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## Debye modelIn thermodynamics and solid state physics, the ## Additional recommended knowledge
## DerivationThe 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 where where in which The approximation that the frequency is inversely proportional to the wavelength (giving a constant speed of sound) is good for low-energy phonons but not for high-energy phonons. (See the article on phonons.) This is one of the limitations of the Debye model. Let's now compute the total energy in the box where is the number of phonons in the box with energy Now, this is where It is reasonable to assume that the minimum wavelength of a phonon is twice the atom separation, as shown in the lower figure. There are making the maximum mode number This is the upper limit of the triple energy sum For slowly-varying, well-behaved functions, a sum can be replaced with an integral (also known as Thomas-Fermi approximation) So far, there has been no mention of , the number of phonons with energy Because a phonon has three possible polarization states (one longitudinal and two transverse) which do not affect its energy, the formula above must be multiplied by 3 Substituting this into the energy integral yields The ease with which these integrals are evaluated for photons is due to the fact that light's frequency, at least semi-classically, is unbound. As the figure above illustrates, this is not true for phonons. In order to approximate this triple integral, Debye used spherical coordinates and boldly approximated the cube by an eighth of a sphere where so we get: The substitution of integration over a sphere for the correct integral introduces another source of inaccuracy into the model. The energy integral becomes Changing the integration variable to , To simplify the look of this expression, define the We then have the specific internal energy: where Differentiating with respect to These formulae give the Debye model at all temperatures. The more elementary formulae given further down give the asymptotic behavior in the limit of low and high temperatures. ## Debye's derivationActually, Debye derived his equation somewhat differently and more simply. Using the solid mechanics of a continuous medium, he found that the number of vibrational states with a frequency less than a particular value was asymptotic to in which if the vibrational frequencies continued to infinity. This form gives the Debye knew that this assumption was not really correct (the higher frequencies are more closely spaced than assumed), but it guarantees the proper behavior at high temperature (the Dulong-Petit law). The energy is then given by: - where
*T*_{D}is*h*ν_{m}/*k*.
- where
- = 3
*N**k**T**D*_{3}(*T*_{D}/*T*)
- = 3
where ## Low temperature limitThe temperature of a Debye solid is said to be low if , leading to This definite integral can be evaluated exactly: In the low temperature limit, the limitations of the Debye model mentioned above do not apply, and it gives a correct relationship between (phononic) heat capacity, temperature, the elastic coefficients, and the volume per atom (the latter quantities being contained in the Debye temperature). ## High temperature limitThe temperature of a Debye solid is said to be high if This is the Dulong-Petit law, and is fairly accurate although it does not take into account anharmonicity, which causes the heat capacity to rise further. The total heat capacity of the solid, if it is a conductor or semiconductor, may also contain a non-negligible contribution from the electrons. ## Debye versus Einstein
So how closely do the Debye and Einstein models correspond to experiment? -- Surprisingly close, but Debye is correct at low temperatures whereas Einstein is not. How different are the models? To answer that question one would naturally plot the two on the same set of axes... except one can't. Both the Einstein model and the Debye model provide a is ε / which means that plotting them on the same set of axes makes no sense. They are two models of the same thing, but of different scales. If one defines then one can say and, to relate the two, we must seek the ratio The Einstein solid is composed of which makes the and the sought ratio is therefore Now both models can be plotted on the same graph. Note that this ratio is the cube root of the ratio of the volume of one octant of a 3-dimensional sphere to the volume of the cube that contains it, which is just the correction factor used by Debye when approximating the energy integral above. ## Debye temperature tableEven 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
## See also## References**^**'Zur Theorie der spezifischen Warmen',*Annalen der Physik*39(4), p. 789 (1912)
*CRC Handbook of Chemistry and Physics*, 56th Edition (1975-1976)- Schroeder, Daniel V.
*An Introduction to Thermal Physics*. Addison-Wesley, San Francisco, Calif. (2000). Section 7.5.
Categories: Condensed matter physics | Thermodynamics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Debye_model". A list of authors is available in Wikipedia. |