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{{short description|Mathematical 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{{AS ref|27|998}} where B_n is the n-th Bernoulli number.

Limiting values

If Gamma is the gamma function and zeta is the Riemann zeta function, then, for x gg 0,

Derivative

The derivative obeys the relation where B(x)=x/(e^x-1) is the Bernoulli function.

Applications in solid-state physics

The Debye model

The Debye model has a density of vibrational states 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 componentux of the vector u that describes the displacement of atoms from their equilibrium positions.Assuming harmonicity and developing into normal modes,Ashcroft & Mermin 1976, App. L,one obtains Inserting the density of states from the Debye model, one obtains From the above power series expansion of D_1 follows that the mean square displacement at high temperatures is linear in temperature The absence of hbar indicates that this is a classical result. Because D_1(x) goes to zero for xtoinfty it follows that for T=0

References

{{Reflist}}

Further reading

• {{AS ref|27|998}}
• "Debye function" entry in MathWorld, defines the Debye functions without prefactor n/xn

Implementations

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