Editing Debye model (section)

== Derivation ==
The Debye model treats [[atomic vibration]]s as [[phonon]]s 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 <math>L</math>, the resonating modes of the sonic disturbances (considering for now only those aligned with one axis), treated as [[particle in a box|particles in a box]], have [[wavelength]]s given as
:<math>\lambda_n = {2L\over n}\,,</math>
where <math>n</math> is an integer. The energy of a phonon is given as
:<math>E_n\ =h\nu_n\,,</math>
where <math>h</math> is the [[Planck constant]] and <math>\nu_{n}</math> is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength,
:<math>E_n=h\nu_n={hc_{\rm s}\over\lambda_n}={hc_sn\over 2L}\,,</math>
in which <math>c_s</math> is the speed of sound inside the solid. In three dimensions, energy can be generalized to
:<math>E_n^2={p_n^2 c_{\rm s}^2}=\left({hc_{\rm s}\over2L}\right)^2\left(n_x^2+n_y^2+n_z^2\right)\,,</math>
in which <math>p_n</math> is the [[norm (mathematics)|magnitude]] of the [[Three-dimensional space|three-dimensional]] [[momentum]] of the phonon, and <math>n_x</math>, <math>n_y</math>, and <math>n_z</math> 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, <math>U</math>, is given by
:<math>U = \sum_n E_n\,\bar{N}(E_n)\,,</math>
where <math>\bar{N}(E_n)</math> is the number of phonons in the box with energy <math>E_n</math>; 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:
:<math>U = \sum_{n_x}\sum_{n_y}\sum_{n_z}E_n\,\bar{N}(E_n)\,.</math>

The Debye model and Planck's law of black body radiation differ here with respect to this sum. Unlike [[electromagnetic radiation|electromagnetic photon radiation]] in a box, there are a finite number of [[phonon]] [[energy state]]s 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:

[[Image:Debye limit.svg|400px]]

It is reasonable to assume that the minimum [[wavelength]] of a [[phonon]] is twice the atomic separation, as shown in the lowest example. With <math>N</math> atoms in a cubic solid, each axis of the cube measures as being <math>\sqrt[3]{N}</math> atoms long. Atomic separation is then given by <math>L/\sqrt[3]{N}</math>, and the minimum wavelength is
:<math>\lambda_{\rm min} = {2L \over \sqrt[3]{N}}\,,</math>
making the maximum mode number <math>n_{max}</math>:
:<math>n_{\rm max} = \sqrt[3]{N}\,.</math>

This contrasts with photons, for which the maximum mode number is infinite. This number bounds the upper limit of the triple energy sum
:<math>U = \sum_{n_x}^{\sqrt[3]{N}}\sum_{n_y}^{\sqrt[3]{N}}\sum_{n_z}^{\sqrt[3]{N}}E_n\,\bar{N}(E_n)\,.</math>

If <math>E_n</math> is a [[Function (mathematics)|function]] that is slowly varying with respect to <math>n</math>, the sums can be [[Thomas-Fermi approximation|approximated]] with [[integral]]s: <math>U \approx\int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}} E(n)\,\bar{N}\left(E(n)\right)\,dn_x\, dn_y\, dn_z\,.</math>

To evaluate this integral, the function <math>\bar{N}(E)</math>, the number of phonons with energy <math>E\,,</math> must also be known. Phonons obey [[Bose–Einstein statistics]], and their distribution is given by the Bose–Einstein statistics formula:
:<math>\langle N\rangle_{BE} = {1\over e^{E/kT}-1}\,.</math>

Because a phonon has three possible polarization states (one [[longitudinal wave|longitudinal]], and two [[transverse wave|transverse]], which approximately do not affect its energy) the formula above must be multiplied by 3,
:<math>\bar{N}(E) = {3\over e^{E/kT}-1}\,.</math>

Considering all three polarization states together also means that an effective sonic velocity <math>c_{{\rm eff}}</math> must be determined and used as the value of the standard sonic velocity <math>c_s.</math> The Debye temperature <math>T_{\rm D}</math> defined below is proportional to <math>c_{{\rm eff}}</math>; more precisely, <math>T_{\rm D}^{-3}\propto c_{{\rm eff}}^{-3}:=\frac{1}{3}c_{{\rm long}}^{-3}+\frac{2}{3}c_{{\rm trans}}^{-3}</math>, where longitudinal and transversal [[Sound wave|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 <math>\bar{N}(E)</math> into the energy integral yields
:<math>U  = \int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}} E(n)\,{3\over e^{E(n)/kT}-1}\,dn_x\, dn_y\, dn_z\,.</math>

These integrals are evaluated for [[photon]]s 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]],
:<math>\ (n_x,n_y,n_z)=(n\sin \theta \cos \phi,n\sin \theta \sin \phi,n\cos \theta )\,,</math>
and approximated the cube with an eighth of a [[sphere]],
:<math>U \approx\int_0^{\pi/2}\int_0^{\pi/2}\int_0^R E(n)\,{3\over e^{E(n)/kT}-1}n^2 \sin\theta\, dn\, d\theta\, d\phi\,,</math>
where <math>R</math> is the radius of this sphere. As the energy function does not depend on either of the angles, the equation can be simplified to
:<math>\,3 \int_0^{\pi/2}\int_0^{\pi/2}\sin\theta\, d\theta\, d\phi\,\int_0^R E(n)\,\frac{1}{e^{E(n)/kT}-1}n^2 dn\, = \frac{3\pi}{2} \int_0^R E(n)\,\frac{1}{e^{E(n)/kT}-1}n^2 dn\,</math>
The number of particles in the original cube and in the eighth of a sphere should be equivalent. The volume of the cube is <math>N</math> [[unit cell]] volumes,
:<math>N = {1\over8}{4\over3}\pi R^3\,,</math>
such that the radius must be
:<math>R = \sqrt[3]{6N\over\pi}\,.</math>

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 <math>E(n)\,</math>, the energy integral becomes
:<math>U = {3\pi\over2}\int_0^R \,{hc_sn\over 2L}{n^2\over e^{hc_{\rm s}n/2LkT}-1} \,dn</math>.

Changing the integration variable to <math>x = {hc_{\rm s}n\over 2LkT}</math>,
:<math>U = {3\pi\over2} kT \left({2LkT\over hc_{\rm s}}\right)^3\int_0^{hc_{\rm s}R/2LkT} {x^3\over e^x-1}\, dx.</math>

To simplify the appearance of this expression, define the Debye temperature <math>T_{\rm D}</math>
:<math>T_{\rm D}\ \stackrel{\mathrm{def}}{=}\  {hc_{\rm s}R\over2Lk} = {hc_{\rm s}\over2Lk}\sqrt[3]{6N\over\pi} = {hc_{\rm s} \over 2k} \sqrt[3]{{6\over\pi}{N\over V}}</math>
where <math>V</math> is the volume of the cubic box of side-length <math>L</math>.

Some authors<ref name="Kittel">{{cite book |last=Kittel |first=Charles |title=[[Introduction to Solid State Physics]] |edition=8 |publisher=John Wiley & Sons |year=2004 |isbn=978-0471415268}}</ref><ref>Schroeder, Daniel V. "An Introduction to Thermal Physics" Addison-Wesley, San Francisco (2000). Section 7.5</ref> describe the Debye temperature as shorthand for some constants and material-dependent variables. However, <math> kT_{\rm D}</math> 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:
:<math>\frac{U}{Nk} = 9T \left({T\over T_{\rm D}}\right)^3\int_0^{T_{\rm D}/T} {x^3\over e^x-1}\, dx = 3T D_3 \left({T_{\rm D}\over T}\right)\,,</math>
where <math>D_3(x)</math> is the third [[Debye function]]. Differentiating this function with respect to <math>T</math> produces the dimensionless heat capacity:
:<math> \frac{C_V}{Nk} = 9 \left({T\over T_{\rm D}}\right)^3\int_0^{T_{\rm D}/T} {x^4 e^x\over\left(e^x-1\right)^2}\, dx\,.</math>

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]] <math>E(\nu )</math> at low frequencies, and corresponds to the exact [[Density of states#Density of wave vector states|density of states]] <math display="inline">(\int g(\nu ) \, d\nu\equiv 3N)</math> at high temperatures, concerning the number of vibrations per frequency interval.{{Original research inline|date=January 2024}}