Einstein Model of Solid Lecture 12a Einstein Model of Solid Model of crystal The partition function Low and high temperature limits Thermodynamic functions Problems

Real crystal - atoms connected by springs Total potential energy of a crystal of N atoms at T = 0 c - coordination number r0 - equilibrium separation between atoms u(r0) = u0 = energy of all bonds between a single atom and its neighbors At T > 0 there are also vibrations

Einstein Model - atoms connected by springs to fixed points Approximation - each atoms moves as the rest of the atoms are fixed There is only a single frequency and 3N vibrational modes (3 per each atom) Where is the quantized vibrational energy of ith vibrational mode

Partition function Partition function - distinguishable oscillators (sites)

Partition function limiting values Low T high T

Free energy, energy and entropy prove

Heat capacity and chemical potential prove For solid G is about the same as F since PV is small by comparison with E, thus the chemical potential is

Problem - obtain low and high temperature limit of heat capacity