Relation to other functions edit

The Debye functions are closely related to the polylogarithm.

Series expansion edit

They have the series expansion^[1]

${\displaystyle D_{n}(x)=1-{\frac {n}{2(n+1)}}x+n\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k+n)(2k)!}}x^{2k},\quad |x|<2\pi ,\ n\geq 1,}$ 

where ${\displaystyle B_{n}}$  is the n-th Bernoulli number.

Limiting values edit

${\displaystyle \lim _{x\to 0}D_{n}(x)=1.}$ 

If ${\displaystyle \Gamma }$  is the gamma function and ${\displaystyle \zeta }$  is the Riemann zeta function, then, for ${\displaystyle x\gg 0}$ ,

${\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}\,dt}{e^{t}-1}}\sim {\frac {n}{x^{n}}}\Gamma (n+1)\zeta (n+1),\qquad \operatorname {Re} n>0,}$ ^[2]

Derivative edit

The derivative obeys the relation

${\displaystyle xD_{n}^{\prime }(x)=n(B(x)-D_{n}(x)),}$ 

where ${\displaystyle B(x)=x/(e^{x}-1)}$  is the Bernoulli function.

The Debye model edit

The Debye model has a density of vibrational states

${\displaystyle g_{\rm {D}}(\omega )={\frac {9\omega ^{2}}{\omega _{\rm {D}}^{3}}}}$  for ${\displaystyle 0\leq \omega \leq \omega _{\rm {D}}}$ 

with the Debye frequency ω_D.

Internal energy and heat capacity edit

Inserting g into the internal energy

${\displaystyle U=\int _{0}^{\infty }d\omega \,g(\omega )\,\hbar \omega \,n(\omega )}$ 

with the Bose–Einstein distribution

${\displaystyle n(\omega )={\frac {1}{\exp(\hbar \omega /k_{\rm {B}}T)-1}}}$ .

one obtains

${\displaystyle U=3k_{\rm {B}}T\,D_{3}(\hbar \omega _{\rm {D}}/k_{\rm {B}}T)}$ .

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

${\displaystyle \exp(-2W(q))=\exp(-q^{2}\langle u_{x}^{2}\rangle }$ ).

In this expression, the mean squared displacement refers to just once Cartesian component u_x of the vector u that describes the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes,^[3] one obtains

${\displaystyle 2W(q)={\frac {\hbar ^{2}q^{2}}{6Mk_{\rm {B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\rm {B}}T}{\hbar \omega }}g(\omega )\coth {\frac {\hbar \omega }{2k_{\rm {B}}T}}={\frac {\hbar ^{2}q^{2}}{6Mk_{\rm {B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\rm {B}}T}{\hbar \omega }}g(\omega )\left[{\frac {2}{\exp(\hbar \omega /k_{\rm {B}}T)-1}}+1\right].}$ 

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

${\displaystyle 2W(q)={\frac {3}{2}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\rm {D}}}}\left[2\left({\frac {k_{\rm {B}}T}{\hbar \omega _{\rm {D}}}}\right)D_{1}\left({\frac {\hbar \omega _{\rm {D}}}{k_{\rm {B}}T}}\right)+{\frac {1}{2}}\right]}$ .

From the above power series expansion of ${\displaystyle D_{1}}$  follows that the mean square displacement at high temperatures is linear in temperature

${\displaystyle 2W(q)={\frac {3k_{\rm {B}}Tq^{2}}{M\omega _{\rm {D}}^{2}}}}$ .

The absence of ${\displaystyle \hbar }$  indicates that this is a classical result. Because ${\displaystyle D_{1}(x)}$  goes to zero for ${\displaystyle x\to \infty }$  it follows that for ${\displaystyle T=0}$ 

${\displaystyle 2W(q)={\frac {3}{4}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\rm {D}}}}}$  (zero-point motion).

• 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

• 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