Rotations in nuclei — a semiclassical description

Rotations in nuclei — a semiclassical description

NUCLEAR PHYSICS A Nuclear Physics A571 (1994) 518-540 North-Holland Rotations in nuclei - a semiclassical description * K. Bencheikh a,b a Instit...

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NUCLEAR PHYSICS A

Nuclear Physics A571 (1994) 518-540 North-Holland

Rotations

in nuclei - a semiclassical description

*

K. Bencheikh a,b a Institut de Physique,

Universite’ de S&j Algeria IN2P3-CNRS, Bat. 104-108, 91405 Orsay Campus, France

b C.S.N.S.M.,

P. Quentin C.S.N.S.M.,

I~ZP3-~NRS,

B&t. 104-108, 9140.5 Orsay Campas, France

J. Bartel C.R.N., IN2P3-CNRS and Universite’ Louis Pasteur, 67037 Stmsbourg Ceder, France

Received 23 April 1993 (Revised 8 October 1993)

Abstract An explicit formulation of extended Thomas-Fermi density functionals relevant to a microscopic description of rotating nuclei within the Skyrme Hartree-Fock formalism, is presented up to order h’. Simple analytical expressions have been obtained for the dynamical moments of inertia within this semiclassical framework. Phenomena identical to Landau orbital diamagnetism and Pauli spin paramagnetism have been exhibited upon using standard Skyrme force parametrizations. Both effects almost cancel. As a result, the Thomas-Fermi moments of inertia, which assume explicitly the rigid-body expression, turn out to represent a rather good approximation of the semiclassical results. Finally the validity of the lnglis formula is discussed in this context.

1. Introduction The development of heavy-ion beam facilities and the availability of very powerful gamma multidetector arrays makes it now possible to perform a systematic study of nuclei at very high spins. The stability of a piece of saturated rotating nuclear matter has been considered years ago by Cohen, Plasil and Swiatecki [l]. These authors have extensively studied the various equilibrium shapes of a given nucleus as a function of the angular momentum. They have also provided an estimate (along the beta-stabili~ valley) of the limiting angular velocity w (or * Work partially performed through funds granted by a French-Algerian and DRS. 0375-9474/94/$07.~ 0 1994 - Elsevier Science B.V. AU rights reserved SSDI 037.5-9474(93)E0.516-B

agreement between CNRS

K Benc~j~

et al. / Rotations

in nuclei

519

angular momentum I) which a nucleus can stand without experiencing centrifugal fission. Such an approach does suffer however from some a priori limitations. First of all, the liquid-drop approach might not be very well suited to an accurate description of scissioning shapes or more generally shapes implying an important domain of non-saturated nucleon densities [Z] where the use of a proximity potential [3] might constitute a significant improvement [4-61. To concentrate merely on the rotational aspect of the approach of ref. [l] now, one may point out that: (i) The relevance of w-independent liquid-drop parameters should be questioned. (ii} The validity of the rigid-body moment of inertia J,, should be assessed. The formalism which we develop here is capable of dealing (and will deal) with both problems as well as providing a satisfactory microscopic account of the proximity potential treatment. Nevertheless, in the present study, its use will be limited to the mere consideration of the validity of the J,, hypothesis, A traditional and simplified (even classical) way of treating nuclear rotations is constituted by the so-called routhian approach consisting in “cranking” a nucleus around one given fixed axis. As well known (see e.g. ref. [7]), the classical “hamiltonian” associated with body-fixed coordinates writes H’=H-w*(Z-+-s),

(1)

where o denotes the angular velocity, 1 is the total orbital ~gular-momentum operator, s is the total spin operator and H assumes the formal expression of the hamiltonian in the (galilean) laboratory frame. In quantum mechanics, the routhian approach, which may be derived either by changing the reference frame or within the framework of a constrained stationary Schriidinger equation, amounts to the solution of the following variational problem: S( H’) = 0.

(2)

Many groups have worked within this framework upon using phenomenological one-body hamiltonians after the seminal works of refs. [8,9]. These approaches are based on the Strutinsky shell-~rrection method [lOI slightly adapted to the routhian formalism [8,11]. More microscopic approaches have also been developed first in a restricted variational space (in the so-called inert-core approximation using simplified interactions [12]) as well as in more realistic calculations without 113,141and with a treatment of pairing correlations [15-171. It is worth mentioning also the relativistic approach of ref. [18]. The physical interest of the latter approaches is commensurate with their microscopic character, i.e. their more or less fundamental rooting in a description of the nucleon-nucleon interaction. Accordingly the numerical task which is involved may be rather heavy, yielding in some cases a lack of clarity of their results in terms of the microscopic input. This is why, without denying the importance of such approaches which are much desired and are in some instances

520

K Bencheikh et al. / Rotations

in nuclei

without credible substitutes, we think that simpler, hence more transparent, approaches of a semiclassical type are worth attempting. “Semiclassical” should be understood here in the Wigner sense [19] of an A-expansion, beyond the lowestorder term referred to here as the Thomas-Fermi (TF) approximation. Long ago, Bloch noted [20] that for a simple one-body hamiltonian the TF value of the moment of inertia was merely given by the rigid-body expression. At this TF order, much later, a discussion of some aspects of the equilibrium density has been made in ref. [21]. Various authors [22-241 have performed semiclassical calculations beyond the Thomas-Fermi order. They have all dealt with rather simple one-body potentials (of the harmonic-oscillator or Woods-Saxon type) without effective mass and most importantly without the Thouless-Valatin [25] type of corrections due to the self-consistent response of the mean field to the time-odd part of the density generated by the cranking piece of the hamiltonian. The electromagnetic analogy of the microscopic routhian is rather well known. It has been explicitly exploited in refs. [23,26]. In the present paper we will tackle the most general problem encountered when using a one-body routhian derived from usual Skyrme force parametrizations, i.e. including effective mass, Thouless-Valatin terms and full spin contributions arising from both the spin-orbit and spin-spin two-body interactions. We will do that in the framework of the extended Thomas-Fermi (ETF) approach of Grammaticos and Voros [27,281 up to fi2-terms. We will not take pairing correlations into account here. This simpli~ing assumption is, of course, rather drastic at low angular velocity. Our present approach should therefore be understood as a first step towards a more complete microscopic study, or as a better macroscopic description in any routhian calculations a la Strutinsky. This paper will be organized as follows. Sect. 2 will be devoted to the derivation of ETF functional expressions for various density functions in terms of the diagonal (local) part p of the spin-scalar density matrix. These results will then be used in sect. 3 to evaluate the semiclassical energy of a rotating nucleus in the laboratory frame, providing thus an explicit expression for the dynamical moments of inertia. The results of partially self-consistent semiclassical calculations of a large sample of nuclei within and beyond the beta stability valley will also be presented in sect. 3, while sect. 4 will collect some including remarks.

2. Extended Thomas-Fermi density functionals in the rotational case

In this section we will develop the ETF functional formalism for rotating nuclei from where it has been left in the study by Grammaticos and Voros [27,28]. There

II Bencheikh et al. / Rotations in nuclei

521

the Baraff-Borowitz semiclassical method 1291 has been used rather than the Kirkwood partition function approach [30]. For the nucleon-nucleon effective force we use the Skyrme parametrization [31] which yields naturally a functional expression for the energy in terms of p and some other densities carrying the necessary informations on the non-local part of the density matrix. Furthermore, as well known, due to the analytical expression of its central interaction, some of the corresponding Fock potential terms simply renormalize the local part of the Hartree-Fock field while the remainder results in the presence of a r-dependent effective mass. In the present work, we will mostly use the set of parameters corresponding to the so-called SkM* force [32] which has been shown to yield excellent bulk and surface nuclear properties. The minimization of the routhian H’ as in Eq. (2) yields for a Slater determinant the following one-body routhian h, whose action on single-particle wave functions is given by:

-iha,

I

-V cp(r, up

4)

+ Ch(S,-i~W,XV).(aIaIa’)cp(r,

u’, q).

(3)

CT’

In Eq. (3) the index u (q resp.) refers to the spin (charge resp.) state. The various form factors appearing in h, are specified now in terms of coefficients Bi (defined in ref. [14] from the usual Skyrme force parameters and listed in Table 1) and of various densities which will be reviewed below: Effective mass form factor fq:

(4)

Table 1 Expression of the coefficients Skyrme force parameters.

BI B2 B3 B4

BS B6 B7

-

Bi appearing

in the form factors

defined

in Eqs. (3)-(8)

$,(l + $0)

B8

-

$,(f

B9

-two

+x,)

;.+I +x,)1 - &[3r,(1 + $n,) - t,(1 + $,)I $3t,c+ +x,) + t& +x,)1 -

$[t&1

+

-

+[t,c;

+x,1

gt,c1

+

;xl>

ix3)

+

-

t,(1 +

12c+

tfoX0

B,o B 11 B 12 B 13

$t,(i

-4 ‘t 0 &3X3 -1

24t3

in terms of usual

+x3)

K Bencheikh et al. / Rotations in nuclei

522

Central one-body potential u, =2(BiP

+ iV*j) +B4(~q + iV*jq) + 2(B,Ap +B6Apq)

+B2Pq) +Bs(r

+(2 + a)B,pl+a

+

U,:

C@-l

+B+Wi(

9

+W&?]

+B07~J+V~JJ

(4,~~ +&,CP%1-

(5)

4

Cranking field form factor aq:

(6)

“u=rxo-;(B,j+B,j,)+;(Vxp+Vxp,). Spin field form factor S,: as, = B,( V Xj + V U,)

+ 2( B,,p + B,,p,)

+ 2~3 B,,p + B,,p,)

- +.L (7)

Spin-orbit

w,=

form factor W,:

-$(Vp+Vp,).

(8)

In all the preceding equations (4)-(8), the explicit mention of the r-dependence the densities and form factors has been omitted. The various densities are given below: Kinetic energy density rq:

c, 4) 1’.

~~(r) = C lbk(r, ka

of

(9)

Current density j,: r,

U,

q)Vqk(r,

ff,

4)

-rp,(r,

c7

q)VrpZ(r,

*P

Q)l.

(10)

Spin-vector density p,: p,(r)

= C &( r, (+, q)(~lol~‘>~k(rP kau’

a’9 4).

(II)

523

K Bencheikh et al. / Rotations in nuclei

Spin-orbit

density (called also spin density in ref. [31]) Jq:

cr’, C?)b,*(r,

fly

411 (12)

In the above definitions of the form factors and throughout the whole paper, whenever any density function appears without any charge index, we refer to the total density as e.g. in P = CP,. 4

Some further remarks can be made about the above expressions of h,. The additional potential term, proportional to Af4, added to the usual U, term stems from the specific expression of the kinetic-energy term with an effective mass form factor, which has been chosen here. Indeed, as pointed out in ref. 1271, such an expression yields a very simple form for the Wigner transform of h,, namely (omitting the r-dependences)

(13) The cranking field (Ye includes two contributions, one from the orbital part of the constraint ( - w * I) and another which is the self-consistent field generated by the time-odd part of the density matrix. Whereas the former has been shown in ref. [33] (see also ref. 1341) to correspond to the Inglis cranking formula [35], the latter is nothing but the Thouless-Valatin self-consistency corrective terms [WI. In what follows, we wilI note as (Y@the part of OLwhich is due to the orbital constraint ol,=rXw.

(14)

As suggested in ref. [14] and for the same reasons, we have not included in the hamiltonian density some small contributions. A part of the latter results from a tensor coupling between spin and gradient vectors; the remainder implies terms involving the spin-vector kinetic energy density (defined e.g. in ref. 1341).Finally, it is worth recalling that the spin-dependent part of h, has three different origins. Indeed, the S,-part of the spin mean field comes from the spin-orbit and spin-spin parts of the Skyrme interaction as well as from the spin part of the constraint. To this, one should add a spin-orbit mean field [the (W, X V) * u term] which finds its origin, of course, in the two-body spin-orbit interaction.

524

K Bencheikh et al. / Rotations in nuclei

In refs. [27,28], Grammaticos and Voros have given functional relations for the r4, j,, pa and Jq densities in terms of pB directly or indirectly through the fq, U,, cu4, S, and W4 form factors. Our aim however is to get functional expressions of the Iatter in terms of the pks only. To disent~gle this complicated system of equations and thus go beyond the above quoted works, we will proceed in the following way, taking stock of the rather general yet crucial remark that in complicated functional expressions at fi2-order it is sufficient to replace all quantities, for instance CQ, by their Thomas-Fermi approximations. We will then first evaluate such a TF appro~mation for o+ from which we will derive explicit functional expressions for p4 and then j4 and ([email protected] will obtain, finally, similar relations for TVand Jq. As noted in ref. [271, through a trivial change of variable in the momentum space

q=p+mol f

(15)

one restores even in the cranked case a hamiltonian which is spherically symmetric in Q, from which one trivially gets for the Thomas-Fermi current (whose Wigner transfo~ involves p Iinearly)z j,=

--

ma9

hfq pq*

One is therefore left at the Thomas-Fermi order with a 4 X 4 system of linear equations in the jq’s and the 0~~‘scomposed of Eq. (16) and a9 =aOf(Zjq+ejp,

(17)

where d and e are defined in the appendix and p is the other charge state than the one referred to by 4. The latter equation is deduced from Eqs. (6) and (14) upon neglecting the p-contribution since it is of first order in 21as compared to the rest of the hamiltonian (see e.g. ref. [28]). As demonstrated in the appendix, the Thomas-Fermi result is indeed astonishingly simple: =TF _

q -fq%

(18)

where all the effect of the Thouless-Valatin terms consists in renormalizing the constraining field form factor by the effective-mass form factor. As an important

K Bencheikh et al. / Rotations in nuclei

525

consequence of the latter, it results from Eq. (16) that the Thomas-Fermi pattern is nothing but the rigid-body result:

current

(19) retrieving in a more general context (f, # 1) the result of ref. [20]. The crucial role of the factor f, in the expression of az is to be emphasized and will be later encountered in sect. 3 when discussing the Inglis cranking approximation. Let us now turn to the spin-vector part of the density pq. As established in ref. [28], one gets 3m(3aZ) -2’3 Pq= -

l/30 P4 4’

(20)

with D, defined as

Dq=Sq- I14Wqxay

(21)

f*

Due to the first-order character in A of pq, one may use the Thomas-Fermi approximations [Eqs. (18), (19)] of j, and CX~to obtain the spin-vector densities at leading order in Eq. (20), yielding 3m(37r2) -2’3

hfq

Pq= -

c3

i

mB’[v(P+P,) fi2

x%--vx

[(P+Pqh]]

(22) In the above equation, the term in &?,/A* VP+Pq)

XTl--vx

[(P+P*h]

From Eq. (22) one therefore

is simply given by: =WP+P*).

(23)

gets the following linear system of equations for the

p,‘s:

p4

1+y

i

w

Py3

-[uGo+41) f,

+Ppyf2p pfi/3 (B,,

+Pav31*+43)l i

4

+ p*B,,)

K Bencheikh et al. / Rotations

526

in nuclei

where 3m(3&) P=

-2’3

h

(25)

*

As a very interesting consequence of Eq. (24), one notices that the spin-vector densities are proportional to o. This “alignment” property would not be exactly realized if one had not neglected small contributions in the Skyrme hamiltonian density as mentioned above. Exploiting the analogy of our problem with magnetism, one may define spin susceptibilities ,y4 as P4 = hXqeJ.

(26)

Now, the key question is to assess the sign of these susceptibilities and decide whether the corresponding alignment is of a “Pauli paramagnetic” character or not. To answer this question in a relevant yet simplified context, we will now consider the following model of isoscalar nuclear matter (Vq; p4 = &I), assuming also a sharp cut-off density distribution. In that case one gets for the susceptibility x (which is the value of each of the x4’s) x=&

(

1+;3w,p

i

with 2 = $(37fVZ)‘/3~~-

V3 + &(2x,

- 1) + &t,(*x,

- 1)p”

Pv

in terms of the usual Skyrme force parameters t,, x,,, t,, x3, (Yand W,. In Eqs. (27) and (28), p is the total density (Z&p,) whereas f is the effective-mass form factor for either of the charge states (Vq; f, =f>. It is noticeable that the velocitydependent parts of the interaction (the terms involving c,, xi, t,, x2, namely) do not contribute to x in our case. This is due to the omission of some small terms in the Skyrme hamiltonian density as above specified. For a variety of Skyrme force parametrizations @III [36], Ska [37], RATP [38], SGI and SGII ([39], SkP 1401,SkM [41] identical for that matter to SkM* [32]) we have evaluated the corresponding susceptibility as listed in Table 2. For all these forces, the sign of x is found positive, yielding thus a spin polarization of the Pauli paramagnetic type as suggested years ago by Dabrowski in a simple model of non-interacting nucleons [26]. Analyzing in more details the various contributions to x, one finds, as demonstrated in Table 2, that for all effective interactions having an effective mass in the

521

K. Bencheikh et al. / Rotations in nuclei

Table 2 Spin susceptibilities x (in MeV-’ fm-3) for various Skyrme forces (whose refs. are given in the text). Also listed are the nuclear-matter effective mass m * in units of the nucleon mass m, the nuclear matter density p (in fmm3), the dimensionless term in the numerator of Eq. (27) depending on the spin-orbit strength parameter Wa (in MeV . fm5), the first term D in P [Eq. (28)] depending linearly on the effective mass (in MeV fm3) and the remainder D’ of P due to the spin-spin part of the force (in MeV. fm3). Also listed is th e ratio g/x where X is the spin susceptibility obtained when the spin-spin part of the force is omitted.

SGI Ska RATP SIR SGII SkM SkP

m*m

P

3mWOp/ii2

D

D’

X(X103)

X/X

0.61 0.61 0.67 0.76 0.79 0.79 1.00

0.155 0.155 0.160 0.145 0.158 0.160 0.163

1.29 1.40 1.39 1.26 1.20 1.51 1.18

255.4 254.1 229.1 208.5 195.9 194.2 152.7

287.3 135.0 146.2 113.0 121.6 63.3 - 62.5

1.1 1.6 1.6 1.8 1.7 2.4 6.0

2.1 1.5 1.6 1.5 1.6 1.3 0.6

physical domain (m * /m N 0.7-0.8) the major contribution to x is due to the constraint and the spin-orbit part of the force. The spin-spin part of the force decreases the value x obtained in its absence by N 50% for those interactions. This feature is lost for interactions whose effective-mass parameters are either too small (m*/m - 0.6) or too large (m*/m = 1). If we may consider that the latter values are ruled out, we have put some limits on acceptable values of the spin-spin contribution to x as opposed to the situation which prevailed at the time of refs. [8,421 where a broad range for this contribution was considered and even its sign was not deemed as being established. We therefore conclude that the actual figure for x resulting from Skyrme interactions having the “right” effective mass is about 2 x lop3 MeV-’ fmP3. Let us turn now to the evaluation of the functional relation concerning the currents j,. From refs. [27,281 we know that j,=

-Fy

+(Sj,),+(Sj,),, 4

where (Sj), and (Sj>, are h2-corrections beyond Thomas-Fermi (Sj,),=

‘3T:hTm [ pi’3

4

V(V*cx,)-V*cK,+

given by

-#-q+vfq.v)CYq 4

V)Vq] ++2[vqwfc7x.,)l 4 +$bqX f(VqXaq) 4 i(VXaw)4 II

-P

- aq + Qq .

(30)

528

K Bencheikh et al. / Rotations

in nuclei

and 3m2(3d) WA,

=

-2’3

fif4”

Pi”<

oq

X

wq).

(31)

We will now insert in Eq. (30) the lowest-order expression of (Ye and in Eq. (31) the lowest-order expression for D4, implying in the latter case [see Eqs. (7), (21)] similarly the lowest-order expressions for CQ j, and pq. One thus gets after some straightforward calculations

(Sj,),=

Or~h~m [wXV(f,P~/‘)].

(32)

4

To evaluate (Sj,>, now, one obtains readily

2mB, + F(P

+pq)w-

;w

(33)

where Wq is given in terms of the p,‘s by Eq. (8) and the pq’s are the solutions of the linear system of Eqs. (24). From the latter, one gets (Sj,), through Eq. (31). Let us discuss the result found in Eq. (32) for the corrective Z1*-termsdue to the orbital motion only, in the model case of an isoscalar (t/q, pq = :p> spherically symmetrical piece of nuclear matter. Writing the effective mass form factor

Qq;

fq=f=l+YP,

(34)

with y > 0 since m* G m, one gets

(Sj),=

(i2T~~~2’3mp-*/3(l +4yp)~ulXr. r

(35)

This corrective term clearly corresponds to a surface-peaked counter-rotation with respect to the rigid-body current proportional to o X r with a positive proportionality constant. We retrieve here the famous Landau diamagnetism characteristic of a finite Fermi gas. It also may be added that this expression exhibits in a compact analytical form the result obtained numerically in ref. [24]. With the expressions of the currents j, given by Eq. (291 [with the corrective terms of Eqs. (31)-(3311 and the expressions of pq resulting from the linear system

K. Be~c~eikh et al. / Rotations

in nuclei

529

of Eqs. (241, one gets the cranking fieid form factors atg at second order in A from Eq. (6). The kinetic energy density rq can be written [27,28] in a notation similar to the one of Eq. (29):

with +?9being the usual ETF functional (see e.g. Ref. 1271)in the non-rotating case, whereas the supplemental Thomas-Fe~i contribution at finite o writes upon using Eq. (18):

The second-order

contributions

Upon implementing gets simply

to rs in the rotating case are given [27,28] by:

in Eq. (38) the TF approximation

for aq of Eq. (181, one

-2/3m2

(sTq)t =

(3T2)&2 ( p;&2

-t -+x 4

ti) - [v( f4pi’3) x WI) ‘

(40)

K. Bencheikh et al. / Rotations

530

in nuclei

The spin corrections to TV at order fi2 are similarly obtained from Eq. (39) using the lowest-order expressions of jq, p4 and (Ye, particularly using Eq. (33) for W, x 0,. Finally, the so-called spin-orbit density Jq was given in ref. [28] by

-2mp,

Jq = ~

w + 3m2(37?-2’3

fq

q

p;‘“(a,

““f4’

(41)

XD,).

Here again the ETF functional expression of Jq will be obtained by taking jq, pq and 0~~ in their lowest order in the latter equation.

3. Dynamical

moments

of inertia

3.1. Explicit formulae for the moments of inertia The energy density in the laboratory interaction constants Bi as in sect. 2:

xc

system can be written using the same

A2 2m

cfqrq 4

+ T-

B3j2 - B,Cjj 4

+h(j*(VxP)-J.Vp+

where the contribution

c[i,*(VXp,)-J;Vp,]) 4

7 to A?’

~=BIP~+B~CP;+B~PV~P+B~C~~V~~~+ 4 4

contains terms depending only on pg. Using the definition of the cranking form factor QL~defined in Eq. (6) one finds B3j2+B4C.ii= 4

-iz[ha,-B,Vx(p+p,) 4

-ha,]

*j,.

(44)

K Eencheikh et al. / Rotationsin nuclei

531

Eq. (42) can be written now as

-~A~cu,*j,+22*~Jq*Wq 4 4

+$C[hS,+$ho-&VX(j+j,)].p,, 4

(45)

where the expressions of the form factors Wq and S, given in Eqs. (7), (8) have been used. Noting that for vectors fields A and B vanishing at infinity, the integral of the scalar product of A with V x B equals the integral of the scalar product of B with V X-4, one can remove in the hamiltonian density [Eq. (45)] the two terms explicitly depending on B, to get z=

&~~qTq+Y++~~[(Cxq 4

--(~a) .jq+ (Sq+ $)*pq] 4

+A2zJq. 4

W,.

(46)

Writing the current density as in Eq. (29) where QL~is given by Eq. (6), one gets 2 ( B3+Bq)jq-2B3jP+BgVX(p+Pq)+Aa0]

jq=-?$[-

+Sj,.

(47)

4 In the above equation 6 j, is the sum of the orbital (Sj,), and spin (Sjq)S second-order currents of Eqs. (31), (32). Upon grouping all terms involving explicitly the currents j, and j, on one side of the equation, summing over the charge states 4 and, since [see Eq. (4)]: -2[(&

+B,)P,+B3Pp]

= Z(l

-f,),

(48)

one gets for the total current Cj,=

If$ CP,[VX(P+ pq)]- f&3Cpq +Cf,Sj,.

4

One therefore C(a,-ao) 4

9

4

(49)

4

obtains -j,= Ca;j,TF+ 4

9

Cp,a0*[VX (p + pq)]+ FaiCp,, 4

4

(50)

532

I(. Re~ck~~kk et al. f Rotations in nuclei

where j,” is the first term in the r.h.s. of Eq. (29). It is worth noting here that the superscript TF refers to the functional dependence of the current on 0~~ at the Thomas-Fermi order and not to the Thomas-Fermi current, i.e. at h2-order one should obviously insert in the functionai expression of j;f” the form factor 0~~ at the same order. One also notices using Eqs. (291, (37) that

(51) One can thus write now the total hamiltonian LZ’), defined by

where %‘I is the non-rotating

spin-independent

density as a sum of an orbital part

hamiltonian density given by

and a spin part (Z’), defined by

+$h(

S, +

$.a)*pq+ ""Jq *W,

(54) The part of (Z’), that depends on w, contains a classical centrifugal energy and a term involving the AZ-corrective terms of Eq. (40) to the kinetic energy density. The latter can be rewritten since, integrating by parts, one finds that I( r X w) - [ V( fqpy3) One therefore

X o] d3r = -2w2/f,$”

d3r.

(55)

gets for (SQ:

(56) where rL is the distance of the point represented

by r to the rotation axis.

K Bencheikh et al. / Rotations in nuclei

533

We will now evaluate the various contributions to (Z), in Eq. (54) replacing consistently (Ye by its Thomas-Fermi expression of Eq. (18). The first term may be written [see Eqs. (20,39)] as

$f,(%L =FpqWt h2m

- +hp, * [ Dq - 2m(Wq

X

ao)].

(57)

4

The second term gives [see Eq. (2111: $h pq * (s, + 30) = $ip,

* [ Dq + m(W, X ao) + &so].

The third term which involves the spin-orbit

(58)

density of Eq. (411, writes

2h2m A2J;Wq=

- -pqWq2

fq

- hpq .m(W,

Finally the fourth term may be transformed C~P+P,)(vxPq)= 4

X a~).

(59)

by first noting that

(60)

cP,p(P+P,)] 4

and that

;~9~j%~(P +p,>(Vxp,)

d3r

9

= -T~/(P+P~)(P~*~)

d3r- $AC/p;m(W,Xq,) d3r, (61)

4

9

where the definition of W, in Eq. (8) has been used. Summing all these contributions one gets

(~)S=~+W2C[ah2-m8g(P+Pq)]xq, 4

(62)

where the susceptibilities x4 have been defined by Eqs. (26) and result from the solution of the linear system of equations (24) for the p,‘s. One obtains now the dynamical moment of inertia Y(‘) by identifying in A%? the w2-dependent terms:

(3”2)-2’3fqpfi/3 +

& +W,(p +pq)

534

K Bencheikh et al. / Rotations in nuclei

The Thomas-Fermi term which comes from the orbital motion turns out to be the rigid-body moment of inertia. Semiclassical corrections of order A2 come from both the orbital motion (YJ$,) and from the spin degrees of freedom
=

I(l+s>l

(64)

0

is identical to the dynamical moment of inertia calculated above. To get an idea about the importance of the Thouless-Valatin self-consistency terms [25,33,34] we have also calculated the moment of inertia in the ThomasFermi approximation but omitting this time the Thouless-Valatin terms in the expression of the cranking potential CU*,Eq. (6). This leads to the following expressions for the kinetic energy and the current densities (with +q being the density for vanishing w-values):

(65) and j,=

-fpqZ.

(66)

fq

Writing the hamiltonian density as

SY= ~~Tq+B3(pT-j2)+B4~(pq~q-j~)+AX, 4

(67)

4

where AZ does not depend on r4’s or jq’s, hence on o at the Thomas-Fermi order, one gets m2 ~=i+tmu:,CP4+B32h2U:ICPqPp

4 f4”

4

(68)

K Bencheikh et al. / Rotations in nuclei

53.5

From the latter one deduces as before the dynamical moment of inertia in the Inglis cranking limit (10

(69) Apart from the corrective term in pppqI’ one clearly sees that the first term in Eq. (62) - which appears to be the leading term - yields as expected [34] for m * < m (i.e. in realistic cases), a smaller moment of inertia than the quantity containing the Thouless-Valatin corrections. It is also worth noting that in this approximate case the kinematic moment of inertia is

(70) where j, is given by Eq. (66). It is quite clear that this estimate of Y(l) is very different from the dynamical moment of inertia of Eq. (69). 3.2. Results of non-fully self-consistent calculations To investigate the impo~ance of the different contributions to the total moment of inertia we have considered 31 nuclei including I%, 56Ni, %Zr, 14’Ce, 240Pu and three isotopic series for Ca (A = 36-50), Sn (A = 100-132) and Pb (A = 186-216). These semiclassical calculations have assumed the spherical symmetry in the ETF approach up to rti”corrections, corresponding to a minimization of the semiclassical energy with respect to the density profiles of both neutrons and protons for the SkM* interaction [32]. These density profiles have been chosen to have the analytic form of generalized Fermi functions allowing for an asymmetry in the diffuse nuclear surface. They have indeed been shown 144,451 to minimize the semiclassical energy extremely well (i.e. to be good approximations of exact variational densities). These densities which have been determined selfconsistently for the non-rotating nuclei have then been used as such to calculate the moment of inertia of the rotating nuclear systems. In this respect our approach is clearly not fully self-consistent. The results of such calculations are reported on Fig. 1. One immediately notices the absence of any significant isovector dependence. This fact will allow us to give a very comprehensive and compact approximate analytic parametrization of 9 as shown below. The good reproduction of the total ETF moment of inertia by its Thomas-Fermi (rigid-body) value is also quite striking. As mentioned above, the spin and orbital semiclassical corrections are not small individually but cancel each other to a large extent. To demonstrate this behaviour, the ETF moments obtained

536

K Bencheikh et al. / Rotations in nuclei

MeV-’ 1 120 i 100

80 I

t

60 --

A 0

100

200

300

Fig. 1. Calculated moments of inertia 9 (divided by h2 and expressed in MeV-‘1 as functions of the nuclear number A. Extended Thomas-Fermi results correspond to black dots f*) whereas those obtained upon neglecting the spin degrees of freedom are represented by crosses (x). Thomas-Fermi (i.e. rigid-body) calculated moments are plotted as open circles (0). Finally, plus signs f+) refer to Inglis cranking dynamical moments of inertia.

by neglecting the spin degrees of freedom are also shown. The corresponding reduction with respect to the Thomas-Fermi moments is found to vary from 43% in I60 to about 6% in “‘Pu which is consistent with the relatively smaller importance of surface effects for heavier nuclei. Including also the spin contribution leads finally to a small increase of 4 as compared to the rigid-body value. The Inglis cranking approach performed at the Thomas-Fermi level underestimates the kinematic moment of inertia substantially, by as much as 25% in heavy nuclei, demonstrating in this way the importance of the Thouless-Valatin self-consistency terms. If instead of the kinematic moment of inertia of Eq. (701, one considers the dynamical moment of inertia of Eq. (69) - in both cases in the Thomas-Fermi and Inglis cranking limits - this underestimation would be even larger by about a factor of two. Let us describe now a crude estimate for the semiclassical moment of inertia, i.e. including the semiclassical corrections of orbital and spin degrees of freedom.

K Bencheikh et al. / Rotations

in nuclei

537

For the orbital corrections alone, one can evaluate the moment of inertia for spherical liquid-drop type densities having a sharp cut-off radius of R = ~,A’J’~ where any isovector dependence is ignored (i.e. assuming pn = pP = $1. One gets

where (m*/mjNM is the nuclear-matter effective mass, which for the Skyrme force SkM * used in our calculations is 0.79. In the above Eq. (711, the moment of inertia in front of the bracket is, of course, nothing but the rigid-body estimate JirF(a ~o~esponding to our model density. In the same model, for spin corrections only, one would get

where the quantity ,Z has been defined in Eqs. (271, (28). For the Skyrme SkM* force, one gets

with qr = - 1.36 and n, = 3.19. To include now correctly the surface effects, one can perform a fit of the various parameters entering Eq. (73) to the numerical results displayed on Fig. 1. We have first made a fit of the parameter to obtain r0 = 1.213 fm, which is quite satisfactory. A separate fit of q1 and ns for r0 given as above yields: n1= - 1.94,

ns = 2.63,

(74)

whereas a fit of the total correction would lead to q1 + qs = 0.67.

(75)

With those values, one finds for a typical rare-earth nucleus (A = 170) that the total corrective term is equal to 2.2% of the rigid-body value, resulting from a - 6.3% correction for the orbital motion and a 8.5% correction for the spin degree of freedom.

4. Summary and conclusions Before summarizing the main results obtained here, let us recall what was known before our work. The rigid-body character of the Thomas-Fermi motion

K Bencheikh et al. / Rotationsin nuclei

538

was known since the early Bloch’s work [20] but only in cases where m*/m = 1. The orbital correction to the latter (at second order) was already studied in rather simple models [22-24,261 and without an explicit expression of the moment of inertia in terms of the density function. The spin correction (at lowest order) was sketched in refs. [23,26] in analogy with the electron-gas problem. However, as noted in ref. 1421,its magnitude (and maybe even its sign) was strongly dependent upon the relative contribution of the spin-spin and spin-orbit parts of the interaction, which indeed stressed the need for a realistic description of both through well studied phenomenological forces as those used here. Finally, to the best of our knowledge, no study of the Thouless-Valatin corrections over the Inglis cranking formula has been attempted previously. Within the formalism of the semiclassical ETF approach, we have provided for the most general Skyrme forces (up to the neglection of some small terms in the energy functional though) rather simple mathematical formulae to express the moments of inertia in terms of the local densities. This has allowed us to assess the relative importance of spin and orbital degrees of freedom as well as of the so-called Thouless-Valatin self-consistency terms. In the realistic case where the effective mass is not equal to the nucleonic mass in infinite nuclear matter, it has in particular been shown that the Inglis cranking formula provides a very poor estimate. From both densities (current and spin densities for that matter) and moments of inertia, we have retrieved in this case the competition between an orbital Landau diamagnetism and a spin Pauli paramagnetism which is found for a confined electron gas. Both effects almost cancel, allowing the rigid-body ansatz, which corresponds to the Thomas-Fermi order, to be indeed a rather good approximation. Further works presently under completion are performed along two main directions. We are making fully self-consistent semiclassical calculations [46] to study equilibrium shapes of rotating nuclei. We are also extending the present approach to finite temperature and also to including pairing correlated solutions. It is a pleasure to acknowledge fruitful discussions with J. Meyer, E. Chabanat, I.N. Mikhailov, J. Libert and R.K. Bhaduri. 5. Appendix Explicit functional Fermi case

expressions of the fields (Ye and the currents j,

in the Thomas-

Let us consider the 4 x 4 linear system for the unknown quantitites

cuq and j,:

a9 =c+Gi,+ejp,

(A.la)

J4 = aqaq,

(A.lb)

K Benche~kh et al. / Rotatiam

in nuclei

539

where c does not depend neither on a4 nor on j4 and where 4 is a charge state and p the other charge state. The coefficients d and e are given in terms of the Skyrme force coefficients Bi of ref. [141 (see also Table 1) by: d-

-2(B, +4)

(A.2a)



h

- 2B,

(A.2b)

e=--X-Upon defining D as D = 1 - (a, + a,)d

+ a,a,(d2

- e’)

(A.31

the solution of the system of Eqs. (A.11 writes Dj,=en,[l-+a,(e-d)],

(A.4a)

Dai, = c[l + aP( e - d)] .

(A.4b)

Using explicit expressions for the a,‘~: Lz4= -

mp9

(A.5)

% and the &‘s in the notation of Eq. (A.la):

(A-6) one finds 1 -da,=

1+ ea,f, f,

+

(A-7)

The quantity D of Eq. (A.31 can then be rewritten as

(A.81 In the Thomas-Fermi case, the form factors CL~and the currents j4 are indeed given by Eqs. (Ala, b) with c being exactly LYE[as defined in Eq. (1411. Upon using Eqs. (A.7)-(A.81 one gets

otw-q -f*%

(A.91

K. Bencheikh et al. / Rotations in nuclei

540

6. References [l] [2] [3] [4] [5] [6] [7] [8] [9]

[lo] [ll]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46]

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