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". 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 "u" 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