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Home , Octahedral symmetry, Point group, Rotational symmetry, Space group, Tetrahedral symmetry

The Cubic Groups

Baccalaureate Thesis in Electrical Engineering

Author: Supervisor: Sana Zunic Dr. Wolfgang Herfort 0627758

Vienna University of Technology May 13, 2010

Contents

1 Concepts from Algebra 4 1.1 Groups ...... 4 1.2 Subgroups ...... 4 1.3 Actions ...... 5

2 Concepts from 6 2.1 Space Groups and their Classification ...... 6 2.2 Motions in R3 ...... 8 2.3 Cubic Lattices ...... 9 2.4 Space Groups with a Cubic ...... 10

3 The Octahedral Groups 11 3.1 The Elements of O and Oh ...... 11 3.2 A Presentation of Oh ...... 14 3.3 The Subgroups of Oh ...... 14

2 Abstract After introducing basics from (mathematical) crystallography we turn to the description of the groups – the symmetry (s) of a .

Preface

The intention of this account is to provide a description of the octahedral sym- metry groups – (s) of the cube. We first give the basic idea (without proofs) of mathematical crystallography, namely that the 219 space groups correspond to the 7 systems. After this we come to describing cubic lattices – such ones that are built from “cubic cells”. Finally, among the cubic lattices, we discuss briefly the ones on which O and Oh act. After this we provide lists of the elements and the subgroups of Oh. A presentation of Oh in terms of generators and relations – using the Dynkin diagram B3 is also given. It is our hope that this account is accessible to both – the mathematician and the engineer. The picture on the title page reflects Ha¨uy’sidea of [4].

3 1 Concepts from Algebra 1.1 Groups Definition 1.1 A group G is an ordered pair (G, ·) where G is a set and ‘·’ a binary operation on G satisfying the following axioms: Closure: ∀a, b ∈ G : a · b ∈ G. Associativity: ∀a, b, c ∈ G :(a · b) · c = a · (b · c). Identity element: ∃e ∈ G ∀a ∈ G : a · e = e · a = a.

Inverse element: ∀a ∈ G ∃b ∈ G : a · b = b · a = e.

Example 1.2 Every vector space V is an with vector addition the group operation. In particular, (Rn, +), is an additive group. Example 1.3 Given a vector space V , the general linear group GL(V ) is the group of invertible linear transformations T : V → V with composition the group operation.

1.2 Subgroups Definition 1.4 A subset H of a group G, where (H, ·) also forms a group is a subgroup of G. We denote it with H ≤ G.

Example 1.5 Given a basis B of a finite dimensional real vector space V , P B L := hBiZ := { b∈B z(b)b | z ∈ Z } is a discrete subgroup of the additive group of V , called a lattice with basis B.

Example 1.6 The O(V ) is defined as the set of all elements T in GL(V ) for which hT x|T yi = hx|yi is valid for any x, y ∈ V . O(V ) is a compact subgroup of GL(V ). A subgroup G of O(V ) satisfies a crystallographic condition if gL ⊆ L holds ∀g ∈ G where L is a lattice in V .

Proposition 1.7 Every subgroup G of O(V ) satisfying the crystallographic con- dition is finite.

Proof: We can assume that G is a closed subgroup of O(V ) and hence is compact. The condition gL = L for g ∈ G implies the existence of a matrix U(g) with integral coefficients and gB = BU(g) where B is a basis of the lattice L. Then U(g) = B−1gB holds for all g ∈ G. Therefore the compact subgroup B−1GB can intersect the discrete set of all matrices with integral entries in a finite set only. Hence G is finite. 

Proposition 1.8 If R is a in R3 that leaves a 3-dimensional lattice invariant then the angle of rotation is a multiple of either 60◦ or 90◦.

4 Proof: Similarly as in the proof of the preceding proposition we can find a regular 3 × 3-matrix B so that B−1RB is a matrix with integral entries. Now, comparing traces and observing that traceR = 2 cos φ + 1 where φ is the angle of rotation of R, we infer that 2 cos φ ∈ Z. This is only possible if φ is a multiple ◦ ◦ of either 60 or 90 .  Definition 1.9 When G is a group and there are subgroups H and N with N/G, G = NH and N ∩ H = 1 then we say that G is the of H with N. As common, we denote the semidirect product by G = N×H.

Example 1.10 For a nonsingular n × n matrix A and an element b in a fi- nite dimensional real vector space V define TA,b(x) := Ax + b. The set of all such transformations forms the affine group A(V ). There is a presentation by matrices of the form  A b  0 1

Then the matrix presentation corresponding to composition TA,b ◦ TA0,b0 is ob- tained by multiplying the matrix presentations of TA,b and TA0,b0 . The trans- lations x 7→ x + b constitute a T of G and we can present G = T ×GL(V ) as a semidirect product. Likewise the group of is the semidirect product T ×O(V ) of the group of all translations x 7→ x + b for b ∈ V and the group of .

1.3 Actions Definition 1.11 An action of a group G on a set X is a function µ : G×X → X so that for all g, h ∈ G and all x ∈ X 1) µ(g, µ(h, x)) = µ(gh, x);

2) and µ(1G, x) = x. It will be convenient to write gx instead µ(g, x). A mor- phism of actions φ :(G, X) → (H,Y ) consists of a homomorphism α : G → H and a map β : X → Y so that β(gx) = α(g)β(x).

Example 1.12 With X := V and G := A(V ) the pair (A(V ),V ) is an action.

Example 1.13 The group of isometries Iso(V ) is the subgroup of all TA,b in the affine group A(V ) with hAx + b|Ay + bi = hx|yi. Again, with X := V we find that (Iso(V ),V ) is an action.

Proposition 1.14 Let φ : V → V be continuous and preserve length, i.e, for all x, y ∈ V we have |φ(x) − φ(y)| = |x − y|. Then φ is a rigid motion. Conversely, every is length preserving.

Proof: Clearly, every isometry is length preserving since |Ox+b−(Oy+b)|2 = |O(x − y)|2 = (x − y)T OT O(x − y) = |x − y|2. Suppose now that φ is length preserving. Then we can assume φ(0) = 0 by replacing φ by the map x 7→ φ(x) − φ(0). In fact, |φ(x) − φ(0)| = |x − 0| = |x|

5 1 2 2 2 and so |φ(x)| = |x|. Since hx|yi = 2 (|x + y| − |x| − |y| ) we can deduce from this hφ(x)|φ(y)i = hx|yi to hold for all x, y ∈ V . For z ∈ V we have:

hφ(x + y) − φ(x) − φ(y)|φ(z)i = hφ(x + y)|φ(z)i − hφ(x)|φ(z)i − hφ(y)|φ(z)i = hx + y|zi − hx|zi − hy|zi = 0.

Since φ is surjective and since z ∈ V can be chosen arbitrarily we conclude that φ(x + y) = φ(x) + φ(y). Then φ(mx) = mφ(x) follows by induction for all x ∈ V and m ∈ N. Then φ((−m)x) = φ(m(−x)) = mφ(−x) = −mφ(x) follows. 1 1 1 Replacing x by k y one can deduce φ( k y) = k φ(y). So φ(rx) = rφ(x) holds for every rational r and, by the continuity of φ it also holds for all r ∈ R. Thus φ is linear and hence φ ∈ O(V ). 

2 Concepts from Crystallography 2.1 Space Groups and their Classification The classification of crystal amounts to classifying so-called space groups – groups of isometries of V that leave invariant a lattice in V .

Definition 2.1 A discrete subgroup G of Iso(V ) is a provided that V/G is compact.

Notation 2.2 For a space group G there is a maximal subgroup of translations x 7→ x + l for l ∈ L contained in it. They constitute a normal subgroup G ∩ TL of G. The factor group K := G/G ∩ TL is the of G.

Example 2.3 Let G be generated by the symmetries of 2-dimensional space equipped with a pattern of the form ... ∆∇∆∇ .... Then G ∩ TL would consist of all translations moving multiples of a unit ∆∇. On the other hand the rotation by 180◦ around the midpoint of a unit gives rise to a subgroup of order 2 inside G. Here G can be understood as a semidirect product of its translation group and its point group. Let G be generated by a glide reflexion on 2-dimensional space with a pattern like ... ΓLΓL ... . Then G ∩ TL would consist of all translations a multiple of units ΓL. Obviously G/G ∩ TL is isomorphic to a group with 2 elements. Here the point group cannot be realized as a subgroup of G.

Proposition 2.4 There is an induced action of the point group K on L. In particular K is finite.

Proof: Every g ∈ G can be written in the form g = TA,b. Now l ∈ L gives rise −1 to the translation TE,l. Then g TE,lg = TE,A−1l, as calculation with matrices shows. Note that K satisfies the crystallographic condition – it acts on a lattice. Therefore it has to be finite by Prop. 1.7. 

6 According to [2] the classification of space groups is intimately connected with an extension problem

1 → L → G → K → 1 as follows:

1. First all possible point groups K, up to isomorphism, are classified. One finds 32 of them and each of them determines a crystal class. For every K there is exactly one GL3(Q)-conjugacy class. Therefore one also speaks of a geometric crystal class.

2. Now, for every K, one determines the GL3(Z)-conjugacy classes. This gives, in total, 73 arithmetic crystal classes. 3. According to the types of point groups 7 crystal systems appear.

7 Relations of one system to another are shown below.

4. Finally, for understanding the possible extensions of L by K one uses cohomological methods, namely, for each (K,L) with L a lattice and K acting on it, one studies the action of NAut(L)(K) upon the first cohomol- ogy H1(K, V/L). Each orbit then corresponds to a space group. There are in total 219 such groups, and, when is taken into account, one finds all the 230 space groups. They exactly correspond to the classi- fication given by Hermann-Maugin.

2.2 Motions in R3 Every rigid motion x 7→ Ax + b with A ∈ O(V ) and b ∈ V can, up to isometry, be described as follows (according to [3]). 1. Identity The identity E (“Einheit”) fixes all points in space. 2. Reflection A rigid motion whose set of fixed points is a plane is a reflection. We shall denote a reflection by σ. Note that σ2 = E.  1 0 0  A =  0 1 0  0 0 −1

3. Rotation Here the fixed set is a one dimensional subspace of the affine R3, the axis of rotation.  cos ϕ −sin ϕ 0  A =  sin ϕ cos ϕ 0  0 0 1

8 4. Rotary reflection An isometry with exactly one fixed point is a rotary reflexion. Any such motion can be understood as the composition of a rotation and a reflexion whose fixed set is orthogonal to the axis of rotation.  cos ϕ −sin ϕ 0  A =  sin ϕ cos ϕ 0  0 0 −1

5. No fixed points

a. Translation x 7→ x + b for a given b ∈ R3. b. Screw rotation We have a composition of a translation and a rota- tion. c. operation We have the composition of a reflection at a plane followed by translation parallel to that plane.

2.3 Cubic Lattices A cubic lattice has three base vectors, also known as the primitive vectors. They span the lattice. When such a lattice is contained in the space group we speak of a cubic . According to [5] one can have: Primitive centering (P): there are lattice points only at the cell’s corners; Body-centered (I): an additional point at the center of the cell is added;

Face-centered (F): an additional point at center of every face of the cell is added; In this way 3 cubic lattices appear:

1. Simple cubic (P) This system consists of one lattice point on each corner of a cube. Each atom at the lattice points is then shared equally between eight adjacent .

A simple cubic lattice and its primitive vectors are shown on the left. The three vectors can be taken an orthonormal basis of R3. The points P and Q are given with

P = ~a2 + ~a3,Q = ~a1 + ~a3.

9 2. Body-centered cubic (I) This system has one lattice point in the center (where space diagonals dissect) in addition to the eight corner points.

Three primitive vectors of the body-centered cubic lattice are shown on the left. The point P is given by

P = − ~a1 − ~a2 + 2 ~a3.

3. Face-centered cubic (F) In addition to the corner lattice points, this sys- tem has one extra lattice point at the center of every face.

A set of primitive vectors for the face-centered lattice is shown. The labelled points are:

P = ~a1 + ~a2 + ~a3,Q = 2 ~a2

R = ~a2 + ~a3,S = − ~a1 + ~a2 + ~a3.

It should be noted, that this set of vectors does not depend on which point is singled out as the origin (O). The face-centered and the body-centered cubic lattices are of great impor- tance, since an enormous variety of solids crystallize in these forms with an atom (or ion) at each lattice site. Silver, aluminum, copper and nickel are some of many examples of a face-centered cubic lattice. On the other hand, bar- ium, iron, lithium and sodium crystallize in a body-centered cubic structure. The corresponding simple cubic form is very rare, the alpha phase of being the only known example among the elements under normal conditions.

2.4 Space Groups with a Cubic Lattice Below is the associated with the five point groups of cubic lattice. The point groups are noted in Sch¨onfliesnotation and the space groups in Hermann-Maugin notation. In each of the five lists below the first three groups are split extensions of the lattice by its point group. Such groups do not contain any glide plane operations or screw rotations.

10 • The tetrahedral group T (23) with centerings (P, I and F) gives:

P 23,F 23,I23,P 213,I213

Numbers with subscript denote screw rotation: the first number indicates the screw axis n, where the angle of rotation ϕ = 2π/n; the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, ◦ 21 is a 180 rotation followed by a translation of 1/2 of the lattice vector.

• The full group Td(43m) with centerings gives:

P 43m, F 43m, I43m, P 43n, F 43c, I43d

Symmetries, where rotation through an angle ϕ = 2π/n followed by inver- sion preserves the original shape are denoted with n (the space diagonal of a cube is 3). When a reflection plane either contains or is to an axis of rotation of order m then, as a symbol, one adds ‘m’ after the axis’ number. If one of these planes is orthogonal to the axis then a slash is placed between the axis number and the ‘m’. An axis that comes from the composition of elements of higher order will not be taken into account in this notation. Therfore Oh is denoted by m3m. The letter ‘n’ indicates the appearence of a glide plane, gliding along half of a diagonal of a face. A d-glide is a glide along a quarter of either a face or space diagonal of the ; a-, b- and c-glides are plane glides parallel to the the axis indicated by the prefixed letter.

• The pyritohedral group Th(m3) with centerings gives:

P m3, F m3, Im3, P n3, P a3, F d3, Ia3

• The octahedral group O(432) with centerings gives:

P 432,F 432,I432,P 4132,P 4232,P 4332,F 4132,I4132

• The full octahedral symmetry group Oh(m3m) with centerings gives:

P m3m, F m3m, Im3m, P n3n, P m3n, P n3m, F m3c, F d3m, F d3c, Ia3d

That is in total 5 + 6 + 7 + 8 + 10 = 36 cubic space groups.

3 The Octahedral Symmetry Groups

3.1 The Elements of O and Oh The group consisting of all proper rotations leaving invariant a cubic lattice is the rotational octahedral symmetry group, denoted by O. If, in addition, we

11 allow reflections, we arrive at the so-called octahedral symmetry group which in Sch¨onfliesnotation is denoted by Oh. The elements of Oh are all the isometries of a cube which fix the intersection of the space diagonals. Here is the list of the elements of O which can be found in [7].

◦ • One identity isometry E (can be interpreted as a rotation by 360 → C1). • Nine rotations about 4-fold axes (from the center of a face to the center of the opposite face). The image below shows only one; C5C6, possible also C1C2 and C3C4. Altogether then:

– three rotations by 90◦ (C4) – three rotations by 180◦ 2 (C4) – three rotations by 270◦ 3 (C4)

• Eight rotations about 3-fold axes (space diagonals). We illustrate only the one; about AG. Altogether we have:

– four rotations by 120◦ (C3) – four rotations by 240◦ 2 (C3)

• Six rotations about 2-fold axes (from the midpoint of an edge to the mid- point of the opposite edge) → C2.

12 – the picture on the left shows only M11M12, other possible 2-fold axes are M1M2, M3M4, M5M6, M7M8 and M9M10, thus altogether six

It is clear that ord(O) = 1 + 9 + 8 + 6 = 24 and there can be no more rigid motions of the cube. We can see this as follows: there are six ways to choose which face is at the bottom, and when chosen, we can still rotate the cube into four different positions. Thus there are 6.4 = 24 elements in O. One of the rotational group’s formal notations is also Hermann-Maugin notation; 432. All rotational symmetries are denoted by n (here 4, 3 and 2), given by the angle of rotation ϕ = 2π/n. By convention, the numbers ‘n’ are written down in decreasing order, any axis with largest m is called a principal axis. The group of rotations O permutes the four space diagonals. Closer inspec- tion shows that O becomes isomorphic with S4 the symmetry group of a set with four elements. By composing each rotation with inversion, we get additional 24 elements. They are listed below. • One inversion I = EI

• Six reflections σv; each of them is a reflection at a plane through a space diagonal and a 4-fold axis of rotation. One can show that C2I = σv.

• Three reflections σh; each of them is a reflection at a plane through the cube’c center parallel to one of its faces.

2 2 4 C4I = C4 C2 σh = C4 σh = σh |{z} 2 |{z} C4 E

◦ • Three rotary reflections S4 (rotations about 4-fold axes through 90 followed by reflection in orthogonal plane);

3 3 4 C4I = C4 C2 σh = C4 C4σh = S4 |{z} 2 |{z} C4 E

3 ◦ • Three rotary reflections S4 (rotations about 4-fold axes through 270 fol- lowed by reflection in orthogonal plane);

3 3 C4I = C4 C2 σh = C4σh = S4 |{z} 2 C4

13 • Four rotary reflections S6 (each of them is the composition of a rotation by 60◦ around one of the 3-fold axes followed by a reflexion at a plane perpendicular to the axes of rotation through the cube’s center). In this way each of the C3-axes becomes a S6-axis.

5 ◦ • Four rotary reflections S6 (rotations about 6-fold axes through 300 fol- lowed by reflection at the same orthogonal plane as in the preceding item). Some reflection planes and rotation axes are shown below.

It is clear that ord(Oh) = 24 + 24 = 48. It is a subgroup of O(3). Remark that Oh is also a symmetry group of the . Its Hermann-Maugin notation is m3m.

3.2 A Presentation of Oh A subgroup of motions generated by reflections is a reflection group. Such groups have been extensively studied and can be classified by Dynkin diagrams [1]. It has been shown in [2] that Oh appears just as the Weyl group of the Dynkin diagram B3. Therefore a presentation for Oh can be derived from B3.

√ √ • 2 • 2 +3 •1

One can read this diagram by letting Oh being generated by 3 elements r, s, t each a reflection, i.e., having order 2. Now, according to the sort of bonds we have respectively (rs)3 = 1, (st)4 = 1 and (rt)2 = 1.

3.3 The Subgroups of Oh

Our next task is to list all subgroups of Oh as in [7]. The order of every subgroup must divide 48. Find below the order of the respective subgroup in brackets.

• Oh(48) and E(1) Every group has the group itself and the group containing only identity element as subgroups.

14 • C4(4), C3(3) and C2(2) The uniaxial group Cn(n) is a group that contains only rotations, and those ◦ about a single n-fold rotation axis. C4 is generated by a 90 rotation, C3 ◦ ◦ by a 120 rotation, C2 by a 180 rotation, and there are no other cube rotations. All groups of this type are cyclic.

• C4h(8), C3h(6) and C2h(4) Adjoining σh to Cn genereates Cnh(2n), which is abelian. This group is called prismatic symmetry group.

• C4v(8), C3v(6) and C2v(4) Adjoining σh to Cn generates the pyramidal symmetry group Cnv(2n).

• Cs(2) The group Cs is a special case of Cnh and Cnv for n = 1. This group is called bilateral symmetry group.

• D4(8),D3(6) and D2(4) The Dn(2n) is a group that has an n-fold axis of rotation and a set of 2-fold axes perpendicular to it. The situation for D3 is shown below.

Dn contains 2n elements; n n-fold rotations and n 2-fold rotations. The dihedral group is the rotation group of a regular .

• D4h(16) and D2h(8) Here we proceed similarly as with Cn; we adjoin a reflection σh to D4 and D3 respectively.

• D3d(12) and D2d(8) If we add a vertical reflection plane through a space diagonal, then the 2-fold rotations generate a total of n vertical reflection planes. The group obtained in this way is the antiprismatic symmetry group Dnd (d for diag- onal). The principal axis is also a 2n-fold rotary reflection axis. Because of the crystallographic condition, Prop. 1.8 we can have such groups only when n = 2 or 3 leading to elements of order 4 and 6 in Oh, as we have seen earlier.

15 • S6(6),S4(4) and S2(2) A symmetry group containing only rotary reflections (rotation about an n-fold axis followed by reflection in an orthogonal plane) is denoted with Sn(n).

• O(24),T (12),Td(24) and Th(24) We have already encountered O, the group of octa- hedron. The tetrahedral group T contains only rotations. A regular can be seen to be made up by only black dots in one of the 8 subcubes in the drawing of sodium chloride on page 17. Possible rotation axes coincide with the space diagonals of the cube.

If we adjoin σh to T , we get the full tetrahedral symmetry group Td. It describes all symmetries of a tetrahedron. T with σv generates the pyrothedral symmetry group Th. The subscript h means that the plane is horizontal relative to a 2-fold axes, but a plane bisects the angles between the 3-fold axes, and thus converts them into a 6-fold rotary reflection axes just as in the Oh case. It is also the symmetry group of the pyritohedron [8].

16 An example of a lattice with the symmetry group Oh is a sodium chloride lattice (NaCl). It can be described as a face-centered cubic lattice with a basis consisting of a sodium ion at the origin and a chlorine ion at the center of the conventional cubic cell. The structure of NaCl is shown below; one type of ion is represented by black balls, the other type by white ones.

17 References

[1] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups. Springer-Verlag, Berlin-G¨ottingen-Heidelberg (1957) [2] G. Maxwell, The Crystallography of Coxeter Groups, J.Alg. 35 159–177 (1975) [3] M. Senechal, Crystalline Symmetries: An Informal Mathematical Introduc- tion, 23–27 (1990) [4] R. J. Ha¨uy, Trait´ede Crystallographie (1822). [5] Wikipedia – Cubic Crystal Systems

[6] Wikipedia – Space Group [7] Wikipedia – Octahedral Symmetry [8] Wikipedia – Pyritohedron

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