1. 16
Heat Capacity

2. 16.1. Classical (Atomistic) Theory of Heat Capacity
Average energy of one-dimensional harmonic oscillator
Average energy per atom ( three-dimensional harmonic oscillator )
It was shown that average kinetic energy of a particle is expressed as
(from previous chapter)

3. So the total energy of vibrating atom is expressed as Etot=Epot.+Ekin.
Average potential energy of vibrating atom has the same average magnitude as the kinetic energy.
So the total energy of vibrating atom is expressed as Etot=Epot.+Ekin.
Namely, the total internal energy per mole
Finally, the molar heat capacity is
Inserting numerical values for N0 and kB
Simple model can readily explain the experimentally observed heat capacity except that the calculated heat capacity turned out to be temperature independent - need QM explanation.

4. 16.2. Quantum Mechanical Considerations-The Phonon
Einstein Model
The energy of the i th energy level of a harmonic oscillator
In 1907, Einstein postulated that the energies of the above-mentioned classical oscillators should be quantized, i.e., he postulated that only certain vibrational modes should be allowed – these lattice vibration quanta were called phonons.
Phonon waves propagate through the crystal with the speed of sound.
They are elastic waves vibrating in a longitudinal and/or in a transverse mode.

5. Boson: any particle with an integral number of spins
Boson: any particle with an integral number of spins. Photon and phonon are boson particles
The average energy of an isolated oscillator.

6. Heat capacity at a constant volume
Einstein temperature is
So,

7. Debye Model
We now refine the Einstein model by taking into account that the atoms in a crystal interact with each other oscillators are thought to vibrate interdependently.
 Einstein model considered only one frequency of vibration wD.
When interactions between the atoms occur, many more frequencies are thought to exist, which range from about the Einstein frequency down to the acoustical modes of oscillation.
Debye modified the Einstein equation by replacing the 3N0 oscillators of a single frequency with the number of modes in a frequency interval, dw, and by summing up over all allowed frequencies.
Applying periodic boundary conditions over N3 primitive cells within a cube of side length L,

8. 16.2.2 Debye Model Total energy of vibration for the solid
Heat capacity at a constant volume
Or indicating with Debye temperature

9. 16.3. Electronic Contribution to The Heat Capacity
Thermal energy at given temperature
(consider electrons as non-interacting particles)
The Heat capacity of the electrons
Population density at EF
For E < EF
So far, we assumed that the thermally excited electrons behave like a classical gas.

10. In reality, the excited electrons must obey the Pauli principle. So,
If we assume a monovalent metal in which we can reasonably assume one free electron per atom, N* can be equated to the number of atoms per mole.
Below the Debye temperature, the heat capacity of metals is sum of electron and phonon contributions.

11. Thermal effective mass