Patterson, Bailey - Solid State Physics Introduction to theory

14 1 Crystal Binding and Structure

1.2.1 Definition and Simple Properties of Groups (AB)

There are two basic ingredients of a group: a set of elements G = {g1, g2,…} and an operation (*) that can be used to combine the elements of the set. In order that the set form a group, there are four rules that must be satisfied by the operation of combining set elements:

1. Closure. If gi and gj are arbitrary elements of G, then gi g j G

( means “included in”).

2. Associative Law. If gi, gj and gk are arbitrary elements of G, then (gi g j ) gk = gi (g j gk ) .

3.Existence of the identity. There must exist a ge G with the property that for any

gk G , ge gk = gk ge = gk . Such a ge is called E, the identity.

4. Existence of the inverse. For each gi G there exists a gi−1 G such that gi gi−1 = gi−1 gi = E ,

Where gi−1 is called the inverse of gi.

From now on the * will be omitted and gi * gj will simply be written gi gj.

3

	P3	P2
		A
1			2
		P1

Fig. 1.6. The equilateral triangle

An example of a group that is small enough to be easily handled and yet large enough to have many features of interest is the group of rotations in three dimensions that bring the equilateral triangle into itself. This group, denoted by D3, has six elements. One thus says its order is 6.

1.2 Group Theory and Crystallography 15

In Fig. 1.6, let A be an axis through the center of the triangle and perpendicular to the plane of the paper. Let g1, g2, and g3 be rotations of 0, 2π/3, and 4π/3 about A. Let g4, g5, and g6 be rotations of π about the axes P1, P2, and P3. The group multiplication table of D3 can now be constructed. See Table 1.2.

Table 1.2. Group multiplication table of D3

D3	g1	g2	g3	g4	g5	g6
g1	g1	g2	g3	g4	g5	g6
g2	g2	g3	g1	g6	g4	g5
g3	g3	g1	g2	g5	g6	g4
g4	g4	g5	g6	g1	g2	g3
g5	g5	g6	g4	g3	g1	g2
g6	g6	g4	g5	g2	g3	g1

The group elements can be very easily described by indicating how the vertices are mapped. Below, arrows are placed in the definition of g1 to define the notation. After g1, the arrows are omitted:

			1		2	3					1 2 3						1	2	3
g		=		↓ ↓ ↓				,	g				,	g							,
	1									2	=				3	=
																	3	1	2
				1	2	3					2 3 1
											1 2 3								3
g			1 2 3					,	g				,	g			1	2
	4	=								5	=				6	=				.
																	3	2	1
			2 1			3					1 3 2

Using this notation we can see why the group multiplication table indicates that g4 g2 = g5:6

g		g		1		2	3 1		2	3		1	2	3		= g		.
	4		2	=								=					5
					2	1		2	3	1			3	2
							3					1

The table also says that g2 g4 = g6. Let us check this:

g		g		1		2	3 1			2	3	1		2	3		= g		.
	2		4	=								=						6
					2	3	1		2	1			3	2	1
											3

In a similar way, the rest of the group multiplication table was easily derived. Certain other definitions are worth noting [61]. A is a proper subgroup of G if A

is a group contained in G and not equal to E (E is the identity that forms a trivial group of order 1) or G. In D3, {g1, g2, g3}, {g1, g4}, {g1, g5}, {g1, g6} are proper subgroups. The class of an element g G is the set of elements {gi−1ggi} for all gi

G. Mathematically this can be written for g G, Cl(g) = {gi−1ggi| for all gi G}.

6 Note that the application starts on the right so 3 → 1 → 2, for example.

16 1 Crystal Binding and Structure

Two operations belong to the same class if they perform the same sort of geometrical operation. For example, in the group D3 there are three classes:

{g1}, {g2 , g3}, and {g4 , g5, g6} .

Two very simple sorts of groups are often encountered. One of these is the cyclic group. A cyclic group can be generated by a single element. That is, in a cyclic group there exists a g G, such that all gk G are given by gk = gk (of course one must name the group elements suitably). For a cyclic group of order N with generator g, gN ≡ E. Incidentally, the order of a group element is the smallest power to which the element can be raised and still yield E. Thus the order of the generator (g) is N.

The other simple group is the abelian group. In the abelian group, the order of the elements is unimportant (gigj = gjgi for all gi, gj G). The elements are said to commute. Obviously all cyclic groups are abelian. The group D3 is not abelian but all of its subgroups are.

In the abstract study of groups, all isomorphic groups are equivalent. Two groups are said to be isomorphic if there is a one-to-one correspondence between the elements of the group that preserves group “multiplication.” Two isomorphic groups are identical except for notation. For example, the three subgroups of D3 that are of order 2 are isomorphic.

An interesting theorem, called Lagrange’s theorem, states that the order of a group divided by the order of a subgroup is always an integer. From this it can immediately be concluded that the only possible proper subgroups of D3 have order 2 or 3. This, of course, checks with what we actually found for D3.

Lagrange’s theorem is proved by using the concept of a coset. If A is a subgroup of G, the right cosets are of the form Agi for all gi G (cosets with identical elements are not listed twice) − each gi generates a coset. For example, the right cosets of {g1, g6} are {g1, g6}, {g2, g4}, and {g3, g5}. A similar definition can be made of the term left coset.

A subgroup is normal or invariant if its right and left cosets are identical. In D3, {g1, g2, g3} form a normal subgroup. The factor group of a normal subgroup is the normal subgroup plus all its cosets. In D3, the factor group of {g1, g2, g3} has elements {g1, g2, g3} and {g4, g5, g6}. It can be shown that the order of the factor group is the order of the group divided by the order of the normal subgroup. The factor group forms a group under the operation of taking the inner product. The inner product of two sets is the set of all possible distinct products of the elements, taking one element from each set. For example, the inner product of {g1, g2, g3} and {g4, g5, g6} is {g4, g5, g6}. The arrangement of the elements in each set does not matter.

It is often useful to form a larger group from two smaller groups by taking the direct product. Such a group is naturally enough called a direct product group. Let

G = {g1 … gn} be a group of order n, and H = {h1 … hm} be a group of order m. Then the direct product G H is the group formed by all products of the form

gi hj. The order of the direct product group is nm. In making this definition, it has been assumed that the group operations of G and H are independent. When this is

1.2 Group Theory and Crystallography 17

not so, the definition of the direct product group becomes more complicated (and less interesting − at least to the physicist). See Sect. 7.4.4 and Appendix C.

1.2.2 Examples of Solid-State Symmetry Properties (B)

All real crystals have defects (see Chap. 11) and in all crystals the atoms vibrate about their equilibrium positions. Let us define ideal crystals as real crystals in which these complications are not present. This chapter deals with ideal crystals. In particular we will neglect boundaries. In other words, we will assume that the crystals are infinite. Ideal crystals exhibit many types of symmetry, one of the most important of which is translational symmetry. Let m1, m2, and m3 be arbitrary integers. A crystal is said to be translationally symmetric or periodic if there exist three linearly independent vectors (a1, a2, a3) such that a translation by m1a1 + m2a2 + m3a3 brings one back to an equivalent point in the crystal. We summarize several definitions and facts related to the ai:

1.The ai are called basis vectors. Usually, they are not orthogonal.

2.The set (a1, a2, a3) is not unique. Any linear combination with integer coefficients gives another set.

3. By parallel extensions, the ai form a parallelepiped whose volume is V = a1 · (a2 × a3). This parallelepiped is called a unit cell.

4.Unit cells have two principal properties:

a)It is possible by stacking unit cells to fill all space.

b)Corresponding points in different unit cells are equivalent.

5.The smallest possible unit cells that satisfy properties a) and b) above are called primitive cells (primitive cells are not unique). The corresponding basis vectors (a1, a2, a3) are then called primitive translations.

6.The set of all translations T = m1a1 + m2a2 + m3a3 form a group. The group is of infinite order, since the crystal is assumed to be infinite in size.7

The symmetry operations of a crystal are those operations that bring the crystal back onto itself. Translations are one example of this sort of operation. One can find other examples by realizing that any operation that maps three noncoplanar points on equivalent points will map the whole crystal back on itself. Other types of symmetry transformations are rotations and reflections. These transformations are called point transformations because they leave at least one point fixed. For example, D3 is a point group because all its operations leave the center of the equilateral triangle fixed.

7One can get around the requirement of having an infinite crystal and still preserve translational symmetry by using periodic boundary conditions. These will be described later.

18 1 Crystal Binding and Structure

We say we have an axis of symmetry of the nth order if a rotation by 2π/n about the axis maps the body back onto itself. Cn is often used as a symbol to represent the 2π/n rotations about a given axis. Note that (Cn)n = C1 = E, the identity.

A unit cell is mapped onto itself when reflected in a plane of reflection symmetry. The operation of reflecting in a plane is called σ. Note that σ2 = E.

Another symmetry element that unit cells may have is a rotary reflection axis. If a body is mapped onto itself by a rotation of 2π/n about an axis and a simultaneous reflection through a plane normal to this axis, then the body has a rotary reflection axis of nth order.

If f(x, y, z) is any function of the Cartesian coordinates (x, y, z), then the inversion I through the origin is defined by I[f(x, y, z)] = f(−x, −y, −z). If f(x, y, z) = f(−x, −y, −z), then the origin is said to be a center of symmetry for f. Denote an nth order rotary reflection by Sn, a reflection in a plane perpendicular to the axis of the rotary reflection by σh, and the operation of rotating 2π/n about the axis by Cn. Then Sn = Cnσh. In particular, S2 = C2σh = I. A second-order rotary reflection is the same as an inversion.

	2
1	3
	b
	4

a 6

5

7

8

Fig. 1.7. The cubic unit cell

To illustrate some of the point symmetry operations, use will be made of the example of the unit cell being a cube. The cubic unit cell is shown in Fig. 1.7. It is obvious from the figure that the cube has rotational symmetry. For example,

C2	1	2	3	4	5	6	7	8
	=	7	6	5	4	3	2	1
	8

obviously maps the cube back on itself. The rotation represented by C2 is about a horizontal axis. There are two other axes that also show two-fold symmetry. It turns out that all three rotations belong to the same class (in the mathematical sense already defined) of the 48-element cubic point group Oh (the group of operations that leave the center point of the cube fixed and otherwise map the cube onto itself or leave the figure invariant).

1.2 Group Theory and Crystallography 19

The cube has many other rotational symmetry operations. There are six fourfold rotations that belong to the class of

C		1		2	3	4	5	6	7	8
	4	=									.
			4	3	7	8	1	2	6	5

There are six two-fold rotations that belong to the class of the π rotation about the axis ab. There are eight three-fold rotation elements that belong to the class of 2π/3 rotations about the body diagonal. Counting the identity, (1 + 3 + 6 + 6 + 8) = 24 elements of the cubic point group have been listed.

It is possible to find the other 24 elements of the cubic point group by taking the product of the 24 rotation elements with the inversion element. For the cube,

1	2	3	4	5	6	7	8
I =	8	5	6	3	4	1	2	.
7

The use of the inversion element on the cube also introduces the reflection symmetry. A mirror reflection can always be constructed from a rotation and an inversion. This can be seen explicitly for the cube by direct computation.

IC2		1	2	3	4	5	6	7	8		1	2	3	4	5	6	7	8
	=	7	8	5	6	3	4	1	2			7	6	5	4	3	2	1
											8
		1	2	3	4	5	6	7	8		= σ h .
	=	2	1	4	3	6	5	8	7

It has already been pointed out that rotations about equivalent axes belong to the same class. Perhaps it is worthwhile to make this statement somewhat more explicit. If in the group there is an element that carries one axis into another, then rotations about the axes through the same angle belong to the same class.

A crystalline solid may also contain symmetry elements that are not simply group products of its rotation, inversion, and translational symmetry elements. There are two possible types of symmetry of this type. One of these types is called a screw-axis symmetry, an example of which is shown in Fig. 1.8.

The symmetry operation (which maps each point on an equivalent point) for Fig. 1.8 is to simultaneously rotate by 2π/3 and translate by d. In general a screw axis is the combination of a rotation about an axis with a displacement parallel to

d d d

Fig. 1.8. Screw-axis symmetry

20 1 Crystal Binding and Structure

the axis. Suppose one has an n-fold screw axis with a displacement distance d. Let a be the smallest period (translational symmetry distance) in the direction of the axis. Then it is clear that nd = pa, where p = 1, 2,…, n – 1. This is a restriction on the allowed types of screw-axis symmetry.

An example of glide plane symmetry is shown in Fig. 1.9. The line beneath the d represents a plane perpendicular to the page. The symmetry element for Fig. 1.9 is to simultaneously reflect through the plane and translate by d. In general, a glide plane is a reflection with a displacement parallel to the reflection plane. Let d be the translation operation involved in the glide-plane symmetry operation. Let a be the length of the period of the lattice in the direction of the translation. Only those glide-reflection planes are possible for which 2d = a.

When one has a geometrical entity with several types of symmetry, the various symmetry elements must be consistent. For example, a three-fold axis cannot have only one mirror plane that contains it. The fact that we have a three-fold axis automatically requires that if we have one mirror plane that contains the axis, then we must have three such planes. The three-fold axis implies that every physical property must be repeated three times as one goes around the axis. A particularly interesting consistency condition is examined in the next Section.

d			d	d

Fig. 1.9. Glide-plane symmetry

1.2.3 Theorem: No Five-fold Symmetry (B)

Any real crystal exhibits both translational and rotational symmetry. The mere fact that a crystal must have translational symmetry places restrictions on the types of rotational symmetry that one can have.

The theorem is:

A crystal can have only one-, two-, three-, four-, and six-fold axes of symmetry.

The proof of this theorem is facilitated by the geometrical construction shown in Fig. 1.10 [1.5, p. 32]. In Fig. 1.10, R is a vector drawn to a lattice point (one of the points defined by m1a1 + m2a2 + m3a3), and R1 is another lattice point. R1 is chosen so as to be the closest lattice point to R in the direction of one of the translations in the (x,z)-plane; thus |a| = |R − R1| is the minimum separation distance between lattice points in that direction. The coordinate system is chosen so that the z-axis is parallel to a. It will be assumed that a line parallel to the y-axis and passing through the lattice point defined by R is an n-fold axis of symmetry.

1.2 Group Theory and Crystallography 21

Strictly speaking, one would need to prove one can always find a lattice plane perpendicular to an n-fold axis. Another way to look at it is that our argument is really in two dimensions, but one can show that three-dimensional Bravais lattices do not exist unless two-dimensional ones do. These points are discussed by Ashcroft and Mermin in two problems [21, p. 129]. Since all lattice points are equivalent, there must be a similar axis through the tip of R1. If θ = 2π/n, then a counterclockwise rotation of a about R by θ produces a new lattice vector R r. Similarly a clockwise rotation by the same angle of a about R1 produces a new lattice point R1r. From Fig. 1.10, R r − R1r is parallel to the z-axis and R r − R1r = p|a|. Further, |pa| = |a| + 2|a| sin(θ − π/2) = |a| (1 − 2 cosθ). Therefore p = 1 − 2 cosθ or |cosθ| = |(p − 1)/2| ≤ 1. This equation can be satisfied only for p = 3, 2, 1, 0, −1 or θ = ±(2π/1, 2π/2, 2π/3, 2π/4, 2π/6). This is the result that was to be proved.

x		p|a|
|a|			|a|
R r	θ	a	θ
R	R1r	R1
			z

y

Fig. 1.10. The impossibility of five-fold symmetry. All vectors are in the (x,z)-plane

The requirement of translational symmetry and symmetry about a point, when combined with the formalism of group theory (or other appropriate means), allows one to classify all possible symmetry types of solids. Deriving all the results is far beyond the scope of this chapter. For details, the book by Buerger [1.5] can be consulted. The following sections (1.2.4 and following) give some of the results of this analysis.

Quasiperiodic Crystals or Quasicrystals (A)

These materials represented a surprise. When they were discovered in 1984, crystallography was supposed to be a long dead field, at least for new fundamental results. We have just proved a fundamental theorem for crystalline materials that forbids, among other symmetries, a 5-fold one. In 1984, materials that showed relatively sharp Bragg peaks and that had 5-fold symmetry were discovered. It was soon realized that the tacit assumption that the presence of Bragg peaks implied crystalline structure was false.

22 1 Crystal Binding and Structure

It is true that purely crystalline materials, which by definition have translational periodicity, cannot have 5-fold symmetry and will have sharp Bragg peaks. However, quasicrystals that are not crystalline, that is not translationally periodic, can have perfect (that is well-defined) long-range order. This can occur, for example, by having a symmetry that arises from the sum of noncommensurate periodic functions, and such materials will have sharp (although perhaps dense) Bragg peaks (see Problems 1.10 and 1.12). If the amplitude of most peaks is very small the denseness of the peaks does not obscure a finite number of diffraction peaks being observed. Quasiperiodic crystals will also have a long-range orientational order that may be 5-fold.

The first quasicrystals that were discovered (Shechtman and coworkers)8 were grains of AlMn intermetallic alloys with icosahedral symmetry (which has 5-fold axes). An icosahedron is one of the five regular polyhedrons (the others being tetrahedron, cube, octahedron and dodecahedron). A regular polyhedron has identical faces (triangles, squares or pentagons) and only two faces meet at an edge. Other quasicrystals have since been discovered that include AlCuCo alloys with decagonal symmetry. The original theory of quasicrystals is attributed to Levine and Steinhardt.9 The book by Janot can be consulted for further details [1.12].

1.2.4 Some Crystal Structure Terms and Nonderived Facts (B)

A set of points defined by the tips of the vectors m1a1 + m2a2 + m3a3 is called a lattice. In other words, a lattice is a three-dimensional regular net-like structure. If one places at each point a collection or basis of atoms, the resulting structure is called a crystal structure. Due to interatomic forces, the basis will have no symmetry not contained in the lattice. The points that define the lattice are not necessarily at the location of the atoms. Each collection or basis of atoms is to be identical in structure and composition.

Point groups are collections of crystal symmetry operations that form a group and also leave one point fixed. From the above, the point group of the basis must be a point group of the associated lattice. There are only 32 different point groups allowed by crystalline solids. An explicit list of point groups will be given later in this chapter.

Crystals have only 14 different possible parallelepiped networks of points. These are the 14 Bravais lattices. All lattice points in a Bravais lattice are equivalent. The Bravais lattice must have at least as much point symmetry as its basis. For any given crystal, there can be no translational symmetry except that specified by its Bravais lattice. In other words, there are only 14 basically different types of translational symmetry. This result can be stated another way. The requirement that a lattice be invariant under one of the 32 point groups leads to symmetrically specialized types of lattices. These are the Bravais lattices. The types of symmetry

8See Shechtman et al [1.21].

9See Levine and Steinhardt [1.15]. See also Steinhardt and Ostlund [1.22].

1.2 Group Theory and Crystallography 23

of the Bravais lattices with respect to rotations and reflections specify the crystal systems. There are seven crystal systems. The meaning of Bravais lattice and crystal system will be clearer after the next Section, where unit cells for each Bravais lattice will be given and each Bravais lattice will be classified according to its crystal system.

Associating bases of atoms with the 14 Bravais lattices gives a total of 230 three-dimensional periodic patterns. (Loosely speaking, there are 230 different kinds of “three-dimensional wall paper.”) That is, there are 230 possible space groups. Each one of these space groups must have a group of primitive translations as a subgroup. As a matter of fact, this subgroup must be an invariant subgroup. Of these space groups, 73 are simple group products of point groups and translation groups. These are the so-called symmorphic space groups. The rest of the space groups have screw or glide symmetries. In all cases, the factor group of the group of primitive translations is isomorphic to the point group that makes up the (proper and improper – an improper rotation has a proper rotation plus an inversion or a reflection) rotational parts of the symmetry operations of the space group. The above very brief summary of the symmetry properties of crystalline solids is by no means obvious and it was not produced very quickly. A brief review of the history of crystallography can be found in the article by Koster [1.14].

1.2.5 List of Crystal Systems and Bravais Lattices (B)

The seven crystal systems and the Bravais lattice for each type of crystal system are described below. The crystal systems are discussed in order of increasing symmetry.

1.Triclinic Symmetry. For each unit cell, α ≠ β, β ≠ γ, α ≠ γ, a ≠ b, b ≠ c, and a ≠ c, and there is only one Bravais lattice. Refer to Fig. 1.11 for nomenclature.

a

b

α γ

β c

Fig. 1.11. A general unit cell (triclinic)