The proton densities in 40Ca and 48Ca in the Hartree-Fock approximation

Nuclear Physics Al44 (1971) 49-55; Not to be reproduced

by photoprint

THE PROTON

DENSITIES

IN THE HARTREE-FOCK

NORDfTA,

@ ~or~h-~oiia~

~fl~l~s~in~ Co., Amsterdam

or microfilm without written permission from the pubtisher

IN *%?a AND *%a APPROXIMATION

A. LANDE Cope~h~~e~, Denmark and

3. P. SVENNE

Received 23 November

1970

Abstract: In Hartree-Fock calcnlations on ‘%a and 48Ca, the rms proton radius of the heavier isotope is found to be smaller than that of the lighter, in agreement with the results of electronscattering and ,u-mesic X-ray experiments. In addition, the surface thickness of 4*Ca is found to be smaller than that in 40Ca, again in agreement with experiment. The detailed shape of the calculated density differs considerably from the Fermi distribution used in fitting the data. A discussion of this discrepancy is given.

1. Introduction Nuclear matter, to quite a good approximation, is incompressible. As a consequence, nuclear radii on the average are proportional to the cube root of the mass number, R = ro&, where r. is a constant of the order of 1.3 fm. However, there are significant deviations from this simple rule, in particular for isotopes of the same element. For 40Ca and 48Ca, the rms radius of the heavier isotope is actually smaller than that of the lighter one. This has been observed in both the electron scattering experiments ‘$“) and in p-mesic X-rays “). The most recent experiment of the former kind “) yields the value -0.31 % for the ratio Ar/r = (~~~--r~~)/r~~ (the radii here are the rms charge radii). This is to be contrasted with a value drJr = -1-6.27 % expected from the A* rule. In addition, these experiments show that this decrease in *%a radius relative to that of 40Ca is entirely due to a corresponding decrease in surface thickness. In this work, we present results of Hartree-Fock calculations for these nuclei which reproduce qualitatively this behaviour. 2, Hartree-Fock calculations The Hartree-Fock (HF) calculation was carried out in the oscillator basis 4J“) with both 40Ca and 48Ca assumed to be spherical. In that case the HF single-particle wave functions are sums over the radial quantum number only of the harmonic-os~i~Iator 49

50

A. LANDE AND J. P. SVENNE

functions having the same orbital and total angular momentum quantum numbers (Z,j). The basis consists of four oscillators in the s+ states, and three in each of the p, d and f4 states, i.e. at least two more oscillator functions than there are occupied states for each I and j. Since the basis is finite, the HF solution depends on the oscillator frequency hw. The appropriate value of ho is determined by the minimum in the HF total energy “). The values of ho will in general differ from isotope to isotope. In this way the theoretical values for E and r, = J{r2)c,-,arge for each nucleus are determined, and in turn the isotopic shift in r,. In the main part of this work we use the non-local potential of Tabakin ‘). It is a sum of attractive and repulsive separable terms,

v(r, r’) = C [-C.Ji(r)gj(r’)+hi(r)hj(r’)l, i,j

where i and j run over the allowed two-body relative states for relative I 5 2. The parameters were chosen ‘) to fit the low- and medium-energy nucleon-nucleon phase shifts. We use the Tabakin potential as modified by Clement and Baranger r5), who have adjusted the parameters in the ‘P, state to reduce the second-order corrections without affecting the phase-shift fit. In order to obtain a measure of the sensitivity of our results to the form of the potential, we repeat the calculations with the HamadaJohnston “) potential, using its shell-model reaction-matrix elements. These are calculated as in Kuo ‘) by a combination of the separation and reference spectrum methods. Besides the rms radius, another parameter showing significant variation in the Ca isotopes is the surface thickness. Since the shape of the density obtained by the HF calculation differs considerably from the Fermi shape (see fig. 2), the usual definition of the surface thickness as the distance from 90 ‘A to 10 ‘A of the central density does not apply to the HF case. Instead, we consider +

R

~

d(Ap)/dr= dpldr

dp,, ---

dp,,

dr

dr

~

+ ~ dp4s dr

1 *

(2)

This ratio, evaluated at r, (more precisely, where dp/dr is a maximum), is a measure of the surface thickness. In the case of the Fermi distribution (eq. (3) with w = 0), it is exactly equal to AZ/Z = (z48 -z,,)/% Calculations were performed over a wide range of values of ho, and the binding energy E and rms radii r, (proton) and r,,, (mass) were obtained as functions of ho. Fig. 1 shows a plot for the Tabakin interaction, and we observe that the minimum E in 48Ca occurs at a smaller value of r, than in 40Ca , in accord with the smaller charge 7 We are indebted to Prof. Ben Mottelson

for making this suggestion.

*ro*4sCa PROTON

51

DENSITIES

radius of 48Ca observed experime~t~~~~. Our value of &Jr, is more negative (- 2.8 “/‘o> than the experimental value (-0.31 %). F5r the Hamada-Johnston reaction-matrix elements we obtain similar results but much larger values of E and more rapid varia-

r.m.s. ( fm)

Fig. 1, Calculated binding energy versus rms radius for the Tabakin potential, The numbers at points on the curve indicate the value of ?6w fin MeV) giving that solution, The calculated isotopic shift is marked on the graph as AT,. TABLE 1 Parameters exp.

(Th)

exp (4*Ca-40Ca)

Tabakin HJ

of the charge density in “%a and %Y& c

Z

3.6758

0.5851

AC/C

h/z

1.87

-10.19

-9.7 -1.2

W -0.1017 Awjw

0.0917

-13.2 - 2.8

rc 3.4869 Ard+-,

-0.31

-3.3 -0.7

In the case of &OCathe values given are the actual values of the parameters in units af fm (except for w which is dimensionless). For the isotopic differences the table gives LQ/~A~ in “6,, where B is the Parameter given in the heading. The experimental numbers are taken from ref. 2),

52

A. LANDE AND J. P. SVENNE TABLE 2

Hartree-Fock

solutions in ‘Wa and @%a, at a value of b2 = Iti/Mco, near the minimum energy (b2 = 3.0 fm’ in both Tabakin cases, b2 = 3.456 in Hamada-Johnston) 40Ca n and p

4sCa --.

--

n Tabakin 1%

- 84.45

Ip+

-53.22

1pt

-39.08

Id+

-25.44

2%

- 17.24

1%

-

7.17

‘4

-

2.53

2p*

.&IF

2.44

-174.22 2.76 rms Hamada-Johnston &IF -342.22 2.75 rms

0.984

0.177 0.028 0.001 0.975 0.215 0.050 0.982 0.179 0.053 0.977 0.197 0.085 -0.178 0.966 0.140 0.127 0.993 0.029 0.114 0.989 0.058 0.133 -0.211 0.974 -0.079

-93.30

-60.54

-49.54

-30.29

-21.69

-15.13

-

5.06

0.915

---204.96 (mass) 2.8 1 -399.00 2.87

P -. 0.985 0.173 0.017 -0.008 0.970 0.236 0.049 0.975 0.217 0.05 1 0.968 0.236 0.089 -0.171 0.955 0.20s 0.130 0.982 0.150 0.111 0.984 0.116 0.132 -0.235 0.972 -0.021

-98.10

-65.57

-53.60

-35.61

-26.31

- 19.72

-

9.67

-

2.16

(proton)

0.986 0.164 0.014 0.006 0.973 0.227 0.040 0.980 0.195 0.034 0.966 0.245 0.081 -0.161 0.955 0.221 0.115 0.984 0.156 0.088 0.979 0.163 0.121 -0.229 0.973 0.031

2.65

2.73

In the case of the Tabakin results, the single-particle energies and wave functions are shown, each s.p. energy is followed by the coefficients in the expansion of the wave function in spherical harmonicoscillator functions. In 40Ca, neutron and proton wave functions are different, as shown. The anomalous depression of the proton energies below the corresponding neutron ones is a result of our having neglected the Coulomb interaction.

lion of E with rc. These differences, especially the latter one, are due in part to our incorrect use of an approximate reaction matrix as an effective interaction for HF calculations and the resultant lack of self-consistency in the reaction matrix. However, the HJ interaction also gives a negative value of Arc/r,. These results are summarized in table 1.

The ratia R [see eq. (2)] is also seen to be negative and in fair agree~~lent with the experimental value. However, the detailed shape of the density distribution differs sig~i~~~n~~~ from the Fermi shape fitted to the e~p~rin~~nta~ data (see fig. 2). We return to thk ia the next section, For ~~~~l~~~~ess, we list in table 2 d~~a~~sof the SF solution for ‘%a and 40Ca at the values of tzw yielding the energy minima. These ~aIcu~ations were performed ~~~~ecting the ~o~l~rnb int~racti~~~ ~rno~g protons. When included (in B cakulatiofi with a smaller basis) the results on isotopic differences were not aEeeted. 3, Shape of the Hartwe-Fock

density

Although the eaiculated values of dr,/rC and R are in qualitative agreement with experiment, the detailed density ~stribntion is not, The rms radii art: too small, a result that may be attributed to the force not saturating nuclear matter properIy_ In so far as isotopic di%erences are concerned; this is perhaps not a serious defect. In common with other HI? calculations we were also unable ta reproduce the observed “)

Fig, 2, Comparison of tha Hartree-Fock (HF) and parabolic Fermi (F) density distributions for 40Ca. The WF density plotted is that for the best value of GO using the Tabakin potential. The Fermi density is the experimental one (eq. (3)) except that the parameter c has arbitrarily been reduced to 2S21 Fm so that both densities have tbe same rms radius (indicated OR the graph). The dashed curve

54

A. LANDE AND J. P. SVENNE

r* weighted difference distributions. We obtain instead a shape similar to the density

difference which results when two harmonic-oscillator potential wells are filled, with their frequencies adjusted to give the rms radii obtained in the HF calculations. For oscillator shell-model densities (in fact for any p(r) ccf(r/r,)/r:) one may show trivially that R,,, = -4Ar,/r,. As shown in table 1, R,,, is fairly close to the ratios obtained for our HF densities. The HF density also displays a hump near the origin, in contrast to the parabolic Fermi distribution PO

p(r) =

[l+w(;)*I

,

l+exp

I--C [

Z

1

(3)

that is used in analysing the electron scattering data (see fig. 2). Similar structure in p(r) has also emerged in single-particle models lo, 11) and in HF calculations I2913). That the central density must be flattened, if not a detailed mechanism achieving this, is now fairly well established by the 750 MeV data. Recent calculations by Negele 13) and Vautherin and Brink 14) demonstrate how the dependence of the potential on local density (plus a suitable extension of the HF theory to incorporate variations in V(p)) contribute to depressing the central density. While Negele’s HF densities go a long way towards fitting the electron scattering data, they also fail to reproduce the observed difference in densities, yielding in fact Arc/r= M + 1 %. We conclude that the isotope effect observed in the HF approximation is due primarily to the density-independent part of the interaction and is relatively insensitive to the discrepancies in the densities and bulk properties of the individual isotopes. For Arc/r= and R we obtain shifts in the right direction. However, when configuration mixing of particle-hole states into the 40Ca ground state [see refs. lo* I’, “)I is more carefully incorporated our results may turn out to be merely fortuitous.

References 1) K. J. van Oostrum, R. Hofstadter, C. K. Noldeke, M. R. Yearian, B. C. Clark, R. Herman and D. G. Ravenhall, Phys. Rev. Lett. 16 (1966) 528; J. P. Bellicard, P. Bounin, R. F. Frosch, R. Hofstadter, J. S. McCarthy, F. J. Uhrhane, M. R. Yearian, B. C. Clark, R. Herman and D. G. Ravenhall, Phys. Rev. Lett. 19 (1967) 527 2) R. F. Frosch, R. Hofstadter, J. S. McCarthy, F. K. Niildeke, K. J. Van Oostrum, M. R. Yearian, B. C. Clark, R. Herman and D. G. Ravenhall, Phys. Rev. 174 (1968) 1380 3) R. D. Ehrlich, Phys. Rev. 173 (1968) 1089 4) A. K. Kerman, J. P. Svenne and F. M. H. Villars, Phys. Rev. 147 (1966) 710 5) R. M. Tarbutton and K. T. R. Davies, Nucl. Phys. A120 (1968) 1 6) T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 7) F. Tabakin, Ann. of Phys. 30 (1964) 51 8) W. H. Bassichis, A. K. Kerman and J. P. Svenne, Phys. Rev. 160 (1967) 746; W. H. Bassichis, B. A. Pohl and A. K. Kerman, Nucl. Phys. All2 (1968) 360

40*48Ca PROTON DENSITIES

55

9) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71 10) L. R. B. Elton, Proc. Int. Conf. on electromagnetic sizes of nuclei, Carleton Univ., Ottawa, Canada, May 22-24, 1967; Rosemary R. Shaw, A. Swift and L. R. B. Elton, Proc. Phys. Sot. 86 (1965) 531; L. R. B. Elton and A. Swift, Nucl. Phys. A94 (1967) 52; L. R. B. Elton and S. J. Webb, Phys. Rev. Lett. 24 (1970) 145 11) B. F. Gibson and K. J. Van Oostrum, Nucl. Phys. A90 (1967) 159 12) D. Vautherin and M. Veneroni, paper 7.54 in Int. Conf. on properties of nuclear states, Montreal, Canada, Aug. 25-30, 1969 13) J. W. Negele, Phys. Rev. Cl (1970) 1260 14) D. Vautherin and D. M. Brink, Phys. Lett. 32B (1970) 149 15) D. M. Clement and E. U. Baranger, Nucl. Phys. A108 (1968) 27