Debye function

Mathematical properties

Relation to other functions

The Debye functions are closely related to the polylogarithm.

Series expansion

They have the series expansion D_n(x) = 1 - \frac{n}{2(n+1)} x + n \sum_{k=1}^\infty \frac{B_{2k}}{(2k+n)(2k)!} x^{2k}, \quad |x| where B_n is the n-th Bernoulli number.

Limiting values

\lim_{x \to 0} D_n(x) = 1. If \Gamma is the gamma function and \zeta is the Riemann zeta function, then, for x \gg 0, D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n,dt}{e^t-1} \sim \frac{n}{x^n}\Gamma(n + 1) \zeta(n + 1), \qquad \operatorname{Re} n 0,

Derivative

The derivative obeys the relation x D^{\prime}_n(x) = n \left(B(x) - D_n(x)\right), where B(x) = x/(e^x-1) is the Bernoulli function.

Applications in solid-state physics

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 Debye frequency ωD.

Internal energy and heat capacity

Inserting g 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 ux of the vector u 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

Implementations

• Fortran 77 code
• Fortran 90 version
• C version of the GNU Scientific Library

References

1. {{AS ref. 27. 998
2. (2015). "Table of Integrals, Series, and Products". [[Academic Press, Inc.]].
3. Ashcroft & Mermin 1976, App. L,