 corresponding to a bare nucleon. This is consistent with the order of the perturbation expansion considered. If j~b),j(u°) and j(d) denote the currents associated with the remaining graphs in fig. 2, and ./J(q) = *'t*i(a)_{_j(b) q-j,;(¢)-rj," ;(d), one can already verify that the sum of./~(q) andjuN(q) satisfies the differential charge conservation law

qU. (j~(q)+jN(q))

= 0.

(10)

The matrix element in configuration space of the interaction energy H (a), associated to j(a)(q), is obtained by a procedure similar to the one used above. The non-relativistic limit is again considered. The product of the nucleon and meson propagators appearing in eq. (9) is written in the form

[((p'l +q)2-MZ+ie)((p'2-p2)2-1~2+ie)]-l

~

f' dz[s2+zu2-ie] -2,

(11)

0

where s = P'2-Pz-pl(1-z). The reduction of Dirac spinors to Pauli spinors is made with the help of eq. (6) and the following formulae: w(Pl)?o(? " (Pi + q) + M)y5 W(pl) #(P',)Y(Y " (P'I + q) + M)Ys w(p,)

"~

iz +[a" (Pl + q -- Pl)]Z,

..~ ~

pl)]z.

i

Z +[2p'i a"

(p', -- PI)

(12)

We substitute then these results into eq. (9) for the current j(a)(q) and use eq. (4) to derive the matrix element of H (") in configuration space. To simplify the equations we have made three reasonable approximations.

454

M. CHEMTOB

(i) The terms O(q2), proportional to the second derivative of the external field, are ignored. (ii) The non-local terms are dropped. This amounts to keep the terms depending only on the m o m e n t u m variable s. (iii) In the argument of the exponential in eq. (4), the center-of-mass m o m e n t u m P = Pl+P2 is neglected with respect to the relative m o m e n t u m p = ½(Pl-P2). This restriction makes the matrix element diagonal with respect to the center-ofmass coordinate. With these approximations, we get

(x~x'zlH(")[XlX2) = i e 8re#

×a(g-g') fo L dz

½[z+(1)z_(2)+z3(2)+z3(1)z3(2)]

b(r-- zr'){2Miao(x'l)(a 1 • V,,) + a(x'l)

• [i(a 1 • q)V,,+ ia,(V,," q ) - ( V , , x q)--2V,,(a 1 • Vr,)]}(a 2 • V,,)e - " ' / i / , where z± = Zx+_izy; R, r and R', r' are the center-of-mass and relative coordinates associated to x~, x2 and x'~, x; (R = ½(x~ +x2), r = x 1 - x 2 . . . . ) and q is equivalent to iVx, ,, acting on the external field. We recall that the contribution from the remaining processes in fig. 2 are readily inferred from the above result. The expression of the nucleonic interaction energy is clearly rather complicated• The reason is that in the 75 model, the relativistic effects due to nucleon recoil and negative energy states are mixed up with more important effects such as the propagation of rcN isobars and the three-field rcN 7 point interaction• The static model separates naturally these effects. This we show below. 2.2. NON-RELATIVISTIC MODEL In this treatment, the dynamics of the system of two nucleons with l a b e l s j ( j = 1, 2) interacting with the electromagnetic field a(r) and the pion field ~ ( r ) is considered from the outset in the non-relativistic limit. One starts from the following Hamiltonian 18):

H = H o + H 1, H~ =

HMN--ef drj(r) "a(r),

where Ho is the Hamiltonian for the free fields and coupling of two nucleons at points x 1 and xz

G

E

HMN the Hamiltonian for n N

fdxp(x-xJ)(trJ'Vx)('gJ"

HMN = 2M j = l , 2 v

(14)

tI~(x)).

(15)

The current j(r) is given by j(r) = j~(r)+J~N(r)-kjs(r). It contains the mesonic current, the nucleonic current and the three-field interaction current which read

NUCLEARMAGNETICMOMENTS

455

respectively

j~(r)

--

((~2V~l -- ~)1V(~2),

jN(r) -- E I+z3(j)(~ j=

j=N(r)

1

2

1,2

_

2 M (o'j x V,)

(16)

)

6 ( r - x j),

E p(r--xj)aj(¢j x ~(r)) a.

G

(17) (18)

2Mj=1,2

Let us consider the effect of each of these three currents. We restrict to the onemeson exchange approximation, with a point nN coupling: p(x-xj) = 6(x-xj). The currents j =(r) and j=N(r) lead to the processes pictured by the Feynman diagrams of figs. 1 and 3 respectively. The rules for calculating their contributions are similar to those of the relativistic theories 19). Let H=(xl, x2) and H=~(xl, x2) denote their contributions to the diagonal matrix elements in configuration space of the interaction energy. The non-diagonal matrix elements vanish. For the mesonic energy H=(xl, x2), one finds a result identical to that of the relativistic model, given by equation (8). The result for H=N(xl, x2) is the following:

e#(G) z H~N(xl, x2) = 4n ~ (x(1)× x(Z))a[-(a 1 • r)(a2" a(x2))+ (a 1 • a(x,))(a2" r)] X (1"~ ~ )

(19)

e -r2~ r

This energy is only a part of the full nucleonic energy H N discussed above. The expression found here is however much simpler.

t

+ (1~ 2)

..... 1

2

Fig. 3. Process involving the point ~N 7 coupling. The notation (1 ~+_2) stands for the same diagram where particles 1 and 2 are interchanged.

+ (1~. 2)

.... 1

2

Fig. 4. Lowest order contribution from the nucleon current in the static model.

Consider now the nucleon currentjN(r). In the static no-recoil model it contributes only starting from order (f/l~)~ by means of processes such as the one given in fig. 4.

456

M. CHEMTOB

Now the existence of the (za, ~) 2 resonance suggests the consideration of the class of processes of the same type represented in fig. 5. The vertex attached to the particle 1 is the nN7 scattering amplitude off the energy shell.

1

2

Fig. 5. Processes contributing to the two-body nucleonic interaction current.

Miyazawa developed a convenient method to handle these processes 7,19), The formalism is particularly simple when one considers only the isovector part of the nucleon current. We make this restriction in the following. In this method, one starts with the electromagnetic current of a physical nucleon. Its isovector part reads in the static model e -

-

%

f(o) 7p--Tn

2M

f

Fl(q2)i(tT x q),

(20)

2

where ~p and 7n are the intrinsic magnetic moments of the proton and neutron and

f(o)/f is the ratio of bare to physical nN pseudovector coupling constants (f/# = G/2M). The isoscalar part would involve (?p+?.) which is a factor 5 smaller than (Tp--?n)" Furthermore this part cannot excite the (23-,-~) resonance. It is therefore well justified to keep only the isovector part of the nucleon current. Let 7% denote the state of a physical nucleon and 7Jk,(-)j the outgoing state of a physical nucleon+ one meson with momentum k, isospin j. The nN~ scattering amplitude is found by taking the matrix element of the current (20) between these states. Now this matrix element is clearly related to the amplitude for the scattering of a pion, from state (k',f) to state (k,j) by a nucleon

• k'),A

(2t)

which can be expressed in terms of the nN scattering cross sections. We give no detail of this derivation, since the method is fully described in reference 2 o). If only the dominant (-~, 1) cross section is kept, the following relation holds (cok =

(k2+.27) 4i (2C0k)~'(~P(k]]l(a X q)z 31~o) = ~ (A(0)#a)[~s zj%( ~" k)(~ x q) +2[½zj%(3k x q - 2 ( a " k)(a x q))+ 6ja(a • k)(a + (a j3 --½~'jz3)(3k x q - ( t r . k)(tr x q))], A(0) = .t~

fo '~ sin 2 63a(P) 0.11/#3. dp o)2p2 ,~

x

q)] (22) (23)

NUCLEAR MAGNETIC MOMENTS

457

Let us now write the contribution of the process in fig. 5 to the interaction energy (the notation (1 ~-- 2) stands for the term obtained from the one explicitly written by interchangir g particles 1 and 2)

HN(x,, x2)

(

(2c%)~2Mie ~)p--~)n2 f(o)f Fz(qz)<%T]l(~r, x q)v3(1)[~o>

3 dk

• a(XI)e-'k'("-'~)[~(,z•k)ty(2)]

k 2 + lz2 - ie

+(1~.~-2).

(24)

....

Insertion of (22) into this equation leads to the result

HN(xl, X2) = 3(A(0)#3) e 7p--7, F2(q2)(Vx, x a ( x l ) ) 2M

57z

2

• [4z3(2)V,-(~(1)x'c(2))a(a I x V,)](a2 • %7,)

e-/~r l"

+(1 ~ 2).

(25)

To close this discussion based on the static model, we consider the effect due to nucleon recoil• It can be introduced in the static model in a phenomenological way 21) by adding to the unperturbed Hamiltonian H o the kinetic energy term: (p21+p~)/2M. The adiabatic approximation is meaningful here only for the time-ordered graphs of fig. 6. We restrict therefore to these diagrams. Their contribution to the interaction

1

2

1

2

Fig. 6. Time-ordered diagrams contributing to the nucleonic recoil interaction current. energy H R, is given by the third-order term in the perturbation expansion of the reaction matrix t

t

R



t

t

=
1

Wo - H o

(f -e

drjN(r)'a(r)

)1

Wo-Ho HMNIXl X2>. -

-

(26) The procedure is again similar to the one used above. The current p j M within iN(r) is neglected, because it leads to non-local terms. One finds for the diagonal matrix element of H a n"(x,,

,(°)2{

x2) - 8~z2M 2M

[__ (tTa•V')zJ(1)

1+%(1)~l•(Vx'×a(x1))2

458

M. CHEMTOB

where K o is the Bessel function of the second kind 1 f

ei~ - x

K°(PX) = ~ J dk(k2 +/~2)~ "

(28)

3. Interaction magnetic moment The nuclear Hamiltonian Hx, associated to the interaction currents, is given by the sum over all distinct pairs of nucleons of the potentials H,j considered in the previous section Hx = ~. H, a • (29) i
Clearly, Hx must be added to the ordinary one-body potential. In this connection, let us emphasize two other important points. (i) The Siegert theorem z2) indicates that the electric multipole moments and therefore the nuclear charge density are not affected by the interaction currents. The independence of the mesonic energy H = (cf. eq. (8)) with respect to the scalar component ao(x) verifies this result. This is not however the case for the nucleonic energy (cf. eq. (13)). (ii) The individual form factors of the nucleon are altered when the nucleon is immersed in the nuclear matter. This is referred to as the exclusion principle effect 12). The modification is due to the exclusion of nucleon intermediate states occupied in the nucleus, in the one-body mesonic current process. Being related to violation of Pauli principle in intermediate states, this effect is exactly compensated by the twobody mesonic current process. We shall discuss its contribution in sect. 5. We consider in this section the interaction dipole magnetic moment. Let us first sketch the main results of the phenomenological approach, since it allows a convenient classification of the various operational forms for this moment. To the N N potential having a scalar and tensor part

V = Z [W+BP~j+MpM+HPTjpM]V¢(r) +*i "*JS,iVT(r),

(30)

i
the phenomenological theory associates, without arbitrariness, the Sachs interaction moment

Jgs = ½ie ~. "c3(i)--z3(J) (r x R)[(M + HPS)V¢(r)P~ + 2Sii Vv(r)P,~], ~
(31)

where the tensor operator is defined as S~i : 3(hi" P)(aj" P)- (a~. aj) and the centerof-mass and relative coordinates associated to q, r 2 are denoted by R, r. The main source of ambiguity of the phenomenological approach is due to the fact that any arbitrary function of V × a(r) can be introduced in the nuclear Hamiltonian without affecting its invariance by the gauge transformation. It is known however that the theory is irleonsistent without the introduction of these solenoidal

NUCLEAR MAGNETIC MOMENTS

459

2 3 ) . Twelve distinct forms are possible for the solenoidal moments (also called spin-exchange moments) if they are asked to (i) transform as axial vectors, (ii) be symmetrical under permutations of two particles, (iii) change sign under time reversal and (iv) be invariant under space translations and velocity-independent. The following list follows closely the classification of Foldy and Berger 4): terms

d/l, z = ½e E z3(i)--za(J)(ai--aj)r2 ~fx(r) MI • i
za(i)--za(J)2 [ (ai-aj)" ~- a,.--a/.] rE [f4(r)P~ fa(r) 1J

•~5 = ½e • z3(i)+z3(J)(ai+aj)rZfs(r), i
[(ai+aj)" PP--ai;

aj]

r2f6(r),

(ai+aj)r 2 [fT(r) • 1 [fs(r)Pifl '

J[/{9 lo = ½e ~ 1--z3(i)z3(J) [(ai+aj). ~-- ai3ai ] r 2 t f9(r) , t ' i
=

½e Z

i
1+

z3(i)z3(j) (ai+ aj)r2f, i(r), 2

~¢glz=½e ~<~ l+z3(i)z3(J)2 [(ai+ aj)" P ? - - a i +a / J 3 rZfl2(r)"

(32)

Note that only the terms J/g2, rig4, JCga and ~ ( l o can be associated to the charge exchange part (i.e. the part [rx(i)zx(j)+~y(i)ry(j)] ) of the N N potential2). All the other terms, except dg s and ~g6, are allowed if the interaction moments are to be proportional to any part of the NN potential. The mesic approach predicts unique expressions for the radial dependencesf~(r). To obtain these functions, it is sufficient to consider the results of the previous section in the case of a uniform magnetic field Of', and take for the vector potential a(r) = k3~ X r. The magnetic moment is then obtained by identifying the energy H x to -~//¢ • o~t'. We shall give no detail of this derivation here, since the calculations are lengthy but not difficult. One first finds that the Sachs moment is present in the mesonic current, given by eq. (8). Its expression is similar to eq. (31), with the OPEP replacing the N N in-

460

M. CHEMTOB

teraction V ~t's -

2 3 4re \ ~ 1 x

[

i73

--4(1---}Pij) "

2

(

P ~~j + 2 S ~ j P ~ j~

-X

1 + -3 + X

3

)5:1 -

,

(33)

with x = #r. Note that the same relation holds also for the two-pion exchange potential 24). Apart from this term, the results of the relativistic and non-relativistic models differ. The mesonic current which is common to both models, has non-translationally invariant terms. These terms are however compensated by corresponding ones coming from the currentj"N(r) (cf. eq. (19)). Furthermore, the nucleonic interaction energy obtained in the relativistic model (cir. eq. (13)) is non-diagonal in configuration space; so that its magnetic moments cannot be included in the classification given above. For these reasons, we shall restrict from now on to the expressions of the nonrelativistic model. Contributions (8) and (19) of the mesonic current j~(r) and the interaction current j~N(r) are grouped together in a single term. This term is called mesonic current (j~N(r) is, in fact, often classified as a mesonic current2°).) The contributions from eqs. (25) and (27) are respectively identified as isovector-nucleonic and nucleonic recoil currents. Let us now give the results in terms of the non-zero radial functions f , ( r ) . (i) Mesonic current: g = ~

~ 11.17 MeV,

(

.12 = 29 2 -

X2

'

J4 = - 2 9

(

1+

X2

(34)

(ii) Isovector-nucleonic current: 9'-

2 #2 5g M (A(0)#3)(yP - y " ) ~ 1.357 MeV,

x3 , f , = --f2 = f 5 ,

f3 = --39'

( , 30e1+ x- +

x3 '

f3 = 2f, = f 6 .

(The above results were already derived in previous works 7).) (iii) Nucleonic recoil current: 9"

-

rcM4n

"~

1.058 MeV,

(35)

NUCLEAR MAGNETIC MOMENTS

fl =

g" K x) 2x 2

fl = -lfz

K x)

f3 =

,

= f 5 = - f 7 = ½f8 = f l l ,

461

--K2(x) 2x 2

f3 = f * = f 6

=

--f9

=

½fl0 = f 1 2 "

(36)

Here Kl(x ) and K2('~ ) are Bessel functions of the second kind. The magnetic moment responsible for the exclusion principle effect is associated to the mesonic energy, given by eq. (8). Let us write its contribution to the spinexchange moments. If the terms depending on the center-of-mass coordinate are omitted, only the moments all//2 and dr/4 appear. They have the following radial functions: f2 = ~ g

(1 - le"

f, =-g

X2 ~

e-"

1+

X2

(37)

For the sake of completeness, we finally consider the interaction moments due to velocity-dependent N N potentials. We restrict to the spin-orbit force, since its existence is well confirmed by two-body scattering data, vLS : E VLs(r)|" S. i
The relative orbital momentum is defined by l = 1(r2-r2)× ( P l - P 2 ) , and S = ½(a~+a2). To obtain the electromagnetic interaction one makes the standard transformation p ~ p - e½(1 + 2"3) tI (r). This leads to the following magnetic moment: 'Ls = e , E < , VLS(r){z3(i) 2 " c 3 ( j ) [ ( t r i + t r j )

r" R--r(ffi+ttj)"

R]

+ l (l + z3(i)2z3(J)) [(tri+trj)rZ--r(tri+aj)"

(38)

In the numerical applications of the next sections, we shall use the spin-orbit part of the Hamada-Johnston potential 25). 4. Magnetic moments of odd-nuclei

In this section, we calculate the interaction magnetic moments for the class of oddnuclei described by a core saturated in t h e j j coupling sense + one valence nucleon. In the extreme single-particle shell model (Schmidt model) the ground-state wave function ~jm is the antisymmetric product of the wave functions of the valence particle ~bjmand of the core ~0 ~'ljm = I// jm It10 .

(39)

Consider the expectation value with respect to this wave function of a two-body operator, tensor of rank 2 j / t z ] = ~ v.vij .Z/t~] . (40) i
462

M. C H E M T O B

For the dipole moment, 2 = 1, Since the operator is not a scalar, only the interaction of the valence particle with the core remains

<%ll~"CZJll~j> = [- ~ ,l]/(--) j - m

0

"

de c o r e

We took on the left hand side the Racah reduced matrix element of ~/tz~. The conventions used for spherical harmonics, 3-j symbols, etc. are the same as in ref. 26). The sum over ~ on the right hand side of eq. (41) extends to all neutrons and protons in the core. The contribution to this sum from a completely filled sub-shell with angular momentum Jc is clearly independent of the magnetic components. One finds • .2j~+ 1 • [2] .-2j + <.t.t= (O)Jll../t' IlJJo ° '(O)j> = ~

(-)K'+j+J°+2[(2K+I)(2K'+I)]&

K, K'

Jo When the sub-shellj~ is partially filled with n~ identical nucleons coupled to angular momentum zero, the formula is very similar: one needs simply multiply the r.h.s. of (42) by n j ( 2 j o + 1). On the other hand, the sum over a filled sub-shell with fixed orbital momentum l~; that is withjc = l~+½ and L = lc-½, has the following more explicit expression: (ljlZot~+a(S = O, L =

O)jll,.CltXlllljl~Zo+~(S =

0, L = 0)j> = -)~jz

(" i) 20

L,L',S,~,

lo L S' ½

½ 2~



x <(ll¢)L[ IJt't2°~l [(ll~)L'> a ,

(43)

with the notation a = ( 2 7 + 1 ) ~. We have set ~¢tal = [dlt2oax.A, t2.3]t2~, where ..¢t't2°1 and .~gt2,1 operate respectively on the space and spin variables. The case of a pure j" configuration of identical nucleons is simply related to the case n = 1 considered here. This is true provided that the contribution inside the j" configuration is ignored. For the moments having the isospin dependences [~3(i)-~3(j)] or [ 1 - ~3(i)~3(])], this contribution actually vanishes. The following equation can then be proved: < j"(aJ)j'~¢(~¢ O)JII Jtd:q[lj"(aJ)j~°(g¢ O)J> _

no (jjzJo+l(O)jll..lltZ3lljj~j~+,(O)j>, 2j¢+1

(44)

where U t2~ is the Racah unit matrix 27) U txJ = ~. u[ ~1, i

= 6jj,.

(45)

NUCLEAR

MAGNETIC

MOMENTS

463

For a j" configuration of seniority 1 : = 1. The relation between the matrix elements of ~,grx~ in conjugate particle and hole configurations is also simply deduced from the following property of the U matrix:

=

(-)~+~.

(46)

To evaluate the two-body matrix elements of the interaction magnetic moments, it is convenient to introduce the center-of-mass and relative coordinates associated to nucleons i and j. We reduce therefore the operators ~g~)l considered in the previous section, into a linear combination of irreducible tensors acting on r, R and the spin ~,g})~. Lrr//r~.l E~olX ~ ' rs~p i nlA r~l • (47) x ~t' cr~,l L ~w rel . m. ] The two-body matrix elements of _..,~ . ~ U are then expressed in terms of those of ~'~z'~, d t 'rx"~ and ~g~x'J. To do this, we first transform the matrix elements f r o m j j coupling to Is coupling, and second introduce the Moshinsky-Brody transformation. The following formulae give the expansion of the operators appearing in eqs. (32), (33) and (38) in irreducible tensors:

(~ x~)lal ---- _ i~zx/~[ytal(~ ) × yttl(~)]t,1,

(~ ×/~)uJs u = - i ~8~ [(0~'~ +,/ioE?~+,/~o~+20~ ~) × [~, × ~j]E=~]E,~,

(4S)

[(~, + ~D(-~" ~)-- (~, + ~D" ~ ] u _

4~ [(_2~o~_ /iOEI~+~/iOC?~)×(~,+.#,~]E,~, 3~/3

where the operators ~E~1 stand for O~"1 = [yE,J(~) x yrx~(/~)lEml. Some general properties of the two-body interaction moments are worth to mention. These are of great help to understand the numbers we give in the following section. One remarks from eqs. (34), (35)and (36)that the strengths of the moments associated to the three types of interaction currents are quite different. It is clear that the moments of mesonic origin are predominant. The selection rules they obey are also simply deduced from the formulae in (48). With the help of (43) one verifies that the contribution from an occupied spin-orbit doublet in the core vanishes if the spin operator has 2= = 2. This is the case of the Sachs moment associated to the tensor potential. It can also be proven that the moments proportional to [(a~-t-a~)'~P-(ai+_aj)/3] give no contribution to s~ nuclei.

464

M. CHEMTOB

TABLE 1 (a) Contribution o f interaction currents to dipole magnetic moments of odd-proton (l(b)) and oddneutron (1 (a)) nuclei Element Conf.

170

(a)

(b)

(c)

~Schmidt

#exp --#Schmidt

#pert

Total

(d)

0.019

0.16

--0.101

0.028

0dl

--0.084

--0.006

--0.011

--1.913

zgSi

ls½

--0.112

0.084

--0.010

--1.913

1.358

0.73

--0.038

--0.001

a3S

0dj

--0.097

0.067

0.002

1.148

--0.505

--0.53

--0.028

--0.023

41Ca

0fk

--0.195

0.022

--0.004

--1.913

0.318

0.28

--0.177

0.025

aTSr

0g]

--0.265

0.065

--0.004

--1.913

0.820

0.66

--0.204

0.014

91Zr

ld i

--0.109

0.085

--0.006

--1.913

0.610

0.41

--0.030

0.010

115Sn

2s½

--0.084

0.142

--0.008

--1.913

0.995

0.81

0.050

--0.001

157Gd

2p]

--0.063

0.108

--0.003

--1.913

2.236

1.72

0.042

0.002

xaTOs

2P½

--0.040

0.024

--0.006

0.638

--0.572

--0.09

--0.022

--0.005

z°TPb

2P½

--0.084

0.033

--0.008

0.638

--0.048

--0.01

--0.059

--0.014

TABLE l(b) Element Conf.

15N

0P½

(a)

(b)

0.127

--0.086

(c)

--0.013

#Sehmidt

#exp /~pert --flSehmidt

--0.251

--0.032

--0.10

Total

(d)

0.028

0.034

27A1

0d~

0.221

--0.056

--0.028

4.753

--1.112

--0.93

0.137

--0.034

alp

ls½

0.182

--0.082

--0.032

2.753

--1.621

--0.99

0.068

--0.027

aTCl

0d4}

0.210

--0.037

--0.002

0.148

0.536

0.31

0.171

0.034

agK

0d~

0.204

--0.024

--0.002

0.148

0.243

--0.26

0.179

0.031

5aCo

0fz

0.321

--0.069

--0.027

5.753

--1.173

--1.03

0.225

--0.036

~tGa

lpj

0.170

--0.103

--0.036

3.753

--1.191

--1.53

0.031

--0.044

aSRb

0f~

0.175

0.011

0.009

0.891

0.462

STRb

lp.~

0.300

--0.112

--0.038

3.753

--1.002

0.195

0.048

--1.10

0.150

--0.043

agy

lp~

0.022

--0.040

--0.009

--0.251

0.114

0.02

--0.027

0.044

nsIn

0g~

0.442

--0.092

--0.032

6.753

--1.218

--0.89

0.318

--0.050

i53Eu

ld~

0.444

--0.126

--0.039

4.753

--3.223

--1.07

0.279

--0.039

I~Au

ld~

0.061

0.047

0.014

0.148

0.112

0.20

0.122

0.045

~°ST1

2s½

0.247

--0.165

--0.053

2.753

--1.125

--0.61

0.029

--0.028

2°9Bi

0h~

0.384

0.048

0.017

2.657

1.423

0.53

0.449

0.051

Column labelled "total" is the sum of the contribution from the mesonic (column (a)), nucleonic (column (b)) and nucleonic recoil (column (c)) moments. Column (d) gives the contribution from the two-body spin-orbit force. The Schmidt moment, PSehrnidt, is corrected (for odd-proton nuclei) by the 0.04 n.m. reduction of the intrinsic moment of relativistic origin (cf. ref. 12)]. Column labelled Ppert gives the perturbative correction, as calculated in refs. a0,31), to the one-body moment.

NUCLEAR MAGNETIC MOMENTS

465

5. Discussion of the results

The numerical evaluation of the formulae derived in the previous section was made with radial functions of the harmonic oscillator well. The parameter e = (M~o/h)~ was defined in such a way so as to fit the nuclear radius. Our results are given in tables l(a) and l(b) for odd-neutron and odd-proton nuclei respectively. Let us mention here that a calculation of the interaction moments, using the shell model was made by Ichimura and Yazaki 16) for nuclei 15N, 170 and 4aca. Our values however differ on certain points f r o m theirs, for a reason we do not understand clearly. Inspection of table 1 suggests immediately that the interaction moments are not the predominant correction. This correction causes essentially a small shift of both Schmidt lines; upwards for odd-proton nuclei and downwards for odd-neutron nuclei. A few satisfying points are however worth to mention. (i) The very small correction for Z°TPb ( - 0 . 0 7 3 n.m.) as compared to the much larger one for 2°9Bi (0.50 n.m.), both going in the right direction. (ii) The interaction moments for d~ nuclei, like arC1, 39K and 199Au are of the same order of magnitude as the Schmidt moments. (iii) We get corrections equal to - 0 . 1 7 n.m. and +0.14 n.m. for 3He and 3H respectively if these nuclei are described by an s~ configuration of the shell model with e ~ 0.74 f m It should be recalled here that the one-body magnetic m o m e n t is very sensitive to configuration mixing. This is indicated by the numerous calculations using improved ground state nuclear wave functions 28- 33). The same conclusion is also suggested by the phenomenological analysis of Blin-Stoyle 34) where the matrix elements of the one-body m o m e n t are determined from the matrix elements of the axialvector interaction in the associated beta-decay processes. To have an idea of the overall agreement with experiment, we give in table 1 the Schmidt values and the perturbation corrections to the one-body moment, taken from refs. 30, 31). The magnetic m o m e n t due to the two-body spin-orbit interaction is also shown in table 1. It can be roughly parametrized for odd-proton nuclei by the formula, +~(2j+l)/(2j+2), for j = l _ ½ with ~ ~ 0.05. This is approximately ~ of the contribution f r o m the one-body spin-orbit force 28), ~ ~ 0.25. For odd-neutron nuclei, the m o m e n t is slightly smaller but of opposite sign, ~ ~ - 0 . 0 3 . The interaction moments depend sensitively on the range chosen for the nuclear well. (In fact, the values adopted for the oscillator parameter c~ are, for light nuclei, slightly smaller than those deduced from the empirical formula: h a ) ~ 40 A -~ MeV, i.e. c~ ~ 0.982 A -"~ f m - a . ) This is particularly true for the m o m e n t associated to the function fz(r) in the mesonic current (cf. eq. (34)): the reason is that this function changes sign in the nuclear internal region. To gain a quantitative estimate of this effect and to illustrate the remark relative to short range effects mentioned in sect. 2. l, we plotted in fig. 7 the mesonic m o m e n t of iv O as a function of the radial

466

M.

CHEMTOB

range parameter. Note the cancellation of the m o m e n t between 0 and/~ (# ~ 0.707 fro-x), which reflects the effect of compensation relative to f2(r). It is also seen on this figure that when the range decreases from/~-1 to (3p)-1, the m o m e n t is reduced by a factor ~ 10. Another striking feature of the interaction moments is their relative independence with respect to the mass number A 14,~5,22). This is the result of two competing effects when A increases. Then, the number of possible interactions between the valence nucleon and the core nucleons increases, but the ratio of the range of the operators, which stays constant ~ ro, to the nuclear radius ~ ro A'r, decreases. For

Moment (017) x 102

10.

,

,

• ~_

Fig. 7. Variation of the mesonic magnetic moment (in n.m.) of xTOwith the radial range parameter. # is the inverse meson Compton wave length. this reason, the interaction moments should not change appreciably when two nucleons of charge opposite to that of the valence nucleon are added to the core. We find that the difference of the moments between 39K and 41K and between 113In and l~SIn is just + 0.043 and - 0 . 0 0 4 n.m. The modification could however be important when the two nucleons have the same charge as the valence nucleon, or as in the case of 85Rb and 87Rb, when the angular m o m e n t u m of the nucleus changes its value. The independence with respect to the mass number suggests also that the limit where the core of the odd nucleus is quantized in an infinite volume (i.e. the Fermigas model) may not be too unrealistic an approximation for the process considered here. Let us search for the effective one-body operators associated to the interaction moments. Take, for example, the moments proportional to (ai++_aj) in the case of an ls saturated core (i.e. with L = 0, S = 0), and neglect antisymmetrization

NUCLEAR MAGNETIC MOMENTS

467

between valence and core. The following equalities are seen to hold: i
(a~+_aj)f(r,j) =

~

(a,+_aj)f(rlj) = av ~ f(r~i),

i =valence (v)

(49)

je core

j~ c o r e

which indicates the existence of an average field 6g~ s affecting the intrinsic m o m e n t of the valence nucleon. In the exchange terms, the spin operator s would be multiplied by a non-local operator. For the m o m e n t s of the type: [ ( a ~ + a j ) - ? P - ( a ~ + a j ) / 3 ] , the average one-body potential contains both s and the operator (s.p)p, involving the m o m e n t u m p of the valence nucleon. One can also prove that the Sachs moment, associated to the central N N potential, has an average effect proportional to the orbital m o m e n t u m 3gll 35). TABLE 2 The effect o f the Sachs m o m e n t (associated to the central p a r t o f the N N potential) and o f the spin exchange m o m e n t ( ~ - - t ~ ) are expressed as modifications of isovector orbital and spin gyromagnetic ratios, ~gz and ~gs, o f the odd nucleon ($gz

6gs

6g'8

170

0.067 0.066

0.027 0.055

--0.318 --0.290

41Ca

0.068 0.068

0.080 0.098

--0.242 --0.224

1~N

0.115 0.114

0.255 0.285

--0.420 --0.389

49K

0.106 0.105

0.218 0.243

--0.352 --0.326

Element

Lower figures correspond to the introduction of a hard core cut-off radius: r e = 0.485 fm. The correction 6g"s to the spin gyromagnetic ratio is associated to the exclusion principle effect.

In table 2, we give for 170, 41Ca, 15N and 39K the values of fig, and 6gs due to the dominant interaction moments: the Sachs m o m e n t and the mesonic spin-exchange m o m e n t associated to the functionf2(r) in eq. (34). We also show the effect of a hard core in the radial dependences, introduced by means of a cut-off radius, r e ~ 0.485 fm. Note that (i) 6g, and bgs remain practically constant for t h e j = l+_½ cases, but they vary appreciably from one case to the other and (ii) the effect of the hard core is most effective for 6gs. The corresponding values for an infinite nucleus are rather close. For a Fermi m o m e n t u m k v ~ 1.25 fm -1, we find fig, ~ 0.10 and 6g~ ~ 0.13. If the central OPE potential is replaced by some phenomenologicM N N potential the actual values of 6g, would be multiplied by a factor ~ 3. This remark raises, of course, the important question concerning the renormalization of interaction currents in nuclear matter as). Consider finally the exclusion principle effect. It is associated to the function f2(r) occurring in eq. (37). In the Fermi-gas model, previous calculations 6,12) give

468

M. CHEMTOB

fig~ ~ - 0.51. But according to ref. 13), this value falls down to fg~ g - 0.18 when the effect of the hard core is included by means of a separation distance method. In the last column of table 2, we show the corresponding values of fig" obtained with the shell model. The comparison of fg~ with figs indicates clearly how the exclusion principle effect is strongly compensated by the moment associated to the three-field zrN7 interaction. 6. Perturbative corrections In this section we evaluate the perturbative corrections to the mesonic moments of nuclei 15N, 39K, 170 and 41Ca. We restrict to the first-order corrections to the ground state wave functions, represented by the diagrams in fig. 8. Their contribution

+

-7 Fig. 8. First-order perturbation corrections to the expectation value o f the interaction m o m e n t ~ ' .

reads as a sum over intermediate states [~i) with two particles and one hole < ~jll&d'tXll[ t/'j> = 2 ~ < ~jl I~d'tZlll~e><4~lVI ~J> ,

i,j

(50)

AE

where V is the effective NN interaction. Only intermediate states corresponding to one particle-hole excitation modes of the core are considered

• . a+a s, ~ jm,)" 1~i) = ~
(51)

Mi

The energy denominators AE are then given by the difference between the energies of the particle and hole: AE -- e h - e p. For nuclei 170 and 15N, AE is obtained from the experimental single particle energies. For nuclei 41Ca and 39K however, we consider all the possible particle-hole excitations (there are 52 in all), and use for AE, the approximate value: [AE[ ~ 2h~ ~ 23 MeV. Performing the sum over magnetic quantum numbers in eq. (50) gives 2 j~.~or~ - AE

x

h K'

"(-)

K

[(2K + I)(2K'+ 1)] "i-

P} <(hj¢)KIl'lltz~[l(pj~)K'>a]

x [~ (-)s(2S+l){~

~

~} (,(jh)JlVl(jp)d>aJ

+ 2 y,[(2K+l)(2J+l)]½(_)s+h+j+;~ {jj h AE J, K x ((jh)Kll,,CttZ~l I(jp)J)a(,(jh)J[ V[(jp)J)a.

(52)

469

NUCLEAR MAGNETIC MOMENTS

This result refers to a particle configuration. For a hole configuration, the first term is multiplied by ( - ) ~ + 1 and the second term by ( - ) 4 , provided we interchange in eq. (52) the roles o f p and h. The numerical values we find for this correction are given in table 3. The secondorder perturbation corrections to the one-body moment, evaluated in ref. 31), are also shown in this table. Both calculations use the Kallio-Kolltveit potential for V. Since we are merely interested here in orders of magnitude, a further restriction was made which saves a lot of computing time. It consisted to impose a limiting value to the relative orbital momentum l < 2, in the two-body matrix elements of the interaction moments. TABLE 3

Comparison with experiment of magnetic moments of nuclei saturated in the ls sense, taking into account perturbative corrections to the one-body and two-body moments Perturbation Perturbation correction correction Interaction Spin-orbit (one-body (mesonic Element moment moment operator) moment) 15N

0.028

170

--0.101

0.028

0.16

0.013

39K

0.179

0.031

--0.26

--0.011

--0.061

--0.177

0.025

0.28

--0.010

0.118

41Ca

0.034

0.10

0.024

Total

~exp //Schmidt --#Schmidt

0.014

--0.251

--0.032

0.101

--1.913

0.019

0.148

0.243

1.913

0.318

It is seen from table 3 that, except for ISN, corrections are 6 % to 13 % of the moments obtained in the zero-order approximation. They are of opposite sign for nuclei ~70 and 39K. G o o d quantitative agreement with experimental value is reached for the moment of 15N. The agreement is however less good for 170 and 41Ca, and remains poor for 39K. 7. Conclusions The main conclusions of this work are that (i) the nuclear interaction currents obtained in the meson-theoretical approach have expressions which lend themselves to practical calculations, (ii) their effect on nuclear static magnetic moments is generally much too small to fill the gap between the Schmidt values and experiment, but it is not intrinsically weak. This suggests to look for their role in dynamic properties of nuclei, such as 7 transitions, electron scattering, etc. I am grateful to Dr. V. Gillet for his constant criticisms and suggestions and to Dr. M. Rho for numerous comments. I have greatly benefited from discussions with Drs. D. Bessis, R. Stora and C. W. Wong whom I want to thank. I am also grateful to Mrs. N. Tichit for her efficient help in the numerical part of this work.

470

M. CHEMTOB

N o t e added in proof" A r e n o r m a l i z e d p e r t u r b a t i o n e x p a n s i o n c a n b e c o n s t r u c t e d f o r t h e m o d e l o f t w o d y n a m i c a l n u c l e o n s , u s e d a t t h e e n d o f s u b s e c t . 2.2 t o t r e a t t h e i n t e r a c t i o n c u r r e n t d u e t o n u c l e o n recoil. I n t h e a d i a b a t i c l i m i t , it c a n b e s h o w n t h a t t h e ren o r m a l i z e d c o u p l i n g c o n s t a n t h a s t h e s a m e v a l u e as i n t h e s i n g l e - n u c l e o n m o d e l . I n s u c h a case, t h e r e w o u l d c o m e a n e x t r a - f a c t o r Z 2 / Z 1 = f / f ( o ) ,,~ 0.65 i n t h e r.h.s, o f eq. (27).

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35)

A. J. F. Siegert, Phys. Rev. 52 (1937) 787 R. K. Osborne and L. L. Foldy, Phys. Rev. 79 (1950) 795 R. G. Sachs, Nuclear theory (Addison Wesley, Cambridge, 1953) .L M. Berger, Phys. Rev. 94 (1954) 1698 F. Villars, Phys. Rev. 72 (1947) 257; Phys. Rev. 86 (1952) 476 H. Miyazawa, Prog. Theor. Phys. 5 (1951) 801 Y. Hara and E. Kuroboshi, Prog. Theor. Phys. 20 (1958) 163; Erratum, Prog. Theor. Phys. 21 (1959) 768 H. A. Bethe and J. S. Levinger, Phys. Rev. 78 (1950) 115 J. G. Brennan, W. M. Frank and D. W. Padgett, Nucl. Phys. 73 (1965) 424 R. J. Adler, Phys. Rev. 141 (1965) 1499 H. P. Noyes, Nucl. Phys. 74 (1965) 508 S. D. Drell and J. D. Walecka, Phys. Rev. 120 (1960) 1069 E. M. Nyman, Nucl. Phys. B1 (1967) 535 M. Ross, Phys. Rev. 88 (1952) 935 A. Russek and L. Spruch, Phys. Rev. 87 (1952) 1111 M. Ichimura and K. Yazaki, Nucl. Phys. 63 (1965) 401 S. S. Schweber, Relativistic quantum field theory (Harper and Row, New York, 1961) S. D. Drell and E. M. Henley, Phys. Rev. 88 (1952) 1053 14. Miyazawa, Phys. Rev. 104 (1956) 1741 G. F. Chew and F. E. Low, Phys. Rev. 101 (1955) 1570, 1579 L. M. Frantz, Phys. Rev. 111 (1957) 346 N. Austern and R. G. Sachs, Phys. Rev. 81 (1951) 705 R. H. Dalitz, Phys. Rev. 95 (1954) 799 S. Hatano and T. Kaneno, Prog. Theor. Phys. 15 (1956) 63 T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 A. Messiah, M6canique quantique (Dunod, Paris, 1960) A. de Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1962) R. J. Blin-Stoyle, Theories of nuclear moments (Oxford, London, 1957) 14. Noya, A. Arima and H. Horie, Supp. Prog. Theor. Phys. 8 (1958) 33 H . A . Mavromatis, L. Zamick and G. E. Brown, Nucl. Phys. 80 (1966) 545 H. A. Mavromatis and L. Zamick, Nucl. Phys. A104 (1967) 17; Phys. Lett. 20 (1966) 171 G. Ripka and L. Zamick, Phys. Lett. 23 (1966) 347 A. B. Migdal, JETP (Sov. Phys.) 46 (1964) 1680 R. J. Blin-Stoyle, Selected topics in nuclear spectroscopy, compiled by B. J. Verhaar (NorthHolland, Amsterdam, 1964) J. S. Bell, Nucl. Phys. 4 (1957) 295