#ABSOLUTE ENTROPY DBYE FULL#

This T 3 dependence of the specific heat at very low temperatures agrees with experiment for nonmetals. This makes it plausible to approximate the integral by increasing the limit to infinity to make use of the standard integral Which is the Dulong-Petit result from classical thermodynamics.įor 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 justifies using the approximation to the exponential from the exponential series,Į x = 1 + x. The energy expression from the Debye theory of specific heat is of the formĮven though this integral cannot be evaluated in closed form, the low and high temperature limits can be assessed.įor the high temperature case where T>T D, the value of x is very small throughtout the range of the integral. But if they are scaled to T/T D, they look very similar and are very close to the Debye theory.

At left below, the specific heats of four substances are plotted as a function of temperature and they look very different. 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. It shows that the specific heat fits the Debye model at both low and high temperatures. The data for silver shown at left is from Meyers. This expression may be evaluated numerically for a given temperature by computer routines. The specific heat expression which arises from Debye theory can be obtained by taking the derivative of the energy expression above.

#ABSOLUTE ENTROPY DBYE FULL#

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 Dulong-Petit Law at high temperatures and the characteristic T 3 behavior at very low temperatures. The Debye specific heat expression is the derivative of this expression with respect to T. When all the constants are put in, the integral takes the form It is called the Debye temperature and is a constant associated with the highest allowed mode of vibration. Where the constant T D is here introduced. The usual form of the integral is obtained by making the substitutionĪnd the limit on the integral in terms of x is obtained from 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 3 p/2. Then you get a factor of 4 p 2 from integrating over the angular coordinates, treating the mode number n as the radius vector. First, there are 3 modes associated with each mode number n: one longitudinal mode and two transverse modes. Here the factor 3 p/2 comes from three considerations. This can be expressed in terms of the phonon modes by expressing the integral in terms of the mode number n. Treating them with Einstein-Bose statistics, the total energy in the lattice vibrations is of the form 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. In this expression, v s is the speed of sound in the solid. With the modes and summing over the modes, Debye was able to find an expression for the energy as a function of temperature and derive an expression for the specific heat of the solid. Like the photons of electromagnetic energy, they obey Bose-Einstein statistics.Ĭonsidering 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. 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'. So the oscillators can accept or lose energy only in discrete units of energy hυ. Quantum harmonic oscillators have equally spaced energy levels with separation ΔE = hυ. The vibrational energies of molecules, e.g., a diatomic molecule, are quantized and treated as quantum harmonic oscillators. Such propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material. The vibrations take the form of collective modes which propagate through the material. But they are tied together with bonds, so they can't vibrate independently. Phonons and the Debye Specific Heat What is a Phonon?Ĭonsidering the regular lattice of atoms in a uniform solid material, you would expect there to be energy associated with the vibrations of these atoms.

Absolute entropy dbye