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 L {\displaystyle L} , 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
λ n = 2 L n , {\displaystyle \lambda _{n}={2L \over n}\,,}
where n {\displaystyle n} is an integer. The energy of a phonon is given as
E n = h ν n , {\displaystyle E_{n}\ =h\nu _{n}\,,}
where h {\displaystyle h} is the Planck constant and ν n {\displaystyle \nu _{n}} is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength,
E n = h ν n = h c s λ n = h c s n 2 L , {\displaystyle E_{n}=h\nu _{n}={hc_{\rm {s}} \over \lambda _{n}}={hc_{s}n \over 2L}\,,}
in which c s {\displaystyle c_{s}} is the speed of sound inside the solid. In three dimensions, energy can be generalized to
E n 2 = p n 2 c s 2 = ( h c s 2 L ) 2 ( n x 2 + n y 2 + n z 2 ) , {\displaystyle E_{n}^{2}={p_{n}^{2}c_{\rm {s}}^{2}}=\left({hc_{\rm {s}} \over 2L}\right)^{2}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)\,,}
in which p n {\displaystyle p_{n}} is the magnitude of the three-dimensional momentum of the phonon, and n x {\displaystyle n_{x}} , n y {\displaystyle n_{y}} , and n z {\displaystyle n_{z}} 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, U {\displaystyle U} , is given by
U = ∑ n E n N ¯ ( E n ) , {\displaystyle U=\sum _{n}E_{n}\,{\bar {N}}(E_{n})\,,}
where N ¯ ( E n ) {\displaystyle {\bar {N}}(E_{n})} is the number of phonons in the box with energy E n {\displaystyle E_{n}} ; 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:
U = ∑ n x ∑ n y ∑ n z E n N ¯ ( E n ) . {\displaystyle U=\sum _{n_{x}}\sum _{n_{y}}\sum _{n_{z}}E_{n}\,{\bar {N}}(E_{n})\,.}
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 N {\displaystyle N} atoms in a cubic solid, each axis of the cube measures as being N 3 {\displaystyle {\sqrt[{3}]{N}}} atoms long. Atomic separation is then given by L / N 3 {\displaystyle L/{\sqrt[{3}]{N}}} , and the minimum wavelength is
λ m i n = 2 L N 3 , {\displaystyle \lambda _{\rm {min}}={2L \over {\sqrt[{3}]{N}}}\,,}
making the maximum mode number n m a x {\displaystyle n_{max}} :
n m a x = N 3 . {\displaystyle n_{\rm {max}}={\sqrt[{3}]{N}}\,.}
This contrasts with photons, for which the maximum mode number is infinite. This number bounds the upper limit of the triple energy sum
U = ∑ n x N 3 ∑ n y N 3 ∑ n z N 3 E n N ¯ ( E n ) . {\displaystyle 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})\,.}
If E n {\displaystyle E_{n}} is a function that is slowly varying with respect to n {\displaystyle n} , the sums can be approximated with integrals: U ≈ ∫ 0 N 3 ∫ 0 N 3 ∫ 0 N 3 E ( n ) N ¯ ( E ( n ) ) d n x d n y d n z . {\displaystyle 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}\,.}
To evaluate this integral, the function N ¯ ( E ) {\displaystyle {\bar {N}}(E)} , the number of phonons with energy E , {\displaystyle E\,,} must also be known. Phonons obey Bose–Einstein statistics, and their distribution is given by the Bose–Einstein statistics formula:
⟨ N ⟩ B E = 1 e E / k T − 1 . {\displaystyle \langle N\rangle _{BE}={1 \over e^{E/kT}-1}\,.}
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,
N ¯ ( E ) = 3 e E / k T − 1 . {\displaystyle {\bar {N}}(E)={3 \over e^{E/kT}-1}\,.}
Considering all three polarization states together also means that an effective sonic velocity c e f f {\displaystyle c_{\rm {eff}}} must be determined and used as the value of the standard sonic velocity c s . {\displaystyle c_{s}.} The Debye temperature T D {\displaystyle T_{\rm {D}}} defined below is proportional to c e f f {\displaystyle c_{\rm {eff}}} ; more precisely, T D − 3 ∝ c e f f − 3 := 1 3 c l o n g − 3 + 2 3 c t r a n s − 3 {\displaystyle T_{\rm {D}}^{-3}\propto c_{\rm {eff}}^{-3}:={\frac {1}{3}}c_{\rm {long}}^{-3}+{\frac {2}{3}}c_{\rm {trans}}^{-3}} , 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 N ¯ ( E ) {\displaystyle {\bar {N}}(E)} into the energy integral yields
U = ∫ 0 N 3 ∫ 0 N 3 ∫ 0 N 3 E ( n ) 3 e E ( n ) / k T − 1 d n x d n y d n z . {\displaystyle 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}\,.}
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,
( n x , n y , n z ) = ( n sin θ cos ϕ , n sin θ sin ϕ , n cos θ ) , {\displaystyle \ (n_{x},n_{y},n_{z})=(n\sin \theta \cos \phi ,n\sin \theta \sin \phi ,n\cos \theta )\,,}
and approximated the cube with an eighth of a sphere,
U ≈ ∫ 0 π / 2 ∫ 0 π / 2 ∫ 0 R E ( n ) 3 e E ( n ) / k T − 1 n 2 sin θ d n d θ d ϕ , {\displaystyle 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 \,,}
where R {\displaystyle R} is the radius of this sphere. As the energy function does not depend on either of the angles, the equation can be simplified to
3 ∫ 0 π / 2 ∫ 0 π / 2 sin θ d θ d ϕ ∫ 0 R E ( n ) 1 e E ( n ) / k T − 1 n 2 d n = 3 π 2 ∫ 0 R E ( n ) 1 e E ( n ) / k T − 1 n 2 d n {\displaystyle \,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\,}
The number of particles in the original cube and in the eighth of a sphere should be equivalent. The volume of the cube is N {\displaystyle N} unit cell volumes,
N = 1 8 4 3 π R 3 , {\displaystyle N={1 \over 8}{4 \over 3}\pi R^{3}\,,}
such that the radius must be
R = 6 N π 3 . {\displaystyle R={\sqrt[{3}]{6N \over \pi }}\,.}
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 E ( n ) {\displaystyle E(n)\,} , the energy integral becomes
U = 3 π 2 ∫ 0 R h c s n 2 L n 2 e h c s n / 2 L k T − 1 d n {\displaystyle U={3\pi \over 2}\int _{0}^{R}\,{hc_{s}n \over 2L}{n^{2} \over e^{hc_{\rm {s}}n/2LkT}-1}\,dn} . Changing the integration variable to x = h c s n 2 L k T {\displaystyle x={hc_{\rm {s}}n \over 2LkT}} ,
U = 3 π 2 k T ( 2 L k T h c s ) 3 ∫ 0 h c s R / 2 L k T x 3 e x − 1 d x . {\displaystyle U={3\pi \over 2}kT\left({2LkT \over hc_{\rm {s}}}\right)^{3}\int _{0}^{hc_{\rm {s}}R/2LkT}{x^{3} \over e^{x}-1}\,dx.}
To simplify the appearance of this expression, define the Debye temperature T D {\displaystyle T_{\rm {D}}}
T D = d e f h c s R 2 L k = h c s 2 L k 6 N π 3 = h c s 2 k 6 π N V 3 {\displaystyle T_{\rm {D}}\ {\stackrel {\mathrm {def} }{=}}\ {hc_{\rm {s}}R \over 2Lk}={hc_{\rm {s}} \over 2Lk}{\sqrt[{3}]{6N \over \pi }}={hc_{\rm {s}} \over 2k}{\sqrt[{3}]{{6 \over \pi }{N \over V}}}}
where V {\displaystyle V} is the volume of the cubic box of side-length L {\displaystyle L} . Some authors describe the Debye temperature as shorthand for some constants and material-dependent variables. However, k T D {\displaystyle kT_{\rm {D}}} 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:
U N k = 9 T ( T T D ) 3 ∫ 0 T D / T x 3 e x − 1 d x = 3 T D 3 ( T D T ) , {\displaystyle {\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=3TD_{3}\left({T_{\rm {D}} \over T}\right)\,,}
where D 3 ( x ) {\displaystyle D_{3}(x)} is the third Debye function. Differentiating this function with respect to T {\displaystyle T} produces the dimensionless heat capacity:
C V N k = 9 ( T T D ) 3 ∫ 0 T D / T x 4 e x ( e x − 1 ) 2 d x . {\displaystyle {\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\,.}
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 E ( ν ) {\displaystyle E(\nu )} at low frequencies, and corresponds to the exact density of states ( ∫ g ( ν ) d ν ≡ 3 N ) {\textstyle (\int g(\nu )\,d\nu \equiv 3N)} 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
n ∼ 1 3 ν 3 V F , {\displaystyle n\sim {1 \over 3}\nu ^{3}VF\,,}
in which V {\displaystyle V} is the volume and F {\displaystyle F} is a factor that he calculated from elasticity coefficients and density. Combining this formula with the expected energy of a harmonic oscillator at temperature T {\displaystyle T} (already used by Einstein in his model) would give an energy of
U = ∫ 0 ∞ h ν 3 V F e h ν / k T − 1 d ν , {\displaystyle U=\int _{0}^{\infty }\,{h\nu ^{3}VF \over e^{h\nu /kT}-1}\,d\nu \,,}
if the vibrational frequencies continued to infinity. This form gives the T 3 {\displaystyle T^{3}} behaviour which is correct at low temperatures. But Debye realized that there could not be more than 3 N {\displaystyle 3N} 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 ν m {\displaystyle \nu _{m}} chosen so that the total number of states is
3 N = 1 3 ν m 3 V F . {\displaystyle 3N={1 \over 3}\nu _{m}^{3}VF\,.}
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
U = ∫ 0 ν m h ν 3 V F e h ν / k T − 1 d ν , = V F k T ( k T /