Question

A single crystal of calcite, CaCO3, is trigonal with point group 3m. The measured values of the thermal expansion of calcite at 25°C are: α11--5.6x10% (K1) and α33- 25x10-6 (K-1) Using Neumann's Law deduce the thermal expansion matrix for this point group.

Answer 1

The thermal expansion matrix for calcite (CaCO3) in the trigonal system with point group 3m is a diagonal matrix with coefficients α11 and α33 representing the expansion in the 'ab' plane and along the 'c' axis, respectively.

Step-by-step explanation:

The thermal expansion matrix for a calcite crystal with a trigonal structure and point group 3m can be deduced using Neumann's Law, which correlates the symmetry of a crystal with its physical properties. In the trigonal system with point group 3m, the thermal expansion coefficients α11 and α33 correspond to the diagonal components of the thermal expansion tensor. The tensor for thermal expansion in such a crystal system is diagonal due to its symmetry, meaning that all off-diagonal components are zero.

The thermal expansion matrix for calcite at 25°C with the given coefficients α11 = -5.6x10-6 K-1 and α33 = 25x10-6 K-1 can be written as:

[α] = [ α11 0 0 0 α11 0 0 0 α33 ]

Expanded Matrix

[ -5.6x10-6 0 0 0 -5.6x10-6 0 0 0 25x10-6 ]

This matrix explains the anisotropic nature of calcite's thermal expansion, where expansion along the 'c' axis (corresponding to α33) is different from expansion within the 'ab' plane (corresponding to α11).