Difference between revisions of "Thermal expansion"
From Online Dictionary of Crystallography
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=== Other languages === | === Other languages === | ||
| − | Dilatation thermique (''Fr''). Wärmeausdehnung (''Ge''). Dilatación tèrmica (''Sp''). | + | <Font color="blue">Dilatation thermique </Font>(''Fr''). <Font color="red">Wärmeausdehnung</Font> (''Ge''). <Font color="green">Dilatación tèrmica </Font>(''Sp''). <Font color="purple">термическое расширение </Font>(''Ru''). |
| − | термическое расширение (''Ru''). | ||
| − | + | = 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''. | 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''. | ||
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α''<sub>ij</sub>'' = ''u<sub>ij</sub>'' / Δ ''T''. | α''<sub>ij</sub>'' = ''u<sub>ij</sub>'' / Δ ''T''. | ||
| − | + | = Volume thermal expansion = | |
The volume thermal expansion, β, relates the relative variation of volume Δ ''V''/''V'' to Δ ''T'': | The volume thermal expansion, β, relates the relative variation of volume Δ ''V''/''V'' to Δ ''T'': | ||
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β = Δ ''V''/''V'' Δ ''T'' = α ''<sub>11</sub>'' + α ''<sub>22</sub>'' + α ''<sub>33</sub>''. | β = Δ ''V''/''V'' Δ ''T'' = α ''<sub>11</sub>'' + α ''<sub>22</sub>'' + α ''<sub>33</sub>''. | ||
| − | + | = 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: | 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: | ||
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For details see Sections 1.4.2 and 2.1.2.8 of ''International Tables Volume D''. | 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''. | 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''<br> | |
| − | + | Chapters 1.4 and 2.1, ''International Tables of Crystallography, Volume D''<br> | |
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| − | |||
[[Category:Physical properties of crystals]] | [[Category:Physical properties of crystals]] | ||
Revision as of 08:26, 27 February 2006
Contents
Thermal expansion
Other languages
Dilatation thermique (Fr). Wärmeausdehnung (Ge). Dilatación tèrmica (Sp). термическое расширение (Ru).
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:
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 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 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, 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.
See also
Chapters 2.3 and 5.3, International Tables Volume C
Chapters 1.4 and 2.1, International Tables of Crystallography, Volume D