A calculation of the lead isotopes using a realistic residual interaction

1.D.1

[

Nuclear Physics 89 (1966) 145--153; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A CALCULATION OF T H E LEAD I S O T O P E S USING A REALISTIC RESIDUAL INTERACTION DAVID M. CLEMENT and ELIZABETH U. BARANGER

University of Pittsburgh, Pittsburgh, Pa.t Received 6 June 1966 Abstract: The non-local Tabakin potential is used as the residual interaction in a shell-model calculation of ~°epb, in which only configurations consisting of two holes in the closed shell are included. The 0 + ground state is lowered by only half the experimental amount, although good agreement is obtained for the lowest 2 +, 4 + and 6+ states. A quasi-particle (BCS) calculation of the lighter lead isotopes yields an odd-even mass difference which is too small. When additional configuration mixing is included using a "pairing phonon", reasonable agreement is attained with the experimental odd-even mass differences, with the low-energy spectra of the odd isotopes and with separation energies. These results are very similar to those obtained in the tin isotopes.

1. Introduction The nuclear shell model has proved to yield a fruitful and successful method for calculating properties of nuclear states. Its derivation from first principles has been beset by difficulties due to the fact that the two-nucleon data seem to imply the existence of a hard core. Although progress has been made in finding methods to treat this hard core, the computational problems are still enormous. An alternative approach is to assume that a velocity-dependent or non-local potential can be found which fits the free two-nucleon scattering data but is not singular and is suitable for use in an ordinary Hartree-Fock calculation. Such potentials have been proposed ~' 2) and Hartree-Fock calculations are in progress 3,4). If the Hartree-Fock approach can be justified then the proper shell-model calculation to be performed is self-evident: i.e., the single-particle energies and wave functions should be those resulting from the Hartree-Fock calculation. The residual interaction should be the original two-nucleon potential. As a preliminary step toward a more complete study, we have performed a shell-model calculation of the lead isotopes, using harmonic oscillator wave functions as our single-particle wave functions and using experimental values for the single-particle energies, under the assumption that these will not differ greatly from the values obtained from a Hartree-Fock calculation. We have used the non-local interaction proposed by Tabakin 2) as our residual shell-model interaction. It is a sum of separable potentials and yields an excellent fit to the S,- P- and D-wave phase parameters up to 320 MeV without generat* Research supported by the National Science Foundation. 145

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D , M, C L E M E N T A N D

E. U . B A R A N G E R

ing strong short-range correlations. It does not yield very good results in nuclear matter, i.e. the minimum energy per particle, which occurs for kF equal to 1.6 f m - ~, is only --8 MeV when calculated in first order. The second-order Goldstone corrections are small except in the 3S~ + 3D 1 state. Hartree-Fock calculations have been performed for light nuclei using this potential 4) and do not predict enough binding. Although this potential is imperfect, it is the best one available at the present time. The fact that the binding energy results are not very accurate is perhaps not too relevant to our shell-model calculations. We feel that the singlet-even and triplet-odd potentials will yield more reliable results in a shell-model calculation than the triplet-even potential since the second-order corrections in nuclear matter are smaller. An extensive comparison has been made 5) o f shell-model matrix elements in the tin region between those calculated with the Tabakin potential and those calculated by Kuo and Brown 6) using the Hamada-Johnston potential. Remarkably good agreement is obtained for the singlet-even and triplet-odd contributions 5). (The agreement for the singlet-odd and triplet-even contributions is much poorer, t) For these various reasons we have some confidence in the use of the singlet-even and triplet-odd parts of Tabakin's potential as a residual interaction in shell-model calculations. The calculations presented here follow closely those performed by Lee et al. 7) in their study of the nuclei 42Ca, 92Zr and ~80 and those of Kuo et al. tt in the tin isotopes. These calculations show that more than the conventional amount of configuration mixing is necessary in order to obtain agreement with experiment. In other words, it is necessary to include core excitation. We have extended these calculations to the lead region to see if the same type of result is obtained for such a heavy nucleus and to see if core excitation is necessary and reasonable even for such a good closed shell as z°spb. We will refer to KBB for all the necessary formalism used in the calculation and will present only results. In sect. 2 of this paper, we discuss a shell-model calculation of 2°6pb, where we restrict ourselves to those configurations involving two neutron holes in the 2°apb core. While the energies of the lowest 2 +, 4 + and 6 + levels agree remarkably well with the experimental values, the energy of the ground state is lowered to only half the experimental value. In sect. 3, we present the results of a BCS calculation applied to the nuclei 197Pb to 2°5pb, where the nucleons are distributed among the usual six-single-hole levels. The odd-even mass difference is found to be only 70 ~o of the experimental value. Excitation of the neutron closed shell is then included via a pairing phonon, and when this is done reasonable agreement is obtained with the experimental odd-even mass differences, the low-lying spectra of the odd isotopes and the neutron separation energies. There have been many earlier shell-model calculations of the lead isotopes; for example, the calculation of 2°6pb by True and Ford s). Recently BCS calculations of the lighter Pb isotopes have been done 9). These calculations use phenomenological residual interactions, ranging from pairing-plus-quadrupole interactions to fairly t We wish to thank T. T. S. Kuo for furnishing us some unpublished matrix elements. tt Ref. 5) referred to hereafter as KBB.

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RESIDUAL INTERACTION

realistic interactions of Gaussian form, and are restricted to the usual five or six single-hole levels. The use of these interactions probably compensates for the restricted amount of configuration mixing included. The calculations presented in this paper differ from these in that the residual interaction used is more "realistic". An attempt has been made through the pairing phonon to include additional configuration mixing; i.e., excitation of zero-coupled pairs from shells previously considered closed and into "empty" shells, and this produces important effects. However, this is obviously only one possible type of core excitation and other types should be included in a more complete calculation.

2. The Energy Levels of 2°6pb The neutron deficient lead isotopes 197Dh sz-,,~15 to 207DK s2-,,1~5 are usually treated in shell-model calculations as having 82 protons and 126 neutrons in inert closed shells, and I-11 neutron holes distributed among the following six single-hole levels: 3p~, 2f~, 3p~, li~, 2f~, l h p The single-hole energies, determined from the 2°Tpb spectrum are listed in table 1. We use the Tabakin interaction as the residual interaction. The parameters of this force as well as its functional dependence are given in Tabakin's TABLE 1

Single-hole energies 10) State

Energy (MeV)

3pt 2f~r 3p~ li~

0 0.57 0.89 1.63

2f~ lh~r

3.47

2.34

paper 2) and in KBB, table 1 and eqs. (2)-(4). We use this interaction to calculate the two-body matrix elements , where labJM> is an antisymmetrized normalized wave function of angular momentum JM with two identical particles inthe single-particle states a and b. The letters a and b represent all the quantum numbers, except the magnetic quantum number, necessary to specify the state. We choose harmonic oscillator wave functions as our single-particle wave functions with the size parameter, b = (h/mco)~, being 2.33 fm, the value used by True and Ford a, lO). The matrix element is expanded in terms of the matrix elements <(nl, S)JIVl(n'l', S)J>, where the states I(nl, S)a¢> are functions of the spincoordinates and the relative coordinate. Here 1 and 1' represent the relative orbital angular momentum a~d a¢ the total angular momentum in the centre-of-mass sys-

148

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AND

E.

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tern. Because of the separability of the Tabakin interaction, these matrix elements are particularly easy to calculate. The expressions for ((n/, S)JIVI(n'I', S ) J ) are summarized in eqs. (7) and (8) of KBB, while the relationship between (abJMIVI cdJM) and ((nl, S)JlVl(n'l', S ) J ) is given by eq. (5). The resulting two-body matrix elements are similar to those calculated in the tin region (KBB). As was found there, the contribution of the 3P I potential to the diagonal J = 0 elements is large and repulsive and is sometimes larger than the attractive contribution of the 1S o potential. Since the interaction is invariant under rotations and reflections, our Hamiltonian

19

(?)"

~z

I~/2 ÷

57

57

~!_.2_ + 313

~ - 313\

1313

313 513

~

-

3 2 113 53

~,

I

13

~

_

113

+

-x_~+

+

+

~-~.t_+ -

4-

\_+

+

0

I

2

3

4

J

5

6

7

8

9

Fig. 1. The spectrum o f 2°ePb up to 3.5 MeV. Here J is the angular momentum o f the level, -k or -is the parity. In the left hand column for each value of J is the spectrum with no residual interaction, in the centre the result calculated using the Tabakin interaction, and in the right hand column the experimental spectrum s0). The darker lines are those experimental levels whose spins and parities are given without parentheses in ref. ,o). The quantity (j~]~) is the dominant configuration o f a given level. The zero of energy is the (3P~r)-z level with no interaction. A 10+ and 124 level (not shown) are predicted at 3.24 and 3.25 MeV, respectively..

matrix is reduced to a set of submatrices corresponding to the allowed values of J~. Each submatrix is diagonalized; the eigenvectors show the amount of configuration mixing, and the eigenvalues yield the spectrum of 2°6pb. The latter are displayed in fig. 1. The zero of energy is taken as the energy of the (3p~) -2 state with no interaction. The experimental ground state energy is located at - 0 . 6 4 MeV, since the interaction energy of the ground state is equal to the difference between the separation energies of 2 ospb and 20Tpb" It is seen that the interaction has lowered the calculated ground state energy to - 0 . 3 4 MeV, about half the experimental amount. This lower-

RESIDUAL INTERACTION

149

ing is mostly due to configuration mixing. Lee et al. 7), who used the Tabakin interaction to calculate the spectra of 42Ca, 92Zr and 1SO, obtained similar results. The lowest 2 + and 4 + states have been brought down a considerable amount by configuration mixing and are almost in exact agreement with experiment. Lee et al. found rather poor agreement for the 2 + and 4 + levels. The lowering of the 6 + level is due to the large diagonal matrix element (2f÷2f~6MlVl2f~2f~6M). The lowest 5-, 6and 7- levels are in fairly good agreement with experiment. The low-lying experimental 3- state is known to be a core-excited state. The lack of agreement of the lowest state with the experimental value is probably due to not including enough configuration mixing, rather than to using the Tabakin potential as the residual interaction. Bertsch it) and Kuo et al. 6) have shown in their calculations of ~sO using the Kallio-Kolltveit and Hamada-Johnston potentials respectively that the three-particle-one-hole states make an enormous contribution and improve greatly the agreement with experiment. 3. Results of the BCS Calculation

The BCS approximation as applied to nuclei has been discussed by several authors ~. Briefly, the Bogoliubov-Valatin quasi-particle transformation is made and the Hamiltonian is rewritten in terms of quasi-particle operators. The elimination of the so-called "dangerous terms" gives rise to the gap equations. In this paper we neglect the effect of residual quasi-particle interactions. In other words, we assume that the ground state of an even nucleus is the BCS state and that the states of an odd nucleus are pure one-quasi-particle (or 1QP) states, whose energies are obtained by solving the gap equations (see eq. (13) of KBB). The residual quasi-particle interaction affects greatly the 2QP and 3QP states and can cause mixing between the 1QP and 3QP states. However, several calculations 13, 14) have shown that in single-closedshell nuclei the admixture of 3QP states into 1QP states of low excitation is small. When we compare our results with experiment, we consider only states of low excitation in the odd nuclei and only the ground state in the even nuclei. In fig. 2 we present the experimental odd-even mass differences defined as

Pn(A) = ½[Sn(A+ 1 ) - S.(A)], with A odd, where S,(A) is the neutron separation energy. This is approximately equal to Emi,(A), which is the lowest 1QP energy for a given nucleus. Curve 1 gives the results of a calculation using the BCS approximation with the single-hole energies and wave-functions discussed in sect. 2. The theoretical value of Emi,(207) is taken as half the calculated ground state interaction energy of 2°6pb. (See sect. 2.) The results are consistently too small. One would like to be able to increase the amount of configuration mixing included. The simplest way to do this is to solve the gap equations using an increased number of single-particle levels. In this way the excitation of * See ref. 12) and other papers cited there.

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D. M. CLEMENT AND E. U. BARANGER

zero-coupled pairs of particles out of the closed shells or into the empty shells would be included. This was in fact done in the calculation of the tin isotopes 5). However, the number of matrix elements to be calculated is enormous if one wishes to include the levels in the major shell below ( N = 50 to 82) and above (N = 126 to 184) the filling shell already included. Instead of calculating these elements, we approximate the additional ones by the matrix elements of a pairing force, i.e. = ( - 1) la+~o+ l ½g(2ja• + 1)½(2jc + 1) ~,

where one of the labels a or c refers to any level and the other label to any o n e of the distant levels only. As described in more detail in KBB, this is in fact equivalent to 1.0

>

2

I, 0

201

l

I

203

205

I 207 A

Fig. 2. Odd-even mass differences. The curves 1, 2, and 3 give Emt~(A) for P0 = 0.0, 0.007 and 0.012 MeV. Here Emla (207) is half the ground state interaction energy of ~°ePb described in sect. 2. The dashed curve connects the experimental values of Pn(A) based on the experimental separation energies given by ref. 1D. solving the gap equations for the filling levels just as before, but using the "effective" matrix elements, - ( - 1)ta+ t°(2jo + 1)½(2jc + 1)½e0, where

Here the sum is taken over the distant levels only, and Ec is the quasi-particle energy of level c. We estimate the magnitude of Po by taking g about half the value used by Kisslinger and Sorensen ~3), or 0.06 MeV, since we expect the matrix elements from the distant levels to be smaller than elements between states in the same major shell, and because their value already includes the effects of the neglected configurations. The energies Ec are all taken to be 7 MeV. This yields a value of Po = 0.007 MeV. I f g is increased to 0.075 ,Po becomes 0.012 MeV. In curves 2 and 3 of fig. 2, we com-

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pare Emi,(A) calculated using these two values of P0 with that computed using no pairing phonon and with the experimental odd-even mass differences. The addition of the pairing phonon has the effect of raising the values of Emt, and giving better agreement with experiment. KBB find a similar effect in their calculation of the tin isotopes. Fig. 3 shows the excitation energies of the states of the odd lead isotopes. The experimental values are plotted as dots. The heavy line is the result of a 1QP calculation with no pairing phonon, the dashed line is the result with Po = 0.007 MeV. With the |t3/2

1.5 ~\ Is÷z 1.0

-

.~ z

'~, ~.~. ~.

P',,';~.S<_" I 207

13+ 3/2

'~

5;i

~/2 I

5~z I

5/e I

°/2 I

205

203

201

199

_

~/2 / 197

A

Fig. 3. Excitation energies of low-lying states. Solid lines connect 1QP energies obtained for the case P0 ----0.0 MeV, the dashed line for Po = 0.007 MeV. The dots are experimental points taken from refs. xs-~o).

exception of the 13_+ level, only levels below 0.7 MeV are shown. Since the lowest excited state in the even isotopes lies at about 800 keV, the lowest 3QP states will lie at about this energy and the IQP approximation will be valid only for states considerably lower than this. However, the ~ + state should still be a 1QP state at high energies since the lowest 3QP states have opposite parity and cannot mix with it. One notes that on the whole the results with and without the pairing phonon are very similar and agree fairly well with experiment. With no pairing phonon, the {~- level lies lowest for 197 ~ A < 205. This is partly due to the crossing of the p~ and f~

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D. M. CLEMENT A N D E. U . BARANGER

single-particle levels between A = 205 and 207. The pairing phonon, while raising the ~- state above the ½- at A = 205, gives better agreement with experiment for the ~-- and ½- states. Both calculations give the same good agreement for the ~_a_÷ states. The ~- and ~- 1QP states lie considerably above the )-~-+ and are roughly parallel to it. In fig. 4 we have plotted experimental and theoretical values of ,.S,,(A+ 1) =

½[S.(A+2)+S.(A+I)] =

~ [BIEI (~ + 2) - BIE] (~) ],

where B.E.(A) is the binding energy of the nucleus for A = 206, 204 and 202. The BCS ground state energy gives the theoretical value for - B . E . ( A ) . As usual 15) the experimental values of the average neutron separation energies decrease as neutrons are added to the nucleus. The theoretical values show very little decrease but are not

7.0 /

1

Id / / f

7.5

f f

I 203

I 205

1 207

A

Fig. 4. Average separation energies. The dashed curve connects experimental values x~). The curves 1 and 2 are calculated values with P0 = 0.0 and 0.007 MeV, respectively. in gross disagreement with the data. These results were also obtained in the tin isotopes 5), where again the theoretical separation energies decrease very little with additional neutrons, and thus this seems to be a general characteristic of the Tabakin potential. It may be connected to the fact that Tabakin's potential does not yield saturation in nuclear matter at the correct density. On the other hand, additional core excitation could affect these results, as could effects due to the changing of the core. Most of the computer programs used in this calculation are modifications of ones developed by T. T. S. Kuo while he was at the University of Pittsburgh. We obtained the program which calculates the Moshinsky brackets from K. T. R. Davies ~6) and thank him for it. In performing the calculations reported in this paper the authors made use of the University of Pittsburgh Computing Centre, supported in part by the National Science Foundation under Grant No. G-11309.

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153

References 1) M. Razavy, G. Field and J. S. Levinger, Phys. Rev. 125 (1962) 269; A. M. Green, Nuclear Physics 33 (1962) 218 2) F. Tabakin, Ann. of Phys. 30 (1964) 51 3) R. Muthukrishnan and M. Baranger, Phys. Lett. 18 (1965) 160; R. Muthukrishnan, thesis, Carnegie Institute of Technology (1965); K. T. R. Davies, S. J. Krieger and M. Baranger, to be published 4) A. K. Kerman, J. P. Svenne and F. M. H. Villars, Phys. Rev. 147 (1966) 710; J. P. Svenne, thesis, Massachusetts Institute of Technology (1965) 5) T. T. S. Kuo, E. Baranger and M. Baranger, Nuclear Physics 81 (1966) 241 6) T. T. S. Kuo and G. E. Brown, Phys. Lett. 18 (1965) 54, and to be published 7) C. W. Lee and E. Baranger, Nuclear Physics 79 (1966) 385 8) W. W. True and K. W. Ford, Phys. Rev, 109 (1958) 1675 9) L. S. Kisslinger and R. A. Sorensen, Math. Phys. Medd. Dan. Ved. Selsk. 32, No. 9 (1960); R. Arvieu et M. Veneroni, Phys. Lett. 5 (1963) 142; A. Plastino, R. Arvieu and S. A. Moszkowski, to be published 10) V. Gillet, A. M. Green and E. A. Sanderson, Phys. Lett. 11 (1964) 44 and to be published 11) G. Bertsch, Nuclear Physics 74 (1965) 234 12) M. Baranger, Phys. Rev. 120 (1960) 957 13) L. S. Kisslinger and R. A. Sorensen, Revs. Mod. Phys. 35 (1963) 853 14) T. T. S. Kuo, E. U. Baranger and Michel Baranger, Nuclear Physics 79 (1966) 513 15) I. Talmi, Revs. Mod. Phys. 34 (1962) 704 16) M. Baranger and K. T. R. Davies, Nuclear Physics 79 (1966) 403 17) J. H. E. Mattauch, W. Thiele and A. H. Wapstra, Nuclear Physics 67 (1965) 1, 33 18) I. BergstrOm, J. Herrlander, P. Thieberger and J. Uhler, Ark. Fys. 20 (1961) 93 19) I. BergstrOm, Nuclear Physics 36 (1962) 31 20) Nuclear Data Sheets (U. S. Government Office, Washington D.C. 1960)