Debye function

Template:Short description 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 propertiesEdit

Relation to other functionsEdit

The Debye functions are closely related to the polylogarithm.

Series expansionEdit

They have the series expansion<ref>Template:AS ref</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 Template:Mvar-th Bernoulli number.

Limiting valuesEdit

<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">Template:Cite book</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>

DerivativeEdit

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 physicsEdit

The Debye modelEdit

The Debye model has a 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 Template:Em Template:Math.

Internal energy and heat capacityEdit

Inserting Template:Math 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 displacementEdit

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 Template:Math of the vector Template:Math 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 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 motion).

ReferencesEdit

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