ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 11, с. 1415-1426
ЯДРА
GAMMA RADIATION IN NON-MARKOVIAN FERMI SYSTEMS
© 2014 V. M. Kolomietz1)*, S. V. Radionov1), B. V. Reznychenko2)
Received December 13, 2013; in final form, March 2, 2014
The gamma-quanta emission is considered within the framework of the non-Markovian kinetic theory. It is shown that the memory effects have a strong influence on the spectral distribution of gamma quanta in the case of long-time relaxation regime. It is shown that the gamma radiation can be used as a probe for both the time-reversible hindrance force and the dissipative friction caused by the memory integral.
DOI: 10.7868/S0044002714100109
1. INTRODUCTION
The dynamics and the dissipative properties of the many-body Fermi system depend in many aspects on the dynamic distortion of the Fermi surface in momentum space. As is well known [1], the presence of Fermi surface distortion allows the description of so-called collisional mechanism of relaxation and gives rise to the damping of collective motion. An additional one-body mechanism of relaxation exists in the finite system where the particles are placed into the external mean field. The origin of this damping is the collision of the particles with moving potential wall [2]. We will consider below both of them.
On the other hand, relaxation of collective motion implies fluctuations in the corresponding collective variables, as follows from the fluctuation-dissipation theorem. Furthermore, the fluctuations in a particle density imply an accelerated motion of charges inside the charged system like a nucleus and lead, therefore, to radiation. The spectral distribution of this fluctu-ational radiation depends on the relaxation (dissipation) properties of the collective motion, in particular, on the dynamic distortion of the Fermi surface. We therefore suggest that a study of the shape of the radiation spectrum emitted from the heated system provides an opportunity to obtain information on the effects of temperature on dissipative properties and on the transition from the low-temperature (quantum) to the high-temperature (classical) regime in a finite many-body system.
In the present paper, we are interested in the spectrum of fluctuations in shape variables. The precise form of such spectra can be expected to depend on the parameters of the model, such as the collision time, and, especially, on the memory effects. Here, we
!)Institute for Nuclear Research, Kiev, Ukraine.
2)Taras Shevchenko National University of Kyiv, Ukraine. E-mail: vkolom@kinr.kiev.ua
want to study these dependencies as one step to our ultimate goal of determining the model parameters from a comparison with experimental data, as might be possible due to a relation of the above-mentioned spectra to y spectra.
In what follows, we combine the thermal and quantum fluctuations by means of the fluctuation-dissipation theorem. Such an approach presents a convenient connection between different regimes of collective motion such as the quantum zero-sound regime at zero temperature and the collisional firstsound regime in a hot system. Such an approach presents a convenient connection between different regimes of radiation such as the quantum regime at zero's temperature and the thermal black body radiation of a hot system.
This paper is organized as follows. In Section 2 we suggest a proof of the Langevin equation for the macroscopic collective variables starting from the collisional Landau—Vlasov kinetic equation, including the memory effects in the collisional integral. In Section 3 we review the classical approach to the fluctuational radiation. We adopt a Langevin equation with a random force as a source of the fluctuations. The main features of the dynamic distortion of the Fermi surface are taken into account. In Section 4 we apply the results of Section 2 to the analysis of the spectral density of the fluctuational radiation. Concluding remarks are presented in Section 5.
2. SURFACE FLUCTUATIONS
IN A FINITE FERMI SYSTEM
To consider the fluctuations which accompany the collective motion in many-body Fermi system, one can start from the collisional kinetic equation in presence of a random perturbation y [1,3]
d_ p
dt m
Vrf -VrU -Vpf = St[f]+y, (1)
where f = f (r, p; t) is the phase-space distribution function, U = U(r, p; t) is the self-consistent mean field and St[f] is the collision integral. The momentum distribution is distorted during the time evolution of the system and takes the following form
f (r, p; t) = feq(r, p)+Sf (r, p; t) = = fsph(r p; t)+ ^ flm(r, p;t),
lm,l>i
where fsph(r, p; t) describes the spherical distribution in momentum space, l is the multipolarity of the Fermi-surface distortion, 5flm is the component of the l, m is multipolarity in p space of the variation 5f, and feq(r, p) is the equilibrium distribution function. We point out that the traditional time-dependent Thomas—Fermi (TDTF) approximation is obtained from Eq. (1) if one takes the distribution function f(r, p; t) in the following restricted form fTF(r, p; t) = fsph(r, p; t)+Sfi=i(r, p; t) instead of Eq. (2), see [4, 5]. Below we will extend the TDTF approximation taking into consideration the dynamic Fermi surface distortion up to multipolarity l = 2 only and assume
Sf = -
9/
de
l=2
Y,Sfim(r,t)Vim(p). (2)
eq
d_ dt
Sf + Löf = SSt[f ]+y,
(3)
where SSt[f] is the collision integral linearized in Sf = / — feq and the operator L represents the drift term
Löf = — • Vröf - Vrf/eq • VpSf - VrÖU • Vp/{
m
PJ eq-
The collision integral SSt[f ] depends on the transition probability of the two-nucleon scattering with initial momenta (pi, p2) and final momenta (pi, p'2). At low temperatures T < eF the momenta (pi, p2) and (pi,p2) are localized near the Fermi surface and the relaxation time approximation can be used, see [1, 3],
¿St[/] = --5/|г>!,
(4)
where t is the collisional relaxation time. The notation l > 1 means that the perturbation 5f |l>i in the collision integral includes only Fermi-surface distortions with a multipolarity l > 1 in order to conserve the particle number in the collision processes [1]. The inclusion of the l = 1 harmonic in the collision integral of Eq. (4), at variance with the isoscalar case [6], is due to nonconservation of the isovector current, i.e. due to a collisional friction force between counterstreaming neutron and proton flows [7]. The relaxation time t depends on the temperature and contains, in the general case, retardation effects (u dependence) [8—12]:
г = г ) =
4 n2ßh
(M2 + ZT2'
(5)
Here, e is the single-particle energy and (df/de)eq ~ ~ 5(e — eF), where eF is the Fermi energy [1]. Below we will restrict ourselves to the azimuthally symmetric case (longitudinal perturbation) where 5flm is m independent.
We will consider a linear response to the external random perturbation y. The linearized kinetic equation (1) is given by
where f3 and Z are constants which are derived by the in-medium nucleon—nucleon scattering. Note that the well-known Landau's prescription [8] assumes Z = 4n2. The parameter /3 in Eq. (5) is rather badly established. It depends mainly on the in-medium nucleon—nucleon scattering cross section aNN. For example, this value was calculated in [13] and [14] with the results between 3 = 2.4 and 3 = 19.3 for different assumptions about the scattering cross section O NN.
Evaluating the first three moments of Eq. (3) in p space and taking into account the condition (2), we can derive a closed set of equations for the following moments of the distribution function, namely, local particle density p, velocity field uv, and pressure tensor in the form of the continuity and Euler-like equations (for details, see Appendix and [3, 4]). We will restrict ourselves by the shape fluctuations of Fermi liquid assuming an incompressible and irrotational flow, i.e.,
Vv Uv = 0 (6)
and assuming also a sharp particle distribution in r
space
p = pc© [R(t) — r]. (7)
For the description of small amplitude oscillation of a certain multipolarity L we specify the surface as
r = R(t) = Ro
1 + aLu(t)YLM(0,
M
. (8)
The basic continuity and Euler-like equations can be then reduced to the following Langevin equation (see Appendix)
—u2mLaLM,u, + (CiiLD) + C'L (u))aLM,w — (9)
— iuYL (u)aLM,^ = Zlm,u ,
where the index u means the Fourier transformation for the corresponding values and ZLM^ is the random
m
force which occurs due to the random perturbation y in Eq. (1). Note that the u dependency of the friction, YL(u), and the stiffness, CL(u), coefficient in Eq. (9) (see below) occurs because of the memory effects in the Fermi liquid. It was shown in [3] that both the Fermi-surface distortion and the relaxation on the distorted Fermi surface provide the non-Markovian character of the collective motion and thereby the memory effects. Note also that the origin of such kind of memory effect is significantly different than the retardation effect (u dependency) in the relaxation time of Eq. (5).
The left part of Eq. (9) derives the eigenfre-quency of surface eigenvibrations of the incompressible Fermi-liquid drop. Namely, the corresponding secular equation reads
—u2mL + C(LD) + C'L (u) — iuYL(u) = 0. (10)
In Eq. (9), the mass coefficient mL is given by (we assume the irrotational motion of liquid for strong collectivized Giant Multipole Resonances (GMR), see [15])
mL = m J drpe^y^ \abM,v |
3
4nL
and the static stiffness coefficient is derived
from the elastic properties of the system
C
(LD)
= ^(L_l)(L + 2 )bsA2/s-
4n
L-l Z2 2L + 1 CA1/з:
bs = 17.2
3 e
Tl, - T2
T2it + T2
5/4
[MeV],
6c = ^-(1- xCT') » 0.55(1 - xcTz) [MeV], where rc = 1-24 fm [15], the parameter xc was cho-
sen as xC = 0.76 x 10-3 MeV"2 and Tcrit = 18 MeV is taken as the critical temperature Tcrit for infinite nuclear Fermi liquid [16]. The nuclear Fermi liquid does
not exist for temperatures T > Tcrit. Using Eq. (12),
one can find a limiting tem
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