1. a0a0  (r) r Effects due to anharmonicity of the lattice potential Frequencies become volume dependent Frequency change modifies internal energy Grueneisen parameter linear thermal expansion coefficient

2. Detailed approach: Remember differential of Helmholtz free energyHelmholtz free We consider expansion of the sample in a stress-free state where p=0 used to calculate expansion coefficientexpansion coefficient Statistical physics provides relation between free energy and partition function Let’s consider a single oscillator and later generalize to 3d sample

3. vibrational contribution to free energy Total free energy F value of the potential energy in equilibrium In the anharmonic case time-averaged position of the oscillator no longer given by a 0. a0a0 atom longer at positions r>a 0 harmonic case:  (r) r an in anharmonic case

4. For our 1d problem p=0 where Average thermal energy of the oscillator

5. Linear expansion coefficient From With and 1d ->3d V B T