Mean field approximation to the Wigner distribution function of atomic nuclei

ANNALS

OF PHYSICS

1%.

1-30

(1984)

Mean Field Approximation to the Wigner Distribution Function. of Atomic Nuclei J. MARTORELL Dept.

Fisica

Atomica

i Nuclear,

Unioersitat

Palma

de Mallorca.

Spain

AND E.MOYA Dept.

DE GUERRA

Fisica Atomica y Nuclear, Unioersidad de Extremadura, Received

December

Facultad de Ciencias. Badajor. Spain 12. 1983

The Wigner one body distribution function of atomic nuclei is studied in the mean field approximation. It is shown how to select the minimal number of independent variables required to characterize the Wigner transform depending on the symmetries obeyed by the system. For selected double magic nuclei the Wigner transform is computed for wavefunctions given by the density dependent Hartree-Fock method. Several approximations to this method based on the density matrix expansion (DME) are also studied and reformulated in terms of approximations to the Wigner transform. Their suitability in reproducing properties of this function is discussed. The case of Wigner transforms corresponding to harmonic oscillator wavefunctions is also discussed and analytic expressions for nuclei like 48Ca. “‘Zr and 20sPb are explicitly given using a completely general method derived for closed orbitals. In addition. two simpler models-that of completely filled harmonic oscillator shells and that of smoothed occupations of Sprung-Bhaduri-are also studied and used to give detailed comparisons of the accuracy of DME approximations. 6% 1984 Academic PESS, IW.

I.

INTRODUCTION

In the past years descriptions of the structure of the nucleus based on the ideas of the shell model, i.e., independent particle motion in an average field determined selfconsistently, have been extremely successful in reproducing separation and total binding energies as well as the distribution of the nucleons in r-space [ 11.The density dependent Hartree-Fock (DDHF) approximations [2-4] are the most remarkable examples of these approaches and its successes:not only overall features like size and deformations are well predicted but also the liner details of the shape of charge distributions are described. In these theories the one body density matrix plays a central role, and it is then satisfactory that its diagonal part appears to be in agreement with experiment. Much less is known experimentally however on the nondiagonal part, expected to be more sensitive to effects like two body correlations 0003.4916184

$7.50

Copynght ‘C 1984 by Academic Press. Inc All rights of reproduction in any form rcser\~rd.

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not included in the above models. Closely related to it is the momentum distribution of the nucleons, also poorly known, but for which heavy ion reactions [5] offer a possibility to obtain new information in addition to that already existent from (e, e’p) and similar reactions 161. It appears then useful to complement the study of the one body density matrix in r-space with that of its Wigner transform (explicitly defined in Eq. (2.1)) which provides the link to the momentum distribution. In addition the Wigner transform of the one body density is the starting point for a number of approximations to the nuclear many body problem. Two of these are particularly interesting: semiclassical methods [7,8], whose comparison with the results presented in this article will be made in a separate publication [9], and the density matrix expansion (DME) [lo] that we discuss in detail here. Explicit use of the Wigner transform provides a natural way to introduce the DME and discuss its suitability for different applications [ 111. It also allows us to address the closely related problem of the suitability of Skyrme effective interactions for the description of high energy dynamics. Following these ideas, in the present work we focus our attention on the Wigner transform as predicted from DDHF wavefunctions and discuss possible approximations to it, some based on the various versions of the DME, and others on the harmonic oscillator model. Comparison with these approximations allows us to give interpretations for the physically more relevant properties of this function. We restrict ourselves here to the study of static Wigner transforms, leaving for future work the extension of our study to the widely considered problem of the dynamics of heavy ion reactions [ 12-151, and to the simpler case of collective modes such as giant resonances [ 16, 171. In both cases the TDHF equations of motion admit formulations in terms of Wigner transforms which again appear as the natural tools to study properties related to the momentum distributions. Furthermore we consider mainly even-even spin saturated spherical nuclei. Some results for deformed nuclei will be discussed in a future publication [ 181. The plan of the article is as follows: In Section II, after defining the Wigner transform, we summarize briefly some of its properties and introduce its moments in p-space, as they are relevant for the discussion of approximations. We discuss then the restrictions imposed by the symmetries of the system on the number of independent variables needed to characterize the Wigner transform, and a theorem is given that shows how these should be chosen for the case of Slater determinants. In Section III we start by presenting a method to compute the exact Wigner transform for DDHF or any other single particle wavefunctions given in configuration space. We then study the Wigner transform for harmonic oscillator wavefunctions. We give explicit analytic expressions not only for nuclei with closed harmonic oscillator shells like I60 and 4oCa, but also for nuclei with only completely filled orbitals such as 48Ca, 90Zr and *“Pb. We give also the corresponding analytic expressions for the case of pure harmonic oscillator occupations with a sharp Fermi surface (closed oscillator shell) and for the model of smeared occupancies of Bhaduri and Sprung [ 191. After summarizing its most relevant features, we reformulate the DME in pspace and discuss the particular cases of the Negele-Vautherin [lo] and

WIGNER DISTRIBUTION

3

OF ATOMIC NUCLEI

Section IV contains a summary of our Campi-Bouyssy [ 111 approximations. numerical results for the DDHF wavefunctions given by the G - 0 force of Sprung and Banerjee [20] and comparison with the different DME approximations and the harmonic oscillator models. Also, using the analytic results of the harmonic oscillator model, we discuss the ability of the various versions of the DME to reproduce the exact moments, both for the case of a sharp Fermi surface and for the model of Bhaduri and Sprung [ 191. Section V summarizes our conclusions.

II. PROPERTIES

OF WIGNER

TRANSFORMS

II. 1. General Definitions The definition

used here for the Wigner transform of the one body density is [8] f(R, p) = 1 ds e-‘P.‘p(R + s/2, R - s/2)

(2.1)

where, in terms of the one body density matrix [8],

and R = (r, + r,)/2, s = rl - r2. Units of h = c = 1 are used everywhere. As is well known, f is not directly accessible to experiment, but some of its integrals are. The simplest example is the momentum distribution, which is given by integration off over the R variable. Other examples of weighted integrals off on the same R variable are discussed in Ref. [5]. Similarly, integration over p leads to distributions in rspace. The momentum weighted integrals are called moments [ 111 1 -(2x)3 1 dp(p)“f(R,

Mn=

P).

(2.3)

Using the notation W”

P(R)

=

K-V,

-

WW

p(rl

1 rdl

r, =

q=

R

(2.4 1

they can be simply rewritten as

M, = @InP(R).

(2.51

The first of these moments have simple physical interpretations Mo = P(R) Ml =j(R)

(2.6)

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i.e., the r-space density and current. Furthermore M, = z(R) - $“p(R)

(2.7)

where t is the kinetic energy density, defined as

r(R) = [(VI . V2)p(r,, f2)lr,=Q=R.

P9

When f is an even function of p, which is usually the case for static Wigner transforms, the odd moments vanish and f is characterized only by the even M,. As done for M, one proceeds for the others, i.e., M, = w(R) - $V25(R) + &(V’)’

p(R)

(2.9)

with 4R)

(2.10)

= 105 . V2)2~h~r2)l,l=.=R

and so on. It will be shown in Section IV that the moments play a key role in the approximations to the one body density matrix generated from the DME. Although in principle f is a function of R and p, a total of six independent variables, in the cases usually found in practice this number can be reduced. In particular f must be a scalar when the nucleus has total spin zero and then only scalar combinations of the R, p variables can appear. The simplest independent combinations are R2, p2 and R . p, therefore there are only three independent variables. The requirement of time reversal invariance imposes in addition that only even powers of R Bp appear. 11.2. Wigner Transforms of Slater Determinant In this case the one body density can be explicitly particle wavefunctions, cpi, of the nucleons

Wavefunctions

written in terms of the single

(2.11) and similarly the moments and related quantities introduced previously admit simple expressions in terms of the vi and their derivatives. For even-even J= 0 nuclei with each orbital and its time reversed occupied (i.e., j, m and j, -m inj -j coupling), the conditions discussed in the previous paragraph hold and the number of independent variables for f is reduced to three (similarly p(r,, r2) depends also on the three independent variables: R2, s2 and (R . s)). Furthermore, as it is well known, the properties of Slater determinants P2=P

trace p = A [h,pl = 0

(2.12)

WIGNER DISTRIBUTION

3

OF ATOMIC NUCLEI

with h the single particle hamiltonian, lead to the following properties off(R. f(R

P) =f(R,

p) expli@)f(R

(2:)’ !’

do

P) =fVk

I

. ~P./-R

P) cd%‘2VR

PI

P) = A

P) (2.13a) (2. I3b)

p) = 0.

(2.13c)

AZB = A”BAB = (Vi . V; - V; V;) AB

(2.14)

{ {K,f}} = 2&(R, p) sin z/Zf(R. Here and in what follows we use the notation (8 ]

for A and B any pair of functions of R and p. Furthermore we denote with a tilde. P(R, p), the Wigner transform of any given single particle operator fc’(r,, r?). In analogy to Eq. (2.1) 1 /PI 2r*) = -(2*)3 i dpdR/“(R,p)e-

ip”rlp ‘z’d(R - (r, + rJ2).

(2.15)

This allows us to generalize relation (2.13~) to any pair of commuting single particle operators, since for any pair of operators y;. r, the Wigner transform of their commutator is given by (2.16) where we use the notation introduced in Eqs. (2.13~) and (2. 14)

({A, B}} = 2iA(sini/2)

B = 2i (sin /1 IA/2 )AB.

(2.17)

The above relations can be used to prove the following THEOREM. Let p and h be the static one bo& densitv mark and single particle hamiltonian of a system of A independent fermions, and let {/‘i} be the set of hermitian single particle, commuting operators associated with the symmetries of the sq’stem. Define S = (F,] as the complete set of commuting single particle operators including h. Then the Wigner transform of p, f (R, p), depends on R and p only through its dependence on the Wigner transforms of the operators belonging to the set S.

The proof of this theorem is given in Appendix I. A corollary that follows from this theorem is that in the case we wish to study here (spin saturated, time reversal invariant, spherically symmetric ground states) if the single particle hamiltonian (h) is that of a harmonic oscillator, f (R, p) will depend only on &R, p) and ll(R, p) (the Wigner transforms of the single particle hamiltonian and orbital angular momentum):

f CR,PI

= cd& p I.

(2.18)

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Later we shall make use of this property to derive an analytical expression of the Wigner transform corresponding to a closed harmonic oscillator orbital. In the case of the harmonic oscillator the proof of the above theorem becomes particularly simple because the “commutators” ( (&,f}} reduce to simple Poisson brackets. We should also stress that the above corollary follows from the very specific properties of the harmonic oscillator potential and does not necessarily stand for a general potential. To understand this consider for instance the operator @a= h2 that belongs to the above defined set 5’. The Wigner transform of @, is

whose R and p dependence is not necessarily equal to that of (&*. To be more specific consider the case h”=p2/2m + g(R) then f;’ = (6)’ - (Vi g(R))/4m. If g(R) is a harmonic oscillator potential the second term in the above equation is just a constant and the R and p dependence of /i2 is that of (hr2, while if g(R) is, for instance, a coulomb potential the second term in that equation is a Dirac delta function of R. Analogous considerations apply to other possible operators belonging to the set S that may be constructed from the primary operators h, b.

III. EXPRESSIONS OF THE WIGNER TRANSFORMS IN DIFFERENT APPROXIMATIONS

In this section we compile the expressions of Wigner transforms used for the computation of the numerical results presented and discussed in Section IV. First (Subsection III. 1) we derive a formula for the Wigner transform of the exact one body density matrix corresponding to DDHF wavefunctions. In Subsection III.2 the harmonic oscillator (H.O.) approximation is considered, and analytical expressions of the Wigner transforms for closed H.O. orbitals are given as functions of h”(H.0.) and p. Finally, in Subsection 111.3, the Wigner transform of the formal density matrix expansion (DME) of Negele and Vautherin [lOI is presented and discussed together with the truncated expansions proposed by Negele and Vautherin [ 101 and by Campi and Bouyssy [ 111. III. 1. Density Dependent Hartree-Fock

Here we choose to calculate the matrix obtained from DDHF, or in single particle states in a harmonic et al. [21] have developed an configuration space). We write

(DDHF)

Wigner transform of the exact one body density general from any potential well, by expanding the oscillator basis (we notice that recently Prakash alternative method which works directly in

Ii>DDHF=~n,‘,~iml)=Ca~~~,,In~~i~imi)H.D.’

h'

(3.1)

WIGNER

DISTRIBUTION

OF ATOMIC

NUCLEI

In terms of the (real) overlaps az;jiji the density matrix defined in Eq. (2.11) has the form

with (3.3)

and v1fiy;‘)(r)

= (2n!/r(n

+ I + !)] “* rre-‘Z’ZL~+ “‘(I-~) Y;(t)

(3.4)

where r, L,, Yf, are the usual gamma functions, Laguerre polynomials and spherical harmonics, respectively, and r is in units of the oscillator length parameter a, = [ l/Mw,]“*. With this expansion the Wigner transform can be obtained extending the method used in Ref. [ 141 for pure H.O. wavefunctions: first the Talmi transformation allows us to write

where the transformation brackets are defined following Brody and Moshinsky In writing Eq. (3.5) we used the fact that --P (-1)” m

Then the calculation of the transforms

(I-

mh~L0)

= (-1)’ Ju+l

s,,,.

[ 22 1.

(3.6)

of the Wigner transform of the density reduces to the calculation

AN,,,&

P> = 1 ds eeip’ ‘-4,1.,,(~R

s/&j.

(3.7)

These can be easily worked out. Use of Rayleigh’s expansion for the exponential, orthogonality of the spherical harmonics and the relation [7] I omdt ecf’2L~(t) t”“Ja(G)

= (-1)” xaf2emX”L;(x)

(3.8)

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leads to the result

xNl,JR, p) = &‘/*(-])“+W

n!N! ;)T(N+Z+

e-(a+b)/2 T(n+Z+

x (21+ l)(ab)“*

L; i’*(u) L;

l/2 1) I

l”(b) P,(cos 6)

(3.9)

with f? the angle between R and p, and a, b defined as a = 2p2ai,

b = 2R2/ai.

(3.10)

Using the result (3.9), together with the symmetry properties of the transformation brackets 122 ] and the energy conservation selection rule n + N + I = n, + n2 + I,, we finally write the Wigner distribution function as

X (1 - +&NjNn)(-l)‘~+n+“2 (,,)r’2 112 (nZ, NZ, O)n,l;, n,Z,, O)P,(cos

n! N!(2Z + 1) r(n+Z+;>T(N+Z+;>

x ILf:

“*(a) Lg l”(b)

6)

3 + (-,)“I+“,

L;

1’2(b) Lc “‘(a)\.

(3.1 I)

This expression generalizes Eq. (28) of Ref. [ 141 to the case of arbitrary single particle wavefunctions expanded in a H.O. basis, and in particular it can also be used for the wavefunctions given by a Woods-Saxon potential [9]. We notice that as expected f depends only on R2, p2 and cos’ 0 and therefore we can restrict the calculation to values of 0 in the interval [O,lr/2]. The numerical computation of fnnnr is then performed in the following steps: (i) A value of the H.O. parameter, a,, is chosen, currently the prescription a, =A ‘16fm. is used. Then the DDHF single particle wavefunctions are expanded in the H.O. basis and the overlaps are determined. The results presented in Section IV correspond to an expansion with 10 H.O. functions for each state, which is enough to guarantee the stability of the Wigner transform against an increase of the basis. We have also checked that varying the chosen a, by amounts of 0.1 fm does not change signiticantly the final numerical results. (ii) The required Moshinsky-Brody coefficients are obtained using the formula given by Trlifaj [23]; which allows a very fast computation. Still this is the more time-consuming part of the calculation. After checking that their contribution was not significant, we set to zero the Moshinsky-Brody coefficients with I> 14, to save computation time. (iii) Finally, for selected values of R, p and 0 the values of the Laguerre and Legendre polynomials are computed numerically with standard routines, and the value off is obtained. Results for i60, 40Ca and 208Pb are presented and discussed in Section IV.

WIGNER

DISTRIBUTION

OF ATOMIC

111.2. Harmonic 111.2.1. Harmonic

Oscillator

with Orbital

9

NUCLEI

Oscillator

Occupations

Determined from DDHF

What we call harmonic oscillator with orbital occupations determined from DDHF is the harmonic oscillator approximation to the DDHF Wigner transform: i.e., the density matrix is constructed as the sum over the set (i) = {ni, ii, j,} of DDHF filled orbitals with the single particle wavefunctions corresponding to a harmonic oscillator potential. Therefore the density matrix is a sum over the filled j-subshells of the quanrz), and the relevant quantities to consider are the tities (2j + 1)/(2I + 1) AnlJrl, Wigner transforms of the closed orbitals defined as

f,,,,(R,

According

p) =i

ds ePip”-4,, ,,,. .,,,(R + s/2. R - s/2).

to the results in Subsection

(3.12)

111.1 these are given by

(nl,Nl,Oln,l,,n,I,,O)~~~,.,,(R.p)

(3.13)

and from the definition of &,,JR, p), seeEq. (3.9), the symmetry properties of the Moshinsky-Brody coefficients and the energy conservation selection rule N + n + I = 2n, + 1, (which implies that N + n -I, = even) one recovers the well known property that for a closed H.O. orbital the Wigner distribution function is completely symmetrical in R and p (note that this is not the case for the DDHF Wigner transform, Eq. (3.11)). This is consistent with the fact that, as shown in Section II, this Wigner transform should only depend on I? = (R x p)’ and 6. which is proportional to R2 +p2 in this case. Since the Wigner distribution function is dimensionless, it is more convenient to use the variables e = h, = a + b = 4@w,, d = ab sin2 8 = 41?

(3.14)

with a and b the dimensionlessvariables defined in Eq. (3.10). We are going to take advantage of these properties to derive an explicit analytic expression off,,(R, p) in terms of these variables. To this end, we first take the angle average on Eq. (3.13). after which only the much simpler I= 0 Moshinsky-Brody brackets remain and we are left with a symmetric function of a and b. Then we rewrite this function as a function of a + b and ab from which we can construct f(R, p) as a function of e and d.

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The angle average of Eq. (3.13) leads to (see Eq. (3.9)) fnll,(R, p) = 4n’~Ze-~“tb~~2 ;

(-l)~‘+n(nO,NO,OJn,I,,n,z,,O)

112 n! N! (21, + 1) L;‘*(a) L;*(b). r(n + 3/2) r(N + 3/2) I

’

(3.15)

Since Eq. (3.15) is completely symmetric under interchange of a and b we can always write it as a function of a + b and ab. To find the coefficients of this new function we make use of the explicit form of the Laguerre polynomials and Moshinsky-Brody coefficients [ 231

Ly2(x)= ,g, (-1Y (“,yf)

f

(3.16)

(nO,NO,Oln,~,,n,~,,O) = (-1)v \/; X n,!

(1/2)‘1+’ (-I)‘]

r(n, + I, + 3/2)

n! (21, + 1) N! r(n t 3/2) T(N + 3/2)

112 6 n+N,K x

1

*I*2

W/2)“+‘*

(I, + t, t t2)! r(l, t t, + t, t 3/2) ’ t,! t,! (n, - tl)! (n, - t2)! (I, + t, t t, -N)! Z-(tl + I, t 3/2) Z-(f2 t I, + 3/2)

(3.17) with K=2n, + 1,. Use of Eqs. (3.16), (3.17) allows us to writeTn,I:,l,l (I?, p) as the following function of a t b and ab fnlr,(R,p)

= (-l)K 8(1/2)‘l (211 + 1) e-‘a+b”2 K IW-m)/Zl a,,,,(m s)(a + b)” WY xc c m=o s=o

(3.18)

where the numerical coefficients aK,“, (m, S) are given by the recurrence relation K-Ii-S a,,,,(m,

s) =

2

N=S

PZ,n,(m

+ s, s, -

2

p=1

a,,,,(m

t 2P, f3-P)

i

t:T

1

(3.19)

with (-l)” PE,n,(m + sv s) =

(3

(3

2m+2s c

(2(m + s) t l)!! (2s + l)!! 1, t t,

I(

(-l,2)t

t,.t2

t

t, + l/2 t2

I +* 2 (“‘)

t, I1 + r; + ‘*) (3.20)

WIGNER

In the particular

DISTRIBUTION

ATOMIC

11

NUCLEI

case n, = 0, Eq. (3.20) has the much simpler form

P;.n,=ocm +.%s>= Furthermore

OF

(--llrn(1)(z,:j(;j (2(m+s)+

2m+2s

1)!!(2s+

l)!!

.

(3.21)

using the relation

we can write Eq. (3.19) for the case n, = 0 as (-1)“2”

,“,, x-T+) t I( (2(m + s) + l)!! (2s + l)!!

a K,nl=o(m? s) =

-

<-

m + 2p

PTl t m+p

1

a,&

+ 2p, s -p).

(3.23)

With Eq. (3.19) (or Eq. (3.23)) the computation of the coefficients a,+,,(m, S) in Eq. (3.18) is rather simple. For given K, n, values (or equivalently n, , I, values) one first computes the coefficients a,,,,(m, 0) for all m from 0 to K; once these are known one can calculate the coefficients atiqn, (m, 1) for all m from 0 to K - 2, and so on. Finally the desired expression for f,,,, (R, p) can be obtained exploiting the fact that this must be a function of h, = e and /i = d (see Eq. (3.14)). Noting that by definition &,(RP)=

l/2 j+‘f,,,,(R,p)d(cos ‘-1

0)

(3.24)

q!2Y (2q + I)!!

(3.25)

and that l/2 .il

1

e’dqd(cos 0) = (a + b)’ (cz~)~

we find that the one body Wigner distribution be given by

function for a H.O. closed orbital must

f,,r,@~ P) = g,,r,(e, d) = (--I)”

8(1/2)‘1

(2E, + 1) exp(-e/2)

1 ItIT-”

K‘ s- = 0

~~,~,(m, S) em&

(3.26)

with

I&,@% s>= (2s;s’l)!! %,,,(Ws).

(3.27)

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In particular, from Eq. (3.26) one can also derive the well known expression for the Wigner distribution function of a closed harmonic oscillator shell K [K/21

SK@, P)=

C

P)= (-I)”

fnl,,,=~-2n,(k

n,=o

8e-ho’2Li(h0).

(3.28)

Equation (3.26) is our final analytic result for fn,l,(R, p); i.e., an exponential in ho times a polynomial in both e = ho and d. This result is very useful in practice, and although the previous equations allow a very easy computation of the coefficients for any orbital, we give explicitly in Appendix II the expressions for selected doubly magic nuclei that follow from these: ?Ja 90Zr and 2osPb. The expression for fnll,(R, p) has been’checked with our computer program used for the general evaluation of Wigner transforms. This provides an additional contirmation of the theorem stated in Section II. 111.2.2. Pure (Bhaduri-Sprung

Harmonic Model)

Oscillator

with

Sharp

and

Smeared

Occupations

Within the H.O. model there are two particularly simple cases which can be easily solved and are particularly interesting to illustrate the discussion on the density matrix expansions and on the origin of the structure of the Wigner transform. These are the standard case of a pure H.O. with sharp Fermi surface (s.F.), with the nucleons fully occupying the lowest N H.O. shells, and the model of smeared occupancies (so.) considered by Bhaduri and Sprung ] 191. According to the previous results, in both cases the Wigner transform will depend only on ho. With equal numbers of neutrons and protons these two model density matrices are respectively given by &.F(rl,r2)=4

N

[K/21

2 K=O

c n=o

(3.29)

A,l,n,hr2)Bl.K-2n

and (3.30)

where the A functions are as defined in Eq. (3.3). The occupation probability parameter, t, and the number of shells N in Eqs. (3.29), (3.30) are fixed by the condition that j p(R) dR = number of nucleons, i.e., 2(N + l)(N + 2)(N + 3)/3 and 4(1 - t)-3, respectively. We note that a somewhat more realistic model would be one where the smearing is governed by a Strutinsky type of function, which is, however, harder to handle [24]. The Wigner transforms of the two model densities are easily calculated with the help of the previous results. From Eq. (3.28) we get f,.,.(R,p)

= 32e-ho’2 2 K=O

(-l)K

Li(h,)

(3.3 1)

WIGNER

which and

is a polynomial

DISTRIBUTION

OF ATOMIC

of degree N in h, modulated

f,,,.(R,p)

= 32ech0j2 E

13

NUCLEI

by the decreasing

(-t)”

L&J

X=0

32

= (1

(3.32)

-(1/2ho(l

+q

exponential,

-tJ/(t

ff))

e

which is a pure exponential in ho. The expression for the density matrix, pS.O, (R. s) given in Ref. [ 191 can be obtained taking the inverse transform of Eq. (3.32) p,.,(R, s) = p(R) e-(4’4’[s’0(112

(3.33)

with

P(R)‘~

4 0

exp~-Wao)2/P~ (1 - ty2 (3.34)

p=-..- 1+t 1 -t’ III.3 Density Matrix Density

matrix expansions

Expansions

are best discussed

starting from the identity

&R + s/2, R - s/2) = eiS’ @p(R)

(3.35)

with the operator 6 = -i/2(V, - V,) defined in Eq. (2.4). In what follows we shall use for brevity the notation p(R, s) for P(R + s/2, R - s/2). Following Negele and Vautherin [ 101 we average over the s direction and approximate p(R, s) by p(R, s) = $, Furthermore

dQ,e’“‘$$R)

=j,(sj)p(R).

(3.36)

they use the identity jo(ab) = g

(4n + 3)(-l)”

fl=O

valid for any k such that -1
j2,+dak> ak

P2,+SW) b/k

(3.37)

< 1, to write Eq. (3.36) in the form

p(R, s) = 5 (4n + 3)(-1)” n=o

j2n;;r(sk)

Pzn+

,Wk) Clk

P(R).

(3.38)

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DE GUERRA

A difticulty that is immediately seen in this formal expansion is that the condition -1
where the ai are the coefficients of the Legendre polynomial, P,,+,(x) = x’(‘-~‘+ ‘, we see that the equivalent to the condition -1 < b/k < 1 would (3.40a)

IW@k2)l G 1 M,,lP

n>2

= W*/P)",

(3.40b)

so that even if condition (3.40a) is imposed, condition (3.40b) will in general fail. An exact expansion of (3.36) would be

(-1)” p(R,s) = ‘? ,eo (2n + l)! S2nM2n*

(3.41)

The problem with this expansion is that it cannot be truncated for large s values if all even moments are different from zero. At this point, the next approximation of Negele and Vautherin consists in keeping the first two terms of expansion (3.38) and fix k = k, (the Fermi momentum corresponding to the local density). Alternatively, the so called local Fermi momentum approximation of Campi and Bouyssy consists in keeping only the first term of Eq. (3.38) but with k fixed by the condition that the second term be identically zero. To avoid confusion we shall denote by pDMEthe density in Eq. (3.38), by pNV the truncated density of Negele and Vautherin and by pee that of Campi and Bouyssy: more explicitly P~(R,~)=

3jlWF) sk

F

dR)

-

V3WF) 5 9 zsk F

- 3p(R) F

1

(3.42)

with k, = k,(R) = [3n*p(R)/2]

1’3

and ~ce(R,s)=3

.il (4 -yg-P(R)

(3.43)

WIGNER DISTRIBUTION

15

OF ATOMIC NUCLEI

with

k”=k”(R)= L M2(R) [ 3 ,,i-I”‘.

(3.44)

To order s2 both approximations coincide with the exact expansion (3.41), but are quite different to higher order in s. Still we see that since, in the approximation of Campi and Bouyssy, k is fixed so that condition (3.40a) is satisfied it would be a better approximation to the exact expansion in the particular case that condition (3.40b) was approximately satisfied. The Wigner transform of the density matrices in Eqs. (3.38), (3.41) and (3.43) is easily obtained by using the relation e(k - PJ

(3.45)

Using definition (2.1) and the above relation we get for the Wigner transform of pDME, pNv and pcs the results j-i,&&

P>= g

.&(R,P)=$[ fca(R,

PI =$

T (4n + 3) “f0

P*n+l(Plk)

Plk

P*n+lwk)

p(R)(j(k-p)

b/k

,,++(5$-3)(5$-3p)]

O(kp-p)

(3.46)

. 3p&-p).

These Wigner transforms are therefore approximations to the exact Wigner distribution function that have a discontinuity at a given p value. The Campi-Bouyssy distribution is a pure step function in phase space, different for different R values, while the NV distribution is a first degree polynomial in (p/k,.)’ for p values inside the step function (p < k,(R)). For a distribution to approximate a given function it is usually required that the moments of the distribution are the sameas those of the function and it is along these lines that. in the next section, comparison of the f s given by the DME with fnntlF will be made. IV. RESULTS AND COMPARISON

OF DIFFERENT

APPROACHES

IV. 1. The Wigner Transform as a Function of R and p IV. 1.1. The DDHF Wigner Transform As already discussed in Section II, f depends on three independent variables. To present our results we have chosen for these R, p and 8, the angle between R and p.

16

MARTORELL

AND

MOYA

DE GUERRA

FIG. 1. The Wigner transform fDDHF of ‘“0 at fixed R and 8. Continuous line: 0 = 0, dashed line: B = 42. Values of R = 1, 2 and 3 fm. For comparison the local Thomas-Fermi approximation is also shown. The momenta p are in fin-‘.

Since we want to stress the p dependence we draw always f as a function of p for fixed values of R and 8. For simplicitly, we show only plots for the two extreme values 13= 0 and 742 (see sub-section 11.2), although it should be stressed that the variation off with 8 for fixed R and p is not always monotonic, and sizeable oscillations might occur for heavy nuclei. Also for simplicity we have chosen only and will present results three representative nuclei: ‘60, 40Ca and *“Pb corresponding only to the G - 0 interaction of Sprung-Banerjee [20]. As already discussed, the method is equally applicable to any other Slater determinant wavefunctions and a systematic survey of results for more nuclei and different types of interactions used in Hartree-Fock calculations will be presented in a separate internal report. We comment here only that the Wigner transforms for different effective interactions are very similar, and their differences not significant at the level of the present discussion. We performed also two types of Hartree-Fock calculations with the G - 0 interaction: complete DDHF calculations as described in Ref. [3] and Hartree-Fock calculations, as described in Ref. [25], with the exchange terms in the potential approximated using the DME of Negele-Vautherin. We found the transforms in the two cases to be so similar that on the scale of the present figures the differences between the Wigner transforms were practically negligible. In Figs. 1, 2 and 3 we show the results for the nuclei 160, 40Ca and *08Pb. For

FIG. 2. Same as Fig. 1 for “‘?a.

WIGNER

FIG.

3.

DISTRIBUTION

OF ATOMIC

Same as Fig. 1 for 208Pb. The values

17

NUCLEI

of R are 1, 3 and 5 fm.

oxygen and calcium the differences between 0 = 0 and n/2 are rather small, whereas for 208Pb they are quite noticeable (this is easily understood on the basis of the H.O. model with DDHF occupations in which 160 and 40Ca are closed shells, whereas ‘08Pb has only closed orbitals). The results presented show explicitly the anisotropy off: Notice, however, that this anisotropy is small around the Fermi surface, and that even in the interior the overall shape off is the same for both angles. This supports the first of the two approximations leading to DME, which leads to a Wigner transform independent of the angle between R and p. On the same figures, for comparison, for each R the Thomas-Fermi distribution corresponding to the local density (p(R)) is also shown. Clearly the Thomas-Fermi approximation is very poor as a description of the “local momentum” distribution. Figure 4, where p*f(R, p) is presented, makes it clear, however, that what is really missing in Thomas-Fermi is a more appropriate description around the local Fermi level (the surface in p space), while on the average the behaviour at lower p (the interior in p space) is satisfactorily given. It is well known that the surface diffuseness off is a finite size effect (in nuclear matter the Thomas-Fermi f is exact at the Hartree-Fock level), and it is also well known that the internal oscillations are a manifestation of shell effects. Indeed as shown in Ref. (24) for the case of the harmonic oscillator, smeared occupancies generated via a Strutinsky smoothing reduce to a great extent the internal oscillations; see Fig. 1 of Ref. [24]. In Fig. 5 we show the I60 results for the Sprung-Bhaduri model of smeared occupancies (which correspond to a much more

L&=Ec!kt. P

FIG. 4. The *‘*Pb DDHF Wigner transform and 5 fm. For comparison the local Thomas-Fermi

1

P

times p2 for fixed R and 0 = 0. The values approximation is also shown.

1

P

of R are 1. 3

18

MARTORELL

AND MOYA DE GUERRA

FIG. 5. The Wigner transform for I60 with occupancies given by the model of Bhaduri and Sprung.

drastic smoothing) to further illustrate this point. As Eq. (3.32) shows, essentially the same purely gaussian shape and lack of oscillations will be obtained for any other number of nucleons in this model. In heavy nuclei these shell effects are somewhat less pronounced due to the interferences between the contributions corresponding to protons and to neutrons. To visualize this fact in Fig. 6 we show separately the proton and neutron contributions for “‘Pb. The smoothing in the sum can be viewed again as due to a smearing of occupations: Some of the levels occupied for neutrons are empty for protons. These effects, and the structure of the oscillations, are further clarified when comparison is made with the Wigner transforms of the H.O. model with DDHF occupations. On the contrary they are, by their very nature, impossible to describe in semiclassical approaches which cannot account for shell effects. IV.1.2. Comparison with the H.O. Model with DDHF Occupations As shown in Subsection 111.2, for the H.O. model the Wigner transforms are analytic and can be obtained very easily. To compare with DDHF, the H.O. parameter a, = v’has to be fixed. We choose to determine its value by

FIG. 6. The *“Pb DDHF Wigner transforms for neutrons (continuous line) and protons (dashed line) plotted for 0 = 0 as a function of p. R = 1 fm.

WIGNER

DISTRIBUTION

FIG. 7. The I60 Wigner transform the harmonic oscillator (dashed line).

OF ATOMIC

19

NUCLEI

at 0 = 0 and R = 1. 2 and 3 fm. DDHF

(continuous

line)

versus

imposing that the H.O. r.m.s. matter radius be equal to the DDHF r.m.s. radius for the nucleus considered. The results are presented in Figs. 7 to 9. Not unexpectedly in the two light nuclei the agreement with DDHF is extremely good, while for “*Pb the differences are more apparent, specially around the Fermi surface. It is remarkable. however, the very good description of the internal oscillations given by the H.O. Wigner transform. This clearly shows that these are mostly determined by bulk properties of the single particle potential, essentially the geometrical size of the well. It is important, however, to keep in mind that these small differences between the two Wigner transforms are responsible for the much more apparent differences in the integral properties: the moments. In particular the density p(R), the kinetic energy density t(R), etc. The comparison between the two models can be made in a more global way using the fact that, at 0 = 0, fH.o. depends on R and p only through its dependence on ho = 2(p2ai + R2/ai). In Fig. 10 the 208Pb DDHF Wigner transform for fixed values of R from 1 to 8 fm is plotted as function of ho. For pure H.O. wavefunctions all lines would be superimposed in a single one. Again it is seen that although there are differences in the detail, on the qualitative level the overall behaviour is very much like that of a harmonic oscillator. Confirming the previous discussion, Fig. 10 also makes clear that the shapes and in

FIG.

8.

Same as Fig. 7 for ?a.

20

MARTORELL

AND MOYA DE GUERRA

FIG. 9. Same as Fig. 7 for 20sPb. The values of R are 1, 3 and 5 fm.

particular the oscillations off as a function of p for fixed R shown in Figs. l-3 are not due to “local” effects (i.e., behaviour of the potential around that R), but that they have the same origin for all R, and are essentially the same when h, is used instead of p. Use of the explicit analytic expressions of the H.O. Wigner transforms allows us to give a mathematical interpretation of the observed oscillations. As Eq. (3.26) shows, when t9= 0, f is a product of a polynomial in e = h, of degree K,,, and an exponential. Real zeroes of the polynomial will lead to oscillations off with change of sign, while complex zeroes will correspond to oscillations where the axis is not reached and f does not change sign. The exponential term is responsible for the damping at high p, whereas the scale on the x-axis is fixed by a, and thus by the r.m.s. radius. Since h, is an even function of p, when f is plotted as a function of momentum the number of oscillations will also be given by the degree of the polynomial in II,,. Therefore in *08Pb, for protons (lh,,,z) K,,, = 5 and for neutrons (lh2) Knax = 6. This is in agreement with what is shown in Fig. 9 except for the fact that the first oscillation is so close to the origin that since the numerical calculations shown start at R = 1 fm., leading to an already too high value of ho, they miss that oscillation completely.

FIG. 10. The “‘Pb DDHF Wigner transforms for neutrons and protons plotted for 19= 0 as a function of h, (Eq. (3.14)), with a, = 2.45 fm.

WIGNER

DISTRIBUTION

OF ATOMIC

21

NUCLEI

IV. 1.3. Comparison with the DME Although we have already stressed that fDME is a distribution that goes to zero discontinuously (Eq. (3.46)), and thus the comparison withf,,,, is only meaningful when made for the moments, as will be done later, it is still interesting to see whether for smaller p the shape off resembles that of the DDHF Wigner transform. The answer is negative for both the Negele-Vautherin and the Campi-Bouyssy approximations. For the latter case this is obvious since by construction it gives an f. Eq. (3.46), which is just proportional to the step function. For the Negele-Vautherin case it is also apparent that the quadratic dependence on p introduced by the additional term in the expansion is not of much use in fitting the shape of the actual Wigner transform. The importance of the higher order terms in the DME expansion can be better studied in the simple model case of I60 described by H.O. wavefunctions. As will be shown in the next subsection the moments are analytic, and it is very easy then to include, via Eq. (3.46) truncated to the appropriate order n = nmax3 as many terms in the DME expansion as desired. In Fig. 1la the results for fDME (Thomas-Fermi), fDME (Negele-Vautherin) and fDME (nmax = 2) are compared to the exact value. In Fig. 1 lb on a much reduced scale are shown the fDME for n msx= 3 and 4. Clearly the trend is to have more and more oscillations with bigger and bigger amplitudes. Intuitively the reason for this is clear when use is made of the -

ra

i

t

!I’

,

,

P

/

P

j

FIG. 11. The I60 Wigner transform for harmonic osccillator wavefunctions (a, = 1.77 I fm). Continuous line: exact result. DME approximation: (a) dashed line: nmax = 1, dash dotted line: rzma, = 2. (b) dashed line: nmar = 3. dash dotted line: nmax = 4. Notice the different scales (factor 10) in (a) and (b).

22

MARTORELL

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DE GUERRA

result to be proved in the next section, that the first 2rrmsX moments of the truncated (a) the degree of the DME polynomial f DME are equal to the exact moments off,,,,: (Eq. (3.46)) is increasing as 2n,BX, and so the number of possible maxima and minima increases too; (b) the zeroth moment is just the sum of the areas (with their signs) of the positive and the negative oscillations, so that these must balance each other; (c) the higher nmax, the higher the number of moments that in addition will have to agree with the DDHF ones. Since fDDHF has a nonvanishing tail beyond the Fermi level, the higher moments are determined by this behaviour. The only way for f,,, to reproduce these is to take very high values for p just below k. Therefore (b) + (c) explain the overall trend of the oscillations. It is then clear that it is the tail of fDDHF, which by definition fDME cannot give for finite k, that determines also the behaviour of the oscillations off,,,. Therefore there is no way to have the same shape as fDDHF. Indeed it is clear that when the number of forfDhW terms in the DME expansion tends to infinity, for finite k, f is not converging towards a continuous function even in the interval below the truncation point of p. We have indeed tried to see if better agreement with DDHF could be obtained by increasing the value of k in Eq. (3.46). As a check we took k values of 3k, and lOk,. The shapes off,,, look quite different in each case, but the disagreement with f,,,, is as bad as with the original prescription k = kF. Finally we would like to comment on another consequence of the vanishing of f,,, for p higher than k,: Negele and Vautherin have provided a justification for the use of the well known Skyrme interaction. They showed that in the static case (nuclear ground state) the Skyrme force leads to an energy density functional that can also be obtained by simplifying that given by their particular choice of pDME. This is a situation where high momenta are not very important, as shown by Fig. 4, and thus using an f which ignores them is reasonably justified. It is an open question, however, whether in dynamical processes like heavy ion reactions, where high momentum components are much more important, it is still a sensible approximation, to use Skyrme-like forces. IV.2. Moments of the Wigner Transform in DSfferent Approaches It can be easily proved that the moments of the DME distribution are those of the exact f (R,p) for any arbitrary positive k. This can be verified by means of the integral (2L + I)!! A! dx = 2” (A - n)! (2(A + n) + 3)!! from which, calling MyAME the 2L moment of the DME distribution &fyAME= k2a &-p(R)+ P(R) 3/k

2L(21-2)...(2J-2n+2) (U + 3)(2L + 5) s-s (U + 2n + 3)

i: (4n+3) II=1

x P2n+l(B/k)

1

=B2A~W

we get

=

M,,(R).

(4.2)

WIGNER

DISTRIBUTION

OF ATOMIC

NUCLEI

23

This is a very interesting result for it not only shows that the moments of the DME distribution are the same as the exact moments, but also the way one should truncate to get a given number of moments correctly. Truncation of fDME, Eq. (3.46), to a given order n =li guarantees that all U-moments, with 2 < li, of the corresponding distribution in phase space coincide with the moments of the exact distribution function, independently of the choice of the k value. One can also see from Eq. (3.46) that the other alternative to get the lowest U exact moments (with 1 G/i) would be to truncate to order /i - 1 fixing k so as to make the /i term zero. These are the two different alternatives taken by Negele and Vautherin and by Campi and Bouyssy, respectively, for the case A = 1. Any other even moments 2A > 2 of the NV and CB distributions have indeed little to do with the exact moments. However it should be mentioned that (as said before) since the U moments of the CB distribution are given by

2a

I 1

3 -5M, 2A + 3 3p

MCB = ~

.’

p

in the particular case that condition (3.40b) would be approximately satisfied one would have

MCB = 21

M2,I

*

(4.4)

so that the CB distribution would give a better approximation to the exact higher order moments than the DME. An improvement on the above approaches would be to keep the first two terms in Eq. (3.46) fixing k so as to make (P,(#/k) k/p^)p(R) = 0. This would lead to exact 21 moments for A< 2. This condition, however, is much harder to implement in practical calculations and furthermore one can find examples (e.g., 160 with harmonic oscillator wavefunctions) where for large R this equation is satisfied only by complex k, which makes it meaningless. The points we have been discussing are further illustrated using the two models introduced in Subsection 111.3.2.Let us first consider the case of smearedoccupancy. In this case the moments are given by n P(R)

(4.5)

which satisfy M,,/p

= (M2/p)”

(2n ;” ‘)!!

(4.6)

24

MARTORELL

AND

MOYA

DE

GUERRA

and we can write the exact expansion of p ,.,.(A, s) (see Eqs. (3.41), (3.33)) in terms of the second and first moments O” W” C-1)” M2” =qR) &.o.(R, s) = nIzo pn + 1)’ For this model the Campi-Bouyssy

g

(-1)”

mG{W

.

n=O

approximation

(4.7)

gives

(4.8) approximating pa@,

the exact density matrix (3.33), (3.34) by s> = 3p(R) d$p(R)

f

(-1)”

;;+‘;;

(s*$I.

fl=O

Then we see that the ratio between the coefftcients of Sag in the Campi-Bouyssy exact expansions is

(4.9) and

(4.10) or R, = 1, 1, 5/7, 25/63 ,..., for rr = 0, 1,2,3 ,... . The first zero in s of pee occurs at s* N 2(6p/M,), at which the exact density matrix is already a small fraction of p(R), P,.,.@, 4 = dW*. C orrespondingly the exact Wigner distribution function (3.32) which is a pure gaussian in phase space f,.,.(R, is approximated

p> = p(R) (%)

3 e--g2’u*,

u* = 5

(4.11)

by the step function fee@, P> = P(R)

“* (dw”

%/?a

(4.12)

-1’1

whose moments (4.13) are the fraction [3 . 5”/(2n f 3)!!] of the exact moments. The Negele-Vautherin approximation gives for this model fdh~)=35

&

($$--

1) - ($-+)]

W,-PI

(4.14)

WIGNER

DISTRIBUTION

OF ATOMIC

25

NUCLEI

whose moments 5!! &fNV = 2n (2n+3)(2n+5)k:’

(4.15)

are quite different from the exact moments for n > 1. Again the ratio between the NV and exact moments is the same as the ratio between the coefficients of s2” in the NV and exact expansions of p,.,.(R, s), this being a general property as can be seen from Eq. (3.41). To order s4 this comparison was already made in Ref. [ 19 1. It is worth pointing out that the comparison of the different approaches for density matrix expansions becomes much simpler in terms of the moments of the Wigner distribution functions. allowing us to compare with little effort the different expansions to any order in s’. We shall now consider the other extreme case of a model density matrix, that of a sharp Fermi surface. To simplify the analysis we shall restrict ourselves to the I60 case in which the Wigner distribution function is simply fS.F.(R, p) = 32e-ho’2(h,

- 2)

(4.16)

with moments M,,

‘2;2+2;;!! [1+rig(R)]

=P@)

(4.17)

a0

where (4.18) (4.19) The corresponding

density matrix is m (-1)”

PR s)= ‘c

,tcl

pn

(s2)”

+ l)!

M,,=p(R)

1 --&g(R) 0

For this model density matrix the Campi-Bouyssy

approximation

I

e-s”4a:.

(4.20)

gives (4.2 1)

and the exact Wigner distribution function (3.31) which, in phase space, is a first order degree polynomial in p2 modulated by a gaussian of width Ilao, .fs.,.(R P) =dR)(

--I3 ‘f

(1 + (g-G)

g(R)) .c-~*‘~’

(4.22)

26 is approximated

MARTORELLANDMOYADEGUERRA

by the step function (3.46), whose moments

~:,S=m&&2(1 +m]” 0

are, for n > 1, quite different from the exact moments, except at g(R) < 1 (or R* s ai/2) where the comparison becomes analogous to that encountered for the model density pSeO.discussed above. The Negele-Vautherin moments for this model are 5!! k;‘“-“(R) My’ = p(R) (2n + 3)(2n + 5)

[$

( 1 + g(R)) 0

- (n - 1) k;(R)

1

(4.24)

which, for n > 1, have a completely different R dependence than the exact moments (4.17). In conclusion we see that for the case of smeared occupancy, in which condition (3.40b) is approximately satisfied, the Campi-Bouyssy expansion is much better than the Negele-Vautherin expansion. For the other extreme case neither of these seems to work, although still looking at the moments, the Campi-Bouyssy expansion looks more reasonable.

V. SUMMARY AND CONCLUSIONS In this work we have discussed some properties of the Wigner transform of the one body nuclear density, restricting our study to the static ground states of spin saturated spherical nuclei. The one body density has been approximated by that given by the most refined version of the present mean field theories: the so called density dependent Hartree-Fock approximation. In addition for reference we have also utilized one body densities generated with harmonic oscillator wavefunctions. In that framework, we have given a theorem to determine from the symmetries of the average field, the minimal number of independent variables required to characterize J This theorem is particularly useful in the case of the oscillator potential, as it allows us to derive simple analytic expressions for the Wigner transforms of completely filled orbitals. As an application we have given in Appendix II results for some selected cases: lf7,*, lg,,,, I!z,,,~ and lii3,* and the nuclei 48Ca, 90Zr and *O’Pb. For the general case of DDHF wavefunctions we have computed numerically the Wigner transforms, and shown explicit results for doubly magic nuclei, plotting f for fixed R as a function of p and the relative angle. We have discussed the origin of the surface diffuseness and the internal oscillations and shown that their behaviour can be qualitatively and almost quantitatively understood using the simpler expressions of the harmonic oscillator model. To study also the behaviour associated with smoothing of the occupations of the single particle levels, we have introduced two simpler models: that of the pure

WIGNER

DISTRIBUTION

OF ATOMIC

NUCLEI

27

harmonic oscillator with a sharp Fermi surface and that of smeared occupancies of Bhaduri and Sprung, showing with analytic expressions the results forf and the effect of the smoothing of the occupation numbers on the Wigner transform. We have finally left open the question of the validity of energy density functionals derived from a truncated DME (and further approximations to it like the well known Skyrme forces) in dynamical problems like high energy heavy ion collisions, where the approximation of setting f to zero above k, does not appear to be as justified as in the static case. I

APPENDIX

The proof of the theorem given in Section II is based on the equivalence between the quantum mechanical commutation relations

(A.11

[e,YPl =o [@,,F,] =o

(A.21

valid for any operators r”, , P, belonging to the set S = (Pa } as defined in Sub-section 11.2. and the relations in phase space

wLf11 =o

(A.3) (A.41

MA3~~=0

for any r$, p0 belonging to s” = {ga}. We must prove that in order to satisfy (A.3),f(R, p) can only be a function of the (cpa’s.To prove this let us call Zj = Zj(R, p) (with j = l,..., N) the set of independent variables on whichf(R, p) depends and write f(R, p) =f(Z,)

= exp [i 6 zjpj] ,r,

f(0)

(A.5)

with Pj defined as P,f(Z,)

= -i &f(Z,). J

Then

(A.61

28

MARTORELL

and in order to satisfy condition

AND

MOYA

DE

GUERRA

(A.2) the whole set of conditions

=o =o

{{&~zj,ll U&J&Zj,ll

(A-7) {{~~,zj,zj*"'zj,}}=o

with j, , j, ,..., j, varying from 1 to N and any 3a E g must be satisfied. Now since by definition {{A, BC}} = 2iA(sinX/2)

BC

= 2i sin(llAB/2 + /IAC/2)ABC

G43)

=2i[cos/1AB/2)B(sinAAC/2)AC + (cos AAC/2) C (sin /iAB/2) AB]

we can easily see that any of conditions (A.7) can be expressed as a sum of terms each of which contains a factor of the form 3, (sin i/2) Zj, for any j = l,..., N and any & E g, and therefore the whole set of conditions (A.7) will be satisfied if all the independent variables Zj belong to the set s= {#a}, as follows from condition (A.4). On the other hand, if any of the Zis (for instance Z,) does not belong to this set (or more generally is not a function of the &‘s) then at least for one $B E 3, { {3D, Z, )} # 0 and at least part of the conditions (A.7) will fail, so that f will not satisfy condition (A.3). Although to simplify the notation in the proof we have implicitly assumed spin saturated systems so that f(R, p) is spin independent, the theorem is quite general and can be equally demonstrated for the Wigner transform of spin dependent operators and density matrices p(r, o; r’, 0’). Obviously in spin space the commutation relations satislied by &‘s and p are also satisfied by their Wigner transforms (see, for instance Ref. [ 181).

APPENDIX

II

The expressions for the oscillator closed shells can be obtained from Eq. (3.28), those for the single particle orbitals outside closed shells are f[ lf7,J = -64ewh0”

l-h,+fh:-+:+d

..

)I

WIGNER DISTRIBUTION

= -96e- ho’2 1 -Ih,,++h:-&h;+&h;---hi 3

f[ lh,,,,]

29

OF ATOMIC NUCLEI

1 ll!!

..

(A.9) ;-+h,+&h:-=

f[ li,,,,]

= 1 12eeho’* +d

(

&-Ah,,

II!!

l-2h,+h:-&h;+&h:-&h:,+&h;

2-$h,+$h;-Lh:,+xh;

693

Lho++$:) 231

..

13!!

..

1

+d3&].

These lead to the following result for the nuclei of interest: f(““Ca)

= 16e- ho’2 4-2h,+fh;+$h;+d

f(gOZr) = 16eeho’*

..

(

-;+%ho

8 )1

-7++h,-2h:++h;+&h:,

[ +d (4;ho+-&h:)

+-&d*]

”

(A. 10) f(*“pb)

= l6e-ho’2

[

-2+5h,-;h;++$h:,-$h;:+p 6-1jz-ho+~h;-&h;+mh; +d3$#

405 8. ll!!

hi

210

”

ACKNOWLEDGMENTS This work Tecnica-Spain). Nuclear-Spain,

has been supported by the CAICYT (Comision Asesora de Investigation Cientifica y We are also very thankful to the Grupo de Altas Energias of the Junta de Energia for allowing and helping us to use their computing facilities. This work was started

30

MARTORELL

AND

MOYA

DE GUERRA

during a summer visit of both authors to McMaster University (Canada), we thank the members of the Theory Group of the Physics Department, and particularly Professor D. W. L. Sprung, for their interest in this work and useful discussions. One of the authors (J. M.) also thanks J. Treiner for clarifying discussions on the use of Wigner transforms in semiclassical methods.

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