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

2. 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

3. 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

4. Partition function
Partition function - distinguishable oscillators (sites)

5. Partition function limiting values
Low T high T

6. Free energy, energy and entropy
prove

7. 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

8. Problem - obtain low and high temperature limit of heat capacity