Crystal Basics

A. Reciprocal Lattice A crystalline solid consists of a regular lattice of atoms. These atoms scatter photons and electrons. For instance, an injected electron would see a potential U(r) which is periodic in space. This potential affects the motion of the electron in the solid, giving rise bands of allowed energies. You should understand how electron energy bands come about by the end of this course. Similarly, the scattering of photons gives rise to regular patterns in the diffracted waves. X-ray crystallography makes use of this as an experimental probe of the crystal structure. The periodicity of a crystalline lattice can be described by specifying how each unit cell repeats and what is in each unit cell. I will focus on just one atom per unit cell. Then the positions of the atoms in the lattice can be written as: R = m1a1 +m2a2 +m3a3 (mi integers). (1) More importantly, the relative displacement between any two atoms can also be written in this form. As mentioned above, these atoms present a periodic potential U(r) to electrons in this solid. The periodicity of the potential means that U(r) has the mathematical property: U(r + R) = U(r) for all lattice vectors R (2)