Editing Debye model (section)

== Temperature limits ==

The temperature of a Debye solid is said to be low if <math>T \ll T_{\rm D}</math>, leading to
:<math> \frac{C_V}{Nk} \sim 9 \left({T\over T_{\rm D}}\right)^3\int_0^{\infty} {x^4 e^x\over \left(e^x-1\right)^2}\, dx.</math>

This [[definite integral]] can be evaluated exactly:
:<math> \frac{C_V}{Nk} \sim {12\pi^4\over5} \left({T\over T_{\rm D}}\right)^3.</math>

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).

The temperature of a Debye solid is said to be high if <math>T \gg T_{\rm D}</math>. Using <math>e^x - 1\approx x</math> if <math>|x| \ll 1</math> leads to
:<math> \frac{C_V}{Nk} \sim 9 \left({T\over T_{\rm D}}\right)^3\int_0^{T_{\rm D}/T} {x^4 \over x^2}\, dx </math>
which upon integration gives 
:<math>\frac{C_V}{Nk} \sim 3\,.</math>
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 [[electrical conductor|conductor]] or [[semiconductor]], may also contain a non-negligible contribution from the electrons.