Debye function

In mathematics, the family of Debye functions is defined by $${\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt.}$$

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

Relation to other functions

The Debye functions are closely related to the polylogarithm.

Series expansion

They have the series expansion^[1] $${\displaystyle D_{n}(x)=1-{\frac {n}{2(n+1)}}x+n\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k+n)(2k)!}}x^{2k},\quad |x|<2\pi ,\ n\geq 1,}$$ where ${\displaystyle B_{n}}$ is the n-th Bernoulli number.

Limiting values

$${\displaystyle \lim _{x\to 0}D_{n}(x)=1.}$$ If ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle \zeta }$ is the Riemann zeta function, then, for ${\displaystyle x\gg 0}$,^[2] $${\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}\,dt}{e^{t}-1}}\sim {\frac {n}{x^{n}}}\Gamma (n+1)\zeta (n+1),\qquad \operatorname {Re} n>0,}$$

Derivative

The derivative obeys the relation $${\displaystyle xD_{n}^{\prime }(x)=n\left(B(x)-D_{n}(x)\right),}$$ where ${\displaystyle B(x)=x/(e^{x}-1)}$ is the Bernoulli function.

Applications in solid-state physics

The Debye model

The Debye model has a density of vibrational states $${\displaystyle g_{\text{D}}(\omega )={\frac {9\omega ^{2}}{\omega _{\text{D}}^{3}}}\,,\qquad 0\leq \omega \leq \omega _{\text{D}}}$$ with the Debye frequency ω_D.

Internal energy and heat capacity

Inserting g into the internal energy $${\displaystyle U=\int _{0}^{\infty }d\omega \,g(\omega )\,\hbar \omega \,n(\omega )}$$ with the Bose–Einstein distribution $${\displaystyle n(\omega )={\frac {1}{\exp(\hbar \omega /k_{\text{B}}T)-1}}.}$$ one obtains $${\displaystyle U=3k_{\text{B}}T\,D_{3}(\hbar \omega _{\text{D}}/k_{\text{B}}T).}$$ 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 $${\displaystyle \exp(-2W(q))=\exp \left(-q^{2}\langle u_{x}^{2}\rangle \right).}$$ 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 $${\displaystyle 2W(q)={\frac {\hbar ^{2}q^{2}}{6Mk_{\text{B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\text{B}}T}{\hbar \omega }}g(\omega )\coth {\frac {\hbar \omega }{2k_{\text{B}}T}}={\frac {\hbar ^{2}q^{2}}{6Mk_{\text{B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\text{B}}T}{\hbar \omega }}g(\omega )\left[{\frac {2}{\exp(\hbar \omega /k_{\text{B}}T)-1}}+1\right].}$$ Inserting the density of states from the Debye model, one obtains $${\displaystyle 2W(q)={\frac {3}{2}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\text{D}}}}\left[2\left({\frac {k_{\text{B}}T}{\hbar \omega _{\text{D}}}}\right)D_{1}{\left({\frac {\hbar \omega _{\text{D}}}{k_{\text{B}}T}}\right)}+{\frac {1}{2}}\right].}$$ From the above power series expansion of ${\displaystyle D_{1}}$ follows that the mean square displacement at high temperatures is linear in temperature $${\displaystyle 2W(q)={\frac {3k_{\text{B}}Tq^{2}}{M\omega _{\text{D}}^{2}}}.}$$ The absence of ${\displaystyle \hbar }$ indicates that this is a classical result. Because ${\displaystyle D_{1}(x)}$ goes to zero for ${\displaystyle x\to \infty }$ it follows that for ${\displaystyle T=0}$ $${\displaystyle 2W(q)={\frac {3}{4}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\text{D}}}}}$$ (zero-point motion).