научная статья по теме SEMICLASSICAL SHELL-STRUCTURE MOMENT OF INERTIA FOR EQUILIBRIUM ROTATION OF A SIMPLE FERMI SYSTEM Физика

Текст научной статьи на тему «SEMICLASSICAL SHELL-STRUCTURE MOMENT OF INERTIA FOR EQUILIBRIUM ROTATION OF A SIMPLE FERMI SYSTEM»

ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 8, с. 1442-1449

ЯДРА

SEMICLASSICAL SHELL-STRUCTURE MOMENT OF INERTIA FOR EQUILIBRIUM ROTATION OF A SIMPLE FERMI SYSTEM

©2010 A. G.Magner1), A. S. Sitdikov2)*, A. A. Khamzin2), J. Bartel3), A. M. Gzhebinsky1)

Received December 28, 2009

Semiclassical shell-structure components of the collective moment of inertia are derived within the mean-field cranking model in the adiabatic approximation in terms of the free-energy shell corrections through those of a rigid body for the statistically equilibrium rotation of a Fermi system at finite temperature by using the nonperturbative extended Gutzwiller periodic-orbit theory. Their analytical structure in terms of the equatorial and 3-dimensional periodic orbits for the axially symmetric harmonic oscillator potential is in perfect agreement with the quantum results for different critical bifurcation deformations and different temperatures.

1. INTRODUCTION

The collective rotations of nuclei were successfully described within several theoretical approaches, in particular, the cranking model [1—5]. It was shown [4, 5] that the moments of inertia (MI) can be presented as a sum of the smooth classical rigid-body term with h corrections of the Extended Thomas-Fermi approach (ETF) [6-8] and shell corrections [4] given by the shell-correction method (SCM) [9, 10] adjusted to the rotational problem. Exact analytical solutions for any rotational frequency were derived for the harmonic oscillator potential [11], and extended to finite temperatures in [12]. It is worth to apply [12-16] the semiclassical periodic-orbit theory (POT) [6, 17, 18] as one of the powerful and fruitful theoretical tools for a deeper understanding and analytical analysis of the main features of the shell structure in a finite rotating Fermion system. For the collective rotations of deformed nuclei this structure was considered semiclassically in [16] by using the perturbation theory of Creagh [6, 19] within the spheroid cavity model.

In the present work the shell-structure corrections to the MI for the collective rotation are derived within the cranking model in terms of the free-energy shell corrections in the adiabatic approximation by using the nonperturbative POT based on the semiclassical Gutzwiller expansion for the Green's function [17], but extended here to systems of higher symmetries [6, 18]. Explicit analytical results are obtained for the deformed harmonic oscillator potential.

'-'Institute for Nuclear Research, NAS of Ukraine, Kyiv.

2)Kazan State Power-Engineering University, Russia.

3)Institut Pluridisciplinaire Hubert Curien, CNRS/IN2P3,

Universite Louis Pasteur, Strasbourg, France.

E-mail: airat_vm@rambler.ru

2. NUCLEAR-ROTATION CRANKING MODEL Within the cranking model, the nuclear rotation around the x axis perpendicular to the symmetry z axis of the axially symmetric mean-field potential V(r) can be described by solving the eigenvalue problem for the single-particle (s.p.) Hamiltonian in the body-fixed rotating coordinate system, which is usually called the Routhian [3-5],

H — H — u£x, (1)

{£x = ^ ni {£x X)i — Ix • i

Here, £x is the operator of the angular momentum projection onto the x axis, and ds is the spin (spinisospin) degeneracy. The Lagrangian multiplier u (rotation frequency of the body-fixed coordinate system) is defined through the constraint on the nuclear angular momentum Ix evaluated as the quantum average of the operator £x, as done in Eq. (1) yielding U — u(Ix).

The particle number conservation determines the chemical potential A through the Fermi occupation numbers ni in the s.p. state i, N — ds^2i ni, where ni = n (ei) — {1 + exp [(ei — A)/T]}_1, with the eigenvalues ei of Hamiltonian H and the temperature T. For the MI 8x one has [14, 15, 20, 21],

©x — (d{£x)„/du)u=0 — {d2E(u)/du2)^=Q , (2)

where E(u) — {Hw) + uIx is the energy of the rotating Fermi system. The yrast line E(Ix) can be determined by eliminating the frequency u through the definition of the kinematical MI ©x — Ix/u (equivalent to the dynamical MI, Eq. (2), in the adiabatic approximation) yielding at zero temperature E(Ix) — E(0) + I2x/1©x. In the case of the

deformed harmonic oscillator (HO) potential the spectrum of the Hamiltonian of Eq. (1) is given by £i = hu± (N±i + 1)+hwz (Nzi + 1/2) ,where N±i = = Nxi + Nyi, with NKi and uK (k = x, y, z) are the HO quantum numbers and partial frequencies, respectively. In this case one finds [2, 3] for the MI (see also [11, 12])

e* =

ds h 2u±uz

(uz - u±Y

+ <jJz

+

(uz + u±y

Lü_L - LOz

(Xz - Xy)

= n (NKi + 1/2 )

In the case of a statistically equilibrium rotation

U*X* = Uy Xy = Uz Xz,

e*g = m J drp (r) (y2 + z2) = = dsh (Xy/u± + Xz/Uz),

representation through the Green's functions G [12, 21],

<X>

ex = (2ds/n) J den(e) x

(7)

(Xy + Xz )+ (3)

(4)

the moment of inertia (3) equals the rigid-body value given by

(5)

x J dT! J dr2£x(r1)£x(r2)Re [G (n, r2; e)] x

x Im [G (ri, T2; e)],

where n(e) are the Fermi occupation numbers n(ei) at ei = e and £x(r1) and £x(r2) are the s.p. angular-momentum projections onto the perpendicular rotation x axis at the spatial points r1 and r2, respectively. With the usual energy-spectral representation for the one-body Green's function G in the mean-field approximation, one obtains from (7) the well-known second-order perturbation result of the cranking model [3—5] including the diagonal terms. For the Green's function G we shall use in (7) the semiclas-sical Gutzwiller trajectory expansion [17] extended to the symmetries of Hamiltonian [6, 18, 22],

G (ri, r2; e) = £ Ga = (8)

where p (r) is the particle density and m is the nucleon mass. The second term for 6x in Eq. (3) corresponds to transitions between s.p. levels inside a major N shell, AN = 0, in contrast to the first component related to the coupling of s.p. levels through shells, AN = 2 [3]. In the spherical limit, this term is reduced identically to the diagonal alignment moment of inertia 6z, 6x — 6z — — -d^i(dni/dsi)\{i\£x\i)\2 [12].

For the shell correction calculations, it is convenient to re-write the MI 6x, Eq. (3), for the inertia 6x in terms of the free energy F of the HO system at

finite temperature T and the rigid-body inertia ©xg, Eq. (5) by eliminating the quantum numbers and . Finally, from Eq. (3), one obtains the explicit expressions of S©x and S©z in terms of the SCM free-energy shell corrections SF [12],

S©x = [(1 + n2) /3ul] SF, (6)

S©z = (2/3u\) SF, n = ^±/uz

with uz = and the obvious spherical limit S©x — S©z at n — 1.

3. SEMICLASSICAL SHELL-STRUCTURE APPROACH

For the derivations of shell effects within the POT, it turns out to be helpful to use the coordinate

^ Aa (ri, r2; e) exp

i ci n in

h

2

The summation index a runs over all classical isolated paths that, for a given energy e, connect the two spatial points r1 and r2 inside the potential well V(r). Here, Sa is the classical action along such a trajectory a, and ¡ia denotes the phase associated with the Maslov index determined by the number of caustic and turning points along the path a [6, 18]. The amplitudes Aa of the Green's function depend on the classical stability factors and trajectory degeneracy, due to the symmetries of that potential [6, 17, 18, 22].

Among all classical trajectories a in Eq. (8), we may single out one a0 which connects directly r1 and r2 without intermediate turning points. Thus, for the Green's function G, Eq. (8), one has a separation,

G = Gao + Gi ~ Go + Gi, (9)

where Ga0 is associated with the a0 component of sum (8),

Ga0 ~ Go = —

r — ri|

m

2nh2 s

exp

-sp (r)

(10)

p (r) = y/2m[e - V(r)\

for points that are spatially close r1 — r2 — r in the nearly local approximation [21, 23, 24].

The second fluctuating term G1 in Eq. (9) is determined by all other trajectories a in (8) besides

o

a

a

s

ftŒPHAfl OH3HKA TOM 73 № 8 2010

8*

of a0 [18, 20, 22]. According to the approximate separation (9) one obtains from (7)

8x = ex° + BS1 + ©X° + eX*, (ii)

with

eXf = (2ds/n) J den(e) x (12)

x J dri J dr2ix (ri) tx (r2) Re [Gv (ri, r2; e)] x

x Im [Gv' (ri, r2; e)].

As shown in [21], the first component of Eq. (11), eX°, averaged over the phase-space variables in the local approximation, leads to the Thomas—Fermi rigid-body moment of inertia because such an averaging removes not only nonlocal oscillating terms, but also the h corrections of the ETF approach [7, 8]. The shell-structure component ¿eX1 of eX1 in (12) for the MI can be related semiclassically to the shell correction 5p(r) of the particle density p(r) through

that of the rigid body MI ¿eXg. Indeed, calculating eX1, Eq. (12), through using the nearly local approximation Go (v = 0) for Ga0, Eq. (10), and Gi (v' = = 1) we select the shell component ¿eX1 of eX1 just like the free energy shell corrections ¿F. Using the transformation of ri and r2 to the center-of-mass and relative coordinates, r = (ri + r2)/2 and s = r2 — ri, as well as the spherical system of coordinates for s, ds = s2ds sin dsddsd^s one obtains for an almost equilibrium rotation

¿eX1 = — [dsm/(nh)2] y de¿n(e) x (13)

x J dr j sds J sin dsddsdystX (r — s/2) x

x tX (r + s/2) cos [p(r)s/h] x x Im [Gi (r — s/2, r + s/2; e)] w ¿eXg =

= m J dr (y2 + z2) ¿p (r).

For the classical angular-momentum projection in the integrand of the first equation we used the approximation tX (r — s/2)tX (r + s/2) w tX (r) = (y2 + + z2)p2(r). The classical angular-momentum projection tX(r) in the rotating frame is caused in the adiabatic approach [3, 12] by the global rotation rather than by the motion of particles along the trajectories a inside the nucleus. We then integrated in Eq. (13) explicitly over s and found sine squared in the integrand over e and r with mean value 1/2 for the averaging over energies e. Adding and subtracting

identically this value 1/2 from the sine squared we first integrate over spherical angles 6S and ys in the term related to this 1/2, writing simply 4n

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