On the Influence of Magnetic Field on the Probability of Diffusing Particle Capture by Absorbing Traps

Abstract

The influence of a magnetic field on the capture of diffusing particles into traps is considered. It is shown that the survival probability for particles in a medium with adsorbing traps in a magnetic field depends on the magnetic field strength; expressions for temporal asymptotic forms of the survival probabilities in media with traps in a magnetic field are derived. The magnetic field influence on the survival probability is determined by a change in the curvature of diffusion trajectories due to rotation of particles in the magnetic field under the action of the Lorentz force.

Effect of External Fields on the Temporal Asymptotic Form of the Survival Probability in Media with Absorbing Traps

The electric field is introduced into the problem of diffusion in media with absorbing traps in the standard manner as anisotropy in the direction along and against the electric field. Accordingly, the diffusion equation in the electric field assumes the form

$$\frac{{\partial W(x,t)}}{{\partial t}} = D\frac{{{{\partial }^{2}}W(x,t)}}{{\partial {{t}^{2}}}} - {v}\frac{{\partial W(x,t)}}{{\partial x}}.$$

((A.1))

Here, ${v}$ = μ_E is the drift velocity of particles in electric field E and μ is the particle mobility. The initial and boundary conditions are imposed analogously to conditions (7) formulated above. Accordingly, we seek the solution as before in the form

$$W(x,t;E) = \sum\limits_{n = 0}^\infty {{{c}_{n}}{{\varphi }_{n}}\exp ( - {{E}_{n}}t).} $$

((A.2))

The expressions for eigenfunctions change as follows in this case:

$${{\varphi }_{n}} = \exp \left( {\frac{{{v}(x - {{x}_{i}})}}{{2D}}} \right)\sin ({{k}_{n}}(x - {{x}_{i}})),$$

((A.3))

and eigenvalues are defined as

$${{E}_{n}} = Dk_{n}^{2} + \frac{{{{{v}}^{2}}}}{{4D}}.$$

((A.4))

Accordingly, in the mean field approximation, we obtain

$$\bar {W}(t;E) \propto \exp \left( { - \frac{{{{{v}}^{2}}t}}{{4D}}} \right).$$

((A.5))

Over long time intervals, the asymptotic form of the survival probability of particles diffusing in a medium with traps is determined by fluctuations of the trap concentration, i.e., by regions free of traps due to fluctuations. It is the existence of such fluctuation regions that determines the temporal asymptotic forms of the particle survival probability in media with traps. According to [4], we must average expression (A.2) derived above over the random distribution of traps. (In this study, this distribution is assumed to be of the Poisson type.) Thus, the asymptotic form of the solution in the fluctuation region is defined by the following expression (see [18] for details):

$$W(t;E)\, \approx \,\sum\limits_{n = 0}^\infty {\int\limits_0^\infty {{\text{exp}}\left( { - D\frac{{{{\pi }^{2}}{{{(2n\, + \,1)}}^{2}}}}{{{{l}^{2}}}}t\, - \frac{{{v}l}}{{4D}}\, - \,cl} \right)dl.} } $$

((A.6))

Thus, the introduction of external field into the problem in question leads to a new exponential factor exp(–${v}$l/4D).