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Difference between revisions of "Twinning"

From Online Dictionary of Crystallography
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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''; 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''.
 
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''.
  
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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''.
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* '''symmetry of a twin'''
  
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* '''isotropic media'''
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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''. The Mallard's law states that the ''twin element'' (i.e. the geometrical element relative to which the twining operation is defined) is restricted to a direct lattice element: lattice nodes (''twin centres''), lattice rows (''twin axes'') and lattice planes (''twin planes'').
  
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the linear coefficient of thermal expansion, α, relates the relative variation
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In most twins the symmetry of a twin (''twin point group'') is that of the individual point group augmented by the symmetry of the twinning operation; however, a symmetry element that is oblique to the twinning element of symmetry is absent in the twin (e.g., ''spinel twins'').
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(Δℓ/ℓ) of the length ℓ of a bar to the temperature variation Δ''T''. In the first order approximation it is given by:
 
  
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α = (Δ ℓ/ℓ) /Δ ''T''
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* '''twin law'''
  
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* '''anisotropic media'''
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The ''twin law'' is indicated by the symbol of the twinning element of symmetry: -1, [uvw] and (''hkl'') for the centre of symmetry, direction of the rotation axis and Miller indeces of the mirror plane, in the order.
  
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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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expansion is represented by a rank 2 tensor, &#945;''<sub>ij</sub>'', given by:
 
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&#945;''<sub>ij</sub>'' = ''u<sub>ij</sub>'' / &#916; ''T''.
 
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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>
 
  
  
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Revision as of 13:09, 18 April 2006

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.

  • symmetry of 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. The Mallard's law states that the twin element (i.e. the geometrical element relative to which the twining operation is defined) is restricted to a direct lattice element: lattice nodes (twin centres), lattice rows (twin axes) and lattice planes (twin planes).

In most twins the symmetry of a twin (twin point group) is that of the individual point group augmented by the symmetry of the twinning operation; however, a symmetry element that is oblique to the twinning element of symmetry is absent in the twin (e.g., spinel twins).

  • twin law

The twin law is indicated by the symbol of the twinning element of symmetry: -1, [uvw] and (hkl) for the centre of symmetry, direction of the rotation axis and Miller indeces of the mirror plane, in the order.

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