# Debye function

In mathematics, the family of Debye functions is defined by

${\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt.}$

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

### Relation to other functions

The Debye functions are closely related to the Polylogarithm.

### Series Expansion

According to,[1]

${\displaystyle D_{n}(x)=1-{\frac {n}{2(n+1)}}x+n\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k+n)(2k)!}}x^{2k},\quad |x|<2\pi ,\ n\geq 1.}$

### Limiting values

For ${\displaystyle x\rightarrow 0}$ :

${\displaystyle D_{n}(0)=1.}$

For ${\displaystyle x\ll 1}$ : ${\displaystyle D_{n}}$ is given by the Gamma function and the Riemann zeta function:

${\displaystyle D_{n}(x)\propto \int _{0}^{\infty }{\rm {d}}t{\frac {t^{n}}{\exp(t)-1}}=\Gamma (n+1)\zeta (n+1).\quad [\Re \,n>0]}$[2]

### Derivative

The derivative obeys the relation

${\displaystyle xD_{n}^{\prime }(x)=n\left(B(x)-D_{n}(x)\right),}$

where ${\displaystyle B(x)=x/(e^{x}-1)}$ is the Bernoulli function.

## Applications in solid-state physics

### The Debye model

${\displaystyle g_{\rm {D}}(\omega )={\frac {9\omega ^{2}}{\omega _{\rm {D}}^{3}}}}$ for ${\displaystyle 0\leq \omega \leq \omega _{\rm {D}}}$

with the Debye frequency ωD.

### Internal energy and heat capacity

Inserting g into the internal energy

${\displaystyle U=\int _{0}^{\infty }{\rm {d}}\omega \,g(\omega )\,\hbar \omega \,n(\omega )}$

with the Bose–Einstein distribution

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

one obtains

${\displaystyle U=3k_{\rm {B}}T\,D_{3}(\hbar \omega _{\rm {D}}/k_{\rm {B}}T)}$.

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

${\displaystyle \exp(-2W(q))=\exp(-q^{2}\langle u_{x}^{2}\rangle }$).

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

${\displaystyle 2W(q)={\frac {\hbar ^{2}q^{2}}{6Mk_{\rm {B}}T}}\int _{0}^{\infty }{\rm {d}}\omega {\frac {k_{\rm {B}}T}{\hbar \omega }}g(\omega )\coth {\frac {\hbar \omega }{2k_{\rm {B}}T}}={\frac {\hbar ^{2}q^{2}}{6Mk_{\rm {B}}T}}\int _{0}^{\infty }{\rm {d}}\omega {\frac {k_{\rm {B}}T}{\hbar \omega }}g(\omega )\left[{\frac {2}{\exp(\hbar \omega /k_{\rm {B}}T)-1}}+1\right].}$

Inserting the density of states from the Debye model, one obtains

${\displaystyle 2W(q)={\frac {3}{2}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\rm {D}}}}\left[2\left({\frac {k_{\rm {B}}T}{\hbar \omega _{\rm {D}}}}\right)D_{1}\left({\frac {\hbar \omega _{\rm {D}}}{k_{\rm {B}}T}}\right)+{\frac {1}{2}}\right]}$.

## References

1. ^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 27". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 998. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
2. ^ Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "3.411.". In Zwillinger, Daniel; Moll, Victor Hugo. Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. pp. 355ff. ISBN 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.
3. ^ Ashcroft & Mermin 1976, App. L,