In a crystal a small strain produces an electric polarisation

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Index notation and Cartesian tensors

Problem

In a crystal a small strain

e_{jk}

produces an electric polarisation

\mathbf P

. The response is written provisionally as

P_i=d_{ijk}e_{jk}

, with

27

array entries

d_{ijk}

in one set of axes; how many of them measurements can fix is part of the question. (Here

d_{ijk}

couples polarisation to strain, not to stress as the tabulated

d

coefficients usually do.)

\mathbf P

is a vector and the strain is a symmetric rank-two tensor,

e_{jk}=e_{kj}

. The relation holds in every set of axes and for every strain the crystal can be given, but no experiment can apply a strain that is not symmetric. ((a)) Follow the quotient-theorem argument for rank two and show that

B_{ijk}e'_{jk}=0

for every symmetric

e'_{jk}

, where

B_{ijk}=d'_{ijk}-\mathsf R_{il}\mathsf R_{jm}\mathsf R_{kn}d_{lmn}

. ((b)) The rank-two argument ended by taking the input along each axis in turn. Show by an example that here

B_{ijk}e_{jk}=0

for every symmetric

e_{jk}

does not force

B_{ijk}=0

, and find exactly what it does force. ((c)) Conclude which combination of the

d_{ijk}

obeys the rank-three tensor law, and how many independent coefficients the measurements can determine.

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