Debye model

The Debye model describes a method to calculate the contribution to the heat capacity of a crystalline solid, the quantized vibrations in crystal lattices, called phonons revealed; it turns out, inter alia, that this is the main contribution in the rule. The 1911 and 1912 developed by Peter Debye " theory of the specific heat of crystals " applies also one of the first theoretical confirmations of 1900 by Max Planck first introduced in quantum theory.

Fundamentals of the Model

Compared with the Einstein model ( 1906 ), which assumes independent oscillators with the same frequency, the Debye model (1912 ) assumes a large number of possible frequencies and different from zero propagation velocity of all waves or phonons from. However, throughout the Langwellennäherung is assumed, that is, it is assumed for simplicity that up to a cutoff frequency, the so-called Debyefrequenz, always strict proportionality between frequency and wave vector (ie, a linear dispersion relation ), where a longitudinal and two transverse acoustic waves degrees of freedom are required.

Remarkable in this approach is that (apart from the non-existence of longitudinal light waves ) is identical with the assumptions used to calculate the Planck's black-body radiation, when the speed of sound is replaced by the speed of light. Therefore produces a radiant cavity ( → Planck's radiation law ) formulas with the same structure as for a heated solid, swinging at the particles in a lattice -like arrangement. In both cases, follow characteristic " T3 - laws." (see below).

But phonons exist only up to a maximum frequency ( in the Debye model that is up to ). This maximum frequency is given by the sum of all possible modes of vibration, as their total number can be at most equal to three times the number of the vibrating lattice (atoms ). It also follows that in principle is somewhat lower ( with the designation used below ) than that of a corresponding harmonic oscillator (see picture) without frequency limitation.

Results of the model

Temperature ranges

The model makes correct predictions about the temperature dependence of the heat capacity, both in the low temperature limit as well as for the high-temperature limit.

In the low temperature range, that is for ( the so-called Debye temperature) is valid for the phonon content of the specific heat capacity:

Where is the reduced Planck's quantum of action, the Boltzmann constant and the number of atoms in the crystal.

The Debye temperature is proportional to the effective velocity of sound, to contribute to the transverse acoustic waves to 2 /3 the longitudinal acoustic waves of 1/3; and that applies precise

In the high temperature range, ie for, applies to the internal energy, the relationship and thus for the heat capacity

In this limit, therefore, results, like with the Einstein model, the law of Dulong and Petit.

The density of states according to the Debye model is derived from the following calculation:. Or rewritten. Well true but in general, so it is in the room and by the Debye model for a total of:

In two borderline cases, the theory is fully correct, while the intermediate behavior is described by the Debye theory only in the sense of "reasonable interpolation " which can be improved if necessary (see below). Namely, the low-temperature behavior is therefore correct, because in the limit matches the Debye approximation with the exact; the high-temperature behavior is therefore correct because the Debye approximation by constructionem the sum rule

Met.

In support of the stated results

The Debye model approaches the dispersion relation of phonons in the manner linearly. The calculation that can be performed also for the elementary ( realistischen! ) case that longitudinal and transverse sound velocity vary considerably, takes a long time, so that omitted details here only for reasons of space.

Since in a solid state at most three times as many vibrational modes as atoms may be present, the density of states diverges for high, however, the density must be at a certain ( material-dependent ) frequency (in the Debye approximation: at ) are cut off.

Based on the exact formula of the vibrational energy U

Results above heat capacity explicitly by performing the integral and differentiating with respect to the temperature T,

Here, the number of vibrational modes with angular frequencies

Note that instead of the above- Debye approximation is the exact, precise and takes the maximum frequency. In Tieftemperaturnäherung to use, that can be replaced by the upper limit of integration in this approximation, and that the lowest non-trivial to match terms of the Taylor expansion of g and. For the high-temperature behavior you just replace the denominator the term by x and calculates the remaining integral with the sum rule. The density of states ( which is needed explicitly for the Tieftemperaturnäherung ) can be specified in the Debye model, which is to.

However, the concrete above the Debye approximation beyond calculating the density G is not generally analytically, numerically or only for part of the temperature range is approached detachably as described above for low temperatures. Here are the above-indicated potential improvements for the intermediate behavior.

Generalization to other quasiparticles

The Debye method can be easily carried out for other bosonic quasi-particles in the solid state in an analogous manner, for example in ferromagnetic systems for so-called magnons instead of phonons. It now has different dispersion relations for, for example, in the case referred to, and other sum rules, eg in this way results in ferromagnets at low temperatures a Magnonenbeitrag for heat capacity, compared to the phonon dominated. In metals, on the other hand, the main contribution comes, of course, of the electrons. He is fermionisch and is calculated using other methods, which go back to Arnold Sommerfeld.