International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 61-62

Section 1.4.4.1. Crystallographic orbits

B. Souvignierd

1.4.4.1. Crystallographic orbits

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Since the operations of a space group provide symmetries of a crystal pattern, two points X and Y that are mapped onto each other by a space-group operation are regarded as being geometrically equivalent. Starting from a point [X \in {\bb E}^3], infinitely many points Y equivalent to X are obtained by applying all space-group operations [\ispecialfonts{\sfi g}=({\bi W}, {\bi w})] to X: [\ispecialfonts Y={\sfi g}(X)=({\bi W}, {\bi w})X] = [({\bi W}X+{\bi w})].

Definition

For a space group [{\cal G}] acting on the three-dimensional space [{\bb E}^3], the (infinite) set[\ispecialfonts{\cal O} = {\cal G}(X):= \{ {\sfi g}(X) | {\sfi g} \in {\cal G} \}]is called the orbit of X under [{\cal G}].

The orbit of X is the smallest subset of [{\bb E}^3] that contains X and is closed under the action of [{\cal G}]. It is also called a crystallographic orbit.

Every point in direct space [{\bb E}^3] belongs to precisely one orbit under [{\cal G}] and thus the orbits of [{\cal G}] partition the direct space into disjoint subsets. It is clear that an orbit is completely determined by its points in the unit cell, since translating the unit cell by the translation subgroup [{\cal T}] of [{\cal G}] entirely covers [{\bb E}^3].

It may happen that two different symmetry operations [\ispecialfonts\sfi g] and [\ispecialfonts\sfi h] in [{\cal G}] map X to the same point. Since [\ispecialfonts{\sfi g}(X) = {\sfi h}(X)] implies that [\ispecialfonts{\sfi h}^{-1} {\sfi g}(X) = X], the point X is fixed by the nontrivial operation [\ispecialfonts{\sfi h}^{-1} {\sfi g}] in [{\cal G}].

Definition

The subgroup [\ispecialfonts{\cal S}_X ={\cal S}_{\cal G}(X): = \{ {\sfi g} \in {\cal G} | {\sfi g}(X) = X \}] of symmetry operations from [{\cal G}] that fix X is called the site-symmetry group of X in [{\cal G}].

Since translations, glide reflections and screw rotations fix no point in [{\bb E}^3], a site-symmetry group [{\cal S}_X] never contains operations of these types and thus consists only of reflections, rotations, inversions and rotoinversions. Because of the absence of translations, [{\cal S}_X] contains at most one operation from a coset [\ispecialfonts{\cal T} {\sfi g}] relative to the translation subgroup [{\cal T}] of [{\cal G}], since otherwise the quotient of two such operations [\ispecialfonts{\sfi t} {\sfi g}] and [\ispecialfonts{\sfi t'} {\sfi g}] would be the non-trivial translation [\ispecialfonts{\sfi t} {\sfi g} {\sfi g}^{-1} {\sfi t}'^{-1} = {\sfi t} {\sfi t}'^{-1}] (see Chapter 1.3[link] for a discussion of coset decompositions). In particular, the operations in [{\cal S}_X] all have different linear parts and because these linear parts form a subgroup of the point group [{\cal P}] of [{\cal G}], the order of the site-symmetry group [{\cal S}_X] is a divisor of the order of the point group of [{\cal G}].

The site-symmetry group of a point X is thus a finite subgroup of the space group [{\cal G}], a subgroup which is isomorphic to a subgroup of the point group [{\cal P}] of [{\cal G}].

Example

For a space group [{\cal G}] of type [P\bar{1}], the site-symmetry group of the origin [X = \pmatrix{0 \cr 0 \cr 0 }] is clearly generated by the inversion in the origin: [\{\overline{1}|0\} (X)= X ]. On the other hand, the point [Y = \pmatrix{\textstyle{{1 \over 2}} \cr 0 \cr \textstyle{{1 \over 2}} }] is fixed by the inversion in Y, i.e.[\{\overline{1}|1,0,1\}(Y)=\pmatrix{\bar{1} & 0 & 0 \cr 0 & \bar{1} & 0 \cr 0 & 0 & \bar{1} } \pmatrix{\textstyle{{1 \over 2}} \cr 0 \cr \textstyle{{1 \over 2}} }+ \pmatrix{1 \cr 0 \cr 1 } = \pmatrix{ \textstyle{{1 \over 2}} \cr 0 \cr \textstyle{{1 \over 2}} } = Y.]The symmetry operation [\{\overline{1}|1,0,1\}] also belongs to [{\cal G}] and generates the site-symmetry group of Y. The site-symmetry groups [{\cal S}_X=\{\{1|0\}, \{\overline{1}|0\}\}] of X and [{\cal S}_Y=\{\{1|0\}, \{\overline{1}|1,0,1\}\}] of Y are thus different subgroups of order 2 of [{\cal G}] which are isomorphic to the point group of [{\cal G}] (which is generated by [\overline{1}]).

The order [|{\cal S}_X|] of the site-symmetry group [{\cal S}_X] is closely related to the number of points in the orbit of X that lie in the unit cell. An application of the orbit–stabilizer theorem (see Section 1.1.7[link] ) yields the crucial observation that each point [\ispecialfonts Y = {\sfi g}(X)] in the orbit of X under [{\cal G}] is obtained precisely [|{\cal S}_X|] times as an orbit point: for each [\ispecialfonts{\sfi h} \in {\cal S}_X] one has [\ispecialfonts{\sfi g} {\sfi h}(X)] = [\ispecialfonts{\sfi g}(X) = Y] and conversely [\ispecialfonts{\sfi g}'(X) = {\sfi g}(X)] implies that [\ispecialfonts{\sfi g}^{-1} {\sfi g}' =] [\ispecialfonts {\sfi h} \in {\cal S}_X] and thus [\ispecialfonts{\sfi g}' = {\sfi g} {\sfi h}] for an operation [\ispecialfonts{\sfi h}] in [{\cal S}_X].

Assuming first that we are dealing with a space group [{\cal G}] described by a primitive lattice, each coset of [{\cal G}] relative to the translation subgroup [{\cal T}] contains precisely one operation [\ispecialfonts\sfi g] such that [\ispecialfonts{\sfi g}(X)] lies in the primitive unit cell. Since the number of cosets equals the order [|{\cal P}|] of the point group [{\cal P}] of [{\cal G}] and since each orbit point is obtained [|{\cal S}_X|] times, it follows that the number of orbit points in the unit cell is [|{\cal P}| / |{\cal S}_X|].

If we deal with a space group with a centred unit cell, the above result has to be modified slightly. If there are k − 1 centring vectors, the lattice spanned by the conventional basis is a sublattice of index k in the full translation lattice. The conventional cell therefore is built up from k primitive unit cells (spanned by a primitive lattice basis) and thus in particular contains k times as many points as the primitive cell (see Chapter 1.3[link] for a detailed discussion of conventional and primitive bases and cells).

Proposition

Let [{\cal G}] be a space group with point group [{\cal P}] and let [{\cal S}_X] be the site-symmetry group of a point X in [{\bb E}^3]. Then the number of orbit points of the orbit of X which lie in a conventional cell for [{\cal G}] is equal to the product [k\times |{\cal P}| / |{\cal S}_X|], where k is the volume of the conventional cell divided by the volume of a primitive unit cell.








































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