Editing Debye function

{{short description|Mathematical function}}
In [[mathematics]], the family of '''Debye functions''' is defined by
<math display="block">D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^t - 1}\,dt.</math>

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<ref>{{AS ref|27|998}}</ref>
<math display="block">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 \ge 1,</math>
where <math>B_n</math> is the {{mvar|n}}-th [[Bernoulli number]].

=== Limiting values ===

<math display="block">\lim_{x \to 0} D_n(x) = 1.</math>
If <math>\Gamma</math> is the [[gamma function]] and <math>\zeta</math> is the [[Riemann zeta function]], then, for <math>x \gg 0</math>,<ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik |chapter=3.411. |pages=355ff}}</ref>
<math display="block">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,</math>

=== Derivative ===

The derivative obeys the relation
<math display="block">x D^{\prime}_n(x) = n \left(B(x) - D_n(x)\right),</math>
where <math>B(x) = x/(e^x-1)</math> is the Bernoulli function.

== Applications in solid-state physics ==

=== The Debye model ===

The [[Debye model]] has a [[Density of states|density of vibrational states]]
<math display="block">g_\text{D}(\omega) = \frac{9\omega^2}{\omega_\text{D}^3} \,, \qquad 0\le\omega\le\omega_\text{D}</math>
with the {{em|Debye frequency}} {{math|''ω''<sub>D</sub>}}.

=== Internal energy and heat capacity ===

Inserting {{math|''g''}} into the internal energy 
<math display="block">U = \int_0^\infty d\omega\,g(\omega)\,\hbar\omega\,n(\omega)</math>
with the [[Bose–Einstein distribution]]
<math display="block">n(\omega) = \frac{1}{\exp(\hbar\omega / k_\text{B} T)-1}.</math>
one obtains
<math display="block">U = 3 k_\text{B}T \, D_3(\hbar\omega_\text{D} / k_\text{B}T).</math>
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
<math display="block">\exp(-2W(q)) = \exp\left(-q^2\langle u_x^2\rangle\right).</math>
In this expression, the [[mean squared displacement]] refers to just once Cartesian component {{math|''u<sub>x</sub>''}} of the vector {{math|'''u'''}} that describes the displacement of atoms from their equilibrium positions.
Assuming harmonicity and developing into normal modes,<ref>Ashcroft & Mermin 1976, App. L,</ref>
one obtains
<math display="block">2W(q) = \frac{\hbar^2 q^2}{6M k_\text{B}T} \int_0^\infty d\omega \frac{k_\text{B}T}{\hbar\omega}g(\omega) \coth\frac{\hbar\omega}{2k_\text{B}T}=\frac{\hbar^2 q^2}{6M k_\text{B}T} \int_0^\infty d\omega \frac{k_\text{B}T}{\hbar\omega} g(\omega) \left[\frac{2}{\exp(\hbar\omega/k_\text{B}T)-1}+1\right].</math>
Inserting the density of states from the Debye model, one obtains
<math display="block">2W(q) = \frac{3}{2} \frac{\hbar^2 q^2}{M\hbar\omega_\text{D}} \left[2\left(\frac{k_\text{B}T}{\hbar\omega_\text{D}}\right) D_1{\left(\frac{\hbar\omega_\text{D}}{k_\text{B}T}\right)} + \frac{1}{2}\right].</math>
From the above [[power series]] expansion of <math>D_1</math> follows that the mean square displacement at high temperatures is linear in temperature
<math display="block">2W(q) = \frac{3 k_\text{B}T q^2}{M\omega_\text{D}^2}.</math>
The absence of <math>\hbar</math> indicates that this is a [[Classical Physics|classical]] result. Because <math>D_1(x)</math> goes to zero for <math>x \to \infty</math> it follows that for <math>T = 0</math>
<math display="block">2W(q)=\frac{3}{4}\frac{\hbar^2 q^2}{M\hbar\omega_\text{D}}</math> ([[Zero-point energy|zero-point motion]]).

==References==
{{Reflist}}

==Further reading==

* {{AS ref|27|998}}
* [http://mathworld.wolfram.com/DebyeFunctions.html "Debye function" entry in MathWorld], defines the Debye functions without prefactor ''n''/''x<sup>n</sup>''

== Implementations ==
* {{cite journal|first1=E. W.|last1=Ng| first2=C. J. |last2=Devine|title=On the computation of Debye functions of integer orders|journal=Math. Comp.|year=1970|volume=24|issue=110 |pages=405–407|doi=10.1090/S0025-5718-1970-0272160-6|mr=0272160|doi-access=free}}
* {{cite journal|first1=I.|last1=Engeln|first2=D.|last2=Wobig| title=Computation of the generalized Debye functions delta(x,y) and D(x,y)|journal=Colloid & Polymer Science|volume=261|year=1983|pages=736–743|doi=10.1007/BF01410947|s2cid=98476561 }}
* {{cite journal|first1=Allan J.|last1=MacLeod|title=Algorithm 757: MISCFUN, a software package to compute uncommon special functions|journal=ACM Trans. Math. Software|year=1996|volume=22|number=3|pages=288–301|doi=10.1145/232826.232846|s2cid=37814348 |doi-access=free}} [http://www.netlib.org/toms/757 Fortran 77 code]
* [http://www.csit.fsu.edu/~burkardt/f_src/toms757/toms757.html Fortran 90 version]
* {{cite journal|first1=Leonard C.|last1=Maximon|title=The dilogarithm function for complex argument|journal=Proc. R. Soc. A|year=2003|volume=459|number=2039| pages=2807–2819| doi=10.1098/rspa.2003.1156|bibcode=2003RSPSA.459.2807M |s2cid=122271244 }}
* {{cite journal|first1=I. I.|last1=Guseinov| first2=B. A. |last2=Mamedov|title=Calculation of Integer and noninteger n-Dimensional Debye Functions using Binomial Coefficients and Incomplete Gamma Functions|year=2007|journal=Int. J. Thermophys.|volume=28|issue=4 |pages=1420–1426|doi=10.1007/s10765-007-0256-1|bibcode=2007IJT....28.1420G |s2cid=120284032 }}
* [https://www.gnu.org/software/gsl/doc/html/specfunc.html#debye-functions C version] of the [[GNU Scientific Library]]

[[Category:Special functions]]
[[Category:Peter Debye]]