Debye function

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In mathematics, the family of Debye functions is defined by

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]

Limiting values[edit]

For  :

For  : is given by the Gamma function and the Riemann zeta function:



The derivative obeys the relation

where is the Bernoulli function.

Applications in solid-state physics[edit]

The Debye model[edit]

The Debye model has a density of vibrational states


with the Debye frequency ωD.

Internal energy and heat capacity[edit]

Inserting g into the internal energy

with the Bose–Einstein distribution


one obtains


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


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

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



  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,

Further reading[edit]