The Fokker-Planck Approach to Thermalization

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows  

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where 

In the preceding F = fc(r, r), H = tc(r, vt)G = k(vt, v) and the normalization constant C is fixed by equating the volume integral of n to unity. For further tractability, Sano and Mozumder expand (r v) in a Taylor s series and retain the first two terms only. The validity of this procedure can be established a posteriori in a given situation. At first, the authors obtain equations for the time derivatives of the expectation values and the correlations of dynamical variables. Then, for convenience of closure and computer calculation, these are transformed into a set of six equations, which are solved numerically. The first of these computes lapse time through the relation 

With this distribution, the escape probability is calculated using the Onsager (1938) formula (see Chapter 9), 

Over the temperature interval 165 K to 300 K, the calculations of Silinsh and Jurgis (1985) indicate that the thermalization rate in pentacene decreases from 3 X 1012 to 0.8 X 1012 s 1. The trend is opposite to what would be expected in liquid hydrocarbons and may be attributed to the rapid increase of mcff with temperature. The calculated mean thermalization distance increases with incident photon energy fairly rapidly, from 3 nm at 2.3 eV to 10 nm at 2.9 eV, both at 204 K. With increasing temperature, (r) decreases somewhat. These thermalization distances have been found to be consistent with the experimental photogeneration quantum efficiency when Onsager s formula for the escape probability is used. 

---