Nuclear variational calculations with improved mixed-density approximation

Nuclear Physics Al66 (1971) 317-329; Not to be reproduced by photoprint or

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C~CULATI~NS

MIXED-DENSITY

APPROXIMATION

V. R. PANDHARIPANDE t The Niels Bohr fnstitute, Copenhagen, Denmark Received 2 December 1970 (Revised 11 February 1971) Abstract: The Slater mixed-density approximation is modiied so as to retain effects of finite interaction range in the approximate exchange energy. A correction for finite nmber of particles makes the proposed exchange-energy approximation exact for 4He and infinite matter. Thomas-Fermi and effective single-particle Hamiltonian calculations are carried out with this appro~mation in %Za and zosPb. The results of these compare very favourably with those of Hartree-Fock calculations in coordinate space.

1. htroduction In simpler approximations to the Hartree-Fock (HF) theory, the Slater ‘) approximation is commonly used for the Dirac mixed density

The number of fermions, either neutrons or protons, is n. The Slater mixed density is essentiaIIy based on the Thomas-Fermi (TF) approximation, which describes the true wave functions 4;(r) around r’, by a set of plane waves $i(~; r 25r’) .-

l&l S kF

kF = (3n2p(r’))*.

and

(2)

In the above TF wave function, the number of particles is kept fixed and the box volume SEis varied to change the density: P2= 2

kF & d3k s 0 (243

gives Q

3rc2n

n (3)

=k:=i* 7 On study leave from Tata Institute

of Fundamental 317

Research, Bombay&

In&a.

318

V. R. PANDHARIPANDE

? (rl

Fig. 1. Schematic density distribution. A

rl = 3.6875 0.10

_

0.00

*$ 2

0.06

ol

0.01

0.02

0 0

2

0

6

4

r2 (fm)

0.10 ,

I

I

I

1

2 r

lfm)

Fig. 2. Comparison of mixed density (13) with true mixed density for zosPb neutrons. In fig. 2A the thick line gives p,,(r), the three thin and dotted lines show respectively ptr and pspp (rl, r2; 0 = 0) against rz for three values of r. Fig. 2B displays ptr and pap. (rl, r2 = t-1; 8 + 0) as a function of r.

MIXED-DENSITY

APPROXIMATION

319

This is conceptually different from varying the number of particles to change the density in a box of fixed volume, and is more convenient for studying the mixed density, since it retains the same number of single-particle wave functions at r1 and r2 even when p(rl) # p(rz). In the Slater method, TF wave functions corresponding to the center-of-mass point R are used to approximate 4i(r) in the region of r1 and r2. The sum in (1) is replaced by an integration over k, assuming n to be large; and the Slater mixed density is obtained from $i(r; r w R) as follows: Pslater(rl v 4

= =

p(R)3.0{sin (kFr)-kFr

cos (kFr))

ki r3

p(R)S(br),

r = Irl--r2/,

(4) R

=

&(r,+r2).

Equation (4) also defines the Slater function S(k,r). The Slater approximation seriously overestimates the mixed density at nuclear surface. Consider for example the density distribution in fig. 1, where p(r,) is zero and accordingly all cJi(rz) are zero. The true p(r,, rz) is zero but pSlater(rl,r2) can be significant if p(R) # 0. Moreover with this approximation the exchange-energy integral can be reduced from its six-dimensional form

s

W,, = 3 VrMr,9 to a three-dimensional

412d3rld3r2

(5)

integral

s

w,,= t k(@))d3R U,,(p)

= 1 V(r)p2S2(kF

r)4nr2dr.

(6) (7)

At semi-infinite sharp surface, p(x, y, z) equals p when x is negative and is zero for positive values of x, the true mixed density is zero when r1 or r2 have positive x. In the Slater approximation, however, the exchange-energy density U,,(p) is calculated assuming constant density and integrating (7) over all space. Thus at a semi-infinite sharp surface the U,,(p) has double its true value due to integration over non-existent density, and consequently when the exchange force is dominant and attractive, the use of the Slater approximation in TF calculations leads to discontinuous density distributions “). This difficulty was not noticed in atomic TF calculations since the Coulomb interaction between electrons is repulsive and hence a variational calculation would not tend to overestimate U&(R)) in atoms.

320

V. R. PANDHARIPANDE

The expression for exchange energy (6) effectively approximates the exchange force by a density-dependent &function interaction, for which the TF density distribution corresponds to a sphere with sharp surface and the TF approximation breaks down. This problem can be avoided by re-introducing the range of V(r) in the effective exchange interaction “):

f(b) =

~,,(&I)

(9)

s

V(r)& kr)4zr2dr ’

K,

=

s

v~~f(r)P(v,)p(r2)d3r,

d3r2.

(10)

However, as r2 -+ rl, the p(r,, r2) approaches p(r,) and V::‘(r) should go over to V(r). The V,exff(r)in (8) does not have this property, and since f (kF) is less than unity, it underestimates,the exchange at small r. Thus, (8) gives too large a surface energy ‘). In sect. 2, the Mater mixed-density approximation is modified to overcome these drawbacks. TF and self-consistent single-particle Hamiltonian “) (SPH) calculations with the new mixed-density approximation are presented in sects. 4 and 5. Sect. 6 gives a comparison with the results of HF calculations in coordinate space. 2. The mixed-density approximation The Slater method uses approximate wave functions corresponding to a constant density p(R), whereas the true local density and the amplitudes of TF plane waves (2) are varying with r. This variation can be incorporated in the approximate wave functions ~i(r; r X rl, r2) as follows:

(11) We still retain the simplicity of plane waves by using a mean k, defined below: kr(ri,

4 = (1.5n2Ep(r,)+p(r2)l)*.

(12)

The single set of plane waves with varying amplitudes (11) thus retains many features of the two TF sets (2) for r x rl and r x r2. It gives proper diagonal elements of the density matrix, and the mixed density with (11) is Pap&1 ,4

= ~~(~M4~(IcF

4

(13)

This mixed density has proper asymptotic behaviour as r2 --) rl, and also as p(rJ or p(r2) tend to zero. Fig. 2 compares the mixed density (13) with the true mixed density calculated from single-particle wave functions for *‘*Pb neutrons. These wave func-

MIXED-DENSITY

APPROXIMATION

321

tions are discussed in sect. 5. The three chosen values of rl. correspond to full, half, and one-sixth mean central density. The radial variation of p(r,, r,; 8 = 0), 8 being the angle between r1 and rZ, is well represented by (13) except when p(rl) is too small and rl is at the outer edge of the nucleus. For the angular variation, p(rI, r,; r2 = rl, 8 # 0), (13) is a good approximation at all densities. With (13) the exchange energy can be written as follows by adding and subtracting unity:

s

W,, = 3 ~(Mrlb(r2){l 41 -

S”U+r)))d3rld3r2.

(14)

We note that, in an g-particle, the exchanges are between spin (or isobaric spin) members having identical spatial wave functions, and hence the (1 -S2(kFr)) term does not contribute. Thus the Slater function may be inadequate to describe the unique self, spin (or isobaric spin) exchange in finite nuclei. Approximating the sum over i in (1) by an integral over k effectively reduces the probability of these exchanges to zero. Their contribution can be easily calculated from wave function (11) and is

We retain the Slater function to describe other exchanges and normalize it by taking the limit r -+ 0. This gives the following finite number correction to Wex:

The coefficient of the (1 - S2(k,r)) term is just the ratio of the number of exchanges exclusive of self, spin (or isobaric spin) exchange to all exchanges. In the case of neutron-proton interaction, n is the larger of the number of neutrons or protons. 3. The nuclear energy An effective interaction 3), with density-dependent a-function repulsion and Yukawa attraction, is used in the present work. The approximate potential energy of a doubly even nucleus is given by + ~~~(r)(PN(rl)PN(r2)PN’(rl)PN’(r2))i

y

(1 -S(k,,

$(k,,.r)))]

d3r, d3r2

The four terms in (17) are respectively the direct, exchange, density-dependent

repul-

V. R. PANDHARIPANDE

322

sion, and Coulomb energies. Here and throughout the rest of this paper, kFN is a function ofpN(rl) and pN(rz) as defined by (12). Subscripts N and N’ stand for neutron (n) and proton (p), and n, is 2, (A-Z), and the larger of Z and (A -2) for the p-p, n-n, and n-p interactions, respectively, mass and charge numbers for the nucleus being A and 2. The direct and exchange forces are

V$(r) = V&N(r)= V:?(r)

;t(v31-l-3v33)v(P), gv31-3Y33)qr),

= 3(3F3 + Y3’ + V’l +3V33)V(v(r),

Vzw(r) = 4(3P3 -t V31- VI1 -3V33)V(r), -1.61t1-rz/ V(r) =

1.6jr, -r,l The density-dependent

(18)

ve

’

S-function interaction is

A,(A’3P’3+A31P31)Cp~(R)+P~,(R)I~.S(Y1-r2), ANN = $A,A3’, ANN’= *AD(3A13 + A31). The P(ZTf1)(2S+1f are projection

operators

(19)

for state (Z’, S) and the parameters

J.AZT+~)(~S+~) , P, A(ZT+l)(zS+l) and A, are given in ref. “). TABLE 1

Comparison

of energies calculated from local density (equations (17) and (21)) with vah.res of Slater determinants (in MeV)

‘%a 208Pb

936.9 5226.3

965.3 5285.3

633.1 3877.7

expectation

631.5 3773.8

TABLE 2 Binding

energies

and

radii

(in

MeV

and

fm).

The

E 4QCa

TF SPH HF EXP

?-OsPb

TF SPH HF EXP

“) Subtract NN50 MeV for L * s.

323.68 332.26 335.63 342 1482.32 1442.84 1449.66 1637 “)

experimental

Mm=)

data

are from

&Arms)

3.242 3.372 3.305

3.271 3.427 3.353 3.49

5.544 5.601 5.560

5.446 5.466 5.438 5.5

refs. *-r’)

MIXED-DENSITY

Approximation

323

APPROXIMATION

(16) is exact for 4He and infinite matter, and is compared with true

W calculated from wave functions of 4oCo and ‘08Pb (sect. 5) in table 1. Correction

for a finite number of nucleons increases the magnitude of W by 4.1 and 0.9 % in 40Ca and “‘Pb, respectively. The true and TF kinetic energies in table 1 are defined as follows:

(20) TTF = c N

0.3 & s

(37r2/IN(r))+&‘)d3?‘.

(21)

N

No attempt has been made here to modify the TF approximation for kinetic energy. SPH calculations, however, do not use this approximation and have exact kinetic energy. 4. Thomas-Fermi

calculations

The nuclear energy is completely expressed in terms of its local density distribution with expressions (17) and (21) for its potential and kinetic parts. Minimising this energy with respect to variations in pN(r) gives two coupled TF equations for neutron and proton densities:

aw

?f!?

apN(r)

+

(h2pN(r))+

FN.

=

(22)

2MN

The Fermi energies FN are obtained from

s

pN(r)d3r = Z

and the

G’(r, ,

(23)

aw/apN(r) is given by

-.Y?!C = ~j (vEr(r)pN,(r2) aPN(r)

(A-Z),

or

+

,.(rJ,;l,,p.(rJ]‘cF(rl ,r2)) d3r2

vEN'(r)

N’

PN(r2)-PN(rI)

1

%hN

PN(r2)

+PN(r~)

r)S(kFN’

2pNh)

PN(rl)

+PN(‘2)

r,

“‘“k(kFflr, S(k,,. i). 1

(25)

FN

The fourth term in (24) arises from a change in the strength of the density-dependent repulsion (19) with variation of pN(rl). The above equations are solved by an iterative method 5), in which a W/apN( r ) is calculated from a trial density distribution. Equations (22) are used to calculate new pN(r) with FN’s chosen to satisfy (23) and

324

V. R. PAN~~ARIPANDE

this process is iterated to achieve self-consistency. All results are given in sect. 6. The appearance of [PN(rl)]* in the denominator of (24) simply relates the decay of neutron and proton densities at the nuclear surface. To simplify the calculations, the when p&r) is less than 0.003 fme3. Similarly, at ratio ~~t~)/~~(Y) is set to (A-2)/Z small densities, CE(r,, r2) approaches unity. 5, Effective single-particle Hamiltooian In this calculation the TF approximation is eliminated and expressions (17) and (20) are used for potential and kinetic energies with L%(r) =f 7

(26)

I4NiCr)l”.

The single-particle wave functions 4&r) are varied to minimize the total energy. The center-of-mass correction in (20) is neglected at this stage and reinserted in the calculation of the binding energy. This variation gives an effective single-particle Hamiltonian 4): i3W

HSjP&(r)= - Udr)-

(27)

Gs’

The above Ha~ltonian is solved set-consistently by a Hartree-type iterative procedure “) to calculate cpNi(r).Single-particle binding energies are then given by the expectation values eNi =

3

T

(#Ni

+N’j 9 Vij[$Ni

CbN’j-@N’j&Ni])+

f$

(4Ni

7 v24Ni),

(28)

N

and the nuclear binding energy is calculated from it,

(29) Table 1 gives a comparison of energies obtained from (28) and (29) with those from pN(r) given by (26) and expressions (17) and (21). Alternatively, the eNifs can be calculated with the approximate mixed density (13) as follows: eNi =

&I

[ v~~‘(r)r~,i(r,)12PN’(r2)f

vEN’(r)

(30)

MIXED-DENSITY

325

APPROXIMATION

However, at least in closed-shell nuclei, the above expression does not offer a significant simplification over (28). Results with both (28) and (30) are given in sect. 6. The density-dependent part of the interaction (19) has different strengths in nuclei with mass numbers A and A - 1, and hence the single-particle energies en, do not represent separation energies as measured by (p, 2p) type experiments. If the change in the wave functions &(r) is neglected, the single-particle separation energies are given by “) eii ~1 -eNi-&ANN s

[P^N(r)I’[(2PN(r))3-(2~N(r))31d3~

_ ANN’

s

c31)

~N(r)~N’(r)[(pN(r)+PN’(r))~-(~N(r)+~N’(r))*ld3ry

where j&(r) is the density distribution of the residual (A - 1) nucleus: bNcr>

=

PN(r)-

(32)

14Ndr)1’*

The above SPH calculation is much simpler than a HF one, and is similar to those in which energies are calculated with a Slater determinant,of single-particle wave functions in an external well (Woods-Saxon, for example), and the parameters of the well are varied to minimise the binding energy. However, it eliminates the hunting for a minimum by a rapidly converging iterative procedure, while still allowing an arbitrary shape and possible shell fluctuations for the well. 6. Comparison with Hartree-Fock

calculations

In the HF method, the binding energy N (+Ni 3 v2+Ni) +&&,

z (4Ni 4N’j 9hj[$Ni

4N’j_4N’j

4Nil>

is minimized with respect to variations in &i(r). This gives a state-dependent particle Hamiltonian vzr(r)PN’(r2)

+

f

(33)

single-

vEN’(r)

z ANN’pNr(rl){[pN(rl) +PN'(rl)]"+~PN(rl)[PN(rl)

+PN'(rl>l-'f

+ ~oul(rl)BN~ ’

The state-dependent single-particle potential in (34) has poles at the nodes of the wave functions 4Ni(r1). Following Vautherin and Veneroni 6), these poles are averaged over a radial distance of 0.6 fm and the Hamiltonian (34) is solved self-consistently by iterations. Single-particle and binding energies are calculated from the wave functions, as discussed in sect. 5.

V. R. PANRHARIPAND~

326

TABLE 3 40Ca single-particle separation energies, eNLS(in MeV) SPH(‘)

SPH(‘)

HF

EXP

SPH

HF

OP (P) Od (P) ‘s (P)

44.11 28.62 12.99 10.61

44.22 28.51 13.00 10.27

44.44 28.84 12.85 10.03

50fll 34&6 12.6 “) 11.6

30.17 20.56 10.12 8.70

42.88 27.95 12.26 8.85

Os OP ‘d ‘s

51.90 36.10 20.14 17.84

52.06 36.04 20.18 17.42

52.40 36.47 20.13 17.13

19.8 “) 18.2

37.81 27.89 17.16 15.76

50.81 35.56 19.50 16.15

OS(PI

tn) (n) (n) (n)

SPH(r) and SPH(2) are with expressions (28) and (30), respectively, and the eigenvames of single-particle Hamiltonians (27) and (34) are listed in the last two columns. Experimental data are from refs. r1-r3). “) Center of mass of d+ and d+. TABLE 4 losPb

OS(P)

single-particle

separation

energies

(in

MeV).

SPH(‘)

SPH(2)

HF

OP Od ‘s Of ‘P Og ‘d zs Oh

(P) (P) (P) (P) (P) (P) (P) (P) (P)

47.34 40.30 32.67 31.36 24.35 21.01 15.45 10.49 9.32 6.09

47.41 40.26 32.47 30.80 24.02 20.49 15.02 10.12 8.83 5.64

47.73 40.59 32.93 31.44 24.54 21.14 15.53 10.50 8.94 6.02

Os ‘P Od Is Of ‘P Og ‘d ‘s Oh If ‘P Oi

(n) (n) (n) (n) (n) (n) (n) (n) (n) (n) (n) (n) (n)

58.65 51.31 43.36 42.07 34.81 31.92 25.74 21.48 20.27 16.28 11.13 9.24 6.61

58.69 51.22 43.13 41.59 34.44 31.38 25.28 21.10 19.92 15.82 11.06 9.39 6.22

58.83 51.42 43.48 42.13 34.93 32.06 25.80 21.45 19.92 16.25 11.00 8.92 6.47

“) Subtract 1.7 MeV for I * s. b, Subtract 2.0 MeV for 2 - s.

See footnote

of

table 3 for

notation

EXP

SPH

HF

9.2 8.0 9.4 “)

32.01 27.95 23.02 21.14 17.41 14.96 11.23 8.30 6.93 4.53

47.26 40.31 32.65 30.80 24.26 20.76 15.26 10.22 8.31 5.78

8.9 8.0 9.0 b)

41.94 37.49 32.22 30.55 26.31 24.04 19.84 17.16 16.00 12.90 9.88 8.19 5.53

58.41 51.16 43.24 41.58 34.69 31.76 25.58 21.24 19.38 16.06 10.82 8.60 6.31

MIXED-DENSITY

APPROXIMATION

327

Tables 2, 3 and 4 give the energies from TF, SPH, and HF calculations for 40Ca and “*Pb. The SPH energies in table 2 are calculated from expectation values (28), and the errors due to the neglect of 1 - s coupling in ’ 08Pb are estimated with a singleparticle coupling de*.,

= -0.1 x $

(I - s) MeV.

(35)

The 40Ca proton, “‘Pb neutron and proton distributions and potentials are shown in figs. 3 and 4.

"Ca

I

0

I

2

3

4

5 r

Fig. 3. 40Ca proton

6

7

8

9

10

(fm)

and zosPb neutron and proton distributions. The broken, full, and dashed lines show results of TF, SPH, and HF calculations, respectively.

V. R. PANDHARIPANDE

328

7. Conclusions

The two approximations to HF compare very well with it when the mixed density is approximated by (13). Particularly the slopes of TF, SPH and HF density distributions at the nuclear surface are almost identical. Recently, Moszkowski ‘) investigated the TF approximation with a simple interaction and found that it gives smaller surface thickness and energy. In TF theory, the surface thickness is entirely governed by the interaction range. For example, with a saturating b-function interaction, quantum effects in HF or SPH give a finite surface thickness and energy, whereas TF gives zero for both. However, if the interaction range is sufficient to give the TF density distribution a reasonable surface thickness then, as is found in the present work, TF should be a good approximation. It may be mentioned that the interaction used here

I

2

0

1

2

3

4

5

6

I”

_/ / 0

I

I

I

I

2

4

6

6

10

r (fm) Fig. 4. 40Ca proton, zosPb neutron and proton potentials Y&) = aW/ap&r). dashed lines show TF and SPH results.

The full and

MIXED-DENSITY

APPROXIMATION

329

is rather short ranged, and it gives about 10 % too small surface thickness in all calculations (w 2 fm instead of 2.3 fm for “*Pb protons). Thus with a more realistic interaction TF may even be better. The SPH results are very close to HF and for many purposes this simpler calculation may be used to substitute HF. The large discrepancies between HF and SPH recently reported by Nemeth 14) are most likely due to the use of the Slater approximation. Single-particle Hamiltonians (27) and (34) include the rearrangement effect (31) approximately as a derivative of the strength of density-dependent repulsion (19) and in HF calculations the eigenvalues eNiare indeed in fair agreement with separation energies (tables 3 and 4). The effective single-particle Hamiltonian (27), on the other hand, can only be interpreted at the Fermi surface, the eNkFof (27) being exactly the separation energy for a nucleon at Fermi surface of infinite matter. In finite nuclei however the eigenvalues of (27) are M 1 MeV lower than the HF eNi of equation (34) (tables 3 and 4). Due to shell bunching the maximum kinetic energy in closed-shell nuclei is generally smaller than (;) times the mean kinetic energy which a local kF is expected to describe. Consequently the true exchange potential for nucleons of the last shell of a closed shell nucleus is about 1 MeV deeper than that computed from local kF. The single-particle energies computed with approximate mixed density (3) compare very well with exact results (tables 3 and 4). Binding energies with (30) are 332.26 and 1408.3 MeV respectively for 4oCo and *“Pb. There is no compelling reason to define k, for mixed-density calculation from the algebraic mean of PN(rl) and &r2) (eq. 12). It may be possible to improve this mixeddensity approximation by a better definition of kF. With an improved kF wave functions (11) could also be used to calculate the kinetic energy. It should be noted that in *“Pb the error in TF kinetic energy (table 1) is much larger than that in W. The author wishes to thank the Niels Bohr Institute for kind hospitality, and the Danish International Development Agency for financial support. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)

J. C. Slater, Phys. Rev. 81 (1951) 385 P. Siemens, Phys. Rev. Cl (1970) 98 V. R. Pandharipande, Nucl. Phys. A135 (1969) 419 V. R. Pandharipande and K. G. Prasad, Nucl. Phys. Al47 (1970) 193 R. G. Seyler and C. H. Blanchard, Phys. Rev. 131 (1963) 355 D. Vautherin and M. Veneroni, Phys. Lett. 25B (1967) 175 S. A. Moszkowski, Phys. Rev. C2 (1970) 402 J. 8. Bellicard and K. J. van Oostrum, Phys. Rev. Lett. 19 (1967) 2242 R. F. Frosch, R. Hofstadter, J. S. McCarthy, G. K. Noldeke, K. J. van Oostrum, M. R. Yearian, B. C. Clark, R. Herman and D. J. Ravenhall, Phys. Rev. 174 (1968) 1380 J. H. E. Mattauch, W. Thiele and A. H. Wapstra, Nucl. Phys. 67 (1965) 1 L. R. B. Elton and A. Swift, Nucl. Phys. A94 (1967) 52 A. N. James, P. T. Andrews, P. Kirkby and B. G. Lowe, Nucl. Phys. Al38 (1969) 145 E. Rost, Phys. Lett. 26B (1968) 184 J. Nemeth, Nucl. Phys. Al56 (1970) 183