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 = 5.6x10-⁶(K-¹) and α₃₃= 25x10-⁶ (K-¹)

Using Neumann's Law deduce the thermal expansion matrix for this point group.

Answer 1

Neumann's Law can be used to determine the thermal expansion matrix for a crystal with point group 3m. The thermal expansion matrix can be obtained by considering the irreducible representations of the group and their corresponding coefficients. For a crystal with point group 3m, the thermal expansion matrix is given by [ alpha₁₁ 0 0 ][ alpha₃₃ 0 0 ].

Step-by-step explanation:

Neumann's Law states that the thermal expansion tensor for a crystal with point group 3m can be obtained by considering the irreducible representations of the group and their corresponding coefficients. For a crystal with point group 3m, there are two irreducible representations: A₁ and E. The thermal expansion matrix for this point group can be written as:

[ alpha₁₁ 0 0 ][ alpha₃₃ 0 0 ]

Where alpha₁₁ and alpha₃₃ are the measured values of the thermal expansion at 25°C. Plugging in the given values, the thermal expansion matrix becomes:

[ 5.6e-6 0 0 ][ 25e-6 0 0 ]