THC Science

In mathematics, the family of Debye functions is defined by

D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt.

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]

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

Limiting values[edit]

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

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

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

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

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

Applications in solid-state physics[edit]

The Debye model[edit]

The Debye model has a density of vibrational states

g_{\rm {D}}(\omega )={\frac {9\omega ^{2}}{\omega _{\rm {D}}^{3}}}

for

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

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

with the Bose–Einstein distribution

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

.

one obtains

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

\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

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

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 $D_{1}$ follows that the mean square displacement at high temperatures is linear in temperature

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

.

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

2W(q)={\frac {3}{4}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\rm {D}}}}

(zero-point motion).

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,

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

[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.

[Zwillinger_2014-2] 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,

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