The Physical Interpretation of Mean Free Path and the Integral Method

Concepts

Natural SciencesPhysicsCondensed Matter PhysicsQuantum MechanicsSolid State PhysicsElectron TransportIntegral Equation FormulationsIntegral EquationsTransport PropertiesElectron Mean Free PathElectron Transport In SolidsElectronic TransportLinear Integral EquationsLinear Transport TheoryMean Free PathMean Free Path AnalysisTransport Mean Free Path

Abstract

In previous papers, general expressions for the linear electronic transport constants of solids were obtained in terms of a conjugate function ψ † related, by a linear inhomogeneous integral equation, to the function (of electron state) ψ measured by the “flux.” It is now shown that τψ † =∫ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ∞ ψ|t) dt, where the integrand is the expectation of ψ for an electron which at time t earlier was in the specified state (of which ψ † is a function) and 1/τ is the collision frequency. In particular, the vector mean free path τv † is: “the limit, after a virtually infinite time, of the mean displacement, in Brownian motion, of the position of an electron initially in the specified state.” If there is a force (e.g. that due to a magnetic field) accelerating the electrons between collisions, then a linear transport constant is the same functional of an “extended conjugate” ψ † as it is of ψ † in the absence of the force. It is shown that τψ †e is obtained (instead of τψ † ) when the integrand in the integral above is replaced by the “expectation after time t ” as modified by the accelerations between collisions. The relation of the present formalism to the Shockley-Chambers theory is discussed.