Configuration mixing and the coulomb energy anomaly

1 .D .1 : 1 .E.7

Nuclear Phystca A293 (1977) 397-409 ; © North-Holland PubILrhlng Co., Anrsterdant Not to be raproduoed by photoprint or mkxoSlm without written permission from tha publisher

CONFIGURATION MIXING AND THE COULOMB ENERGY ANOMALY A . POVES, A. L . CEDILLO and J . M . G . GbMEZ Departamento de F~sica Tedrica, Universidad Autdnoma de Madrid, Canto Blanco, Madrid-34, Spain Received 4 August 1977 Abstrrct : The Coulomb displacement energy of the mirror nuclei "Sc-"Ca, "Ca-' 9K, "F-"O and 's0-'eN is studied . Shell-model calculations for the ground states are carried out in the If z ld 3r2 and lp  =ld sr,2s, r2 spaces, considering up to 4p-4h excitations. The configuration mixing effect on the Coulomb energy shift is calculated and the change of the charge radius due to core excitations is taken into aooount . The effect is particularly important in the calcium region and the Coulomb energy anomaly is decreased in particle nuclei, but increased in hole nuclei . Hartree=Fock calculations using the Skyrme III force are presented and the model dependence on the interaction is discussed in comparison with the HF results of other authors. Other corrections are also taken into account . We finally estimate the discrepancy in the Coulomb displacement energy obtaining approximately, 0.24, 0 .50, 0 .19 and 0 .32 MeV for the above-mentioned mirror nuclei, respectively . These numbers are very similar to the prediction using the charge symmetry breaking potential of Okamoto and Pack . 1 . Introduction

Coulomb energies in nuclei have long been used as a source of information about the nuclear wave function, and in recent years considerable attention has been devoted to an anomaly, often called the Nolen and Schißer anomaly'), which seems to make realistic wave functions incompatible with the experimental Coulomb energy shifts. To make a short outline of the trends in the history of this anomaly. let us recall that when reasonable Woods-Saxon wave functions are used, requiring the singleparticle potential to reproduce the experimental data on the charge radius and the neutron separation energies, the calculated Coulomb displacement energies in minor nuclei become too small in comparison with experiment t). Nolen and Schißer included the one-body electromagnetic spin-orbit force and also did rough estimates of other corrections, which later have been calculated in more detail z-s) without getting any significant reduction of the anomaly. Besides the spin-0rbit term and the finite proton size, an important correction was considered by Auerbach, Kahana and Weneser 6): the core polarization induced by the Coulomb potential, giving rise to a neutron-proton density dißerence in the core (AKW eßect) . Some Hartree-Fork calculations' ~ °) yield an anomaly similar to that obtained with phenomenological single particle potentials, and it was Negele') who suggested that a small charge symmetry breaking (CSR) potential in the nuclear force could be the source of the anomaly in the Coulomb energy shifts . However, calculations 397

398

A . POVES et al.

by Shlomo') showed that, because of the short-range character of CSB potentials due to the electromagnetic mixing of mesons, a potential capable of explaining the anomaly in 3He-3H could not resolve the discrepancy, for example, in 4'Sc4'Ca. A common factor of many of the CSB potentials suggested in the literature is that the contribution to the Coulomb energy shift of minor nuclei is larger for isospin doublets with A = A° -1 than for isospin doublets with A = A° + l, where A° is the mass number of a doubly closed shell nucleus. Hereafter, these types of nuclei will be called hole nuclei and particle nuclei, respectively . On the other hand, the required energy shift to solve the anomaly seemed to be of the opposite type, i.e., larger for particle nuclei than for hole nuclei . However, Negele') has pointed out that when core polarization (i .e., the neutron-proton core density dif%rence) is taken into account, the discrepancy of the calculated Coulomb energy shift with experimental data becomes as large lIl 39Ca-39 K as in 4'Sc4'Ca . In this paper we present calculations of the Coulomb displacement energies of mirror nuclei starting from the assumption that part of the required additional contributions to the Coulomb shift can be obtained by improving the nuclear wave functions, rather than from some additional correction terms not considered up to now. We shall try it by taking into account core excitations of particle-hole type which are not contained in Hartree-Fork (HF) calculations . Recently several authors e- ") have considered this problem under simplifying assumptions that allowed them to work in small subspaces. In ref. ") we have investigated the influence of 2p-2h excitations in the 4'Sc41Ca Coulomb energy shift, and it has been shown that it is possible to find nuclear wave functions that can reproduce simultaneously the experimental charge radii and the Coulomb displacement energy of these nuclei, although the amplitudes required to achieve this result seem to be rather unphysical . More realistic ones give however an insu~cient but non-negligible contribution . Following the same idea, in sect. 2 we present large shell-model calculations of the ground state wave functions in the ' 60 and 4°Ca regions, including up to 4p-4h core excitations and considering t effect both in the Coulomb interaction and the electromagnetic spin-orbit potent. As a result, the correction to the Coulomb energy shift of mirror nuclei becomes very substantial around ~°Ca and less important around 160, with the pecularity that the discrepancy is increased in hole nuclei and is decreased in panicle nuclei . The possibility of an explanation of the anomaly by means of CSB potentials becomes then more plausible, as we discuss later on. In sect . 3 we discuss HF calculations of Coulomb energy shifts, using the Skyrme III(SIIn interaction, with particular emphasis on the core polarization of%cts . Finally, in sect . 4 we present a general summary and discussion of the results. 2. Configuration mixing effects

For the sake of simplicity, the ground states of the isobaric doublets ' s 0-1sN, ~~F-1'O, 39~+ -39K and aim aim have been selected for this investigation of the

CONFIGURATION MIXING

39 9

Coulomb energy shifts . In most investigations on the Coulomb energy anomaly, mirror pairs ofparticle nuclei ôr hole nuclei have been studied and it has been assumed that a Slater determinant is a good approximation to the wave function . However, configuration mining calculations have shown 12 -1 s) for example that 160 and a°Ca, in spite of their character of doubly magic nuclei, are quite open to core excitation mixing in the ground state. These shell-model calculations have been carried out for 160 by Zuker, Buck and McGrory 1i), Zuker 13) and Reehal and Wildenthal 1a) in the configuration space defined by the lp,~, ld} and 2s} orbits, and for a°Ca by Zuker 1s) in the ld~., lf~ configuration space. All these nuclear calculations give a good description of the ground and low-lying states of nuclei in these two regions, and the closed shell configuration in 160 and a°Ca represents only about 60--70 of the ground state wave function . Therefore it seems convenient to study in detail the effect of configuration mixing on the Coulomb displacement energies . As mentioned in sect. 1, several authors have recently considered this problem. Sato 9) has studied the effect on the Coulomb term making some simplifications, for example neglecting the cross terms in the expectation value of the Coulomb potential. An important shortcoming in his calculations is that he did not take into account that configuration mixing modifies the rms radius of the charge distribution and therefore his wave function would not reproduce the experimental charge radius . This shortcoming appears also in the work of Barroso 1 °). )~ is the residual interaction, and the ~ and the Coulomb potential Vc are treated in first order perturbation theory . Here we improve on these calculations basically in three ways : (i) We write H = HN + V~ where HN is the nuclear Hamiltonian, and H is diagonalized in the shell-model spaces mentioned above including up to 4p-4h excitations. (ü) The effect of configuration mixing on the charge radius is taken into account. (iii) The effect on other terms besides the Coulomb potential is also considered. 2.1 . DESCRIPTION OF COULOMB ENERGY CALCULATIONS

To construct and diagonalize the large matrices involved in the lp~ld}2st and ld~.lf~ spaces, we use the Oak Ridge-Rochester code 16) for multishell calculations adapted to work in the neutron-proton formalism. For HN we use the interaction of Reehal and Wildenthal 1a) in the 160 region, and Zuker's interaction 1s) in the ao~ region . Using other interactions 1z .1s) the results are very similar. Two types ofcalculation have been carried out. In one case HN is diagonalized and then V~ is treated in first order perturbation theory . In the other case, H = HN + Vc is diagonalized. The differences between both calculations of the Coulomb displacement energies are of a few keV, and the perturbatioe calculations are good to about 0.2 ~ of the Coulomb energy shift.

40 0

A. DOVES et al.

We now present the results of the diagonalization of H. Since we work in the neutron-proton formalism, the Coulomb interaction is simply added to the nuclear proton-proton two-body matrix elements and proton single particle energies which are an input of the code. The Coulomb matrix elements have been calculated with harmonic oscillator wave functions. It is quite clear that these wave functions are good enough, considering that we are now looking for corrections to the Coulomb energy shifts arising from configuration mixing, and not for an accurate value of the Coulomb displacement energy itself. As is usual in shell-model calculations, a constant HO well is used to generate the single particle wave functions ¢, in each mass region for the particle nuclei, core and hole nuclei . The size parameter is chosen to reproduce the experimental rms charge radii of' 60 and 4°Ca, for the calculations in the two regions, 2.73 and 3.48 fm, respectively "). The mean square radius for point protons is given by 1 «i~i~i = Z ~ ni «tir2i~,i, r

where n;` is the occupation number of protons in orbit i. Since
Occupation numbers of protons and charge radii in fm isN n`(IPi~z) n`(Id sn ) R`(2s~ii) ~ r ~~iti~ ~ r 7~ t l= ew

0.91 0.08 0 .01 2 .69 2 .65 2 .70(3) 2 .58(3)

~sO I .67 0.31 0.02 2 .72

~bO

mO

mF

1 .64 0 .34 0.02 2 .73 2 .73(3)

1 .58 0.38 0 .03 2 .74 2 .72(3) ')

1 .63 1 .33 0 .03 2 .81

') Taken to be 0 .996 times the radius of' 60, following Singhal et al.' 9 ).

CONFIGURATION MIXING

40 1

TABLE 2 Occupation numbers of protons and neutrons and charge radii in fm 39K n"(ld3, Z}- ~

2.69

(rz)ëz (r2):ö

3.45

n`(lfZ) n`(lf,~Z)

0.31 0.54

3.429(18)

39~

40Ca

41

3 .38 0.62 0.55

3.32 0.68 0 .29 3.48

Ca

3 .46 0.54 1 .26 3.48

3.48 3.476(7)

sl~ 3 .67 1 .33 0.43 3.51

in the ld~ lf~ space. This last truncation was required by computer space limitations, but according to calculations of Sakakura et ul. 18 ), the ground state of 4°Ca is quite stable under the inclusion of higher particle-hole excitations . In tables 1 and 2 we present the results of proton occupation numbers and charge radii calculated for all the nuclei involved . For comparison, the experimental charge radii ") are also quoted . The n* values give an idea 'of the amount of excitation . We may also mention that the lowest configuration represents about 60 ~, of the wave functions in all these nuclei, in table 2 the neutrôn occupation numbers n"(lf7 ) are also given. Comparison with nx(lf~) shows for example that n" in 39 K is different from nx in 39 Ca, i.e. the isospin is not a good quantum number in our calculation because the Coulomb force is included in the Hamiltonian. Let us denote by ~ the shell-model solution of H = HN + Vcincluding configuration mixing, and by ~s the wave function assuming a closed core, i.e . a Slater determinant. Then the Coulomb egergy shift of an isospin doublet is dEc(iG) - <~(T= _ -i)IVcIIG(TZ = -i)i - <~(T= = i)IVcIIY(T= = i)~, and similarly for ~s. The effect of configuration mixing is then given by b~(mix) = dEc(~r)-dEc(~s).

(3)

In table 3 the . values of dEc and b~(mix) are given. The correction b~(mix) is quite important in the 4° Ca region, but is rather small around ' 6 0. Note that it is positive for the particle nuclei and negative for the hole nuclei. TABLE 3 Comparison of shell-model Coulomb energy shifts with experiment (MeV)

dE(exp) d Ec(~s) dE~{~) b~(mix)

1 sal sN

1vF- 1,0

39~-39K

41~-41~

3.54 3.29 3.28 -0 .01

3.54 3.35 3.38 0.03

7.30 6.93 6.83 -0.10

7.28 6.72 6.82 0.10

402

A. DOVES et al.

v

u W `r

0~

Y

Fig . 1 . The differences of charge densifia in the mirror nuclei "So-" Ca (a) and as~_s9K (b) ~ arbitrary units. The dashed Gne is calculated under the assumption of a closed shell core . The solid Gne is the result including core excitations.

CONFIGURATION MIXING

40 3

Fig. 1 shows the modification in the charge density difFerence between members of an isobaric pair when configuration mixing is included . There is a shift towards the origin in the case of particle nuclei, and away from the origin in the case of hole nuclei. Thus the signs of the .correction to the Coulomb displacement energies are illustrated by the figure, since the main part of the Coulomb energy shift is given by the direct term, dE~direct) =

J

LP~b(T= _

-i)- P~b(Tz

= ~)]Vcd3r .

2 .2 . AN APPROXIMATIVE APPROACH : EFFECT ON THE ELECTROMAGNETIC SPIN-

ORBIT TERM

The results of the last subsection show that the effect of configuration mixing on the main term of the Coulomb displacement energy is only a small percentage of its value. This could lead us to think that other small charge dependent terms contributing to the displacement energies would not be seriously ati'ected by the inclusion of configuration mixing . We shall now show that this is not always true. Let HN be the nuclear Hamiltonian and V an interaction which is a perturbation to HN. If ~N is the ground state eigenvector, perturbation theory gives dE =

Cll/p~V~WN~'

In a model space with two active shells a and b, an approximate expression for d E is dE = n;e;+n;e;+n6Eb+n~b+Zna(na-i)1â`+Zn;(n;-1)f;' +Znb(nb-1)ßâ +~nb(nb-1)~y +non;l7,xar+nbnb176*tir + nan61~an 6. + nanb ~,rbr + nan61~,,~br + nbn; Yb+~Yr, where the s are the modifications to the single particle energies due to V, and the ~ are the usual shell-model averages of two-body matrix elements, _ , 1 ~(2J+1)CfâJIVLIâJi, Ja~2.lo + 1) .r

~~

_ -

1 .je.llVhaJrrfi' 2I+1)~(2J+1)<,j, ~,,(

As is well known, Va represents the average interaction energy of two particles in the a-shell when this shell is filled . In eq . (6) the interaction of n particles in a given orbit, for example, is approximated by ~ times the number of pairs ~n(n-1), even though the occupation number n may not be an integer .

404

A . POVES et al.

Thecontribution of V to the energy difference ofthe members ofan isobaric doublet can be written as With the above expressions and Vc as the perturbation V, we find about 90 agreement with the results of the exact calculation. Moreover, we find that the main contribution to 8E comes from the modifications in the single particle energies . Hence a reasonable approximation is 8E(~N)

= a(ES)+dn~x~(Eb-Eb)-(Er-Ea)~ "

(lU)

Here SE(~r~ is the effect on the binding energy difference due to V when configuration mixing is . not included, and dn~x = nexlTs = - ~) - nex(Tz = z),

(11)

where "ex~ means excited from shell a to shell 6. Looking at the results of our configuration mixing calculations, which give values for dn~x of 0.1-0.2, it is evident that there will be relatively important effects due to configuration mixing when the matrix elements of the intéraction between the core and each ofthe active orbits are very different . The Coulomb interaction, which has a very smooth behaviour from orbit to orbit, is affected in a very small degree as we have shown. Another term of the electromagnetic interaction whose contribution has to be taken into account in the analysis ofthe Nolen and Schiffer anomaly is the spin-orbit potential, .o. _ - 3 r I' ~ez9iQi+et9zQZ - zetez(Qt+az)~~+

(12)

where e is 1 for a proton an 0 for a neutron. In table 4 we present the results for this potential. Because ofthe strong state dependence of this potential, the configuration mixing effect S,.o.(mix) = 8,.o.(~GN)-S,.o.(~Gs) is a substantial correction to the e.m. spin-orbit energy . Adding the configuration mixing effects on the Coulomb interaction and the e.m. spin-orbit potential we finally obtain a correction to the Coulomb displacement energy of 136 keV in 4t Sc4 1Ca, -165 keV m 39Ca-3'K, 36 keV in t'F-"O and TABLE 4

F.ft'ed of configuration mixing on the e.m . spin-0rbit interaction (keV)

aE,A c a~

~8 53 -25 ö, ., .(mix)

bE,,, (~,,,)

» F_i~ 0

39~ . _39K

" Ix_41~

-71 -63 8

1 l0 50 -60

- lo~ -71 36

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405

31 keV in i s 0- 1 s N. Thus the corrections are about 1 ~ in the ' 60 region and about 2 ~ in the 4° Ca region, and therefore it is important to take them into account. We remark, however, that the Coulomb energy anomaly is decreased in particle nuclei, but is increased in hole nuclei . The comparison with experiment is analysed in sect. 4. -

3. Hartree-Fork calculations In order to discuss the Coulomb energy anomaly including the results of the last section, it is necessary to decide upon the accurate size of the discrepancy between theory and experiment, using the best available models. We shall use the single particle wave functions obtained from HF theory . There are several available HF calculations ofthe Coulomb energy shift in the litterature. Negele s) used the densitydependent Hartree-Fork theory (DDHF). Giai et ul . a) used Skyrme forces (SI and SII) . More recently the density matrix expansion (DME) has been used by Negele') and Sato 9). Since different authors include different corrections in their final numbers, quoting from their results it is not easy to extract accurately the amount of model dependence of the discrepancy due to the different Hartree-Fock approaçhes used. However, Sato 9) has performed two identical calculations for all the isobaric pairs that we are studying, using the DME and SII forces, and the existence ofsome model dependence can be inferred from his calculations. To see it more vividly, we also present here the results of our calculations with the SIII interaction, which is one of the best phenomenological forces for HF calculations . For a detailed discussion on the SIII interaction and other Skyrme parametrizations we refer to the work of Beiner et al. s°). The simplest way to obtain the Coulomb displacement energy, including the core polarization effects, is by performing a HF calculation for each of the nuclei of an isospin doublet and subtracting the binding energies. But it should be noted that when the parameters of the SIII interaction were determined Z° ), a local approximation was used for the exchange part of the Coulomb energy, namely the Slater approximation. Therefore we "use the HF theory for the SIII force: in the standard way, i.e. with the approximation mentioned for the exchange Coulomb term included in the self~onsistent process and then we use ti}e wave functions so obtained to calculate the Coulomb energy without any approximation for the exchange teen . For the quantities dE~{HF~o«) defined as in eq . (2), but using closed shell core selfconsistent wave functions, we obtain 6.ß3 MeV for °'Sc a' Ca, 6.94 MeV for 39 Ca-39 K, 3.39 MeV for "F-"O and 3.29 MeV for 1 s0- 1s N. The rms charge radii calculated using SIII are 3.48 fm for a°Ca and 2.69 fm for ' 60, as compared to the experimental values 3.48 and 2.73 fm, respectively .

406

A . POVFS et al. T~e~e 5 Values of dF.c(HF~,) from several HF calculadone (MeV)

SII (Sato) DME (Sato) SIII (present work)

Isa1sN .)

17r- l7O ~)

D9+~- 79K6)

3 .24 3 .23 3 .24

3 .20 3 .21 3 .34

6 .91 6 .90 6.94

41~41~6) 6 .69 6.70 6 .83

') Results renormalized to rab(` 6 0) = 2 .73 fm . b) Results renotmaliud to r~b(4°Cs) = 3 .48 fm . in table 5 we present the dE~HF~)-values of the SIII force in comparison with Sato's results. However, since the SIII, SII and DME forces give different charge radii, the dE~{HF~,a) values of the table are all renormalized by the factors

using the charge radii ofa°Ca and 160. In this way the model dependence, or effective force dependence, is best visualized, and it can be seen that it is not negligible. T~e~e 6 Values of ö(c.p .) from several HF calculations (keV) 1 sal sN SII (Sato) DME(Sato) SIII (present work)

-130 -140 -87

39~ - 79K

' 7 F-' 7 0 55 l5 +SO

41~41

-155 -205 -132

140 70 177

In table 6 we give the values ofthe core polarization effect, defined by the quantity (14) where dE(HF)

o

~Y'HF( Ts =

-~)IHI~~T~ _ -~)>-~~HF(T:

_ ~)IHI~HF(T>< _ ~)>~

(ls)

The model dependence of S(c.p.) is not negligible either. 4. Discusedoo and con~~lusioos

From the time that Nolen and Schiffer pointed out the systematic discrepancy between experimental and calculated Coulomb displacement energies of mirror nuclei, there have been many calculations of the main terms in the Coulomb displacement energy or corrections to this term . Part of this process will be found in refs. z-6). Later settlements of the "state of the art are those of Negele') for the 39 Ca- 39 K and 41 Sc41 Ca pairs, and Shlomo and Riska 21) and Sato 9) for these two pairs and

CONFIGURATION MIXING

407

t sN and t'F- t 'O. The main conclusions of these papers are : (i) The existence of an important discrepancy in all those nuclei, which is almost equal for the pairs of particle nuclei and hole nuclei near a given closed shell core. (ü) The unsatisfactory results in the attempts to explain the discrepancies by means of charge symmetry breaking potentials constructed from meson theory . We shall now show that the above conclusions may be substantially modified when the conüguration mixing effects calculated in sect. 2 and the model dependence considered in sect . 3 are taken into account. ~ s0-

TABLE

7

Summary of the Coulomb energy shift calculation (Mew Sa13N Coulomb (SIII) Core polarization (SIII) Configuration mixing Short-range oorrclations Proton finite size Neutron finite size Magnetic interactions Dynamic n-p mass dilFerence + vacuum polarization theory experiment

17F - 17O

39~- 39K

41SL.-41~ .

3.24 -0 .09 -0 .03 0.10 -0 .09 -0 .02 0 .10

3.34 0.05 0.04 0.08 -0 .07 -0 .01 -0 .05

6.94 -0 .13 -0 .16 0.13 -0 .10 -0 .02 0 .14

6.83 0.I8 0.14 0.12 -0.08 -0.02 -0.08

3.25 3S4

3.43 3.54

6.86 7.30

7.16 7.28

0.04

0.05

0.06

0.07

Let us present in table 7 a summary, similar to that of Negele or Shlomo and Risks, of the terms contributing to the Coulomb displacement energy . Such a tablé will be very helpful in understanding where the difference of our final numbers with those of previous work comes from . The Coulomb term, including exchange, is taken . from tâble 5 and contains therefore the correction by the factor (13) to take into account the difference between the calculated and experimental charge radii. The core polarization is taken from table 6. This effect is sometimes called "core density difference" (Negele) or "isospin mixing in the core" (Shlomo and Risks) . The configuration mixing effects are those ofsect . 2. For the short-range correations we quote the results of Sato 9). This effect has also been calculated by Shlomo and Risks 2'), who obtain the same result for A = 15 and a few tens of keV more for the other isobaric pairs here considered. The proton and neutron finite size effect on the electric interaction is calculated using the potentials derived by Schneider and Thaler s2) from the experimental form factors. In the magnetic interactions we have included not only the spin-0rbit potential, but all the magnetic two-body terms deduced from the Breit equation 23). The remaining corrections are the conventional dynamic neutron-proton mass difference and the vacuum polarization . Once all the contributions are added, the comparison with experiment shows a

408

A . POVES et al.

resonable agreement in the particle nuclei, but an important discrepancy in the hole nuclei which does not seem able to be explained from model dependence. TABU : H

Resulting discrepancies with SIII and DME compared with the contribution of the Okamoto and Pask CSB potential (MeV)

~ sa~ sN

m F_m0

a9Ca _~g K

SIII DME average

0 .29 0 .35 0 .32

0. l 1 0.27 0.19

0 .44 0 .56 0 .50

0.12 0.36 0.24

CSB

0 .31

0. I8

0 .34

0.24

ay_sya

In table 8 the discrepancy between theory and experiment is given for the two extremes of the model dependence, the SIII and DME forces, giving the smallest and largest discrepancies, respectively . It should be noted that the only difference between both sets of numbers comes from the Coulomb and core polarization terms (see tables 5 and 6). The results ofother authors a .') forthese two terms are in between. We also give in table 8 the average of SIII and DME, which we consider quite representative numbers of the discrepancy, with an uncertainty of the order of f 0.10 MeV coming from the model dependence of the force used in the Hartree-Fork calculations . There is probably some additional non-negligible uncertainty coming from the short-range correlations. Finally, in table 8 we quote from Shlomo s) the contribution to the Coulomb energy shift that is obtained from the charge symmetry breaking potential ofOkamoto and Pask Za) . This potential contains exchange of pco~+urn+yn, and it explains approximately the discrepancy in the Coulomb displacement energy of 3He-'H . There is a striking similarity of the discrepancies and the contribution of the mentioned CSB potential. We consider that this similarity is significant, in spite ofthe uncertainties in the numbers, because the Okamoto and Pask potential does not contain any parameter adjusted to reproduce the discrepancies that we have studied in this paper. Therefore, we believe that the charge symmetry breaking potential arising from indirect electromagnetic effects in meson theory is a reasonable way of explaining the Coulomb energy anomaly. We want to acknowledge many helpful discussions with J. Martorell. We thank Professor A . M . Lane for useful comments on an earlier version of the manuscript . Thanks are due to Professor S. S. M. Wong for providing us with the shell-model code.

CONFIGURATION MIXING

409

Reterences 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 1I) 12) 13) 14) IS) 16) 17) 18) 19) 20) 21) 22) 23) 24)

J . A . Nolen and J . P . Schiffer, Ann . Rev . Nucl . Sci . 1 9 (1969) 471 N . Auerbach, J . Hûfner, A . K . Kerman and C. M . Shakin, Phys . Rev . Lett . 23 (1969) 484 1, W . Negele, Nucl . Phys . A165 (1971) 305 N . V . Giai, D . Vautherin, M . Veneroni and D . M . Brink, Phys . Lett . 35B (1971) 135 S . Shlomo, Phys . Lett. 42B(1972) 146 E. H . Auerbach, S. Kahana and J . Weneser, Phys . Rev . Lett. 23 (1969) 1253 J . W . Negele, Proc . Conf. on nuclear structure, Amsterdam, 1974 J . M . G. Gourez and A . Poves, Proc. Int . Symp. on highly excited states in nuclei, Jûlich 1975, vol . 1, ed. A . Faessler, p . 11 I H . Sato, Nucl . Phys. A269 (1976) 378 A . Barroso, Nucl . Phys . A281 (1977) 267 l . M . G. Gourez and A . Poves, Z . Phys, to be published A . P. Zuker, B . Buck and J . B. McGrory, Phys . Rev . Lett . 21 (1968) 39 A . P. Zuker, Phys . Rev. Lett . 23 (1969) 983 B . S. Reehal and B . H . Wildenthal, Part . and Nucl . 6 (1973) 137 A . P. Zuker, Proc. Conf. on the structure of If ~ nuclei, ed . R . A . Ricci (Editrice Compositori, Bologna, 1971) p . 95 J . B . French, E . C. Halbert, J . B . McGrory and S . S. M . Wong, Adv . in Nucl . Phys . 3 (1969) 193 C . W . de Jager, H . de Vries and C. de Vries, Atomic Data and Nucl . Data Tables 14 (1974) 479 M . Sakakura, A. Arima and T. Sebe, Phys. Lett . 61B (1976) 335 R. P. Singhah J . R . Moreira and H . S . Caplan, Phys . Rev . Lett . 24 (1970) 73 M . Beiner, H . Flocard, N . V . Giai and P . Quentin, Nucl . Phys . A238 (1975) 29 S . Shlomo and D. O. Riska, Nucl . Phys . A2S4 (1975) 281 R . E . Schneider and R . M . Thaler, Phys . Rev . 137 (1965) 874 F . E . Close and H . Osborn, Phys . Rev. D2 (1970) 2127 K . Okamoto and C . Pask, Phys. Lett . 36B (1971) 317