Primitive cells, Wigner Seitz cells, and 2D lattices Chapter One Crystal Structures (Lattices, bases, and unit cells) Main Concepts ! To describe crystal structures, we need to understand 2 concepts 1[.]
Chapter One: Crystal Structures (Lattices, bases, and unit cells) Main Concepts ! ! To describe crystal structures, we need to understand concepts: Lattice (what is the underlying symmetry? Cubic? Hexagonal? Something else?) Basis (what are the contents of the unit cell?) The lattice is defined by fundamental translation vectors a1, a2, a3 such that the atomic arrangements look the same in every respect when viewed from the point r as when viewed from r' r' = r + u1 a1 + u2 a2 + u3 a3 Where u1, u2, and u3 are integers An example of a lattice: the rectangular lattice in 2D ! ! Define translation vectors a1 and a2 such that any point can be reached by adding u1 a1 + u2 a2 (where u1 and u2 are integers) By “point” we mean any position where the lattice looks the same (ie If the lattice was infinite, you couldn’t tell if you moved or not) A 2D lattice: rectangular lattice in 2D } 1a a2 a1 One can reach any point in space by adding an integer number of translation vectors a1 and a2 r' = r + a1 + a2 Where u1 = and u2 = 2 a1 Crystal structure = lattice + basis A lattice a2 a1 A basis A crystal structure We need to identify the symmetry (lattice vectors) and the lattice contents (basis) to fully describe a structure (1) How we choose lattice vectors? (This will lead us to thinking about unit cells) The Unit Cell ! ! ! ! ! The unit cell is defined by the translational vectors a1, a2 and a3 This is the basic building block of the crystal structure (it fills space) The choice of origin is arbitrary The choice of unit cells is arbitrary as well! How we choose unit cells? How we pick lattice vectors a1, a2 and a3? ! ! ! ! ! ! The choice of the origin for unit cells is arbitrary! (See overhead example for 2D square lattice) The choice of the unit cell is arbitrary as well! (See overhead example for 2D square lattice) Note: I can pick vectors a1 and a2 such that the square lattice looks like the rectangular lattice The only requisite is that the lattice must look the same when you translate by a crystal translational vector T: T = u1 a1 + u2 a2 + u3 a3 Where u1, u2, and u3 are integers Answer: Choose primitive unit cells ! ! A primitive unit cell is made of primitive translation vectors a1, a2, and a3 such that there is no cell of smaller volume that can be used as a building block for crystal structures A primitive unit cell will fill space by repetition of suitable crystal translation vectors This is defined by the parallelpiped a1, a2 and a3 The volume of a primitive unit cell can be found by V = | a1 • a x a | (vector products) a3 a2 a1 Cubic cell: Volume = a3 (homework: show this using the eqn!) Primitive unit cells can have angles between vectors that are not 90° ! ! ! Example: Monoclinic unit cells (eg Like the monoclinic crystal of gypsum shown in last class) The equation still works, but a3 is displaced from the zaxis by an angle β Homework: Prove that the formula still applies and find the volume (hint: define a vector which is a2 x a3, and use vector identities) z-axis β a3 a2 a1 Important points There is only one lattice point/primitive cell There can be different choice for a1, a2 and a3, but the volumes of these cells are all the same