Debye model

The Debye model describes a method with which the contribution of the quantized vibrations in crystal lattices , the phonons , to the heat capacity of a crystalline solid can be calculated; it turns u. a. found out that this is usually the main contribution.

This “theory of the specific heat of crystals”, developed by Peter Debye in 1911 and 1912, is considered to be one of the first theoretical confirmations of the quantum thesis presented by Max Planck in 1900 .

Basics

Dispersion relation compared with the result of simple harmonic oscillators

Compared to the Einstein model of 1906, which assumes independent oscillators with identical frequencies , the Debye model assumes a large number of possible frequencies and a non-zero propagation speed of all waves or phonons. ${\displaystyle N}$

However, the long-wave approximation is assumed throughout; H. For the sake of simplicity, it is assumed that the angular frequency and the wave vector below a material- specific cut-off frequency, the Debye frequency , are always strictly proportional to each other ( i.e. a linear dispersion relation applies). One longitudinal and two transversal sound waves - degrees of freedom are assumed. ${\displaystyle \omega }$ ${\displaystyle {\vec {k}}}$ ${\displaystyle \omega _{D}}$

What is remarkable about this approach is that (apart from the non-existence of longitudinal light waves ) it is identical to Planck's assumptions for calculating the cavity radiation if the speed of sound is replaced by the speed of light . This results in formulas for a radiating cavity (→ Planck's law of radiation ) with the same structure as for a heated solid in which particles vibrate in a grid-like arrangement. In both cases, characteristic “ T ³ laws” follow (see below).

Phonons only exist up to a maximum frequency (in the Debye model up to ). This results from the sum of all possible vibration modes , since their total number can be at most three times the number of vibrating lattice particles ( atoms ). It also follows that in principle there is something less than the maximum frequency of a corresponding harmonic oscillator (see picture) without frequency limitation: . ${\displaystyle \omega _{D}}$ ${\displaystyle \omega _{D}}$ ${\displaystyle \omega _{D}\lessapprox \omega _{\rm {max}}}$

Results

Debye temperatures of various materials

	${\displaystyle \Theta _{\mathrm {D} }}$ in K
diamond	1850
chrome	610
α-iron	464
aluminum	428
copper	345
silver	215
gold	165
sodium	160
lead	95

The Debye model correctly predicts the temperature dependence of the heat capacity in both the low and high temperature limits .

The intermediate behavior, i.e. H. the mean temperature range is only described by the Debye theory in terms of a “reasonable interpolation ”, which can be improved if necessary (see below). ${\displaystyle T\approx \Theta _{\mathrm {D} }}$

Low temperature range

In the low temperature range, i.e. H. for ( is the Debye temperature ), the following applies to the phonon part of the heat capacity: ${\displaystyle T\ll \Theta _{\mathrm {D} }}$ ${\displaystyle \Theta _{\mathrm {D} }}$

${\displaystyle C_{V}={\frac {12\,\pi ^{4}}{5}}\,N\,k_{\mathrm {B} }\cdot \underbrace {\left({\frac {T}{\Theta _{\mathrm {D} }}}\right)^{3}} _{\ll 1}}$

With

• the number of atoms in the crystal ${\displaystyle N}$
• the Boltzmann constant ${\displaystyle k_{\mathrm {B} }}$
• ${\displaystyle \Theta _{D}={\frac {\hbar \,\omega _{D}}{k_{\mathrm {B} }}}}$
  • the Debye (circle) frequency ${\displaystyle \omega _{D}}$
  • the reduced Planckian quantum of action . ${\displaystyle \hbar }$

The Debye temperature is proportional to an effective speed of sound , to which the transverse sound waves contribute 2/3 and the longitudinal sound waves 1/3: ${\displaystyle c_{\rm {eff}}}$

${\displaystyle {\frac {1}{\Theta _{\mathrm {D} }^{3}}}\propto {\frac {1}{c_{\rm {eff}}^{3}}}:={\frac {1}{3}}\left({\frac {2}{c_{t}^{3}}}+{\frac {1}{c_{l}^{3}}}\right).}$

The low temperature behavior is correct because in the Limes the Debye approximation agrees with the exact one (see below). ${\displaystyle \omega \ll \omega _{D}}$ ${\displaystyle g(\omega )}$

High temperature range

In the high temperature range, i. H. for , applies to the internal energy and thus to the heat capacity ${\displaystyle T\gg \Theta _{D}}$ ${\displaystyle U=3\,N\,k_{\mathrm {B} }\,T}$

${\displaystyle C_{V}=3\,N\,k_{\mathrm {B} }\,}$.

In this Limes, as with the Einstein model, the law of Dulong-Petit results .

The high temperature behavior is correct because the Debye approximation per constructionem also includes the sum rule

${\displaystyle \int _{0}^{\omega _{\rm {max}}}g(\omega )\,{\rm {d}}\omega \equiv 3N}$

Fulfills.

Density of states

The density of states results from the Debye model:

${\displaystyle g(\omega )\,d\omega =g(k)\,dk}$.

${\displaystyle \Leftrightarrow g(\omega )=g(k)\cdot {\frac {d(k)}{d(\omega )}}}$

with the circular wavenumber . ${\displaystyle k}$

But now it generally holds in k-space : ${\displaystyle g(k)={\frac {L}{\pi }}}$

and according to the Debye model :, well ${\displaystyle \omega =v_{s}\cdot k}$ ${\displaystyle {\frac {d(k)}{d(\omega )}}={\frac {1}{v_{s}}}}$

and thus overall:

${\displaystyle g(\omega )={\frac {L}{\pi }}\cdot {\frac {1}{v_{s}}}.}$

Reason

The Debye model approximates the dispersion relation of phonons linearly in the given way. The calculation, which can also be carried out elementarily for the (realistic!) Case that the longitudinal and transversal speed of sound differ considerably, takes a long time, so that details are omitted here only for reasons of space.

Since there can be at most three times as many oscillation modes as atoms in a solid , but the density of states for high ones diverges, the density must be cut off at a certain material-dependent frequency (in the Debye approximation at ). ${\displaystyle \omega }$ ${\displaystyle \omega _{\rm {max}}}$ ${\displaystyle \omega _{\rm {max}}=\omega _{D}}$

Based on the exact formula for the vibration energy :

${\displaystyle U=\int _{0}^{\omega _{\rm {max}}}{\frac {g(\omega )\,\hbar \,\omega }{e^{\frac {\hbar \,\omega }{k_{\mathrm {B} }\,T}}-1}}\,\mathrm {d} \omega }$

with the number of oscillation modes with circular frequencies ${\displaystyle g(\omega )\,\mathrm {d} \omega }$ ${\displaystyle \in [\omega ,\omega +\mathrm {d} \omega ]\,.}$

the above heat capacity results explicitly by performing the integral and differentiation according to the temperature: ${\displaystyle C_{v}}$

${\displaystyle C_{v}={\frac {\partial U}{\partial T}}\,.}$

Note that instead of the Debye approximation, the exact one is shown above and the exact maximum frequency is not used. ${\displaystyle g_{D}}$ ${\displaystyle g(\omega )}$ ${\displaystyle \omega _{D}}$

In the low-temperature approximation one uses that in this approximation the upper integration limit can be replaced by and that the lowest non-trivial terms of the Taylor expansions of  g and at coincide. ${\displaystyle \infty }$ ${\displaystyle g_{D}}$ ${\displaystyle \omega \to 0}$

For the high-temperature behavior, replace the term in the denominator with  x and calculate the remaining integral using the sum rule. ${\displaystyle e^{x}-1}$

The density of states (which explicitly required for Tieftemperaturnäherung is) can be specified in the Debye model, to be. ${\displaystyle g(\omega )}$ ${\displaystyle \omega _{\rm {max}}}$ ${\displaystyle \omega _{D}}$

The specific calculation of the density of states g , which goes beyond the Debye approximation,  cannot be solved analytically in general, but only numerically or approximated for parts of the temperature scale, as above for low temperatures. This is where the above-mentioned improvement possibilities for the intermediate behavior lie.

Generalization to other quasiparticles

Debye's method can be carried out in an analogous manner for other bosonic quasiparticles in the solid, e.g. B. in ferromagnetic systems for magnons instead of phonons. One now has other dispersion relations for , e.g. B. in the case mentioned, and other sum rules, z. B. In this way, in ferromagnets at low temperatures there is a magnon contribution to the heat capacity which dominates over the phonon contribution . In metals, however, the main contribution comes from the electrons. It is fermionic and is calculated using other methods that go back to Arnold Sommerfeld . ${\displaystyle \omega \to 0}$ ${\displaystyle \omega \propto k^{2}}$ ${\displaystyle \int _{0}^{\omega _{\rm {max}}}\,g(\omega ){\rm {d}}\omega =N\,.}$ ${\displaystyle \propto T^{3/2}}$ ${\displaystyle \,\,\propto T^{3}\,,}$ ${\displaystyle \propto T\,,}$

Individual evidence

1. ^ Peter Debye (1884–1966): Nobel Prize Winner for Chemistry
2. ↑ For a detailed classical derivation see z. B .: Georg Joos : textbook of theoretical physics . 12th edition. Akademische Verlagsgesellschaft, Frankfurt am Main 1970 - on the one hand Debye's theory of the specific heat of solid bodies , p. 566 ff., Or on the other hand, Planck's radiation law , p. 580 ff.
3. ↑ Further details can be found e.g. B. in Werner Döring : Introduction to Theoretical Physics , Vol. 5, §14. Göschen Collection, De Gruyter, Berlin 1957