Editing Debye model (section)

== Another derivation ==
First the [[vibrational frequency]] distribution is derived from Appendix VI of Terrell L. Hill's ''An Introduction to Statistical Mechanics''.<ref>{{cite book|last1=Hill|first1=Terrell L.|title=An Introduction to Statistical Mechanics|date=1960|publisher=Addison-Wesley Publishing Company, Inc.|location=Reading, Massachusetts, U.S.A.|isbn=9780486652429|url-access=registration|url=https://archive.org/details/introductiontost0000hill}}</ref> Consider a [[Three-dimensional space|three-dimensional]] [[Isotropy|isotropic]] [[elastic solid]] with N atoms in the shape of a [[rectangular parallelepiped]] with side-lengths <math>L_x, L_y, L_z</math>. The [[Elastic Wave|elastic wave]] will obey the [[wave equation]] and will be [[plane waves]]; consider the [[wave vector]] <math>\mathbf{k} = (k_x, k_y, k_z)</math> and define <math>l_x=\frac{k_x}{|\mathbf{k}|}, l_y=\frac{k_y}{|\mathbf{k}|}, l_z=\frac{k_z}{|\mathbf{k}|}</math>, such that
{{NumBlk|:|<math> l_x^2 + l_y^2 + l_z^2 = 1.</math>|{{EquationRef|1}}}}

Solutions to the [[wave equation]] are
:<math> u(x,y,z,t) = \sin(2\pi\nu t)\sin\left(\frac{2\pi l_x x}{\lambda}\right)\sin\left(\frac{2\pi l_y y}{\lambda}\right)\sin\left(\frac{2\pi l_z z}{\lambda}\right) </math>
and with the [[boundary conditions]] <math>u=0</math> at <math>x,y,z=0, x=L_x, y=L_y, z=L_z</math>,
{{NumBlk|:|<math> \frac{2l_xL_x}{\lambda}=n_x; \frac{2l_yL_y}{\lambda}=n_y; \frac{2l_zL_z}{\lambda}=n_z </math>|{{EquationRef|2}}}}
where <math>n_x,n_y,n_z</math> are [[positive integers]]. Substituting ({{EquationNote|2}}) into ({{EquationNote|1}}) and also using the [[dispersion relation]] <math>c_s=\lambda\nu</math>,
:<math> \frac{n_x^2}{(2\nu L_x/c_s)^2} + \frac{n_y^2}{(2\nu L_y/c_s)^2} + \frac{n_z^2}{(2\nu L_z/c_s)^2} = 1. </math>

The above equation, for fixed [[frequency]] <math>\nu</math>, describes an eighth of an ellipse in "mode space" (an eighth because <math>n_x,n_y,n_z</math> are positive). The number of modes with frequency less than <math>\nu</math> is thus the number of integral points inside the ellipse, which, in the limit of <math>L_x,L_y,L_z \to\infty</math> (i.e. for a very large parallelepiped) can be approximated to the volume of the ellipse. Hence, the number of modes <math>N(\nu)</math> with frequency in the range <math>[0,\nu]</math> is
{{NumBlk|:|<math> N(\nu) = \frac{1}{8}\frac{4\pi}{3}\left(\frac{2\nu}{c_{\mathrm{s}}}\right)^3L_xL_yL_z = \frac{4\pi\nu^3V}{3c_{\mathrm{s}}^3},</math>|{{EquationRef|3}}}}
where <math>V=L_xL_yL_z</math> is the volume of the parallelepiped. The wave speed in the longitudinal direction is different from the transverse direction and that the waves can be polarised one way in the longitudinal direction and two ways in the transverse direction and ca be defined as <math>\frac{3}{c_s^3} = \frac{1}{c_\text{long}^3} + \frac{2}{c_\text{trans}^3}</math>.

Following the derivation from ''A First Course in Thermodynamics'',<ref>{{cite book|last1=Oberai|first1=M. M.|last2=Srikantiah|first2=G|title=A First Course in Thermodynamics | date=1974|publisher=Prentice-Hall of India Private Limited|location=New Delhi, India|isbn=9780876920183}}</ref> an upper limit to the frequency of vibration is defined <math>\nu_D</math>; since there are <math>N</math> atoms in the solid, there are <math>3N</math> quantum harmonic oscillators (3 for each x-, y-, z- direction) oscillating over the range of frequencies <math>[0,\nu_D]</math>. <math> \nu_D </math> can be determined using
{{NumBlk|:|<math> 3N = N(\nu_{\rm D}) = \frac{4\pi\nu_{\rm D}^3V}{3c_{\rm s}^3} </math>.|{{EquationRef|4}}}}
By defining <math>\nu_{\rm D} = \frac{kT_{\rm D}}{h}</math>, where ''k'' is the [[Boltzmann constant]] and ''h'' is the [[Planck constant]], and substituting ({{EquationNote|4}}) into ({{EquationNote|3}}),
{{NumBlk|:|<math> N(\nu) = \frac{3Nh^3\nu^3}{k^3T_{\rm D}^3},</math>|{{EquationRef|5}}}}this definition is more standard; the energy contribution for all [[Oscillation|oscillators]] oscillating at [[frequency]] <math>\nu</math> can be found. [[Quantum harmonic oscillator]]s can have energies <math>E_i = (i+1/2)h\nu</math> where <math>i = 0,1,2,\dotsc</math> and using [[Maxwell-Boltzmann statistics]], the number of particles with energy <math>E_i</math> is
:<math>n_i=\frac{1}{A}e^{-E_i/(kT)}=\frac{1}{A}e^{-(i+1/2)h\nu/(kT)}.</math>

The energy contribution for [[Oscillation|oscillators]] with frequency <math>\nu</math> is then
{{NumBlk|:|<math> dU(\nu) = \sum_{i=0}^\infty E_i\frac{1}{A}e^{-E_i/(kT)}</math>.|{{EquationRef|6}}}}

By noting that <math>\sum_{i=0}^\infty n_i = dN(\nu)</math> (because there are <math>dN(\nu)</math> modes oscillating with [[frequency]] <math>\nu</math>),
:<math>\frac{1}{A}e^{-1/2h\nu/(kT)}\sum_{i=0}^\infty e^{-ih\nu/(kT)} = \frac{1}{A}e^{-1/2h\nu/(kT)}\frac{1}{1-e^{-h\nu/(kT)}} = dN(\nu) .</math>

From above, we can get an expression for 1/A; substituting it into ({{EquationNote|6}}),
:<math>\begin{align}
dU &= dN(\nu)e^{1/2h\nu/(kT)}(1-e^{-h\nu/(kT)})\sum_{i=0}^\infty h\nu(i+1/2)e^{-h\nu(i+1/2)/(kT)} \\ \\
&=dN(\nu)(1-e^{-h\nu/(kT)})\sum_{i=0}^\infty h\nu(i+1/2)e^{-h\nu i/(kT)} \\
&=dN(\nu)h\nu\left(\frac{1}{2}+(1-e^{-h\nu/(kT)})\sum_{i=0}^\infty ie^{-h\nu i/(kT)}\right) \\
&=dN(\nu)h\nu\left(\frac{1}{2}+\frac{1}{e^{h\nu/(kT)}-1}\right).
\end{align}</math>

Integrating with respect to ν yields
:<math>U = \frac{9Nh^4}{k^3T_{\rm D}^3}\int_0^{\nu_D}\left(\frac{1}{2}+\frac{1}{e^{h\nu/(kT)}-1}\right)\nu^3 d\nu.</math>