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

  1. Feb 15, 2012 #1
    1. The problem statement, all variables and given/known data

    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.

    3. 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.
     
  2. jcsd
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