Twinning

Twins

Other languages

Macles (Fr). Geminati (It)

Oriented association and twinning

Crystals (also called individuals) belonging to the same phase form an oriented association if they can be brought to the same crystallographic orientation by translation, rotation or reflection. Individuals related by a translation form a parallel association; strictly speaking these individuals have the same orientation even without applying a translation. Individuals related either by a reflection (mirror plane or centre of symmetry) or a rotation form a twin.

An element of symmetry crystallographically relating differently oriented crystals cannot belong to the individual. The element of symmetry that relates the indivduals of a twin is called twinning element of symmetry and the connected operation is a twinning operation of symmetry.

• 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 u_ij and the coefficient of thermal expansion is represented by a rank 2 tensor, α_ij, given by:

α_ij = u_ij / 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 α _ij:

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 c^V/V

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

• anisotropic media

γ_ij = c_ijkl^T α_kl V/c^V

where the Grüneisen parameter is now represented by a second rank tensor, κ_ij, and c_ijkl^T 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.

See also

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