Editing Debye model (section)

=== Derivation with the actual dispersion relation ===
[[File:phonon k 3k.gif|right|thumb|250px|Because only the [[Discretization|discretized]] points matter, two different waves could render the same physical manifestation (see [[Phonon]]).]]
This problem could be made more applicable by relaxing the assumption of linearity of the dispersion relation. Instead of using the dispersion relation <math> \omega = v_{\rm s} k </math>, a more accurate dispersion relation can be used. In classical mechanics, it is known that for an equidistant chain of masses which interact harmonically with each other, the dispersion relation is<ref name=":0" />

<math display="block"> \omega (k) = 2 \sqrt {\frac {\kappa}{m}}\left|\sin\left(\frac {k a}{2}\right)\right| ,</math>

with <math> m</math> being the mass of each atom, <math> \kappa</math> the spring constant for the [[harmonic oscillator]], and <math> a</math> still being the spacing between atoms in the ground state. After plotting this relation, Debye's estimation of the cut-off wavelength based on the linear assumption remains accurate, because for every wavenumber bigger than <math>\pi / a </math> (that is, for <math> \lambda </math> is smaller than <math>2 a</math>), a wavenumber that is smaller than <math>\pi / a</math> could be found with the same angular frequency. This means the resulting physical manifestation for the mode with the larger wavenumber is indistinguishable from the one with the smaller wavenumber. Therefore, the study of the dispersion relation can be limited to the first [[Brillouin zone]] <math display="inline"> k \in \left[-\frac{\pi}{a},\frac{\pi}{a}\right] </math> without any loss of accuracy or information.<ref>{{Cite book|last=Srivastava|first=G. P. |url=https://books.google.com/books?id=XiCmDwAAQBAJ&q=brillouin+zone+debye+model+book&pg=PA44|title=The Physics of Phonons |date=2019-07-16|publisher=Routledge|isbn=978-1-351-40955-1|language=en}}</ref> This is possible because the system consists of [[Discretization|discretized]] points, as is demonstrated in the animated picture. Dividing the dispersion relation by <math>k</math> and inserting <math>\pi / a</math> for <math>k</math>, we find the speed of a wave with <math>k = \pi / a</math> to be
<math display="block"> v_{\rm s}(k = \pi / a) = \frac {2 a}{\pi} \sqrt {\frac {\kappa}{m}} .</math>

By simply inserting <math>k = \pi/a</math> in the original dispersion relation we find
<math display="block"> \omega(k = \pi / a) = 2 \sqrt {\frac {\kappa}{m}} = \omega_{\rm D} .</math>

Combining these results the same result is once again found
<math display="block"> \omega_{\rm D} = \frac {\pi v_{\rm s}}{a} .</math>

However, for any chain with greater complexity, including diatomic chains, the associated cut-off frequency and wavelength are not very accurate, since the cut-off wavelength is twice as big and the dispersion relation consists of additional branches, two total for a diatomic chain. It is also not certain from this result whether for higher-dimensional systems the cut-off frequency was accurately predicted by Debye when taking into account the more accurate dispersion relation.