9780030839931, Chapter 16, Problem 2P

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Chapter 16, Problem 2P Problem A metal is perturbed by a spatially uniform electric field and temperature gradient. Making the relaxation-time approximation (16.9) (where is the local equilibrium distribution appropriate to the imposed temperature gradient), solve the Boltzmann equation (16.13) to linear order in the field and temperature gradient, and verify that the solution is identical to (13.43). Step-by-step solution Step of 4 Let us define the total rate at which the distribution function changes due to collisions be Also, the change in the number of electrons per unit volume with wave vectors in the volume element dk about k in the time interval dt, due to all collisions is Since electrons can be scattered either into or out of dk by collisions, is simply the sum of and and is given as (1) Step of 4 In the relaxation-time approximation, If no collisions occurred, then r and k coordinates of every electron would evolve according to the semi classical equations of motion: hk = =F(r,k) An electron at r, k at time must have been at at time t-dt. In the absence of collisions, this is the only point electrons at r,k could have come from, and every electron at this point will reach the point (2) Step of 4 Taking collisions into account, the two correction terms must be added to equation (2). The right-hand side is wrong because it assumes that all electrons are obtained from vdt, to k in the time. Adding these corrections, the leading order in dt is (3) dt In the limit dt 0, equation (3) reduces to or a h = (4) The above equation is known as the Boltzmann equation. Step of 4 Consider a small fixed region of the solid within which the temperature is effectively The thermal current density is just the product of the temperature with the entropy current density, From the thermodynamic identity: In terms of current densities: (5) Here, the energy and the number of current densities are given as j" 1 (6) Combine (5) and (6) to find thermal current density: = The distribution function appears in equation (7), which is evaluated at H = 0 in the presence of a uniform static electric field and temperature gradient: (8) Here, E Vµ e From the above observation, it can be concluded that the Boltzmann equation is given by equation (4), to linear order in the field and temperature gradient, and the solution is identical to equation (8).