A general introduction to space groups

Souvignier, B.

doi:10.1107/97809553602060000921

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

International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 22-41
https://doi.org/10.1107/97809553602060000921

Chapter 1.3. A general introduction to space groups

B. Souvignier^a ^*

^a Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: [email protected]

This chapter provides an introduction to the structure and classification of crystallographic space groups. Viewing space groups as groups of affine mappings which leave a crystal pattern invariant as a whole suggests a natural decomposition of a space group into its translation subgroup and its point group, since an affine mapping is composed from a linear part and a translation part. Starting with the translation part, one observes that the vectors by which the translations in the space group move the crystal pattern form a lattice, and we discuss fundamental concepts concerned with lattices, such as the metric tensor, the unit cell and the distinction into primitive and centred lattices. We then proceed to the point groups, which are formed by the linear parts of the affine mappings contained in a space group. A crucial point is that the point group acts on the translation lattice and the interplay between point groups and lattices is discussed in detail. In particular, the distinction between symmorphic and non-symmorphic groups is explained. The final part of this chapter deals with various schemes in which crystallographic space groups are classified. The most important of these are the classification into space-group types, geometric crystal classes and Bravais types of lattices.

Keywords: space groups; space-group types; affine mappings; point groups; lattices; reciprocal lattice; unit cells; classification of space groups; Bravais types of lattices; coset decomposition; geometric crystal classes; arithmetic crystal classes; crystal families; crystal systems; lattice systems.

1.3.1. Introduction

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We recall from Chapter 1.2 that an isometry is a mapping of the point space $[{\bb E}^n]$ which preserves distances and angles. From the mathematical viewpoint, $[{\bb E}^n]$ is an affine space in which two points differ by a unique vector in the underlying vector space $[{\bb V}^n]$ . The crucial difference between these two types of spaces is that in an affine space no point is distinguished, whereas in a vector space the zero vector plays a special role, namely as the identity element for the addition of vectors. After choosing an origin Mathematical symbol , the points of the affine space $[{\bb E}^n]$ are in one-to-one correspondence with the vectors of $[{\bb V}^n]$ by identifying a point with the difference vector $[\overrightarrow{OP}]$ .

A crystallographic space-group operation is an isometry that maps a crystal pattern onto itself. Since isometries are invertible and the composition of two isometries leaves a crystal pattern invariant as a whole if the two single isometries do so, the space-group operations form a group $[{\cal G} ]$ , called a crystallographic space group.

As a mapping of points in an affine space, a space-group operation is an affine mapping and is thus composed of a linear mapping of the underlying vector space and a translation. Once a coordinate system has been chosen, space-group operations are conveniently represented as matrix–column pairs $[({\bi W}, {\bi w})]$ , where $[{\bi W}]$ is the linear part and $[{\bi w}]$ the translation part and a point with coordinates $[{\bi x}]$ is mapped to $[{\bi W} {\bi x} + {\bi w} ]$ (cf. Section 1.2.2 ).

A translation is a matrix–column pair of the form $[({\bi I}, {\bi w}) ]$ , where $[{\bi I}]$ is the unit matrix and all translations taken together form the translation subgroup $[{\cal T}]$ of $[{\cal G}]$ . The translation subgroup is an infinite group that forms an abelian normal subgroup of $[{\cal G}]$ . The factor group $[{\cal G}/{\cal T} ]$ is a finite group that can be identified with the group of linear parts of $[{\cal G}]$ via the mapping $[({\bi W}, {\bi w}) \,\mapsto\, {\bi W} ]$ , which simply forgets about the translation part. The group $[{\cal P} = \{ {\bi W} \mid ({\bi W}, {\bi w}) \in {\cal G}\} ]$ of linear parts occurring in $[{\cal G}]$ is called the point group $[{\cal P}]$ of $[{\cal G}]$ .

The representation of space-group operations as matrix–column pairs is clearly adapted to the fact that space groups can be built from these two parts, the translation subgroup and the point group. This viewpoint will be discussed in detail in Section 1.3.3 . It allows one to treat space groups in many aspects analogously to finite groups, although, due to the infinite translation subgroup, they are of course infinite groups.

1.3.2. Lattices

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A crystal pattern is defined to be periodic in three linearly independent directions, which means that it is invariant under translations in three linearly independent directions. This periodicity implies that the crystal pattern extends infinitely in all directions. Since the atoms of a crystal form a discrete pattern in which two different points have a certain minimal distance, the translations that fix the crystal pattern as a whole cannot have arbitrarily small lengths. If $[{\bf v}]$ is a vector such that the crystal pattern is invariant under a translation by $[{\bf v}]$ , the periodicity implies that the pattern is invariant under a translation by $[m {\bf v}]$ for every integer m. Furthermore, if a crystal pattern is invariant under translations by $[{\bf v}]$ and $[{\bf w}]$ , it is also invariant by the composition of these two translations, which is the translation by $[{\bf v} + {\bf w}]$ . This shows that the set of vectors by which the translations in a space group move the crystal pattern is closed under taking integral linear combinations. This property is formalized by the mathematical concept of a lattice and the translation subgroups of space groups are best understood by studying their corresponding lattices. These lattices capture the periodic nature of the underlying crystal patterns and reflect their geometric properties.

1.3.2.1. Basic properties of lattices

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The two-dimensional vector space $[{\bb V}^2]$ is the space of columns $[\pmatrix{ x \cr y }]$ with two real components $[x,y \in {\bb R}]$ and the three-dimensional vector space $[{\bb V}^3 ]$ is the space of columns $[\pmatrix{ x \cr y \cr z }]$ with three real components $[x,y,z \in {\bb R}]$ . Analogously, the Mathematical symbol -dimensional vector space $[{\bb V}^n]$ is the space of columns $[{\bf v} = \pmatrix{v_1 \cr \vdots \cr v_n }]$ with n real components.

For the sake of clarity we will restrict our discussions to three-dimensional (and occasionally two-dimensional) space. The generalization to n-dimensional space is straightforward and only requires dealing with columns of n instead of three components and with bases consisting of n instead of three basis vectors.

Definition

For vectors $[{\bf a}, {\bf b}, {\bf c}]$ forming a basis of the three-dimensional vector space $[{\bb V}^3]$ , the set $[ {\bf L}: = \{ l {\bf a} + m {\bf b} + n {\bf c} \mid l,m,n \in {\bb Z} \} ]$ of all integral linear combinations of $[{\bf a}, {\bf b}, {\bf c} ]$ is called a lattice in $[{\bb V}^3]$ and the vectors $[{\bf a}, {\bf b}, {\bf c}]$ are called a lattice basis of $[{\bf L}]$ .

It is inherent in the definition of a crystal pattern that the translation vectors of the translations leaving the pattern invariant are closed under taking integral linear combinations. Since the crystal pattern is assumed to be discrete, it follows that all translation vectors can be written as integral linear combinations of a finite generating set. The fundamental theorem on finitely generated abelian groups (see e.g. Chapter 21 in Armstrong, 1997 ) asserts that in this situation a set of three translation vectors $[{\bf a}, {\bf b}, {\bf c} ]$ can be found such that all translation vectors are integral linear combinations of these three vectors. This shows that the translation vectors of a crystal pattern form a lattice with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ in the sense of the definition above.

By definition, a lattice is determined by a lattice basis. Note, however, that every two- or three-dimensional lattice has infinitely many bases.

Example

The square lattice $[{\bf L} = {\bb Z}^2 = \left\{ \pmatrix{ m \cr n } \mid m, n \in {\bb Z}\right\} ]$ in $[{\bb V}^2]$ has the vectors $[{\bf a} = \pmatrix{ 1 \cr 0 },\quad {\bf b} = \pmatrix{ 0 \cr 1 } ]$ as its standard lattice basis. But $[{\bf a}' = \pmatrix{ 1 \cr -2 },\quad {\bf b}' = \pmatrix{ -2 \cr 3 } ]$ is also a lattice basis of $[{\bf L}]$ : on the one hand $[{\bf a}' ]$ and $[{\bf b}']$ are integral linear combinations of $[{\bf a}, {\bf b} ]$ and are thus contained in $[{\bf L}]$ . On the other hand $[-3 {\bf a}' - 2 {\bf b}' = \pmatrix{ -3 \cr 6 } + \pmatrix{ 4 \cr -6 } = \pmatrix{ 1 \cr 0 } = {\bf a} ]$ and $[-2 {\bf a}' - {\bf b}' = \pmatrix{ -2 \cr 4 } + \pmatrix{ 2 \cr -3 } = \pmatrix{ 0 \cr 1 } = {\bf b}, ]$ hence $[{\bf a}]$ and $[{\bf b}]$ are also integral linear combinations of $[{\bf a}', {\bf b}']$ and thus the two bases $[{\bf a}, {\bf b} ]$ and $[{\bf a}', {\bf b}']$ both span the same lattice (see Fig. 1.3.2.1 ).

Figure 1.3.2.1| top | pdf |

Conventional basis $[{\bf a}, {\bf b}]$ and a non-conventional basis $[{\bf a}', {\bf b}']$ for the square lattice.

The example indicates how the different lattice bases of a lattice $[{\bf L}]$ can be described. Recall that for a vector $[{\bf v}]$ = $[x {\bf a} + y {\bf b} + z {\bf c} ]$ the coefficients Mathematical symbol are called the coordinates and the vector $[\pmatrix{ x \cr y \cr z }]$ is called the coordinate column of $[{\bf v}]$ with respect to the basis $[{\bf a}, {\bf b}, {\bf c}]$ . The coordinate columns of the vectors in $[{\bf L}]$ with respect to a lattice basis are therefore simply columns with three integral components. In particular, if we take a second lattice basis $[{\bf a}', {\bf b}', {\bf c}']$ of $[{\bf L}]$ , then the coordinate columns of $[{\bf a}']$ , $[{\bf b}']$ , $[{\bf c}']$ with respect to the first basis are columns of integers and thus the basis transformation $[{\bi P}]$ such that $[({\bf a}', {\bf b}', {\bf c}') = ({\bf a}, {\bf b}, {\bf c}) {\bi P} ]$ is an integral 3 × 3 matrix. But if we interchange the roles of the two bases, they are related by the inverse transformation $[{\bi P}^{-1} ]$ , i.e. $[({\bf a}, {\bf b}, {\bf c}) = ({\bf a}', {\bf b}', {\bf c}') {\bi P}^{-1} ]$ , and the argument given above asserts that $[{\bi P}^{-1}]$ is also an integral matrix. Now, on the one hand $[\det {\bi P}]$ and $[\det {\bi P}^{-1} ]$ are both integers (being determinants of integral matrices), on the other hand $[\det {\bi P}^{-1} = 1 / \det {\bi P}]$ . This is only possible if $[\det {\bi P} = \pm 1]$ .

Summarizing, the different lattice bases of a lattice $[{\bf L}]$ are obtained by transforming a single lattice basis $[{\bf a}, {\bf b}, {\bf c} ]$ with integral transformation matrices $[{\bi P}]$ such that $[\det {\bi P} = \pm 1]$ .

1.3.2.2. Metric properties

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In the three-dimensional vector space $[{\bb V}^3]$ , the norm or length of a vector $[{\bf v} = \pmatrix{ v_x \cr v_y \cr v_z }]$ is (due to Pythagoras' theorem) given by $[ |{\bf v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}. ]$ From this, the scalar product $[ {\bf v} \cdot {\bf w} = v_x w_x + v_y w_y + v_z w_z \ {\rm for }\ {\bf v} = \pmatrix{ v_x \cr v_y \cr v_z }, {\bf w} = \pmatrix{ w_x \cr w_y \cr w_z } ]$ is derived, which allows one to express angles by $[ \cos \angle ({\bf v}, {\bf w}) = {{{\bf v} \cdot {\bf w}}\over{| {\bf v} | \, | {\bf w} |}}. ]$

The definition of a norm function for the vectors turns $[{\bb V}^3]$ into a Euclidean space. A lattice $[{\bf L}]$ that is contained in $[{\bb V}^3]$ inherits the metric properties of this space. But for the lattice, these properties are most conveniently expressed with respect to a lattice basis. It is customary to choose basis vectors a, b, c which define a right-handed coordinate system, i.e. such that the matrix with columns a, b, c has a positive determinant.

Definition

For a lattice $[{\bf L} \subseteq {\bb V}^3]$ with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ the metric tensor of $[{\bf L}]$ is the 3 × 3 matrix $[ {\bi G} = \pmatrix{ {\bf a} \cdot {\bf a} & {\bf a} \cdot {\bf b} & {\bf a} \cdot {\bf c} \cr {\bf b} \cdot {\bf a} & {\bf b} \cdot {\bf b} & {\bf b} \cdot {\bf c} \cr {\bf c} \cdot {\bf a} & {\bf c} \cdot {\bf b} & {\bf c} \cdot {\bf c} }. ]$ If $[{\bi A}]$ is the 3 × 3 matrix with the vectors $[{\bf a}, {\bf b}, {\bf c} ]$ as its columns, then the metric tensor is obtained as the matrix product $[{\bi G} = {\bi A}^{\rm T}\cdot {\bi A}]$ . It follows immediately that the metric tensor is a symmetric matrix, i.e. $[{\bi G}^{\rm T} = {\bi G} ]$ .

Example

Let $[{\bf a} = \pmatrix{ 1 \cr 1 \cr 1 },\quad {\bf b} = \pmatrix{ 1 \cr 1 \cr 0 },\quad {\bf c} = \pmatrix{ 1 \cr -1 \cr 0 } ]$ be the basis of a lattice $[{\bf L}]$ . Then the metric tensor of $[{\bf L}]$ (with respect to the given basis) is $[{\bi G} = \pmatrix{ 3 & 2 & 0 \cr 2 & 2 & 0 \cr 0 & 0 & 2 }.]$

With the help of the metric tensor the scalar products of arbitrary vectors, given as linear combinations of the lattice basis, can be computed from their coordinate columns as follows: If $[{\bf v} = x_1 {\bf a} + y_1 {\bf b} + z_1 {\bf c} ]$ and $[{\bf w} = x_2 {\bf a} + y_2 {\bf b} + z_2 {\bf c}]$ , then $[ {\bf v} \cdot {\bf w} = (x_1 \, y_1 \, z_1) \cdot {\bi G} \cdot \pmatrix{ x_2 \cr y_2 \cr z_2 }. ]$

From this it follows how the metric tensor transforms under a basis transformation $[{\bi P}]$ . If $[({\bf a}', {\bf b}', {\bf c}') = ({\bf a}, {\bf b}, {\bf c}) {\bi P} ]$ , then the metric tensor $[{\bi G}']$ of $[{\bf L}]$ with respect to the new basis $[{\bf a}', {\bf b}', {\bf c}']$ is given by $[ {\bi G}' = {\bi P}^{\rm T} \cdot {\bi G} \cdot {\bi P}. ]$

An alternative way to specify the geometry of a lattice in $[{\bb V}^3 ]$ is using the cell parameters, which are the lengths of the lattice basis vectors and the angles between them.

Definition

For a lattice $[{\bf L}]$ in $[{\bb V}^3]$ with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ the cell parameters (also called lattice parameters, lattice constants or metric parameters) are given by the lengths $[ a = | {\bf a} | = \sqrt{ {\bf a} \cdot {\bf a} }, \quad b = | {\bf b} | = \sqrt{ {\bf b} \cdot {\bf b} }, \quad c = | {\bf c} | = \sqrt{ {\bf c} \cdot {\bf c} } ]$ of the basis vectors and by the interaxial angles $[ \alpha = \angle ({\bf b}, {\bf c}), \quad \beta = \angle ({\bf c}, {\bf a}), \quad \gamma = \angle ({\bf a}, {\bf b}). ]$

Owing to the relation $[{\bf v} \cdot {\bf w} = | {\bf v} | \, | {\bf w} | \, \cos \angle ({\bf v}, {\bf w}) ]$ for the scalar product of two vectors, one can immediately write down the metric tensor in terms of the cell parameters: $[ {\bi G} = \pmatrix{ a^2 & a b \cos \gamma & a c \cos \beta \cr a b \cos \gamma & b^2 & b c \cos \alpha \cr a c \cos \beta & b c \cos \alpha & c^2 }. ]$

1.3.2.3. Unit cells

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A lattice $[{\bf L}]$ can be used to subdivide $[{\bb V}^3]$ into cells of finite volume which all have the same shape. The idea is to define a suitable subset $[{\bf C}]$ of $[{\bb V}^3]$ such that the translates of $[{\bf C}]$ by the vectors in $[{\bf L}]$ cover $[{\bb V}^3]$ without overlapping. Such a subset $[{\bf C}]$ is called a unit cell of L, or, in the more mathematically inclined literature, a fundamental domain of $[{\bb V}^3]$ with respect to $[{\bf L} ]$ . Two standard constructions for such unit cells are the primitive unit cell and the Voronoï domain (which is also known by many other names).

Definition

Let $[{\bf L}]$ be a lattice in $[{\bb V}^3]$ with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ .

(i) The set $[{\bf C}: = \{ x {\bf a} + y {\bf b} + z {\bf c} \mid 0 \leq x,y,z \,\lt\, 1 \} ]$ is called the primitive unit cell of $[{\bf L}]$ with respect to the basis $[{\bf a}, {\bf b}, {\bf c}]$ . The primitive unit cell is the parallelepiped spanned by the vectors of the given basis.
(ii) The set $[{\bf C}: = \{ {\bf w} \in {\bb V}^3 \mid |{\bf w}| \leq |{\bf w}-{\bf v}| \ {\rm for\ all }\ {\bf v} \in {\bf L} \} ]$ is called the Voronoï domain or Dirichlet domain or Wigner–Seitz cell or Wirkungsbereich or first Brillouin zone (for the case of reciprocal lattices in dual space, see Section 1.3.2.5 ) of $[{\bf L}]$ (around the origin).

The Voronoï domain consists of those points of $[{\bb V}^3]$ that are closer to the origin than to any other lattice point of $[{\bf L}]$ .

See Fig. 1.3.2.2 for examples of these two types of unit cells in two-dimensional space.

Figure 1.3.2.2| top | pdf |

Voronoï domains and primitive unit cells for a rectangular lattice (a) and an oblique lattice (b).

It should be noted that the attribute `primitive' for a unit cell is often omitted. The term `unit cell' then either denotes a primitive unit cell in the sense of the definition above or a slight generalization of this, namely a cell spanned by vectors a, b, c which are not necessarily a lattice basis. This will be discussed in detail in the next section. If a unit cell in the even more general sense of a cell whose translates cover the whole space without overlap (thus including e.g. Voronoï domains) is meant, this should be indicated by the context.

The construction of the Voronoï domain is independent of the basis of $[{\bf L}]$ , as the Voronoï domain is bounded by planes bisecting the line segment between the origin and a lattice point and perpendicular to this segment. In two-dimensional space, the Voronoï domain is simply bounded by lines, in three-dimensional space it is bounded by planes and more generally it is bounded by (n − 1)-dimensional hyperplanes in n-dimensional space.

The boundaries of the Voronoï domain and its translates overlap, thus in order to get a proper fundamental domain, part of the boundary has to be excluded from the Voronoï domain.

The volume V of the unit cell can be expressed both via the metric tensor and via the cell parameters. One has $[ \eqalign{V^2 &= \det {\bi G} \cr&= a^2 b^2 c^2 (1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos \alpha \cos \beta \cos \gamma) }]$ and thus $[ V = a b c \sqrt{ 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos \alpha \cos \beta \cos \gamma }. ]$ Although the cell parameters depend on the chosen lattice basis, the volume of the unit cell is not affected by a transition to a different lattice basis $[{\bf a}', {\bf b}', {\bf c}']$ . As remarked in Section 1.3.2.1 , two lattice bases are related by an integral basis transformation $[{\bi P}]$ of determinant $[\pm 1]$ and therefore $[\det {\bi G}' = \det ({\bi P}^{\rm T} \cdot {\bi G} \cdot {\bi P}) = \det {\bi G} ]$ , i.e. the determinant of the metric tensor is the same for all lattice bases.

Assuming that the vectors $[{\bf a}, {\bf b}, {\bf c}]$ form a right-handed system, the volume can also be obtained via $[ V = {\bf a} \cdot ({\bf b} \times {\bf c}) = {\bf b} \cdot ({\bf c} \times {\bf a}) = {\bf c} \cdot ({\bf a} \times {\bf b}). ]$

1.3.2.4. Primitive and centred lattices

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The definition of a lattice as given in Section 1.3.2.1 states that a lattice consists precisely of the integral linear combinations of the vectors in a lattice basis. However, in crystallographic applications it has turned out to be convenient to work with bases that have particularly nice metric properties. For example, many calculations are simplified if the basis vectors are perpendicular to each other, i.e. if the metric tensor has all non-diagonal entries equal to zero. Moreover, it is preferable that the basis vectors reflect the symmetry properties of the lattice. By a case-by-case analysis of the different types of lattices a set of rules for convenient bases has been identified and bases conforming with these rules are called conventional bases. The conventional bases are chosen such that in all cases the integral linear combinations of the basis vectors are lattice vectors, but it is admitted that not all lattice vectors are obtained as integral linear combinations.

To emphasize that a basis has the property that the vectors of a lattice are precisely the integral linear combinations of the basis vectors, such a basis is called a primitive basis for this lattice.

If the conventional basis of a lattice is not a primitive basis for this lattice, the price to be paid for the transition to the conventional basis is that in addition to the integral linear combinations of the basis vectors one requires one or more centring vectors in order to obtain all lattice vectors. These centring vectors have non-integral (but rational) coordinates with respect to the conventional basis. The name centring vectors reflects the fact that the additional vectors are usually the centres of the unit cell or of faces of the unit cell spanned by the conventional basis.

Definition

Let $[{\bf a}, {\bf b}, {\bf c}]$ be linearly independent vectors in $[{\bb V}^3]$ .

(i) A lattice $[{\bf L}]$ is called a primitive lattice with respect to a basis $[{\bf a}, {\bf b}, {\bf c}]$ if $[{\bf L}]$ consists precisely of all integral linear combinations of $[{\bf a}, {\bf b}, {\bf c} ]$ , i.e. if $[{\bf L}]$ = $[{\bf L}_P]$ = $[\{ l {\bf a} + m {\bf b} + n {\bf c} \mid l,m,n \in {\bb Z}\} ]$ .
(ii) A lattice $[{\bf L}]$ is called a centred lattice with respect to a basis $[{\bf a}, {\bf b}, {\bf c} ]$ if the integral linear combinations $[{\bf L}_P]$ = $[\{ l {\bf a} + m {\bf b} + n {\bf c} \mid l,m,n \in {\bb Z}\}]$ form a proper sublattice of $[{\bf L}]$ such that $[{\bf L}]$ is the union of $[{\bf L}_P]$ with the translates of $[{\bf L}_P]$ by centring vectors $[{\bf v}_1, \ldots, {\bf v}_s]$ , i.e. $[{\bf L} = {\bf L}_P \cup ({\bf v}_1 + {\bf L}_P)\ \cup]$ $[ \ldots \cup ({\bf v}_s + {\bf L}_P) ]$ .

Typically, the basis $[{\bf a}, {\bf b}, {\bf c}]$ is a conventional basis and in this case one often briefly says that a lattice $[{\bf L}]$ is a primitive lattice or a centred lattice without explicitly mentioning the conventional basis.

Example

A rectangular lattice has as conventional basis a vector $[{\bf a}]$ of minimal length and a vector $[{\bf b}]$ of minimal length amongst the vectors perpendicular to $[{\bf a}]$ . The resulting primitive lattice $[{\bf L}_P]$ is indicated by the filled nodes in Fig. 1.3.2.3 . Now consider the lattice $[{\bf L}]$ having both the filled and the open nodes in Fig. 1.3.2.3 as its lattice nodes. One sees that $[{\bf a}' = \textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf b}]$ , $[{\bf b}' = -\textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf b}]$ is a primitive basis for $[{\bf L}]$ , but it is more convenient to regard $[{\bf L} ]$ as a centred lattice with respect to the basis $[{\bf a}, {\bf b} ]$ with centring vector $[{\bf v} = \textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf b} ]$ . The filled nodes then show the sublattice $[{\bf L}_P]$ of $[{\bf L}]$ , the open nodes are the translate $[{\bf v} + {\bf L}_P]$ and $[{\bf L}]$ is the union $[{\bf L}_P \cup ({\bf v} + {\bf L}_P) ]$ .

Figure 1.3.2.3| top | pdf |

Primitive rectangular lattice (only the filled nodes) and centred rectangular lattice (filled and open nodes).

Recalling that a lattice is in particular a group (with addition of vectors as operation), the sublattice $[{\bf L}_P]$ spanned by the basis of a centred lattice is a subgroup of the centred lattice $[{\bf L}]$ . Together with the zero vector $[{\bf v}_0 = {\bf 0}]$ , the centring vectors form a set $[{\bf v}_0, {\bf v}_1, \ldots, {\bf v}_s]$ of coset representatives of $[{\bf L}]$ relative to $[{\bf L}_P]$ and the index [i] of $[{\bf L}_P]$ in $[{\bf L}]$ is s + 1. In particular, the sum of two centring vectors is, up to a vector in $[{\bf L}_P]$ , again a centring vector, i.e. for centring vectors $[{\bf v}_i]$ , $[{\bf v}_j]$ there is a unique centring vector $[{\bf v}_k]$ (possibly 0) such that $[{\bf v}_i + {\bf v}_j = {\bf v}_k + {\bf w} ]$ for a vector $[{\bf w} \in {\bf L}_P]$ .

The concepts of primitive and centred lattices suggest corresponding notions of primitive and centred unit cells. If $[{\bf a}, {\bf b}, {\bf c}]$ is a primitive basis for the lattice $[{\bf L}]$ , then the parallelepiped spanned by $[{\bf a}, {\bf b}, {\bf c}]$ is called a primitive unit cell (or primitive cell ); if $[{\bf a}, {\bf b}, {\bf c}]$ spans a proper sublattice $[{\bf L}_P]$ of index [i] in $[{\bf L}]$ , then the parallelepiped spanned by $[{\bf a}, {\bf b}, {\bf c}]$ is called a centred unit cell (or centred cell ). Since translating a centred cell by translations from the sublattice $[{\bf L}_P]$ covers the full space, the centred cell contains one representative from each coset of the centred lattice $[{\bf L}]$ relative to $[{\bf L}_P]$ . This means that the centred cell contains [i] lattice vectors of the centred lattice and due to this a centred cell is also called a multiple cell. As a consequence, the volume of the centred cell is [i] times as large as that of a primitive cell for $[{\bf L}]$ .

For a conventional basis $[{\bf a}, {\bf b}, {\bf c}]$ of the lattice $[{\bf L}]$ , the parallelepiped spanned by $[{\bf a}, {\bf b}, {\bf c} ]$ is called a conventional unit cell (or conventional cell ) of $[{\bf L}]$ . Depending on whether the conventional basis is a primitive basis or not, i.e. whether the lattice is primitive or centred, the conventional cell is a primitive or a centred cell.

Remark : It is important to note that the cell parameters given in the description of a crystallographic structure almost always refer to a conventional cell. When in the crystallographic literature the term `unit cell' is used without further attributes, in most cases a conventional unit cell (as specified by the cell parameters) is meant, which is a primitive or centred (multiple) cell depending on whether the lattice is primitive or centred.

Example (continued)

In the example of a centred rectangular lattice, the conventional basis $[{\bf a}, {\bf b}]$ spans the centred unit cell indicated by solid lines in Fig. 1.3.2.4 , whereas the primitive basis $[{\bf a}' = \textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf b}]$ , $[{\bf b}' = -\textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf b}]$ spans the primitive unit cell indicated by dashed lines. One observes that the centred cell contains two lattice vectors, $[{\bf o}]$ and $[{\bf a'} ]$ , whereas the primitive cell only contains the zero vector $[{\bf o} ]$ (note that due to the condition $[0 \leq x,y \,\lt\, 1]$ for the points in the unit cell the other vertices $[{\bf a}', {\bf b}', {\bf b} ]$ of the cell are excluded). The volume of the centred cell is clearly twice as large as that of the primitive cell.

Figure 1.3.2.4| top | pdf |

Primitive cell (dashed line) and centred cell (solid lines) for the centred rectangular lattice.

Figures displaying the different primitive and centred unit cells as well as tables describing the metric properties of the different primitive and centred lattices are given in Section 3.1.2 .

Examples

(i) The conventional basis for a primitive cubic lattice (cP) is a basis $[{\bf a}, {\bf b}, {\bf c}]$ of vectors of equal length which are pairwise perpendicular, i.e. with $[|{\bf a}| = |{\bf b}| = |{\bf c}|]$ and $[{\bf a} \cdot {\bf b} = {\bf b} \cdot {\bf c} = {\bf c} \cdot {\bf a} = 0 ]$ . As the name indicates, this basis is a primitive basis.
(ii) A body-centred cubic lattice (cI) has as its conventional basis the conventional basis $[{\bf a}, {\bf b}, {\bf c} ]$ of a primitive cubic lattice, but the lattice also contains the centring vector $[{\bf v} = \textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf b} + \textstyle{{1}\over{2}} {\bf c} ]$ which points to the centre of the conventional cell. If we denote the primitive cubic lattice by $[{\bf L}_P]$ , then the body-centred cubic lattice $[{\bf L}_I]$ is the union of $[{\bf L}_P]$ and the translate $[{\bf v} + {\bf L}_P = \{ {\bf v} + {\bf w} \mid {\bf w} \in {\bf L}_P \}]$ . Since $[{\bf L}_P]$ is a sublattice of index 2 in $[{\bf L}_I ]$ , the ratio of the volumes of the centred and the primitive cell of the body-centred cubic lattice is 2.

A possible primitive basis for $[{\bf L}_I]$ is $[{\bf a}' = {\bf a}]$ , $[{\bf b}' = {\bf b} ]$ , $[{\bf c}' = \textstyle{{1}\over{2}} ({\bf a} + {\bf b} + {\bf c})]$ . With respect to this basis, the metric tensor of $[{\bf L}_I]$ is $[a^2 \cdot \pmatrix{ 1 & 0 & \textstyle{{1}\over{2}} \cr 0 & 1 & \textstyle{{1}\over{2}} \cr \textstyle{{1}\over{2}} & \textstyle{{1}\over{2}} & \textstyle{{3}\over{4}} } ]$ (where $[a = {\bf a} \cdot {\bf a}]$ ). However, it is more common to use a primitive basis with vectors of the same length and equal interaxial angles. Such a basis is $[{\bf a}'' = \textstyle{{1}\over{2}} (-{\bf a} + {\bf b} + {\bf c}) ]$ , $[{\bf b}'' = \textstyle{{1}\over{2}} ({\bf a} - {\bf b} + {\bf c})]$ , $[{\bf c}'' = \textstyle{{1}\over{2}} ({\bf a} + {\bf b} - {\bf c})]$ (cf. Fig. 1.5.1.3 ), and with respect to this basis the metric tensor of $[{\bf L}_I]$ is $[{{a^2}\over{4}} \cdot \pmatrix{ 3 & -1 & -1 \cr -1 & 3 & -1 \cr -1 & -1 & 3 } .]$
(iii) The conventional basis for a face-centred cubic lattice () is again the conventional basis $[{\bf a}, {\bf b}, {\bf c} ]$ of a primitive cubic lattice, but the lattice also contains the three centring vectors $[{\bf v}_1 = \textstyle{{1}\over{2}} {\bf b} + \textstyle{{1}\over{2}} {\bf c} ]$ , $[{\bf v}_2 = \textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf c}]$ , $[{\bf v}_3 = \textstyle{{1}\over{2}} {\bf a} + \textstyle{{1}\over{2}} {\bf b}]$ which point to the centres of faces of the conventional cell.

The face-centred cubic lattice $[{\bf L}_F]$ is the union of the primitive cubic lattice $[{\bf L}_P]$ with its translates $[{\bf v}_i + {\bf L}_P]$ by the three centring vectors. The ratio of the volumes of the centred and the primitive cell of the face-centred cubic lattice is 4. In this case, the centring vectors actually form a primitive basis of $[{\bf L}_F]$ . With respect to the basis $[{\bf a}' = \textstyle{{1}\over{2}} ({\bf b} + {\bf c})]$ , $[{\bf b}' = \textstyle{{1}\over{2}} ({\bf a} + {\bf c}) ]$ , $[{\bf c}' = \textstyle{{1}\over{2}} ({\bf a} + {\bf b})]$ (cf. Fig. 1.5.1.4 ) the metric tensor of $[{\bf L}_F]$ is $[{{a^2}\over{4}} \cdot \pmatrix{ 2 & 1 & 1 \cr 1 & 2 & 1 \cr 1 & 1 & 2 } .]$
(iv) In the conventional basis of a primitive hexagonal lattice, the basis vector c is chosen as a shortest vector along a sixfold axis. The vectors a and b then are shortest vectors along twofold axes in a plane perpendicular to c and such that they enclose an angle of 120°. The corresponding metric tensor has the form $[\let\normalbaselines\relax\openup1pt\pmatrix{ a^2 & -\displaystyle{{a^2}\over{2}} & 0 \cr -\displaystyle{{a^2}\over{2}} & a^2 & 0 \cr 0 & 0 & c^2 } .]$
(v) In the unit cell of the primitive hexagonal lattice $[{\bf L}_P]$ , a point with coordinates $[\textstyle{{2}\over{3}}, \textstyle{{1}\over{3}}, z ]$ is mapped to the points $[-\textstyle{{1}\over{3}}, \textstyle{{1}\over{3}}, z]$ and $[-\textstyle{{1}\over{3}}, -\textstyle{{2}\over{3}}, z]$ under the threefold rotation around the c axis. Both of these points are translates of $[\textstyle{{2}\over{3}}, \textstyle{{1}\over{3}}, z]$ by lattice vectors of $[{\bf L}_P ]$ . This means that a centring vector of the form $[\textstyle{{2}\over{3}} {\bf a} + \textstyle{{1}\over{3}} {\bf b} + z {\bf c} ]$ will result in a lattice which is invariant under the threefold rotation. Choosing $[{\bf v}_1 = \textstyle{{1}\over{3}} (2 {\bf a} + {\bf b} + {\bf c})]$ as centring vector, the lattice generated by $[{\bf L}_P]$ and $[{\bf v}_1 ]$ contains $[{\bf L}_P]$ as a sublattice of index 3 with coset representatives $[{\bf 0}]$ , $[{\bf v}_1]$ and $[2 {\bf v}_1 = \textstyle{{1}\over{3}} (4 {\bf a} + 2 {\bf b} + 2 {\bf c}) ]$ . The coset representative $[2 {\bf v}_1]$ is commonly replaced by $[{\bf v}_2 = \textstyle{{1}\over{3}} ({\bf a} + 2 {\bf b} + 2 {\bf c})]$ and the centred lattice $[{\bf L}_R]$ with centring vectors $[{\bf v}_1]$ and $[{\bf v}_2]$ so obtained is called the rhombohedrally centred lattice (hR). The ratio of the volumes of the centred and the primitive cell of the rhombohedrally centred lattice is 3.

For this lattice, the primitive basis of $[{\bf L}_R]$ consisting of three shortest non-coplanar vectors which are permuted by the threefold rotation is also regarded as a conventional basis. With respect to the above lattice basis of the primitive hexagonal lattice, this basis can be chosen as $[{\bf a}' = \textstyle{{1}\over{3}} (2 {\bf a} + {\bf b} + {\bf c}) ]$ , $[{\bf b}' = \textstyle{{1}\over{3}} (-{\bf a} + {\bf b} + {\bf c})]$ , $[{\bf c}' = \textstyle{{1}\over{3}} (-{\bf a} - 2 {\bf b} + {\bf c})]$ . The metric tensor with respect to this basis is $[\let\normalbaselines\relax\openup1pt{{1}\over{9}} \cdot \displaystyle\pmatrix{ 3 a^2 + c^2 & -\displaystyle{{3}\over{2}} a^2 + c^2 & -\displaystyle{{3}\over{2}} a^2 + c^2 \cr -\displaystyle{{3}\over{2}} a^2 + c^2 & 3 a^2 + c^2 & -\displaystyle{{3}\over{2}} a^2 + c^2 \cr -\displaystyle{{3}\over{2}} a^2 + c^2 & -\displaystyle{{3}\over{2}} a^2 + c^2 & 3 a^2 + c^2 }. ]$

Details about the transformations between hexagonal and rhombohedral lattices are given in Section 1.5.3.1 and Table 1.5.1.1 (see also Fig. 1.5.1.6 ).

Remark : In three-dimensional space $[{\bb V}^3]$ , the conventional bases have been chosen in such a way that any isometry of a centred lattice maps the sublattice generated by the conventional basis to itself. This means that the matrices of the isometries of the lattice are not only integral with respect to a primitive basis, but also when written with respect to the conventional basis. The advantage of the conventional basis is that the matrices are much simpler.

In dimensions $[n \geq 4]$ , such a choice of a conventional basis is in general no longer possible. For example, one will certainly regard the standard orthonormal basis $[{\bf a} = \pmatrix{ 1 \cr 0 \cr 0 \cr 0 }\quad {\bf b} = \pmatrix{ 0 \cr 1 \cr 0 \cr 0 }\quad {\bf c} = \pmatrix{ 0 \cr 0 \cr 1 \cr 0 } \quad {\bf d} = \pmatrix{ 0 \cr 0 \cr 0 \cr 1 }]$ of the four-dimensional hypercubic lattice as a conventional basis. The body-centred lattice with centring vector $[\textstyle{{1}\over{2}} ({\bf a} + {\bf b} + {\bf c} + {\bf d}) ]$ is invariant under all the isometries of the hypercubic lattice, but the body-centred lattice itself allows isometries that do not leave the hypercubic lattice invariant. Thus, not all isometries of the body-centred lattice are integral with respect to the conventional basis of the hypercubic lattice.

1.3.2.5. Reciprocal lattice

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For crystallographic applications, a lattice $[{\bf L}^*]$ related to $[{\bf L}]$ is of utmost importance. If the atoms are placed at the nodes of a lattice $[{\bf L}]$ , then the diffraction pattern will have sharp Bragg peaks at the nodes of the reciprocal lattice $[{\bf L}^* ]$ . More generally, if the crystal pattern is invariant under translations from $[{\bf L}]$ , then the locations of the Bragg peaks in the diffraction pattern will be invariant under translations from $[{\bf L}^*]$ .

Definition

Let $[{\bf L} \subset {\bb V}^3]$ be a lattice with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ . Then the reciprocal basis $[{\bf a}^*, {\bf b}^*, {\bf c}^* ]$ is defined by the properties $[{\bf a} \cdot {\bf a}^* = {\bf b} \cdot {\bf b}^* = {\bf c} \cdot {\bf c}^* = 1]$ and $[{\bf b} \cdot {\bf a}^* = {\bf c} \cdot {\bf a}^* = {\bf c} \cdot {\bf b}^* = {\bf a} \cdot {\bf b}^* = {\bf a} \cdot {\bf c}^* = {\bf b} \cdot {\bf c}^* = 0 , ]$ which can conveniently be written as the matrix equation $[ \pmatrix{ {\bf a} \cdot {\bf a}^* & {\bf a} \cdot {\bf b}^* & {\bf a} \cdot {\bf c}^* \cr {\bf b} \cdot {\bf a}^* & {\bf b} \cdot {\bf b}^* & {\bf b} \cdot {\bf c}^* \cr {\bf c} \cdot {\bf a}^* & {\bf c} \cdot {\bf b}^* & {\bf c} \cdot {\bf c}^* } = \pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } = {\bi I}_3. ]$

This means that $[{\bf a}^*]$ is perpendicular to the plane spanned by $[{\bf b}]$ and $[{\bf c}]$ and its projection to the line along $[{\bf a}]$ has length $[1/|{\bf a}|]$ . Analogous properties hold for $[{\bf b}^*]$ and $[{\bf c}^*]$ .

The reciprocal lattice $[{\bf L}^*]$ of $[{\bf L}]$ is defined to be the lattice with lattice basis $[{\bf a}^*, {\bf b}^*, {\bf c}^* ]$ .

In three-dimensional space $[{\bb V}^3]$ , the reciprocal basis can be determined via the vector product. Assuming that $[{\bf a}, {\bf b}, {\bf c} ]$ form a right-handed system that spans a unit cell of volume V, the relation $[{\bf a} \cdot ({\bf b} \times {\bf c}) = V]$ and the defining conditions $[{\bf a} \cdot {\bf a}^* = 1]$ , $[{\bf b} \cdot {\bf a}^* = {\bf c} \cdot {\bf a}^* = 0 ]$ imply that $[{\bf a}^* = \textstyle{{1}\over{V}} ({\bf b} \times {\bf c})]$ . Analogously, one has $[{\bf b}^* = \textstyle{{1}\over{V}} ({\bf c} \times {\bf a}) ]$ and $[{\bf c}^* = \textstyle{{1}\over{V}} ({\bf a} \times {\bf b})]$ .

The reciprocal lattice can also be defined independently of a lattice basis by stating that the vectors of the reciprocal lattice have integral scalar products with all vectors of the lattice: $[ {\bf L}^* = \{ {\bf w}^* \in {\bb V}^3 \mid {\bf v} \cdot {\bf w}^* \in {\bb Z}\ {\rm for\ all }\ {\bf v} \in {\bf L} \}. ]$

Owing to the symmetry $[{\bf v} \cdot {\bf w} = {\bf w} \cdot {\bf v}]$ of the scalar product, the roles of the basis and its reciprocal basis can be interchanged. This means that $[({\bf L}^*)^* = {\bf L}]$ , i.e. taking the reciprocal lattice $[({\bf L}^*)^*]$ of the reciprocal lattice $[{\bf L}^*]$ results in the original lattice $[{\bf L}]$ again.

Remark : In parts of the literature, especially in physics, the reciprocal lattice is defined slightly differently. The condition there is that $[{\bf a}_i \cdot {\bf a}^*_j = 2 \pi]$ if Mathematical symbol and 0 otherwise and thus the reciprocal lattice is scaled by the factor 2π as compared to the above definition. By this variation the exponential function $[\exp(-2\pi i \, {\bf v} \cdot {\bf w})]$ is changed to $[\exp(-i \, {\bf v} \cdot {\bf w})]$ , which simplifies the formulas for the Fourier transform.

Example

Let $[{\bf a}, {\bf b}, {\bf c}]$ be the lattice basis of a primitive cubic lattice. Then the body-centred cubic lattice $[{\bf L}_I]$ with centring vector $[\textstyle{{1}\over{2}} ({\bf a} + {\bf b} + {\bf c})]$ is the reciprocal lattice of the rescaled face-centred cubic lattice $[2 {\bf L}_F ]$ , i.e. the lattice spanned by $[2 {\bf a}, 2 {\bf b}, 2 {\bf c}]$ and the centring vectors $[{\bf b} + {\bf c}]$ , $[{\bf a} + {\bf c} ]$ , $[{\bf a} + {\bf b}]$ .

This example illustrates that a lattice and its reciprocal lattice need not have the same type. The reciprocal lattice of a body-centred cubic lattice is a face-centred cubic lattice and vice versa. However, the conventional bases are chosen such that for a primitive lattice with a conventional basis as lattice basis, the reciprocal lattice is a primitive lattice of the same type. Therefore the reciprocal lattice of a centred lattice is always a centred lattice for the same type of primitive lattice.

The reciprocal basis can be read off the inverse matrix of the metric tensor $[{\bi G}]$ : We denote by $[{\bi P}^*]$ the matrix containing the coordinate columns of $[{\bf a}^*, {\bf b}^*, {\bf c}^*]$ with respect to the basis $[{\bf a}, {\bf b}, {\bf c}]$ , so that $[{\bf a}^* = P^*_{11} {\bf a} + P^*_{21} {\bf b} + P^*_{31} {\bf c} ]$ etc. Recalling that scalar products can be computed by multiplying the metric tensor $[{\bi G}]$ from the left and right with coordinate columns with respect to the basis $[{\bf a}, {\bf b}, {\bf c}]$ , the conditions $[\pmatrix{ {\bf a} \cdot {\bf a}^* & {\bf a} \cdot {\bf b}^* & {\bf a} \cdot {\bf c}^* \cr {\bf b} \cdot {\bf a}^* & {\bf b} \cdot {\bf b}^* & {\bf b} \cdot {\bf c}^* \cr {\bf c} \cdot {\bf a}^* & {\bf c} \cdot {\bf b}^* & {\bf c} \cdot {\bf c}^* } = {\bi I}_3]$ defining the reciprocal basis result in the matrix equation $[{\bi I}_3 \cdot {\bi G} \cdot {\bi P}^* = {\bi I}_3 ]$ , since the coordinate columns of the basis $[{\bf a}, {\bf b}, {\bf c} ]$ with respect to itself are the rows of the identity matrix $[{\bi I}_3 ]$ , and $[{\bi P}^*]$ was just defined to contain the coordinate columns of $[{\bf a}^*, {\bf b}^*, {\bf c}^*]$ . But $[{\bi G} \cdot {\bi P}^* = {\bi I}_3 ]$ means that $[{\bi P}^* = {\bi G}^{-1}]$ and thus the coordinate columns of $[{\bf a}^*, {\bf b}^*, {\bf c}^*]$ with respect to the basis $[{\bf a}, {\bf b}, {\bf c}]$ are precisely the columns of the inverse matrix $[{\bi G}^{-1}]$ of the metric tensor $[{\bi G}]$ .

From $[{\bi P}^* = {\bi G}^{-1}]$ one also derives that the metric tensor $[{\bi G}^*]$ of the reciprocal basis is $[ {\bi G}^* = {{\bi P}^*}^{\rm T} \cdot {\bi G} \cdot {\bi P}^* = {\bi G}^{-1} \cdot {\bi G} \cdot {\bi G}^{-1} = {\bi G}^{-1}. ]$ This means that the metric tensors of a basis and its reciprocal basis are inverse matrices of each other. As a further consequence, the volume Mathematical symbol of the unit cell spanned by the reciprocal basis is $[V^* = V^{-1} ]$ , i.e. the inverse of the volume of the unit cell spanned by $[{\bf a}, {\bf b}, {\bf c} ]$ .

Of course, the reciprocal basis can also be computed from the vectors $[{\bf a}_i]$ directly. If $[{\bi B}]$ and $[{\bi B}^*]$ are the matrices containing as ith column the vectors $[{\bf a}_i]$ and $[{\bf a}^*_i]$ , respectively, then the relation defining the reciprocal basis reads as $[{\bi B}^{\rm T} \cdot {\bi B}^* = {\bi I}_3]$ , i.e. $[{\bi B}^* = ({\bi B}^{-1})^{\rm T}]$ . Thus, the reciprocal basis vector $[{\bf a}^*_i]$ is the ith column of the transposed matrix of $[{\bi B}^{-1}]$ and thus the ith row of the inverse of the matrix $[{\bi B}]$ containing the $[{\bf a}_i]$ as columns.

The relations between the parameters of the unit cell spanned by the reciprocal basis vectors and those of the unit cell spanned by the original basis can either be obtained from the vector product expressions for $[{\bf a}^*]$ , $[{\bf b}^*]$ , $[{\bf c}^*]$ or by explicitly inverting the metric tensor $[{\bi G}]$ (e.g. using Cramer's rule). The latter approach would also be applicable in n-dimensional space. Either way, one finds $[ \displaylines{a^* = {{bc \sin \alpha}\over{V}}, \quad b^* = {{ca \sin \beta}\over{V}},\quad c^* = {{ab \sin \gamma}\over{V}},\cr \sin \alpha^* = {{V}\over{abc \sin \beta \sin \gamma}},\quad \cos \alpha^* = {{\cos \beta \cos \gamma - \cos \alpha}\over{\sin \beta \sin \gamma}}, \cr \sin \beta^* = {{V}\over{abc \sin \gamma \sin \alpha}},\quad \cos \beta^* = {{\cos \gamma \cos \alpha - \cos \beta}\over{\sin \gamma \sin \alpha}}, \cr \sin \gamma^* = {{V}\over{abc \sin \alpha \sin \beta}}, \quad \cos \gamma^* = {{\cos \alpha \cos \beta - \cos \gamma}\over{\sin \alpha \sin \beta}}. }]$

Examples

(i) The lattice $[{\bf L}]$ spanned by the vectors $[{\bf a} = \pmatrix{ 1 \cr 1 \cr 1 },\quad {\bf b} = \pmatrix{ 1 \cr 1 \cr 0 },\quad {\bf c} = \pmatrix{ 1 \cr -1 \cr 0 } ]$ has metric tensor $[{\bi G} = \pmatrix{ 3 & 2 & 0 \cr 2 & 2 & 0 \cr 0 & 0 & 2 }.]$ The inverse of the metric tensor is $[ {\bi G}^* = {\bi G}^{-1} = {{1}\over{2}} \pmatrix{ 2 & -2 & 0 \cr -2 & 3 & 0 \cr 0 & 0 & 1 }. ]$ Interpreting the columns of $[{\bi G}^{-1}]$ as coordinate vectors with respect to the original basis, one concludes that the reciprocal basis is given by $[ {\bf a}^* = {\bf a} - {\bf b}, \quad {\bf b}^* = \textstyle{{1}\over{2}} (-2 {\bf a} + 3 {\bf b}), \quad {\bf c}^* = \textstyle{{1}\over{2}} {\bf c}. ]$ Inserting the columns for a, b, c, one obtains $[ {\bf a}^* = \pmatrix{ 0 \cr 0 \cr 1 }, \quad {\bf b}^* = {{1}\over{2}} \pmatrix{ 1 \cr 1 \cr -2 }, \quad {\bf c}^* = {{1}\over{2}} \pmatrix{ 1 \cr -1 \cr 0 }. ]$

For the direct computation, the matrix $[{\bi B}]$ with the basis vectors $[{\bf a}, {\bf b}, {\bf c}]$ as columns is $[{\bi B} = \pmatrix{ 1 & 1 & 1 \cr 1 & 1 & -1 \cr 1 & 0 & 0 } ]$ and has as its inverse the matrix $[{\bi B}^{-1} = {{1}\over{2}} \pmatrix{ 0 & 0 & 2 \cr 1 & 1 & -2 \cr 1 & -1 & 0 }.]$ The rows of this matrix are indeed the vectors $[{\bf a}^*]$ , $[{\bf b}^*]$ , $[{\bf c}^*]$ as computed above.
(ii) The body-centred cubic lattice $[{\bf L} ]$ has the vectors $[{\bf a} = {{1}\over{2}} \pmatrix{ -1 \cr 1 \cr 1 }, \quad {\bf b} = {{1}\over{2}} \pmatrix{ 1 \cr -1 \cr 1 },\quad {\bf c} = {{1}\over{2}} \pmatrix{ 1 \cr 1 \cr -1 } ]$ as primitive basis.

The matrix $[{\bi B} = {{1}\over{2}} \pmatrix{-1 & 1 & 1 \cr 1 &-1 & 1 \cr 1 & 1 &-1 } ]$ with the basis vectors $[{\bf a}, {\bf b}, {\bf c}]$ as columns has as its inverse the matrix $[{\bi B}^{-1} = \pmatrix{ 0 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & 0 } .]$ The rows of $[{\bi B}^{-1}]$ are the vectors $[ {\bf a}^* = \pmatrix{ 0 \cr 1 \cr 1 }, \quad {\bf b}^* = \pmatrix{ 1 \cr 0 \cr 1 }, \quad {\bf c}^* = \pmatrix{ 1 \cr 1 \cr 0 }, ]$ showing that the reciprocal lattice of a body-centred cubic lattice is a face-centred cubic lattice.

1.3.3. The structure of space groups

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1.3.3.1. Point groups of space groups

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The multiplication rule for symmetry operations $[ ({\bi W}_2, \, {\bi w}_2) ({\bi W}_1, \, {\bi w}_1) = ({\bi W}_2 {\bi W}_1, \, {\bi W}_2 {\bi w}_1 + {\bi w}_2) ]$ shows that the mapping $[\Pi: ({\bi W}, {\bi w})\, \mapsto\, {\bi W}]$ which assigns a space-group operation to its linear part is actually a group homomorphism, because the first component of the combined operation is simply the product of the linear parts of the two operations. As a consequence, the linear parts of a space group form a group themselves, which is called the point group of $[{\cal G}]$ . The kernel of the homomorphism Π consists precisely of the translations $[({\bi I}, {\bi t}) \in {\cal T} ]$ , and since kernels of homomorphisms are always normal subgroups (cf. Section 1.1.6 ), the translation subgroup $[{\cal T}]$ forms a normal subgroup of $[{\cal G}]$ . According to the homomorphism theorem (see Section 1.1.6 ), the point group is isomorphic to the factor group $[{\cal G} / {\cal T} ]$ .

Definition

The point group $[{\cal P}]$ of a space group $[{\cal G} ]$ is the group of linear parts of operations occurring in $[{\cal G} ]$ . It is isomorphic to the factor group $[{\cal G} / {\cal T}]$ of $[{\cal G}]$ by the translation subgroup $[{\cal T}]$ .

When $[{\cal G}]$ is considered with respect to a coordinate system, the operations of $[{\cal P}]$ are simply 3 × 3 matrices.

The point group plays an important role in the analysis of the macroscopic properties of crystals: it describes the symmetry of the set of face normals and can thus be directly observed. It is usually obtained from the diffraction record of the crystal, where adding the information about the translation subgroup explains the sharpness of the Bragg peaks in the diffraction pattern.

Although we have already deduced that the translation subgroup $[{\cal T} ]$ of a space group $[{\cal G}]$ forms a normal subgroup in $[{\cal G} ]$ because it is the kernel of the homomorphism mapping each operation to its linear part, it is worth investigating this fact by an explicit computation. Let $[\ispecialfonts{\sfi t} = ({\bi I}, \, {\bi t})]$ be a translation in $[{\cal T}]$ and $[\ispecialfonts{\sfi W} = ({\bi W}, \, {\bi w})]$ an arbitrary operation in $[{\cal G}]$ , then one has $[\ispecialfonts \eqalign{ {\sfi W} {\sfi t} {\sfi W}^{-1} &= ({\bi W}, \, {\bi w}) ({\bi I}, \, {\bi t}) ({\bi W}^{-1}, \, -{\bi W}^{-1} {\bi w}) \cr &= ({\bi W}, \, {\bi W} {\bi t} + {\bi w}) ({\bi W}^{-1}, \, -{\bi W}^{-1} {\bi w}) \cr &= ({\bi I}, \, -{\bi w} + {\bi W} {\bi t} + {\bi w}) = ({\bi I}, \, {\bi W} {\bi t}), }]$ which is again a translation in $[{\cal G}]$ , namely by $[{\bi W} {\bi t} ]$ . This little computation shows an important property of the translation subgroup with respect to the point group, namely that every vector from the translation lattice is mapped again to a lattice vector by each operation of the point group of $[{\cal G}]$ .

Proposition. Let $[{\cal G}]$ be a space group with point group $[{\cal P}]$ and translation subgroup $[{\cal T}]$ and let $[{\bf L} = \{ {\bi t} \mid ({\bi I}, {\bi t}) \in {\cal T} \} ]$ be the lattice of translations in $[{\cal T}]$ . Then $[{\cal P} ]$ acts on the lattice $[{\bf L}]$ , i.e. for every $[{\bi W} \in {\cal P} ]$ and $[{\bi t} \in {\bf L}]$ one has $[{\bi W} {\bi t} \in {\bf L} ]$ .

A point group that acts on a lattice is a subgroup of the full group of symmetries of the lattice, obtained as the group of orthogonal mappings that map the lattice to itself. With respect to a primitive basis, the group of symmetries of a lattice consists of all integral basis transformations that fix the metric tensor of the lattice.

Definition

Let $[{\bf L}]$ be a three-dimensional lattice with metric tensor $[{\bi G}]$ with respect to a primitive basis $[{\bf a}, {\bf b}, {\bf c} ]$ .

(i) An automorphism of $[{\bf L}]$ is an isometry mapping $[{\bf L}]$ to itself. Written with respect to the basis $[{\bf a}, {\bf b}, {\bf c} ]$ , an automorphism of $[{\bf L}]$ is an integral basis transformation fixing the metric tensor of $[{\bf L}]$ , i.e. it is an integral matrix $[{\bi W} \in {\rm GL}_3({\bb Z})]$ with $[{\bi W}^{\rm T} \cdot {\bi G} \cdot {\bi W} = {\bi G} ]$ .
(ii) The group $[ {\cal B}: = Aut({\bf L}) = \{ {\bi W} \in {\rm GL}_3({\bb Z}) \mid {\bi W}^{\rm T} \cdot {\bi G} \cdot {\bi W} = {\bi G} \} ]$ of all automorphisms of $[{\bf L}]$ is called the automorphism group or Bravais group of $[{\bf L}]$ . Note that $[Aut({\bf L}) ]$ acts on the coordinate columns of $[{\bf L}]$ , which are simply columns with integral coordinates.

Since the isometries in the Bravais group of a lattice preserve distances, the possible images of the vectors in a basis are vectors of the same lengths as the basis vectors. But due to its discreteness, a lattice contains only finitely many lattice vectors up to a given length. This means that a lattice automorphism can only permute the finitely many vectors up to the maximum length of a basis vector. Thus, there can only be finitely many automorphisms of a lattice. This argument proves the following important fact:

Theorem. The Bravais group of a lattice is finite. As a consequence, point groups of space groups are finite groups.

As subgroups of the Bravais group of a lattice, point groups can be realized as integral matrix groups when written with respect to a primitive basis. For a centred lattice, it is possible that the Bravais group of a lattice contains non-integral matrices, because the centring vector is a column with non-integral entries. However, in dimensions two and three the conventional bases are chosen such that the Bravais groups of all lattices are integral when written with respect to a conventional basis.

Information on the Bravais groups of the primitive lattices in two- and three-dimensional space is displayed in Tables 1.3.3.1 and 1.3.3.2 . The columns of the tables contain the names of the lattices, the metric tensor with respect to the conventional basis (with only the upper half given, the lower half following by the symmetry of the metric tensor), the Hermann–Mauguin symbol for the type of the Bravais group and generators of the Bravais group (given in the shorthand notation introduced in Section 1.2.2.1 and the corresponding Seitz symbols discussed in Section 1.4.2.2 ).

Table 1.3.3.1| top | pdf |
Automorphism groups of two-dimensional primitive lattices

Lattice	Metric tensor	Bravais group
Lattice	Metric tensor	Hermann–Mauguin symbol	Generators
Oblique	$[\pmatrix{g_{11} & g_{12}\cr & g_{22}} ]$	2	2: $[\bar x, \bar y]$
Rectangular	$[\pmatrix{g_{11} & 0 \cr & g_{22} } ]$	2mm	2: $[\bar x, \bar y]$
	$[\pmatrix{g_{11} & 0 \cr & g_{22} } ]$		m₁₀: $[\bar x, y]$
Square	$[\pmatrix{g_{11} & 0 \cr & g_{11} } ]$	4mm	4⁺: $[\bar y, x]$
	$[\pmatrix{g_{11} & 0 \cr & g_{11} } ]$		m₁₀: $[\bar x, y]$
Hexagonal	$[\pmatrix{g_{11} & -{{1}\over{2}} g_{11} \cr & g_{11} } ]$	6mm	6⁺:
	$[\pmatrix{g_{11} & -{{1}\over{2}} g_{11} \cr & g_{11} } ]$		m₂₁: $[\bar x, \bar x + y]$

Table 1.3.3.2| top | pdf |
Automorphism groups of three-dimensional primitive lattices

Lattice	Metric tensor	Bravais group
Lattice	Metric tensor	Hermann–Mauguin symbol	Generators
Triclinic	$[\pmatrix{ g_{11} & g_{12} & g_{13} \cr & g_{22} & g_{23} \cr & & g_{33} } ]$	$[\bar{1}]$	$[\bar{1}{:}\ \bar x,\bar y,\bar z]$
Monoclinic	$[\pmatrix{ g_{11} & 0 & g_{13} \cr & g_{22} & 0 \cr & & g_{33} } ]$	2/m	$[2_{010}{:}\ \bar x,y,\bar z]$
			$[m_{010}{:}\ x,\bar y,z]$
Orthorhombic	$[\pmatrix{ g_{11} & 0 & 0 \cr & g_{22} & 0 \cr & & g_{33} } ]$	mmm	$[m_{100}{:} \ \bar x,y,z ]$
			$[m_{010}{:}\ x,\bar y,z]$
			$[m_{001}{:} \ x,y,\bar z]$
Tetragonal	$[\pmatrix{ g_{11} & 0 & 0 \cr & g_{11} & 0 \cr & & g_{33} } ]$	4/mmm	$[4_{001}{:}\ \bar y,x,z]$
			$[m_{001}{:} \ x,y,\bar z]$
			$[m_{100}{:} \ \bar x,y,z]$
Hexagonal	$[\pmatrix{ g_{11} & -{{1}\over{2}} g_{11} & 0 \cr & g_{11} & 0 \cr & & g_{33} } ]$	6/mmm	$[6_{001}{:}\ x-y,x,z]$
			$[m_{001}{:}\ x,y,\bar z]$
			$[m_{100}{:}\ \bar x+y,y,z]$
Rhombohedral	$[\pmatrix{ g_{11} & g_{12} & g_{12} \cr & g_{11} & g_{12} \cr & & g_{11} } ]$	$[\bar{3}m]$	$[\bar{3}_{111}{:}\ \bar z,\bar x,\bar y]$
			$[m_{1\bar{1}0}{:}\ y,x,z ]$
Cubic	$[\pmatrix{ g_{11} & 0 & 0 \cr & g_{11} & 0 \cr & & g_{11} } ]$	$[m\bar{3}m]$	$[m_{001}{:} \ x,y,\bar z]$
			$[\bar{3}_{111}{:}\ \bar z,\bar x,\bar y]$
			$[m_{110}{:} \ \bar y,\bar x,z]$

The finiteness and integrality of the point groups has important consequences. For example, it implies the crystallographic restriction that rotations in space groups of two- and three-dimensional space can only have orders 1, 2, 3, 4 or 6. On the one hand, an integral matrix clearly has an integral trace.¹ But a matrix $[{\bi W}]$ with the property that $[{\bi W}^k = {\bi I} ]$ can be diagonalized over the complex numbers and the diagonal entries have to be kth roots of unity, i.e. powers of $[\zeta_k = \exp({2\pi i/k} )]$ . Since diagonalization does not change the trace, the sum of these kth roots of unity still has to be an integer and in particular these roots of unity have to occur in complex conjugate pairs. In dimension 2 this means that the two diagonal entries are complex conjugate and the only possible ways to obtain an integral trace are $[\zeta_1 + \zeta_1^{-1} = 2]$ , $[\zeta_2 + \zeta_2^{-1} = -2]$ , $[\zeta_3 + \zeta_3^{-1} = -1]$ , $[\zeta_4 + \zeta_4^{-1} = 0]$ and $[\zeta_6 + \zeta_6^{-1} = 1]$ . In dimension 3 the third diagonal entry does not have a complex conjugate partner, and therefore has to be $[\pm 1]$ . Thus the possible orders in dimension 3 are the same as in dimension 2.

A much stronger result was obtained by H. Minkowski (1887 ). He gave an explicit bound for the maximal power Mathematical symbol of a prime p which can divide the order of an n-dimensional finite integral matrix group. In dimension 2 this theorem implies that the orders of the point groups divide 24 and in dimension 3 the orders of the point groups divide 48. The Bravais groups 4mm (of order 8) and 6mm (of order 12) of the square and hexagonal lattices in dimension 2 and the Bravais group $[m\bar{3}m]$ (of order 48) of the cubic lattice in dimension 3 show that Minkowski's result is the best possible in these dimensions.

1.3.3.2. Coset decomposition with respect to the translation subgroup

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The translation subgroup $[{\cal T}]$ of a space group $[{\cal G} ]$ can be used to distribute the operations of $[{\cal G}]$ into different classes by grouping together all operations that differ only by a translation. This results in the decomposition of $[{\cal G}]$ into cosets with respect to $[{\cal T}]$ (see Section 1.1.4 for details of cosets).

Definition

Let $[{\cal G}]$ be a space group with translation subgroup $[{\cal T} ]$ .

(i) The right coset $[\ispecialfonts{\cal T} {\sfi W}]$ of an operation $[\ispecialfonts{\sfi W} \in {\cal G} ]$ with respect to $[{\cal T}]$ is the set $[\ispecialfonts\{ {\sfi t} {\sfi W} \mid {\sfi t} \in {\cal T} \} ]$ .

Analogously, the set $[\ispecialfonts{\sfi W} {\cal T} = \{ {\sfi W} {\sfi t} \mid {\sfi t} \in {\cal T} \} ]$ is called the left coset of $[\ispecialfonts{\sfi W}]$ with respect to $[{\cal T}]$ .
(ii) A set $[\ispecialfonts\{ {\sfi W}_1, \ldots, {\sfi W}_m \} ]$ of operations in $[{\cal G}]$ is called a system of coset representatives relative to $[{\cal T}]$ if every operation $[\ispecialfonts{\sfi W}]$ in $[{\cal G}]$ is contained in exactly one coset $[\ispecialfonts{\cal T} {\sfi W}_i ]$ .
(iii) Writing $[{\cal G}]$ as the disjoint union $[ \ispecialfonts{\cal G} = {\cal T} {\sfi W}_1 \cup \ldots \cup {\cal T} {\sfi W}_m ]$ is called the coset decomposition of $[{\cal G}]$ relative to $[{\cal T}]$ .

If the translation subgroup $[{\cal T}]$ is a subgroup of index [i] in $[{\cal G}]$ , a set of coset representatives for $[{\cal G} ]$ relative to $[{\cal T}]$ consists of [i] operations $[\ispecialfonts{\sfi W}_1, {\sfi W}_2, \ldots, {\sfi W}_{[i]}]$ , where $[\ispecialfonts{\sfi W}_1 ]$ is assumed to be the identity element $[\ispecialfonts{\sfi e}]$ of $[{\cal G} ]$ . The cosets of $[{\cal G}]$ relative to $[{\cal T}]$ can be imagined as columns of an infinite array with [i] columns, labelled by the coset representatives, as displayed in Table 1.3.3.3 .

Table 1.3.3.3| top | pdf |
Right-coset decomposition of $[{\cal G}]$ relative to $[{\cal T}]$

$[\ispecialfonts{\sfi W}_1 = {\sfi e}]$	$[\ispecialfonts{\sfi W}_2]$	$[ \ispecialfonts{\sfi W}_3]$	$[\ldots]$	$[\ispecialfonts{\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_1]$	$[\ispecialfonts{\sfi t}_1 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_1 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_1 {\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_2]$	$[\ispecialfonts{\sfi t}_2 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_2 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_2 {\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_3]$	$[\ispecialfonts{\sfi t}_3 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_3 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_3 {\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_4]$	$[\ispecialfonts{\sfi t}_4 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_4 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_4 {\sfi W}_{[i]}]$
$[\vdots]$	$[\vdots]$	$[\vdots]$		$[\vdots]$

Remark : We can assume some enumeration $[\ispecialfonts{\sfi t}_1, {\sfi t}_2, {\sfi t}_3, \ldots ]$ of the operations in $[{\cal T}]$ because the translation vectors form a lattice. For example, with respect to a primitive basis, the coordinate vectors of the translations in $[{\cal G}]$ are simply columns $[\pmatrix{ l \cr m \cr n }]$ with integral components Mathematical symbol . A straightforward enumeration of these columns would start with $[\eqalign{&\pmatrix{ 0 \cr 0 \cr 0 }, \, \pmatrix{ 1 \cr 0 \cr 0 }, \, \pmatrix{ 0 \cr 1 \cr 0 }, \, \pmatrix{ 0 \cr 0 \cr 1 }, \, \pmatrix{ \bar{1} \cr 0 \cr 0 }, \, \pmatrix{ 0 \cr \bar{1} \cr 0 }, \, \pmatrix{ 0 \cr 0 \cr \bar{1} }, \cr & \pmatrix{ 1 \cr 1 \cr 0 }, \, \pmatrix{ 1 \cr 0 \cr 1 }, \, \pmatrix{ 0 \cr 1 \cr 1 } \ldots }]$

Writing out the matrix–column pairs, the coset $[{\cal T} ({\bi W}, {\bi w}) ]$ consists of the operations of the form $[({\bi I}, {\bi t}) ({\bi W}, {\bi w}) = ({\bi W}, {\bi w} + {\bi t}) ]$ with $[{\bi t}]$ running over the lattice translations of $[{\cal T}]$ . This means that the operations of a coset with respect to the translation subgroup all have the same linear part, which is also evident from a listing of the cosets as columns of an infinite array, as in the example above.

Proposition

Let $[\ispecialfonts{\sfi W} = ({\bi W}, {\bi w})]$ and $[\ispecialfonts{\sfi W}' = ({\bi W}', {\bi w}') ]$ be two operations of a space group $[{\cal G}]$ with translation subgroup $[{\cal T}]$ .

(1) If $[{\bi W} \neq {\bi W}']$ , then the cosets $[\ispecialfonts{\cal T} {\sfi W} ]$ and $[\ispecialfonts{\cal T} {\sfi W}']$ are disjoint, i.e. their intersection is empty.
(2) If $[{\bi W} = {\bi W}']$ , then the cosets $[\ispecialfonts{\cal T} {\sfi W}]$ and $[\ispecialfonts{\cal T} {\sfi W}']$ are equal, because $[\ispecialfonts{\sfi W} {\sfi W}'^{-1}]$ has linear part $[{\bi I}]$ and is thus an operation contained in $[{\cal T}]$ .

The one-to-one correspondence between the point-group operations and the cosets relative to $[{\cal T}]$ explicitly displays the isomorphism between the point group $[{\cal P}]$ of $[{\cal G}]$ and the factor group $[{\cal G}/{\cal T}]$ . This correspondence is also exploited in the listing of the general-position coordinates. What is given there are the coordinate triplets for coset representatives of $[{\cal G}]$ relative to $[{\cal T} ]$ , which correspond to the first row of the array in Table 1.3.3.3 . As just explained, the other operations in $[{\cal G} ]$ can be obtained from these coset representatives by adding a lattice translation to the translational part.

Furthermore, the correspondence between the point group and the coset decomposition relative to $[{\cal T}]$ makes it easy to find a system of coset representatives $[\ispecialfonts\{ {\sfi W}_1, \ldots, {\sfi W}_m \}]$ of $[{\cal G}]$ relative to $[{\cal T}]$ . What is required is that the linear parts of the $[\ispecialfonts{\sfi W}_i]$ are precisely the operations in the point group of $[{\cal G}]$ . If $[{\bi W}_1, \ldots, {\bi W}_m]$ are the different operations in the point group $[{\cal P}]$ of $[{\cal G}]$ , then a system of coset representatives is obtained by choosing for every linear part $[{\bi W}_i]$ a translation part $[{\bi w}_i]$ such that $[\ispecialfonts{\sfi W}_i = ({\bi W}_i, {\bi w}_i)]$ is an operation in $[{\cal G} ]$ .

It is customary to choose the translation parts $[{\bi w}_i]$ of the coset representatives such that their coordinates lie between 0 and 1, excluding 1. In particular, if the translation part of a coset representative is a lattice vector, it is usually chosen as the zero vector $[{\bi o}]$ .

Note that due to the fact that $[{\cal T}]$ is a normal subgroup of $[{\cal G}]$ , a system of coset representatives for the right cosets is at the same time a system of coset representatives for the left cosets.

1.3.3.3. Symmorphic and non-symmorphic space groups

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If a coset with respect to the translation subgroup contains an operation of the form $[({\bi W}, {\bi w})]$ with $[{\bi w}]$ a vector in the translation lattice, it is clear that the same coset also contains the operation $[({\bi W}, {\bi o})]$ with trivial translation part. On the other hand, if a coset does not contain an operation of the form $[({\bi W}, {\bi o}) ]$ , this may be caused by an inappropriate choice of origin. For example, the operation $[(-{\bi I}, (1/2,1/2,1/2))]$ is turned into the inversion $[(-{\bi I}, (0,0,0))]$ by moving the origin to Mathematical symbol (cf. Section 1.5.1.1 for a detailed treatment of origin-shift transformations).

Depending on the actual space group $[{\cal G}]$ , it may or may not be possible to choose the origin such that every coset with respect to $[{\cal T}]$ contains an operation of the form $[({\bi W}, {\bi o})]$ .

Definition

Let $[{\cal G}]$ be a space group with translation subgroup $[{\cal T} ]$ . If it is possible to choose the coordinate system such that every coset of $[{\cal G}]$ with respect to $[{\cal T}]$ contains an operation $[({\bi W}, {\bi o})]$ with trivial translation part, $[{\cal G}]$ is called a symmorphic space group, otherwise $[{\cal G}]$ is called a non-symmorphic space group.

One sees that the operations with trivial translation part form a subgroup of $[{\cal G}]$ which is isomorphic to a subgroup of the point group $[{\cal P}]$ . This subgroup is the group of operations in $[{\cal G}]$ that fix the origin and is called the site-symmetry group of the origin (site-symmetry groups are discussed in detail in Section 1.4.4 ). It is the distinctive property of symmorphic space groups that they contain a subgroup which is isomorphic to the full point group. This may in fact be seen as an alternative definition for symmorphic space groups.

Proposition. A space group $[{\cal G}]$ with point group $[{\cal P}]$ is symmorphic if and only if it contains a subgroup isomorphic to $[{\cal P}]$ . For a non-symmorphic space group $[{\cal G}]$ , every finite subgroup of $[{\cal G}]$ is isomorphic to a proper subgroup of the point group.

Note that every finite subgroup of a space group is a subgroup of the site-symmetry group for some point, because finite groups cannot contain translations. Therefore, a symmorphic space group is characterized by the fact that it contains a site-symmetry group isomorphic to its point group, whereas in non-symmorphic space groups all site-symmetry groups have orders strictly smaller than the order of the point group.

Symmorphic space groups can easily be constructed by choosing a lattice $[{\bf L}]$ and a point group $[{\cal P}]$ which acts on $[{\bf L} ]$ . Then $[{\cal G} = \{ ({\bi W}, {\bi w}) \mid {\bi W} \in {\cal P}, {\bi w} \in {\bf L} \} ]$ is a space group in which the coset representatives can be chosen as $[({\bi W}, \, {\bi o})]$ .

Non-symmorphic space groups can also be constructed from a lattice $[{\bf L}]$ and a point group $[{\cal P}]$ . What is required is a system of coset representatives with respect to $[{\cal T}]$ and these are obtained by choosing for each operation $[{\bi W} \in {\cal P}]$ a translation part $[{\bi w}]$ . Owing to the translations, it is sufficient to consider vectors $[{\bi w}]$ with components between 0 and 1. However, the translation parts cannot be chosen arbitrarily, because for a point-group operation of order k, the operation $[({\bi W}, {\bi w})^k ]$ has to be a translation $[({\bi I}, {\bi t})]$ with $[{\bi t} \in {\bf L} ]$ . Working this out, this imposes the restriction that $[ ({\bi W}^{k-1} + \ldots + {\bi W} + {\bi I}) {\bi w} \in {\bf L}. ]$ Once translation parts $[{\bi w}]$ are found that fulfil all these restrictions, one finally has to check whether the space group obtained this way is (by accident) symmorphic, but written with respect to an inappropriate origin. A change of origin by $[{\bi p}]$ is realized by conjugating the matrix–column pair $[({\bi W}, {\bi w})]$ by the translation $[({\bi I}, -{\bi p})]$ (cf. Section 1.5.1 on transformations of the coordinate system) which gives $[ ({\bi I}, -{\bi p}) ({\bi W}, {\bi w}) ({\bi I}, {\bi p}) = ({\bi W}, {\bi W} {\bi p} + {\bi w} - {\bi p}) = ({\bi W}, {\bi w} + ({\bi W} - {\bi I}) {\bi p}). ]$ Thus, the space group just constructed is symmorphic if there is a vector $[{\bi p}]$ such that $[({\bi W} - {\bi I}) {\bi p} + {\bi w} \in {\bf L} ]$ for each of the coset representatives $[({\bi W}, {\bi w})]$ .

The above considerations also show how every space group can be assigned to a symmorphic space group in a canonical way, namely by setting the translation parts of coset representatives with respect to $[{\cal T}]$ to $[{\bi o} ]$ . This has the effect that screw rotations are turned into rotations and glide reflections into reflections. The Hermann–Mauguin symbol (see Section 1.4.1 for a detailed discussion of Hermann–Mauguin symbols) of the symmorphic space group to which an arbitrary space group is assigned is simply obtained by replacing any screw rotation symbol N_m by the corresponding rotation symbol N and every glide reflection symbol a, b, c, d, e, n by the symbol m for a reflection. A space group is found to be symmorphic if no such replacement is required, i.e. if the Hermann–Mauguin symbol only contains the symbols 1, 2, 3, 4, 6 for rotations, $[\bar{1}]$ , $[\bar{3}]$ , $[\bar{4}]$ , $[\bar{6}]$ for rotoinversions and m for reflections.

Example

The space groups with Hermann-Mauguin symbols P4mm, P4bm, P4₂cm, P4₂nm, P4cc, P4nc, P4₂mc, P4₂bc are all assigned to the symmorphic space group with Hermann–Mauguin symbol Mathematical symbol .

1.3.4. Classification of space groups

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In this section we will consider various ways in which space groups may be grouped together. For the space groups themselves, the natural notion of equivalence is the classification into space-group types, but the point groups and lattices from which the space groups are built also have their own classification schemes into geometric crystal classes and Bravais types of lattices, respectively.

Some other types of classifications are relevant for certain applications, and these will also be considered. The hierarchy of the different classification levels and the numbers of classes on the different levels in dimension 3 are displayed in Fig. 1.3.4.1 .

Figure 1.3.4.1| top | pdf |

Classification levels for three-dimensional space groups.

1.3.4.1. Space-group types

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The main motivation behind studying space groups is that they allow the classification of crystal structures according to their symmetry properties. Since many properties of a structure can be derived from its group of symmetries alone, this allows the investigation of the properties of many structures simultaneously.

On the other hand, even for the same crystal structure the corresponding space group may look different, depending on the chosen coordinate system (see Chapter 1.5 for a detailed discussion of transformations to different coordinate systems). Because it is natural to regard two realizations of a group of symmetry operations with respect to two different coordinate systems as equivalent, the following notion of equivalence between space groups is natural.

Definition

Two space groups $[{\cal G}]$ and $[{\cal G}']$ are called affinely equivalent if $[{\cal G}']$ can be obtained from $[{\cal G} ]$ by a change of the coordinate system.

In terms of matrix–column pairs this means that there must exist a matrix–column pair $[({\bi P}, {\bi p})]$ such that $[ {\cal G}' = \{ ({\bi P}, {\bi p})^{-1} ({\bi W}, {\bi w}) ({\bi P}, {\bi p}) \mid ({\bi W}, {\bi w}) \in {\cal G} \}. ]$ The collection of space groups that are affinely equivalent with $[{\cal G}]$ forms the affine type of $[{\cal G}]$ .

In dimension 2 there are 17 affine types of plane groups and in dimension 3 there are 219 affine space-group types. Note that in order to avoid misunderstandings we refrain from calling the space-group types affine classes, since the term classes is usually associated with geometric crystal classes (see below).

Grouping together space groups according to their space-group type serves different purposes. On the one hand, it is sometimes convenient to consider the same crystal structure and thus also its space group with respect to different coordinate systems, e.g. when the origin can be chosen in different natural ways or when a phase transition to a higher- or lower-symmetry phase with a different conventional cell is described. On the other hand, different crystal structures may give rise to the same space group once suitable coordinate systems have been chosen for both. We illustrate both of these perspectives by an example.

Examples

(i) The space group $[{\cal G}]$ of type Pban (50) has a subgroup $[{\cal H}]$ of index 2 for which the coset representatives relative to the translation subgroup are the identity $[\ispecialfonts{\sfi e}{:}\ x,y,z]$ , the twofold rotation $[\ispecialfonts{\sfi g}{:}\ {-}x,y,-z]$ , the n glide $[\ispecialfonts{\sfi h}{:}\ x+\textstyle{{1}\over{2}},y+\textstyle{{1}\over{2}},-z ]$ and the b glide $[\ispecialfonts{\sfi k}{:}\ {-}x+{{1}\over{2}},y+{{1}\over{2}},z ]$ . This subgroup is of type Pb2n, which is a non-conventional setting for Pnc2 (30). In the conventional setting, the coset representatives of Pnc2 are given by $[\ispecialfonts{\sfi g}'{:}\ {-}x,-y,z]$ , $[\ispecialfonts{\sfi h}'{:}\ {-}x,y+\textstyle{{1}\over{2}},z+\textstyle{{1}\over{2}}]$ and $[\ispecialfonts{\sfi k}'{:}\ x,-y+\textstyle{{1}\over{2}},z+\textstyle{{1}\over{2}} ]$ , i.e. with the z axis as rotation axis for the twofold rotation. The subgroup $[{\cal H}]$ can be transformed to its conventional setting by the basis transformation $[({\bf a}', {\bf b}', {\bf c}') = ({\bf c}, {\bf a}, {\bf b}) ]$ . Depending on whether the perspective of the full group $[{\cal G} ]$ or the subgroup $[{\cal H}]$ is more important for a crystal structure, the groups $[{\cal G}]$ and $[{\cal H}]$ will be considered either with respect to the basis $[{\bf a}, {\bf b}, {\bf c}]$ (conventional for $[{\cal G}]$ ) or to the basis $[{\bf a}', {\bf b}', {\bf c}']$ (conventional for $[{\cal H}]$ ).
(ii) The elements carbon, silicon and germanium all crystallize in the diamond structure, which has a face-centred cubic unit cell with two atoms shifted by 1/4 along the space diagonal of the conventional cubic cell. The space group is in all cases of type $[Fd\bar{3}m]$ (227), but the cell parameters differ: a_C = 3.5668 Å for carbon, a_Si = 5.4310 Å for silicon and a_Ge = 5.6579 Å for germanium (measured at 298 K). In order to scale the conventional cell of carbon to that of silicon, the coordinate system has to be transformed by the diagonal matrix $[a_{\rm Si} / a_{\rm C} \cdot {\bi I}_3 \approx \pmatrix{ 1.523 & 0 & 0 \cr 0 & 1.523 & 0 \cr 0 & 0 & 1.523 } .]$

By a famous theorem of Bieberbach (see Bieberbach, 1911 , 1912 ), affine equivalence of space groups actually coincides with the notion of abstract group isomorphism as discussed in Section 1.1.6 .

Bieberbach theorem

Two space groups in n-dimensional space are isomorphic if and only if they are conjugate by an affine mapping.

This theorem is by no means obvious. Recall that for point groups the situation is very different, since for example the abstract cyclic group of order 2 is realized in the point groups of space groups of type P2, Pm and $[P\bar{1}]$ , generated by a twofold rotation, reflection and inversion, respectively, which are clearly not equivalent in any geometric sense. The driving force behind the Bieberbach theorem is the special structure of space groups having an infinite normal translation subgroup on which the point group acts.

In crystallography, a notion of equivalence slightly stronger than affine equivalence is usually used. Since crystals occur in physical space and physical space can only be transformed by orientation-preserving mappings, space groups are only regarded as equivalent if they are conjugate by an orientation-preserving coordinate transformation, i.e. by an affine mapping that has a linear part with positive determinant.

Definition

Two space groups $[{\cal G}]$ and $[{\cal G}']$ are said to belong to the same space-group type if $[{\cal G}']$ can be obtained from $[{\cal G}]$ by an orientation-preserving coordinate transformation, i.e. by conjugation with a matrix–column pair $[({\bi P}, {\bi p})]$ with $[\det {\bi P}\,\gt\, 0]$ . In order to distinguish the space-group types explicitly from the affine space-group types (corresponding to the isomorphism classes), they are often called crystallographic space-group types.

The (crystallographic) space-group type collects together the infinitely many space groups that are obtained by expressing a single space group with respect to all possible right-handed coordinate systems for the point space.

Example

We consider the space group $[{\cal G}]$ of type Mathematical symbol (80) which is generated by the right-handed fourfold screw rotation $[\ispecialfonts{\sfi g}{:}\ {-}y, x+1/2, z+1/4]$ (located at ), the centring translation $[\ispecialfonts{\sfi t}{:}\ x+1/2, y+1/2, z+1/2]$ and the integral translations of a primitive tetragonal lattice. Conjugating the group $[{\cal G}]$ to $[{\cal G}' = \ispecialfonts{\sfi m} {\cal G}{\sfi m}^{-1}]$ by the reflection $[\ispecialfonts{\sfi m}]$ in the plane Mathematical symbol turns the right-handed screw rotation $[\ispecialfonts{\sfi g}]$ into the left-handed screw rotation $[\ispecialfonts{\sfi g}'{:}\ {-}y, x+1/2, z-1/4 ]$ , and one might suspect that $[{\cal G}']$ is a space group of the same affine type but of a different crystallographic space-group type as $[{\cal G}]$ . However, this is not the case because conjugating $[{\cal G} ]$ by the translation $[\ispecialfonts{\sfi n} = t(0,1/2,0)]$ conjugates $[\ispecialfonts{\sfi g}]$ to $[\ispecialfonts{\sfi g}'' = {\sfi n} {\sfi g} {\sfi n}^{-1}{:}\ {-}y+1/2,]$ Mathematical symbol . One sees that $[\ispecialfonts{\sfi g}'']$ is the composition of $[\ispecialfonts{\sfi g}' ]$ with the centring translation $[\ispecialfonts{\sfi t}]$ and hence $[\ispecialfonts{\sfi g}'' ]$ belongs to $[{\cal G}']$ . This shows that conjugating $[{\cal G} ]$ by either the reflection $[\ispecialfonts{\sfi m}]$ or the translation $[\ispecialfonts{\sfi n} ]$ both result in the same group $[{\cal G}']$ . This can also be concluded directly from the space-group diagrams in Fig. 1.3.4.2 . Reflecting in the plane z = 0 turns the diagram on the left into the diagram on the right, but the same effect is obtained when the left diagram is shifted by $[\textstyle{{1}\over{2}}]$ along either a or b.

Figure 1.3.4.2| top | pdf |

Space-group diagram of Mathematical symbol (left) and its reflection in the plane z = 0 (right).

The groups $[{\cal G}]$ and $[{\cal G}']$ thus belong to the same crystallographic space-group type because $[{\cal G}]$ is transformed to $[{\cal G}']$ by a shift of the origin by $[\textstyle{{1}\over{2}} {\bf b} ]$ , which is clearly an orientation-preserving coordinate transformation.

Enantiomorphism

The 219 affine space-group types in dimension 3 result in 230 crystallographic space-group types. Since an affine type either forms a single space-group type (in the case where the group obtained by an orientation-reversing coordinate transformation can also be obtained by an orientation-preserving transformation) or splits into two space-group types, this means that there are 11 affine space-group types such that an orientation-reversing coordinate transformation cannot be compensated by an orientation-preserving transformation.

Groups that differ only by their handedness are closely related to each other and share many properties. One addresses this phenomenon by the concept of enantiomorphism.

Example

Let $[{\cal G}]$ be a space group of type Mathematical symbol (76) generated by a fourfold right-handed screw rotation $[(4^+_{001}, (0,0,1/4)) ]$ and the translations of a primitive tetragonal lattice. Then transforming the coordinate system by a reflection in the plane z = 0 results in a space group $[{\cal G}']$ with fourfold left-handed screw rotation $[(4^-_{001}, (0,0,1/4)) = (4^+_{001}, (0,0,-1/4))^{-1}]$ . The groups $[{\cal G} ]$ and $[{\cal G}']$ are isomorphic because they are conjugate by an affine mapping, but $[{\cal G}']$ belongs to a different space-group type, namely Mathematical symbol (78), because $[{\cal G}]$ does not contain a fourfold left-handed screw rotation with translation part $[\textstyle{{1}\over{4}} {\bf c}]$ .

Definition

Two space groups $[{\cal G}]$ and $[{\cal G}']$ are said to form an enantiomorphic pair if they are conjugate under an affine mapping, but not under an orientation-preserving affine mapping.

If $[{\cal G}]$ is the group of isometries of some crystal pattern, then its enantiomorphic counterpart $[{\cal G}']$ is the group of isometries of the mirror image of this crystal pattern.

The splitting of affine space-group types of three-dimensional space groups into pairs of crystallographic space-group types gives rise to the following 11 enantiomorphic pairs of space-group types: Mathematical symbol (76/78), (91/95), (92/96), (144/145), (151/153), (152/154), (169/173), (170/172), (178/179), (180/181), (212/213). These groups are easily recognized by their Hermann–Mauguin symbols, because they are the primitive groups for which the Hermann–Mauguin symbol contains one of the screw rotations Mathematical symbol , , , , , , or . The groups with fourfold screw rotations and body-centred lattices do not give rise to enantiomorphic pairs, because in these groups the orientation reversal can be compensated by an origin shift, as illustrated in the example above for the group of type .

Example

A well known example of a crystal that occurs in forms whose symmetry is described by enantiomorphic pairs of space groups is quartz. For low-temperature α-quartz there exists a left-handed and a right-handed form with space groups Mathematical symbol (152) and (154), respectively. The two individuals of opposite chirality occur together in the so-called Brazil twin of quartz. At higher temperatures, a phase transition leads to the higher-symmetry β-quartz forms, with space groups (181) and (180), which still form an enantiomorphic pair.

1.3.4.2. Geometric crystal classes

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We recall that the point group of a space group is the group of linear parts occurring in the space group. Once a basis for the underlying vector space is chosen, such a point group is a group of 3 × 3 matrices. A point group is characterized by the relative positions between the rotation and rotoinversion axes and the reflection planes of the operations it contains, and in this sense a point group is independent of the chosen basis. However, a suitable choice of basis is useful to highlight the geometric properties of a point group.

Example

A point group of type 3m is generated by a threefold rotation and a reflection in a plane with normal vector perpendicular to the rotation axis. Choosing a basis $[{\bf a}, {\bf b}, {\bf c}]$ such that c is along the rotation axis, a is perpendicular to the reflection plane and b is the image of a under the threefold rotation (i.e. b lies in the plane perpendicular to the rotation axis and makes an angle of 120° with a), the matrices of the threefold rotation and the reflection with respect to this basis are $[\pmatrix{ 0 & -1 & 0 \cr 1 & -1 & 0 \cr 0 & 0 & 1 } \ {\rm and} \ \pmatrix{ -1 & 1 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 }.]$

A different useful basis is obtained by choosing a vector $[{\bf a}' ]$ in the reflection plane but neither along the rotation axis nor perpendicular to it and taking $[{\bf b}']$ and $[{\bf c}']$ to be the images of $[{\bf a}']$ under the threefold rotation and its square. Then the matrices of the threefold rotation and the reflection with respect to the basis $[{\bf a}', {\bf b}', {\bf c}' ]$ are $[\pmatrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 }\ {\rm and}\ \pmatrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 }.]$

Different choices of a basis for a point group in general result in different matrix groups, and it is natural to consider two point groups as equivalent if they are transformed into each other by a basis transformation. This is entirely analogous to the situation of space groups, where space groups that only differ by the choice of coordinate system are regarded as equivalent. This notion of equivalence is applied at both the level of space groups and point groups.

Definition

Two space groups $[{\cal G}]$ and $[{\cal G}']$ with point groups $[{\cal P}]$ and $[{\cal P}']$ , respectively, are said to belong to the same geometric crystal class if $[{\cal P}]$ and $[{\cal P}' ]$ become the same matrix group once suitable bases for the three-dimensional space are chosen.

Equivalently, $[{\cal G}]$ and $[{\cal G}']$ belong to the same geometric crystal class if the point group $[{\cal P}']$ can be obtained from $[{\cal P}]$ by a basis transformation of the underlying vector space $[{\bb V}^3]$ , i.e. if there is an invertible 3 × 3 matrix $[{\bi P}]$ such that $[ {\cal P}' = \{ {\bi P}^{-1} {\bi W} {\bi P} \mid {\bi W} \in {\cal P} \}. ]$

Also, two matrix groups $[{\cal P}]$ and $[{\cal P}']$ are said to belong to the same geometric crystal class if they are conjugate by an invertible 3 × 3 matrix $[{\bi P}]$ .

Historically, the geometric crystal classes in dimension 3 were determined much earlier than the space groups. They were obtained as the symmetry groups for the set of normal vectors of crystal faces which describe the morphological symmetry of crystals.

Note that for the geometric crystal classes in dimension 3 (and in all other odd dimensions) the distinction between orientation-preserving and orientation-reversing transformations is irrelevant, since any conjugation by an arbitrary transformation can already be realized by an orientation-preserving transformation. This is due to the fact that the inversion $[-{\bi I}]$ on the one hand commutes with every matrix W, i.e. $[(-{\bi I}) {\bi W} = {\bi W} (-{\bi I}) ]$ , and on the other hand $[\det(-{\bi I}) = -1]$ . If $[{\bi P}]$ is orientation reversing, one has $[\det {\bi P} \,\lt\, 0]$ and then $[(-{\bi I}) {\bi P} = -{\bi P}]$ is orientation preserving because $[\det(-{\bi P}) = -\det {\bi P}\,\gt\, 0]$ . But $[(-{\bi P})^{-1} {\bi W} (-{\bi P}) = {\bi P}^{-1} {\bi W} {\bi P} ]$ , hence the transformations by $[{\bi P}]$ and $[-{\bi P}]$ give the same result and one of $[{\bi P}]$ and $[-{\bi P}]$ is orientation preserving.

Remark : One often speaks of the geometric crystal classes as the types of point groups. This emphasizes the point of view in which a point group is regarded as the group of linear parts of a space group, written with respect to an arbitrary basis of $[{\bb R}^n]$ (not necessarily a lattice basis).

It is also common to state that there are 32 point groups in three-dimensional space. This is just as imprecise as saying that there are 230 space groups, since there are in fact infinitely many point groups and space groups.

What is meant when we say that two space groups have the same point group is usually that their point groups are of the same type (i.e. lie in the same geometric crystal class) and can thus be made to coincide by a suitable basis transformation.

Example

In the space group P3 the threefold rotation generating the point group is given by the matrix $[{\bi W} = \pmatrix{ 0 & -1 & 0 \cr 1 & -1 & 0 \cr 0 & 0 & 1 },]$ whereas in the space group R3 (in the rhombohedral setting) the threefold rotation is given by the matrix $[{\bi W}' = \pmatrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 }.]$ These two matrices are conjugate by the basis transformation $[{\bi P} = {{1}\over{3}} \pmatrix{ 1 & 0 & -1 \cr 0 & 1 & -1 \cr 1 & 1 & 1 } ,]$ which transforms the basis of the hexagonal setting into that of the rhombohedral setting. This shows that the space groups P3 and R3 belong to the same geometric crystal class.

The example is typical in the sense that different groups in the same geometric crystal class usually describe the same group of linear parts acting on different lattices, e.g. primitive and centred. Writing the action of the linear parts with respect to primitive bases of different lattices gives rise to different matrix groups.

1.3.4.3. Bravais types of lattices and Bravais classes

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In the classification of space groups into geometric crystal classes, only the point-group part is considered and the translation lattice is ignored. It is natural that the converse point of view is also adopted, where space groups are grouped together according to their translation lattices, irrespective of what the point groups are.

We have already seen that a lattice can be characterized by its metric tensor, containing the scalar products of a primitive basis. If a point group $[{\cal P}]$ acts on a lattice $[{\bf L}]$ , it fixes the metric tensor $[{\bi G}]$ of $[{\bf L}]$ , i.e. $[{\bi W}^{\rm T} \cdot {\bi G} \cdot {\bi W} = {\bi G} ]$ for all $[{\bi W}]$ in $[{\cal P}]$ and is thus a subgroup of the Bravais group $[Aut({\bf L})]$ of $[{\bf L}]$ . Also, a matrix group $[{\cal B}]$ is called a Bravais group if it is the Bravais group $[Aut({\bf L})]$ for some lattice $[{\bf L}]$ . The Bravais groups govern the classification of lattices.

Definition

Two lattices $[{\bf L}]$ and $[{\bf L}']$ belong to the same Bravais type of lattices if their Bravais groups $[Aut({\bf L})]$ and $[Aut({\bf L}')]$ are the same matrix group when written with respect to suitable primitive bases of $[{\bf L}]$ and $[{\bf L}']$ .

Note that in order to have the same Bravais group, the metric tensors of the two lattices $[{\bf L}]$ and $[{\bf L}']$ do not have to be the same or scalings of each other.

Example

The mineral rutile (TiO₂) has a space group of type Mathematical symbol (136) with a primitive tetragonal cell with cell parameters a = b = 4.594 Å and c = 2.959 Å. The metric tensor of the translation lattice L is therefore $[{\bi G} = \pmatrix{ 4.594^2 & 0 & 0 \cr 0 & 4.594^2 & 0 \cr 0 & 0 & 2.959^2 } ]$ and the Bravais group of the lattice is generated by the fourfold rotation $[\pmatrix{ 0 & -1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 }]$ around the z axis, the reflection $[\pmatrix{ -1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 }]$ in the plane x = 0 and the reflection $[\pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & -1 } ]$ in the plane z = 0.

The silicate mineral cristobalite also has (at low temperatures) a primitive tetragonal cell with a = b = 4.971 Å and c = 6.928 Å, and the space-group type is Mathematical symbol (92). In this case the metric tensor of the translation lattice $[{\bf L}' ]$ is $[{\bi G}' = \pmatrix{ 4.971^2 & 0 & 0 \cr 0 & 4.971^2 & 0 \cr 0 & 0 & 6.928^2 } ]$ and one checks that the Bravais group of $[{\bf L}']$ is precisely the same as that of L. Therefore, the translation lattices L for rutile and $[{\bf L}' ]$ for cristobalite belong to the same Bravais type of lattices.

The different Bravais types of lattices, their cell parameters and metric tensors are displayed in Tables 3.1.2.1 (dimension 2) and 3.1.2.2 (dimension 3): in dimension 2 there are 5 Bravais types and in dimension 3 there are 14 Bravais types of lattices.

It is crucial for the classification of lattices via their Bravais groups that one works with primitive bases, because a primitive and a body-centred cubic lattice have the same automorphisms when written with respect to the conventional cubic basis, but are clearly different types of lattices.

Example

The silicate mineral zircon (ZrSiO₄) has a body-centred tetragonal cell with cell parameters a = b = 6.607 Å and c = 5.982 Å. The body-centred translation lattice $[{\bf L}']$ is spanned by the primitive tetragonal lattice $[{\bf L}]$ with basis $[{\bf a}, {\bf b}, {\bf c} ]$ with $[\alpha = \beta = \gamma = 90^\circ]$ and the centring vector $[{\bf v} = \textstyle{{1}\over{2}} ({\bf a} + {\bf b} + {\bf c})]$ . A primitive basis of $[{\bf L}']$ is obtained as $[({\bf a}', {\bf b}', {\bf c}') = ({\bf a}, {\bf b}, {\bf c}) {\bi P} ]$ with $[{\bi P} = {{1}\over{2}} \pmatrix{ -1 & 1 & 1 \cr 1 & -1 & 1 \cr 1 & 1 & -1 } ,]$ i.e. $[{\bf a}' = \textstyle{{1}\over{2}} (-{\bf a} + {\bf b} + {\bf c}) = -{\bf a} + {\bf v} ]$ , $[{\bf b}' = \textstyle{{1}\over{2}} ({\bf a} - {\bf b} + {\bf c}) = -{\bf b} + {\bf v} ]$ , $[{\bf c}' = \textstyle{{1}\over{2}} ({\bf a} + {\bf b} - {\bf c}) = -{\bf c} + {\bf v} ]$ and the metric tensor $[{\bi G}']$ of $[{\bf L}']$ with respect to the primitive basis $[{\bf a}', {\bf b}', {\bf c}']$ is $[ \eqalign{{\bi G}' &= {\bi P}^{\rm T} \pmatrix{ 6.607^2 & 0 & 0 \cr 0 & 6.607^2 & 0 \cr 0 & 0 & 5.982^2 } {\bi P} \cr&= \pmatrix{ 5.547^2 & -12.880 & -8.946 \cr -12.880 & 5.547^2 & -8.946 \cr -8.946 & -8.946 & 5.547^2 }. }]$ The Bravais group of the primitive tetragonal lattice $[{\bf L}]$ is generated (as in the previous example) by $[\displaylines{{\bi W}_1 = \pmatrix{ 0 & -1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 },\quad {\bi W}_2 = \pmatrix{ -1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \cr {\rm and} \ {\bi W}_3 = \pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & -1 } ,}]$ and these matrices also generate the Bravais group of the body-centred tetragonal lattice $[{\bf L}']$ , but written with respect to the primitive basis $[{\bf a}', {\bf b}', {\bf c}']$ these matrices are transformed to $[\eqalign{{\bi W}_1' &= {\bi P}^{-1} {\bi W}_1 {\bi P} = \pmatrix{ 0 & 1 & 0 \cr 0 & 1 & -1 \cr -1 & 1 & 0 } ,\cr {\bi W}_2' &= {\bi P}^{-1} {\bi W}_2 {\bi P} = \pmatrix{ 1 & 0 & 0 \cr 1 & 0 & -1 \cr 1 & -1 & 0 }\ {\rm and}\cr {\bi W}_3' &= {\bi P}^{-1} {\bi W}_3 {\bi P} = \pmatrix{ 0 & -1 & 1 \cr -1 & 0 & 1 \cr 0 & 0 & 1 } .}]$

That the primitive and the body-centred tetragonal lattices have different types ultimately follows from the fact that the body-centred lattice $[{\bf L}']$ does not have a primitive basis consisting of vectors $[{\bf a}'', {\bf b}'', {\bf c}'' ]$ which are pairwise perpendicular and such that $[{\bf a}'']$ and $[{\bf b}'']$ have the same length. This would be required to have the matrices $[{\bi W}_1]$ , $[{\bi W}_2]$ and $[{\bi W}_3]$ in the Bravais group of $[{\bf L}']$ .

As we have seen, the metric tensors of lattices belonging to the same Bravais type need not be the same, but if they are written with respect to suitable bases they are found to have the same structure, differing only in the specific values for certain free parameters.

Definition

Let $[{\bf L}]$ be a lattice with metric tensor $[{\bi G}]$ with respect to a primitive basis and let $[{\cal B}]$ = $[Aut({\bf L})]$ = $[ \{ {\bi W} \in {\rm GL}_3({\bb Z}) \mid {\bi W}^{\rm T} \cdot {\bi G} \cdot {\bi W} = {\bi G} \} ]$ be the Bravais group of $[{\bf L}]$ . Then $[\displaylines{ {\bf M}({\cal B}): = \{ {\bi G}' \ {\rm symmetric\ 3 \times 3\ matrix} \mid\hfill\cr\hfill {\bi W}^{\rm T} \cdot {\bi G}' \cdot {\bi W} = {\bi G}' \ {\rm for\ all }\ {\bi W} \in {\cal B} \} }]$ is called the space of metric tensors of $[{\cal B}]$ . The dimension of $[{\bf M}({\cal B})]$ is called the number of free parameters of the lattice $[{\bf L}]$ .

Analogously, for an arbitrary integral matrix group $[{\cal P}]$ , $[\displaylines{\quad{\bf M}({\cal P}): = \{ {\bi G}'\ {\rm symmetric\ 3 \times 3\ matrix} \mid \hfill\cr\hfill{\bi W}^{\rm T} \cdot {\bi G}' \cdot {\bi W} = {\bi G}' \ {\rm for\ all }\ {\bi W} \in {\cal P} \} }]$ is called the space of metric tensors of $[{\cal P}]$ . If $[\dim {\bf M}({\cal P}')]$ = $[\dim {\bf M}({\cal P})]$ for a subgroup $[{\cal P}']$ of $[{\cal P}]$ , the spaces of metric tensors are the same for both groups and one says that $[{\cal P}']$ does not act on a more general lattice than $[{\cal P}]$ does.

It is clear that $[{\bf M}({\cal B})]$ contains in particular the metric tensor $[{\bi G}]$ of the lattice $[{\bf L}]$ of which $[{\cal B}]$ is the Bravais group. Moreover, $[{\cal B}]$ is a subgroup of the Bravais group of every lattice with metric tensor in $[{\bf M}({\cal B}) ]$ .

Example

Let $[{\bf L}]$ be a lattice with metric tensor $[\pmatrix{ 17 & 0 & 0 \cr 0 & 17 & 0 \cr 0 & 0 & 42 } ,]$ then $[{\bf L}]$ is a tetragonal lattice with Bravais group $[{\cal B}]$ of type 4/mmm generated by the fourfold rotation $[{\bi W}_1 = \pmatrix{ 0 & -1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 }]$ and the reflections $[{\bi W}_2 = \pmatrix{ -1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \ {\rm and}\ {\bi W}_3 = \pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & -1 }.]$ The space of metric tensors of $[{\cal B}]$ is $[ {\bf M}({\cal B}) = \left\{ \pmatrix{ g_{11} & 0 & 0 \cr 0 & g_{11} & 0 \cr 0 & 0 & g_{33} } \mid g_{11},g_{33} \in {\bb R} \right\} ]$ and the number of free parameters of $[{\bf L}]$ is 2.

For every lattice $[{\bf L}']$ with metric tensor $[{\bi G}']$ in $[{\bf M}({\cal B})]$ such that $[g_{11} \neq g_{33}]$ , one can check that the Bravais group of $[{\bf L}']$ is equal to $[{\cal B} ]$ , hence these lattices belong to the same Bravais type of lattices as $[{\bf L}]$ . On the other hand, if it happens that $[g_{11} = g_{33}]$ in the metric tensor $[{\bi G}']$ of a lattice $[{\bf L}']$ , then the Bravais group of $[{\bf L}']$ is the full cubic point group of type $[m\bar{3}m]$ and $[{\cal B}]$ is a proper subgroup of the Bravais group of $[{\bf L}']$ . In this case the lattice $[{\bf L}']$ is of a different Bravais type to $[{\bf L}]$ , namely cubic.

The subgroup $[{\cal P}]$ of $[{\cal B}]$ generated only by the fourfold rotation $[{\bi W}_1]$ has the same space of metric tensors as $[{\cal B}]$ , thus this subgroup acts on the same types of lattices as $[{\cal B}]$ (i.e. tetragonal lattices). On the other hand, for the subgroup $[{\cal P}']$ of $[{\cal B}]$ generated by the reflections $[{\bi W}_2 ]$ and $[{\bi W}_3]$ , the space of metric tensors is $[ {\bf M}({\cal P}') = \left\{ \pmatrix{ g_{11} & 0 & 0 \cr 0 & g_{22} & 0 \cr 0 & 0 & g_{33} } \mid g_{11}, g_{22}, g_{33} \in {\bb R} \right\} ]$ and is thus of dimension 3. This shows that the subgroup $[{\cal P}']$ acts on more general lattices than $[{\cal B}]$ , namely on orthorhombic lattices.

Remark : The metric tensor of a lattice basis is a positive definite² matrix. It is clear that not all matrices in $[{\bf M}({\cal B})]$ are positive definite [if $[{\bi G} \in {\bf M}({\cal B})]$ is positive definite, then $[-{\bi G}]$ is certainly not positive definite], but the different geometries of lattices on which $[{\cal B}]$ acts are represented precisely by the positive definite metric tensors in $[{\bf M}({\cal B})]$ .

The space of metric tensors obtained from a lattice can be interpreted as an expression of the metric tensor with general entries, i.e. as a generic metric tensor describing the different lattices within the same Bravais type. Special choices for the entries may lead to lattices with accidental higher symmetry, which is in fact a common phenomenon in phase transitions caused by changes of temperature or pressure.

One says that the translation lattice $[{\bf L}]$ of a space group $[{\cal G}]$ with point group $[{\cal P}]$ has a specialized metric if the dimension of the space of metric tensors of $[{\cal B} = Aut({\bf L}) ]$ is smaller than the dimension of the space of metric tensors of $[{\cal P}]$ . Viewed from a slightly different angle, a specialized metric occurs if the location of the atoms within the unit cell reduces the symmetry of the translation lattice to that of a different lattice type.

Example

A space group $[{\cal G}]$ of type P2/m (10) with cell parameters a = 4.4, b = 5.5, c = 6.6 Å, $[\alpha = \beta = \gamma = 90^\circ]$ has a specialized metric, because the point group $[{\cal P}]$ of type 2/m is generated by $[{\bi W} = \pmatrix{ -1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & -1 }]$ and $[{\bi -I}]$ , and has $[{\bf M}({\cal P}) = \left\{ \pmatrix{ g_{11} & 0 & g_{13} \cr 0 & g_{22} & 0 \cr g_{13} & 0 & g_{33} } \mid g_{11}, g_{22}, g_{33}, g_{13} \in {\bb R} \right\} ]$ as its space of metric tensors, which is of dimension 4. The lattice $[{\bf L}]$ with the given cell parameters, however, is orthorhombic, since the free parameter $[g_{13}]$ is specialized to $[g_{13} = 0 ]$ . The automorphism group $[Aut({\bf L})]$ is of type mmm and has a space of metric tensors of dimension 3, namely $[\left\{ \pmatrix{ g_{11} & 0 & 0 \cr 0 & g_{22} & 0 \cr 0 & 0 & g_{33} } \mid g_{11}, g_{22}, g_{33} \in {\bb R} \right\} .]$

The higher symmetry of the translation lattice would, for example, be destroyed by an atomic configuration compatible with the lattice and represented by only two atoms in the unit cell located at 0.17, 1/2, 0.42 and 0.83, 1/2, 0.58. The two atoms are related by a twofold rotation around the b axis, which indicates the invariance of the configuration under twofold rotations with axes parallel to b, but in contrast to the lattice L, the atomic configuration is not compatible with rotations around the a or the c axes.

By looking at the spaces of metric tensors, space groups can be classified according to the Bravais types of their translation lattices, without suffering from complications due to specialized metrics.

Definition

Let $[{\bf L}]$ be a lattice with metric tensor $[{\bi G}]$ and Bravais group $[{\cal B} = Aut({\bf L})]$ and let $[{\bf M}({\cal B}) ]$ be the space of metric tensors associated to $[{\bf L}]$ . Then those space groups $[{\cal G}]$ form the Bravais class corresponding to the Bravais type of $[{\bf L}]$ for which $[{\bf M}({\cal P}) = {\bf M}({\cal B})]$ when the point group $[{\cal P}]$ of $[{\cal G}]$ is written with respect to a suitable primitive basis of the translation lattice of $[\cal G]$ . The names for the Bravais classes are the same as those for the corresponding Bravais types of lattices.

The Bravais groups of lattices provide a link between lattices and point groups, the two building blocks of space groups. However, although the Bravais group of a lattice is simply a matrix group, the fact that it is expressed with respect to a primitive basis and fixes the metric tensor of the lattice preserves the necessary information about the lattice. When the Bravais group is regarded as a point group, the information about the lattice is lost, since point groups can be written with respect to an arbitrary basis. In order to distinguish Bravais groups of lattices at the level of point groups and geometric crystal classes, the concept of a holohedry is introduced.

Definition

The geometric crystal class of a point group $[{\cal P}]$ is called a holohedry (or lattice point group, cf. Chapters 3.1 and 3.3 ) if $[{\cal P}]$ is the Bravais group of some lattice $[{\bf L}]$ .

Example

Let $[{\cal P}]$ be the point group of type $[\bar{3}m]$ generated by the threefold rotoinversion $[{\bi W}_1 = \pmatrix{ 0 & 1 & 0 \cr -1 & 1 & 0 \cr 0 & 0 & -1 }]$ around the z axis and the twofold rotation $[{\bi W}_2 = \pmatrix{ 1 & -1 & 0 \cr 0 & -1 & 0 \cr 0 & 0 & -1 },]$ expressed with respect to the conventional basis $[{\bf a}, {\bf b}, {\bf c} ]$ of a hexagonal lattice. The group $[{\cal P}]$ is not the Bravais group of the lattice $[{\bf L}]$ spanned by $[{\bf a}, {\bf b}, {\bf c} ]$ because this lattice also allows a sixfold rotation around the z axis, which is not contained in $[{\cal P}]$ . But $[{\cal P} ]$ also acts on the rhombohedrally centred lattice $[{\bf L}']$ with primitive basis $[{\bf a}' = \textstyle{{1}\over{3}} (2 {\bf a} + {\bf b} + {\bf c}) ]$ , $[{\bf b}' = \textstyle{{1}\over{3}} (-{\bf a} + {\bf b} + {\bf c})]$ , $[{\bf c}' = \textstyle{{1}\over{3}} (-{\bf a} - 2 {\bf b} + {\bf c})]$ . With respect to the basis $[{\bf a}', {\bf b}', {\bf c}']$ the rotoinversion and twofold rotation are transformed to $[{\bi W}_1' = \pmatrix{ 0 & 0 & -1 \cr -1 & 0 & 0 \cr 0 & -1 & 0 } \ {\rm and} \ {\bi W}_2' = \pmatrix{ 0 & -1 & 0 \cr -1 & 0 & 0 \cr 0 & 0 & -1 } ,]$ and these matrices indeed generate the Bravais group of $[{\bf L}' ]$ . The geometric crystal class with symbol $[\bar{3}m]$ is therefore a holohedry.

Note that in dimension 3 the above is actually the only example of a geometric crystal class in which the point groups are Bravais groups for some but not for all the lattices on which they act. In all other cases, each matrix group $[{\cal P}]$ corresponding to a holohedry is actually the Bravais group of the lattice spanned by the basis with respect to which $[{\cal P}]$ is written.

1.3.4.4. Other classifications of space groups

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In this section we summarize a number of other classification schemes which are perhaps of slightly lower significance than those of space-group types, geometric crystal classes and Bravais types of lattices, but also play an important role for certain applications.

1.3.4.4.1. Arithmetic crystal classes

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We have already seen that every space group can be assigned to a symmorphic space group in a natural way by setting the translation parts of coset representatives with respect to the translation subgroup to $[{\bi o}]$ . The groups assigned to a symmorphic space group in this way all have the same translation lattice and the same point group but the different possibilities for the interplay between these two parts are ignored.

If we want to collect together all space groups that correspond to symmorphic space groups of the same type, we arrive at the classification into arithmetic crystal classes. This can also be seen as a classification of the symmorphic space-group types. The distribution of the space groups into arithmetic classes, represented by the corresponding symmorphic space-group types, is given in Table 2.1.3.3 .

The crucial observation for characterizing this classification is that space groups that correspond to the same symmorphic space group all have translation lattices of the same Bravais type. This means that the freedom in the choice of a basis transformation of the underlying vector space is restricted, because a primitive basis has to be mapped again to a primitive basis. Assuming that the point groups are written with respect to primitive bases, this means that the basis transformation is an integral matrix with determinant $[\pm 1]$ .

Definition

Two space groups $[{\cal G}]$ and $[{\cal G}']$ with point groups $[{\cal P}]$ and $[{\cal P}']$ , respectively, both written with respect to primitive bases of their translation lattices, are said to lie in the same arithmetic crystal class if $[{\cal P}']$ can be obtained from $[{\cal P}]$ by an integral basis transformation of determinant $[\pm 1 ]$ , i.e. if there is an integral 3 × 3 matrix $[{\bi P} ]$ with $[\det {\bi P} = \pm 1]$ such that $[ {\cal P}' = \{ {\bi P}^{-1} {\bi W} {\bi P} \mid {\bi W} \in {\cal P} \}. ]$

Also, two integral matrix groups $[{\cal P}]$ and $[{\cal P}']$ are said to belong to the same arithmetic crystal class if they are conjugate by an integral 3 × 3 matrix $[{\bi P}]$ with $[\det {\bi P} = \pm 1 ]$ .

Example

Let $[\displaylines{{\bi M}_1 = \pmatrix{ -1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } ,\quad {\bi M}_2 = \pmatrix{ 1 & 0 & 0 \cr 0 & -1 & 0 \cr 0 & 0 & 1 } \cr {\rm and} \ {\bi M}_3 = \pmatrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 } }]$ be reflections in the planes x = 0, y = 0 and x = y, respectively, and let $[{\cal P}_1 = \langle {\bi M}_1 \rangle ]$ , $[{\cal P}_2 = \langle {\bi M}_2 \rangle]$ and $[{\cal P}_3 = \langle {\bi M}_3 \rangle ]$ be the integral matrix groups generated by these reflections. Then $[{\cal P}_1]$ and $[{\cal P}_2]$ belong to the same arithmetic crystal class because they are transformed into each other by the basis transformation $[{\bi P} = \pmatrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 }]$ interchanging the x and y axes. But $[{\cal P}_3]$ belongs to a different arithmetic crystal class, because $[{\bi M}_3]$ is not conjugate to $[{\bi M}_1]$ by an integral matrix $[{\bi P}]$ of determinant $[\pm 1]$ . The two groups $[{\cal P}_1]$ and $[{\cal P}_3]$ belong, however, to the same geometric crystal class, because $[{\bi M}_1]$ and $[{\bi M}_3]$ are transformed into each other by the basis transformation $[{\bi P} = \pmatrix{ {{1}\over{2}} & -{{1}\over{2}} & 0 \cr {{1}\over{2}} & {{1}\over{2}} & 0 \cr 0 & 0 & 1 } ,]$ which has determinant $[\textstyle{{1}\over{2}}]$ . This basis transformation shows that $[{\bi M}_1]$ and $[{\bi M}_3]$ can be interpreted as the action of the same reflection on a primitive lattice and on a C-centred lattice.

As explained above, the number of arithmetic crystal classes is equal to the number of symmorphic space-group types: in dimension 2 there are 13 such classes, in dimension 3 there are 73 arithmetic crystal classes. The Hermann–Mauguin symbol of the symmorphic space-group type to which a space group $[{\cal G}]$ belongs is obtained from the symbol for the space-group type of $[{\cal G}]$ by replacing any screw-rotation axis symbol N_m by the corresponding rotation axis symbol N and every glide-plane symbol a, b, c, d, e, n by the symbol m for a mirror plane.

It is clear that the classification into arithmetic crystal classes refines both the classifications into geometric crystal classes and into Bravais classes, since in the first case only the point groups and in the second case only the translation lattices are taken into account, whereas for the arithmetic crystal classes the combination of point groups and translation lattices is considered. Note, however, that for the determination of the arithmetic crystal class of a space group $[{\cal G}]$ it is not sufficient to look only at the type of the point group and the Bravais type of the translation lattice. It is crucial to consider the action of the point group on the translation lattice.

Example

Let $[{\cal G}]$ and $[{\cal G}']$ be space groups of types P3m1 (156) and P31m (157), respectively. Since $[{\cal G}]$ and $[{\cal G}']$ are symmorphic space groups of different types, they must belong to different arithmetic classes. The point groups $[{\cal P}]$ and $[{\cal P}']$ of $[{\cal G}]$ and $[{\cal G}' ]$ both belong to the same geometric crystal class with symbol 3m and the translation lattices of both space groups are primitive hexagonal lattices, and thus of the same Bravais type. It is the different action on the translation lattice which causes $[{\cal G}]$ and $[{\cal G}']$ to lie in different arithmetic classes:

In the conventional setting, the point group $[{\cal P}]$ of $[{\cal G}]$ contains the threefold rotation $[{\bi R} = \pmatrix{ 0 & -1 & 0 \cr 1 & -1 & 0 \cr 0 & 0 & 1 } ]$ and the reflections $[\displaylines{{\bi M}_1 = \pmatrix{ 0 & -1 & 0 \cr -1 & 0 & 0 \cr 0 & 0 & 1 } ,\quad {\bi M}_2 = \pmatrix{ -1 & 1 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 }\cr {\rm and}\ {\bi M}_3 = \pmatrix{ 1 & 0 & 0 \cr 1 & -1 & 0 \cr 0 & 0 & 1 } , }]$ whereas the point group $[{\cal P}']$ of $[{\cal G}']$ contains the same rotation $[{\bi R}]$ and the reflections $[\displaylines{ {\bi M}_1' = \pmatrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 } ,\quad {\bi M}_2' = \pmatrix{ 1 & -1 & 0 \cr 0 & -1 & 0 \cr 0 & 0 & 1 }\cr {\rm and}\ {\bi M}_3' = \pmatrix{ -1 & 0 & 0 \cr -1 & 1 & 0 \cr 0 & 0 & 1 } . }]$ Since the threefold rotation is represented by the same matrix in both groups, the lattice basis for both groups can be taken as the conventional basis $[{\bf a}, {\bf b}, {\bf c}]$ of a hexagonal lattice, with a and b of the same length and enclosing an angle of 120° and c perpendicular to the plane spanned by a and b. One now sees that in $[{\cal P}' ]$ the reflection planes of $[{\bi M}_1']$ , $[{\bi M}_2']$ and $[{\bi M}_3']$ contain the vectors $[{\bf a} + {\bf b}]$ , a and b, respectively, whereas in $[{\cal P}]$ these vectors are just perpendicular to the reflection planes. In the so-called hexagonally centred lattice with primitive basis $[{\bf a}' = \textstyle{{1}\over{3}} {\bf a} + \textstyle{{2}\over{3}} {\bf b} ]$ , $[{\bf b}' = -\textstyle{{2}\over{3}} {\bf a} - \textstyle{{1}\over{3}} {\bf b}]$ , $[{\bf c}' = {\bf c}]$ , the vectors $[{\bf a}']$ and $[{\bf b}']$ are perpendicular to the vectors a and b. The group $[{\cal G}']$ can thus be regarded as the action of $[{\cal G} ]$ on the hexagonally centred lattice, showing that $[{\cal G}]$ and $[{\cal G}']$ are actions of the same group on different lattices which therefore belong to different arithmetic crystal classes.

As we have seen, the assignment of a space group to its arithmetic crystal class is equivalent to the assignment to its corresponding symmorphic space group, which in turn can be seen as an assignment to the combination of a point group and a lattice on which this point group acts. This correspondence between arithmetic crystal classes and point group/lattice combinations is reflected in the symbol for an arithmetic crystal class suggested in de Wolff et al. (1985 ), which is the symbol of the symmorphic space group with the letter for the lattice moved to the end, e.g. 4mmP for the arithmetic crystal class containing the symmorphic space groups of type P4mm (99) and the non-symmorphic groups derived from this symmorphic group, i.e. the groups of space-group type P4bm, P4₂cm, P4₂nm, P4cc, P4nc, P4₂mc and P4₂bc (100–106).

Recall that the members of one arithmetic crystal class are space groups with the same translation lattice and the same point group, possibly written with respect to different primitive bases. If the point group happens to be the Bravais group of the translation lattice, this is independent of the chosen primitive basis and thus being a Bravais group is clearly a property of the full arithmetic crystal class.

Definition

The arithmetic crystal class of a space group $[{\cal G}]$ is called a Bravais arithmetic crystal class if the point group of $[{\cal G}]$ is the Bravais group of the translation lattice of $[{\cal G}]$ .

The arithmetic crystal class of an integral matrix group $[{\cal P}]$ is a Bravais arithmetic crystal class if $[{\cal P}]$ is maximal among the integral matrix groups with the same space of metric tensors $[{\bf M}({\cal P})]$ , i.e. if for any integral matrix group $[{\cal P}']$ properly containing $[{\cal P}]$ as a subgroup, the space of metric tensors $[{\bf M}({\cal P}') ]$ is strictly smaller than that of $[{\cal P}]$ . This amounts to saying that $[{\cal P}']$ must act on a lattice with specialized metric.

Note that in the previous edition of IT A the shorter term Bravais class was used as a synonym for Bravais arithmetic crystal class. However, in this edition the term Bravais class is reserved for the classification of space-group types according to their lattices (see Section 1.3.4.3 ).

Since the lattice types are characterized by their Bravais groups, the Bravais arithmetic crystal classes are in one-to-one correspondence with the Bravais types of lattices. The 14 Bravais arithmetic crystal classes (given by the symbol for the arithmetic class, with the number of the associated symmorphic space-group type in brackets) and the corresponding lattice types are: $[\bar{1}]$ P (2), triclinic; 2/mP (10), primitive monoclinic; 2/mC (12), centred monoclinic; mmmP (47), primitive orthorhombic; mmmC (65), single-face-centred orthorhombic; mmmF (69), all-face-centred orthorhombic; mmmI (71), body-centred orthorhombic; 4/mmmP (123), primitive tetragonal; 4/mmmI (139), body-centred tetragonal; $[\bar{3}mR]$ (166), rhombohedral; 6/mmmP (191), hexagonal; $[m\bar{3}mP]$ (221), primitive cubic; $[m\bar{3}mF]$ (225), face-centred cubic; and $[m\bar{3}mI]$ (229), body-centred cubic.

Bravais flocks

In the classification of space groups according to their translation lattices, the point groups play only a secondary role (as groups acting on the lattices). From the perspective of arithmetic crystal classes, this classification can now be reformulated in terms of integral matrix groups. The crucial point is that every arithmetic crystal class can be assigned to a Bravais arithmetic crystal class in a natural way: If $[{\cal P}]$ is a point group, there is a unique Bravais arithmetic crystal class containing a Bravais group $[{\cal B}]$ of minimal order with $[{\cal P} \leq {\cal B}]$ . Conversely, a Bravais group $[{\cal B}]$ acting on a lattice $[{\bf L}]$ is grouped together with its subgroups $[{\cal P}]$ that do not act on a more general lattice, i.e. on a lattice $[{\bf L}']$ with more free parameters than $[{\bf L} ]$ . This observation gives rise to the concept of Bravais flocks, which is mainly applied to matrix groups.

Definition

Two integral matrix groups $[{\cal P}]$ and $[{\cal P}']$ belong to the same Bravais flock if they are both conjugate by an integral basis transformation to subgroups of a common Bravais group, i.e. if there exists a Bravais group $[{\cal B}]$ and integral 3 × 3 matrices $[{\bi P}]$ and $[{\bi P}']$ such that $[{\bi P} {\bi W} {\bi P}^{-1} \in {\cal B} ]$ for all $[{\bi W} \in {\cal P}]$ and $[{\bi P}' {\bi W}' {\bi P}'^{-1} \in {\cal B} ]$ for all $[{\bi W}' \in {\cal P}']$ . Moreover, $[{\cal P}]$ , $[{\cal P}']$ and $[{\cal B}]$ must all have spaces of metric tensors of the same dimension.

Each Bravais flock consists of the union of the arithmetic crystal class of a Bravais group $[{\cal B}]$ and the arithmetic crystal classes of the subgroups of $[{\cal B}]$ that do not act on a more general lattice than $[{\cal B}]$ .

The classification of space groups into Bravais flocks is the same as that according to the Bravais types of lattices and as that into Bravais classes. If the point groups $[{\cal P} ]$ and $[{\cal P}']$ of two space groups $[{\cal G}]$ and $[{\cal G}']$ belong to the same Bravais flock, then the space groups are also said to belong to the same Bravais flock, but this is the case if and only if $[{\cal G}]$ and $[{\cal G}']$ belong to the same Bravais class.

Example

For the body-centred tetragonal lattice the Bravais arithmetic crystal class is the arithmetic crystal class 4/mmmI and the corresponding symmorphic space-group type is I4/mmm (139). The other arithmetic crystal classes in this Bravais flock are (with the number of the corresponding symmorphic space group in brackets): 4I (79), $[\bar{4}I]$ (82), 4/mI (87), 422I (97), 4mmI (107), $[\bar{4}m2I]$ (119) and $[\bar{4}2mI]$ (121).

1.3.4.4.2. Lattice systems

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It is sometimes convenient to group together those Bravais types of lattices for which the Bravais groups belong to the same holohedry.

Definition

Two lattices belong to the same lattice system if their Bravais groups belong to the same geometric crystal class (which is thus a holohedry).

Remark : The lattice systems were called Bravais systems in earlier editions of this volume.

Example

The primitive cubic, face-centred cubic and body-centred cubic lattices all belong to the same lattice system, because their Bravais groups all belong to the holohedry with symbol $[m\bar{3}m]$ .

On the other hand, the hexagonal and the rhombohedral lattices belong to different lattice systems, because their Bravais groups are not even of the same order and lie in different holohedries (with symbols 6/mmm and $[\bar{3}m]$ , respectively).

From the definition it is obvious that lattice systems classify lattices because they consist of full Bravais types of lattices. On the other hand, the example of the geometric crystal class $[\bar{3}m]$ shows that lattice systems do not classify point groups, because depending on the chosen basis a point group in this geometric crystal class belongs to either the hexagonal or the rhombohedral lattice system.

However, since the translation lattices of space groups in the same Bravais class belong to the same Bravais type of lattices, the lattice systems can also be regarded as a classification of space groups in which full Bravais classes are grouped together.

Definition

Two Bravais classes belong to the same lattice system if the corresponding Bravais arithmetic crystal classes belong to the same holohedry.

More precisely, two space groups $[\cal G]$ and $[{\cal G}']$ belong to the same lattice system if the point groups $[{\cal P}]$ and $[{\cal P}']$ are contained in Bravais groups $[{\cal B}]$ and $[{\cal B}']$ , respectively, such that $[{\cal B}]$ and $[{\cal B}']$ belong to the same holohedry and such that $[{\cal P}]$ , $[{\cal P}']$ , $[{\cal B}]$ and $[{\cal B}']$ all have spaces of metric tensors of the same dimension.

Every lattice system contains the lattices of precisely one holohedry and a holohedry determines a unique lattice system, containing the lattices of the Bravais arithmetic crystal classes in the holohedry. Therefore, there is a one-to-one correspondence between holohedries and lattice systems. There are four lattice systems in dimension 2 and seven lattice systems in dimension 3. The lattice systems in three-dimensional space are displayed in Table 1.3.4.1 . Along with the name of each lattice system, the Bravais types of lattices contained in it and the corresponding holohedry are given.

Table 1.3.4.1| top | pdf |
Lattice systems in three-dimensional space

Lattice system	Bravais types of lattices	Holohedry
Triclinic (anorthic)	aP	$[\bar{1}]$
Monoclinic	mP, mS	2/m
Orthorhombic	oP, oS, oF, oI	mmm
Tetragonal	tP, tI	4/mmm
Hexagonal	hP	6/mmm
Rhombohedral	hR	$[\bar{3}m]$
Cubic	cP, cF, cI	$[m\bar{3}m]$

1.3.4.4.3. Crystal systems

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The point groups contained in a geometric crystal class can act on different Bravais types of lattices, which is the reason why lattice systems do not classify point groups. But the action on different types of lattices can be exploited for a classification of point groups by joining those geometric crystal classes that act on the same Bravais types of lattices. For example, the holohedry $[m\bar{3}m]$ acts on primitive, face-centred and body-centred cubic lattices. The other geometric crystal classes that act on these three types of lattices are 23, $[m\bar{3}]$ , 432 and $[\bar{4}3m]$ .

Definition

Two space groups $[{\cal G}]$ and $[{\cal G}']$ with point groups $[{\cal P}]$ and $[{\cal P}']$ , respectively, belong to the same crystal system if the sets of Bravais types of lattices on which $[{\cal P}]$ and $[{\cal P}']$ act coincide. Since point groups in the same geometric crystal class act on the same types of lattices, crystal systems consist of full geometric crystal classes and the point groups $[{\cal P}]$ and $[{\cal P}']$ are also said to belong to the same crystal system.

Remark : In the literature there are many different notions of crystal systems. In International Tables, only the one defined above is used.

In many cases, crystal systems collect together geometric crystal classes for point groups that are in a group–subgroup relation and act on lattices with the same number of free parameters. However, this condition is not sufficient. If a point group $[{\cal P}]$ is a subgroup of another point group $[{\cal P}']$ , it is clear that $[{\cal P}]$ acts on each lattice on which $[{\cal P}']$ acts. But $[{\cal P}]$ may in addition act on different types of lattices on which $[{\cal P}']$ does not act.

Note that it is sufficient to consider the action on lattices with the maximal number of free parameters, since the action on these lattices implies the action on lattices with a smaller number of free parameters (corresponding to metric specializations).

Example

The holohedry of type 4/mmm acts on tetragonal and body-centred tetragonal lattices. The crystal system containing this holohedry thus consists of all the geometric crystal classes in which the point groups act on tetragonal and body-centred tetragonal lattices, but not on lattices with more than two free parameters. This is the case for all geometric crystal classes with point groups containing a fourfold rotation or rotoinversion and that are subgroups of a point group of type 4/mmm. This means that the crystal system containing the holohedry 4/mmm consists of the geometric classes of types 4, $[\bar{4}]$ , 4/m, 422, 4mm, $[\bar{4}2m]$ and 4/mmm.

This example is typical for the situation in three-dimensional space, since in three-dimensional space usually all the arithmetic crystal classes contained in a holohedry are Bravais arithmetic crystal classes. In this case, the geometric crystal classes in the crystal system of the holohedry are simply the classes of those subgroups of a point group in the holohedry that do not act on lattices with a larger number of free parameters.

The only exceptions from this situation are the Bravais arithmetic crystal classes for the hexagonal and rhombohedral lattices.

Example

A point group containing a threefold rotation but no sixfold rotation or rotoinversion acts both on a hexagonal lattice and on a rhombohedral lattice. On the other hand, point groups containing a sixfold rotation only act on a hexagonal but not on a rhombohedral lattice. The geometric crystal classes of point groups containing a threefold rotation or rotoinversion but not a sixfold rotation or rotoinversion form a crystal system which is called the trigonal crystal system. The geometric crystal classes of point groups containing a sixfold rotation or rotoinversion form a different crystal system, which is called the hexagonal crystal system.

The classification of the point-group types into crystal systems is summarized in Table 1.3.4.2 .

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Crystal systems in three-dimensional space

Crystal system	Point-group types
Triclinic	$[\bar{1}]$ , 1
Monoclinic	2/m, m, 2
Orthorhombic	mmm, mm2, 222
Tetragonal	4/mmm, $[\bar{4}2m]$ , 4mm, 422, 4/m, $[\bar{4}]$ , 4
Hexagonal	6/mmm, $[\bar{6}2m]$ , 6mm, 622, 6/m, $[\bar{6}]$ , 6
Trigonal	$[\bar{3}m]$ , 3m, 32, $[\bar{3}]$ , 3
Cubic	$[m\bar{3}m]$ , $[\bar{4}3m]$ , 432, $[m\bar{3}]$ , 23

Remark : Crystal systems can contain at most one holohedry and in dimensions 2 and 3 it is true that every crystal system does contain a holohedry. However, this is not true in higher dimensions. The smallest counter-examples exist in dimension 5, where two (out of 59) crystal systems do not contain any holohedry.

1.3.4.4.4. Crystal families

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The classification into crystal systems has many important applications, but it has the disadvantage that it is not compatible with the classification into lattice systems. Space groups that belong to the hexagonal lattice system are distributed over the trigonal and the hexagonal crystal system. Conversely, space groups in the trigonal crystal system belong to either the rhombohedral or the hexagonal lattice system. It is therefore desirable to define a further classification level in which the classes consist of full crystal systems and of full lattice systems, or, equivalently, of full geometric crystal classes and full Bravais classes. Since crystal systems already contain only geometric crystal classes with spaces of metric tensors of the same dimension, this can be achieved by the following definition.

Definition

For a space group $[{\cal G}]$ with point group $[{\cal P}]$ the crystal family of $[{\cal G}]$ is the union of all geometric crystal classes that contain a space group $[{\cal G}']$ that has the same Bravais type of lattices as $[{\cal G}]$ .

The crystal family of $[{\cal G}]$ thus consists of those geometric crystal classes that contain a point group $[{\cal P}']$ such that $[{\cal P}]$ and $[{\cal P}']$ are contained in a common supergroup $[\cal B]$ (which is a Bravais group) and such that $[{\cal P}]$ , $[{\cal P}']$ and $[\cal B]$ all act on lattices with the same number of free parameters.

In two-dimensional space, the crystal families coincide with the crystal systems and in three-dimensional space only the trigonal and hexagonal crystal system are merged into a single crystal family, whereas all other crystal systems again form a crystal family on their own.

Example

The trigonal and hexagonal crystal systems belong to a single crystal family, called the hexagonal crystal family, because for both crystal systems the number of free parameters of the corresponding lattices is 2 and a point group of type $[\bar{3}m]$ in the trigonal crystal system is a subgroup of a point group of type 6/mmm in the hexagonal crystal system.

A space group in the hexagonal crystal family belongs to either the trigonal or the hexagonal crystal system and to either the rhombohedral or the hexagonal lattice system. A group in the hexagonal crystal system cannot belong to the rhombohedral lattice system, but all other combinations of crystal system and lattice system are possible. The distribution of the space groups in the hexagonal crystal family over these different combinations is displayed in Table 1.3.4.3 .

Table 1.3.4.3| top | pdf |
Distribution of space-group types in the hexagonal crystal family

Crystal system	Geometric crystal class	Lattice system
Crystal system	Geometric crystal class	Hexagonal	Rhombohedral
Hexagonal	6/mmm	P6/mmm, P6/mcc, ,
	$[\bar{6}2m]$	$[P\bar{6}m2]$ , $[P\bar{6}c2]$ , $[P\bar{6}2m]$ , $[P\bar{6}2c]$
	6mm	P6mm, P6cc, ,
	622	P622, , , , ,
	6/m	P6/m,
	$[\bar{6}]$	$[P\bar{6}]$
	6	P6, , , , ,
Trigonal	$[\bar{3}m]$	$[P\bar{3}1m]$ , $[P\bar{3}1c]$ , $[P\bar{3}m1]$ , $[P\bar{3}c1]$	$[R\bar{3}m]$ , $[R\bar{3}c]$
	3m	P3m1, P31m, P3c1, P31c	R3m, R3c
	32	P312, P321, , , ,	R32
	$[\bar{3}]$	$[P\bar{3}]$	$[R\bar{3}]$
	3	P3, ,	R3

Remark : Up to dimension 3 it seems exceptional that a crystal family contains more than one crystal system, since the only instance of this phenomenon is the hexagonal crystal family consisting of the trigonal and the hexagonal crystal systems. However, in higher dimensions it actually becomes rare that a crystal family consists only of a single crystal system.

For the space groups within one crystal family the same coordinate system is usually used, which is called the conventional coordinate system (for this crystal family). However, depending on the application it may be useful to work with a different coordinate system. To avoid confusion, it is recommended to state explicitly when a coordinate system differing from the conventional coordinate system is used.

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International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 22-41
https://doi.org/10.1107/97809553602060000921