Historical
(a) Classical
Dulong and Petit (1819)
Cv=3Nk, Correct at high temperature
(b) Einstein
Based on Planck's quantum hypothesis (1901)
Quantised energy, Showed exponential dependence of Cv
(c) Debye
Showed complete dependence (1912)
In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to T3 – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong-Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.
· Uses wide spectrum of frequencies to describe the complicated pattern of lattice vibrations. [It is assumed that hypothetical oscillators generate simple sine waves throughout the crystal and these will displace the atoms away from their equilibrium positions by an amount equal to the amplitude of the sine wave at that point. If we have a whole set of such oscillators generating sine waves of certain frequencies and amplitudes then we might hope that the superposition of such waves will simulate the complicated pattern of the actual atomic vibrations.]
· Assume that the distribution of oscillators is quasi-continuous in w, and so we may use integration instead of summation. [This could not be done in the derivation of the mean energy of a single oscillator where the individual quantum steps might be large compared to kT.]
· Each frequency (mode) contributes an Einstein-like term to C.
The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical.
Consider a cube of side L. From the particle in a box article, the resonating modes of the sonic disturbances inside the box (considering for now only those aligned with one axis) have wavelengths given by
In order to go further there are two problems to solve:
· A density of states function is required
· Need to set the range of frequencies over which the integration is to be performed, i.e. the cut-off or limiting frequency needs to be determined
Neglecting the zero point energy, the mean thermal energy will be given by
Consider a cube of side L. From the particle in a box article, the resonating modes of the sonic disturbances inside the box (considering for now only those aligned with one axis) have wavelengths given by
In order to go further there are two problems to solve:
· A density of states function is required
· Need to set the range of frequencies over which the integration is to be performed, i.e. the cut-off or limiting frequency needs to be determined
Neglecting the zero point energy, the mean thermal energy will be given by
for each frequency. The Debye specific heat will take the form,
Debye temperature table
Even though the Debye model is not completely correct, it gives a good approximation for the low temperature heat capacity of insulating, crystalline solids where other contributions (such as highly mobile conduction electrons) are negligible. For metals, the electron contribution to the heat is proportional to T, which at low temperatures dominates the Debye T3 result for lattice vibrations. In this case, the Debye model can only be said to approximate the lattice contribution to the specific heat.
Debye's Contribution to Specific Heat Theory
Einstein's oscillator treatment of specific heat gave qualitative agreement with experiment and gave the correct high temperature limit (the Law of Dulong and Petit). The quantitative fit to experiment was improved by Debye's recognition that there was a maximum number of modes of vibration in a solid. He pictured the vibrations as standing wave modes in the crystal, similar to the electromagnetic modes in a cavity which successfully explained blackbody radiation. The density of states for these modes, which are called "phonons", is of the same form as the photon density of states in a cavity.
To impose a finite limit on the number of modes in the solid, Debye used a maximum allowed phonon frequency now called the Debye frequency uD.
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