Debye function

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In mathematics, the family of Debye functions is defined by

[math]\displaystyle{ 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[1]

[math]\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| \lt 2\pi,\ n \ge 1, }[/math]

where [math]\displaystyle{ B_n }[/math] is the n-th Bernoulli number.

Limiting values

[math]\displaystyle{ \lim_{x \to 0} D_n(x) = 1. }[/math]

If [math]\displaystyle{ \Gamma }[/math] is the gamma function and [math]\displaystyle{ \zeta }[/math] is the Riemann zeta function, then, for [math]\displaystyle{ x \gg 0 }[/math],

[math]\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 \gt 0, }[/math][2]

Derivative

The derivative obeys the relation

[math]\displaystyle{ xD^{\prime}_n(x) = n(B(x)-D_n(x)), }[/math]

where [math]\displaystyle{ 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 vibrational states

[math]\displaystyle{ g_{\rm D}(\omega)=\frac{9\omega^2}{\omega_{\rm D}^3} }[/math] for [math]\displaystyle{ 0\le\omega\le\omega_{\rm D} }[/math]

with the Debye frequency ωD.

Internal energy and heat capacity

Inserting g into the internal energy

[math]\displaystyle{ U=\int_0^\infty d\omega\,g(\omega)\,\hbar\omega\,n(\omega) }[/math]

with the Bose–Einstein distribution

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

one obtains

[math]\displaystyle{ U=3 k_{\rm B}T\, D_3(\hbar\omega_{\rm D}/k_{\rm 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]\displaystyle{ \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 ux of the vector u that describes the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes,[3] one obtains

[math]\displaystyle{ 2W(q)=\frac{\hbar^2 q^2}{6M k_{\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}{6M k_{\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]. }[/math]

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

[math]\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] }[/math].

From the above power series expansion of [math]\displaystyle{ D_1 }[/math] follows that the mean square displacement at high temperatures is linear in temperature

[math]\displaystyle{ 2W(q)=\frac{3 k_{\rm B}T q^2}{M\omega_{\rm D}^2} }[/math].

The absence of [math]\displaystyle{ \hbar }[/math] indicates that this is a classical result. Because [math]\displaystyle{ D_1(x) }[/math] goes to zero for [math]\displaystyle{ x\to\infty }[/math] it follows that for [math]\displaystyle{ T=0 }[/math]

[math]\displaystyle{ 2W(q)=\frac{3}{4}\frac{\hbar^2 q^2}{M\hbar\omega_{\rm D}} }[/math] (zero-point motion).

References

  1. ↑ Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "Chapter 27". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 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. pp. 998. LCCN 65-12253. ISBN 978-0-486-61272-0. http://www.math.sfu.ca/~cbm/aands/page_998.htm. 
  2. ↑ "3.411." (in English). Table of Integrals, Series, and Products (8 ed.). Academic Press, Inc.. 2015. pp. 355ff. ISBN 978-0-12-384933-5. 
  3. ↑ Ashcroft & Mermin 1976, App. L,

Further reading

Implementations



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