Symmetry of special projections

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 33-35

Section 2.2.14. Symmetry of special projections

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail: hahn@xtl.rwth-aachen.de

2.2.14. Symmetry of special projections

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Projections of crystal structures are used by crystallographers in special cases. Use of so-called `two-dimensional data' (zero-layer intensities) results in the projection of a crystal structure along the normal to the reciprocal-lattice net.

Even though the projection of a finite object along any direction may be useful, the projection of a periodic object such as a crystal structure is only sensible along a rational lattice direction (lattice row). Projection along a nonrational direction results in a constant density in at least one direction.

2.2.14.1. Data listed in the space-group tables

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Under the heading Symmetry of special projections, the following data are listed for three projections of each space group; no projection data are given for the plane groups.

(i) The projection direction. All projections are orthogonal, i.e. the projection is made onto a plane normal to the projection direction. This ensures that spherical atoms appear as circles in the projection. For each space group, three projections are listed. If a lattice has three kinds of symmetry directions, the three projection directions correspond to the primary, secondary and tertiary symmetry directions of the lattice (cf. Table 2.2.4.1). If a lattice contains less than three kinds of symmetry directions, as in the triclinic, monoclinic and rhombohedral cases, the additional projection direction(s) are taken along coordinate axes, i.e. lattice rows lacking symmetry.

The directions for which projection data are listed are as follows: $[\openup 2pt\eqalign{&\left.\matrix{\hbox{Triclinic}\hfill\cr \hbox{Monoclinic}\hfill\cr \hbox{(both settings)}\hfill\cr \hbox{Orthorhombic}\hfill\cr}\right\}{\hskip 3.5pt{[001]}}{\hskip 12.5pt{[100]}} {\hskip 12pt{[010]}} \cr &\matrix{\hbox{Tetragonal}\hfill &\ [001] &\ [100] &\ [110]\cr \hbox{Hexagonal}\hfill &\ [001] &\ [100] &\ [210]\cr \hbox{Rhombohedral}\hfill &\ [111] &\ [1\bar{1}0] &\ [2\bar{1}\bar{1}]\cr \hbox{Cubic}\hfill &\ [001] &\ [111] &\ [110]\cr}}]$
(ii) The Hermann–Mauguin symbol of the plane group resulting from the projection of the space group. If necessary, the symbols are given in oriented form; for example, plane group pm is expressed either as p1m1 or as p11m.

(iii) Relations between the basis vectors a′, b′ of the plane group and the basis vectors a, b, c of the space group. Each set of basis vectors refers to the conventional coordinate system of the plane group or space group, as employed in Parts 6 and 7 . The basis vectors of the two-dimensional cell are always called a′ and b′ irrespective of which two of the basis vectors a, b, c of the three-dimensional cell are projected to form the plane cell. All relations between the basis vectors of the two cells are expressed as vector equations, i.e. a′ and b′ are given as linear combinations of a, b and c. For the triclinic or monoclinic space groups, basis vectors a, b or c inclined to the plane of projection are replaced by the projected vectors $[{\bf a}_{p},{\bf b}_{p},{\bf c}_{p}]$ .

For primitive three-dimensional cells, the metrical relations between the lattice parameters of the space group and the plane group are collected in Table 2.2.14.1. The additional relations for centred cells can be derived easily from the table.

Table 2.2.14.1| top | pdf |
Cell parameters a′, b′, γ′ of the two-dimensional cell in terms of cell parameters a, b, c, α, β, γ of the three-dimensional cell for the projections listed in the space-group tables of Part 7

Projection direction	Triclinic	Monoclinic		Orthorhombic	Projection direction	Tetragonal
Projection direction	Triclinic	Unique axis b	Unique axis c	Orthorhombic	Projection direction	Tetragonal
[001]	$[a' = a \sin \beta]$	$[a' = a \sin \beta]$			[001]
	$[b' = b \sin \alpha]$
	$[\gamma' = 180^{\circ} - \gamma^{*}]$ ^†	$[\gamma' = 90^{\circ}]$	$[\gamma' = \gamma]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$
[100]	$[a' = b \sin \gamma]$		$[a' = b \sin \gamma]$		[100]
	$[b' = c \sin \beta]$	$[b' = c \sin \beta]$
	$[\gamma' = 180^{\circ} - \alpha^{*}]$ ^†	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$
[010]	$[a' = c \sin \alpha]$				[110]	$[a' = (a/2) \sqrt{2}]$
	$[b' = \alpha \sin \gamma]$		$[b' = a \sin \gamma]$
	$[\gamma' = 180^{\circ} - \beta^{*}]$ ^†	$[\gamma' = \beta]$	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$

Projection direction	Hexagonal	Projection direction	Rhombohedral^‡	Projection direction	Cubic
[001]		[111]	$[a' = {\displaystyle{2 \over \sqrt{3}}}\; a \sin (\alpha/2)]$	[001]
			$[b' = {\displaystyle{2 \over \sqrt{3}}} \;a \sin (\alpha/2)]$
	$[\gamma' = 120^{\circ}]$		$[\gamma' = 120^{\circ}]$		$[\gamma' = 90^{\circ}]$
[100]	$[a' = (a/2) \sqrt{3}]$	$[[1\bar{1}0]]$	$[a' = a \cos (\alpha/2)]$	[111]	$[a' = a\sqrt{2/3}]$
					$[b' = a\sqrt{2/3}]$
	$[\gamma' = 90^{\circ}]$		$[\gamma' = \delta]$ ^§		$[\gamma' = 120^{\circ}]$
[210]		$[[\bar{2}11]]$	$[a' = {\displaystyle{1 \over \sqrt{3}}}\; a\sqrt{1 + 2 \cos \alpha}]$	[110]	$[a' = (a/2)\sqrt{2}]$
			$[b' = a \sin (\alpha/2)]$
	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$

^† $[\cos \alpha^{*} = {\displaystyle{\cos \beta \cos \gamma - \cos \alpha \over \sin \beta \sin \gamma}}; \;\cos \beta^{*} = {\displaystyle{\cos \gamma \cos \alpha - \cos \beta \over \sin \gamma \sin \alpha}};\; \cos \gamma^{*} = {\displaystyle{\cos \alpha \cos \beta - \cos \gamma \over \sin \alpha \sin \beta}}.]$
^‡The entry `Rhombohedral' refers to the primitive rhombohedral cell with $[a = b = c, \alpha = \beta = \gamma]$ (cf. Table 2.1.2.1

).
^§ $[\cos \delta = {\displaystyle{\cos \alpha \over \cos \alpha/2\;}}]$ .

(iv) Location of the origin of the plane group with respect to the unit cell of the space group. The same description is used as for the location of symmetry elements (cf. Section 2.2.9).

Example

`Origin at x, 0, 0' or `Origin at $[{1 \over 4},{1 \over 4},z]$ '.

2.2.14.2. Projections of centred cells (lattices)

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For centred lattices, two different cases may occur:

(i) The projection direction is parallel to a lattice-centring vector. In this case, the projected plane cell is primitive for the centring types A, B, C, I and R. For F lattices, the multiplicity is reduced from 4 to 2 because c-centred plane cells result from projections along face diagonals of three-dimensional F cells.

Examples

(1) A body-centred lattice with centring vector $[{1 \over 2}({\bf a} + {\bf b} + {\bf c})]$ gives a primitive net, if projected along [], [ $[\bar{1}11]$ ], [ $[1\bar{1}1]$ ] or [ $[11\bar{1}]$ ].
(2) A C-centred lattice projects to a primitive net along the directions [110] and [ $[1\bar{1}0]$ ].
(3) An R-centred lattice described with `hexagonal axes' (triple cell) results in a primitive net, if projected along [ $[\bar{1}11]$ ], [] or [ $[\bar{1}\bar{2}1]$ ] for the obverse setting. For the reverse setting, the corresponding directions are [ $[1\bar{1}1]$ ], [ $[\bar{2}\bar{1}1]$ ], []; cf. Chapter 1.2 .

(ii) The projection direction is not parallel to a lattice-centring vector (general projection direction). In this case, the plane cell has the same multiplicity as the three-dimensional cell. Usually, however, this centred plane cell is unconventional and a transformation is required to obtain the conventional plane cell. This transformation has been carried out for the projection data in this volume.

Examples

(1) Projection along [] of a cubic I-centred cell leads to an unconventional quadratic c-centred plane cell. A simple cell transformation leads to the conventional quadratic p cell.
(2) Projection along [] of an orthorhombic I-centred cell leads to a rectangular c-centred plane cell, which is conventional.
(3) Projection along [] of an R-centred cell (both in obverse and reverse setting) results in a triple hexagonal plane cell h (the two-dimensional analogue of the H cell, cf. Chapter 1.2 ). A simple cell transformation leads to the conventional hexagonal p cell.

2.2.14.3. Projections of symmetry elements

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A symmetry element of a space group does not project as a symmetry element unless its orientation bears a special relation to the projection direction; all translation components of a symmetry operation along the projection direction vanish, whereas those perpendicular to the projection direction (i.e. parallel to the plane of projection) may be retained. This is summarized in Table 2.2.14.2 for the various crystallographic symmetry elements. From this table the following conclusions can be drawn:

(i) n-fold rotation axes and n-fold screw axes, as well as rotoinversion axes $[\bar{4}]$ , parallel to the projection direction project as n-fold rotation points; a $[\bar{3}]$ axis projects as a sixfold, a $[\bar{6}]$ axis as a threefold rotation point. For the latter, a doubling of the projected electron density occurs owing to the mirror plane normal to the projection direction $[(\bar{6} \equiv 3/m)]$ .

Table 2.2.14.2| top | pdf |
Projections of crystallographic symmetry elements

Symmetry element in three dimensions	Symmetry element in projection
Arbitrary orientation
$[\left.\!\matrix{\hbox{Symmetry centre}\hfill &\bar{1}\hfill\cr \hbox{Rotoinversion axis} &\bar{3} \equiv 3 \times \bar{1}\hfill\cr}\right\}]$	Rotation point 2 (at projection of centre)
Parallel to projection direction
Rotation axis 2; 3; 4; 6	Rotation point 2; 3; 4; 6
$[\!\matrix{\hbox{Screw axis}\hfill& 2_{1}\hfill\cr & 3_{1},3_{2}\hfill\cr & 4_{1},4_{2},4_{3}\hfill\cr & 6_{1},6_{2},6_{3},6_{4},6_{5}\hfill\cr}]$	$[\!\matrix{\hbox{Rotation point}\hfill &2\hfill\cr & 3\hfill\cr & 4\hfill\cr & 6\hfill\cr}]$
$[\!\matrix{\hbox{Rotoinversion axis}\hfill &\bar{4}\hfill\cr & \bar{6} \equiv 3/m\hfill\cr\cr & \bar{3} \equiv 3 \times \bar{1}\hfill\cr}]$	$[\!\matrix{\hbox{Rotation point}\hfill &4\hfill\cr &3, \hbox{with overlap}\hfill\cr & \quad \hbox{of atoms}\hfill\cr &6\hfill\cr}]$
Reflection plane m	Reflection line m
Glide plane with $[\perp]$ component^†	Glide line g
Glide plane without $[\perp]$ component^†	Reflection line m
Normal to projection direction
$[\!\matrix{\hbox{Rotation axis}\hfill &2\semi\ 4\semi \ 6\hfill\cr & 3\hfill\cr}]$	$[\!\matrix{\hbox{Reflection line } m\hfill\cr \hbox{None}\hfill\cr}]$
$[\!\matrix{\hbox{Screw axis}\hfill & 4_{2}\semi\ 6_{2},6_{4}\hfill\cr & 2_{1}\semi\ 4_{1},4_{3}\semi\ 6_{1},6_{3},6_{5}\hfill\cr & 3_{1},3_{2}\hfill\cr}]$	$[\!\matrix{\hbox{Reflection line } m\hfill\cr \hbox{Glide line }g\hfill\cr \hbox{None}\hfill\cr}]$
$[\!\matrix{\hbox{Rotoinversion axis}\hfill &\bar{4}\hfill\cr & \bar{6} \equiv 3/m\hfill\cr\cr\cr &\bar{3} \equiv 3 \times \bar{1}\hfill}]$	$[\!\matrix{\hbox{Reflection line }m \hbox{ parallel to axis}\hfill\cr \hbox{Reflection line }m \hbox{ perpendicular}\hfill\cr\quad\hbox{to axis (through projection of}\hfill\cr\quad\hbox{inversion point)}\hfill\cr \hbox{Rotation point 2 (at projection}\hfill\cr\quad\hbox{of centre)}\hfill\cr}]$

Reflection plane m	None, but overlap of atoms
Glide plane with glide vector t	Translation with translation vector t

^†The term `with $[\perp]$ component' refers to the component of the glide vector normal to the projection direction.

(ii) n-fold rotation axes and n-fold screw axes normal to the projection direction (i.e. parallel to the plane of projection) do not project as symmetry elements if n is odd. If n is even, all rotation and rotoinversion axes project as mirror lines: the same applies to the screw axes $[4_{2}, 6_{2}]$ and $[6_{4}]$ because they contain an axis 2. Screw axes $[2_{1}]$ , $[4_{1}]$ , $[4_{3}]$ , $[6_{1}]$ , $[6_{3}]$ and $[6_{5}]$ project as glide lines because they contain $[2_{1}]$ .
(iii) Reflection planes normal to the projection direction do not project as symmetry elements but lead to a doubling of the projected electron density owing to overlap of atoms. Projection of a glide plane results in an additional translation; the new translation vector is equal to the glide vector of the glide plane. Thus, a reduction of the translation period in that particular direction takes place.
(iv) Reflection planes parallel to the projection direction project as reflection lines. Glide planes project as glide lines or as reflection lines, depending upon whether the glide vector has or has not a component parallel to the projection plane.
(v) Centres of symmetry, as well as $[\bar{3}]$ axes in arbitrary orientation, project as twofold rotation points.

Example: (15, b unique, cell choice 1)

The C-centred cell has lattice points at 0, 0, 0 and $[{1 \over 2},{1 \over 2},0]$ . In all projections, the centre $[\bar{1}]$ projects as a twofold rotation point.

Projection along [001]: The plane cell is centred; $[2 \ \| \ [010]]$ projects as m; the glide component $[(0,0,{1 \over 2})]$ of glide plane c vanishes and thus c projects as m.

Result: Plane group c2mm (9), $[{\bf a}' = {\bf a}_{p}, {\bf b}' = {\bf b}]$ .

Projection along [100]: The periodicity along b is halved because of the C centring; $[2 \ \| \ [010]]$ projects as m; the glide component $[(0,0,{1 \over 2})]$ of glide plane c is retained and thus c projects as g.

Result: Plane group p2gm (7), $[{\bf a}' = {\bf b}/2,\ {\bf b}' = {\bf c}_{p}]$ .

Projection along [010]: The periodicity along a is halved because of the C centring; that along c is halved owing to the glide component $[(0,0,{1 \over 2})]$ of glide plane c; $[2 \ \| \ [010]]$ projects as 2.

Result: Plane group p2 (2), $[{\bf a}' = {\bf c}/2, {\bf b}' = {\bf a}/2]$ .

Further details about the geometry of projections can be found in publications by Buerger (1965) and Biedl (1966).

References

Biedl, A. W. (1966). The projection of a crystal structure. Z. Kristallogr. 123, 21–26.Google Scholar

Buerger, M. J. (1965). The geometry of projections. Tschermaks Mineral. Petrogr. Mitt. 10, 595–607.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 33-35