A folded diagram microscopic calculation of nuclear Coulomb displacement energies

Nuclear Physics A361 Not to be reproduced

(1981) 4 12 - 434 ; @ North-Holland Publishing Co., Amsterdam

by photoprint

or microfilm without written permission

from the publisher

A FOLDED DIAGRAM MICROSCOPIC CALCULATION OF NUCLEAR COULOMB DISPLACEMENT ENERGIES K. C. TAM ‘, H. MUTHER, H. M. SDMMERMANN institut fir

*+ and T. T. S. KU0 tt

Kernphysik der KFd Jiilich, O-5170 Jiilicb, Germany

and AMAND I~tit~t~~r ~heoref~che

FAESSLER

Physik, Vn~vers~t~t ~b~gen,

D-,-7400 T~bingen, Germany

Received 7 July 1980 (Revised I December 1980) Abstract: We have performed a rather extensive microscopic calculation of the *7F-‘7O Coulomb energy differences d-EC. Our main purpose has been to study the effects of(i) folded diagrams, (ii) core polarization, and (iii) the effects due to different nucleon-nucl~n potentials, different single-particle spectra and different radial wave functions. Using a proton-neutron representation we have included higher-order Coulomb corrections like e.g. the coupling of valence particles to collective vibrations of the core. The inclusion of folded diagrams is very important; it is equivaIent to a self-consistent treatment of the Q-box starting energies. It causes a significant suppression of the effect due to core polarization, its contribution to dEc becoming small, about 0.03 MeV. As a consequence of this “self-correcting’* behavior our results for dEc are quite stable with respect to the choice of different single-particle spectra. Two rather different nucleon-nucleon potentials, the Reid soft-core potential and a meson exchange potential of the Bonn-Jtilich group also lead to quite similar results for dE? In the case of j = )*, for example, dEc calculated with the Reid potential ranges, dependmg on other assumptions, from 3.36 to 3.43 MeV, and from 3.36 to 3.40 MeV with the Bonn-Jiilich potential: Both are significantly smaller than the experimental value of 3.54 MeV. Optimizing the radial wave functions in the spirit of a Brueckner-Hartree-Fock theory improves the calculated values of AEc for thej =i f+ state but not forj = #+ or )+. We feel that other processes such as the explicit inclusion of core deformation or mesonic degrees of freedom or both are needed to explain the Coulomb displacement energies.

1, Introduction A very interesting problem in nuclear structure theory is the calculation of Coulomb displacement energies in mirror nuclei such as “F and “0. Many calculations have been performed over a wide range of nuclei and it has been a persistent common feature that the calculated values are generally about IO % too small T On leave from USDOE Contract ++ On Ieave from t Alexander von

the State University of New York at Stony Brook and supported in part by the No. DE-ACO2-76ERl~l. the State University of New York at Albany. H~boldt Foun~tion awardee. 412

K. C. Tam et al.

(il

(iii)

1 Coulomb displacement energies

413

(ii)

(iv)

Fig. 1. Some representative Coulomb energy diagrams. Note that all diagrams have only one Coulomb vertex indicated by a dashed line. The nuclear vertex is indicated by a wavy line, and it excludes the Coulomb interaction.

compared with the experimental values. This discrepancy is referred to as the NolenSchiffer-Okamoto anomaly ’ -3). As discussed in a recent review article by Shlomo 4), indeed, a large amount of work has been carried out to investigate this discrepancy. Many correction terms in addition to the first-order Coulomb interaction [fig. 1 (i)] have been considered, such as the effect of isospin mixing in the core wave function 5, and the introduction of a charge symmetry breaking force ‘26P7).Despite all these efforts, this discrepancy is still far from being resolved. In this paper we will carry out a systematic calculation of the Coulomb displacement energies within the frame work of a folded diagram many-body perturbation theory *-“). From the viewpoint of this theory there seem to be several processes which may be very important for the ~lculation of Coulomb displacement energies but have not yet been investigated. We will discuss these processes in the present section. As our example we consider the pair of mirror nuclei t’F and “0 both of which have one valence nucleon. kn order to calculate the Coulomb displacement energies one usually employs a low-order perturbation theory “). In this way some low-order process like diagrams (i) to (iii) of fig. 1 are included. Diagram (i) represents the Coulomb interaction between the valence proton and the core. Diagram (ii) is the core polarization diagram investigated by Auerbach et al. 5). In both diagrams the valence proton is considered to be entirely in the single-particle orbit j. In reality this proton clearly is not in this orbit at all times. For example, it may be excited into a 2plh state and consequently contribute to the Coulomb displacement energy via diagram 1 (iii). A major function of the folded diagrams, such as diagram 1 (iv), is to properly account for the fact that the valence proton is not in the single-particle orbitj all the time. This correction is quite significant; its effect is mainly to renormalize the non-folded terms by a wound integral factor (1 -K) [with K % 0.15, refs. r1,i2), being the probability for the proton to be absent from orbitj]. Similar renormalization terms also appear for the neutron and must be taken into account likewise. To

414

k C. Tam et al. / Coulomb disp~#~erne~tenergies

our knowledge no one has studied these important effects and it is a primary purpose of the present paper to determine the influence of folded diagrams on Coulomb displacement energies. Our second concern is the inclusion of the core polarization effect, namely the coupling of valence particles with the collective vibrations of the core. It is well known that single-particle energies are strongly influenced by the collective vibrations of the core. Hamamoto and Siemens l’) have pointed out that this influence can reduce the particle-hole gap in the lead region from x 10 MeV to w 7 MeV. The importance of such couplings was also emphasized by Brown et al. “f. They found that such influences on the single-panicle and single-hole energies can explain, to a large extent, the persistent discrepancy between the experimental and calculated positions of the giant dipole resonances in the lead region. The 17F-‘70 Coulomb energy difference is in fact just the difference between the proton and neutron singleparticle energies. Due to the Coulomb interaction protons and neutrons move in different single-particle orbits with different energies. We therefore expect, by analogy, that such couplings with collective core vibrations could play an important role in Coulomb energy calculations. Barroso 13) has already made some investigations of such effects on Coulomb displacement energies. However, neither core polarization diagrams higher than second-order nor folded diagrams had been included in his calculations. In addition he employed a simplified effective interaction as mentioned later in this section. We have carried out in the present work a more extensive investigation of the core polarization effect on the Coulomb displacement energies by including the TDA core polarization diagrams to all orders, using a proton-neutron (pn) formulation, As discussed in sect. 3, this core polarization effect is strongly influenced by the inclusion of folded diagrams. Our third point of interest is the effective interaction to be used in the calculation of Coulomb displacement energies. Simple effective interactions have often been preferred, mainly for the sake of convenience. Barroso 13), for example, employs the Kallio-Koltveit interaction which is neither energy nor density dependent. The same can be said about a number of other calculations 4). On the other hand it has been suggested to use more sophisti~ted interactions, such as an empirical charge s~metry breaking force ‘) and a density-dependent effective interaction la*r5) for the calculation of the Coulomb displacement energy. The choice of the effective interaction is clearly a central problem in such calculations. There is much uncertainty about the importance of mesonic charge symmetry breaking. In the present work we would like to avoid this uncertainty by using a G-matrix effective interaction accurately derived 16) from the Reid soft-core NN interaction 19). Apart from the Coulomb interaction, the free NN force employed in our work does not have any charge symmetry breaking component. The Coulomb force is included in our calculations of, e.g., the core poiarization effects, and leads to different contributions for protons and neutrons (see fig. 2). These core polarization effects can be interpreted as terms in an expansion of an effective pp or nn

K. C. Tam et al.

I Coulomb displacement energies

415

n

n

(d)

n Pt

nt

(In

Fig. 2. Diagrams contributing to the difference between proton-proton and neutron-neutron effective interactions. The nuclear interaction without its Coulomb component is indicated by a wavy line while that with the Coulomb component is represented by a wavy line with dot.

interaction, In this way we will in fact obtain a significant effective charge symmetry breaking force without employing any mesonic exchange mechanisms *). Only after we have performed an accurate many-body calculation determining this effective charge symmetry breaking force, can we make a reliable statement about the possible influence of mesonic charge symmetry breaking effects. The majority of the calculations reported in this paper is carried out with harmonic oscillator wave functions and the Reid soft-core NN potential. How reliabie is this choice? To examine this question we have also performed calculations using restricted B~eckner-~a~r~-F~k wave fictions and ~lc~ations using a recent meson exchange NN potential ““, (to be referred to as the Bonn-J~~ich ~tential). A special feature of this potential is its much weaker tensor force as compared with the tensor force component of the Reid soft-core potential. Somewhat to our surprise our calculated Coulomb energies are remarkably stable with respect to all these variations. We will discuss these processes in some detail in sects. 2 and 3.

416

K. C. Tom et al, 1 Coulomb displacement energies

2. Calculations 2.1. THE COULOMB

DISPLACEMENT

ENERGY

A frequently used procedure 4, for calculating the Coulomb displacement energy AE, between mirror nuclei is to treat it as an expectation value problem, i.e. -for “F and I70 4

= ~~,(‘7~~I~cl~~(“7~~~-~~~(‘70~I~cl~~(’7~~~,

(1)

where Vc is the Coulomb potential and Y, is the ground-state wave function of the nucleus under consideration in the following sense. The assumption has been made that the ground-state spatial spin wave functions of these two nuclei may be taken to be the same as far as the calculation of AE, is concerned. The influence of the Coulomb interaction on the wave function has been neglected. This appears to be plausible since the Coulomb interaction is, in a sense, much weaker than the nuclear interaction. When calculating AE, according to eq. (l), all diagrams contained in AE, will have one Coulomb vertex as shown by diagrams 1 (i) to 1 (iii). Actually, the 17F-“0 Coulomb displacement energy is the quantity given by AEij: = E;-E’,, where E’p and E’, are, respectively, the energies of the j-state of 1‘F and “0 example is thej = 3’ ground state). They satisfy the Schriidinger equations Hi,$ = Eiptjjp, H$j, = E$,b jn,

(2) (an

(3) (3.1)

with H = T+V,+V,.

(3.2)

Here, as mentioned earlier, Vc is the Coulomb potential. V, is the NN potential, which is taken to be the Reid soft-core potential ‘4 or the Bonn-J~lich potential ‘4. $j, and $‘, are the total wave functions of the j-states of 17F and l’0, respectively. There are a number of differences between AE, of eq. (1) and the correct AE; of eq. (2). In a correct calculation we must also take into account all diagrams with more than one Coulomb vertex in order to describe the effect of the Coulomb interaction on the wave functions. In addition, our wave function must have proper normalization. In principle, all this can be done in a folded diagram theory for the Coulomb displacement energy AE&. In this framework the calculation of AEi is in fact just the calculation of the charge symmetry breaking (CSB) effective interaction for a charge symmetric model wave function. This CSB interaction is the difference between the model space effective interaction of a valence proton with the protons in a charge symmetric I60 core and the corresponding effective interaction of a valence neutron with the neutrons in the r60 core [see fig. 2, diagrams (I) and (II)]. If Vc is set equal to zero, these two interactions will become the same

K. C. Tam et al. 1 Coulomb displacement energies

417

and we will not have any Coulomb displacement energy. Even if we include V,, the model space effective interaction of a valence proton with the neutrons in the charge symmetric core will be canceled by the effective interaction of a valence neutron with the core protons. There are many diagrams which will contribute to the above charge symmetry breaking effective interaction. Some of them are given in fig. 2. Diagrams (a) and (b) are part of the pp effective interaction, while (c) and (d) belong to the nn effective interaction. The Coulomb potential is a long-range interaction and thus its main effect is probably to produce an average Coulomb field. Hence it should be important to include all the Coulomb self-energy insertions as shown in fig. 2(a). This gives rise to a significant charge symmetry breaking effect. All proton intermediate states are subject to the average Coulomb field, while the neutrons, of course, are not. The charge structure of the intermediate states in the pp effective interaction is different from the charge structure in the neutron-neutron effective interaction. As a result the above processes will make the effective pp interactions different from those for nn. There are additional physical processes which will distinguish the pp and nn effective interactions. Again because of its long-range nature the Coulomb interaction may participate in an important way in the collective excitations of the core. In fig. 2(b) a valence proton interacts with a pn p-h chain. Here both nuclear and Coulomb interaction participate. Diagram (d) is the corresponding diagram between a valence neutron and a neutron-proton p-h chain. Clearly the above processes will also contribute to the difference between effective pp and nn interactions. In the present work we will include such p-h chains to all orders. Since we include the Coulomb interaction and use different single-particle spectra for protons and neutrons we will use a proton-neutron representation throughout our calculation. For illustration we have pointed out in the above some representative diagrams, involving more than one Coulomb interaction, which may contribute significantly to the Coulomb displacement energies AE,. It is, however, not justifiable to single out only some specific diagrams for AE, and to calculate them alone. For example, there may be other compensating diagrams which we should also include. It is better to manage problems of this kind in a more systematic and even-handed way. A convenient prescription for doing this is the so-called folded diagram method which we will use in the present work. Since the details of this method have been given elsewhere ’ - lo), we will only briefly describe it in this paper. According to this theory, the energies Et and E’, of eqs. (3) and (3.1) can be expressed in a simple and systematic form, namely E::, = &il,+Qjp-Q~SQj,+Q~SQj,SQ~-. E:b = s;+Qj,-Q;jQi,+Q;je’,JQ;-

. .,

(4)

. . .,

(5)

where the energies E are energy differences for protons (p) and neutrons (n), re-

418

K. C. Tam et al. / Coulomb displacement

energies

spectively, defined by ,!?; = E;-E;,

WI

E;, = E&E;,

(54

with EE the ground-state energy of 160. The E’Sare the unperturbed single-particle energies and the symbols j represent the folding operations. Qi, j Qi j Q$ for example, is the collection of all twice-folded diagrams. Q denotes the so-called Q-box which is composed of all one-body irreducible diagrams. All graphs shown in fig. 2 are irreducible and part of the Q-box. Note that we employ here a one-dimensional model space which consists of a closed 160 core plus one valence nucleon in the Od-1s shell. In the above equationsj denotes the quantum numbers of the valence particle. For instance, j = Od, for the 3’ state of “0. The Coulomb displacement energy AE& is given by the difference EL - E$ Its calculation is divided into two steps. First we have to construct the Q-boxes. Then follows the calculation of the folded Q-box series as indicated by eqs. (4) and (5). For a given Q-box, the folded Q-box series can be summed up easily and with high precision using a partial summation method lo). The most difficult part is the evaluation of the Q-boxes. We are able to sum up our series essentially exactly for a given exact Q. Our only approximation lies in the treatment of the Q-box. One possibility is to determine Q by low-order perturbation theory, i.e. we include diagrams Dl to D6 of fig. 3. Since we use a pn formalism the external lines ofdiagram D2, for example, are both Od, neutrons for the +’ calculation of “0 while for the 5’ state of “F they are both Od, protons. We have used several different methods to calculate the Q-boxes and their respective results will be compared in sect. 3. In diagrams D2 to D5 of fig. 3 the effect of core polarization on the Coulomb displacement energies is treated by low-order perturbation theory. To investigate higher-order core polarization processes, we have replaced diagrams D2 to D5 by

DI

02

D3

04

D2’

03’

04’

05

D6

D5’

Fig. 3. Q-box diagrams. The TDA core vibration phonons are represented by hatched blobs.

K. C. Tam et al.

/ Coulomb displacement energies

419

D2’ to DS, where the shaded blobs represent p-h TDA phonons. Another choice is the use of RPA phonons. However, we may then have to include self-screening and vertex renormalization diagrams *l). Our calculation would become much more complicated, and furthermore, there are indications *‘) that RPA results with selfscreening are practically equivalent to those of TDA without screening. Thus, for simplicity, we have chosen to work with TDA phonons. As an attempt to estimate the convergence of our calculations we will compare Coulomb energies obtained from diagrams D2 to DS with those obtained from diagrams D2’ to D5’. Let us now briefly summarize our calculation procedure. We first calculate the Q-boxes using both a low-order perturbation theory and a TDA phonon approximation as indicated by fig. 3. With these Q-boxes the folded diagram series of eqs. (4) and (5) are calculated to all orders using a partial summation method lo). This gives us the separation energies Ej, and ,!?“,of “F and l’0. The difference i?j,-Ej, yields the Coulomb displacement energy. 2.2. CALCULATION

OF DIAGRAMS

In this subsection we will give some details of how we have calculated the various diagrams. A few examples will be given. Usually these types of nuclear energy diagrams are evaluated in an isospin coupled representation and with an energyindependent effective interaction. In the present work we include the Coulomb interaction and thus have chosen a pn representation. In addition we employ an energy-dependent G-matrix effective interaction. Thus we should describe some specific calculational details, such as the treatment of the energy variable of the diagrams in conjunction with the evaluation of folded diagrams. First let us consider the evaluation of diagram D2 in fig. 3. It is given by D2(w) = where the labels a, m, k and I stand for the respective harmonic oscillator quantum numbers as well as the z-component of the isospin. For example, a = (n,, I,, j,, t 3. It is apparent from the diagram, that a, k and I are particles and m holes. o is an energy variable ; it is given here by o = E,, the single-particle energy for state a. w’ is the energy variable for the G-matrix vertex, and in this case it is o’ = o + E,,,. Vc is the Coulomb interaction. The matrix elements in eq. (6) are antisymmetrized, but not normalized, being related to the antisymmetrized and normalized matrix elements (denoted by subscript AN) through
= [Cl +s,,)(l

+&#(am.JlG(o’)+

where si, for example, stands for Z = (n,, 1,,j,).

Vc/ClklJ)AN

(7)

420

K. C. Tam et al. / Coulomb di~p~~cern~~i energies

The G-matrix in the pn representation

G(d) = VN+ V,Q

is calculated by solving the equation 1 o’ - Q( T+ dc)Q QG(e’J”

by way of a momentum space matrix inversion method 23*16) where the Pauli exclusion operator Q is treated in an essentially exact way. We have used a Qoperator specified by (n,, n,, n,) = (3, 10, 21) [ref. r6)], and we have taken d,, the shift of the intermediate state energies of the G-matrix due to the Coulomb field, as a constant. In fact, we use dc = 2 UC,,“,when all external lines of G are protons and dc = ucoU,when only two external lines of G are protons. Clearly, A, = 0 for all external lines being neutrons. ucoUIis an average Coulomb energy felt by a proton in the intermediate states + of the G-matrix. We have estimated u,,,, to be about 2.75 MeV. It may be mentioned that ucou, basically represents an average Coulomb self-energy insertion as shown by fig. 2(a). Such in~rtions are summed up to all orders in eq. (8) *. For a given value of the starting energy w’ the introduction of dc in eq. (8) will cause the pp, pn and nn G-matrices to be different. This will contribute to the charge symmetry breaking component of our effective interaction. Note that VNin eq. (8) is the NN interaction; the Coulomb interaction Vc is treated separately as indicated by eq. (6). In the following we will consider the evaluation of diagram D2’ of fig. 3 to illustrate the calc~ation of diagrams when valence particles are coupled to TDA core vibration phonons. This diagram is given by

x (-)j,+j,,+jp2+jh2
(G(o,) + ~ctp~p~)(p~p~lG(#~)+ t

.I

t

&/ah,> - D2(0),

(9)

+

where D2(cu) is given in eq. (6). In diagram D2’ we in fact include “two” p-h phonons. One is between the particle and hole lines as shown in the figure. In addition we include the phonon involving particle line p3 as well. Hence we have included the lowest-order term in the TDA phonon twice. This term is just diagram D2 and is explicitly removed in eq. (9). The energy variables o1 and w2 of the G-matrices are related to CL) by w1 = w+E~, and o2 = o+ahl. The Xs and E,, are, respectively, ’ Note that the intermediate states of our G-matrix are restricted to orbits above the Of-lp shell. * Our model space involves a Q-projection operator specified by (n,, nb, nc) = (3, 10,21). The present approach then corresponds to a double partition method where for single-particle states from Od,,, to IP 1,2 we include both nuclear and Coulomb self-energy insertions to all orders, while for higher states, which enter into our calculation through the G-matrix, we only include the Coulomb self-energy insertions. This treatment is in accord with nuclear matter results which, for high momenta, predict small contributions from nuclear self-energy insertions, if they are summed to all orders “).

K, C. Tam et al. / Coulomb displacement energies

the wave functions (normalized) given by the equation

421

and the energies of the standard TDA phonons,

(9.1) Note that the p-h interaction A and therefore also the X’s and ES all depend on the energy variable w of diagram Do’. This energy dependence is strictly observed in the present salutation by solving the above equation for a rather wide range of energy variables. To see this dependence, we give the equation for the evaluation of the p-h matrix element : (ph-‘J[Alp’h’-‘J)

= ___

x(-l)

F/,lp’hJ’),

l+jh+jh’-J-J’(ph’J’(~~“)+

(9.2)

with w” = o-et_ + &h+ E,,.. We see that 0” depends directly on the zingle-particle energies E. The Coulomb interaction is included in the calculation of the p-h interaction as shown above and in fig. 2(b). The X-coefftcient is the familiar LS-jj transformation coefftcient, related to the 6-j symbol by _X ‘: i

J’

:

:

J’

0

, = ~~$&j(-)J+J’+b+d

(9.3)

The cross-coupled matrix element 24) in eq. (9) is related to the ordinal particle matrix element by

r--Y Gd,IG+ ~,VJJ =

G---!

-J$$J~X

particle-

j4 j2 J A j, J

! ! J’

J’

0

I?

x(-1>’1tjZfj3tj4(jlj2~G+~c~j3j4).

fi (9.4)

The calculation of diagrams is very much simplified by the use of cross-coupled matrix elements, as has been discussed in ref. *“). In fact, we have employed this method in the present work. The treatment of the energy denominator for the types of diagrams like D3 and D4 should be pointed out. Recently the folded diagram method of Kuo, et al. ‘) has been generalized 25) by way of an equation of motion approach. The generalized method is physically more desirable because it employs a model space corresponding to the addition of valence particles to the true ground state of the core system. In the earlier method we had added valence particles to the unperturbed ground state

422

K. C. Tam et al. / Coulomb displacement energies

of the core system in forming the model space. For the present calculation these two methods give different energy denominators in the diagram evaluation. For example, for diagram D3 we have, using an uncoupled representation for simplicity in illustration,

klm

Please note that we have --o in the energy denominator would method has the convenience of the energy shell. The expression resentation, is written as

Wa) =

-m-(&,-&k-&[)

’

(10)

the denominator. Had we used the earlier method, have been w-(2&, +E,- E~-EJ. The generalized making a larger class of self-energy insertions on for diagram D4, again using an uncoupled rep-

c WlG(~,W)

klm

(~MTw,)l&

>

(11)

Ek-E,

where o3 = sk+ sr and o4 = o + Ed. In the previous formulation the energy denominator would have been W--E,-.s,+ek which leads to singularities in D4 for certain o. This type of singular behavior can be eliminated in the generalized folded diagram method. As a consequence we expect this new theory to exhibit improved convergence and we will use it in the present work. Note that in this method the separation energies are still given by the expressions (4) and (5), except that the Q-box diagrams are now calculated somewhat differently as noted above. We will now describe the calculation of the folded diagrams. Let us denote the tiurn of diagrams Dl to D6 of fig. 3 as Q(o). (For the TDA phonon case, Q(o) will consist of diagrams Dl, D6 and D2’ to D5’.) The various Q-boxes in eqs. (4) and (5) are all evaluated with o = E; and si,, respectively. If we do not include any folded terms we are simply led to E;j, NNE; + Q’,(&‘,), _.

E’, x E;+Q#).

(12) (13)

One may calculate the folded diagrams term by term with, for example, an energy derivative method 1‘3 12). But it is more convenient to use a partial summation method lo). In this way the folded diagrams for a given Q-box can be summed up to all orders, leading to the self-consistent equations

Note well that the summation these equations by an iteration for each of the above equations. with the largest model space

& = E;+ Q#),

(14)

Ej, = si,+ Q@‘,).

(15)

of folded diagrams corresponds to the solution of scheme lo). Thus we will obtain only one solution Generally this solution belongs to the wave function overlap, if the iteration procedure converges. We

K. C. Tam et al. 1 Coulomb displacement energies

423

now see that the structure of our calculation is quite simple. We first calculate the Q-boxes according to fig. 3 for a number of energy values CO.Then one can solve the above eqs. (14) and (15) graphically in an unambiguous way, giving us Ei,, Ej,, and the Coulomb displacement energies AEj,. 2.3. THE SINGLE-PARTICLE

SPECTRUM

In order to calculate the Q-box diagrams as shown by the various equations of the preceding section, we must know the single-particle energies E. Introducing a single-particle potential U to the hamiltonian H of eq. (3.2), we rewrite H as H = HO-+-H, with H;, = T-t U and H, = V, + V,- U. The single-particle energies and wave functions are determined by the unsoured h~iltonian H,,. If N, is treated exactly the solutions of H are clearly ind~ndent of the imposed singleparticle potential U. In this case any U we like must lead to the correct results. But in practice, N, is treated in a ~~urbative fashion. We are therefore interested to obtain some U which will “minimize” H, in the sense that it can be treated accurately by perturbation. To find the best choice for U is a difficult and controversial problem. In the case of nuclear matter - an infinite nuclear system - there have been extensive studies 26) on this problem. Most groups working in that field seem to prefer the following choice. They use a Brueckner-Hartree-Fock (BHF) singleparticle potential for states below the Fermi momentum kr. For states way above k,, one takes a free particle spectrum. The region just above the Fermi surface is less certain. Several years ago it was popular to use a discontinuous single-particle spectrum where one employed a free particle spectrum for all states > k,. This energy spectrum is discontinuous at k,, and in retrospect one sees that this choice was rather artificial. The present consensus seems to be a spectrum which is continuous at the Fermi surface and to treat states with k > kF in the same way as states with k < k,. The proper choice of a single-particle spectrum in finite nuclear systems - nuclei is even less certain. We will use several different choices in the present work and investigate how sensitive our results are with respect to these choices. We will follow rather closely the ideas developed in nuclear matter theory. For shell-model states higher than the of-Ip shell, we use a free particle s~tr~ 16). These high Iying states enter into our calc~ation through the G-matrix in~~ediate states as shown by eq. (8). For the ten shell-model states from OS,to lp, we have three alternative pr~criptions. First a simple harmonic oscillator spectrum with some suitable constant shifts, roughly in accordance with the experimental d, levels. This is denoted as spectrum A and will be discussed in sect. 3. In the other two cases we include lowest-order BHF self-energy insertions to determine a self-consistent spectrum. This is given by

424

K. C. Tam Ed al. / Coulomb displacement

energies

where h is summed over the oscillator states OS+,Op, and Op,. a can be any of the ten oscillator states mentioned above; t is the kinetic energy. In the case of c1being a hole state, it is well known that the G-matrix in eq. (16) can be evaluated on the energy shell, i.e. 0, = E,+Eh

(16.1)

For a being a particle state, however, we are unable to do so. One approximation is to let o, be off the energy shell by a constant amount - an average value Ap. In this way we have o, = ~,+q,+Ap

(16.2)

for a being a particle state. We have taken two different values for Ap. One is Ap = -28 MeV, corresponding to two oscillator quanta excitations +. The other choice is Ap = 0, being in analogy to the continuum choice advocated in nuclear matter theories. The resulting spectra will be denoted respectively as spectrum B(Ap = -28 MeV) and C(Ap = 0). 2.4. CALCULATION

WITH AN EXTENDED

MODEL SPACE

Previous calculations have indicated that the “F-r’0 Coulomb displacement energies are sensitive to the radial wave functions used for the valence nucleons 3). In earlier calculations with empirical effective interactions 4), Woods-Saxon wave functions have been employed which are more realistic than pure oscillator wave functions, especially for valence nucleons. In the formulation described so far, we have used a one-dimensional model space consisting of a closed 160 core plus one valence nucleon in the Od-1s oscillator shell. The effect of core excitations has been included in our Coulomb energy calculations. Clearly, the Q-box diagrams (fig. 3) allow for p-h excitations of the core. When carried to infinite order, our theory guarantess to yield the exact energy of that eigenstate whose overlap with the model space is largest. The fact that only low-order diagrams with a restricted set of intermediate states ++ can be included in our perturbation expansion of Q makes it advisable, however, to explicitly consider corrections due to the admixture of Id, and 2d, components into our Od, shell-model valence wave function. This type of approach may be quite important, as it directly changes the radial wave functions of the valence nucleons and as a result should modify the Coulomb energies in a significant way. To investigate the above radial wave function corrections, we use an approach essentially equivalent to a Brueckner-Hartree-Fock theory. We continue to use the case ofj = 3’ as an illustration of our method. We want to look at the effect due to the admixture of Id, and 2d, wave functions to the Od, wave function. The folded t We use hw = 14 MeV for the harmonic oscillator in the oxygen region. ” The major shells with orbits Id,,,, 2d,,, and higher are not included as possible intermediate states.

K. C. Tam et al. / Coulomb displacement energies

425

diagram theory described in subsects. 2.1 and 2.2 can easily be extended to study such problems. The essential step is to introduce an extended model space composed of a closed 160 core plus one valence nucleon populating any of the M,, Id, and 2d, orbits. In addition to the diagonal Q-box diagrams of fig. 3 we now need off-diagonal Q-box elements as well. For example, the present investigation requires diagram D2 of fig. 3 to be evaluated between the Od, and 2d, states. Within this extended model space the Q-box becomes a 3 x 3 matrix. The self-consistent equations (14) and (15) are replaced by 3 x 3 matrix equations, of the form g?X;

= $‘X;+

c (mlQ’,(B$lm’)X;. m’

(17)

for the calculation of the j = 3’ states of I’F. The sum for m’ extends over Od,, Id, and 2d,. This equation is solved by matrix diagonalization using different values for the starting energy in the self-consistent equation for Q, i.e. the starting energy equal to Sj, is obtained by means of a graphical method. Eq. (17) yields three solutions, denoted by N = 1, 2 and 3. In our Coulomb energy calculation we are mainly interested in the solution with the lowest energy. The extended model space formulations for the 3’ and ++ states are very similar. For the 3’ states we use a three dimensional model space with a Is,, 2s+ or 3s+ nucleon plus a closed 160 core. In the case of j = 3’ our model space again has three dimensions consisting of a Od,, Id, and 2d, nucleon plus a closed 160 core. In summary, we have generalized the scalar self-consistent equations (14) and (15) to three-dimensional matrix equations, allowing the valence nucleons to have a degree of freedom of three major oscillator shells. Our Coulomb displacement energies are given by the differences between the lowest EL” solutions and the lowest g:j,” solutions. If we only include diagram Dl of fig. 3 in the respective Qbox, our formulation is simply equivalent to a restricted Brueckner-Hartree-Fock (BHF) calculation. If we also include diagrams D2 and D3, it may be referred to as a restricted density-dependent Brueckner-Hartree-Fock (DBHF) calculation. The calculations are restricted since the single-particle wave functions of the core are not determined self-consistently and the degree of freedom for the valence particles is limited to three oscillator shells. We therefore have performed one-dimensional and three-dimensional BHF-

Fig. 4. A typical process included in the extended model space calculation.

426

K. C. Tam et al. / Coulomb displacement energies

type calculations. All diagrams that are contained in our Q-box are summed to infinite order through the matrix equation. The diagram in fig. 4, for instance, which has not been included in the one-dimensional case is taken care of to all orders in the three-dimensional formulation.

3. Results and discussion The first step in our calculation of the Coulomb displacement energy AE, is the choice of the single-particle spectrum. To test the sensibility of AE, with respect to this choice we have used three different single-particle spectra; they are shown in table 1. A is a pure harmonic oscillator spectrum with constant shifts for protons and neutrons. B and C are both obtained by a BHF procedure (see subsect. 2.3); they differ in the treatment of the self-energy insertions. These three spectra are significantly different from each other. For example, the neutron Od, energies are -4, 0.37, and - 1.42 MeV for the different cases and one would expect to obtain rather different values for AE, from them. As we will see shortly, however, the results for AE, are very close to each other. In table 2 we show some typical values of the Q-box diagrams of fig. 3 forj = $ + . Their calculation has been discussed in subsect. 2.2. Here we give the results for two values of the diagram energy variable o, -4 and - 1 MeV. We are interested in the difference between diagrams with proton external lines (denoted as p) and those with neutron external lines (denoted as n). They are directly related to the Coulomb energy difference AE,. We observe that the proton diagrams are generally less attractive than the neutron diagrams. This, of course, yields the Coulomb energy difference. If we do not include any folded diagrams, the difference between the above two kinds of diagrams, evaluated at the same energy o, will just give the Coulomb energy difference. One could claim, for example, that diagram D2 in scheme (I) TABLE1 Different single-particle spectra used in the present calculation s112

p3/2

PI

-18 -15

,*

n P

-32 -29

-18 -15

B

n P

- 38.90 - 35.36

- 19.33 - 15.62 - 16.03 - 12.33

C

n P

- 38.66 -35.12

- 15.83 - 12.12

A

- 19.12 -15.41

d 512 -4’ -1

d 312

%,Z

-4 -1

-4 -1

f 712

f 512

P3/2

PI,2

10

10

10

10

13

13

13

13

14.75 17.87

19.49 22.62

13.96 17.25

15.25 18.54

0.37 3.72

5.35 8.70

2.44 5.95

- 1.42

3.44

0.41

13.26

17.85

12.09

13.33

1.93

6.79

3.92

16.39

20.98

15.38

16.62

A represents harmonic oscillator energies while B and C are self-consistent spectra. See subsect. 2.3 for discussions. All entries are in MeV.

K. C. Tam et al. / Coulomb displacement energies

427

TABLE2 Some typical values of the j = 4’ diagonal Q-box diagrams of fig. 3 j =

0

-4

-1

I

4’

n

Dl

D2

D3

D4

D5

Sum

- 25.79 -22.19 3.60

- 2.93 -2.56 0.37

1.36 0.12

- 1.85 -1.47 0.38

1.75 1.45 -0.30

- 27.46 - 23.29 4.17

1.48

II

n P A*

- 25.79 -22.19 3.60

-3.37 -2.81 0.56

1.62 1.82 0.20

- 1.77 -1.40 0.37

1.64 1.32 -0.32

-27.81 - 23.34 4.47

1

n

- 25.99 -22.39 3.60

-3.25 -2.82 0.43

1.26 1.35 0.09

-1.85 -1.47 0.38

1.75 1.45 -0.30

-28.09 -23.87 4.22

II

n

- 25.99 -22.39 3.60

- 3.89 -3.17 0.71

1.45 1.60 0.15

- 1.78 -1.40 0.38

1.64 1.32 -0.32

- 28.70 -24.12 4.58

Single particle spectrum A is used. Two sets of values are given. One is calculated with second-or&r perturbation theory, denoted by I. The other with TDA phonons, denoted by II. A, and A, are, respectively, the differences between proton and neutron diagrams. All entries are in MeV.

with o = -4 MeV contributes the amount A, = +0.37 MeV to AE,. The A, values of diagrams D4 and D5 are of opposite sign, together they contribute + 0.08 MeV to AE,. This value is approximately the same as the result from the AuerbachKahana-Weneser (AKW) core polarization diagram ‘). (Note that we do not use self-consistent wave functions, therefore we should compare the sum of diagrams D4 and D5 with the AKW core polarization diagram.) As another observation we note the sizable differences between the quantities A, and A, in some cases. A, results from the calculation with TDA phonons, i.e. diagrams D2’ to D5’ in fig. 3, while A, is calculated in second-order perturbation theory. It seems as though phonons would make a large contribution to the Coulomb energy difference. Let us now study the importance of the folded diagrams. Even though the Q-box elements do not seem to depend strongly on o in the energy region from -4 to - 1 MeV, their variation with energy nevertheless significantly influences AE, via the folded diagrams. As we have outlined in the previous sections, the inclusion of folded diagrams leads to the self-consistent equations (14) and (15) whose graphical solutions are shown in fig. 5. In the conventional approach without folded diagrams the Coulomb energy difference would be given by the vertical difference between points a’ and c’ in case (I) (without phonons), or between points b’ and d’ if we include theTDAphonons(II).This wouldlead to a large overestimate of thecoulomb displacement energy. Forj = +’ the results range from 4.17 to 4.58 MeV which has to be compared with the experimental value of 3.54 MeV (see table 2). The picture

428

K.

c. Tam

el

al. !

Cautambdisplueement

energies

j = 5/2+

-24I

n)

proton

-26 T E -

w [M&j

Fig. 5. Energy dependence of Q;(w) and Q!(o) for j = $+, calculated with the Reid soft-core potential and the single-particle spectrum sA of subsect. 2.3. The curves denoted by I are calculated with diagrams Dl to D6 of fig. 3 while the curves denoted by II are obtained from the diagrams Dl, D6and D2’ through DS. The self-consistent solutions o = .s’+ Q’(o) are obtained by finding the intersections (a, b and c, d) of the straight line (o-e’) with Q’(w). Neglect of the folded diagrams leads to the equation w = &+Qj(ej). The corresponding solutions for (w-ej) are a’, b’ and c’, d’ for the proton and neutron, respectively. Please note that the energy scale corresponds to a Q-box in which the oscillator potential energy (- uj) is not included.

drastically changes with the inclusion of folded diagrams. AE, is now given by the vertical distances between (a) and (c), or (h) and (d) for the cases (I) and (II), respectively. In the case of j = 3 c, which is depicted in fig. 5, we now obtain values for A& which are (I) 0.17 and (II) 0.14 MeV smaller than the experimental number. In table 3 we show the calculated values of the ‘7F-‘70 Coulomb displacement energy AE, with inclusion of folded diagrams. We have calculated AE, for three different choices of the single particle spectrum and in each case used both prescriptions for the evaluation of the Q-box. We are most interested in the j = 3’ state, for which our calculated values of A EC are about 0.1 to 0.15 MeV smaller than the experimental result. The effect of core polarization, treated here by including TDA phonons, is found to be small, producing an increase of only about 0.03 MeV in AEc. The fact that it is so small is caused by the inclusion of the folded diagrams. Without them the effect of core polarization would appear to be much larger, as discussed earlier in connection with fig. 5. We would like to mention that we have

429

K. C. Tam et al. 1 Coulomb displacement energies TABLE3 The i’F-“0

Coulomb displacement energies AE; calculated with the Reid soft core potential including folded diagrams

4’

EA % %

expt

4’

t+

I

II

I

II

I

II

3.31 3.41 3.41

3.40 3.43 3.43

3.29 3.36 3.36

3.32 3.31 3.38

3.36 3.46 3.46

3.40 3.49 3.50

3.54

3.60

3.17

As discussed in the text, in I the Q-box is calculated with diagrams Dl to D6 of fig. 3 while in II diagrams D2 to D5 are replaced by the respective TDA phonon diagrams D2’ to D5’. Ed, Ed and ec refer to the three single-particle spectra of subsection 2.3. All entries are in MeV.

TABLE3a Correction terms in the calculation of the Coulomb displacement energy AE, for the Od,,, state in “F-“0, together with the first-order Coulomb contribution [fig. 1 (i)] Correction mechanism

Correction 6

first-order Coulomb contribution center of mass motion finite size of the proton finite size of the neutron magnetic interaction vacuum polarization proton-neutron mass difference short-range correlations

3.36 - 0.070 0.095 - 0.035 - 0.060 0.020 0.035 0.045

The values for the correction terms have been obtained from ref. 4).

studied the effect of the spurious 1 - lowest state in the intermediate TDA phonons. Its presence was shown to effect the Coulomb energies to a negligible extent. A remarkable feature of our results is their insensitivity to the differences in the three single-particle spectra q,, sB and ec which we have employed. The spectra .sr, and ec ensure the cancellation of self-energy insertions, and are considered to be more realistic than the more naive harmonic oscillator spectrum q,. As we have seen in table 1, these three spectra are significantly different from each other. Yet table 3 shows that sB and cc give practically identical results for AE,. This again is caused mainly by the inclusion of the folded diagrams. We think that the insensitivity of our value for AE, with respect to the choice of the single-particle spectrum is a strong indication of the reliability of our calculation, as exact manybody calculations are independent of the particular single-particle spectrum in use. In the cases of the j = 3’ and f + states, our results for AE, show considerably larger discrepancies with experiment. For the j = 3’ state, the effect of core

430

K. C. Tam ef al. / Coulomb displacement

energies

polarization improves the calculated results slightly. In the case of the j = 3’ state, however, core polarization processes move the calculated values further away from the experimental number. The discrepancy especially for the j = 3’ state could be due to the sensitivity of the Coulomb energy on the detailed form of the ls+ single-particle wave function. This dependence on the radial wave function has been found also by earlier investigators 3, and we will discuss it later in connection with the extended model space calculation. When comparing our results with the experimental Coulomb energies we should keep in mind that there exist a number of additional corrections which we have not examined in the present work. These correction terms have been discussed extensively in the literature ’ 93- ‘) and were reviewed and summarized by Shlomo 4). For the case of the Od, state in “F-l’0 the pertinent values are listed in table 3a, together with the first-order Coulomb contribution [lig. 1 (i)]. Since we employ G-matrices in our calculation we have considered the effect of short-range correlations. The remaining terms are small and frequently of opposite sign. Therefore it is very unlikely that our results would be changed significantly by the inclusion of these corrections. We have performed an extensive many-body calculation of the Coulomb displacement energy AE,, but even after considering a large number of possible contributing mechanisms the results still show significant disagreement with the experimental values. What are the reasons for this persistent discrepancy? The folded diagram theory which we used in this work is formally exact. The only approximation in our calculation lies in the treatment of the Q-box, which is composed of all irreducible diagrams as discussed in subsect. 2.2. In practice, however, we can only include classes of Q-box diagrams. The use of different single-particle spectra (Ed, es and sc) and the different ways of treating the core polarization (I and II) in our calculation all correspond to different approximation schemes for the Q-box. Our results are seen to be very stable with respect to these different treatments of Q. Another support for the reliability of our calculation may be pointed out. We calculate AE, by first evaluating the “F and “0 separation energies E;P and Ey and then taking the difference between them. To check the consistency of our calculation we show the results for j.?y, i.e. the separation energies of “0, in fig. 6. All six calculations are seen to be in quite good agreement with the experimental values. It is’interesting that the excitation spectrum of “0 is very well reproduced, only the ground-state (3’) binding energy is off by a rather large amount, ranging from - 0.6 to - 1.7 MeV. This ground-state binding energy depends on the NN potential. Let us therefore examine the problem of the NN interaction. So far all our calculations were based on the Reid soft-core potential. Could our failure in reproducing the experimental Coulomb energy differences be due to certain defects of this potential? To investigate this question we have repeated our calculation using a different NN interaction. We employed the meson exchange Bonn-Jtilich potential which is characterized by a considerably weaker tensor force (with respect to Reid soft-core

431

K. C. Tam et al. / Coulomb displacement energies “0

4.60

s/*+4.45

z ; :

4.95

4.79

4.91-

4.71

5.09

1.26 -

I .06

l/2+0.89

1.18

0.94_

0.67

(-2.69)

(-4.14) Expt.

0.32 5/2+-

(-3.16) I

(-3.48)

----(-2.41)

(-2.51)

I

II

L

P

%

_

,

ls

(-2.55)

I

II

\

, CC

Fig. 6. Energy levels of I’0 calculated with the Reid soft-core potential. sA, sr, and sc refer to the singleparticle spectrum of subsect. 2.3. I and II refer to the calculation of the Q-box as explained in the caption of fig. 4.

potential). As a consequence, it contains relatively stronger central components. In short, these two potentials are quite different. Nevertheless we find an amazing agreement for both interactions when we calculate their respective Coulomb energy differences (see table 4). The results are only 0.04 MeV apart, the Bonn-Jtilich potential giving the smaller values for AE,. The “0 spectrum calculated with the Bonn-Jiilich potential is shown in fig. 7, the ground state of “0 is now overbound. Thus there seems to be no direct relation between binding energy and Coulomb energy calculations. While the total binding energy significantly depends on the interaction used, the relative energies, like Coulomb displacement energies and the excitation spectrum of 170, are rather insensitive to this choice. Finally we have studied another possibility for the improvement of AE,. As TABLE4 Dependence of the j = )’ Coulomb displacement energies on nucleon-nucleon potentials

%I

EC

I

II I

II See table 3 for explanations.

Bonn-Jtilich

RSC

Expt

3.36 3.39

3.40 3.43

3.54

3.36 3.40

3.41 3.43

432

K. C. Tam et al. 1 Coulomb displacement

energies

“0 32.22 20.95

I-

0.08

o-

3/2+

-l -1.75 -2-2.69 -3 -

-3.27

-4 -

-4.14 -

-5L Reid

Expt

Ec,II

-4.41

1/2+

-4.55

5/2+

Bonn- Jiilich Ec,lI

Fig. 7. Comparison of energy levels of “0. See the caption of fig. 5 for the meaning of sc and II.

discussed already in subsect. 2.4, we can improve the valence particle radial wave functions by admixing oscillator wave functions with more radial nodes. The results for such a sequence of extended model space calculations are presented in table 5. The values of AE, forj = 3’ are now even smaller than those given in table 3. This is explained by the fact that the d, proton can now spread out over a larger volume, thus reducing the Coulomb energy. We see, however, that the present results are in TABLE5 Effect of improved radial wave functions on the Coulomb displacement energies f

t

f

;

3.35 3.27

3.34 2.97

3.51 3.17

;

3.41 3.34

3.36 3.09

3.46 3.20

3.54

3.60

3.17

BHF

DBHF

expt

As discussed in the text, the Q-box for BHF is calculated with diagram Dl of fig. 3 while for DBHF we use diagrams Dl to D6. a refers to the case of one-dimensional model space while in p we use an extended three-dimensional model space. Calculations were done with the Reid soft core potential and single-particle spectrum ~c.

K. C. Tam et al. / Coulomb

displacement

energies

433

better agreement with experiment than those given in table 3 in the case of j = f ’ . This is an interesting point. The Coulomb energy forj = 4’ is rather sensitive to the radial wave function which we employ. For the j = 3’ case the results are not improved by using an extended model space. There have been extensive discussions of the relation between the Coulomb displacement energy and the r.m.s. (root mean square) radius of the nucleus under consideration. The effect on AE, due to changes of the radial wave function 3, may be viewed from this point. The r.m.s. radius of the core charge distribution - I60 in our case - can be obtained empirically from electron scattering experiments. Nolen and Schiffer ‘) have shown years ago that the calculated value of AE, is too small when we use single-particle wave functions from a (Woods-Saxon) potential well which fit the empirical charge radius. In this case the ratio of the r.m.s. radii for the valence nucleon and the core (r&,)&/(r&)* is found to be 1.37 (“F). A ratio of 1.15 would be necessary to fit the experimental Coulomb energy. It is clear that “squeezing” the nucleus in this way will increase AE,. We would like to emphasize, however, that our purpose is not to derive the Coulomb displacement energies from wave functions whose radii have been artificially adjusted such as to give the correct experimental energies AE,. In the present perturbative calculation the radius of the wave function for state j is not just determined by the single-particle potential U(ho = 41Kf MeV). Instead it is given through the effective one-body operator r& whose diagrammatic structure is similar to the graphs in fig. 3. Microscopically the radius is obtained from a linear combination of one-particle states plus many-particle many-hole excitations which is determined by the NN interaction in use and the Coulomb interaction. It is interesting to see that the unperturbed radii of both, valence nucleon and core, are modified. In fact, our many-body treatment with the inclusion of folded diagrams is, in principle, independent of the auxiliary single-particle potential U(H = (T+ U) + (V- U)). Starting with any U we will obtain the correct wave functions (radius) and energies once convergence of the series is achieved. However, if AE, can be calculated correctly by including only a subset of perturbation terms, then our calculation should also be required to reproduce the experimental charge radius (even though, within the framework of many-body theory, there exists no obvious connection between the accuracy of AE, and rr._. in any given order of the perturbation). In the present sequence of microscopic calculations we have not succeeded in reproducing the experimental “F-“O Coulomb displacement energies. The results of the extended model space calculation indicate the sensitive dependence of AE, on the wave functions in use. It will indeed be a very interesting project to calculate (r2)* for “F using the effective operator many body approach. A calculation of this kind, which is beyond the scope of our present investigation, will be necessary in order to study the connection between Coulomb energies, the nuclear charge

434

K. C. Tam et al. / Coulomb displacement energies

radii and the structure of the radial wave functions from a microscopic point of view. As a summary, the present work has shown that the role of folded diagrams is very important for microscopic Coulomb energy calculations. They should be included in future calculations of this type. All of our calculations reproduce the “0 excitation spectrum quite well. However, they all yield (j = $‘) ground-state Coulomb energy differences which are about 0.1 to 0.15 MeV smaller than the experimental value. The results for AE, did not change when we used different realistic nucleon-nucleon potentials. We have shown that a contribution of only 0.03 MeV is obtained from the coupling of the valence particles to TDA core phonons. We believe that these results are a clear indication that conventional many body calculations using realistic nucleon-nucleon potentials are unlikely to give adequate Coulomb energy differences. We must look for additional physical processes in order to explain this discrepancy. Valence neutrons are more effective than valence protons in polarizing the proton distribution of the core. Therefore, even though TDA phonons could not remove the discrepancy, further improvement may be expected from the explicit inclusion of core deformed states. The authors wish to thank G. E. Brown, E. Osnes and S. Shlomo for helpful discussions. References 1) J. A. Nolen, Jr. and J. P. Schiffer, Ann. Rev. Nucl. Sci. 19 (1969b) 471 2) K. Okamoto, Phys. Lett. 11 (1964) 150 3) K. Okamoto and C. Pask, Ann. of Phys. 68 (1971) 18 4) S. Shlomo, Rep. Prog. Phys. 41 (1978) 957, and references quoted therein 5) E. H. Auerbach, S. Kahana and J. Weneser, Phys. Rev. Lett. 23 (1969) 1253 6) J. Negele, Nucl. Phys. Al65 (1971) 305 7) H. Sato, Nucl. Phys. A269 (1976) 378 8) T. T. S. Kuo, S. Y. Lee and K. F. Ratcliff, Nucl. Phys. Al76 (1971) 65 9) T. T. S. Kuo, Ann. Rev. Nucl. Sci. 24 (1974) 101 10) E. M. Krenciglowa and T. T. S. Kuo, Nucl. Phys. A235 (1974) 171 11) J. Shurpin, D. Strottman, T. T. S. Kuo, M. Conze and P. Manakos, Phys. Lett. B69 (1977) 395 12) J. Shurpin, H. Mtither, T. T. S. Kuo and A. Faessler, Nucl. Phys. A293 (1977) 61 13) A. Barroso, Nucl. Phys. AZ31 (1977) 267 14) J. Damgaard, C. K. Scott and E. Osnes, Nucl. Phys. A154 (1970) 12 15) J. Applegate and G. E. Brown, private communication 16) E. M. Krenciglowa, C. L. Kung, T. T. S. Kuo and E. Osnes, Ann. der Phys. 101(1976) 154 17) I. Hamamoto and P. Siemens, Nucl. Phys. A269 (1976) 199 18) G. E. Brown, J. S. Dehesa and J. Speth, preprint (Kemforschungsanlage Jtilich, 1979) 19) R. Reid, Ann. der Phys. 50 (1968) 411 20) K. Hollinde, R. Machleidt, M. R. Anastasio, A. Faessler and H. Mtither, Phys. Rev. Cl8 (1978) 18 21) M. W. Kirson, Ann. of Phys. 82 (1974) 345 22) J. Blomqvist and T. T. S. Kuo, Phys. Lett. B29 (1969) 544 23) S. F. Tsai and T. T. S. Kuo, Phys. Lett. 39B (1972) 427 24) T. T. S. Kuo, J. Shurpin, K. C. Tam, E. Osnes and P. J. Ellis; preprint (State University of New York at Stony Brook) 25) T. T. S. Kuo and E. M. Krenciglowa; preprint (SUNY Stony Brook), Nucl. Phys. A, to be published 26) J. P. Jeukenne, A. Lejeune, C. Mahaux, Phys. Reports 25C (1976) 83 27) H. A. Bethe, Ann. Rev. Nucl. Sci. 21 (1971) 93