Introduction

Glazer, A. M.; Cox, K. G.

doi:10.1107/97809553602060000633

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.6, pp. 173-174

Section 1.6.7.1. Introduction

A. M. Glazer^a ^* and K. G. Cox^b ^‡

^a Department of Physics, University of Oxford, Parks Roads, Oxford OX1 3PU, England, and ^bDepartment of Earth Sciences, University of Oxford, Parks Roads, Oxford OX1 3PR, England
Correspondence e-mail: glazer@physics.ox.ac.uk

1.6.7.1. Introduction

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The linear photoelastic (or piezo-optic ) effect (Narasimhamurty, 1981) is given by $[P_i^\omega = \varepsilon_o \chi_{ijk\ell}E_j^\omega S_{k\ell}^0]$ , and, like the electro-optic effect, it can be discussed in terms of the change in dielectric impermeability caused by a static (or low-frequency) field, in this case a stress, applied to the crystal. This can be written in the form $[\Delta\eta_{ij} = \pi_{ijk\ell}T_{k\ell}^0. \eqno (1.6.7.1)]$ The coefficients $[\pi_{ijk\ell}]$ form a fourth-rank tensor known as the linear piezo-optic tensor. Typically, the piezo-optic coefficients are of the order of 10⁻¹² m² N⁻¹. It is, however, more usual to express the effect as an elasto-optic effect by making use of the relationship between stress and strain (see Section 1.3.3.2 ), thus $[ T_{k\ell} = c_{k\ell mn}S_{mn}, \eqno (1.6.7.2)]$ where the $[c_{k\ell mn}]$ are the elastic stiffness coefficients. Therefore equation (1.6.7.2) can be rewritten in the form $[\Delta\eta_{ij} = \pi_{ijk\ell}c_{k\ell mn}S_{mn} = p_{ijmn}S_{mn} \eqno(1.6.7.3)]$ or, in contracted notation, $[\Delta\eta_{i} = p_{ij}S_{j}, \eqno(1.6.7.4)]$ where, for convenience, the superscript 0 has been dropped, the elastic strain being considered as essentially static or of low frequency compared with the natural mechanical resonances of the crystal. The $[p_{ijmn}]$ are coefficients that form the linear elasto-optic (or strain-optic) tensor (Table 1.6.7.1). Note that these coefficients are dimensionless, and typically of order 10⁻¹, showing that the change to the optical indicatrix is roughly one-tenth of the strain.

Table 1.6.7.1 | top | pdf |
Symmetry constraints on the linear elasto-optic (strain-optic) tensor $[p_{ij}]$ (contracted notation) (see Section 1.1.4.10.6 )

Triclinic	Orthorhombic	Tetragonal	Trigonal
Point group 1	Point groups 222, mm2, mmm	Point groups $[4, {\bar 4}, 4/m]$	Point groups $[3, {\bar 3}]$
$[\,\pmatrix{ p_{11}& p_{12}& p_{13}& p_{14}& p_{15}& p_{16}\cr p_{21}& p_{22}& p_{23}& p_{24}& p_{25}& p_{26}\cr p_{31}& p_{32}& p_{33}& p_{34}& p_{35}& p_{36}\cr p_{41}& p_{42}& p_{43}& p_{44}& p_{45}& p_{46}\cr p_{51}& p_{52}& p_{53}& p_{54}& p_{55}& p_{56}\cr p_{61}& p_{62}& p_{63}& p_{64}& p_{65}& p_{66}}]$	$[\,\pmatrix{p_{11}& p_{12}& p_{13}&0&0& 0\cr p_{21}& p_{22}& p_{23}&0&0& 0\cr p_{31}& p_{32}& p_{33}&0&0& 0\cr 0&0&0& p_{44}&0& 0\cr 0&0&0&& p_{55}&0\cr 0&0&0&0&0& p_{66}}]$	$[\,\pmatrix{p_{11}& p_{12}& p_{13}&0&0& p_{16}\cr p_{12}& p_{22}& p_{13}&0&0& -p_{16}\cr p_{31}& p_{31}& p_{33}&0&0& 0\cr 0&0&0& p_{44}& p_{45}& 0\cr 0&0&0&-p_{45}& p_{55}&0\cr p_{61}&-p_{61}&0&&0& p_{66}}]$	$[\,\pmatrix{ p_{11}& p_{12}& p_{13}& p_{14}& p_{15}& p_{16}\cr p_{12}& p_{11}& p_{13}&-p_{14}&-p_{15}& -p_{16}\cr p_{31}& p_{31}& p_{33}&0&0& 0\cr p_{41}&-p_{41}&0& p_{44}& p_{45}& -p_{51}\cr p_{51}&-p_{51}&0&-p_{44}& p_{44}& p_{41}\cr -p_{16}& p_{16}&0&-p_{15}& p_{14}& {1\over 2}(p_{11}-p_{12})}]$
Monoclinic		Point groups $[4mm, {\bar 4}2m, 422, 4/mmm]$	Point groups $[3m, {\bar 3}m, 32]$
Point groups $[2, m, 2/m \ (2 \parallel x_2)]$	Point groups $[2, m, 2/m\ (2 \parallel x_3)]$	$[\pmatrix{p_{11}& p_{12}& p_{13}&0&0& 0\cr p_{12}& p_{22}& p_{13}&0&0& 0\cr p_{31}& p_{31}& p_{33}&0&0& 0\cr 0&0&0& p_{44}&0& 0\cr 0&0&0&0& p_{44}& 0\cr 0&0&0&0&0& p_{66}}]$	$[\pmatrix{p_{11}& p_{12}& p_{13}& p_{14}&0& 0\cr p_{12}& p_{11}& p_{13}&-p_{14}&0& 0\cr p_{13}& p_{13}& p_{33}&0&0& 0\cr p_{41}&-p_{41}&0& p_{44}&0& 0\cr 0&0&0&0& p_{44}& p_{41}\cr 0&0&0&0& p_{14}& {1\over 2}(p_{11}-p_{12})}]$
$[\,\pmatrix{p_{11}& p_{12}& p_{13}&0& p_{15}& 0\cr p_{21}& p_{22}& p_{23}&0& p_{25}& 0\cr p_{31}& p_{32}& p_{33}&0& p_{35}& 0\cr 0&0&0& p_{44}&0& p_{46}\cr p_{51}& p_{52}& p_{53}&0& p_{55}&0\cr 0&0&0& p_{64}&0& p_{66}}]$	$[\pmatrix{p_{11}& p_{12}& p_{13}&0&0& p_{16}\cr p_{21}& p_{22}& p_{23}&0&0& p_{26}\cr p_{31}& p_{32}& p_{33}&0&0& p_{36}\cr 0&0&0& p_{44}& p_{45}& 0\cr 0&0&0& p_{54}& p_{55}& 0\cr p_{61}& p_{62}& p_{63}&0&0& p_{66}}]$

Hexagonal	Cubic	Isotropic
Point groups $[6, {\bar 6}, 6/m]$	Point groups $[m{\bar 3}, 23]$	$[\,\pmatrix{p_{11}& p_{12}& p_{12}&0&0&0\cr p_{12}& p_{11}& p_{13}&0&0&0\cr p_{12}& p_{12}& p_{11}&0&0& 0\cr 0&0&0&{1\over 2}(p_{11}-p_{12})&0& 0\cr 0&0&0&0&{1\over 2}(p_{11}-p_{12})&0\cr 0&0&0&0&0& {1\over 2}(p_{11}-p_{12})}]$
$[\,\pmatrix{p_{11}& p_{12}& p_{13}&0&0& p_{16}\cr p_{12}& p_{11}& p_{13}&0&0& -p_{16}\cr p_{31}& p_{31}& p_{33}&0&0& 0\cr 0&0&0& p_{44}& p_{45}& 0\cr 0&0&0&-p_{45}& p_{44}&0\cr -p_{16}& p_{16}&0&0&0& {1\over 2}(p_{11}-p_{12})}]$	$[\,\pmatrix{p_{11}& p_{12}& p_{21}&0&0&0\cr p_{21}& p_{11}& p_{12}&0&0&0\cr p_{12}& p_{21}& p_{11}&0&0& 0\cr 0&0&0& p_{44}&0& 0\cr 0&0&0&0& p_{44}&0\cr 0&0&0&0&0& p_{44}}]$
Point groups $[6mm, {\bar 6}m2, 622, 6/mmm]$	Point groups $[{\bar 4}3m, 432, m{\bar 3}m]$
$[\pmatrix{p_{11}& p_{12}& p_{13}&0&0& 0\cr p_{12}& p_{11}& p_{13}&0&0& 0\cr p_{13}& p_{13}& p_{33}&0&0& 0\cr 0&0&0& p_{44}&0& 0\cr 0&0&0&0& p_{44}& 0\cr 0&0&0&0&0& {1\over 2}(p_{11}-p_{12})}]$	$[\,\pmatrix{p_{11}& p_{12}& p_{12}&0&0&0\cr p_{12}& p_{11}& p_{13}&0&0&0\cr p_{12}& p_{12}& p_{11}&0&0& 0\cr 0&0&0& p_{44}&0& 0\cr 0&0&0&0& p_{44}&0\cr 0&0&0&0&0& p_{44}}]$

The elasto-optic effect can arise in several ways. The most obvious way is through application of an external stress, applied to the surfaces of the crystal. However, strains, and hence changes to the refractive indices , can arise in a crystal through other ways that are less obvious. Thus, it is a common finding that crystals can be twinned, and thus the boundary between twin domains, which corresponds to a mismatch between the crystal structures either side of the domain boundary, will exhibit a strain. Such a crystal, when viewed between crossed polars under a microscope will produce birefringence colours that will highlight the contrast between the domains. This is known as strain birefringence. Similarly, when a crystal undergoes a phase transition involving a change in crystal system, a so-called ferroelastic transition, there will be a change in strain owing to the difference in unit-cell shapes. Hence there will be a corresponding change in the optical indicatrix. Often the phase transition is one going from a high-temperature optically isotropic section to a low-temperature optically anisotropic section. In this case, the high-temperature section has no internal strain, but the low-temperature phase acquires a strain, which is often called the spontaneous strain (by analogy with the term spontaneous polarization in ferroelectrics).

References

Narasimhamurty, T. S. (1981). Photoelastic and electro-optic properties of crystals. New York: Plenum.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.6, pp. 173-174