Abstract
The Boltzmann equation is solved for homogeneous electric and magnetic fields (E and H) and a temperature gradient βT/βr. The solution is presented as an ascending power series in E, H, and βT/βr. In contrast to common procedure this treatment is not restricted to the linearized form of the equation and the relaxation time approximation, and no restricting assumptions are made about E (k) and the transition probabilities. Applications of the general solution are investigated, for zero and nonβzero magnetic field, considering only the first order terms in E and βT/βr. In the second case terms in H of arbitrary order are treated.
In the first case the electrical and thermal conductivity, and the thermoelectric effects, are treated using a unified tensor formalism. In the second case only the Hall effect is investigated, while the treatment of the other galvanomagnetic and thermomagnetic effects will be published in a later paper. In both these cases the Onsager relationships are valid, and it is shown that, up to the secondβorder approximation, it is possible to introduce a relaxation time, which can be expressed in terms of the E (k) dependence and the transition probabilities of the crystal. These results contain all results generally found in the literature as special cases.