International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 529-530

Section 3.4.4.2. Layer groups

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail:  janovec@fzu.cz

3.4.4.2. Layer groups

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An adequate concept for characterizing symmetry properties of simple domain twins and planar domain walls is that of layer groups. A layer group describes the symmetry of objects that exist in a three-dimensional space and have two-dimensional translation symmetry. Typical examples are two-dimensional planes in three-dimensional space [two-sided planes and sectional layer groups (Holser, 1958[link]a,[link]b), domain walls and interfaces of zero thickness], layers of finite thickness (e.g. domain walls and interfaces of finite thickness) and two semi-infinite crystals joined along a planar and coherent (compatible) interface [e.g. simple domain twins with a compatible (coherent) domain wall, bicrystals].

A crystallographic layer group comprises symmetry operations (isometries) that leave invariant a chosen crystallographic plane p in a crystalline object. There are two types of such operations:

  • (i) side-preserving operations keep invariant the normal [{\bf n}] of the plane p, i.e. map each side of the plane p onto the same side. This type includes translations (discrete or continuous) in the plane p, rotations of 360°/n, n = 2, 3, 4, 6, around axes perpendicular to the plane p, reflections through planes perpendicular to p and glide reflections through planes perpendicular to p with glide vectors parallel to p. The corresponding symmetry elements are not related to the location of the plane p in space, i.e. they are the same for all planes parallel to p.

  • (ii) side-reversing operations invert the normal n of the plane i.e. exchange sides of the plane. Operations of this type are: an inversion through a point in the plane p, rotations of 360°/n, n = 3, 4, 6 around axes perpendicular to the plane followed by inversion through this point, 180° rotation and 180° screw rotation around an axis in the plane p, reflection and glide reflections through the plane p, and combinations of these operations with translations in the plane p. All corresponding symmetry elements are located in the plane p.

A layer group [{\cal L}] consists of two parts: [{\cal L}=\widehat{\cal L} \ \cup \ \underline{s}\widehat{\cal L},\eqno(3.4.4.7) ]where [\widehat{\cal L}] is a subgroup of [{\cal L}] that comprises all side-preserving operations of [{\cal L}]; this group is isomorphic to a plane group and is called a trivial layer group or a face group. An underlined character [\underline{s}] denotes a side-reversing operation and the left coset [\underline{s}\widehat{\cal L} ] contains all side-reversing operations of [{\cal L}]. Since [\widehat{\cal L}] is a halving subgroup, the layer group [{\cal L} ] can be treated as a dichromatic (black-and-white) group in which side-preserving operations are colour-preserving operations and side-reversing operations are colour-exchanging operations.

There are 80 layer groups with discrete two-dimensional translation subgroups [for a detailed treatment see IT E (2010)[link], or e.g. Vainshtein (1994[link]), Shubnikov & Kopcik (1974[link]), Holser (1958[link]a)]. Equivalent names for these layer groups are net groups (Opechowski, 1986[link]), plane groups in three dimensions (Grell et al., 1989[link]), groups in a two-sided plane (Holser, 1958[link]a,b[link]) and others.

To these layer groups there correspond 31 point groups that describe the symmetries of crystallographic objects with two-dimensional continuous translations. Holser (1958[link]b) calls these groups point groups in a two-sided plane, Kopský (1993[link]) coins the term point-like layer groups. We shall use the term `layer groups' both for layer groups with discrete translations, used in a microscopic description, and for crystallographic `point-like layer groups' with continuous translations in the continuum approach. The geometrical meaning of these groups is similar and most of the statements and formulae hold for both types of layer groups.

Crystallographic layer groups with a continuous translation group [point groups of two-sided plane (Holser, 1958[link]b)] are listed in Table 3.4.4.1[link]. The international notation corresponds to international symbols of layer groups with discrete translations; this notation is based on the Hermann–Mauguin (international) symbols of three-dimensional space groups, where the c direction is the direction of missing translations and the character `1' represents a symmetry direction in the plane with no associated symmetry element (see IT E , 2010[link]).

Table 3.4.4.1| top | pdf |
Crystallographic layer groups with continuous translations

InternationalNon-coordinate
[1 ] [1 ]
[\bar1 ] [\underline{\bar1} ]
[112 ] [2 ]
[11m ] [\underline{m} ]
[112/m ] [2/\underline{m} ]
[211 ] [\underline{2} ]
[m11 ] [\underline{m} ]
[2/m11 ] [\underline{2}/m ]
[222 ] [\underline{2}\underline{2}2 ]
[mm2 ] [mm2 ]
[m2m ] [m\underline{2}\underline{m} ]
[mmm ] [mm\underline{m}]
[4 ] [4 ]
[\bar4 ] [\underline{\bar4}]
[4/m ] [4/\underline{m}]
[422 ] [4\underline{2}\underline{2} ]
[4mm ] [4mm ]
[\bar42m ] [\underline{\bar4}\underline{2}m ]
[4/mmm ] [4/\underline{m}mm ]
[3 ] [3 ]
[\bar{3}] [\underline{\bar3}]
[32 ] [3\underline{2}]
[3m ] [3m ]
[\bar{3}m] [\underline{\bar3}m ]
[6 ] [6 ]
[\bar6] [\underline{\bar6} ]
[6/m] [6/\underline{m} ]
[622] [6\underline{2}\underline{2} ]
[6mm ] [6mm ]
[\bar{6}m2] [\underline{\bar6}m\underline{2} ]
[6/mmm ] [6/\underline{m}mm]

In the non-coordinate notation (Janovec, 1981[link]), side-reversing operations are underlined. Thus e.g. [\underline{2} ] denotes a [180^{\circ}] rotation around a twofold axis in the plane p and [{\underline m}] a reflection through this plane, whereas 2 is a side-preserving [180^{\circ}] rotation around an axis perpendicular to the plane and m is a side-preserving reflection through a plane perpendicular to the plane p. With exception of [\underline{\bar{1}}] and [{\underline 2}], the symbol of an operation specifies the orientation of the plane p. This notation allows one to signify layer groups with different orientations in one reference coordinate system. Another non-coordinate notation has been introduced by Shubnikov & Kopcik (1974[link]).

If a crystal with point-group symmetry G is bisected by a crystallographic plane p, then all operations of G that leave the plane p invariant form a sectional layer group = [\overline{{\sf G}(p)} ] of the plane p in G. Operations of the group [\overline{{\sf G}(p)}] can be divided into two sets [see equation (3.4.4.7[link])]: [\overline{{\sf G}(p)}=\widehat{{\sf G}(p)} \ \cup \ \underline{g}\widehat{{\sf G}(p)}, \eqno(3.4.4.8) ]where the trivial layer group [\widehat{{\sf G}(p)}] expresses the symmetry of the crystal face with normal [{\bf n}]. These face symmetries are listed in IT A (2005[link]), Part 10[link] , for all crystallographic point groups G and all orientations of the plane expressed by Miller indices [(hkl)]. The underlined operation [\underline{g}] is a side-reversing operation that inverts the normal [{\bf n}]. The left coset [\underline{g}\widehat{{\sf G}(p)}] contains all side-reversing operations of [\overline{{\sf G}(p)}].

The number [n_p] of planes symmetry-equivalent (in G) with the plane p is equal to the index of [\overline{{\sf G}(p)} ] in G: [n_p=[G:\overline{{\sf G}(p)}]=|G|:|\overline{{\sf G}(p)}|. \eqno(3.4.4.9) ]

Example 3.4.4.1.  As an example, we find the sectional layer group of the plane [(010) ] in the group [G=4_z/m_zm_xm_{xy}] (see Fig. 3.4.2.2[link]).[\eqalignno{{4_z/m_zm_xm_{xy}(010)}&=m_x2_ym_z \cup \ \underline{m}_y\{m_x2_ym_z\}\cr &= m_x2_ym_z \cup \ \{\underline{m}_y,\underline{2_z},\underline{\bar 1},\underline{2}_x\}\cr &=m_x\underline{m}_ym_z. &(3.4.4.10)} ]

In this example [n_p=|4_z/m_zm_xm_{xy}|:|m_x\underline{m}_ym_z|=16:8= 2 ] and the plane crystallographically equivalent with the plane [(010) ] is the plane (100) with sectional symmetry [\underline{m}_xm_ym_z ].

References

International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, 5th ed., edited by Th. Hahn. Heidelberg: Springer.
International Tables for Crystallography (2010). Vol. E, Subperiodic groups, 2nd. ed., edited by V. Kopský & D. B. Litvin. Chichester: Wiley.
Grell, H., Krause, C. & Grell, J. (1989). Tables of the 80 Plane Space Groups in Three Dimensions. Berlin: Akademie der Wissenschaften der DDR.
Holser, W. T. (1958a). Relation of symmetry to structure in twinning. Z. Kristallogr. 110, 249–265.
Holser, W. T. (1958b). Point groups and plane groups in a two-sided plane and their subgroups. Z. Kristallogr. 110, 266–281.
Janovec, V. (1981). Symmetry and structure of domain walls. Ferroelectrics, 35, 105–110.
Kopský, V. (1993). Translation normalizers of Euclidean motion groups. J. Math. Phys. 34, 1548–1576.
Opechowski, W. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: North-Holland.
Shubnikov, A. V. & Kopcik, V. A. (1974). Symmetry in Science and Art. New York: Plenum Press.
Vainshtein, B. K. (1994). Modern Crystallography I. Symmetry of Crystals. Berlin: Springer.








































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