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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 8-9

Section 1.1.3.5. Representation surface of a tensor

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.3.5. Representation surface of a tensor

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1.1.3.5.1. Definition

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Let us consider a tensor [t_{ijkl\ldots}] represented in an orthonormal frame where variance is not important. The value of component [t'_{1111\ldots}] in an arbitrary direction is given by [t'_{1111\ldots} = t_{ijkl\ldots}B^{i}_{1}B\hskip1pt^{j}_{1}B^{k}_{1}B^{l}_{1}\ldots,]where the [B^{i}_{1}], [B\hskip1pt^{j}_{1}, \ldots] are the direction cosines of that direction with respect to the axes of the orthonormal frame.

The representation surface of the tensor is the polar plot of [t'_{1111\ldots}].

1.1.3.5.2. Representation surfaces of second-rank tensors

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The representation surfaces of second-rank tensors are quadrics. The directions of their principal axes are obtained as follows. Let [t_{ij}] be a second-rank tensor and let [{\bf OM} = {\bf r}] be a vector with coordinates [x_{i}]. The doubly contracted product, [t_{ij}x^{i}x\hskip1pt^{j}], is a scalar. The locus of points M such that [t_{ij}x^{i}x\hskip1pt^{j} = 1]is a quadric. Its principal axes are along the directions of the eigenvectors of the matrix with elements [t_{ij}]. They are solutions of the set of equations [t_{ij}x^{i} = \lambda x\hskip1pt^{j},]where the associated quantities λ are the eigenvalues.

Let us take as axes the principal axes. The equation of the quadric reduces to [t_{11}(x^{1})^{2} + t_{22}(x^{2})^{2} + t_{33}(x^{3})^{2} = 1.]

If the eigenvalues are all of the same sign, the quadric is an ellipsoid; if two are positive and one is negative, the quadric is a hyperboloid with one sheet; if one is positive and two are negative, the quadric is a hyperboloid with two sheets (see Section 1.3.1[link] ).

Associated quadrics are very useful for the geometric representation of physical properties characterized by a tensor of rank 2, as shown by the following examples:

  • (i) Index of refraction of a medium. It is related to the dielectric constant by [n = \varepsilon^{1/2}] and, like it, it is a tensor of rank 2. Its associated quadric is an ellipsoid, the optical indicatrix, which represents its variations with the direction in space (see Section 1.6.3.2[link] ).

  • (ii) Thermal expansion. If one cuts a sphere in a medium whose thermal expansion is anisotropic, and if one changes the temperature, the sphere becomes an ellipsoid. Thermal expansion is therefore represented by a tensor of rank 2 (see Chapter 1.4[link] ).

  • (iii) Thermal conductivity. Let us place a drop of wax on a plate of gypsum, and then apply a hot point at the centre. There appears a halo where the wax has melted: it is elliptical, indicating anisotropic conduction. Thermal conductivity is represented by a tensor of rank 2 and the elliptical halo of molten wax corresponds to the intersection of the associated ellipsoid with the plane of the plate of gypsum.

1.1.3.5.3. Representation surfaces of higher-rank tensors

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Examples of representation surfaces of higher-rank tensors are given in Sections 1.3.3.4.4[link] and 1.9.4.2[link] .








































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