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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
Relation to other functions

The Debye functions are closely related to the Polylogarithm.
Limiting values

For x -> 0:

\( D_n(0)=1. \)

For x -> &infty;, the functions diverge; the integral without prefactor is given by the Riemann zeta function:

\( \int_0^\infty{\rm d}t\frac{t}{\exp(t)-1} = n! \zeta(n=1). \)

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[1], 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

\( 2W(q)=\frac{3}{2}\frac{\hbar^2 q^2}{M\hbar\omega_{\rm D}}\left[2\left(\frac{k_{\rm B}T}{\hbar\omega_{\rm D}}\right)D_1\left(\frac{\hbar\omega_{\rm D}}{k_{\rm B}T}\right)+\frac{1}{2}\right]. \)


^ Ashcroft & Mermin 1976, App. L,

Further Reading

Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 27", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 998, ISBN 978-0486612720, MR0167642.
"Debye function" entry in MathWorld, defines the Debye functions without prefactor n/xn


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Mathematics Encyclopedia

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