ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 4, с. 677-687
ЯДРА
SEMICLASSICAL INERTIA OF NUCLEAR COLLECTIVE DYNAMICS
© 2007 A. G. Magner*, A. M. Gzhebinsky, S. N. Fedotkin
Institute for Nuclear Research, Kyiv, Ukraine Received April 14,2006
For the low-lying collective excitations in nuclei the transport coefficients, such as the stiffness, the inertia, and the friction, are derived within the periodic-orbit theory in the lowest orders of semiclassical expansion corresponding to the extended Thomas—Fermi approach. The multipole vibrations near the spherical shape are described in the mean-field approximation through the infinitely deep square-well potential and Strutinsky averaging of the transport coefficients. Due to the consistency condition, the collective inertia for increased enough particle numbers and temperatures is essentially larger than that of irrotational flow. The average energies of collective vibrations, reduced friction, and effective damping coefficients are in better agreement with experimental data than those found from the hydrodynamic model.
PACS:21.60.Ev, 21.60.Cs, 24.10.Pa, 24.75.+i
1. INTRODUCTION
Many theoretical approaches were suggested for the description of collective dynamics at low excitation energies of complex nuclei, such as the vibration modes and fission processes [1—3]. One of the most powerful tools for this description is based on the response-function theory [1, 4]. The collective variables are introduced explicitly as deformation parameters of a mean single-particle field. The nuclear excitations are parametrized in terms of the transport coefficients, like stiffness, inertia, and friction parameters, through the adequate collective response functions. The quantum formulation of this problem can be significantly simplified by using the Strutinsky shell-correction method (SCM) [5— 7] in the semiclassical approximation [8], within the periodic-orbit theory (POT) [7, 9—11]. It would be worth to apply the ideas of SCM to the calculations of transport coefficients in the zero-frequency limit [4].
For the energy dissipation rate, assuming absence of two-body dissipation, a simple macroscopic derivation of the famous wall formula for the friction, due to collisions of particles of the perfect Fermi gas with a slowly moving surface of the mean-field potential, was suggested in [12]. From a quite general classical (Thomas—Fermi) and quantum starting point, Koonin and Randrup [13] have rederived this wall formula for average friction with trajectory corrections. For the quantum case, the multiple-reflection expansion of Green function, based on the iteration procedure with zero approximation for free-particle motion by Balian and Bloch [14], was used
E-mail: magner@kinr.kiev.ua
in [13, 15, 16]. Other independent derivations of the wall formula with focus on specific averaging procedures, and its fruitful applications can be found, e.g., in [12, 17, 18]. The SCM averaging procedure and Thomas—Fermi (TF) approach for level density were applied in [18] for relating the wall formula to the averaged quantum friction coefficient.
In the present paper, we have derived the explicit analytical expressions for the average friction and inertia within the Gutzwiller path-integral version of the POT [9, 11] at leading orders in h. In Section 2, following [4] we begin with a general response-function formalism. For the infinitely deep spherical square-well potential, like in [13, 15, 17, 18] for the friction case, we obtained in Section 3 the inertia parameter averaged over many single-particle states near the Fermi surface. With the consistency condition [4, 19], the smooth nuclear vibration energies, the reduced and effective friction coefficients are derived in Section 4. In this section, our analytical results are compared with some experimental data on the low-lying vibration energies [20—22] and on nuclear fission [3, 23, 24]. The semiclassical results are summarized in the conclusion section. Some details of the semiclassical and liquid-drop calculations can be found in Appendices.
2. RESPONSE THEORY AND TRANSPORT COEFFICIENTS
Many-body collective excitations are conveniently described in terms of the nuclear response to an external perturbation Vext = F with a vibration amplitude and one-body operator F. Its quantal
average perturbation 5(F)t at time t can be calculated through the Fourier transform 5(F)u obtained within the linear response theory [1,4]:
S( F )u = -xFF MCt, F =
9V\
9QJq=Qo
(1)
as the coefficients of expansion of the intrinsic response function x(u) in u in the zero-frequency limit, u — 0, for slow enough collective motion [4, 19]:
C (0)
C =
where xFf (u) is the collective response function. The Hamiltonian at qfxt = 0 depends on a collective variable Q as a time-dependent deformation parameter of the mean potential V(Q), Qo is a static deformation. For the vibrations of the axially symmetric nuclear surface with a multipolarity L near the spherical shape one writes R(9,Q) = R[1 + Q(t)Ybo(9)], Q(t) = Que-iWt, in the spherical coordinates r, 9, The unperturbed quantities are zero in this case, e.g., Q0 = 0. We emphasize importance of the consistency condition [4]
M =
Y
1 +
1 + 1 +
x(0)_
x(0) g(0) x(0)
C (0),
)
Y(0),
(5)
M(0) +
7(0)2
X(0)
The intrinsic response x(u), see Eq. (3), can be expressed in terms of the one-body Green function G [8, 16]:
x(w) = — J den(e) J dr\ x 0
ö(F)u = KÖQu, k = -x(0) - C(0), (2) XJ dr2£(ri)F(r2)ImG(ri, r2,e) x
x [G(ri, r2,e — hu) + G(ri, r2,e + hu)]
(6)
where k is the coupling constant; C(0) is the stiffness, C(0) = id"2F/OQ2) q=0; F is the nuclear free energy; x(0) is the isolated susceptibility. With help of this condition, the collective response xFf (u) can be expressed in terms of the so-called intrinsic response function x(u) [1, 4]:
x(u)
G=
iwi
e — £i + ie'
xFF (u) = k
x(u) = —
x(u) + k ' S(F)„
(3)
One dominating peak in the collective strength function Imx^j,(u) of Eq. (3) at low excitation energies is assumed to be well separated from all other solutions of the secular equation x(u) + k = 0. In this case, the response function in the Q mode can be conveniently written in an inverted form:
where n(e) is the Fermi occupation numbers at the energy £ for temperature T, n(e) = {1 + exp[(£ — — A)/T]}-1, A is the chemical potential. The factor of 2 accounts for the spin degeneracy. For the Green function G(r1, r2,£) (bar above G means the complex conjugation), we may use the energy spectral representation of Eq. (6) with eigenvalues £j, eigenfunctions |f), and e — +0 in the mean-field approximation.
With Eq. (6) for the intrinsic response function x(u) in the zero-frequency limit u — 0, one has the friction 7(0) and the inertia M(0), see [4, 16, 19]:
1
1
m
(7)
w=0
xQQ(u) xqq(u)
+ k & —Mu2 — iYu + C, (4)
2h
xQQ(u)
x(u)
de
n
dn(e) de
dri x
(See more detailed explanations of this approach and its applications to the collective nuclear dynamics in [4, 19].) Here, the inverse collective response function for low frequencies is approximated by the corresponding response function of a damped harmonic oscillator with the stiffness C, the inertia M, and the friction 7 parameters, as shown in Eq. (4). Such self-consistent transport coefficients were related in [4, 19] to the auxiliary parameters C(0), 7(0) = = -i\dx/d^=0, and M(0) = [(1/2)d2x/dw%=c
x dr2F(ri) F(r2)[ImG(ri, r2,e)]2
M(0) =
1 fd2x(u)
2 I du2
(8)
w=0
2h2
n
den(e) dri dr2F(ri)F(ri) x
d2
x ImG(ri,r2,e)Tr^ReG(ri,r2,e). de2
2
2
k
With the spectral representation of Eq. (6) for the Green function G, Eq. (8) is equivalent to the well-known cranking model inertia in the mean-field limit, see, e.g., [4].
3. SEMICLASSICAL APPROACH
The intrinsic response function %(w) (6) can be found with help of the semiclassical expansion of Green function G derived by Gutzwiller [7, 9] from the quantum path-integral propagator,
G(ri, r2 ,e) = G a (ri, r2 ,e)
(9)
x exp
2 nh2
i
in
-Sa(ri,r2,e) - — ßa
The index a covers all classical paths inside the potential well which connect the two spatial points ri and r2 for a given energy e, and Sa is the classical action along such a trajectory a. The ¡ia denotes the phase related to the Maslov index through the number of all caustic and turning points of the path a [7]. The Jacobian Ja(p1,ta; r2,e) for transformation from the initial momentum p1 and time ta of the particle motion along the trajectory a to its final coordinate r2 and energy e is derived in Appendix A by means of the technics explained in [7, 9, 11].
Among all classical trajectories a we may single out a0, which directly connects r1 and r2 without reflections from the potential well edge in intermediate points, see Fig. 1. For the Green function G (9), one has then a separation, G = Ga0 + Gosc, which leads to the corresponding splitting of the level density (averaged over the energy spectrum) into a smooth part and its shell correction, see [7, 9—11]. The trace of the first term, Ga0(r1, r2,e) (r2 — r1), corresponds to a smooth term of the level density gETF(e) in the statistical extended Thomas—Fermi (ETF) model [7]. The trace of the second contribution, Gosc (r1, r2 ,e), coming from all other closed trajectories, a = a0, represents its oscillating part gosc (e) in terms of a POT sum over the periodic orbits в, see [10, 11]. The component gosc(e) describes the shell effects in the single-particle spectrum.
The semiclassical friction y(0) and inertia M(0) can be found by substitution of the trajectory expansion of Green function (9) into Eqs. (7) and (8). For the integration over spatial coordinates r1 and r2 in Eqs. (7), (8), we specify the coordinate dependence of the single-particle operator F(r)(1) for the multipole
Fig. 1. Trajectory a0 from the initial point ri to the final point r2, and its characteristics; p1 is the initial momentum; ppi is cylindrical component perpendicular to the axis z'; p2 is the cor
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