Thermal expansion - Revision history

BrianMcMahon: Tidied translations and corrected Spanish (U. Mueller)

2017-11-20T14:02:31Z

Tidied translations and corrected Spanish (U. Mueller)

BrianMcMahon at 13:43, 17 May 2017

2017-05-17T13:43:29Z

BrianMcMahon: Style edits to align with printed edition

2017-05-17T13:43:00Z

Style edits to align with printed edition

MassimoNespolo at 16:34, 14 August 2008

2008-08-14T16:34:37Z

GiovanniFerraris: /* Other languages */

2006-03-16T14:14:31Z

‎Other languages

AndreAuthier at 08:26, 27 February 2006

2006-02-27T08:26:12Z

BrianMcMahon at 15:22, 25 January 2006

2006-01-25T15:22:08Z

BrianMcMahon at 17:04, 9 December 2005

2005-12-09T17:04:16Z

New page

= 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 Δ''T''. 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, α''ij'', given by:

α''ij'' = ''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'''

γ''ij'' = ''cijklT'' α''kl'' ''V''/''cV''

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

=== See also ===

Chapter 1.4, ''International Tables of Crystallography, Volume D''

Section 2.1.2.8, ''International Tables of Crystallography, Volume D''

[[Category:Physical properties of crystals]]

← Older revision		Revision as of 14:02, 20 November 2017
Line 1:		Line 1:
−	= ~~Thermal expansion~~ =	+	<font color="blue">Dilatation thermique</font>(''Fr''). <font color="red">Wärmeausdehnung</font> (''Ge''). <font color="black">Dilatazione termica</font>(''It''). <font color="purple">熱膨張</font>(''Ja''). <font color="brown">термическое расширение</font> (''Ru''). <font color="green">Dilatación térmica</font> (''Sp'').

−	<Font color="blue">Dilatation thermique </Font>(''Fr''). <Font color="red">Wärmeausdehnung</Font> (''Ge''). <Font color="green">Dilatación tèrmica </Font>(''Sp''). <Font color="brown">термическое расширение </Font>(''Ru''). <Font color="black">Dilatazione termica</Font>(''It''). <Font color="purple">熱膨張</Font>(''Ja'').

	= Coefficient of thermal expansion =		= Coefficient of thermal expansion =

@@ Line 61: / Line 61: @@
 = See also =
-*Chapters 2.3 and 5.3, ''International Tables for Crystallography, Volume C''
+*Chapters 2.3 and 5.3 of ''International Tables for Crystallography, Volume C''
-*Chapters 1.4 and 2.1, ''International Tables for Crystallography, Volume D''
+*Chapters 1.4 and 2.1 of ''International Tables for Crystallography, Volume D''
 [[Category:Physical properties of crystals]]

@@ Line 1: / Line 1: @@
 = Thermal expansion =
+<Font color="blue">Dilatation thermique </Font>(''Fr''). <Font color="red">Wärmeausdehnung</Font> (''Ge''). <Font color="green">Dilatación tèrmica </Font>(''Sp''). <Font color="brown">термическое расширение </Font>(''Ru''). <Font color="black">Dilatazione termica</Font>(''It''). <Font color="purple">熱膨張</Font>(''Ja'').
 = 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 &#916; ''T''.
+The coefficient of thermal expansion relates the deformation that takes place when the temperature ''T'' of a solid is varied by the temperature variation &#916; ''T''.
-* '''isotropic media'''
+* '''Isotropic media'''
-the linear coefficient of thermal expansion, &#945;, relates the relative variation
+The linear coefficient of thermal expansion, &#945;, relates the relative variation
 (&#916;&#8467;/&#8467;) of the length &#8467; of a bar to the temperature variation &#916;''T''. In the first order approximation it is given by:
-&#945; = (&#916; &#8467;/&#8467;) /&#916; ''T''
+&#945; = (&#916; &#8467;/&#8467;) /&#916; ''T''.
-* '''anisotropic media'''
+* '''Anisotropic media'''
-the deformation is described by the strain tensor ''u<sub>ij</sub>'' and the coefficient of thermal
+The deformation is described by the strain tensor ''u<sub>ij</sub>'' and the coefficient of thermal
 expansion is represented by a rank 2 tensor, &#945;''<sub>ij</sub>'', given by:
@@ Line 28: / Line 25: @@
 The volume thermal expansion, &#946;, relates the relative variation of volume &#916; ''V''/''V''  to &#916; ''T'':
-* '''isotropic media'''
+* '''Isotropic media'''
-&#946; = &#916; ''V''/''V'' &#916; ''T'' = 3 &#945;,
+&#946; = &#916; ''V''/''V'' &#916; ''T'' = 3 &#945;.
-* '''anisotropic media'''
+* '''Anisotropic media'''
-it is given by the trace of &#945; ''<sub>ij</sub>'':
+It is given by the trace of &#945;''<sub>ij</sub>'':
 &#946; = &#916; ''V''/''V'' &#916; ''T'' = &#945; ''<sub>11</sub>'' + &#945; ''<sub>22</sub>'' + &#945; ''<sub>33</sub>''.
@@ Line 43: / Line 39: @@
 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'''
+* '''Isotropic media'''
-&#946; = &#947; &#954; ''c<sup>V</sup>''/V
+&#946; = &#947; &#954; ''c<sup>V</sup>''/''V''
 where &#947; is the average Grüneisen parameter, &#954; the isothermal compressibility, ''c<sup>V</sup>'' the specific heat at
 constant volume.
-* '''anisotropic media'''
+* '''Anisotropic media'''
 &#947;''<sub>ij</sub>'' = ''c<sub>ijkl</sub><sup>T</sup>'' &#945;''<sub>kl</sub>'' ''V''/''c<sup>V</sup>''
-where the Grüneisen parameter is now represented by a second rank tensor, &#954;''<sub>ij</sub>'', and ''c<sub>ijkl</sub><sup>T</sup>'' is
+where the Grüneisen parameter is now represented by a second-rank tensor, &#947;''<sub>ij</sub>'', and ''c<sub>ijkl</sub><sup>T</sup>'' 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''.
+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 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 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, ''International Tables Volume C''<br>
+*Chapters 2.3 and 5.3, ''International Tables for Crystallography, Volume C''
-Chapters 1.4 and 2.1, ''International Tables of Crystallography, Volume D''<br>
+*Chapters 1.4 and 2.1, ''International Tables for Crystallography, Volume D''
 [[Category:Physical properties of crystals]]

@@ Line 4: / Line 4: @@
 === Other languages ===
-<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''). <Font color="black">Dilatazione termica </Font>(''It'')
+<Font color="blue">Dilatation thermique </Font>(''Fr''). <Font color="red">Wärmeausdehnung</Font> (''Ge''). <Font color="green">Dilatación tèrmica </Font>(''Sp''). <Font color="brown">термическое расширение </Font>(''Ru''). <Font color="black">Dilatazione termica</Font>(''It''). <Font color="purple">熱膨張</Font>(''Ja'')
 = Coefficient of thermal expansion =

@@ Line 4: / Line 4: @@
 === Other languages ===
-<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'').
+<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''). <Font color="black">Dilatazione termica </Font>(''It'')
 = Coefficient of thermal expansion =

@@ Line 59: / Line 59: @@
 the elastic stiffness tensor at constant temperature.
-=== See also ===
+== Measurement ==
-Chapter 1.4, ''International Tables of Crystallography, 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''.
 Section 2.1.2.8, ''International Tables of Crystallography, Volume D''
 [[Category:Physical properties of crystals]]