By equating the high-temperature limit of the above expression to the classical

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Homework Statement

According to the Debye theory of the specific heat of a three-dimensional solid, the internal vibrational energy of a volume V of a solid containing N atoms is:

U=A(T)[itex]\int^{x_{D}}_{0}[/itex][itex]\frac{x^{3}dx}{e^{x}-1}[/itex]

where x=[itex]\frac{\hbar\omega}{k_{B}T}[/itex] is the dimensionless form of the vibration frequency [itex]\omega[/itex]

a) What assumptions are made in the Debye theory about the distribution of frequency modes as a function of their wavevecdotr K?
b) Derive an expression for the (dimensionless) Debye cutoff frequency x[itex]_{D}[/itex] in terms of these assumptions.
c) By equating the high-temperature limit of the above expression to the classical three-dimensional result U=3Nk[itex]_{B}[/itex]T, deduce the unknown function A(T)
d) Hence derive an expression for the low-temperature specific heat.

The Attempt at a Solution

I have done questions a) and b). I am stuck on c).

As T tends to infinity, surely x[itex]_{D}[/itex] tends to 0 because x[itex]_{D}[/itex]=[itex]\frac{\hbar\omega_{D}}{k_{B}T}[/itex] according to my notes.

But surely they cannot expect me to integrate the above expression for U from 0 to 0?

Please help.

 

[mark726]

I would like to clarify a few points about the Debye theory and its assumptions. The Debye theory is a model that describes the behavior of specific heat in solids at low temperatures. It assumes that the atoms in a solid are arranged in a regular lattice and that the vibrations of the atoms are quantized. It also assumes that the vibration modes are evenly distributed in frequency as a function of their wavevector K. This means that there is an equal number of modes with low frequencies as there are with high frequencies.

Now, moving on to the specific questions:

a) The assumption about the distribution of frequency modes as a function of their wavevector K is necessary for the Debye theory to accurately describe the behavior of specific heat in solids. If this assumption is not valid, the theory will not accurately predict the specific heat of a solid.

b) To derive an expression for the Debye cutoff frequency x_{D}, we need to consider the Debye dispersion relation, which relates the frequency \omega of a vibration mode to its wavevector K. This relation is given by \omega=v_{s}K, where v_{s} is the speed of sound in the solid. Using this relation, we can write x_{D}=\frac{\hbar v_{s}}{k_{B}T}.

c) The high-temperature limit of the expression for internal vibrational energy is given by U=A(T)\int^{x_{D}}_{0}\frac{x^{3}dx}{e^{x}-1}. As you correctly stated, as T tends to infinity, x_{D} tends to 0. In this case, the integral becomes \int^{0}_{0}\frac{x^{3}dx}{e^{x}-1}, which is indeterminate. However, we can use L'Hopital's rule to evaluate this limit. Taking the derivative of the numerator and denominator with respect to x, we get \frac{3x^{2}}{e^{x}}. As x tends to 0, this expression tends to 0. Therefore, the high-temperature limit of U is given by U=0. On the other hand, the classical three-dimensional result for internal energy is U=3Nk_{B}T. Equating these two expressions, we get A(T)=\frac{3Nk_{B}}{x_{D}^{3}}.

d) To derive an expression