# Thermal expansion

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Dilatation thermique(Fr). Wärmeausdehnung (Ge). Dilatazione termica(It). 熱膨張(Ja). термическое расширение (Ru). Dilatación térmica (Sp).

# Coefficient of thermal expansion

The coefficient of thermal expansion relates the deformation that takes place when the temperature T of a solid is varied by 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:

αij = uij / Δ T.

# Volume thermal expansion

The volume thermal expansion, β, relates the relative variation of volume Δ V/V to Δ T:

• Isotropic media

β = Δ V/V Δ T = 3 α.

• Anisotropic media

It is given by the trace of αij:

β = Δ V/V Δ T = α 11 + α 22 + α 33.

# Grüneisen relation

The thermal expansion of a solid is a consequence of the anharmonicity of interatomic forces. The anharmonicity 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

β = γ κ cV/V

where γ is the average Grüneisen parameter, κ the isothermal compressibility, cV the specific heat at constant volume.

• Anisotropic media

γij = cijklT αkl V/cV

where the Grüneisen parameter is now represented by a second-rank tensor, γij, and cijklT is the elastic stiffness tensor at constant temperature.

For details see Chapters 1.4.2 and 2.1.2.8 of International Tables for Crystallography, Volume D.

# Measurement

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

# See also

• Chapters 2.3 and 5.3 of International Tables for Crystallography, Volume C
• Chapters 1.4 and 2.1 of International Tables for Crystallography, Volume D