Debye function
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[]
Relation to other functions[]
The Debye functions are closely related to the polylogarithm.
Series Expansion[]
They have the series expansion[1]
where is the n-th Bernoulli number.
Limiting values[]
If is the gamma function and is the Riemann zeta function, then, for ,
Derivative[]
The derivative obeys the relation
where is the Bernoulli function.
Applications in solid-state physics[]
The Debye model[]
The Debye model has a
- for
with the Debye frequency ωD.
Internal energy and heat capacity[]
Inserting g into the internal energy
with the Bose–Einstein distribution
- .
one obtains
- .
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
- ).
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
References[]
- ^ 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.
- ^ 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 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.
- ^ Ashcroft & Mermin 1976, App. L,
Further reading[]
- 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.
- "Debye function" entry in MathWorld, defines the Debye functions without prefactor n/xn
Implementations[]
- Fortran 77 code by from Transactions on Mathematical Software
- Fortran 90 version
- C version of the GNU Scientific Library
- Special functions
- Peter Debye