Characterization of space-group representations

Janssen, T.

doi:10.1107/97809553602060000629

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 49-50

Section 1.2.3.4. Characterization of space-group representations

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.2.3.4. Characterization of space-group representations

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The irreducible representations of space groups are characterized by a star of vectors in the Brillouin zone, and by the irreducible, possibly projective, representations of the point group of one point from that star.

The stars are sets of vectors in the Brillouin zone related mutually by transformations from the point group K of the space group G modulo reciprocal-lattice vectors. To obtain all stars, it is sufficient to take all vectors in the fundamental domain of the Brillouin zone, i.e. a part of the Brillouin zone such that no vectors in the domain are related by point-group elements (modulo $[\Lambda^{*}]$ ) and such that every point in the Brillouin zone is related to a vector in the fundamental domain by a point-group operation.

From each star one takes one point $[{\bf k}]$ and determines the nonequivalent irreducible representations of the point group $[K_{{\bf k}}]$ , the ordinary representations if the group $[G_{{\bf k}}]$ is symmorphic or $[{\bf k}]$ is inside the Brillouin zone, or the projective representations with factor system $[\omega]$ [equation (1.2.3.26)] otherwise. These representations are labelled $[\mu]$ . There are several conventions for the choice of this label, but an irreducible representation of G is always characterized by a pair ( $[{\bf k},\mu)]$ , where $[{\bf k}]$ fixes the star and $[\mu]$ the irreducible point-group representation .

The projective representations of the group of $[{\bf k}]$ , i.e. of $[K_{{\bf k}}]$ , can be obtained from the ordinary representations of a larger group. If the factor system $[\omega (R,R')]$ is of order m, the order of this larger group $[\hat{K}_{{\bf k}\omega}]$ is m times the order of $[K_{{\bf k}}]$ . Then the irreducible representations of the space group are labelled by the vector $[{\bf k}]$ in the Brillouin zone and an irreducible ordinary representation of $[\hat{K}_{{\bf k}\omega}]$ , where $[\omega]$ follows from (1.2.3.26).

Two stars such that one branch of the first one has the same $[K_{{\bf k}}]$ as one branch of the other determine representations that are quite similar. The only difference is the numerical value of the factors $[\exp (i{\bf k}\cdot{\bf a})]$ , the form of the representation matrices being the same. Such irreducible representations of the space group are said to belong to the same stratum. Strata are denoted by a symbol for one vector $[{\bf k}]$ in the Brillouin zone. For example, the origin, conventionally denoted by $[\Gamma]$ , belongs to one stratum that corresponds to the ordinary representations of the point group K. For a simple cubic space group, the point [ $[{\textstyle{1\over2}}, 0,0]$ ] is denoted by X. Its $[K_{{\bf k}}]$ is the tetragonal group [4/mmm] . All points [ $[\xi, 0,0]$ ] with $[\xi \neq 0]$ and $[-{\textstyle{1\over2}} \,\lt\, \xi \,\lt\, {\textstyle{1\over2}}]$ form one stratum with point group [4mm] . This stratum is denoted by $[\Delta]$ etc. The strata can be compared with the Wyckoff positions in direct space. There a Wyckoff position is a manifold in the unit cell for which all points have the same site symmetry, modulo the lattice translations. Here it is a manifold of k vectors with the same symmetry group modulo the reciprocal lattice. The action of $[G_{{\bf k}}]$ does not involve the nonprimitive translations. Therefore, the strata correspond to Wyckoff positions of the corresponding symmorphic space group. The stratum symbols for the various three-dimensional Bravais classes are given in Table 1.2.6.11.

As an example, we consider here the orthorhombic space group [Pnma] . The orthorhombic Brillouin zone has a fundamental domain with volume that is one-eighth of that of the Brillouin zone. The various choices of $[{\bf k}]$ in this fundamental domain, together with the corresponding point groups $[K_{{\bf k}}]$ , are given in Table 1.2.3.1. The vectors $[{\bf k}]$ correspond to Wyckoff positions of the group [Pmmm] .

Table 1.2.3.1 | top | pdf |
Choices of $[{\bf k}]$ in the fundamental domain of [Pnma] and the elements of $[K_{\bf k}]$

$[{\bf k}]$	Wyckoff position	Elements
	a		$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
$[{{1}\over{2}}]$ 00	b		$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
0 $[{{1}\over{2}}]$ 0	e		$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
00 $[{{1}\over{2}}]$	c		$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
0 $[{{1}\over{2}}{{1}\over{2}}]$	g	E	$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
$[{{1}\over{2}} 0{{1}\over{2}}]$	d	E	$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
$[{{1}\over{2}} {{1}\over{2}} 0]$	f	E	$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	h	E	$[m_{x}]$	$[m_{y}]$	$[m_{z}]$	$[\bar{1}]$	2_x	2_y	2_z
$[\xi]$ 00	i	E	$[m_{y}]$	$[m_{z}]$	2_x
$[\xi{{1}\over{2}} 0]$	k	E	$[m_{y}]$	$[m_{z}]$	2_x
$[\xi 0{{1}\over{2}}]$	j	E	$[m_{y}]$	$[m_{z}]$	2_x
$[\xi {{1}\over{2}}{{1}\over{2}}]$	l	E	$[m_{y}]$	$[m_{z}]$	2_x
$[0\eta 0]$	m	E	$[m_{x}]$	$[m_{z}]$	2_y
$[{{1}\over{2}}\eta 0]$	o	E	$[m_{x}]$	$[m_{z}]$	2_y
$[0\eta {{1}\over{2}}]$	n	E	$[m_{x}]$	$[m_{z}]$	2_y
$[{{1}\over{2}} \eta {{1}\over{2}}]$	p	E	$[m_{x}]$	$[m_{z}]$	2_y
$[00\zeta]$	q	E	$[m_{x}]$	$[m_{y}]$	2_z
$[{{1}\over{2}} 0\zeta]$	s	E	$[m_{x}]$	$[m_{y}]$	2_z
$[0{{1}\over{2}} \zeta]$	r	E	$[m_{x}]$	$[m_{y}]$	2_z
$[{{1}\over{2}}{{1}\over{2}}\zeta]$	t	E	$[m_{x}]$	$[m_{y}]$	2_z
$[0\eta\zeta]$	u	E	$[m_{x}]$
$[{{1}\over{2}}\eta\zeta]$	v	E	$[m_{x}]$
$[\xi 0 \zeta]$	w	E	$[m_{y}]$
$[\xi {{1}\over{2}} \zeta]$	x	E	$[m_{y}]$
$[\xi\eta 0]$	y	E	$[m_{z}]$
$[\xi\eta {{1}\over{2}}]$	z	E	$[m_{z}]$
$[\xi\eta\zeta]$	$[\alpha]$	E

In the tables, the vectors $[{\bf k}]$ and their corresponding Wyckoff positions are given for the holohedral space groups. In general, the number of different strata is smaller for the other groups. One can still use the same symbols for these groups, or take the symbols for the Wyckoff positions for the groups that are not holohedral. Consider as an example the group [Pmm2] . Its holohedral space group is [Pmmm] . The strata of irreducible representations can be labelled by the symbols for Wyckoff positions of [Pmm2] as well as those of [Pmmm] . This is shown in Table 1.2.3.2.

Table 1.2.3.2 | top | pdf |
Strata of irreducible representations of [Pmm2] and [Pmmm]

$[{\bf k}]$	Wyckoff position in	Wyckoff positions in	$[K_{{\bf k}}]$
00 $[\zeta]$	a
$[0 {{1}\over{2}}\zeta]$	b
$[{{1}\over{2}} 0\zeta]$	c
$[{{1}\over{2}}{{1}\over{2}}\zeta]$	d
$[\xi 0\zeta]$	e
$[\xi{{1}\over{2}}\zeta]$	f
$[0\eta\zeta]$	g
$[{{1}\over{2}}\eta\zeta]$	h
$[\xi\eta\zeta]$	i	$[y,z,\alpha]$	1

The defining relations for the point group [mmm] are $[A^{2}=B^{2}=(AB)^{2}=C^{2}=E,\quad AC=CA,\quad BC=CB.]$ For the subgroups, the defining relations follow from these. The corresponding expressions in the representation matrices $[\Gamma (A_{i})]$ for the generators of the point groups give expressions $[W_{i}^{{\rm left}}(A_{1},\ldots, A_{r}) = \lambda_{i} W_{i}^{{\rm right}}(A_{1},\ldots, A_{r}),\quad i=1,\ldots.]$ In the example one has $[\matrix{\Gamma (A)^{2}=\lambda_{1}E \hfill & \Gamma (B)^{2}=\lambda_{2}E\hfill\cr \left(\Gamma (A)\Gamma (B) \right)^{2}=\lambda_{3}E\hfill&\Gamma (C)^{2}=\lambda_{4}E\hfill\cr \Gamma (A)\Gamma (C)=\lambda_{5}\Gamma (C)\Gamma (A)\hfill & \Gamma (B)\Gamma (C) =\lambda_{6}\Gamma (C)\Gamma (B).\hfill}]$ The values for $[\lambda_{i}]$ characterize the projective representation factor system and are given in Table 1.2.3.3. They are unity for ordinary representations.

Table 1.2.3.3 | top | pdf |
Characteristic values of $[\lambda_{i}]$ for the projective irreps of $[K_{\bf k}]$ for the point group [mmm]

$[{\bf k}]$	$[A^{2}]$	$[B^{2}]$	$[(AB)^{2} ]$	$[ C^{2} ]$			Representations
$[{\bf k}]$	$[\lambda_{1}]$	$[\lambda_{2}]$	$[\lambda_{3}]$	$[\lambda_{4}]$	$[\lambda_{5}]$	$[\lambda_{6}]$	Number	Dimension
000	1	1	1	1	1	1	8	1
$[{{1}\over{2}}]$ 00	−1	1	−1	1	−1	1	2	2
0 $[{{1}\over{2}}]$ 0	1	−1	1	1	1	1	2	2
00 $[{{1}\over{2}}]$	1	1	1	−1	−1	1	2	2
0 $[{{1}\over{2}}{{1}\over{2}}]$	1	−1	1	−1	−1	1	2	2
$[{{1}\over{2}} 0{{1}\over{2}}]$	−1	1	−1	−1	1	1	8	1
$[{{1}\over{2}} {{1}\over{2}} 0]$	−1	−1	−1	1	−1	1	2	2
$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	−1	−1	−1	−1	1	1	2	2
$[\xi]$ 00	1	1	1				4	1
$[\xi{{1}\over{2}} 0]$	−1	1	−1				4	1
$[\xi 0{{1}\over{2}}]$	1	−1	−1				4	1
$[\xi {{1}\over{2}}{{1}\over{2}}]$	−1	−1	1				4	1
$[0\eta 0]$	1	1	1				4	1
$[{{1}\over{2}}\eta 0]$	−1	1	1				1	2
$[0\eta {{1}\over{2}}]$	1	−1	1				1	2
$[{{1}\over{2}} \eta {{1}\over{2}}]$	−1	−1	1				4	1
$[00\zeta]$	1	1	1				4	1
$[{{1}\over{2}} 0\zeta]$	−1	1	−1				1	2
$[0{{1}\over{2}} \zeta]$	1	−1	1				1	2
$[{{1}\over{2}}{{1}\over{2}}\zeta]$	−1	−1	−1				1	2
$[0\eta\zeta]$	1						2	1
$[{{1}\over{2}}\eta\zeta]$	−1						2	1
$[\xi 0 \zeta]$	1						2	1
$[\xi {{1}\over{2}} \zeta]$	−1						2	1
$[\xi\eta 0]$	1						2	1
$[\xi\eta {{1}\over{2}}]$	−1						2	1
$[\xi\eta\zeta]$							1	1

By putting factors i in front of the representation matrices in the appropriate places, some of the values of $[\lambda_{i}]$ can be changed from [-1] to [+1] . In this way, one obtains either ordinary representations, which are necessarily one-dimensional for these Abelian groups, or projective representations, which are in this case two-dimensional. This is indicated as well in Table 1.2.3.3. The one-dimensional irreducible representations are ordinary representations of the group $[K_{{\bf k}}]$ . The two-dimensional ones are projective representations, but correspond to ordinary representations of the larger groups isomorphic to $[D_{4}\times C_{2}]$ and $[D_{4}]$ .

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 49-50