Isoscalar effective charges with scaling and Hartree-Fock wavefunctions

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17 January 1977

ISOSCALAR EFFECTIVE CHARGES WITH SCALING AND HARTREE-FOCK WAVEFUNCTIONS* L. ZAMICK

Rutgers University, Serin Physics Laboratory, Frelinghuysen Road, Piscataway, New Jersey 08854, USA M. GOLIN

Physics Department, The Florida State University, Tallahassee, Florida 32306, USA and S. MOSZKOWSKI

Department of Physics, Umversity of California, Los Angeles, California 90024, USA Received 12 July 1976 The lSOSCalarE2 charge for a closed shell plus one nucleon is studied by three methods: (a) a trial product wave function with uniform scahng of the oscillator length parameters, (b) Random Phase Approximation (R.P.A.), (c) a less restricted trial product wave function. Whereas the first two methods give quite different results, the last two are essentmUy identical.

We define the isoscalar effective charge correction for a closed shell plus one nucleon as the sum of the neutron correction and the proton correction 8e 0 = 6e n + 8ep. With the popular prescription, 1/2 for the neutron, 1/2 for the proton, 8e 0 would equal

unity. To calculate this quantity we use the Skyrme interaction [1] (with spin dependence suppressed)

H=

r(o+~ •

-toS(rij)+~-[k28(%)+8(rq)k21

~
+ t2k'8(ril)kl + t 3 ~ 8(rzl)8(rik). ) i<1< k We had previously succeeded in obtaining simple results for the mean energies of the isoscalar quadrupole state with the Skyrme interaction [ 2 4 ] Finite Range Energy which becomes, as Wong [5] pointed out, ~/2hw[ (m*/m) 1/2 m mfinxte nuclear matter, where m* is the effectwe mass. * Supported in part by the National Science Foundation. 116

Since the isoscalar effective charge can be calculated by considering the coupling o f the valence nucleon to isoscalar quadrupole states o f the core, we might expect some simple expressions for the effective charge as well. But in pursuit of this goal we run into some surprising obstacles. We will now indicate what they are and how to overcome them. The problem can best be illustrated by considering the zero range version o f the Skyrme interaction (t 1 = t 2 = 0). We can calculate the effective charge not only by the usual R.P.A. method of coupling the valence nucleon to the vibrations of the core, but also by doing a brute force variational calculation for the closed shell plus one system. If we use a product wave function with no further restrictions, this is o f course just the Hartree-Fock method. But the calculation is very simple if we do a restricted calculation, using Harmonic Oscillator wave functions with oscillator length parameters the same for all orbits

b x = b y =b Oe - a / 2 ,

b z =b Oe a.

We call this a scaling solution, or somewhat loosely "scaled variational". With the zero range interaction

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the expectation value of the potential energy with the scaled wave functions is independent o f ct, so the resuits become:

Table 1 A comparison of the "scaled variational" (S.V.) and the R.P.A. results of the isoscalar effective charge correction

n2

Mass of core

E(a) --- constant +2-mm b 2 lea + ( e - 2 a - c a ) / 3 ]

h2

+2rob2 [(Nx + Ny * 1)ea+(Nz+ l/2)e-2~]. In the above ~ = Sum[(N x + 1/2) +(Ny + 1]2) +(N z + 1/2)] over the core, e.g., Y. = 6, 3 6 , 1 2 0 in 4He, 160, and 40Ca. The last term is the valence kinetic energy. Note that N x + Ny + N z = 2n + l, the latter being spherical quantum numbers of the valence nucleon such that

en, 1 = (2n + I + 3/2) tim. It is customary to set N z = 0. This is equivalent to choosing a valence wave function Unt(r) Yl, l(~), so that the equilibrium condition dE/dt~ = 0 leads to e 3a = ( 2 ~ / 3 + 1)/(2~/3 + (2n + l + 1)). The quadrupole moment can be written as the sum o f a core contribution and a valence contribution Q = Qc + a v ,

Qc = (2Y~/3) b2 [e2a - e - a ] ,

Qv = b 2 [ e2a - (2n + 1+ 1) e - a ] . The isoscalar effective charge ts 1 + 8e 0 = [ a c + a v ] a / [ a c + a v ] a = 0 = - ( ( 2 ~ / 3 ) [e 2a - e - a ] + [e 2a - (2n + l + 1) e - a ] ~}](2n + l). Note that if we use the above zero range condition and if we linearize in a we obtain 6e 0 = 1. Thts is essentiaUy the Mottelson condition [6]. We note parenthetically, that the effective charge renormalization comes not only from the core but also from the valence nucleon, the latter beingespecially important in light nuclei like 4He. The core and valence contributions (6ec, 6ev) for mass 4, 16, and 40 are respectively (0.71, 0.24), (0.91,0.06), and (0.96, 0.02). The difficulty that we have previously alluded to arises when one makes a comparison o f the "scaled

17 January 1977

to MeV [fml 3

tl MeV [fml s

Zero

4 16 Range 40

982.0 1045.8 1077.1

0 0 0

4 16 Range 40

1167.1 1292.4 1314.2

821.3 1096.74 1199.11

Finite

t3 MeV [fm] 6

8e ° S.V.

18 794 0.95 19 189 0.98 18 133 0.99 0 0 0

0.48 0.26 0.13

8e° R.P.A. 0.76 1.16 1.71 0.18 0.41 0.57

variational" results with the R.P.A. The results are shown in table 1. Whereas for the zero range interaction the value of 8e 0 is close to unity in the scaling approximation for all nuclei, in the R.P.A. it increases at least the core contribution from 0.76 to 1.71 as one goes from core mass 4 to 40. For the. finite range interaction, in which the two parameters t o and t 1 are adjusted to give the right binding energy at the right radius, we note that 6e ° (scaling) decreases with increasing mass number but 6e 0 (R.P.A.) increases. So the results are indeed quite different. It should be mentioned that in the R.P.A. work, the particle-hole splittings were calculated with the same Skyrme interaction used to get the particle-hole interaction. Thus, whereas for mass 4, 16, and 40, respectively, the value o f 2tim was equal to 3 7 . 3 1 , 2 6 . 7 2 , and 21.20 MeV, the mean values o f e p - e h for the zero range interaction were 32.66, 27.01, and 23.45 MeV; for the finite range interaction they were 45.07, 42.79, and 39.33 MeV. The difference between the "scaled variational" and R.P.A. is worrisome, especially in light of the work o f G.E. Brown [7], who showed that the R.P.A. result for the effective charge could be obtained by linearizing the Hartree-Fock result for the closed shell plus one system. We would like to suggest here that the fault lies with the use o f a scaled oscillator rather than a less restricted trial wave function. We present the following argument to support this. Consider the case of a zero range interaction. Suppose the valence nucleon is a proton with spin up p t . Then, by the Pauli principle, this p t valence particle 117

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will not interact with the pl" particles in the core, and hence will not deform them. The valence particle will interact with the p~ nucleons and will deform them. We thus see already that a uniform scaling trial solution is unreasonable. Likewise, we disagree with the argument that the Mottelson conditions [6] necessarily arise from the use of a zero-range interaction. It is true that if we assume scaling, the zero range interaction will yield the Mottelson conditions. But the above argument shows that a zero range interaction will not lead to scaling. Mottelson himself [6] did not use a zero range interaction to arrive at his conditions but rather he imposed self-consistency of the potential and the density. On the other hand, if our valence space consisted of four nucleons pt, p$, nt, n~ such as 20Ne, then perhaps a scaling solution would be correct. We next note that if we carry out the calculations in a manner close to the way suggested by G.E. Brown, then we do indeed get very close agreement between R.P.A. and Hartree-Fock. Rather than consider the most general case, we look at mass 5. Let the valence particle be pl'. Let the single partlcle potential (arising from a zero range interaction) be U. The core orbit s o becomes deformed. In perturbation theory

(so Ud O) I~k) = I s 0)+

AE

Id°)'

for p~, n$, and n¢ (d o = d state orbit with lz = O) IV) = Iso), for pt. The Hartree-Fock equation becomes

(s o Udo) = (SoS0 Udos 0 ) × 3 + [(sod 0 Vdos 0) + {SoS0 Vdodo) ] 3 ((soUdo)/AE) + (sOP1 VdoP 1) (where Pl denotes the valence orbit with l z = 1) or

{ [(sodogdOso)+(SosoVdodo)] 3}-I (soUd O)= 1 + -AE X (sop 1 VdoP 1) since (SosoUdos O) = O. The factor of 3 comes from the fact that 3/4 of the 4 core particles contribute. The above expression has what looks like a renormalized propagator, just like in the R.P.A. 118

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We obtain the quadrupole moment by summing the quadrupole moments of the various orbits. Without going into further detail, we mention that the mass 5 case was calculated using the above "less restricted Hartree-Fock" wavefunctions for both the zero range and the finite range interactions. The expressions for 8e* are identical to those obtained in the R.P.A. as one would indeed expect from the work of G.E. Brown [7] and the respective results are 0.76 and 0.18. The former happens to agree closely with the scaled variational method result 0.71 (core contribution only). We add a remark concerning the difference of the velocity dependent s state interaction (proportional to t 1) and the corresponding p state interaction (proportional to t2). If t 1 and t 2 are both positive, then they both contribute positive amounts to the finite range energy and therefore act coherently in raising the energy of the isoscalar quadrupole state. However, they act oppositely on the isoscalar effective charge. As we have already seen in table 1, a positive value of t 1 serves to decrease the effective charge. However, a positive value of t 2 will increase the effective charge. (It should be noted that for the interactions used in ref. [1 ], t 2 was small and negative.) For example, if we set t I and t 3 = 0, tl~ereby achieving saturation solely via a p state repulsion, then the result for 160 is t o = 1285.3 MeV fm 3, t 2 = 1444.2 MeV fm 5, 8eOpA = 1.08. Recall that the corresponding value for a velocity dependent S state repulsion was ~e 0 = 0.41. We see from this example that deformation properties can play an important role in limiting the values of the parameters t I and t 2 of the Skyrme interaction.

References [1 ] T.H.R. Skyrme, Phil. Mag. 1 (1956) 1043; Nucl. Phys. 9 (1959) 615; D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626. [2] M. Golin and L. Zamick, Nucl. Phys. A249 (1975) 320. [3] L. Zamick, Lecture Notes in Physics, vol. 40, Effective interactions and operators in nuclei, ed. B.R. Barrett (Springer-Verlag, 1975). [4] K. Goeke, S.A. Moszkowski and S. Krewald, Phys. Lett. to be published. [5 ] C.W. Wong, private commumcation. [6] B.R. Mottelson, The many body problem, Les Houches (John Wiley, New York, 1958). [7 ] G.E. Brown, Facets of Physics, eds. D.A. Bromley and V. Hughes (Academic Press, New York, 1970) p. 141.