Nacle+ar Pbyslca A2'ß (1977) 78-86;© Nord~aHo!land Pwb~hlnp Co., AntrterrfantNot w be reproduced b7 photopriM or microfilm without wrlttm permialon f}om the publisher

THIRD-ORDER CONTRIBUTIONSTO THE BIIVDING ENERGY OF O%YGEN

R. K. TRIPATHIt, S A. 3ANDSRSON sad J. P. ELLIOTTSeArooJ of Matlumotkal asd Pl{yaJaal Sefeaoer, Unloaalty of Sswex, Brighton, Ssrlrex, F.ngJand

Received S August 1976(Revised 4 October 197f7

Abstract: Hindinfi enaray . calculations reported in earlier papers using matrix elements deducedfrom phase shifts aro continued to third order in a perturbative treatment. The total third-order contribution is small due to a cancellation between dia~ams and there are indicationsthat hi8her order terms are also small .

i. I>~rodeetlon

In previous publications' ~~) results of first-and second-orderperturbation calcula-tions using Sussex matrix elements (SME)3) were presented for anumber of nuclei ator near to closed shells. In the present paper we continue this work to third order forthe case of the closed shell nucleus, 160. The purpose is to assess the order by orderconvergence ofperturbation theory, to calculatethe upper bounds onthe energy usingthetechniques givenbyNissimovand Elliott 4) andto investigatethee1%ct in third orderof a number of points ofinterest which arose in the second-order calculations . Theseinclude (i) the slow convergence with energy of the: tensor contributions, due to largematrix elements far offdiagonal, (ü) the importance ofthe hard-core contribution forproducing saturation and (iii) the effect of the modified single-particle spectrum .

Rassis, Mavromatis and Singh have made a number of recent calculations s-s)going beyond second order for the energy of 160 and other nuclei, however, thesehave been strictly limited in scope. By ignoring configurations beyond 2ftm excitationthey were able to test several upper bound and related formulae 4~~),but since 2itmexcitations yield only a small fraction of perturbation corrections their results areincomplete. In addition they have used only the original unsaturating form of theSME which gives rise to unnaturally large Hartree-Fuck (HF) corrections . We returnto this point in sect. 4.

2 Some extlcolitiorttl eietttils

The definition of the SME and a detailed description of the perturbation theoryused has been given previously 2), but to summarize, keeping the same notation, thet Present address : Tats Institute of Ftitndamental Research, Hombay, India .

78

THI1tD-ORDBR CONTRIBUTION3

zero-order l;Iamiltonian is chosen as a single-particle oscillator with shifted energiesHo = Ha�+~I where d _ ~�,~ A.,~~nlj)Ge~j~ . The perturbation is then given by( f~,+g,-d) when ~, originatea fromthe smooth part Y, ofthetwo-nucleon potentialY and includes the c.m. correction,

t<~

t

t<~

while g, was derived 2) from the singular part Y, = Y- Y, of the potential usingstandard reaction matrix methods. Of the family ~) of possible Ya only that corre-sponding to a coie radius c = 0.3 fm is used here although we do compare withthecase of no core, c = 0.The single-particle energy shifts ~,u were calculateä using the formula derived in

ref. ~) :~.r~ _

~

«,n'l'j'~Y,~n1,~,R'i'i') -<nlfl~~zr~l~.i)r,.'!'J'(ooouptea)

where the subscript d (= 4:9 fm for oxygen) indicates that the integral is ova therange r ~ 0 to r = d only. This corresponds to replacing the unrealistic oscillatorsinglo-particle potential (which is positive) by . a cut-off oscillator potential ofvariabledepth and as such is a simple first step towards Hartreo-Focle self-consistency whichat the same time retains the advantage of the oscillator wave function basis. The in-tention is to make Ho more realistic and hence improve the cnvergence of the pertur-bation theory . We shall see how this works out in sect . 3 bnt we note that whereas insecond order the only role of ~ was.to modify the energy denominators, it enters thenumerator in third order. Table 1 contains some of the calculated values of it,~.The third-order contnûutions to the energy of 160 are calculated by following the

formulae given by Kassis 6) for each possible diagram. The prototype diagrams areshown in fig. l using the same numbering as Krisis. There are two modifications that

T~s1a 1single-particle energy sbifte a,,r for 1°O in mils of dim~ 14.4 MeV

~n I 2j a,u n 1 2j ~ e l 2J

0 0 1 -4~33 2 0 1 -2.38 3 0 1 -1.241 2 S -2.67 2 2 S -1.33

0 1 3 -3.91 1 2 3 -2.39 2 2 3 -1.160 1 1 -3.39 0 4 9 -2.99 1 4 9 -1.651 0 1 -3.28 . 0 4 7 -2.32 1 4 7 -1.280 2 S -3.44 2 1 3 -1.76

0 6 13 -2.230 2 3 -3.01 2 1 1 -1.65

0 6 11 -2.04

1 1 3 -3.06 1 3 7 -2.141 1 1 -2.92 1 3 S -1.790 3 7 -3.19 0 5 I1 -2.770 3 5 -2.76 0 5 9 -2.48

80

R IK. TRIPATHI et o~

6

FIB. 1 . Third-ot+de~ dia~amr.

should be noted in comparing our formalism with that of Krisis. First of all in eachdiagram the wavy line nowrepresents ~,+g, with the proviso that diagrams contain-ing ladder components in g, . must be subtracted to avoid double counting of Y,ladders already summed in g, . Secondly because of the one-body operator ~ in ourperturbation, a small modification to Keaeis' formulae is necessary in diagrams 4, S,13 and 14 . The quantity Y(a, b) defined by I~assï.s on his page 212 must have 2,a,, bsubtracted from it, a change which amounts to an insertion-il, in addition to the HFbnbblei attached to a panicle (or hole) line in each ofthese diagrams.Large o$-diagonal tensor matrix etements,<n3Si I YIn"sD i) lead, as in second ôrder,

to a slow convergence with excitatioa energy ofthird-orderterms: These are estimatedusing the same techniques as described in ref. _). For the ladder diagram (no: 1 inßg . 1) extrapolation of the matrix elemrçnts in the 3D1 channel is also necessary. Thisis achieved using the one=pion-exchange potential ïn this channel.

3. lteenlts and üscaesloa

Fn the earlièr paper z) .on 160 the reported a bindiag.energy in first order of Eo+El

_-42.2 MeVwith afurther 40 MoV.coming in.ssx~nd order. Frg.. 2shows . how thissecond-order contribution E2 converges as a function of the cut-off in the sum overintermediate states . Although our energy denominators include the shifts ~i,u, it isconvenientto label a set ofintermediate states by the excitation Ntlw that they wouldhave in as oscillator potential. [The numbersEz are essentially those given in the ear-lier calculation ~) butdiffer marginally from thembecause ofthe more accurate choiceof ~i�ri given in table 1 . Previously ~) we had used an average value ~ in the energydenominators. ] Fig. 2 shows alsôthe result Ez +E3 ofincluding third-order contribu-tions as described in sect. 2. One sees thatthe third-ordercontribution issmall andthatthe convergence with Nis good:

Although..Es, may be small,the contribution from each ofthe separate diagrams of

2 3 4

7 8 9

Tl~l>RD-oxDEx coxr1e18vT1oNS

sl

N

Analysis ofthird~rder oontrlbutions (Me~ up to 4 ilm

>xs. 2. Second and third~rder

ass function ofthe cai~otï label Nifl the sum over informe"dicte Mata.

fig. l . is not small - there is cancellation. It is instructive to study this cancellation andits sensitivity to the choice of the single-particle energies Z�u and to the saturatingeffect of the repulsive ge contributions. For this purpose table 2 shows the separatecontributions from each ofthe diagrams numbered as in fig . 1 and with intermediatestate excitationsN~ 4. The first column refers to the main calculations ofthis paper,as given in fig. 2. The second column shows the eflhct of using harmonic oscillatordenominators, ~�~ = 0. The third column retains the shifted denominators, with ~i,,~taken from table 1 but omits the hard-core component Yo in the. nucleon-ancleonpotential, i.e. c = 0.The last column removes both ofthese efücts, i.e . c a 0 and~,v= 0. There are three features oftable 2 to which we draw attention':

T~ia 2

vatâes in columns wlth c - 0 sT~ow the effect ofne~lectina a~, those. labelled 2 ~ 0 chow the effectof usln~ a pure harmonic oscûlator spectrum.

Diaeram cs0.3~i~0

c~0.32=0

c=0~i~0

c~0~ie0

1 -0.63 - 1.52 - 1.37 - 3.172 , - 1.21 - 2.89 - 1.84 - 4.313 3.06 7.22 3 .74 8.754 -6.65 -37.09 -10.22 -77.385 4.60 63.08 7 .92 86.19

6-E-7 0.02 . 0 .05 . - 0.22 - 0.498-F9 -a01 0.01 1 .01 2.3210 0.02 0.03 - 1.38 = 3.12

11-F12 0.0o aoo - 0.52 - i:1613 -0.31 - 1.23 - 1 .49 -. 9.9814 0.09 1 .07 1 .09 . 1,1 .80

total -1.02 8.73 - 3.28 9.25

82

RK. TRIFATHI st a~

(i) Comparison of cola. 1 and 2 (or 3 and 4) shows how the energy shifts .i,,~ notonly reduce all terms by a factor of approximately ~due to an increase of about ~ ineach energy denominator factor but also greatly reduce diagrams 4, S, 13 and 14 inwhich the operator d enters the numerator . This is, ofcourse, to be expected for theseHartree-Fuck diagrams because the introduction of d is a step towards self-consis-tency.

(ü) Comparison of cola . 1 and 3 (or 2 and 4) shows the way in which all the HFdiagrams 4to 14 are reduced when the hard~on potential Yo is included . This is thesame conclusion as in second order s) and is due to the fact that, with Ya included,the nucleus saturates at the value ofthe siu parameter b = 1 .7 fm which is used in thepresent calculation .

(iü) Even within col. 1, with the desirable features of both Ya and ~.,~ included,there remains some cancellation and it is of some interest to study this in the contextof the diagrammatic grouping normally applied in Rrueckner-Hartree-Fuck calcula-tions. In such calculations diagram 1 is automatically included in the definition ofthereaction matrix anddiagram S is included inthe definition ofthe single-particlepoten-tial for occupied configurations . Diagrams 2, 3 and 4 are the major third-order con-tributionsto three-body correlations in nuclear matter 9). Theremaining HF diagramsfr14 are urn in nuclearmatter and in finite nuclei theyare usually assumed to be-smalland neglected . Our numbers suggest that this last assumption is justified for a saturat-ing pôtential but at the same time indicate that it is dangerous to separate diagrams4and S, especiallyifan oscillator single-particle spectrum is used . It is easy to ace whythis is so. The contributions from diagrams 4 and S are naturally of opposite sign andtake the form ofasingle-particle (arid hole) potential energy perturbation multipliedby the probability for a2p-2h excitation of the core . Recause of the tensor force thelatter is quite large andthe .1�,1, which give a rough estimate of the potential energyshifts, show no sharp discontinuity on passing from hole to particle configurations(see table 1).

4. An upper bound io the energy andcommeob onoontrDwtione fromabove tàh~i order

It is well known that the second- and third-order contributions to the energy maybe combined to give an upper bound and Nssimov and Elliott4) have shownhow tododuce a least upper bound which is independent ofthe choice ofHa. Making use ofthe perturbation results np to 1Vbm this amounts to diagonalizingHin the set ofstatesQ~H~O) when Q, projects on to the sub-apace of states at abco excitation and a =

0, 2, 4, . . . N. The lowest eigenvalue, denoted by E, gives the upper bound and isshown in ßg . 3. Forcomparison, the figure contains also the second- and third-orderenergies, both with the pun oscillator for Ho (i.e. ~l,� = 0) and with the energyshifts ~,�,~included . [For simplicity in making the comparison in fig. 3 we did not usepreciselythe values of .~;,~ given in table 1, but instead used a simple approximation tothose numbers given by ~.,! _ (-4.3+n+j:nbcn. Thus the shifts are taken to be the

THIRD-0RDBIt CONTRIBUTIONS

83

Fia. 3. A comparison between tho upper bound E and the perturbation reeWta, showing also theporttabatlon t+eaWts with harmonic osc~lator denominators (dotted lines).

same for all states of an oscillator shell and for the denominators, ~m is simply re-placed by }itco. As a result ofthis approximation, the curve ofEz +E3 in fig. 3 differsslightly from that in fig. 2. Onefurther point is that, because of the use ofthe reactionmatrix go, the energy E is not strictly an upper bound, but the contribution from gois small.]Forthe oscillator case we see that E is close to the third-order sum~3~ ~ Eo+Et+E2+E3, but the large third-order contribution here suggests that the order-by-order convergence is slow andwe suspect the closeness ofE and~3~ to be fortuitous .With the shifts ~;,~ included, E is significantly above Et3~ and, since E is an upperbound, it is an open question which of these two numbers is a better approximation.For guidance on this point we now set up some schematic models of the many-bodyproblem which exploit the dominant role of the tensor-force in exciting two-particletwo-hole pairs (2p-2h).

In the first of these models we follow the ideas used in obtaining the bound E andset up a 2x2 matrix based on the closed shell ~0) and one other normalized statedefined by ~ 1) = QH~O)/<O~HQH~O)~ where Q is a projection operator which exclu-des the state ~0). The matrix elements of H are then <O~H~1) _ <0/HQH~O)} and<1~H~1) _ <O~HQHQhT~O)/(O~HQH~O) . Now if we assume that the state ~1) lies at asharp excitation energy d of the zero-order Hamiltonian Ho these matrix elementsare simply related to the second- and third-order energies calculated earlier becausethen

Es = COIHQ(H-Ho-Ei)QHIO)/d s = -~lIH-Ho-Etll)Ez/d .

84

RK. TRiPATHI ~t a~

The assumption of a sharp energy is not unreasonable because of the large tensorforce admixtures ~) which have a broad peak with a mean excitation of about 180MeV. This 2 x2 model then gives the Hamiltonian as H = H°+Hl where

with X~ ~ -Esd and Y = -E3dfE~. From the main calculation of this paper weknow Ez = -39.1 MeV andE3 = -1.4 MeV so that it remains only to estimate d.This is done by calculating (O~HQH~O~ with the SME (like the calculation of Ezwithout anyenergy denominators) and then using eq . (2), d = -<O~HQH~OjJE2 a179.4 MeV. Within the 2x 2 modelwe maythen calculate higher order corrections E,and compare with E. Now for a system with only one excited state, E is simply thelowest eigenvalue of the 2 x2 matrix and is given in this case by E = Eci>-33.9,where we have introduced the notation Ec.~ _ ~;~° E, for the sum to ath order. Oathe other hand the linked diagram expansion for the2x2model yields E�_ (-1Y+ ix(Y/dY-ß(X2/d) which may be summed to give E~°°~ = E~l~-XZ/(d +Y) = Etl~-40.5. The contribution from fourth order and above is in this case very small, lessthan 0.1 MeV, but Ec~) difi'ers from E by a large amount, 6.6 MeV. This comes fromthe presence in E of contributions corresponding to unlinked diagrams which nolonger sum precisely to zero because of the truncation of the matrix 1°) . Identifying~ 1 ~. with a 2p-2h excitation our2x2 model is, in fact, a special case ofthe more generalproblem discussed by Padjen and Ripka 11) whoshowed that Ec~) is a better approxi-mation than E so long as d +Y is not small.The problem of truncation may also be eliminated by extending our model to an

infinite matrix, treating the 2p-2h excitation as a boson ~ 1 ~ = at~0~ and writingH= E~l~+(d+Y~ta+X(at+a) where at and a are the boson operators for a one-dimensional oscillator. The transformation bt = at +X/(d+Y)reducesHto the formH= E{l~-X2J(d +Y)+(d +Y~btb which has lowest eigenvalue Ec~~-X~J(d +Y) _Eti~-40.5. This agrees precisely with E~°°~ which means that the linked cluster per-turbation series based on the 2x 2 matrix agrees with the exact eigenvalue of theinfinite matrix (or boson) model. From these model calculations we are therefore ledto conclude that the estimate Ecs) for the energy, plotted in fig. 3, is a better approxi-mation than E and that the contributions from fourth and higher orders are small.It is worth noting that the convergence is governed by the ratio Y/d = -0.04 andnot by the ratio X/(d +Y) = 0.48 that would be suggested by . the 2x2 matrix ifrestriction to linked terms wasnot made. Convergence is distinctly slower ifoscillatordenominators are. used (~i., i~ ~ 0) since then Yjd = 0.46. In this case, the lowesteigenvalue of the boson model is Etl~-39.9 which is quite close to the earlier resultEci~-40.5 but, even with linked terms only, the series is slowly converging, withE_ _ -58.3, E3 = 26.8, E4 = - i2.3, Es = 5.7_and E6 = -2.6.The purity of the shell-model wave function is however governed by the ratio Xld.

From the first-order correction to the wave function the percentage: impurity is given

THI1tD-OADBR CONTRIHUTIONS

8S

by 100X2J(X~+d2) e 18 ~, while for the infinite matrix model thewavefunctionisgiven simply by ~ = e-}`é"t~,o where ~° is the closed shell aced c s -X/(d +I'~ m-0.48. This gives a total impurity of21 ~ ofwhich 19 ~is in the single 2p-2h excita-tion described by at~r°. For the heavier nucleus 4°Ca we found earlier =) that Ez =-111 MeV andwe estimate, for this nucleus also, d s~ 180MeV taking into accountthe larger ~) values of ,l,u . No complete third=order calculations have been made for~°Ca but ifwo assume that.the 160pattern isfollowed, with E, small, then the infinitematrix model suggests that Ecs> is a good approximation to the total energy althoughthe total impurity ofthe wave function would nowbe 46 ~. The fact that the impurityin 4°Ca is much larger than in 160 simply reflects the greaternumber ofparticles and,in the infinite matrix model, the average occupation probability for a closed shell orbitis 97 ~ in both nuclei.

In a recent paper °) Mavromatis is rather pessimistic about the convergence ofperturbation theory in a calculation of the energy of '°Ca, where ho finds fourth-orderterms whichare largerthan second-0rder ones . His calculation maybe criticized,however, on anumber of grounds. First andforemost the interaction he uses does notsaturate at the values of the oscillator length that he uses so that the perturbationcorrections are predominantly Hartree-Fork terms which would bo close to zero fora saturating potential. Secondly his fourth-order contribution of 119 MeY includesan unlinked term of 130 MeV which ought to be excluded . Thirdly he uses the har-monic oscillator as the zero-order Hamiltonian which, as explained above, is a poorapproximation. (It is even worse in calcium than in oxygen since the .~ are larger.)A better treatment of all ofthese points has been made in the present calculation andin each case this results in improved convergence. We do not therefore share hispessimmstm . Finally the contribution from the tensor force which is known ~) to bethe dominant feature in second order is almost entirely neglected by Mavromatissince he restricts to intermediate states at 2iß only.

S. Condasion

We have calculated the third-order energy of 160 and shown it to be very smallwhen the nuclear size parameter is taken at the saturation minimum and the single-particle energies are chosen in a realistic manner . We argue on the basis of simplemodels that higher order corrections may be negligible . If we are correct in this thehan explanation must be found for the discrepancy between our total binding energyof 84 MeV and the experimental figure of 128 MeV. Although large, this di$ier-ence represents only 13 ~ of the total potential energy but this is outside the errorestimates of 5 to 10 ~ on the SME originating from phase shift error bars and theuncertainties in the choice of Ya which could give a change in the binding energyof order 10 MeY. Perhaps the most likely reason for the discrepancy is our use of apurely two-body interaction. The field theory of nuclear forces implies the existenceof three-body forces although their prediction is difficult and the forms suggested are

86

R. K. TRIPATHI et al.

very complicated. Calculations in the three-nucleon systems also suggest the neodfor a small three-body force . Using the same philosophy as in the derivation of theSME we might introduce a three-body force through its oscillator matrix elements.The simplest model is to assume the only non-zero matrix element to be that in whichthe three nucleons have no oscillator quanta in their internal state, analogous to theOs state for the two-body system. A matrix element of magnitude sr 1 MeY wouldbe required to resolve the discrepancy. This value is not inconsistent with results forthe three-nucleon system and it would also correct the low binding calculated s) fornuclei with one valence nucleon.

We are grateful to our former wlleagues Drs. H. A. Mavromatis and B. Singhfor the use of their computer programmes and for the impetus which their earlierwork provided for our calculations. One of us (R.K.T.) wishes to thank The Univer-sity of Sussex for hospitality and support through a Science Research Council grant.

8eferences

1) J. Dey, J. P. Elliott, A. D. Jackson, H. A. Mavromatia, )w A. Sanderson and B. Singh, NucLPhya. A134 (1969) 385

2) E. A. Sandereon, J. P. l3lliott, H. A. Mavromatis and H. Singh, Nucl . Phys. A219 (1974) 1903) J. P. Elliott, A. D. Jackson, H. A. Mavromatie, E. A. Sandarion andB. Sinah, Nucl . Phye. A1Z1

(1968) 2414) H. Niaeimov and J. P. Frlliott, Nucl. Phys. A198 (1972) 13) N. I. Kaisie, H. A. Mavromatis and B. Sinah, Phys. Lett . 37B (1971) 156) N. L Kaseis, Nucl . Phys . A194 (1972) 2037) H. A. Mrvromatie rnd B. Sinah, Phys . Lett . 41B (1972) 2518) H. A. Mavmmatie, Nucl. Phye. A257 (1976) 1099) H. A. Hethe, Ann. Rev. Nucl. Sci . 21 (1971) 93

10) H. A. Mavromatis, Nucl . Phys. A206 (1973) 47711) R. Padjen and G. Ripka, Nucl . Pl~ye. A149 (1970) 273