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.
The Debye functions are closely related to the polylogarithm.
They have the series expansion[1]
is the n-th Bernoulli number.
is the gamma function and
is the Riemann zeta function, then, for
The derivative obeys the relation
is the Bernoulli function.
The Debye model has a density of vibrational states
with the Debye frequency ωD.
Inserting g into the internal energy
ℏ ω
exp ( ℏ ω
The heat capacity is the derivative thereof.
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
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
ℏ ω
g ( ω ) coth
ℏ ω
ℏ ω
exp ( ℏ ω
Inserting the density of states from the Debye model, one obtains
From the above power series expansion of
follows that the mean square displacement at high temperatures is linear in temperature
indicates that this is a classical result.
(zero-point motion).