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| In [[mathematics]], the family of '''Debye functions''' is defined by
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| :<math>D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^t - 1}\,dt.</math>
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| 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]].
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| == Mathematical properties ==
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| === Relation to other functions ===
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| The Debye functions are closely related to the [[Polylogarithm]].
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| === Limiting values ===
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| For <math>x \rightarrow 0</math> :
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| :<math>D_n(0)=1.</math>
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| For <math>x >> 1</math> : <math>D_n</math> is given by the [[Gamma function]] and the [[Riemann zeta function]]:
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| :<math>D_n(x)\propto\int_0^\infty{\rm d}t\frac{t^{n}}{\exp(t)-1} = \Gamma(n + 1) \zeta(n + 1). \quad [\Re \, n > 0]</math> <ref>Gradshteyn, I. S., & Ryzhik, I. M. (1980). Table of integrals. Series, and Products (Academic, New York, 1980), (3.411).</ref>
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| == Applications in solid-state physics ==
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| === The Debye model ===
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| The [[Debye model]] has a [[density of vibrational states]]
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| :<math>g_{\rm D}(\omega)=\frac{9\omega^2}{\omega_{\rm D}^3}</math> for <math>0\le\omega\le\omega_{\rm D}</math> | |
| with the ''Debye frequency'' ''ω''<sub>D</sub>.
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| === Internal energy and heat capacity ===
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| Inserting ''g'' into the internal energy
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| :<math>U=\int_0^\infty{\rm d}\omega\,g(\omega)\,\hbar\omega\,n(\omega)</math>
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| with the [[Bose-Einstein distribution]]
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| :<math>n(\omega)=\frac{1}{\exp(\hbar\omega/k_{\rm B}T)-1}</math>.
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| one obtains
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| :<math>U=3 k_{\rm B}T\, D_3(\hbar\omega_{\rm D}/k_{\rm B}T)</math>.
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| The heat capacity is the derivative thereof.
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| === Mean squared displacement ===
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| The intensity of [[X-ray diffraction]] or [[neutron diffraction]] at wavenumber ''q'' is given by
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| the [[Debye-Waller factor]] or the [[Lamb-Mössbauer factor]].
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| For isotropic systems it takes the form
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| :<math>\exp(-2W(q))=\exp(-q^2\langle u_x^2\rangle</math>). | |
| In this expression, the [[mean squared displacement]] refers to just once Cartesian component
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| ''u<sub>x</sub>'' of the vector '''u''' that describes the displacement of atoms from their equilibrium positions.
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| Assuming harmonicity and developing into normal modes,<ref>Ashcroft & Mermin 1976, App. L,</ref>
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| one obtains
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| :<math>2W(q)=\frac{\hbar^2 q^2}{6M k_{\rm B}T}\int_0^\infty{\rm d}\omega\frac{k_{\rm B}T}{\hbar\omega}g(\omega)\coth\frac{\hbar\omega}{2k_{\rm B}T}=\frac{\hbar^2 q^2}{6M k_{\rm B}T}\int_0^\infty{\rm d}\omega\frac{k_{\rm B}T}{\hbar\omega}g(\omega)\left[\frac{2}{\exp(\hbar\omega/k_{\rm B}T)-1}+1\right].</math> | |
| Inserting the density of states from the Debye model, one obtains
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| :<math>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]</math>.
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * {{AS ref|27|998}}
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| * [http://mathworld.wolfram.com/DebyeFunctions.html "Debye function" entry in MathWorld], defines the Debye functions without prefactor ''n''/''x<sup>n</sup>''
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| == Implementations ==
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| * [http://www.netlib.org/toms/757 Fortran 77 code] by [[Allan MacLeod]] from [[Transactions on Mathematical Software]]
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| * [http://www.csit.fsu.edu/~burkardt/f_src/toms757/toms757.html Fortran 90 version]
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| * [http://www.gnu.org/software/gsl/manual/html_node/Debye-Functions.html C version] of the [[GNU Scientific Library]]
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| [[Category:Special functions]]
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