Editing Debye model (section)

== Debye versus Einstein ==

[[Image:DebyeVSEinstein.jpg|thumb|upright=1.5|'''Debye vs. Einstein'''. Predicted heat capacity as a function of temperature.|right|300px]]

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

The scale of the Debye model is <math>T_{\rm 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 {{Em|not}} the same, that is to say
:<math>{\epsilon\over k} \ne T_{\rm 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 the Einstein condensation temperature as
:<math>T_{\rm E} \ \stackrel{\mathrm{def}}{=}\  {\epsilon\over k}\,,</math>
then one can say
:<math>T_{\rm E} \ne T_{\rm D}\,,</math>
and, to relate the two, the ratio<math>\frac{T_{\rm E}}{ T_{\rm D}} \, </math> is used.

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 the frequency of oscillation must correspond to the minimum wavelength sustainable by the atomic lattice, <math>\lambda_{min}</math>, where
:<math>\nu = {c_{\rm s}\over\lambda} = {c_{\rm s}\sqrt[3]{N}\over 2L} = {c_{\rm s}\over 2}\sqrt[3]{N\over V}</math>,
which makes the Einstein temperature <math>T_{\rm E} = {\epsilon\over k} = {h\nu\over k} = {h c_{\rm s}\over 2k}\sqrt[3]{N\over V}\,,</math> and the sought ratio is therefore
:<math>{T_{\rm E}\over T_{\rm D}} = \sqrt[3]{\pi\over6}\ = 0.805995977...</math>

Using the ratio, both models can be plotted on the same graph. It is the [[cube root]] of the ratio of the volume of one [[Octant (solid geometry)|octant]] of a three-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. Alternatively, the ratio of the two temperatures can be seen to be the ratio of Einstein's single frequency at which all oscillators oscillate and Debye's maximum frequency. Einstein's single frequency can then be seen to be a mean of the frequencies available to the Debye model.