Rock salt ( ) crystallises in the cubic system with point group

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Invariance, Ch Point groups and crystallographic symmetry

Problem

Rock salt (NaCl\mathrm{NaCl}) crystallises in the cubic system with point group OhO_h, which contains C4C_4 axes through the centres of opposite cube faces, along xx, yy and zz. ((a)) Write the 3×33\times3 matrix of the C4C_4 rotation about xx (through 90∘90^\circ). State its trace and determinant and confirm it is a proper rotation through 90∘90^\circ. ((b)) Given that invariance of a rank-two property tensor TT under the C4C_4 about zz already forces T=diag(a,a,b)T=\mathrm{diag}(a,a,b), use the additional C4C_4 about xx to show that b=ab=a, so TT is isotropic. ((c)) State what this implies for NaCl\mathrm{NaCl}'s electrical conductivity.

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