Editing Debye model (section)

=== Definition ===
Assuming the [[dispersion relation]] is
:<math> \omega = v_{\rm s} |\mathbf k| ,</math>
with <math>v_{\rm s}</math> the [[speed of sound]] in the crystal and '''k''' the wave vector, the value of the Debye frequency is as follows:

For a one-dimensional monatomic chain, the Debye frequency is equal to<ref>{{Cite web| url=https://openphysicslums.files.wordpress.com/2012/08/latticevibrations.pdf| title=The one dimensional monatomic solid| access-date=2018-04-27}}</ref>
:<math> \omega_{\rm D} = v_{\rm s} \pi / a = v_{\rm s} \pi N / L = v_{\rm s} \pi \lambda ,</math>
with <math>a</math> as the distance between two neighbouring atoms in the chain when the system is in its [[ground state]] of energy, here being that none of the atoms are moving with respect to one another; <math>N</math> the total number of atoms in the chain; <math>L</math> the size of the system, which is the length of the chain; and <math> \lambda </math> the [[linear density|linear number density]]. For <math>L</math>, <math>N</math>, and <math>a</math>, the relation <math>L = N a</math> holds.

For a two-dimensional monatomic square lattice, the Debye frequency is equal to
:<math> \omega_{\rm D}^2 = \frac {4 \pi}{a^2} v_{\rm s}^2 = \frac {4 \pi N}{A} v_{\rm s}^2 \equiv 4 \pi \sigma v_{\rm s}^2 ,</math>
with <math> A \equiv L^{2} = N a^{2} </math> is the size (area) of the surface, and <math>\sigma</math> the [[surface density|surface number density]].

For a three-dimensional monatomic [[cubic crystal system|primitive cubic crystal]], the Debye frequency is equal to<ref>{{Cite web|url=http://farside.ph.utexas.edu/teaching/sm1/lectures/node71.html|title=Specific heats of solids| last=Fitzpatrick|first=Richard|date=2006|website=Richard Fitzpatrick [[University of Texas at Austin]]|access-date=2018-04-27}}</ref>
:<math> \omega_{\rm D}^3 = \frac {6 \pi^2}{a^3} v_{\rm s}^3 = \frac {6 \pi^2 N}{V} v_{\rm s}^3 \equiv 6 \pi^2 \rho v_{\rm s}^3 ,</math>
with <math> V \equiv L^3 = N a^3 </math> the size of the system, and <math>\rho</math> the [[density|volume number density]].

The general formula for the Debye frequency as a function of <math>n</math>, the number of dimensions for a (hyper)cubic lattice is
:<math> \omega_{\rm D}^n = 2^n \pi^{n/2} \Gamma\left(1+\tfrac{n}{2}\right) \frac {N}{L^n} v_{\rm s}^n ,</math>
with <math>\Gamma</math> being the [[gamma function]].

The speed of sound in the crystal depends on the mass of the atoms, the strength of their interaction, the [[pressure]] on the system, and the [[polarization (waves)|polarisation]] of the spin wave (longitudinal or transverse), among others. For the following, the speed of sound is assumed to be the same for any polarisation, although this limits the applicability of the result.<ref name=":0">{{Cite book|title=The Oxford Solid State Basics|last=Simon|first=Steven H.| publisher=Oxford University Press|isbn=9780199680764|edition= First|location=Oxford|oclc=859577633|author-link=Steven H. Simon | date = 2013-06-20}}</ref>

The assumed [[dispersion relation]] is easily proven inaccurate for a one-dimensional chain of masses, but in Debye's model, this does not prove to be problematic.{{Citation needed|date=January 2024}}