The Debye model The
Debye model has a
density of vibrational states g_\text{D}(\omega) = \frac{9\omega^2}{\omega_\text{D}^3} \,, \qquad 0\le\omega\le\omega_\text{D} with the .
Internal energy and heat capacity Inserting into the internal energy U = \int_0^\infty d\omega\,g(\omega)\,\hbar\omega\,n(\omega) with the
Bose–Einstein distribution n(\omega) = \frac{1}{\exp(\hbar\omega / k_\text{B} T)-1}. one obtains U = 3 k_\text{B}T \, D_3(\hbar\omega_\text{D} / k_\text{B}T). The heat capacity is the derivative thereof.
Mean squared displacement The intensity of
X-ray diffraction or
neutron diffraction at wavenumber
q is given by the
Debye-Waller factor or the
Lamb-Mössbauer factor. For isotropic systems it takes the form \exp(-2W(q)) = \exp\left(-q^2\langle u_x^2\rangle\right). In this expression, the
mean squared displacement refers to just once Cartesian component of the vector that describes the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes, one obtains 2W(q) = \frac{\hbar^2 q^2}{6M k_\text{B}T} \int_0^\infty d\omega \frac{k_\text{B}T}{\hbar\omega}g(\omega) \coth\frac{\hbar\omega}{2k_\text{B}T}=\frac{\hbar^2 q^2}{6M k_\text{B}T} \int_0^\infty d\omega \frac{k_\text{B}T}{\hbar\omega} g(\omega) \left[\frac{2}{\exp(\hbar\omega/k_\text{B}T)-1}+1\right]. Inserting the density of states from the Debye model, one obtains 2W(q) = \frac{3}{2} \frac{\hbar^2 q^2}{M\hbar\omega_\text{D}} \left[2\left(\frac{k_\text{B}T}{\hbar\omega_\text{D}}\right) D_1{\left(\frac{\hbar\omega_\text{D}}{k_\text{B}T}\right)} + \frac{1}{2}\right]. From the above
power series expansion of D_1 follows that the mean square displacement at high temperatures is linear in temperature 2W(q) = \frac{3 k_\text{B}T q^2}{M\omega_\text{D}^2}. The absence of \hbar indicates that this is a
classical result. Because D_1(x) goes to zero for x \to \infty it follows that for T = 0 2W(q)=\frac{3}{4}\frac{\hbar^2 q^2}{M\hbar\omega_\text{D}} (
zero-point motion). ==References==