Thermal expansion in a crystal may be completely described from a phenomenological point of view by a second rank tensor whose elements are defined by

λ_ij=(∂l_ij/∂T)_σ

Or

ε_ij=(∂e_ij/∂T)_σ

Where the l_ij and the e_ij are the elements of the linear Lagrangian and Eulerian strain tensors respectively. These λ_ij and ε_ij have been formulated in terms of crystal cell parameters. For example, for a monoclinic crystal the λ_ij are of the form:

λ₁₁(T) = 1/a₀sinβ₀ d[a(T)sinβ(T)]/dT ,

λ₁₃(T) = ½ (1/a₀sinβ₀ d[a(T)cosβ(T)]/dT - cotβ₀/c₀ dc(T)/dT) ,

λ₂₂(T) = 1/b₀ db(T)/DT ,

and

λ₃₃(T) = 1/c₀ dc(T)/dT

where a₀, b₀, c₀, and β₀ are the crystal’s cell parameters at some reference temperature T₀. By expressing the crystal cell parameters as power series expansions in the temperature, thermal expansion coefficients have been computed for indialite (hexagonal cordierite), emerald and beryl and for the clinopyroxenes: diopside, hedenbergite, jadeite, ureyite, acmite, and spodumene. The extended Grüneisen equation has been used to further examine the nature of the thermal expansion in emerald, beryl, and diopside.

The crystal structures of the synthetic clinopyroxenes CaCoSi₂O₆ (cobalt diopside) and CaNiSi₂O₆ (nickel diopside) have also been determined.