Editing Debye model (section)

=== Polarization dependence ===
In reality, longitudinal waves often have a different wave velocity from that of transverse waves. Making the assumption that the velocities are equal simplified the final result, but reintroducing the distinction improves the accuracy of the final result.

The dispersion relation becomes <math> \omega_i = v_{s,i}|\mathbf k|</math>, with <math> i = 1, 2, 3 </math>, each corresponding to one of the three polarizations. The cut-off frequency <math> \omega_{\rm D} </math>, however, does not depend on <math>i</math>. We can write the total number of modes as <math> \sum_{i}\sum_{\rm modes} 1 </math>, which is again equal to <math>3 N</math>. Here the summation over the modes is now dependent on <math>i</math>.

==== One-dimensional chain in 3D space ====
The summation over the modes is rewritten
:<math> \sum_{i}\sum_{\rm modes} 1 = \sum_i \frac {L}{\pi v_{s,i}} \int_0^{\omega_{\rm D}} d \omega_i = 3 N .</math>

The result is
:<math> \frac {L \omega_{\rm D}}{\pi} (\frac {1}{v_{s,1}} + \frac {1}{v_{s,2}} + \frac {1}{v_{s,3}}) = 3 N .</math>

Thus the Debye frequency is found
:<math> \omega_{\rm D} = \frac{ \pi N}{L} \frac{3}{\frac {1}{v_{s,1}} + \frac {1}{v_{s,2}} + \frac {1}{v_{s,3}}} = \frac {3 \pi N}{L} \frac {v_{s,1} v_{s,2} v_{s,3}}{v_{s,2} v_{s,3} + v_{s,1} v_{s,3} + v_{s,1} v_{s,2}} = \frac{\pi N}{L} v_{\mathrm{eff}}\,. </math>

The calculated effective velocity <math> v_{\mathrm{eff}} </math> is the harmonic mean of the velocities for each polarization. By assuming the two transverse polarizations to have the same phase speed and frequency,
:<math> \omega_{\rm D} = \frac {3 \pi N}{L} \frac {v_{s,t}v_{s,l}}{2v_{s,l} + v_{s,t}} .</math>

Setting <math> v_{s,t} = v_{s,l} </math> recovers the expression previously derived under the assumption that velocity is the same for all polarization modes.

==== Two-dimensional crystal ====
The same derivation can be done for a two-dimensional crystal to find
:<math> \omega_{\rm D}^2 = \frac {4 \pi N}{A} \frac{3}{\frac {1}{v_{s,1}^2} + \frac {1}{v_{s,2}^2} + \frac {1}{v_{s,3}^2}} = \frac {12 \pi N}{A} \frac {(v_{s,1} v_{s,2} v_{s,3})^2}{(v_{s,2} v_{s,3})^2 + (v_{s,1} v_{s,3})^2 + (v_{s,1} v_{s,2})^2} = \frac{4 \pi N}{A} v_{\mathrm{eff}}^2\,.</math>

The calculated effective velocity <math> v_{\mathrm{eff}}</math> is the square root of the harmonic mean of the squares of velocities. By assuming the two transverse polarizations to be the same,
:<math> \omega_{\rm D}^2 = \frac {12 \pi N}{A} \frac {(v_{s,t} v_{s,l})^2}{2 v_{s,l}^2 + v_{s,t}^2} .</math>

Setting <math> v_{s,t} = v_{s,l} </math> recovers the expression previously derived under the assumption that velocity is the same for all polarization modes.

====Three-dimensional crystal====
The same derivation can be done for a three-dimensional crystal to find (the derivation is analogous to previous derivations)
:<math> \omega_{\rm D}^2 = \frac{6 \pi^2 N}{V} \frac{3}{\frac {1}{v_{s,1}^3} + \frac {1}{v_{s,2}^3} + \frac {1}{v_{s,3}^3}} = \frac {18 \pi^2 N}{V} \frac {(v_{s,1} v_{s,2} v_{s,3})^3}{(v_{s,2} v_{s,3})^3 + (v_{s,1} v_{s,3})^3 + (v_{s,1} v_{s,2})^3 } = \frac{6 \pi^2 N}{V} v_{\mathrm{eff}}^3\,.</math>

The calculated effective velocity <math> v_{\mathrm{eff}}</math> is the cube root of the harmonic mean of the cubes of velocities. By assuming the two transverse polarizations to be the same,
:<math> \omega_{\rm D}^3 = \frac {18 \pi^2 N}{V} \frac {(v_{s,t} v_{s,l})^3}{2 v_{s,l}^3 + v_{s,t}^3} .</math>

Setting <math> v_{s,t} = v_{s,l} </math> recovers the expression previously derived under the assumption that velocity is the same for all polarization modes.