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= Twins =
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<font color="blue">Maclage</font> (''Fr''). <font color="red">Zwillingsbildung, Verzwillingung</font> (''Ge''). <font color="black">Geminazione</font> (''It''). <font color="purple">双晶化</Font> (''Ja'').  <font color="brown">двойникование</font> (''Ru''). <font color="green">Maclado (formación de macla)</font> (''Sp'').
  
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= Oriented association and twinning =
  
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=== Other languages ===
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Crystals (also called ''individuals'' or ''components'') or domains belonging to the same phase form an oriented association if they can be brought to coincidence by a translation, rotation, inversion or reflection. Individuals related by a translation form a ''parallel association''; domains related by a translation form ''antiphase domains''. Individuals or domains related by a reflection, inversion or rotation form a twin called, respectively, [[reflection twin]], [[inversion twin]] or [[rotation twin]].
  
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<Font color="blue"> Macles </Font>(''Fr'').  <Font color="black"> Geminati </Font>(''It'')
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A [[mapping]] relating differently oriented crystals cannot be a symmetry operation of the individual: it is called a [[twin operation]] and the [[geometric element]] about which it is performed, associated with this operation, is called a [[twin element]]. [[Mallard's law]] states that the twin element is restricted to a [[direct lattice]] element: it can thus coincide with a lattice node (''twin centre''), a lattice row (''twin axis'') or a lattice plane (''twin plane'').
  
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= Oriented association =
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The symmetry of a twin (''twin point group'') is obtained by extending the intersection [[point group]] of the individuals in their respective orientations by the twin operation.
  
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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. Individuals related either by a rflection (mirror plane or centre of symmetry)or a rotation form a twin.
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= Classification of twins =
  
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* '''isotropic media'''
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Twins are classified following Friedel's ''reticular'' (''i.e.'' lattice) ''theory of twinning'' [see G. Friedel (1926). ''Leçons de Cristallographie'', Nancy, where reference to previous work of the author can be found; see also [[Friedel's law]]]. This theory states that the presence, either in the lattice or a sublattice of a crystal, of (pseudo)symmetry elements is a necessary, even if not sufficient, condition for the formation of twins. In the presence of the necessary reticular conditions, the formation of a twin finally still depends on the matching of the crystal structures at the contact surface between the individuals.
  
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the linear coefficient of thermal expansion, &#945;, relates the relative variation
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The following categories of twins are described under the listed entries:
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(&#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:
 
  
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&#945; = (&#916; &#8467;/&#8467;) /&#916; ''T''
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*[[Twinning by merohedry]]
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*[[Twinning by pseudomerohedry]]
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*[[Twinning by reticular merohedry]]
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*[[Twinning by reticular pseudomerohedry]]
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*[[Twinning by metric merohedry]]
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*[[Twinning by reticular polyholohedry]]
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*[[Hybrid twin]]s
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*[[Allotwin]]s
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*[[Selective merohedry]]
  
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* '''anisotropic media'''
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== See also ==
  
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the deformation is described by the strain tensor ''u<sub>ij</sub>'' and the coefficient of thermal
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*[[Twin index]]
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expansion is represented by a rank 2 tensor, &#945;''<sub>ij</sub>'', given by:
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*[[Twin lattice]]
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*[[Twin law]]
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*[[Twin obliquity]]
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*[[Corresponding twins]]
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*[[Twinning (endemic conditions of)]]
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*[[Twinning (effects of)]]
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*Chapter 1.3 of ''International Tables for Crystallography, Volume C''
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*Chapter 3.3 of ''International Tables for Crystallography, Volume D''
  
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&#945;''<sub>ij</sub>'' = ''u<sub>ij</sub>'' / &#916; ''T''.
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[[Category:Twinning]]
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= Volume thermal expansion =
 
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The volume thermal expansion, &#946;, relates the relative variation of volume &#916; ''V''/''V''  to &#916; ''T'':
 
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* '''isotropic media'''
 
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&#946; = &#916; ''V''/''V'' &#916; ''T'' = 3 &#945;,
 
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* '''anisotropic media'''
 
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it is given by the trace of &#945; ''<sub>ij</sub>'':
 
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&#946; = &#916; ''V''/''V'' &#916; ''T'' = &#945; ''<sub>11</sub>'' + &#945; ''<sub>22</sub>'' + &#945; ''<sub>33</sub>''.
 
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= Grüneisen relation =
 
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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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* '''isotropic media'''
 
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&#946; = &#947; &#954; ''c<sup>V</sup>''/V
 
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where &#947; is the average Grüneisen parameter, &#954; the isothermal compressibility, ''c<sup>V</sup>'' the specific heat at
 
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constant volume.
 
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* '''anisotropic media'''
 
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&#947;''<sub>ij</sub>'' = ''c<sub>ijkl</sub><sup>T</sup>'' &#945;''<sub>kl</sub>'' ''V''/''c<sup>V</sup>''
 
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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
 
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the elastic stiffness tensor at constant temperature.
 
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For details see Sections 1.4.2 and 2.1.2.8 of ''International Tables Volume D''.
 
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= Measurement =
 
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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''.
 
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= See also =
 
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Chapters 2.3 and 5.3, ''International Tables Volume C''<br>
 
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Chapters 1.4 and 2.1, ''International Tables of Crystallography, Volume D''<br>
 
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[[Category:Fundamental crystallography]]
 

Latest revision as of 14:24, 20 November 2017

Maclage (Fr). Zwillingsbildung, Verzwillingung (Ge). Geminazione (It). 双晶化 (Ja). двойникование (Ru). Maclado (formación de macla) (Sp).

Oriented association and twinning

Crystals (also called individuals or components) or domains belonging to the same phase form an oriented association if they can be brought to coincidence by a translation, rotation, inversion or reflection. Individuals related by a translation form a parallel association; domains related by a translation form antiphase domains. Individuals or domains related by a reflection, inversion or rotation form a twin called, respectively, reflection twin, inversion twin or rotation twin.

A mapping relating differently oriented crystals cannot be a symmetry operation of the individual: it is called a twin operation and the geometric element about which it is performed, associated with this operation, is called a twin element. Mallard's law states that the twin element is restricted to a direct lattice element: it can thus coincide with a lattice node (twin centre), a lattice row (twin axis) or a lattice plane (twin plane).

The symmetry of a twin (twin point group) is obtained by extending the intersection point group of the individuals in their respective orientations by the twin operation.

Classification of twins

Twins are classified following Friedel's reticular (i.e. lattice) theory of twinning [see G. Friedel (1926). Leçons de Cristallographie, Nancy, where reference to previous work of the author can be found; see also Friedel's law]. This theory states that the presence, either in the lattice or a sublattice of a crystal, of (pseudo)symmetry elements is a necessary, even if not sufficient, condition for the formation of twins. In the presence of the necessary reticular conditions, the formation of a twin finally still depends on the matching of the crystal structures at the contact surface between the individuals.

The following categories of twins are described under the listed entries:

See also

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