Semiclassical approximations for nuclear Hamiltonians II. Spin-dependent potentials

ANNALS

OF PHYSICS

129,

Semiclassical

153-171 (1980)

Approximations II. Spin-Dependent

for Nuclear Potentials

Hamiltonians

B. GRAMMATICOS+AND A. VOROS* Service de Physique Thsorique, CEA, Centre d’Etudes Nu&aires de Saclay, BP no 2, 91190 Gif-sur-Yvette, France Received January 30, 1980

A systematic semiclassical expansion procedure for physical quantities in nuclei, based on the Thomas-Fermi approximation to the Hat-tree-Fock equations and constructed in a previous work, is extended here to the realistic case where the effective one-body Hamiltonian for nucleons contains spin-dependent terms. Spin-induced corrections to the kinetic energy density and surface energy of the nucleus, and expressions for various spindependent densities inside the nucleus are computed up to fourth order in fi for several nuclear Hamiltonians.

I. INT~00ucT10N In a previous paper [l] (hereafter referred to as I) we have presented a systematic procedure for calculating semiclassical expansions of physically interesting quantities, based on the Wigner transformation. This method is particularly powerful when dealing with one-body Hamiltonians, such as the ones encountered in the individual particle description of many-particle systems. Whenever one can be contented with just the semiclassical or, equivalently, smoothly varying, part of the Hartree-Fock energy of a system of nucleons [2,3] the semiclassical approximations are of great help as they drastically reduce the lengthy microscopic calculations. In I we considered one-body Hamiltonians which can be of interest in nuclear structure calculations. The Hamiltonians considered did not contain any spindependent terms. It is, however, well known, after the success of the traditional shell-model, that the single-particle nuclear Hamiltonians must incorporate a spinorbit part in order to account correctly for the observed nuclear properties. In this paper we shall extend our method to the case of spin-dependent Hamiltonians. It is clear that the spin is an effect of the order of fi and thus gives no contribution at the classical limit. However as soon as one considers &expansions to higher orders the effect of spin starts manifesting itself. The case of spin-dependent Hamiltonians has already been examined by some authors. Jennings et al. [4] have computed the level density for a one-body Hamil* Members of CNRS. + Service de Physique Theorique,

CRN, BP no 20, CRO, 67037 Strasbourg, France.

153 OOO3-4916/80/110153-19$05OO/O Copyright 0 1980 All rights of reproduction

by Academic Press, Inc. in any form reserved.

154

GRAMMATICOS

AND VOROS

tonian including a spin-orbit part. However, in this work the spin-orbit contribution is introduced through an expansion in its coupling constant whereas, the spin-orbit potential being proportional to ti, its contribution appears in a most natural way in the semiclassical expansion as we shall see later. Brack et al. [5] have given the expression of the kinetic energy density in terms of the matter density to second order in fi. Balazs and Pauli [6] have also considered a certain spin-dependent Hamiltonian. Since, however, their spin-orbit potential was of the same order in fi as the rest of the Hamiltonian, they arrived at conclusions which are not directly applicable in the case of a system of nucleons. This paper is organized as follows. In Section II we extend the method introduced in I, to density matrices which operate in spin space as well. Section III is devoted to the study of the Hamiltonian H = &fp2 + V + #iA ’ O, where kA Q, with A = S(r) x p, is the spin-orbit part. An application, as in I, to the case of the surface energy of a system of nucleons makes clear the importance of the spin-orbit force. In Section IV we examine a Hamiltonian which is the most general form of the Hartree-Fock one-body Hamiltonian derived from an interaction of the Skyrme type [7]. Our results are summarized in Section V.

II. OUTLINE OF THE METHOD As we have explained in I, in order to obtain the semiclassical expansion of a given operator A, we begin by taking its Wigner transform. If we consider the matrix elements of the operator in configuration space: (r 1A^ I r’) = A(r, r’, fi) the Wigner transform is given by the expression: A&r, p, h) = 1 A (r - $, r + f , ~3)efper”lWd3r”.

(1I.l)

We shall denote by x = (r, p) a point in the classical phase space. The semiclassical expansion of a is then defined as the expansion of the Wigner transform Aw in powers 0fA: A&,

p, h) = f

Ah-, P) &“,

(11.2)

n=O

and we interpret the term of order zero in fi as the classical limit of the operator A. As in I our interest will be focused on the semiclassical expansion of the density matrix at zero temperature for a system of independent nucleons, whose motion is governed by a one-body Hamiltonian l?. We have simply j3 = B(E - I?), where 6 is the Heaviside step function and E the Fermi energy of the system.

(11.3)

SEMICLASSICAL

155

APPROXIMATIONS

The following assumptions on fi are made in order to derive the semiclassical expansion of 6. - Essential assumption: Hw is a smooth (Cm) function of x = (r, p) and fr, hence it has a semiclassical expansion: &

= f fw, n=O

(17.4)

P) fi”

with H,, being the classical Hamiltonian of the system. In the cases we shall treat here, the semiclassical expansion of H,, will comprise just two terms: Hw = Ho + H,fi

-- We shall later require that H,(r, p) depend on p only through I p / although that condition can be somewhat relaxed (see Section IV below). We have shown in I how to compute the semiclassical expansion of an arbitrary function f(A) of the operator A. Forming the Taylor series of f(A) around the cnumber value H,(x) for some fixed point x in phase space, we have: f(i?)

Evaluating

= f

$-.f”‘(H,(x))(i?

- H,(x))‘.

r=o .

(11.5)

the Wigner transform at the same point x we get (11.6)

where the expansion coefficients gr(x, fi) = [(A -

4 1x,=x

Ho(xWlw(x’,

(11.7)

are universal, i.e., independent of the particular functionf. Each 9, has a full power series in fi. The first values of LkYp to order h2, are [8]: go = 1, Yl = fiH, + h2H2 + O(h3),

(11.8) 93

=

-

7

c

(aqo

lk + a,jfo

. 4&o

(and S’a , IgS ,... = @(As)).

. arkHo . ac+&o

- 2a,,H, . %,Ho . arjPaH,,

* 4.,r,Jfo 1 + wi3)

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GRAMMATICOS

AND

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Once the 9,. are calculated (through the algorithm using Green’s functions developed in I), the Wigner transform for the density matrix follows immediately BW = B(E - A)W = B(E - H,(x))

-

f’ v 7=1

.

S+l)(Ho(x)

_ E),

(11.9)

where 8%) is the nth derivative of the Dirac measure. At this point we shall stress two differences with the treatment given in I. Firstly, the expressions of the L!?~are modified by the presence in Hw of the higher order terms #iH, , @H2 ,... which were absent in I; in particular, g1 is no longer zero. Secondly, since we are interested in a system of spin 4 fermions, the Hamiltonian I? and likewise $ are operators in spin space as well, i.e., 2 x 2 matrices. In the simplest case, the spin-dependent part of I? will be a spin-orbit potential of the general form Vs.0 - h(S(r) x (-im>>s,

(11.10)

where X is a dimensionless constant and s is the nucleon spin (for S(r) = r the familiar 1. s form is recovered). The Wigner transform of this object is simply V s.0.w = ws

x Ph

(II.1 1)

where CI are the dimensionless Pauli matrices. So we note that the usual spin-orbit contribution to the Hamiltonian is of order ?L This suggests defining the extension of the semiclassical treatment of I to spindependent potentials as follows. Any operator of interest A^ is expressed as a 2 x 2 matrix of operators: A = g4, + A . a), (11.12) and the Wigner transformation is defined to leave the 2 x 2 matrix structure intact, i.e., it is taken matrix element by matrix element: A,

= &4,,

+ Aw . a).

(11.13)

The semiclassical expansion is then the power series in h for the matrix Aw . Physically, this means that spin is taken as a purely quanta1 effect, vanishing in the classical limit: with the dependence in fi explicited, the spin term Aw is (at least) linear in fi (another view, less physical and much more difficult to implement computationally, would have been to let the nucleon spin s tend to a non-zero value of some classical internal angular momentum as fi --, 0). A useful feature in our approach is that for physically interesting operators like the Hamiltonian g, the classical term (e.g., Ho) is still a scalar in spin space, and spin effects can be treated perturbatively in fi, like the spin-orbit term (II. 11): V s.0.w -- M(S x p)a = hH, .

(II. 14)

SEMICLASSICAL

157

APPROXIMATIONS

The density matrix 6 to be computed is also a 2

x

2 matrix:

p^= !&I + s . 4.

(11.15)

Besides the usual scalar part jJ = Tr p”, it contains a vector part 15= Tr p”~ (trace is meant over the spin degrees of freedom only). Because Wigner transformation does not atlect spin, and because HO is a scalar in spin space, the expression (11.9) still holds for j?. The %‘k are now matrices in spin space as well and are still given by formula (11.7). The binomial expansion formula for operators, applied to the right-hand side of (11.7), implies that all products of spin operators occurring therein are completely symmetrized in the ordering of factors. With that prescription, the algebraic expression of g7 is the same as if Hr were a scalar in spin space (cf. Eq. (11.8) and Appendix). Starting from this expression for fi we can introduce various physically interesting quantities [7]. The probability density is just the diagonal part of the density matrix operator: p(r) = Tr(r 1j? 1r). In terms of the Wigner transform p&, p) (we shall suppress the index “W” when it is obvious), it is given by: f(r) = 1%

dr,

(I I. 16)

P).

The kinetic energy density operator + = V, . V&,(rl , r2) and probability current operator j = (1/2i)(V, - V,) p&r1 , r2) can also be introduced; their Wigner transforms are: T(r, 10 = *V~h(r9 P) + (p2/fi2) fo(r, P),

(11.17)

j(r, PI = WV dry PI. The usual kinetic energy density and probability partial trace in momentum space: 44

= t V’f(r)

current are recovered through

+ j $ pdr, p) &

a

,

(11.18) i(r) = J i pdr, p) &

Similarly

.

we can introduce the spin density, related to the Wigner transform of 6: (11.19)

The spin kinetic energy density is likewise given by: T(r) = t Vr2pW + I$

p(r, P) &

.

(11.20)

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We can also introduce the spin tensor JJh

3r2> = ww,

- V2)ti PYbl 7 r3,

(11.21)

whose Wigner transform is (11.22)

Juy@, P) = (~$9 pv(rP p), which leads to the spin-tensor density: Judd

= j” 4 dr,

PI &

The vector part of Juy , namely J = (1/2i)(V, - V,) interest and its semiclassical expression is given by:

J(r) = 1 a x dr,

. x

PI &

(11.23)

e(rl , r2), is also of particular

.

(11.24)

As we have explained in I, the integration over the momentum p can be performed algebraically in a closed form if H,, is invariant by rotations in momentum space. In this context it is useful to introduce the symbol ( f) denoting the angular average over p (11.25) and to consider as intermediate

quantities the moments

c49 = Irn P
and

44

= 6

P2(wwXr~

(11.26)

P> dp

(k = 1,2, 3,...), which are now 2 x 2 matrices as well. The algebraic expression for ah(r) is the same as in I (taking into account that now g1 # 0)

49 = IE
wherep,(r)

(11.27)

> 0 is the local Fermi momentum,

defined through the equation:

E = H,(r, pF) - E = 0.

(11.28)

SEMICLASSICAL APPROXIMATIONS

159

Likewise, ak(r) admits the closed form a&) = -

f

$ [ (- &

n=1

.

$)“-‘(

0

p”e2ff$p

“‘)I

0

2)=~p(l)

(11.29)

and the expression of the quantities (11.16)-(11.24) in terms of moments 01~and

ak

is

p(r) = (1/2r2fz3) Tr u,(r), T(r) = (1/27r2fi5)Tr c+,(r) + $V2p(r), j(r)

= (1/2.rr2fi4) Tr a4(r),

p(r) = (1/27r2ii3)Tr c+(r) . Q, T(r) = (1/2.rr2P)

(II.30)

Tr cy&r) . a + tV2p(r),

Juy(r) = (1/27r2fi4)Tr(~4,u,(r) * 4,

J(r) = (1/27r2fi4)Tr(a,(r) x 0) (-=-JAG->= &JAN,

where a4,&) is the ,uth component in momentum

III.

THE

HAMILTONIAN

space of aa(

&fp2 + V + fiA -Q

In this section we consider a one-body Hamiltonian H(r, r’) = - i Ff(q

of the form

) V26(r - r’) + V (q)

- iti (fb X S (q))

6(r - r’)

* V8(r - r’).

(III. 1)

The Wigner transform of (111.1) is readily calculated and gives Hw =

ijfp2 + V + kA . o,

(111.2)

with A = S x p, the classical part being just Ho = &fp2 + V (we have absorbed into the function S(r) the factor X of Eq. (11.14)). As in I we calculate the semiclassical expansion for the density matrix in detail up to second order in rZ. We have, expanding around the classical Hamiltonian, pw = &E - H), - @,6’(H,

= B(E - Ho) - 9,&H,

- E)

- E) - ~~$‘(HO - E) + O(P).

(111.3)

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GRAMMATICOS

AND

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Using expressions (11.8) we obtain 91 = ti(S x p) * a, 9T2 = $ [(p * Vf)” -fVWo]

+ ?v((S x p) . a)’

(111.4) = $ [(p * Vf)” -fV”I&]

93 = - f

+ @(S x py,

[f(vHo)2 -l 2f(P . ?MP * VH,) + f2(P * V)”

m,

where we have used the identity ( Q . A)2 = A2. In order to make the expressions simpler we shall focus on the difference between the case with spin-orbit term and without, i.e., 8’3 = 3 (full Hamiltonian)--S(O) (no spin-orbit). Thus: 83, = h(S x p) * a, 69, = P(S

x p)” = fv(S2p2 - (S . p)“),

(111.5)

68, = 0.

The angular average over p yields:


(111.6)

= 0.

We remark that after the angular average there remains no term linear in (I, i.e., the vector part of the density matrix vanishes: p = 0. This is due to the sphericity of the Fermi sea and is true at higher orders as a simple counting of powers of p can prove. We can now write the matter and kinetic energy density operators, or rather their increments with respect to the spin-independent case: 6p(r, p) = 2(471-/8p3h3)fi2&Y2p2,

(111.7)

ST@, p) = 2(4z-/8n3h3) A2jS2p4.

The factor 2 is due to the trace over the spin part and the 1/87rS3 is the usual phase space factor. Integrating over p we obtain

sp = p - p

l = m4nfia%kp

f”

=

(111.8) ST = 7 - 40) = j(S2/7+2) kg,

SEMICLASSICAL

161

APPROXIMATIONS

where we have put pr = rikr , PF being the local Fermi momentum,

&fpF2+ v = E (PF Substituting

>

solution

of

0).

kF as a function Of p kF = [3,9#‘)]1/3

= [3$@

-

(111.9)

,+)11/3,

we find 87

=

+J

-

kF22ip

=

5;p

-

f

(3,.+)3/3

Q+

(3+w

(the term --kr26p appears in the zeroth order term T(O) = kF5/W stituted by (llT.9)). So at this order we can write T in terms of p: 7 = 53 (3Gy3

= 2 ; p (III.10)

when kF is sub-

1 -(VP>” psi3 + 31 v2p + 36 p +I 6 ~V-VP s

(Vf)” 1 vzf -- 1 __+&Pf++P,p. 12p f”

S2

(III.1 1)

This is the result that Brack et al. [5] have already calculated using the partition function method. In order to make the equivalence more transparent we proceed to reestablish their #P/m factors. A simple comparison of our Hamiltonian with the one of Brack et al. gives the following relations for the effective masses and spin-orbit form factors:

.63= mf, SB= fi?3,

(111.12)

where by fB , SB we denote the quantities used by Brack et al. We can then immediately verify that: fir=2~p=2~~p=~(~)2~p=*7B, The calculation

(III.1 3)

of the tensor JUy proceeds along similar lines. We have JU” = WVPUP”

(111.14)

with p = fi(S x p). Equivalently

Jw =

P,@ x p)v = ~A,~~~P~P~ .

(111.15)

The angular average gives (ITT.16)

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GRAMMATICOS

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where the last equality makes use of the dual tensor S,,, 3 Q,,,S~ . The angular integration then gives

and

J = --2Pcw

(= --EA,“(~u”lf)P).

(III. 18)

Again the equivalence with the result of Brack et al. is immediate

s

2m

SB

(111.19)

J=-2p7=--F212~=JB.

The calculation we made to second order in #&has been extended to fourth order using a computer and the REDUCE language [9] for symbolic manipulations. The semiclassical expansions for coefficients &Y1, of the density matrix can be found in the Appendix. Here we limit ourselves to the expressions of T and Juy as functions of p (the quantities j, p and T being zero by symmetry). We use the same conventions as for the corresponding formula in I: any free-standing gradient operator acts only on the rightmost term, e.g., (0’. V)(Vp . V) means Ci,k aj f. &p . a&p and (Vp . V)” f means CjBk $p . L$p . a& (This convention will not be used in Section IV.)

+

--

IpI,

@2)-2/3

[-

&

Ff

+

+

( v2;;;‘2

q)

1 (CIl;3vYf + J@ (y-j”)” ] 30

+ p-2’3 [&

v4p + + (&

+ p-513[- & (Vp * V) v2p - j&j(v2P)2

(OP. v>2f1 Pf +i&i -P

-

-II,3

.VP12

f” ((VP)")" 45

-!.?1080(Of * Vp) v2p -

] + P-8'3 [&VP)

I

- &j

2v2p

V2(VP)"

SEMICLASSICAL

+

(37?)--213

+

+

(--2S(Vf*

+

+

w(vf)2

+

6p-2/3

t+(vf.

[

p

(s

[+

V)S

+

(S

-

163

APPROXIMATIONS

* 02s + 2(S . V)(VS) + ; V2(S2) + (W2) (S

. Vf)(VS)

. Vf)”

-

-

Vf(S

* qs

-

s2 yf-

(S

* V>“f>

4@3)3)]

~(S.(Vp.v)S+(S~vP)vS)

VP> s2 -

w.

WVP.

(111.20)

w]/

(the term 2(S2/f2)p in the first line, and the last five lines, are the spin-orbit butions of order fi2 and A4, respectively); J,” = - $Y p + (37P)-“i3]Pl/3 [- $(VS,.

contri-

+ V,(V x S)“)

+ -$ (V2f. SW - (S x VI” vd- + V,f P x f9” + (Vf . V) SW)

+ + (- ; W)” SW+ ; V,f(S x Of)” + w,,)] + P-2'3 [& t ~~-W(s

K-VP.

V)&"

+ (V,S

x VP)" + w.

x VP)")

"PN")]/.

(111.21)

Application: The Spin-Orbit Contribution to the Surface Energy of the Nucleus

In I we calculated the surface energy of a system of nucleons using the semiclassical relations between T and p. For that purpose we made use of Swiatecki’s expression [IO] for the surface energy per unit area Es/S,

&IS = j-‘” (~bl - ; Z[p,l) dx. -co

(111.22)

The hypotheses underlying the derivation of Eqs. (111.22) are that the system possesses a plane geometry and is semi-infinite, its density falling more or less rapidly from the value p,, (at which it starts at - 03) to 0. The functional Z’[p] is derived through the assumption that the nucleon interaction is a Skyrme force with Coulomb interaction neglected, and with the restriction to systems with equal numbers of protons and neutrons.

164

GRAMMATICOS

AND

In the presence of the spin-orbit interaction becomes:

X[p, T, J] = ;

VOROS

the expression of the functional

&[p]

T + ; top2 + k ts2 + ; (3t, + 5t,) pi

+ & (St1 - 5t2)(Vp)2 - ; WV . J

(111.23)

where to , tl , t2 , t3, w are the force parameters. The effective mass is given by the expression f = 1 + (b/po)p with b = (3t, + 5t2)/(16ti2/2m) p. , and the spin orbit

form factor by S = Qw . Vp. With the usual caveat (when it comes to employing the semiclassical expansions beyond the classical turning point), we can transform the functional &’ into a functional of p alone, using the relations just derived between T, p and J. As in I the surface energy is obtained through a restricted variation, using for p a simple two-parameter Fermi form: (111.24)

p = po/(l + P).

The parameter p. is the equilibrium density of infinite nuclear matter, while cxis fixed in such a way as to render Es stationary. The analytical (but computer-assisted) calculation of Es gives then (111.25)

Es = S(Acx3+ Bar + C/a)

with

D = 5 - &

852

1

1

B = 2m PO ;iz - G W, + &

9

1 2b1J3- 1 + 31/i arctan 3112 + 6 .7;1,2

[k log 1 y ,t,r:{2,8

po ;

+BWPo3@b

[-

ml

+ %J PO2 &

+

1 E-@B-g-

+ ;

Pt1 -

log(l

+ b) + & + $

1

1

(&

+ +r)

St,)

PO2 ;

- ;] log (1 + $1

165

SEMICLASSICAL APPROXIMATIONS

c = - ; (T)“‘”

pi/3 &;

(1 - log 3 + *)

+ k (3t, + 3,) ; (7)“‘”

- ; &

PO2- k t,pz ;

py (- $ + 3 log 3112- G).

(111.26)

The numerical results have been obtained with the parametrization SIII of the Skyrme interaction. They are summarized in Table I. (The results without spin orbit were already presented in I. The results without effective mass correspond to f = 1, i.e., b = 0; they can be obtained most easily if one puts f = 1 in expressions (111.20), (111.21) and can be verified tediously if one takes the limit b = 0 in expressions (111.26). One remarks that the spin-orbit contribution to the surface energy, far from being negligible, is a l- to 2-MeV effect in accordance with previous calculations [l I-121. Apart from the total energy, the spin-orbit effect on the surface shape is also appreciable but its inclusion in both second and fourth order in +ileads us back to a relatively small LX,i.e., to the surface obtained in the fourth-order calculations without spin-orbit term. TAEILEI Order of computation

Without spin-orbit

fi”

oi = 2.47 E = 19.15 Without (I = 2.50 effective mass I E = 18.90 With OL= 2.51 effective mass I E = 18.82

64

Without a = 2.32 effective mass I E = 19.42 With OL= 2.31 effective mass I E = 19.41

IV. THE CASE OF THE HARTREE-FOCK

HAMILTONIAN

With spin-orbit id. id. OL= 2.86 E = 16.56 a = 2.81 E = 16.86 a = 2.32 E = 18.10 a = 2.26 E = 18.47

FOR A FULL

SKYRME FORCE

In this section we consider the one-body Hamiltonian obtained when a Skyrme interaction together with the Hartree-Fock approximation in the absence of timereversal invariance is employed. The general form of this Hamiltonian has already been derived in Ref. [7]. Its Wigner transform is simply (IV. I)

166

GRAMMATICOS

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The spin-dependent term now contains a spin potential and a spin-effective mass: B = a(r) + A + c(r)p2.

(IV.2)

The spin-orbit potential A . G is in principle the same as in expression (111.2), but in view of an application to nucleons in a rotating frame we will introduce a slightly more general form (w playing the role of the rotational frequency) A = o/2 + S x p.

(IV.3)

As the classical Hamiltonian HO

=

8fp”

+

a ’

p+ V

is not quadratic in p, we perform the change of variable p = q - a/f which transforms it into a Hamiltonian quadratic in the new variable q. In that case the angular averaging, now around q = 0, can be performed in a most straightforward way. By analogy to the quantities ak , ak of Section II we introduce here the moments: Bk = Iorn qk-YpW%

4) 47, (IV.4)

Pk = km 4k-2(wWXr,

4) 4,

and we proceed to express the quantities p, T, etc., in terms of the p’s. We find: p = W77%3) Tr P3W, p = ( 1/27r2h3)Tr p3(r) . Q,

(IV.5)

As in Section III we can compute the quantities &%I , 69,) Sg3 (the contributions to gl, g2, CY3from B # 0): Ml = fiB * Q, &S2 = Pi2B2 (IV.6) , 69, = 0.

SEMICLASSICAL

167

APPROXIMATIONS

In view of the angular averaging we write B as B=~+A+cP~=B+A~+SX(I+~~--C~

f

f”

+ cq2 (IV.7)

= B, + S x q - 2(c/f)a 3q + cq2 where A,, = o/2 - (S x a)/”

(IV.8)

B, = a + A, + c(cu2/f2). The angular integration needed here is:

can now be readily performed.

The new averaging rule

(CC. q)P x 4) = 3q2(Dx 0

(IV.9)

Skipping over the intermediate calculations we give the results for the various densities, after the radial integration over q and the trace over spin have been carried Out (defining qF > 0 by &(r, qF) = E and kF = qF/fi): 1 a' =

2rr2fi2f"

P = -(l/rr2+$)(Bo 6 (T - ;vz,

a2 + 6B,.c

=+fi2kp(2S2+4$ [ kF

- 4(S x a) .f) +fi4kF3.5?],

+ cfi2kF2)kF ,

- Gp)

=&/[3B,2-4+%

x S)+8B,,*c+-]lik,

t 7 [S2 + 6 $ a2 - 4(S x a) . T, + 3B, . c] fi3kF3 + 7c2fi5kF5/, 1 7--pp=-__

kF .rr2fij N kF2 + +-)

4

8(j+$P)

=&[(&I

(B, + cfi2kF2)- ; fi2kF2(y?

X S-2ya)kp+5(c

- 2;

$)I,

x S--2$a)fi2kF3],

The symbol Zipmeans, as previously: 6p = p (with term hB . a) - p(O)(without term AB . G in the Hamiltonian), similarly for the other densities.

and

We can further eliminate kF between p and the other densities and express them all in terms of p. We must of course bear in mind that in the case of calculations with Skyrme forces the potentials a and B themselves depend on densities other than p, so that auxiliary self-consistency problems have to be solved in any practical applica-

168

GRAMMATICOS

AND

VOROS

tion. The elimination results in the following formulas, where we use the convention that every gradient operator acts only on the first position-dependent quantity situated on its right (e.g., V(a * f) = Ck Va, *fk: the parentheses only determine the way vectors are contracted):

1

(77--j3~-7d+6-7-

+ (3r2F3 p1/3 I’i72a* a Pf” I4 +

1

v[-2a(Vf*

V)a

-

$-(Vp

+

-+ $

e

f

p5j3

+

vzf

$ V2(a2)

a2 * V2f +

+ + [CaX OfI + 3(3.rr2)2’3

1

(Vf)”

X

-

t V(aV)a

(a . V)“f+

a).

[V

p [SZ +

3B,

X

-

(a

i (a * V)(Va)

* Vf)

Va

a -S(Vf

. c _

x

+

Vf(a

* V)a]

a)]/

7 OL ’ (cfx

+ 12q

s)

f”

+ 3(379)-a/3 2a(B, x S) + 4(B,, . c) a2 J p1’3[B,2 Pf” I’ f” p=-

f

p - 3(3,+-W

!k’ fif

p1/3,

2s

T = ; v2p - 3(3+)2/3 ic p5/3_ & (3B, _ ~ f fif f j

=

_

+T+

p113 V(Va)

(3+)i2'3

-

I

+ j[(Vy+

Vf.

c x S-2&)p+

J”” = p (4,.

+ y)

V)a

-

(V

.

+

a +

a.

V)

x

+ f2

e) -

3(3+)-W

-a2Bo fi2f”

l/3 P

’

of]

p1i3(B,,

fif2

3(37$73

a

V2a

3(3G)-2/3 f

X

~1,B,,P’/~.

x

S -

?!!!t?

f

a),

(IV.1 1)

Application: Nucleons in a Rotating Frame

As in I we now consider the case of nucleons in a rotating mean field [13]. In that case the one-body Hamiltonian governing the motion of the nucIeus incorporates the

169

SEMICLASSICAL APPROXIMATIONS

constraint --w J, with J being the total angular momentum The Wigner transform of the Hamiltonian is just: ffw

=

vector J = #a + L. (IV.12)

ifp2 + (r x w)p + I/ + fiA . Q

with A defined by Eqs. (IV.3). This Hamiltonian is a special case of the one we have just studied and it can be recovered by the following substitutions: a=rxw,

B, = A, = T - F

c = 0. (“‘13)

and

= 7 _ 3 X (r X a),

The following expressions result: a2 p + 31 v2p + 61 Pfd2 p -r = 33 m2)2’3 p5p + ti”f” +pL+L,q+;,q

4 (37r2)-2'3p1,3 0~' + kf[3(a fi”f” 1 +$(a

X

VfJ2+d(a 3(3n2)-2/3

+$s+

j

=

-

pf”

2Y+

$-

(3+)i.2’3

(a x V)2f]

X~)[2o+$.(Vfx P"~ (Ao2 -

p1/3

_ i

+ [2w t- i(Vf

X w) * Vf-

!.

w

f

x S)

),

iaXV)XVf++[V(/.X(Vfxa)]

f(

X a)] X $--I

2a*(&

a)],/

+ 3(3$p2’3

(A,, x S)

pl/3,

p = _ 3(37?)-213 fif

Ao~l’~,

1 Sxa 7 = a v2p - tf ( 3A, - 2 _fJuy = - 9

p -t 3(3;r'3

1

-

3(3772)-z/3

iiy

p1/3cx2Ao,

~,AOVPV'~

(IV.14)

V. SUMMARY AND CONCLUSIONS This paper has been devoted to the extension to spin-dependent potentials of the methods developed in I. We have actually shown that starting from the Wigner transform of the spatial part of operators and keeping spin as such (to be treated separately), our semiclassical expansion algorithms can be easily generalized to yield expressions for various physically interesting quantities. Such a study has been

170

GRAMMATICOS

AND

VOROS

motivated by the interest of the semiclassical approach for nuclear structure studies, which cannot claim to be realistic unless a spin-orbit potential (an effect of order fi) is included in the single particle Hamiltonian. The importance of the spin-orbit term was exemplified in a study of the nuclear surface where the spin-orbit contribution amounted to as much as 10 % of the surface energy. Several possible extensions of this work suggest themselves at this point. The study of more complicated Hamiltonians including non-local potentials would lie within reach of our methods. Such a study would also be interesting for the investigation of the exchange Hartree-Fock potential. With respect to the latter point, the selfconsistency problem, which exists even for the direct, local, potential but was barely touched upon in this work, would necessitate further studies.

APPENDIX

We list here the coefficients of the semiclassical expansion to order #i4 of the density matrix 8(E - A) for the Hamiltonian H(r, p) = &f(r)p” + V(r) + k@(r) x p) . Q = Ho@, P) + fiA(r, P) . 0: e(E - I?),

= B(E - H,,) -

i

[C?@)(r, p, fi) + 6gn(r, p, h)] 6(n-1)(Ho - E).

?l=l

The coefficients ‘3;‘) correspond to the spin-independent case (A = 0) and have already appeared in the Appendix of I with H denoting there what we now call Ho. We only give here the additional terms ~399~arising from A # 0, which are 2 x 2 matrices in spin space. In this new context, the previous c-numbers ‘3:‘) mean Sfp’ . II, and similarly for all scalar quantities below. 69Fl = +iA . CT (we recall that @” = 0),

63, = @A2 + f 66,

[-2vf

x (p . 0)s

= A3 1[AZ + ; (p . vf)”

-fV2A]o

- ;fV2Ho]

- f OH, x (P * v)S - (S x V)(P * v) H,l

~+fi4]-~(v~x(p.V)S)A+~(S

+;S.(VH,.V)S-;(S*VH,)(VS)-;(S

x V)A-;(S

x vf>@.V)A

X V)Ho

+ f [ - ; (v2s x p)A - k V2(A2) + ; A . V2A +;((p.v)S

[; V2(s2) - S * V2S - (VS)2],

A + ; (P . vHo)(S x vf)

+ ;[(p . vf)(p . v)A - (VH, . v)A -$(pd~)~A/

+ ;

x v)(p.v)A]/,

SEMICLASSICAL

69, = A~[-f2(p

1 V)2 H, + 2f(p .

+ h4{[A2 + #(p . Of)” + 2f[--(VI&,.

vf)(p

171

APPROXIMATIONS . v%)

-

f(VfG21(A . 4

$JV2Ha] A2 + 2@ * VH,,)(S

VIA + (P . W(P

* VA - (Off,

x Vf) . A

x (P . V)S>

- (S x V)(p - V) H,,] . A - f2A(p * V)” A), 62fs = W[-$ff2(p

* V)Z H, + Jyp * Vf)(p . VI&) - *j(Vf&J2!

A-

69J6 = 0.

REFERENCES 1. B. GRAMMATICOS AND A. VOROS, Ann. Phys. (N.Y.) 123 (1979), 359. 2. V. M. STRUTINSKY, Nucl. Phys. A 122 (1968), 1. 3. M. BRACK AND P. QUENTIN, in “Physics and Chemistry of Fission 1973,” Vienna. 4. B. K. JENNINGS, R. BHADURI, AND M. BRACK, Nucl. Phys. A 253 (1975), 5. M. BRACK, B. K. JENNINGS, AND Y. H. CHU, Phys. Lett. B 65 (1976), 1. 6. N. L. BALAZS AND H. C. PAULI, 2. Phys. A 277 (1976), 265. 7. Y. M. ENGEL et al., Nucl. Phys. A 249 (1975), 215. 8. A. VOROS, Ann. Inst. Henri Poincare’ A 26 (1977), 343. 9. A. C. HEARN, “REDUCE User’s Manual,” Univ. of Utah, 1973. 10. W. J. SWIATECKI, Proc. Phys. Sot. London Sect. A 64 (1951), 226. 11. J. COTE AND J. M. PEARSON, Nucl. Phys. A 304 (1978), 104. 12. Y. H. CHU, B. K. JENNINGS, AND M. BRACK, Phys. Lett. B 68 (1977), 407. 13. B. GRAMMATICOS AND K. F. LIU, Nuovo Cimento A 50 (1979), 349.

Vol.

29.

1, p. 353, IAEA