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[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 ,
Derivative[edit]
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
- for
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
References[edit]
- ^ 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 978-0-12-384933-5. LCCN 2014010276.
- ^ Ashcroft & Mermin 1976, App. L,
Further reading[edit]
- 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[edit]
- Ng, E. W.; Devine, C. J. (1970). "On the computation of Debye functions of integer orders". Math. Comp. 24 (110): 405–407. doi:10.1090/S0025-5718-1970-0272160-6. MR 0272160.
- Engeln, I.; Wobig, D. (1983). "Computation of the generalized Debye functions delta(x,y) and D(x,y)". Colloid & Polymer Science. 261: 736–743. doi:10.1007/BF01410947. S2CID 98476561.
- MacLeod, Allan J. (1996). "Algorithm 757: MISCFUN, a software package to compute uncommon special functions". ACM Trans. Math. Software. 22 (3): 288–301. doi:10.1145/232826.232846. S2CID 37814348. Fortran 77 code
- Fortran 90 version
- Maximon, Leonard C. (2003). "The dilogarithm function for complex argument". Proc. R. Soc. A. 459 (2039): 2807–2819. Bibcode:2003RSPSA.459.2807M. doi:10.1098/rspa.2003.1156. S2CID 122271244.
- Guseinov, I. I.; Mamedov, B. A. (2007). "Calculation of Integer and noninteger n-Dimensional Debye Functions using Binomial Coefficients and Incomplete Gamma Functions". Int. J. Thermophys. 28 (4): 1420–1426. Bibcode:2007IJT....28.1420G. doi:10.1007/s10765-007-0256-1. S2CID 120284032.
- C version of the GNU Scientific Library