In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 to estimate phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box in contrast to the Einstein solid model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low-temperature dependence of the heat capacity of solids, which is proportional to the cube of temperature – the Debye T 3 law. Similarly to the Einstein photoelectron model, it recovers the Dulong–Petit law at high temperatures. Due to simplifying assumptions, its accuracy suffers at intermediate temperatures.
Derivation The Debye model treats atomic vibrations as phonons confined in the solid's volume. It is analogous to Planck's law of black body radiation, which treats electromagnetic radiation as a photon gas confined in a vacuum space. Most of the calculation steps are identical, as both are examples of a massless Bose gas with a linear dispersion relation.
For a cube of side-length , the resonating modes of the sonic disturbances (considering for now only those aligned with one axis), treated as particles in a box, have wavelengths given as
where is an integer. The energy of a phonon is given as
where is the Planck constant and is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength,
in which is the speed of sound inside the solid. In three dimensions, energy can be generalized to
in which is the magnitude of the three-dimensional momentum of the phonon, and , , and are the components of the resonating mode along each of the three axes.
The approximation that the frequency is inversely proportional to the wavelength (giving a constant speed of sound) is good for low-energy phonons but not for high-energy phonons, which is a limitation of the Debye model. This approximation leads to incorrect results at intermediate temperatures, whereas the results are exact at the low and high temperature limits.
The total energy in the box, , is given by
where is the number of phonons in the box with energy ; the total energy is equal to the sum of energies over all energy levels, and the energy at a given level is found by multiplying its energy by the number of phonons with that energy. In three dimensions, each combination of modes in each of the three axes corresponds to an energy level, giving the total energy as:
The Debye model and Planck's law of black body radiation differ here with respect to this sum. Unlike electromagnetic photon radiation in a box, there are a finite number of phonon energy states because a phonon cannot have an arbitrarily high frequency. Its frequency is bounded by its propagation medium—the atomic lattice of the solid. The following illustration describes transverse phonons in a cubic solid at varying frequencies:
It is reasonable to assume that the minimum wavelength of a phonon is twice the atomic separation, as shown in the lowest example. With atoms in a cubic solid, each axis of the cube measures as being atoms long. Atomic separation is then given by , and the minimum wavelength is
making the maximum mode number :
This contrasts with photons, for which the maximum mode number is infinite. This number bounds the upper limit of the triple energy sum
If is a function that is slowly varying with respect to , the sums can be approximated with integrals:
To evaluate this integral, the function , the number of phonons with energy must also be known. Phonons obey Bose–Einstein statistics, and their distribution is given by the Bose–Einstein statistics formula:
Because a phonon has three possible polarization states (one longitudinal, and two transverse, which approximately do not affect its energy) the formula above must be multiplied by 3,
Considering all three polarization states together also means that an effective sonic velocity must be determined and used as the value of the standard sonic velocity The Debye temperature defined below is proportional to ; more precisely, , where longitudinal and transversal sound-wave velocities are averaged, weighted by the number of polarization states. The Debye temperature or the effective sonic velocity is a measure of the hardness of the crystal.
Substituting into the energy integral yields
These integrals are evaluated for photons easily because their frequency, at least semi-classically, is unbound. The same is not true for phonons, so in order to approximate this triple integral, Peter Debye used spherical coordinates,
and approximated the cube with an eighth of a sphere,
where is the radius of this sphere. As the energy function does not depend on either of the angles, the equation can be simplified to
The number of particles in the original cube and in the eighth of a sphere should be equivalent. The volume of the cube is unit cell volumes,
such that the radius must be
The substitution of integration over a sphere for the correct integral over a cube introduces another source of inaccuracy into the resulting model.
After making the spherical substitution and substituting in the function , the energy integral becomes .
Changing the integration variable to ,
To simplify the appearance of this expression, define the Debye temperature
where is the volume of the cubic box of side-length .
Some authorsSchroeder, Daniel V. "An Introduction to Thermal Physics" Addison-Wesley, San Francisco (2000). Section 7.5 describe the Debye temperature as shorthand for some constants and material-dependent variables. However, is roughly equal to the phonon energy of the minimum wavelength mode, and so we can interpret the Debye temperature as the temperature at which the highest-frequency mode is excited. Additionally, since all other modes are of a lower energy than the highest-frequency mode, all modes are excited at this temperature.
From the total energy, the specific internal energy can be calculated:
where is the third Debye function. Differentiating this function with respect to produces the dimensionless heat capacity:
These formulae treat the Debye model at all temperatures. The more elementary formulae given further down give the asymptotic behavior in the limit of low and high temperatures. The essential reason for the exactness at low and high energies is, respectively, that the Debye model gives the exact dispersion relation at low frequencies, and corresponds to the exact density of states at high temperatures, concerning the number of vibrations per frequency interval.
Debye's derivation
Debye derived his equation differently and more simply. Using continuum mechanics, he found that the number of vibrational states with a frequency less than a particular value was asymptotic to
in which is the volume and is a factor that he calculated from elasticity coefficients and density. Combining this formula with the expected energy of a harmonic oscillator at temperature (already used by Einstein in his model) would give an energy of
if the vibrational frequencies continued to infinity. This form gives the behaviour which is correct at low temperatures. But Debye realized that there could not be more than vibrational states for N atoms. He made the assumption that in an atomic solid, the spectrum of frequencies of the vibrational states would continue to follow the above rule, up to a maximum frequency chosen so that the total number of states is
Debye knew that this assumption was not really correct (the higher frequencies are more closely spaced than assumed), but it guarantees the proper behaviour at high temperature (the Dulong–Petit law). The energy is then given by
Substituting for ,
where is the function later given the name of third-order Debye function.
Another derivation First the vibrational frequency distribution is derived from Appendix VI of Terrell L. Hill's An Introduction to Statistical Mechanics. Consider a three-dimensional isotropic elastic solid with N atoms in the shape of a rectangular parallelepiped with side-lengths . The elastic wave will obey the wave equation and will be plane waves; consider the wave vector and define , such that
Solutions to the wave equation are
and with the boundary conditions at ,
where are positive integers. Substituting () into () and also using the dispersion relation ,
The above equation, for fixed frequency , describes an eighth of an ellipse in "mode space" (an eighth because are positive). The number of modes with frequency less than is thus the number of integral points inside the ellipse, which, in the limit of (i.e. for a very large parallelepiped) can be approximated to the volume of the ellipse. Hence, the number of modes with frequency in the range is
where is the volume of the parallelepiped. The wave speed in the longitudinal direction is different from the transverse direction and that the waves can be polarised one way in the longitudinal direction and two ways in the transverse direction and ca be defined as .
Following the derivation from A First Course in Thermodynamics, an upper limit to the frequency of vibration is defined ; since there are atoms in the solid, there are quantum harmonic oscillators (3 for each x-, y-, z- direction) oscillating over the range of frequencies . can be determined using
By defining , where k is the Boltzmann constant and h is the Planck constant, and substituting () into (), this definition is more standard; the energy contribution for all oscillators oscillating at frequency can be found. Quantum harmonic oscillators can have energies where and using Maxwell-Boltzmann statistics, the number of particles with energy is
The energy contribution for oscillators with frequency is then
By noting that (because there are modes oscillating with frequency ),
From above, we can get an expression for 1/A; substituting it into (),
Integrating with respect to ν yields
Temperature limits
The temperature of a Debye solid is said to be low if , leading to
This definite integral can be evaluated exactly:
In the low-temperature limit, the limitations of the Debye model mentioned above do not apply, and it gives a correct relationship between (phononic) heat capacity, temperature, the elastic coefficients, and the volume per atom (the latter quantities being contained in the Debye temperature).
The temperature of a Debye solid is said to be high if . Using if leads to
which upon integration gives
This is the Dulong–Petit law, and is fairly accurate although it does not take into account anharmonicity, which causes the heat capacity to rise further. The total heat capacity of the solid, if it is a conductor or semiconductor, may also contain a non-negligible contribution from the electrons.
Debye versus Einstein
The Debye and Einstein models correspond closely to experimental data, but the Debye model is correct at low temperatures whereas the Einstein model is not. To visualize the difference between the models, one would naturally plot the two on the same set of axes, but this is not immediately possible as both the Einstein model and the Debye model provide a functional form for the heat capacity. As models, they require scales to relate them to their real-world counterparts. One can see that the scale of the Einstein model is given by :
The scale of the Debye model is , the Debye temperature. Both are usually found by fitting the models to the experimental data. (The Debye temperature can theoretically be calculated from the speed of sound and crystal dimensions.) Because the two methods approach the problem from different directions and different geometries, Einstein and Debye scales are the same, that is to say