What is a Phonon?Considering the regular lattice of atoms in a uniform solid material, you would expect there to be energy associated with the vibrations of these atoms. But they are tied together with bonds, so they can't vibrate independently. The vibrations take the form of collective modes which propagate through the material. Such propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material. The vibrational energies of molecules, e.g., a diatomic molecule, are quantized and treated as quantum harmonic oscillators. Quantum harmonic oscillators have equally spaced energy levels with separation DE = hu. So the oscillators can accept or lose energy only in discrete units of energy hu. The evidence on the behavior of vibrational energy in periodic solids is that the collective vibrational modes can accept energy only in discrete amounts, and these quanta of energy have been labeled "phonons". Like the photons of electromagnetic energy, they obey BoseEinstein statistics. Considering a solid to be a periodic array of mass points, there are constraints on both the minimum and maximum wavelength associated with a vibrational mode.

Index Reference Schroeder Ch 7.  

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Debye Specific HeatBy associating a phonon energy with the vibrational modes of a solid, where v_{s} is the speed of sound in the solid, Debye approached the subject of the specific heat of solids. Treating them with EinsteinBose statistics, the total energy in the lattice vibrations is of the form This can be expressed in terms of the phonon modes by expressing the integral in terms of the mode number n. Here the factor 3p/2 comes from three considerations. First, there are 3 modes associated with each mode number n: one longitudinal mode and two transverse modes. Then you get a factor of 4p^{2} from integrating over the angular coordinates, treating the mode number n as the radius vector. Finally you constrain the integral to the quadrant in which all the components of n are positive, giving a factor of 1/8: the product of those is 3p/2. The usual form of the integral is obtained by making the substitution and the limit on the integral in terms of x is obtained from where the constant T_{D} is here introduced. It is called the Debye temperature and is a constant associated with the highest allowed mode of vibration. When all the constants are put in, the integral takes the form The Debye specific heat expression is the derivative of this expression with respect to T. The integral cannot be evaluated in closed form, but numerical evaluation of the integral shows reasonably good agreement with the observed specific heats of solids for the full range of temperatures, approaching the DulongPetit Law at high temperatures and the characteristic T^{3} behavior at very low temperatures. The specific heat expression which arises from Debye theory can be obtained by taking the derivative of the energy expression above. This expression may be evaluated numerically for a given temperature by computer routines.
Since the Debye specific heat expression can be evaluated as a function of temperature and gives a theoretical curve which has a specific form as a funtion of T/T_{D}, the specific heats of different substances should overlap if plotted as a function of this ratio. At left below, the specific heats of four substances are plotted as a function of temperature and they look very different. But if they are scaled to T/T_{D}, they look very similar and are very close to the Debye theory.

Index References Schroeder Ch 7. Meyers Ch 5. Kittel Intro. to Solid State, Ch 5.  

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High and Low Temperature Limits

where T_{D} = Debye temperature. 
Even though this integral cannot be evaluated in closed form, the low and high temperature limits can be assessed.
For the high temperature case where T>>T_{D}, the value of x is very small throughtout the range of the integral. This justifies using the approximation to the exponential from the exponential series,
e^{x} = 1 + x. This reduces the energy expression to
which is the DulongPetit result from classical thermodynamics.
For low temperatures where T<< T_{D}, the exponential in the denominator becomes very large before reaching the limit, implying that the integrand is very small near the upper limit. This makes it plausible to approximate the integral by increasing the limit to infinity to make use of the standard integral
Then energy is then
and the specific heat is
This T^{3} dependence of the specific heat at very low temperatures agrees with experiment for nonmetals. For metals the electron specific heat becomes significant at low temperatures and is combined with the above lattice specific heat in the EinsteinDebye specific heat.
Debye specific heat 
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