Investigation of the atom-atom and structural relaxation in liquid alkali metals by means of the memory function formali
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IFFRACTION AND SCATTERING OF IONIZING RADIATIONS
Investigation of the Atom–Atom and Structural Relaxation in Liquid Alkali Metals by Means of the Memory Function Formalism N. M. Blagoveshchenskii, A. G. Novikov, and V. V. Savostin Institute for Physics and Power Engineering, Obninsk, 249033 Russia email: [email protected] Received November 17, 2010
Abstract—An attempt is made to systematize the data on the relaxation characteristics of liquid alkali metals (Li, Na, and K), which were investigated based on neutronscattering data with the application of the two time memory function formalism. DOI: 10.1134/S1063774511070066
INTRODUCTION The motion equation for an atom of mass m in a liquid in a field of external forces U(r, t) (the Newton formula), 2
(1) m d r2 + gradU (r, t ) = 0 , dt can be modified taking into account the effect of the thermostat by consistently supplementing (1) with a friction force which is proportional to the velocity (the limiting transition of the Stokes formula, F = 6πμRV, where μ is the viscosity of the medium, at R ra) and a random external force R(t), which is due to the simultaneous effect of many environmental particles on the atom: 2 (2) m d r2 + mγ dr + gradU (r, t) = R(t) . dt dt The simplest theory of Brownian motion is based on the phenomenological Langevin equation [1] for the atomic velocity V: (3) m dV = −mγV (t) + R(t), dt which coincides with the modified Newton equation (2) in the absence of a potential of external forces. The random component R(t) is such that 〈R(t)〉 = 0 and 〈V(t)R(t)〉 = 0. (4) In other words, the ensembleaveraged R(t) is zero and does not correlate with the atomic velocity. The main drawback of the Langevin equation (3) is that the response of the environment in it is deter mined only by the instantaneous velocity; i.e., at any specified instant of time t, when the atom has a veloc ity V(t), the liquid should instantaneously recover its equilibrium state corresponding to this velocity, thus eliminating this effect of memory with respect to the previous dynamics of the atom. The Langevin equation (3) can be generalized using the formalism of the memory function Γ(t),
which is introduced as a timedependent analog of the friction coefficient in the expression for the environ mental response force in terms of the integral that takes into account the memory about the previous particle dynamics as a result of the convolution of Γ(t) with V(t) [2]: t
∫
mdV / dt = −m Γ(t − τ)V (τ)d τ + R(t ).
(5)
0
Note that this heuristic expression of the force as an integral taking into account the memory about the previous atomic motion via the memory function Γ(t) and the random force R(t), satisfying conditions (4), is exact in reality, as was shown in [3]. The equation for the velocity autocorrelation func V (0)V (t) can be written as tion (VAF) Ψ(t) = V (0)V (0) t
d Ψ(t ) = − Γ(t − τ)Ψ(τ)d τ dt
∫ 0
(6)
t
d Ψ(t ) = − Γ(τ)Ψ(t − τ)d τ. dt
∫
or
0
The kernel of this integrodifferential equation, which is also referred to as the Volterra equation, is the mem
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