# Thermal expansion

## Coefficient of thermal expansion

The coefficient of thermal expansion relates the deformation that takes place when the temperature T of a solid is varied to the temperature variation Δ T.

• isotropic media

the linear coefficient of thermal expansion, α, relates the relative variation (Δℓ/ℓ) of the length ℓ of a bar to the temperature variation ΔT. In the first order approximation it is given by:

α = (Δ ℓ/ℓ) /Δ T

• anisotropic media

the deformation is described by the strain tensor uij and the coefficient of thermal expansion is represented by a rank 2 tensor, αij, given by:

$\alpha_{ij} = \dfrac{u_{ij}}{\Delta T}.$

## Volume thermal expansion

The volume thermal expansion, $$\beta$$, relates the relative variation of volume $$\frac{\Delta{V}}{V}$$ to $$\Delta{T}$$:

• isotropic media

$\beta = \dfrac{\Delta{V}}{V \Delta{T}} = 3\alpha$

• anisotropic media

it is given by the trace of the $$\alpha_{ij}$$ matrix:

$\beta = \dfrac{\Delta{V}}{V \Delta{T}} = \alpha_{11}+\alpha_{22}+\alpha_{33}$

## Grüneisen relation

The thermal expansion of a solid is a consequence of the anharmonicity of interatomic forces, which is most conveniently accounted for by means of the so-called `quasiharmonic approximation', assuming the lattice vibration frequencies to be independent of temperature, but dependent on volume. This approach leads to the Grüneisen relation that relates the thermal expansion coefficients and the elastic constants:

### isotropic media

$\beta = \dfrac{\gamma \kappa c_v}{V}$

where

• $$\gamma$$ is the average Grüneisen parameter,
• $$\kappa$$ the isothermal compressibility,
• $$c_v$$ the specific heat at constant volume.

### anisotropic media

$\beta_{ij} = \dfrac{\gamma_{ijkl}^T \alpha_{kl} V}{c_v}$

where the Grüneisen parameter is now represented by a second rank tensor, κij, and cijklT is the elastic stiffness tensor at constant temperature. For details see Sections 1.4.2 and 2.1.2.8 of International Tables Volume D.

## Measurement

The coefficient of thermal expansion can be measured using diffraction methods (for powder diffraction methods, see Section 2.3 of International Tables Volume C, for single crystal methods, see Section 5.3 of International Tables Volume C), optical methods (interferometry) or electrical methods (pushrod dilatometry methods or capacitance methods). For details see Section 1.4.3 of International Tables Volume D.