Editing Debye model (section)

== 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
:<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 that he calculated from [[Elasticity (physics)|elasticity coefficient]]s and density. Combining this formula with the expected energy of a harmonic oscillator at temperature <math>T</math> (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> behaviour 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 [[Frequency|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 = {1 \over 3} \nu_m^3 V F \,.</math>

Debye knew that this assumption was not really correct (the higher [[Frequency|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
:<math>\begin{align}
U &= \int_0^{\nu_m} \,{h\nu^3 V F\over e^{h\nu/kT}-1}\, d\nu\,,\\
&= V F kT (kT/h)^3 \int_0^{T_{\rm D}/T} \,{x^3 \over e^x-1}\, dx\,.
\end{align}</math>
Substituting <math>T_{\rm D}</math> for <math>h\nu_m/k</math>,
:<math>\begin{align}
U &= 9 N k T (T/T_{\rm D})^3 \int_0^{T_{\rm D}/T} \,{x^3 \over e^x-1}\, dx\,, \\
&= 3 N k T D_3(T_{\rm D}/T)\,,
\end{align}</math>
where <math>D_3</math> is the function later given the name of third-order [[Debye function]].