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The complete reduction of a fourth-rank Cartesian tensor into parts wich are irreducible under the three-dimensional rotation group is presented. Results are given for the general case of a tensor without index permutational symmetry, and also for tensors with specific index symmetry properties. Applications in the realm of crystal physics are indicated with particular reference to four-photon absorption, and a table of irreducible tensor representations is provided for the seven crystal classes.
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Andrews et al. (1982) studied this question.
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