Symmetry of special projections

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000647

International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

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International Tables for Crystallography (2006). Vol. E. ch. 1.2, pp. 16-17 | 1 | 2 |

Section 1.2.14. Symmetry of special projections

V. Kopský^a and D. B. Litvin^b ^*

^a Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610-6009, USA
Correspondence e-mail: u3c@psu.edu

1.2.14. Symmetry of special projections

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1.2.14.1. Data listed in the subperiodic group tables

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Under the heading Symmetry of special projections, the following data are listed for three orthogonal projections of each layer group and rod group and two orthogonal projections of each frieze group:

(i) For layer and rod groups, each projection is made onto a plane normal to the projection direction. If there are three kinds of symmetry directions (cf. Table 1.2.4.1), the three projection directions correspond to the primary, secondary and tertiary symmetry directions. If there are fewer than three symmetry directions, the additional projection direction(s) are taken along coordinate axes.

For frieze groups, each projection is made on a line normal to the projection direction.

The directions for which data are listed are as follows:

(a) Layer groups: $[\matrix{\left.\matrix{{\rm Triclinic/oblique}\hfill\cr {\rm Monoclinic/oblique}\hfill\cr {\rm Monoclinic/rectangular}\hfill\cr {\rm Orthorhombic/rectangular}\hfill}\right\}\hfill&[001] [100] [010] \hfill\cr &\cr\left.\matrix{{\rm Tetragonal/square}}\right.\hfill& [001] [100] [110]\hfill\cr&\cr\left.\matrix{{\rm Trigonal/hexagonal}\hfill\cr {\rm Hexagonal/hexagonal}\hfill}\right\}\hfill&[001] [100] [210]\hfill}]$

(b) Rod groups: $[\matrix{\left.\matrix{{\rm Triclinic}\hfill\cr {\rm Monoclinic/inclined}\hfill\cr {\rm Monoclinic/orthogonal}\hfill\cr {\rm Orthorhombic}\hfill}\right\}\hfill&[001] [100] [010]\hfill\cr&\cr\left.\matrix{\rm Tetragonal}\right.\hfill&[001] [100] [110]\hfill\cr&\cr \left.\matrix{{\rm Trigonal}\hfill\cr {\rm Hexagonal}\hfill}\right\}\hfill&[001] [100] [210]\hfill}]$

(c) Frieze groups: $[\left.\matrix{{\rm Oblique}\hfill\cr {\rm Rectangular}\hfill}\right\}\hskip8pt\quad\quad\quad\quad[10] [01]]$
(ii) The Hermann–Mauguin symbol. For the [001] projection of a layer group, the Hermann–Mauguin symbol for the plane group resulting from the projection of the layer group is given. For the [001] projection of a rod group, the Hermann–Mauguin symbol for the resulting two-dimensional point group is given. For the remainder of the projections, in the case of both layer groups and rod groups, the Hermann–Mauguin symbol is given for the resulting frieze group. For the [10] projection of a frieze group, the Hermann–Mauguin symbol of the resulting one-dimensional point group, i.e. 1 or m, is given. For the [01] projection, the Hermann–Mauguin symbol of the resulting one-dimensional space group, i.e. p1 or pm, is given.

(iii) For layer groups, the basis vectors a′, b′ of the plane group resulting from the [001] projection and the basis vector a′ of the frieze groups resulting from the additional two projections are given as linear combinations of the basis vectors a, b of the layer group. Basis vectors a, b inclined to the plane of projection are replaced by the projected vectors a_p, b_p. For the two projections of a rod group resulting in a frieze group, the basis vector a′ of the resulting frieze group is given in terms of the basis vector c of the rod group. For the [01] projection of a frieze group, the basis vector a′ of the resulting one-dimensional space group is given in terms of the basis vector a of the frieze group.

For rod groups and layer groups, the relations between a′, b′ and γ′ of the projected conventional basis vectors and a, b, c, α, β and γ of the conventional basis vectors of the subperiodic group are given in Table 1.2.14.1. We also give in this table the relations between a′ of the projected conventional basis and a, b and γ of the conventional basis of the frieze group.

Table 1.2.14.1 | top | pdf |
a ′, b′, γ′ (a′) of the projected conventional coordinate system in terms of a, b, c, α, β, γ (a, b, γ) of the conventional coordinate system of the layer and rod groups (frieze groups)

(a) Layer groups.

Projection direction	Triclinic/oblique	Monoclinic/oblique
[001]	$[a'=a\sin\beta]$
	$[b'=b\sin \alpha]$
	$[\gamma'=180^{\circ}-\gamma^{*}]$ ^†	$[\gamma'=\gamma]$
[100]	$[a'=b\sin \gamma]$	$[a'=b\sin \gamma]$
	$[b'=c\sin\beta]$
	$[\gamma'=180^{\circ}-\alpha^{*}]$ ^†	$[\gamma' = 90^{\circ}]$
[010]	$[a'=a\sin\gamma]$	$[a'=a\sin\gamma]$
	$[b'=c\sin\alpha]$
	$[\gamma'=180^{\circ}-\beta^{*}]$ ^†	$[\gamma'=90^{\circ}]$
	Monoclinic/rectangular	Orthorhombic/rectangular
[001]
	$[b'=b\sin\alpha]$
	$[\gamma'=90^{\circ}]$	$[\gamma'=90^{\circ}]$
[100]

	$[\gamma'=\alpha]$	$[\gamma'=90^{\circ}]$
[010]
	$[b'=c\sin\alpha]$
	$[\gamma'=90^{\circ}]$	$[\gamma'=90^{\circ}]$
	Tetragonal/square
[001]

	$[\gamma'=90^{\circ}]$
[100]

	$[\gamma'=90^{\circ}]$
[110]	$[a'=(a/2)(2)^{1/2}]$

	$[\gamma'=90^{\circ}]$
	Trigonal/hexagonal, hexagonal/hexagonal
[001]

	$[\gamma'=120^{\circ}]$
[100]	$[a'=[(3)^{1/2}/2]a]$

	$[\gamma'=90^{\circ}]$
[210]

	$[\gamma'=90^{\circ}]$

(b) Rod groups.

Projection direction	Triclinic	Monoclinic/inclined
[001]	$[a'=a\sin\beta]$
	$[b'=b\sin\alpha]$	$[b'=b\sin\alpha]$
	$[\gamma'=180^{\circ}-\gamma^{*}]$ ^†	$[\gamma'=90^{\circ}]$
[100]	$[a'=c\sin\beta]$
	$[b'=b\sin\gamma]$
	$[\gamma'=180^{\circ}-\alpha^{*}]$ ^†	$[\gamma=\alpha]$
[010]	$[a'=c\sin\alpha]$	$[a'=c\sin\alpha]$
	$[b'=a\sin\gamma]$
	$[\gamma'=180^{\circ}-\beta^{*}]$ ^†	$[\gamma'=90^{\circ}]$
	Monoclinic/orthogonal	Orthorhombic
[001]

	$[\gamma'=\gamma]$	$[\gamma'=90^{\circ}]$
[100]
	$[b'=b\sin\gamma]$
	$[\gamma'=90^{\circ}]$	$[\gamma'=90^{\circ}]$
[010]
	$[b'=a\sin\gamma]$
	$[\gamma'=90^{\circ}]$	$[\gamma'=90^{\circ}]$
	Tetragonal
[001]

	$[\gamma'=90^{\circ}]$
[100]

	$[\gamma'=90^{\circ}]$
[110]
	$[b'=(a/2)(2)^{1/2}]$
	$[\gamma'=90^{\circ}]$
	Trigonal, hexagonal
[001]

	$[\gamma'=120^{\circ}]$
[100]
	$[b'=[(3)^{1/2}/2]a]$
	$[\gamma'=90^{\circ}]$
[210]

	$[\gamma'=90^{\circ}]$

Projection direction	Oblique	Rectangular
[10]	$[a'=b\sin\gamma]$
[01]	$[a'=a\sin\gamma]$

^† $[\cos \alpha^{*}=(\cos\beta\cos\gamma-\cos\alpha)/(\sin\beta\sin\gamma)]$ , $[\cos \beta^{*}=(\cos\gamma\cos\alpha-\cos\beta)/(\sin\gamma\sin\alpha)]$ , $[\cos \gamma^{*}=(\cos\alpha\cos\beta-\cos\gamma)/(\sin\alpha\sin\beta).]$

(iv) Location of the origin of the plane group, frieze group and one-dimensional space group is given with respect to the conventional lattice of the subperiodic group. The same description is used as for the location of symmetry elements (see Section 1.2.9). Example: `Origin at x, 0, 0' or `Origin at x, 1/4, 0'.

1.2.14.2. Projections of centred subperiodic groups

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The only centred subperiodic groups are the nine types of centred layer groups. For the [100] and [010] projection directions, because of the centred layer-group lattice, the basis vectors of the resulting frieze groups are a′ = b/2 and a′ = a/2, respectively.

1.2.14.3. Projection of symmetry elements

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A symmetry element of a subperiodic group projects as a symmetry element only if its orientation bears a special relationship to the projection direction. In Table 1.2.14.2, the three-dimensional symmetry elements of the layer and rod groups and in Table 1.2.14.3 the two-dimensional symmetry elements of the frieze groups are listed along with the corresponding symmetry element in projection.

Table 1.2.14.2 | top | pdf |
Projection of three-dimensional symmetry elements (layer and rod groups)

Symmetry element in three dimensions		Symmetry element in projection
Arbitrary orientation
Symmetry centre $[\bar{1}]$		Rotation point 2 at projection of centre
Parallel to projection direction
Rotation axis	2, 3, 4, 6	Rotation point	2, 3, 4, 6
Screw axis	2₁	Rotation point	2
	3₁, 3₂		3
	4₁, 4₂, 4₃		4
	6₁, 6₂, 6₃, 6₄, 6₅		6
Rotoinversion axis	$[\bar{4}]$	Rotation point	4
	$[\bar{6}\equiv 3/m]$		3 (with overlap of atoms)
	$[\bar{3}\equiv 3\times \bar{1}]$		6
Reflection plane m		Reflection line m
Glide plane with ⊥ component^†		Glide line g
Glide plane without ⊥ component^†		Reflection line m
Normal to projection direction
Rotation axis	2, 4, 6	Reflection line m
Rotation axis	3	None
Screw axis	4₂, 6₂, 6₄	Reflection line m
	2₁, 4₁, 4₃, 6₁, 6₃, 6₅	Glide line g
	3₁, 3₂	None
Rotoinversion axis	$[\bar{4}]$	Reflection line m parallel to axis
	$[\bar{6}\equiv 3/m]$	Reflection line m perpendicular to axis
	$[\bar{3}\equiv 3 \times \bar{1}]$	Rotation point 2 (at projection of centre)
Reflection plane m		None, but overlap of atoms
Glide plane with glide component t		Translation t

^† The term `with ⊥ component' refers to the component of the glide vector normal to the projection direction.

Table 1.2.14.3 | top | pdf |
Projection of two-dimensional symmetry elements (frieze groups)

Symmetry element in two dimensions	Symmetry element in projection
Rotation point 2	Reflection point m
Parallel to projection direction
Reflection line m	Reflection point m
Glide line g	Reflection point m
Normal to projection direction
Reflection line m	None (with overlap of atoms)
Glide line g with glide component t	Translation t

Example: Layer group cm2m (L35)

Projection along [001]: This orthorhombic/rectangular plane group is centred; m perpendicular to [100] is projected as a reflection line, 2 parallel to [010] is projected as the same reflection line and m perpendicular to [001] gives rise to no symmetry element in projection, but to an overlap of atoms. Result: Plane group c1m1 (5) with a′ = a and b′ = b.

Projection along [100]: The frieze group has the basis vector a′ = b/2 due to the centred lattice of the layer group. m perpendicular to [100] gives rise only to an overlap of atoms, 2 parallel to [010] is projected as a reflection line and m perpendicular to [001] is projected as the same reflection line. Result: Frieze group $[{\scr p}11m]$ (F4) with a′ = b/2.

Projection along [010]: The frieze group has the basis vector a′ = a/2 due to the centred lattice of the layer group. The two reflection planes project as perpendicular reflection lines and 2 parallel to [010] projects as the rotation point 2. Result: Frieze group $[{\scr p}2mm]$ (F6) with a′ = a/2.

References

International Tables for Crystallography (2006). Vol. E. ch. 1.2, pp. 16-17