Debye model

in solid-state physics describes the Debye model (after Peter Debye) a method, in order to compute the contribution of the lattice vibrations (phonons) for the thermal capacity of a crystalline solid body.


the Debye model

Dispersionsrelation im Vergleich mit dem Ergebnis von einfachen harmonischen Oszillatoren
the existence of a Dispersionsrelation considers bases of the model Dispersionsrelation in the comparison

with the result of simple harmonious oscillators in relation to the Einstein model, which N accepts independent oscillators with same frequency , i.e. a multiplicity of possible frequencies in dependence of the wave vector.

The phonon dispersion is accepted as linear, up to an upper critical frequency (Debyefrequenz). The upper critical frequency results, if one adds all fashions, since the total number of the fashions must agree with the number of atoms, i.e. that the critical frequency in the Debye model is lower than from a harmonious oscillator beginning (see picture).

results of the model

this model meets correct forecasts over <math> the T^3< /math> - Dependence of the thermal capacity in the low temperature limit. For <math> T \ ll \ Theta_D< /math> (the Debye temperature) C = {applies for 12 r R \ pi^4 \ more over 5

<}> {T^3 \ more over \ Theta_D^3} /math to the phonon portion of the molecular thermal capacity with r atoms per elementary cell< math>

Like the Einstein model the Debye model for math <T> supplies > \ Theta_D< /math> the correct high temperature limit after Dulong Petit. The Debye model brings the Dispersionsrelation of phonons linear <math> \ to omega closer = v_ {sound} * k< /math>, whereby additionally transversal and longitudinal speed were accepted as same. Thus the phonon density becomes

<more over> math g = {DN \ over D \ omega} = {3 \ omega^2 \ {2 \ pi^2 v_ {S} ^3}} </math>

There this density for high <math> \ omega< /math> diverged, the density must with a certain (material-dependent) frequency <math> \ omega_ {max}< /math> cut off become. This critical frequency is often indicated to same suppl. tables as temperature of a thermal suggestion.

Dependence

<math> C= \ partial_T U< /math> with <math> U = 3 \ int_0^ {\ omega_ {max}} {g (\ omega) \ hbar \ omega \ more over e^ {\ hbar \ omega \ more over k_B T} - 1} D \ omega< /math> and <math> \ omega_ {max} = {\ Theta_D k_B \ more over \ hbar}< /math>

is however not generally analytic, but only numeric or for parts of the thermometric scale approached solvable, like above for small temperatures.

Debye temperatures of different materials

<math> \ Theta_D< /math> in K
lead 95
sodium 160
gold 165
silver 225
copper 345
aluminum 428
< math> \ alpha< /math> - iron 464
chrome 610
diamond 1850
 

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