Crystallographic restriction theorem

Last updated March 02, 2024

The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.^[1]

Crystals are modeled as discrete lattices, generated by a list of independent finite translations ( Coxeter 1989 ). Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group (alternatively, the point is the only system allowing for infinite rotational symmetry). The strength of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups.

Dimensions 2 and 3

The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and they can be treated together.

Lattice proof

A rotation symmetry in dimension 2 or 3 must move a lattice point to a succession of other lattice points in the same plane, generating a regular polygon of coplanar lattice points. We now confine our attention to the plane in which the symmetry acts ( Scherrer 1946 ), illustrated with lattice vectors in the figure.

Now consider an 8-fold rotation, and the displacement vectors between adjacent points of the polygon. If a displacement exists between any two lattice points, then that same displacement is repeated everywhere in the lattice. So collect all the edge displacements to begin at a single lattice point. The edge vectors become radial vectors, and their 8-fold symmetry implies a regular octagon of lattice points around the collection point. But this is impossible, because the new octagon is about 80% as large as the original. The significance of the shrinking is that it is unlimited. The same construction can be repeated with the new octagon, and again and again until the distance between lattice points is as small as we like; thus no discrete lattice can have 8-fold symmetry. The same argument applies to any k-fold rotation, for k greater than 6.

A shrinking argument also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original.

Thus the theorem is proved.

The existence of quasicrystals and Penrose tilings shows that the assumption of a linear translation is necessary. Penrose tilings may have 5-fold rotational symmetry and a discrete lattice, and any local neighborhood of the tiling is repeated infinitely many times, but there is no linear translation for the tiling as a whole. And without the discrete lattice assumption, the above construction not only fails to reach a contradiction, but produces a (non-discrete) counterexample. Thus 5-fold rotational symmetry cannot be eliminated by an argument missing either of those assumptions. A Penrose tiling of the whole (infinite) plane can only have exact 5-fold rotational symmetry (of the whole tiling) about a single point, however, whereas the 4-fold and 6-fold lattices have infinitely many centres of rotational symmetry.

Trigonometry proof

Consider two lattice points A and B separated by a translation vector r. Consider an angle α such that a rotation of angle α about any lattice point is a symmetry of the lattice. Rotating about point B by α maps point A to a new point A'. Similarly, rotating about point A by α maps B to a point B'. Since both rotations mentioned are symmetry operations, A' and B' must both be lattice points. Due to periodicity of the crystal, the new vector r' which connects them must be equal to an integer multiple of r:

\mathbf {r} '=m\mathbf {r}

with $m$ integer. The four translation vectors, three of length $r=|\mathbf {r} |$ and one, connecting A' and B', of length $r'=|\mathbf {r} '|$ , form a trapezium. Therefore, the length of r' is also given by:

r'=2r\cos \alpha -r.

Combining the two equations gives:

\cos \alpha ={\frac {m+1}{2}}={\frac {M}{2}}

where $M=m+1$ is also an integer. Bearing in mind that $|\cos \alpha |\leq 1$ we have allowed integers $M\in \{-2,-1,0,1,2\}$ . Solving for possible values of $\alpha$ reveals that the only values in the 0° to 180° range are 0°, 60°, 90°, 120°, and 180°. In radians, the only allowed rotations consistent with lattice periodicity are given by 2π/n, where n = 1, 2, 3, 4, 6. This corresponds to 1-, 2-, 3-, 4-, and 6-fold symmetry, respectively, and therefore excludes the possibility of 5-fold or greater than 6-fold symmetry.

Short trigonometry proof

Consider a line of atoms A-O-B, separated by distance a. Rotate the entire row by θ = +2π/n and θ = −2π/n, with point O kept fixed. After the rotation by +2π/n, A is moved to the lattice point C and after the rotation by -2π/n, B is moved to the lattice point D. Due to the assumed periodicity of the lattice, the two lattice points C and D will be also in a line directly below the initial row; moreover C and D will be separated by r = ma, with m an integer. But by trigonometry, the separation between these points is:

2a\cos {\theta }=2a\cos {\frac {2\pi }{n}}

.

Equating the two relations gives:

2\cos {\frac {2\pi }{n}}=m

This is satisfied by only n = 1, 2, 3, 4, 6.

Matrix proof

For an alternative proof, consider matrix properties. The sum of the diagonal elements of a matrix is called the trace of the matrix. In 2D and 3D every rotation is a planar rotation, and the trace is a function of the angle alone. For a 2D rotation, the trace is 2 cos θ; for a 3D rotation, 1 + 2 cos θ.

Examples

Consider a 60° (6-fold) rotation matrix with respect to an orthonormal basis in 2D.

{\begin{bmatrix}{1/2}&-{{\sqrt {3}}/2}\\{{\sqrt {3}}/2}&{1/2}\end{bmatrix}}

The trace is precisely 1, an integer.

Consider a 45° (8-fold) rotation matrix.

{\begin{bmatrix}{1/{\sqrt {2}}}&-{1/{\sqrt {2}}}\\{1/{\sqrt {2}}}&{1/{\sqrt {2}}}\end{bmatrix}}

The trace is 2/√2, not an integer.

Selecting a basis formed from vectors that spans the lattice, neither orthogonality nor unit length is guaranteed, only linear independence. However the trace of the rotation matrix is the same with respect to any basis. The trace is a similarity invariant under linear transformations. In the lattice basis, the rotation operation must map every lattice point into an integer number of lattice vectors, so the entries of the rotation matrix in the lattice basis – and hence the trace – are necessarily integers. Similar as in other proofs, this implies that the only allowed rotational symmetries correspond to 1,2,3,4 or 6-fold invariance. For example, wallpapers and crystals cannot be rotated by 45° and remain invariant, the only possible angles are: 360°, 180°, 120°, 90° or 60°.

Example

Consider a 60° (360°/6) rotation matrix with respect to the oblique lattice basis for a tiling by equilateral triangles.

{\begin{bmatrix}0&-1\\1&1\end{bmatrix}}

The trace is still 1. The determinant (always +1 for a rotation) is also preserved.

The general crystallographic restriction on rotations does not guarantee that a rotation will be compatible with a specific lattice. For example, a 60° rotation will not work with a square lattice; nor will a 90° rotation work with a rectangular lattice.

Higher dimensions

When the dimension of the lattice rises to four or more, rotations need no longer be planar; the 2D proof is inadequate. However, restrictions still apply, though more symmetries are permissible. For example, the hypercubic lattice has an eightfold rotational symmetry, corresponding to an eightfold rotational symmetry of the hypercube. This is of interest, not just for mathematics, but for the physics of quasicrystals under the cut-and-project theory. In this view, a 3D quasicrystal with 8-fold rotation symmetry might be described as the projection of a slab cut from a 4D lattice.

The following 4D rotation matrix is the aforementioned eightfold symmetry of the hypercube (and the cross-polytope):

A={\begin{bmatrix}0&0&0&-1\\1&0&0&0\\0&-1&0&0\\0&0&-1&0\end{bmatrix}}.

Transforming this matrix to the new coordinates given by

B={\begin{bmatrix}-1/2&0&-1/2&{\sqrt {2}}/2\\1/2&{\sqrt {2}}/2&-1/2&0\\-1/2&0&-1/2&-{\sqrt {2}}/2\\-1/2&{\sqrt {2}}/2&1/2&0\end{bmatrix}}

will produce:

BAB^{-1}={\begin{bmatrix}{\sqrt {2}}/2&{\sqrt {2}}/2&0&0\\-{\sqrt {2}}/2&{\sqrt {2}}/2&0&0\\0&0&-{\sqrt {2}}/2&{\sqrt {2}}/2\\0&0&-{\sqrt {2}}/2&-{\sqrt {2}}/2\end{bmatrix}}.

This third matrix then corresponds to a rotation both by 45° (in the first two dimensions) and by 135° (in the last two). Projecting a slab of hypercubes along the first two dimensions of the new coordinates produces an Ammann–Beenker tiling (another such tiling is produced by projecting along the last two dimensions), which therefore also has 8-fold rotational symmetry on average.

The A4 lattice and F4 lattice have order 10 and order 12 rotational symmetries, respectively.

To state the restriction for all dimensions, it is convenient to shift attention away from rotations alone and concentrate on the integer matrices ( Bamberg, Cairns & Kilminster 2003 ). We say that a matrix A has order k when its k-th power (but no lower), A^k, equals the identity. Thus a 6-fold rotation matrix in the equilateral triangle basis is an integer matrix with order 6. Let Ord_N denote the set of integers that can be the order of an N×N integer matrix. For example, Ord₂ = {1, 2, 3, 4, 6}. We wish to state an explicit formula for Ord_N.

Define a function ψ based on Euler's totient function φ; it will map positive integers to non-negative integers. For an odd prime, p, and a positive integer, k, set ψ(p^k) equal to the totient function value, φ(p^k), which in this case is p^k−p^k−1. Do the same for ψ(2^k) when k > 1. Set ψ(2) and ψ(1) to 0. Using the fundamental theorem of arithmetic, we can write any other positive integer uniquely as a product of prime powers, m = Π_αp_α^k_α; set ψ(m) = Σ_α ψ(p_α^k_α). This differs from the totient itself, because it is a sum instead of a product.

The crystallographic restriction in general form states that Ord_N consists of those positive integers m such that ψ(m) ≤ N.

**Smallest dimension for a given order** OEIS: A080737
m	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
ψ(m)	0	0	2	2	4	2	6	4	6	4	10	4	12	6	6	8	16	6	18	6	8	10	22	6	20	12	18	8	28	6	30

For m>2, the values of ψ(m) are equal to twice the algebraic degree of cos(2π/m); therefore, ψ(m) is strictly less than m and reaches this maximum value if and only if m is a prime.

These additional symmetries do not allow a planar slice to have, say, 8-fold rotation symmetry. In the plane, the 2D restrictions still apply. Thus the cuts used to model quasicrystals necessarily have thickness.

Integer matrices are not limited to rotations; for example, a reflection is also a symmetry of order 2. But by insisting on determinant +1, we can restrict the matrices to proper rotations.

Formulation in terms of isometries

The crystallographic restriction theorem can be formulated in terms of isometries of Euclidean space. A set of isometries can form a group. By a discrete isometry group we will mean an isometry group that maps each point to a discrete subset of R^N, i.e. the orbit of any point is a set of isolated points. With this terminology, the crystallographic restriction theorem in two and three dimensions can be formulated as follows.

For every discrete isometry group in two- and three-dimensional space which includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4 or 6.

Isometries of order n include, but are not restricted to, n-fold rotations. The theorem also excludes S₈, S₁₂, D_4d, and D_6d (see point groups in three dimensions), even though they have 4- and 6-fold rotational symmetry only. Rotational symmetry of any order about an axis is compatible with translational symmetry along that axis.

The result in the table above implies that for every discrete isometry group in four- and five-dimensional space which includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4, 5, 6, 8, 10, or 12.

All isometries of finite order in six- and seven-dimensional space are of order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 or 30 .

Notes

↑ Shechtman et al (1982)

Related Research Articles

In mathematical physics and mathematics, the Pauli matrices are a set of three $2 \times 2$ complex matrices that are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

In linear algebra, an invertible complex square matrix $U$ is unitary if its matrix inverse $U -1$ equals its conjugate transpose $U *$ , that is, if

In mathematics, the orthogonal group in dimension $n$ , denoted $O(n)$ , is the group of distance-preserving transformations of a Euclidean space of dimension $n$ that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of $n \times n$ orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

In mechanics and geometry, the 3D rotation group, often denoted O(3), is the group of all rotations about the origin of three-dimensional Euclidean space $under the operation of composition.$

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

<span class="mw-page-title-main">Root system</span> Geometric arrangements of points, foundational to Lie theory

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

In mathematics, the modular group is the projective special linear group $of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.$

A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. For each wallpaper there corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper group is a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations, tiles and physical wallpaper.

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire $(n - 1)$ -dimensional flat of fixed points in a $n$ -dimensional space.

<span class="mw-page-title-main">Euler angles</span> Description of the orientation of a rigid body

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.

Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections.

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

References

Bamberg, John; Cairns, Grant; Kilminster, Devin (March 2003), "The crystallographic restriction, permutations, and Goldbach's conjecture" (PDF), American Mathematical Monthly , 110 (3): 202–209, CiteSeerX 10.1.1.124.8582 , doi:10.2307/3647934, JSTOR 3647934
Elliott, Stephen (1998), The Physics and Chemistry of Solids, Wiley, ISBN 978-0-471-98194-7
Coxeter, H. S. M. (1989), Introduction to Geometry (2nd ed.), Wiley, ISBN 978-0-471-50458-0
Scherrer, W. (1946), "Die Einlagerung eines regulären Vielecks in ein Gitter", Elemente der Mathematik , 1 (6): 97–98
Shechtman, D.; Blech, I.; Gratias, D.; Cahn, JW (1984), "Metallic phase with long-range orientational order and no translational symmetry", Physical Review Letters , 53 (20): 1951–1953, Bibcode:1984PhRvL..53.1951S, doi: 10.1103/PhysRevLett.53.1951

External links

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Shechtman et al (1982)

[1]