Debye function

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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[edit]

Relation to other functions[edit]

The Debye functions are closely related to the Polylogarithm.

Series Expansion[edit]

According to,[1]

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| < 2\pi,\ n \ge 1.

Limiting values[edit]

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] [2]

Applications in solid-state physics[edit]

The Debye model[edit]

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[edit]

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[edit]

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,[3] 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[edit]

  1. ^ 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 .
  2. ^ Gradshteyn, I. S., & Ryzhik, I. M. (1980). Table of integrals. Series, and Products (Academic, New York, 1980), (3.411).
  3. ^ Ashcroft & Mermin 1976, App. L,

Further reading[edit]

Implementations[edit]