30 Phys520.nb 5 Symmetries and point group in a nut shell 5.1. Basic ideas: 5.1.1. Symmetry operations ◼ Symmetry: A system remains invariant under certain operation. These operations are called symmetry operations Typically symmetry operations in crystals: (lattice) translations, space rotations, mirror reflections, time-reversal, ... ◼ The product of two symmetry operations: Consider two symmetry operations, A and B. If we perform A first and then B, this is defined as the product B A. If we first perform B and then A, then this is called the product A B ◼ The inverse operation: If A undoes B, then A is the inverse of B, A = B-1. It is easy to notice that if A undoes B, then B will also undo A. So A and B are the inverse of each other. A = B-1 and B = A-1 ◼ The identity: doing nothing. If A B = B A = I, we say A = B-1. 5.1.2. Groups A group: a set of elements and defines a product. The product satisfies four conditions ◼ Closure: If A and B are two elements, A B is also an element ◼ Identity: there is one and only one identity element (I), such that A I = I A = A for any element A in the group. ◼ Inverse: For any element A, there is one and only one element A-1, such that A A-1 = A-1 A = I ◼ Associativity: A (B C) = (A B) C Note: in general, we don’t require A B = B A. If this is true, the group is called an abelian group. If not, it is a nonabelian group. Examples: ◼ All integers and “+” form a group ◼ Positive integers and “+” doesn’t form a group ◼ Nonzero rational numbers and “×” from a group ◼ Rational numbers and “×” doesn’t form a group, because 0 has no inverse. ◼ A single translation by a lattice vector a1 (Ta1 ) doesn't form a group, because Ta1 Ta1 is NOT an element. n ◼ All lattice translations parallel to the lattice vector a1 (Ta1 where n is an integer, which means translation by n a1) form a group. ◼ All lattice translations by arbitrary lattice vector n1 a1 + n2 a2 + n3 a3 form a group. 5.1.3. Lattice translations All lattice translations form a group. TX TY = TX+Y (5.1) where X = n1 a1 + n2 a2 + n3 a3 and Y = m1 a1 + m2 a2 + m3 a3 are two lattice vectors. And Phys520.nb 31 X + Y = (n1 + m1) a1 + (n2 + m2) a2 + (n3 + m3) a3 (5.2) is also a lattice vector. This is an abelian group, because X + Y = Y +X. Note: A group formed by translations is abelian. A group formed by rotations may not be abelian. 5.2. Point group For crystals, all rotations, reflections, inversions, and their combinations form a group: the point group. 5.2.1. Rotations and reflections The operations includes: ◼ Identity: E ◼ Rotations Cn: rotation along some axis by 2 π/n n m n-m It is easy to check that Cn = E. And the inverse of Cn is Cn . ◼ Inversion: I (x,y,z becomes -x, -y, -z) Inversion changes parity (or say chirality), i.e. left-handed ↔ right-handed ◼ Mirror reflection: σ. σ = I C2 = C2 I (5.3) where I is inversion and C2 is 180 degree rotation along an axis perpendicular to the mirror plane ◼ Sn = σ Cn A rotation Cn followed by a mirror reflection perpendicular to the rotating axis. 5.2.2. Examples: ◼ C1: only one operator: the identity {E}. Examples: some crystals with triclinic Bravais lattices. (NOT all crystals with triclinic lattices have C1 symmetry. Some of them have Cs) ◼ Cs: two operations:{E, I}. The symmetry of a triclinic Bravais lattice (ignoring any structure inside each unit cell) ◼ C2: two operations:{E, C2}, where C2 is 180-degree rotation along z. Examples: some of the crystals with a monoclinic Bravais lattice. ◼ D2: if we add 180-degree rotation along x to C2, then we must also add 180-degree rotation along y, because a C2 rotation along x followed by a C2 rotation along z is equivalent to a C2 rotation along y. The closure condition requires us to add the fourth element to the group. This group is known as D2. ◼ Oh: largest lattice point group with 48 elements. All symmetry operators for a simple cubic lattice (ignoring any structure inside each unit cell) 5.2.3. Crystallographic restriction theorem In a crystal, the lattice translational symmetry enforces a constrain on rotations Cn. The value of n can only be 1 (identity), 2, 3, 4 or 6. Key implication: it is impossible to have a crystal with 5-fold rotational symmetry. (possible in quasi-crystals but not crystals). Proof: Consider one chain in the Bravais lattice (blue points). Rotate the chain along the perpendicular direction by angle θ and -θ. If rotation by θ is a symmetry operation, -θ rotational is also a symmetry operation (inverse). Image from wikipedia If both θ and -θ are symmetry operators, the green and orange points must also be lattice points of the same Bravais lattice, and thus the distance between a green and orange points (marked as m a) must be an integer times the lattice constant a. Two sides of the triangle has length a. The third side is m a. The top angle is 2 π - 2 θ. Thus geometry tells us that 32 Phys520.nb If both θ and -θ are symmetry operators, the green and orange points must also be lattice points of the same Bravais lattice, and thus the distance between a green and orange points (marked as m a) must be an integer times the lattice constant a. Two sides of the triangle has length a. The third side is m a. The top angle is 2 π - 2 θ. Thus geometry tells us that 1 - cos(π - 2 θ) 1 + cos(2 θ) m a = a2 + a2 - 2 a2 cos(2 π - 2 θ) = 2 a = 2 a = 2 a cos(θ) (5.4) 2 2 m cos θ = (5.5) 2 Because m is an integer and -1 ≤ cosθ ≤ +1, m can only take the following values: -2, -1, 0, +1, +2. So θ can only be π, 2 π/3, π/2, π/3, 0. 5.3. Lattice systems There are multiple ways to classify crystals. In earlier chapters, we introduced the idea of crystal families. There are 6 crystal families. Each family contains 1,2, 3 or 4 Bravais lattices, and there are 14 Bravais lattice in total. After we understand the ideas of point groups, we can introduce a new classification, known as lattice system. There 7 lattice systems, one more than crystal families. Lattice systems are very close to crystal families. The only modification is to split hexagonal crystal family into two lattice systems: hexagonal and rhombohedral. NOTE: there is an hexagonal crystal family and there is a hexagonal lattice system. They are both called “hexagonal”, but they are not the same thing. Th hexagonal lattice system is a subset of the hexagonal crystal family. The idea of lattice systems is as the following: assume that we draw all possible Bravais lattice and we put a sphere at each Bravais lattice point (ignoring details inside each primitive unit). Then we can ask what is the point group for all these lattices? It turns out that there are only 7 possible point groups here. They give us the 7 lattice systems. NOTE: in a crystal, the degeneracy is often higher than the prediction below for 2 reasons (1) many crystal has additional symmetries, e.g. the time-reversal symmetry and (2) when we consider spin-1/2 particles, like electrons, there are additional representations, known as double-group representations. This extra representations are from quantum physics. 5.3.1. Triclinic lattice system with Ci symmetry Triclinic lattice system coincides with the triclinic crystal family. Lattices in this lattice systems has Ci symmetry, identity and inversion. Ci contains 2 elements, and thus 2 1D representations. Because only has 1D representations, no degeneracy is in general expected in this type of crystals. 5.3.2. Monoclinic lattice system with C2 h symmetry Monoclinic lattice system coincides with the monoclinic crystal family. Lattices in this lattice systems has C2 h symmetry. C2 h contains identity, a 2-fold rotation (perpendicular to the non-rectangular surface), a mirror plane (parallel to the non-rectangular surface), and inversion. Phys520.nb 33 C2 h contains 4 elements, and thus 4 1D representations. Because only has 1D representations, no degeneracy is in general expected in this type of crystals. 5.3.3. Orthorhombic lattice system with D2 h symmetry Orthorhombic lattice system coincides with the orthorhombic crystal family. Lattices in this lattice systems has D2 h symmetry, like a cuboid. The D2 h group contains 8 elements: Identity, 2-fold rotations along x, y and z, inversion I, and three mirror planes xy, yz xz. D2 h contains 8 elements, and 8 1D representations. Because only has 1D representations, no degeneracy is in general expected in this type of crystals. 5.3.4. tetragonal lattice system with D4 h symmetry Orthorhombic lattice system coincides with the orthorhombic crystal family. Lattices in this lattice systems has D4 h symmetry, like a square prism The D4 h group contains 16 elements: Identity, 90 and 180 and 270 degree rotations along z, 2-fold rotations along x or y, 2-fold rotations along x+y or x-y, inversion I, mirror planes xy, yz, xz, (x+y)z and (x-y)z, 90 and 270 degree improper rotation along z.