# Thermal expansion

### Coefficient of thermal expansion

The *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 Δ*T*. In the first order approximation it is given by:

α = (Δ ℓ/ℓ) /Δ *T*

**anisotropic media**

the *u _{ij}* and the coefficient of thermal expansion is represented by a rank 2 tensor, α

*, given by:*

_{ij}\[\alpha_{ij} = \dfrac{u_{ij}}{\Delta T}.\]

### 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

**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, κ* _{ij}*, and

*c*is the elastic stiffness tensor at constant temperature. For details see Sections 1.4.2 and 2.1.2.8 of

_{ijkl}^{T}*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 *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*.

### See also

Chapters 2.3 and 5.3, *International Tables Volume C*

Chapters 1.4 and 2.1, *International Tables of Crystallography, Volume D*