Debye function

In mathematics, the family of Debye functions is defined by

D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^t - 1}\,dt.

The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed the heat capacity of what is now called the Debye model.

Mathematical properties

Debye function Maple complex plot

Relation to other functions

The Debye functions are closely related to the Riemann zeta function\zeta(x) and PolylogarithmLi_{m}(x):

D_n(x)=\frac{n*((-1)^n*n!*\zeta(n+1)+\sum_{m=0}^{n} ((-1)^{n-m+1}*n!*x^m*Li_{n-m+1}(e^{x}/m!))}{x^{n+1}}-\frac{n}{n+1}[1]

Series Expansion

According to,[2]

D_n(x) =\int_{0}^{x}\frac{t^n}{\exp(t)-1}dt= 1 - \frac{n}{2(n+1)} x + n \sum_{k=1}^\infty \frac{B_{2k}}{(2k+n)(2k)!} x^{2k}, \quad |x| < 2\pi,\ n \ge 1.

Limiting values

For x \rightarrow 0 :

D_n(0)=1.

For x \ll 1 : D_n is given by the Gamma function and the Riemann zeta function:

D_n(x)\propto\int_0^\infty{\rm d}t\frac{t^{n}}{\exp(t)-1} = \Gamma(n + 1) \zeta(n + 1). \quad [\Re \, n > 0] [3]

Applications in solid-state physics

The Debye model

The Debye model has a density of vibrational states

g_{\rm D}(\omega)=\frac{9\omega^2}{\omega_{\rm D}^3} for 0\le\omega\le\omega_{\rm D}

with the Debye frequency ωD.

Internal energy and heat capacity

Inserting g into the internal energy

U=\int_0^\infty{\rm d}\omega\,g(\omega)\,\hbar\omega\,n(\omega)

with the Bose–Einstein distribution

n(\omega)=\frac{1}{\exp(\hbar\omega/k_{\rm B}T)-1}.

one obtains

U=3 k_{\rm B}T\, D_3(\hbar\omega_{\rm D}/k_{\rm 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(-q^2\langle u_x^2\rangle).

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,[4] one obtains

2W(q)=\frac{\hbar^2 q^2}{6M k_{\rm B}T}\int_0^\infty{\rm d}\omega\frac{k_{\rm B}T}{\hbar\omega}g(\omega)\coth\frac{\hbar\omega}{2k_{\rm B}T}=\frac{\hbar^2 q^2}{6M k_{\rm B}T}\int_0^\infty{\rm d}\omega\frac{k_{\rm B}T}{\hbar\omega}g(\omega)\left[\frac{2}{\exp(\hbar\omega/k_{\rm 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_{\rm D}}\left[2\left(\frac{k_{\rm B}T}{\hbar\omega_{\rm D}}\right)D_1\left(\frac{\hbar\omega_{\rm D}}{k_{\rm B}T}\right)+\frac{1}{2}\right].

References

  1. ↑ A. E. Dubinov, A. A. Dubinova ,Exact integral-free expressions for the integral Debye functions,Technical Physics Letters,December 2008, Volume 34, Issue 12, pp 999-1001
  2. ↑ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 27", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 998, ISBN 978-0486612720, MR 0167642.
  3. ↑ Gradshteyn, I. S., & Ryzhik, I. M. (1980). Table of integrals. Series, and Products (Academic, New York, 1980), (3.411).
  4. ↑ Ashcroft & Mermin 1976, App. L,

Further reading

Implementations