Debye function

In mathematics, the family of Debye functions is defined by
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
where formula_3 is the n-th Bernoulli number.
Limiting values.
If formula_5 is the gamma function and formula_6 is the Riemann zeta function, then, for formula_7,
Derivative.
The derivative obeys the relation
where formula_10 is the Bernoulli function.
Applications in solid-state physics.
The Debye model.
The Debye model has a density of vibrational states
with the "Debye frequency" "ω"D.
Internal energy and heat capacity.
Inserting "g" into the internal energy 
with the Bose–Einstein distribution
one obtains
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
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,
one obtains
Inserting the density of states from the Debye model, one obtains
From the above power series expansion of formula_19 follows that the mean square displacement at high temperatures is linear in temperature
The absence of formula_21 indicates that this is a classical result. Because formula_22 goes to zero for formula_23 it follows that for formula_24