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{{Statistical mechanics|cTopic=Models}}
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]].<ref>{{cite journal |first=Peter |last=Debye |title={{lang|de|Zur Theorie der spezifischen Waerme}} |journal=[[Annalen der Physik]] |location=Leipzig |volume=39 |issue=4 |page=789 |year=1912 |doi= }}</ref> It treats the [[oscillation|vibration]]s of the [[Crystal_structure#Classification|atomic lattice]] (heat) as [[phonon]]s in a box, in contrast to the [[Einstein solid|Einstein model]], which treats the solid as many individual, non-interacting [[quantum harmonic oscillator]]s. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to <math>T^3</math> – the '''Debye T<sup>3</sup> law'''. Just like the [[Einstein solid|Einstein model]], it also recovers the [[Dulong–Petit law]] at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

== 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 in a box|gas of photons in a box]]. The Debye model treats atomic vibrations as [[phonon]]s in a box (the box being the solid). Most of the calculation steps are identical.

Consider a cube of side <math>L</math>. 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

:<math>\lambda_n = {2L\over n}\,,</math>

where <math>n</math> is an integer. The energy of a phonon is

:<math>E_n\ =h\nu_n\,,</math>

where <math>h</math> is [[Planck's constant]] and <math>\nu_{n}</math> is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength, we have:

:<math>E_n=h\nu_n={hc_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 we will use:

:<math>E_n^2={p_n^2 c_s^2}=\left({hc_s\over2L}\right)^2\left(n_x^2+n_y^2+n_z^2\right)\,,</math>

in which <math>p_n</math> 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 [[phonon]]s.) 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,

:<math>E = \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>. 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:

:<math>U = \sum_{n_x}\sum_{n_y}\sum_{n_z}E_n\,\bar{N}(E_n)\,.</math>

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.

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

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 <math>N</math> atoms in a solid. Our solid is a cube, which means there are <math>\sqrt[3]{N}</math> atoms per edge. Atom 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</math> (infinite for [[photon]]s)

:<math>n_{\rm max} = \sqrt[3]{N}\,.</math>

This is 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>

For slowly varying, well-behaved functions, a sum can be replaced with an integral (also known as [[Thomas-Fermi approximation]])

:<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>

So far, there has been no mention of <math>\bar{N}(E)</math>, the number of phonons with energy <math>E\,.</math> Phonons obey [[Bose-Einstein statistics]]. Their distribution is given by the famous Bose-Einstein 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>

(Actually one uses an ''effective sonic velocity'' <math>c_s:=c_{{\rm eff}}</math>, i.e. the Debye temperature <math>T_d</math> (see below) is proportional to <math>c_{{\rm eff}}</math>, more precisely <math>T_D^{-3}\propto c_{{\rm eff}}^{-3}:=(1/3)c_{{\rm long}}^{-3}+(2/3)c_{{\rm trans}}^{-3}</math>, 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

:<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>

The ease with which these integrals are evaluated for [[photon]]s 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 [[phonon]]s. In order to approximate this triple integral, [[Peter Debye|Debye]] used spherical coordinates
:<math>\ (n_x,n_y,n_z)=(n\cos \theta \cos \phi,n\cos \theta \sin \phi,n\sin \theta )</math>
and boldly approximated the cube by 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, 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 <math>N</math> unit-cell volumes,

:<math>N = {1\over8}{4\over3}\pi R^3\,,</math>

so we get:

:<math>R = \sqrt[3]{6N\over\pi}\,.</math>

The substitution of integration over a sphere for the correct integral introduces another source of inaccuracy into the model.

The energy integral becomes

:<math>U = {3\pi\over2}\int_0^R \,{hc_sn\over 2L}{n^2\over e^{hc_sn/2LkT}-1} \,dn</math>

Changing the integration variable to <math>x = {hc_sn\over 2LkT}</math>,

:<math>U = {3\pi\over2} kT \left({2LkT\over hc_s}\right)^3\int_0^{hc_sR/2LkT} {x^3\over e^x-1}\, dx</math>

To simplify the appearance
of this expression, define the '''Debye temperature''' <math>T_D</math>
<blockquote style="border: 1px solid black; padding:10px;">
:<math>T_D\ \stackrel{\mathrm{def}}{=}\ {hc_sR\over2Lk} = {hc_s\over2Lk}\sqrt[3]{6N\over\pi} = {hc_s\over2k}\sqrt[3]{{6\over\pi}{N\over V}}</math>
</blockquote>

Many references<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, Calif. (2000). Section 7.5 </ref> describe the Debye temperature as merely shorthand for some constants and material-dependent variables. However, as shown below, <math> kT_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 (and hence every mode) is excited.

Continuing, we then have the specific internal energy:
<blockquote style="border: 1px solid black; padding:10px;">
:<math>\frac{U}{Nk} = 9T \left({T\over T_D}\right)^3\int_0^{T_D/T} {x^3\over e^x-1}\, dx = 3T D_3 \left({T_D\over T}\right)\,,</math>
</blockquote>

where <math>D_3(x)</math> is the (third) [[Debye function]].

Differentiating with respect to <math>T</math> we get the dimensionless heat capacity:

<blockquote style="border: 1px solid black; padding:10px;">
:<math> \frac{C_V}{Nk} = 9 \left({T\over T_D}\right)^3\int_0^{T_D/T} {x^4 e^x\over\left(e^x-1\right)^2}\, dx\,.</math>
</blockquote>

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]]''<math>E(\nu )</math> at low frequencies, and (ii) corresponds to the exact ''[[Density_of_states#Density_of_wave_vector_states | density of states]]'' <math>(\int g(\nu ) \, {\rm d\nu}\equiv 3N)\,,</math>concerning the number of vibrations per frequency interval.

== Debye's derivation ==

Actually, Debye derived his equation somewhat differently and more simply. Using the [[solid mechanics]] of a [[continuous medium]], he found that the number of vibrational states with a frequency less than a particular value was asymptotic to

:<math> n \sim {1 \over 3} \nu^3 V F\,, </math>

in which <math> V </math> is the volume and <math> F </math> is a factor which he calculated from [[Elasticity (physics)|elasticity coefficient]]s and density. Combining this with the expected energy of a harmonic oscillator at temperature T (already used by [[Einstein solid|Einstein]] in his model) would give an energy of

:<math>U = \int_0^\infty \,{h\nu^3 V F\over e^{h\nu/kT}-1}\, d\nu\,,</math>

if the vibrational frequencies continued to infinity. This form gives the <math>T^3</math> behavior which is correct at low temperatures. But Debye realized that there could not be more than <math>3N</math> 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 <math>\nu_m</math>chosen so that the total number of states is <math>3N</math>:

:<math> 3N = {1 \over 3} \nu_m^3 V F \,.</math>

Debye knew that this assumption was not really correct (the higher frequencies are more closely spaced than assumed), but it guarantees the proper behavior at high temperature (the [[Dulong–Petit law]]). The energy is then given by:

:<math>U = \int_0^{\nu_m} \,{h\nu^3 V F\over e^{h\nu/kT}-1}\, d\nu\,,</math>

::<math> = V F kT (kT/h)^3 \int_0^{T_D/T} \,{x^3 \over e^x-1}\, dx\,,</math>

::where <math>T_D</math> is <math>h\nu_m/k</math>.

::<math> = 9 N k T (T/T_D)^3 \int_0^{T_D/T} \,{x^3 \over e^x-1}\, dx\,,</math>

::<math> = 3 N k T D_3(T_D/T)\,,</math>

where <math>D_3</math> is the function later given the name of third-order [[Debye function]].

== Low temperature limit ==

The temperature of a Debye solid is said to be low if <math>T \ll T_D</math>, leading to

:<math> \frac{C_V}{Nk} \sim 9 \left({T\over T_D}\right)^3\int_0^{\infty} {x^4 e^x\over \left(e^x-1\right)^2}\, dx</math>

This definite integral can be evaluated exactly:
<blockquote style="border: 1px solid black; padding:10px;">
:<math> \frac{C_V}{Nk} \sim {12\pi^4\over5} \left({T\over T_D}\right)^3</math>
</blockquote>

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).

== High-temperature limit ==

The temperature of a Debye solid is said to be high if <math>T \gg T_D</math>. Using <math>e^x - 1\approx x</math> if <math>|x| \ll 1</math>leads to

:<math> \frac{C_V}{Nk} \sim 9 \left({T\over T_D}\right)^3\int_0^{T_D/T} {x^4 \over x^2}\, dx </math>

<blockquote style="border: 1px solid black; padding:10px;">
:<math>\frac{C_V}{Nk} \sim 3\,.</math>
</blockquote>
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 [[electrical conductor|conductor]] or [[semiconductor]], may also contain a non-negligible contribution from the electrons.

== Debye versus Einstein ==

[[Image:DebyeVSEinstein.jpg|thumb|'''Debye vs. Einstein'''. Predicted heat capacity as a function of temperature.|right|300px]]

So how closely do the Debye and Einstein models correspond to experiment? Surprisingly close, but Debye is correct at low temperatures whereas Einstein is not.

How different are the models? To answer that question one would naturally plot the two on the same set of axes... except one can't. Both the Einstein model and the Debye model provide a ''functional form'' for the heat capacity. They are ''models'', and no model is without a scale. A scale relates the model to its real-world counterpart. One can see that the scale of the Einstein model, which is given by

:<math>C_V = 3Nk\left({\epsilon\over k T}\right)^2{e^{\epsilon/kT}\over \left(e^{\epsilon/kT}-1\right)^2}</math>

is <math>\epsilon/k</math>. And the scale of the Debye model is <math>T_D</math>, 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 '''not''' the same, that is to say

:<math>{\epsilon\over k} \ne T_D\,,</math>

which means that plotting them on the same set of axes makes no sense. They are two models of the same thing, but of different scales. If one defines '''Einstein temperature''' as

:<math>T_E \ \stackrel{\mathrm{def}}{=}\ {\epsilon\over k}\,,</math>

then one can say

:<math>T_E \ne T_D\,,</math>

and, to relate the two, we must seek the ratio

:<math>\frac{T_E}{ T_D} = ?</math>

The [[Einstein solid]] is composed of '''single'''-frequency [[quantum harmonic oscillator]]s, <math>\epsilon = \hbar\omega = h\nu</math>. That frequency, if it indeed existed, would be related to the speed of sound in the solid. If one imagines the propagation of sound as a sequence of atoms ''hitting'' one another, then it becomes obvious that the frequency of oscillation must correspond to the minimum wavelength sustainable by the atomic lattice, <math>\lambda_{min}</math>.

:<math>\nu = {c_s\over\lambda} = {c_s\sqrt[3]{N}\over 2L} = {c_s\over 2}\sqrt[3]{N\over V}</math>

which makes the '''Einstein temperature'''

:<math>T_E = {\epsilon\over k} = {h\nu\over k} = {h c_s\over 2k}\sqrt[3]{N\over V}\,,</math>

and the sought ratio is therefore

:<math>{T_E\over T_D} = \sqrt[3]{\pi\over6}\ = 0.805995977...</math>

Now both models can be plotted on the same graph. Note that this ratio is the cube root of the ratio of the volume of one octant of a 3-dimensional sphere to the volume of the cube that contains it, which is just the correction factor used by Debye when approximating the energy integral above.

== Debye temperature table ==
Even though the Debye model is not completely correct, it gives a good approximation for the low temperature heat capacity of insulating, crystalline solids where other contributions (such as highly mobile conduction electrons) are negligible. For metals, the electron contribution to the heat is proportional to <math>T</math>, which at low temperatures dominates the Debye <math>T^3</math> result for lattice vibrations. In this case, the Debye model can only be said to approximate for the lattice ''contribution'' to the specific heat. The following table lists Debye temperatures for several pure elements:<ref name=Kittel/>

{|
|
{| class="wikitable"
| [[Aluminium]]
| 428 K
|-
| [[Beryllium]]
| 1440 K
|-
| [[Cadmium]]
| 209 K
|-
| [[Caesium]]
| 38 K
|-
| [[Carbon]]
| 2230 K
|-
| [[Chromium]]
| 630 K
|-
| [[Copper]]
| 343.5 K
|-
| [[Gold]]
| 170 K
|-
| [[Iron]]
| 470 K
|-
| [[Lead]]
| 105 K
|}
||
{| class="wikitable"
|-
| [[Manganese]]
| 410 K
|-
| [[Nickel]]
| 450 K
|-
| [[Platinum]]
| 240 K
|-
| [[Sapphire]]
| 1047 K
|-
| [[Silicon]]
| 645 K
|-
| [[Silver]]
| 215 K
|-
| [[Tantalum]]
| 240 K
|-
| [[Tin]] (white)
| 200 K
|-
| [[Titanium]]
| 420 K
|-
| [[Tungsten]]
| 400 K
|}
||
{| class="wikitable"
|-
| [[Zinc]]
| 327 K
|}
|
|}
The Debye model's fit to experimental data is often phenomenologically improved by allowing the Debye temperature to become temperature dependent;<ref>{{cite book |title=Solid-State Physics: Introduction to the Theory |first1=James D |last1=Patterson |first2=Bernard C. |last2=Bailey |publisher=Springer |isbn=978-3-540-34933-4 |year=2007 |pages=96–97}}</ref> for example, the value for water ice increases from about 222 K<ref>{{cite doi|10.1051/0004-6361:20031746}}</ref> to 300 K<ref>{{cite doi|10.1063/1.1731497}}</ref> as the temperature goes from [[Absolute zero]] to about 100 K.

== Extension to other quasi-particles ==
For other [[boson]]ic [[quasi-particle]]s, e.g., for [[magnon]]s (quantized spin waves) in ferromagnets instead of the [[phonons]] (quantized sound waves) one easily derives analogous results. In this case at low frequencies one has different [[dispersion relation]]s, e.g., <math>E(\nu )\propto k^2</math> in the case of magnons, instead of <math>E(\nu )\propto k</math> for phonons (with <math>k=2\pi /\lambda </math>). One also has different [[Density_of_states#Density_of_wave_vector_states|density of states]] (e.g., <math>\int g(\nu ){\rm d}\nu \equiv N\,</math>). As a consequence, in ferromagnets one gets a magnon contribution
to the heat capacity, <math>\Delta C_{\,{\rm V|\,magnon}}\,\propto T^{3/2}</math>, which dominates at sufficiently low temperatures the phonon contribution, <math>\,\Delta C_{\,{\rm V|\,phonon}}\propto T^3</math>. In metals, in contrast, the main low-temperature contribution to the heat capacity, <math>\propto T</math>, comes from the electrons. It is [[fermion]]ic, and is calculated by different methods going back to [[Arnold Sommerfeld]].

== See also ==

* [[Bose gas]]
* [[Debye frequency]]
* [[Gas in a box]]
* [[Kinetic theory of solids]]
* [[Grüneisen parameter]]

== References ==

<references />

== Further readings ==

*''CRC Handbook of Chemistry and Physics'', 56th Edition (1975–1976)
*Schroeder, Daniel V. ''An Introduction to Thermal Physics''. Addison-Wesley, San Francisco, Calif. (2000). Section 7.5.

== External links ==
* [http://www.thphys.de/F14_Debye_Experiment.html Experimental determination of specific heat, thermal and heat conductivity of quartz using a cryostat.]

{{DEFAULTSORT:Debye Model}}
[[Category:Condensed matter physics]]
[[Category:Thermodynamic models]]
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en>Mark viking: Changed a wl, added a wl