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:

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-<Font color="blue"> Maclage </Font>(''Fr''). <Font color="red"> Zwillingsbildung </Font>(''Ge''). <Font color="green"> Maclado (formación de macla) </Font> (''Sp''). <Font color="purple"> двойникование </Font> (''Ru''). <Font color="black"> Geminazione </Font>(''It'')
+<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'').
 = 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''.
+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]].
-* '''symmetry of a 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'').
-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 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'': ''m''<math> \bar 3</math>''m'' crystal point group; {111} [[twin law]]; <math> \bar 3</math>/''m'' twin point group.
-'''[[twin law]]'''
+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 ''reticular'' (i.e. lattice) ''theory of twinning'' which indicates the presence, either in the lattice or a sublattice of a crystal, of (pseudo)symmetry elements as necessary, even if not sufficient, condition for the formation of twins. In presence of the reticular necessary conditions, the formation of a twing finally depends on the matching of the crystal structures at the contact surface between the individuals.
+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.
-* '''twinning by merohedry'''
-The twin operation belongs to to the point group of the lattice but not to the point group of the crystal. Therefore, the point group of the crystal must be a subgroup of the point group of the lattice, i.e. the crystal shows only a part (''[[merohedry]]'') of the symmetry elements belonging to the its lattice which, instead, shows ''[[holohedry]]'' (complete symmetry). The twinning element of symmetry may (''Class I of twins by merohedry'') or may not belong to the Laue class of the crystal (''Class II of twins by merohedry''): consequences are discussed under ''solving the crystal structure of twins''. - Examples - Class I: in crystals with point group 2 (Laue group 2/''m'') the mirror plane ''m'' acts as twinning operator .
-Class II: in crystals with point group 4 (Laue group 4/''m'') a mirror plane ''m'' parallel to the foufold axis 4 acts as twinning operator.
-* '''twinning by pseudomerohedry'''
-A lattice is said to be pseudosymmetric if at least a lattice row or/and a lattice plane approximately correspond to elements of symmetry; these elements can act as twinning operators (e.g., a monoclinic lattice with its oblique angle ''close'' to 90° is pseudo-orthorhombic and thus shows two pseudo twofold axes and two pseudo mirror planes).
-* '''twinning by reticular merohedry'''
-In the presence of a sublattice displaying symmetry other than that of the crystal lattice, a symmetry element belonging to the sublattice point group but not to the crystal point group can act as twinning operator. If lattice and sublattice have the same point group but (some of) their elements of symmetry are differently oriented ''twins by polyholohedry'' can form.
-* '''twinning by reticular pseudomerohedry'''
-Substituting lattice with sublattice, the definition of twinning by reticular pseudomerohedry corresponds to that given for the twinning by pseudomerohedry.
-* '''overlap of lattices'''
-By effect of a twinning operation, both the direct and reciprocal lattice of the individuals forming a twin are overlapped. Overlapping (restoration) of nodes belonging to different individuals can be: (i) exact and total (twinning by merohedry); (ii) exact but partial (i.e. only a fraction of the nodes of an individual lattice is restored; twinning by reticular merohedry); (iii) total but approximate (twinning by pseudomerohedry), approximate and partial (twinning by reticular pseudomerohedry).
-* '''twin index'''
-The reciprocal ''n'' of the fraction 1/''n'' of (quasi)restored nodes is called ''twin index''
-* '''twin lattice'''
-The lattice that is formed by the (quasi)restored nodes is the ''twin lattice''. It corresponds to the crystal lattice in twins by (pseudo)merohedry and to a sublattice of the crystal (individual) in twins by reticular (pseudo)merohedry.
-* '''twin obliquity'''
-The ''twin obliquity'' is a measure of the distorsion of a (sub)lattice in twins by (reticular) pseudomerohedry.
-* '''twinning by metric merohedry'''
-* '''corresponding twins'''
-= Other categories of twins =
+The following categories of twins are described under the listed entries:
-= Endemic conditions for twinning =
+*[[Twinning by merohedry]]
+*[[Twinning by pseudomerohedry]]
+*[[Twinning by reticular merohedry]]
+*[[Twinning by reticular pseudomerohedry]]
+*[[Twinning by metric merohedry]]
+*[[Twinning by reticular polyholohedry]]
+*[[Hybrid twin]]s
+*[[Allotwin]]s
+*[[Selective merohedry]]
 == See also ==
-Chapter 1.3 of ''International Tables of Crystallography, Volume C''<br>
+*[[Twin index]]
-Chapter 3.3 of ''International Tables of Crystallography, Volume D''<br>
+*[[Twin lattice]]
+*[[Twin law]]
+*[[Twin obliquity]]
+*[[Corresponding twins]]
+*[[Twinning (endemic conditions of)]]
+*[[Twinning (effects of)]]
+*Chapter 1.3 of ''International Tables for Crystallography, Volume C''
+*Chapter 3.3 of ''International Tables for Crystallography, Volume D''
-[[Category:Fundamental crystallography]]
+[[Category:Twinning]]

Difference between revisions of "Twinning"

From Online Dictionary of Crystallography

Latest revision as of 14:24, 20 November 2017

Oriented association and twinning

Classification of twins

See also