Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

(iii) referred to these axes as coordinate axes, the inertia tensor is diagonal

with diagonal entriesλ 1 ,λ 2 ,λ 3.

Two further examples of physical quantities represented by second-order tensors

are magnetic susceptibility and electrical conductivity. In the first case we have

(in standard notation)

Mi=χijHj, (26.43)

and in the second case

ji=σijEj. (26.44)

HereMis the magnetic moment per unit volume andjthe current density

(current per unit perpendicular area). In both cases we have on the left-hand side

a vector and on the right-hand side the contraction of a set of quantities with

another vector. Each set of quantities must therefore form the components of a

second-order tensor.

For isotropic mediaM∝Handj∝E, but for anisotropic materials such as

crystals the susceptibility and conductivity may be different along different crystal

axes, makingχijandσijgeneral second-order tensors, although they are usually

symmetric.

The electrical conductivityσin a crystal is measured by an observer to have components

as shown:

[σij]=





1

√

√^20

231

011



. (26.45)

Show that there is one direction in the crystal along which no current can flow. Does the

current flow equally easily in the two perpendicular directions?

The current density in the crystal is given byji=σijEj,whereσij,relativetotheobserver’s
coordinate system, is given by (26.45). Since [σij] is a symmetric matrix, it possesses
three mutually perpendicular eigenvectors (or principal axes) with respect to which the
conductivity tensor is diagonal, with diagonal entriesλ 1 ,λ 2 ,λ 3 , the eigenvalues of [σij].
As discussed in chapter 8, the eigenvalues of [σij] are given by|σ−λI|= 0. Thus we
require ∣
∣∣
∣
∣∣

1 −λ

√

√^20

23 −λ 1

011 −λ

∣

∣∣

∣

∣∣=0,

from which we find
(1−λ)[(3−λ)(1−λ)−1]−2(1−λ)=0.

This simplifies to giveλ=0, 1 ,4 so that, with respect to its principal axes, the conductivity
tensor has componentsσ′ijgiven by

[σ′ij]=





400

010

000



.

Sincej′i=σ′ijEj′, we see immediately that along one of the principal axes there is no current
flow and along the two perpendicular directions the current flows are not equal.