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Debye function

From Wikipedia, the free encyclopedia

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]

They have the series expansion[1]

where is the n-th Bernoulli number.

Limiting values[edit]

If is the gamma function and is the Riemann zeta function, then, for ,



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


From the above power series expansion of follows that the mean square displacement at high temperatures is linear in temperature


The absence of indicates that this is a classical result. Because goes to zero for it follows that for

(zero-point motion).


  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. Vol. 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 (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. pp. 355ff. ISBN 978-0-12-384933-5. LCCN 2014010276.
  3. ^ Ashcroft & Mermin 1976, App. L,

Further reading[edit]