diagrammatic expansions for isobaric analog states and the coulomb displacement energy

I 1.D.1 [ NuciearPhysics A151 (1970) 323--330; ~ ) North-l-lollandPublishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

D I A G R A M M A T I C EXPANSIONS FOR ISOBARIC ANALOG STATES

AND T H E C O U L O M B D I S P L A C E M E N T ENERGY

CHUN WA WONG t

Department of Physics, University of California, Los Angeles, California 90024

Received 9 April 1970

Abstract: Certain aspects of diagrammatic expansions for isobaric analog states are discussed. The effect of the charge-distortion potential of Auerbach, Kahana and Weneser on the Coulomb displacement energy is estimated to be about 6 % by relating it to the average isotope shift of the core charge distribution.

1. Introduction

Calculations of Coulomb displacement energies AEc between isobaric analog states using independent-particle models with empirical charge distributions for core protons usually come out to be smaller than observed values. Typical discrepancies are 0.6 to 0.7 MeV in 41Ca [refs. 1,2)] and perhaps 1 MeV in Pb [ref. 3)], these values being 5 to 10 % o f A E c and apparently rather insensitive to neutron excess 2). Recent- ly Auerbach, Kahana and Weneser 1) suggested that a certain diagram involving l p - l h excitation might account for the discrepancy. We would like to discuss in sect. 2 certain problems in diagrammatic expansions for the IAS and to describe a derivation of the A K W diagrams. In sect. 3 we attempt another estimate of the lp - lh contribution to AE c by using a relation to the average isotope shift of the core charge distribution.

2. Diagrammatic expansions for the IAS

For the linked-cluster diagrammatic expansions we study, we are unable to repro- duce the expected charge independence of the interaction energy for charge-independent forces to an arbitrary order of perturbation, except in the trivial case of mirror nuclei, i.e., two nuclei having + M r. In the following discussion we shall ignore this difficulty and concentrate on charge-dependent effects.

A conceptually simple diagrammatic prescription for the IAS can then be made by perturbing from the analog of the unperturbed configuration in the parent state (PS) under the restriction of vector coupling of isospins to (T, T-- 1). In order not to intro- duce any excess-neutron induced impurities in the wave function, of the type mem- tioned by Robson 4), the vacuum is chosen to contain an equal number of neutrons

t This work was partially supported by the National Science Foundation.

323

324 C H U N W A W O N G

and protons in equivalent core orbits, which are charge-independent in the absence of charge-dependent forces. This vacuum has T -- 0, and unlike the usual HF picture, all excess neutrons (44 in 2°8pb) must be put in valence orbits.

Secondly, the interaction of the new proton t_ ln) with any neutron in the IAS should be limited to only T = 1 two-body matrix elements, like that of the originial neutron in PS, whenever the remaining A - 2 nucleons are coupled to ( T - 1 , T - 1 ) . This is the case in the interaction with an excess neutron, so that in the absence of charge-dependent forces but in the presence of a neutron excess, the new proton in the unperturbed configuration sees a neutron potential, if the HF picture were used. (This

(a) (b) (c) (d): Fig. 1. Linked-cluster expansion for a single valence particle.

t --×AH=p + --~ + ~ . . . . ×V

Fig. 2. Definition of the one-body AH insertion.

prescription also enables us to see the equality of all second-order interaction energies in PS and IAS for charge-independent forces.) Finally, since we have used the method of nondegenerate perturbation theory, the diagrammatic procedure only sets up the matrix elements of the Hamiltonian within configurations of a certain degenerate model space s) and does not give the diagonalized interaction energies directly. It is necessary that the intermediate states in the diagrams be outside this degenerate model space.

The usual linked-cluster perturbation expansion then gives the energy diagrams of fig. 1 for the simple case of a single valence nucleon (i.e., I N - Z I = 1). Here h is the unperturbed s.p. Hamiltonian, which must be charge-independent in the absence of charge-dependent forces even when there is a neutron excess; A H = H - h is the perturbation. A Hugenholtz dot vertex gives both direct and exchange nuclear two-body interactions v. The Coulomb two-body interaction v c is represented by a horizontal broken line in the Goldstone convention. Exchange Coulomb interaction diagrams, being relatively small, will not be shown explicitly. One-body operators, in which class we include the entire perturbation A H whenever its two-body parts involve the Har- tree-Fock bubble (and its exchange) and therefore it behaves like a one-body operator (see fig. 2), are represented by a horizontal broken line terminated in a cross with identifying symbols.

As usual, we may impose the requirement that h be so chosen that all one-body A H

insertions vanish, thus giving a self-consistent description for the s.p. orbits. The

ISOBARIC ANALOGUE STATES 325

result is a charge-dependent s.p. Hamiltonian h in which the charge dependence is induced by charge-dependent interactions only and not by excess neutrons:

<ilhnlJ> = <i[Tlj>+ ~ [<ipdvljpc-pj>+(indvljn~-n~j>], (1) c

<ilhplj> = <il TIj> + ~ [<ipolv + vcljPc- PC j> + <in¢[vljn¢- nCj>] c

where = <ilh, lj> + <il Vclj> + <il Vcalj>, (2)

(3) <il Vclj> = ~ <ipdvcljpo-poj>, c

<i[ VcalJ> = ~ {[<Pi polvlpj po- po p~> + <p~ ndv[pj n~- n, p~>] c

-[<niP¢lvlnjp~-pcnj>+<nindvln~n¢-n~nj>']}, (4)

and the sum c covers the equal number Arc = Zc ( < Z ) of protons and neutrons in equivalent orbits in the core. The one-body potential Vc is the Coulomb potential due to the core protons, and VCd is the additional charge-distortion potential due to the distortion of the core wave functions by the Coulomb and other charge-dependent forces. The quantity Vca is the additional potential introduced in ref. 1).

It should be emphasized that no symmetry term appears in the present picture, and all excess neutrons must appear in valence orbits. Thus Robson's objection a) does not apply, and the impurities in the perturbed wave function induced by the charge dependence of the s.p. Hamiltonian h is not spurious. In the self-consistent picture this charge dependence is expressed by the potential Vcd, which in the absence of charge-dependent nuclear forces, is roughly proportional to Zc [see eq. (4) or the proton bubble in fig. lc], and remains relatively insensitive to neutron excess.

Now if the neutron excess is large, the s.p. representation of ref. 1) as sketched here, though very convenient for the description of analog states, may not be so efficacious for the total interaction energy, because the usual H F picture, with its symmetry term, gives the best unperturbed energy in the sense of a variational principle. Would it be useful to try to mix in the H F description somehow, at least for problems for which the spurious impurities in the H F picture noted by Robson are either unimportant or can be isolated and thus explicitly excluded?

In the usual H F picture the vacuum contains Z~ protons and N" (not necessarily equal to Z~) neutrons. A symmetry potential V~ appears. The difference between the symmetry potentials for neutron and for proton orbits contains, as the contribution from those core protons and neutrons in equivalent and therefore equally filled orbits f, the potential

<il VddlJ> = ~ {[-<Pi pflvlpj pf - - Pf Pj> + <Pi nflvlpj n f - nf pj>] f

-- [<n i pflv[nj pf -- pf nj> + <n i r/flv[nj n f - - n f / ' / j > ] } , ( 5 )

which, when summed over the same orbits f = c of eq. (4), differs from VCd only in a symmetry-induced part Vsd, which is spurious. On the other hand,

326 C H U N W A W O N G

the new proton which appears on going to the IAS must still interact with an excess neutron, whether in the core or in a valence orbit, like the original neutron, i.e. with T = 1 two-body matrix elements only, whenever the remaining A - 2 nucleons are coupled to ( T - 1, T - 1). Therefore the new proton must sit in the one-body potential [ref. 1)1 V n + l/c + Vdd (or Vcd)- This picture may be considered a modification of the analog, spin idea 6). By the same argument, if this modification is not made and the new proton still sits in Vp + Vc (where Vp differs from Vn by the symmetry potentials), then the unperturbed energy for IAS will come out near that of the state ( T - 1, T - 1) instead, i.e., very badly.

Unfortunately, the modified analog-spin idea is complicated by the simultaneous appearance of two distinct types of protons - the old, or HF, protons and the new, or analog, proton - in s.p. states which are not mutually orthogonal. The merit of the MacDonald analog-spin idea 6) is clearly that only one orthogonal set of proton states appear. On the other hand, it is necessary either to keep all AH insertion diagrams, which correct for the poor starting point in the expansion, or to ignore these troublesome diagrams but instead readjust the energy by a suitable amount, of the order of the symmetry energy. A formally more satisfactory solution to the complica- tion of the modified analog-spin idea might perhaps be the orthogonalization of the new proton states; for example, by restricting the eigenfunctions of the new proton to outside the HF proton Fermi sea.

Finally, we should mention that the alternative diagrammatic prescription of going to the analog state after the linked-cluster expansion for the PS wave function has not been ruled out yet. For example, we are able to get the AEc diagrams of ref. 1) in this prescription, the important Vca effect now appearing as a Coulomb-induced proton lp - lh correlation of the perturbed wave function in the PS which then interacts with the unperturbed configuration through nuclear forces in different ways in the PS or in the IAS. Again, we are unable to show that the same prescription gives the same interaction energy in both PS and IAS for charge-independent forces to an arbitrary order of perturbation.

3. The Coulomb displacement energy

The diagrams we shall use in the discussion of the Coulomb displacement energy are based on a charge-independent unperturbed Hamiltonian h with parameters chosen such that the unperturbed charge distribution roughly fits the observed charge distribution. (This choice of the unperturbed Hamiltonian reproduces closely the condition ofthe usual independent-particle calculations.)Diagrams of AEc of second order in AH and with at least one Coulomb interaction include those of ref. 1) (figs. 3a and 4) and fig. 5 for one valence nucleon, and fig. 6 for two valence nucleons.

Fig. 3a represents the difference between the IAS and the PS of the core Coulomb energy due to the isotopic or isotonic shift of the core charge distribution. Fig. 3a has the same structure as fig. 3b for the nucleonic shift of the core charge distribution.


As usual, knowledge of the change in the core charge distribution will enable us to estimate the corresponding change in the core Coulomb energy. Therefore on the average

AE~(fig. 3a) ~ - ( A 2 RJAsR¢)(ANRJR~)2E c, (6) where

A2R~ = A z R ~ - A N R ~ , (7)

AzR~ = R¢(IAS)-R~(Z, N - 1), (8)

AsR¢ = R¢(PS)-Re(Z, N - 1), (9)

are respectively the isotonic and isotopic shifts of the rms charge radius of the Z¢ core protons, Ec the total Coulomb energy of these protons, and the factor 2 corrects for the factor ½ in Ec included originally to correct the double counting of the number

p (a) n (b)

Fig. 3. a) Charge-distortion effect in the Coulomb displacement energy, b) Nucleonic shift of the core charge distribution.

Figs. 4-5. Correlational corrections to the Coulomb displacement energy for a single valence proton.

P P

Fig. 6. A correlational correction to the Coulomb displacement energy for two valence protons.

of distinct pairs in the two-body Coulomb interaction. The factor ANRc/R ¢ is the fractional isotope shift, and can be written asf /3A, where (3A)-1 is the usual A + size effect a n d f ~ ½ represents a constant empirically fitted to experimental isotope shifts for spherical nuclei. The factor A2Rc/ANR o depends on the detail of nuclear interactions.

Now the nuclear vertex in both figs. 3a and 3b depends quite sensitively on the detail of the nuclear (reaction-matrix) interaction, especially its state and density dependences. On the other hand, the ratio AzRc/ANRc is likely to be much less sen- sitive. We therefore try to estimate this by using the simple zero-range density-independent interaction of ref. 1) for which it is just - (3v 1 -Vo)/(3v 1 + Vo), where vs is

328 cHtrN WA WONG

the two-body interaction strength in the (ordinary) spin S state. If v0 ~ ½vx, as suggested in ref. ~), we get - ~ for this factor and

where

AEo(fig. 3a) g(Ec/3A ),

O = -2f(a2Ro/A Ro),

(10)

(11)

and the remaining factor Ec/3A is just the absolute value of the isotope change in the core Coulomb energy due to the usual A + size effect, being proportional to Z as indi- cated by the proton bubble in fig. 3a. The choice of Ec = 0.64 ZZA -~ MeV gives, for the isotope effects, a value of 0.6 MeV for 4°Ca and 1.5 MeV for Pb, which are about 8 % of the observed AEc. The estimated value of g then gives a 6 % correction for fig. 3a.

By using an average Au Rc we need not worry about the fact that fig. 3a should be averaged over all excess neutrons, whereas fig. 3b represents the effect of the last neutron only. In addition, if contributions to ANR ~ other than fig. 3b are significant, then the use of the empirical value of ANR¢ in eq. (6) will have included similar contributions in AE c. From this viewpoint, local variations of AzcR¢ might carry some information concerning AEc, although it is not clear how this information can be used. Finally, in the Ca isotopes where most of the isotope shifts in R~ are abnormally small or have the wrong sign, the quantity of interest in eq. (6) is the ANR¢ for 4aCa, which seems to have a normal value corresponding t o f ~ ½ if one is permitted to take half of the empirical increase 7) of 0.03 fm from 4°Ca to 42Ca.

In the other diagrams for AEc, the contribution of fig. 4 is, according to ref. 1), negligible. This is presumably because the Coulomb interaction involves only a single valence proton rather than the Z~ core protons, causing a reduction by a factor 1/Z~ compared to fig. 3a. The proton-proton nuclear vertex is also weaker for realistic interactions; but on the other hand, admixture of deformed components to the unperturbed configuration will enhance it considerably, just as it will enhance the nuclear correlation energy due to 2p-lh admixtures. It seems unlikely that fig. 4 will be much greater than the 100 keV estimated by Bertsch 8) for two protons in the same major shell.

Fig. 5 represents the blocking effect by the new proton on Coulomb-induced 2p-2h proton core correlations. If for the Ca isotopes we assume rather arbitrarily that the nuclear 2p-2h correlation energy is ½ MeV per particle, then the ratio 0.08 of total Coulomb to total nuclear potential energies indicates that the same proportion of the correlation energy, or 1½ MeV, might be Coulomb induced. (In this order-of-magnitude estimate we neglect a possible factor of 2 from a diagram similar to fig. 5 but with a Coulomb interaction first.) Assuming only 2h~o excitations and taking into account the blocking effect of a single valence proton in a major shell capable of holding 20 protons, we obtain - 8 0 keV for fig. 5. Applying the same blocking correction to the second proton ph excitation we would, in this crude picture, estimate for fig. 6 with


two protons outside the core a correction of only + 4 keV. According to Bertsch s), fig. 6 may contribute as much as 50 keV for the J = 0 ground state. This indicates that fig. 5 can contribute several times the effect estimated here, but the correction has a sign opposite to that of the empirical AEc discrepancy.

4. Further discussion

We should point out that the diagrammatic prescription taken as a whole differs from the usual shell-model method using empirical unperturbed energies in that there are A H insertion diagrams, like fig. lb, which, if included, will bring the interaction energy to values more completely consistent with the chosen two-body force. This feature is absolutely necessary in principle, but it is not usually an advantage in prac- tice because of the great sensitivity of the result to the nature of the chosen two-body force. The normal procedure of ignoring all such diagrams usually gives more satisfactory numerical results, but the same procedure can cause difficulty for the analog states (T, T - 1) if the unperturbed particle energies are taken, as usual, from the observed (T-½, T--½) proton s.p. levels of the nucleus (N, Z + 1). The starting point in such calculations is then that of MacDonald analog spin, and the diagonalization of the Hamiltonian in the model space should be such as to eliminate the spurious symmetry energy in the (7", T - 1) states. A perfect cancellation of this spurious energy re- quires that the symmetry effect on the empirical energies be exactly consistent with that from the assumed two-body force, a requirement which is difficult to satisfy unless one knows how much of the empirical energies can be traced to the symmetry effect. This point might be relevant in understanding the difficulty found by Kuo 9) in getting the right energy for the IAS in 2°SBi. (A related problem is that it is not clear in calculations like Kuo's how much of the calculated isospin impurity is real and not spurious. A similar conclusion has been made in ref. 10) with respect to a different calculation.)

The estimates given here for one valence nucleon outside a T = 0 core are only suggestive of the magnitude of a few correlational corrections to AEo one of which (fig. 3a) is considered important in ref. 1), but all of which are considered small in ref. 2). There are, of course, many higher-order effects which can give significant contributions to both the valence-core part and the valence-valence part of AE 0 In calculating all these diagrams it may well be necessary to use reasonably realistic effective (or reaction-matrix) interactions with appropriate state and density dependences. Furthermore if the interaction is not guaranteed to give the right charge radius, then it might be better to ignore all diagrams like fig. ld involving AHinsertions which work through a change of the charge radius from that of the unperturbed configuration. In conclusion, we feel that detailed estimates of high-order as well as low-order correlational corrections to AE c will be helpful in the interpretation of the observed discrepancy from independent-particle calculations.

330 CHUN WA WONG

The author would like to thank Professor George Igo for calling his attention to the problem of Coulomb displacement energies, and both him and Professors S. A. Moszkowski for discussions.

References

1) E. H. Auerbach, S. Kahana and J. Weneser, Phys. Rev. Lett. 23 (1969) 1253 2) J. A. Nolen and J. P. Schiffer, Tech. Report no. 70-035, University of Maryland, September 1969;

J. P. Schiffer, in Nuclear isospin, eds. Anderson e t aL (Academic Press, New York and London, 1969) p. 733

3) D. Vautherin and M. Veneroni, Phys. Lett. 29B (1969) 203 4) D. Robson, in Nuclear isospin, eds. Anderson et al. (Academic Press, New York and London,

1969) p. 385 5) B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771 6) W. M. MacDonald, in Isobaric spin in nuclear physics, eds. J. D. Fox and D. Robson (Aca-

demic Press, New York and London, 1966) p. 173 7) H. R. Collard and R. Hofstadter, in Nuclear radii, ed. H. Schopper (Landolt-BSrnstein: New

Series, Springer-Verlag, Berlin, 1967) 8) G. Bertsch, Phys. Rev. 174 (1968) 1313 9) T. T. S. Kuo, Nucl. Phys. A122 (1968) 325

10) G. L. Payne, J. D. Perez and W. M. MacDonald, Phys. Rev. 187 (1969) 1733

diagrammatic expansions for isobaric analog states and the coulomb displacement energy

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