TABLE OF CONTENTS The atoms in a monatomic crystal continually vibrate around their mean positions, and it is these vibrations that give the crystal its thermal energy. A rigorous quantum theory of the crystal should include the dynamics of all the electrons and the nuclei explicitly. However, the electrons are much lighter than the nuclei and therefore have a much faster reaction time to any disturbance. That is, if a nucleus is given a small displacement, the electrons will readjust their positions in response to the forces between the nucleus and the electrons in a very short time. This is the basis for the adiabatic approximation, which states that the readjustment time is so short compared to the period of vibration of the nuclei that it can be taken as zero. This means that the electrons are in a state that is determined by the instantaneous position of the nuclei.
The adiabatic approximation allows the nuclear and electronic motions to be treated separately because all the electronic information can be absorbed into a potential energy function that is a function of the nuclear coordinates alone. When the atoms are at their mean (equilibrium) positions, the potential energy is a minimum, and it increases as the atoms are displaced from equilibrium. Since the motion of the atoms is constrained to be in the vicinity of their equilibrium positions, the displacements are not extremely large and the potential energy can be expanded in a Taylor series in the displacements. If...
