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 (r). r. a 0. Effects due to anharmonicity of the lattice potential. Grueneisen parameter. Frequencies become volume dependent. Frequency change modifies internal energy . linear thermal expansion coefficient. Detailed approach:. Remember differential of Helmholtz free energy.

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  1. (r) r a0 Effects due to anharmonicity of the lattice potential Grueneisen parameter Frequencies become volume dependent Frequency change modifies internal energy linear thermal expansion coefficient

  2. Detailed approach: Remember differential of Helmholtz free energy We consider expansion of the sample in a stress-free state where p=0 used to calculate expansion 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 (r) value of the potential energy in equilibrium r In the anharmonic case time-averaged position of the oscillator no longer given by a0 . a0 atom longer at positions r>a0 an harmonic case: in anharmonic case

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

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

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