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In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the 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 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, which is proportional to – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures. See M. Shubin and T. Sunada for a rigorous treatment of the Debye model. == Derivation == The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical. Consider a cube of side . From the particle in a box article, the resonating modes of the sonic disturbances inside the box (considering for now only those aligned with one axis) have wavelengths given by : where is an integer. The energy of a phonon is : where is Planck's constant and is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength, we have: : in which is the speed of sound inside the solid. In three dimensions we will use: : in which is the magnitude of the three-dimensional momentum of the phonon. 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 (see the article on phonons.) This is one of the limitations of the Debye model, and corresponds to incorrectness of the results at intermediate temperatures, whereas both at low temperatures and also at high temperatures they are exact. Let's now compute the total energy in the box, : where is the number of phonons in the box with energy . In other words, the total energy is equal to the sum of energy multiplied by the number of phonons with that energy (in one dimension). In 3 dimensions we have: : Now, this is where Debye model and Planck's law of black body radiation differ. Unlike electromagnetic radiation in a box, there is a finite number of phonon energy states because a phonon cannot have infinite frequency. Its frequency is bound by the medium of its propagation—the atomic lattice of the solid. Consider an illustration of a transverse phonon below. ::::: It is reasonable to assume that the minimum wavelength of a phonon is twice the atom separation, as shown in the lower figure. There are atoms in a solid. Our solid is a cube, which means there are atoms per edge. Atom separation is then given by , and the minimum wavelength is : This is the upper limit of the triple energy sum : For slowly varying, well-behaved functions, a sum can be replaced with an integral (also known as Thomas-Fermi approximation) : So far, there has been no mention of , the number of phonons with energy Phonons obey Bose–Einstein statistics. Their distribution is given by the famous Bose–Einstein 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, : (Actually one uses an ''effective sonic velocity'' , i.e. the Debye temperature (see below) is proportional to , more precisely , where one distinguishes longitudinal and transversal sound-wave velocities (contributions 1/3 and 2/3, respectively). The Debye temperature or the effective sonic velocity is a measure of the hardness of the crystal.) Substituting this into the energy integral yields : The ease with which these integrals are evaluated for photons is due to the fact that light's frequency, at least semi-classically, is unbound. As the figure above illustrates, this is not true for phonons. In order to approximate this triple integral, Debye used spherical coordinates : and boldly approximated the cube by an eighth of a sphere : where is the radius of this sphere, which is found by conserving the number of particles in the cube and in the eighth of a sphere. The volume of the cube is unit-cell volumes, : so we get: : The substitution of integration over a sphere for the correct integral introduces another source of inaccuracy into the model. The energy integral becomes : Changing the integration variable to , : To simplify the appearance of this expression, define the Debye temperature
Many references〔Schroeder, Daniel V. "An Introduction to Thermal Physics" Addison-Wesley, San Francisco (2000). Section 7.5〕 describe the Debye temperature as merely shorthand for some constants and material-dependent variables. However, as shown below, 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 (and hence every mode) is excited. Continuing, we then have the specific internal energy:
where is the (third) Debye function. Differentiating with respect to we get 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. As already mentioned, this behaviour is exact, in contrast to the intermediate behaviour. The essential reason for the exactness at low and high energies, respectively, is that the Debye model gives (i) the exact ''dispersion relation'' at low frequencies, and (ii) corresponds to the exact ''density of states'' concerning the number of vibrations per frequency interval. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Debye model」の詳細全文を読む スポンサード リンク
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