Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.8, p. 220

Section 1.8.2. Macroscopic equations

G. D. Mahana*

aDepartment of Physics, 104 Davey Laboratory, Pennsylvania State University, University Park, Pennsylvania, USA
Correspondence e-mail:

1.8.2. Macroscopic equations

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The basic equations of transport are given below (Ziman, 1962[link]; Goldsmid, 1986[link]; Mahan, 1990[link]). (The symbols used in this chapter are defined in Section 1.8.6[link].) [\eqalignno{{\bf J} &= \boldsigma({\bf E} - {\bf S} \buildrel\longrightarrow \over {\nabla T}) &(\cr {\bf J}_Q &= {\bf J} T {\bf S} - {\bf K}\buildrel\longrightarrow \over {\nabla T},&(}%fd1.8.2.2]where [\bf J] and [{\bf J}_Q] are the current density and the heat current, respectively. The three main transport coefficients are the electrical conductivity [\sigma], the thermal conductivity K and the Seebeck coefficient S. The electrical resistivity [\rho] is the inverse of the conductivity, [\rho = 1/\sigma]. In general, the currents, electric field and [\buildrel\longrightarrow \over {\nabla T}] are vectors while [\sigma], S and [ K] are second-rank tensors. The number of independent tensor components is determined by the symmetry of the crystal (see Chapter 1.1[link] ). We assume cubic symmetry, so all of the quantities can be treated as scalars. Onsager relations require that the Seebeck coefficient S is the same in the two equations. A description of the transport properties of most crystals is simply given as a graph, or table, of how each of the three parameters [(\sigma, S, K)] varies with temperature. The range of variation among crystals is enormous.

The above equations assume that there is no magnetic field and have to be changed if a magnetic field is present. This special case is discussed below.


Goldsmid, H. J. (1986). Electronic refrigeration. London: Pion Limited.
Mahan, G. D. (1990). Many-particle physics, 2nd ed. New York: Plenum.
Ziman, J. M. (1962). Electrons and phonons. Oxford University Press.

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