Debye function on arvau

Wikipedia / Debye function

In mathematics, the family of Debye functions

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 is the -th Bernoulli number.

Limiting values

is the gamma function is the Riemann zeta function,

Derivative

is the Bernoulli function.

Applications in solid-state physics

The Debye model

The Debye model has a density of vibrational states with the .

Internal energy and heat capacity

Inserting with the Bose–Einstein distribution 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 In this expression, the mean squared displacement refers to just once Cartesian component of the vector that describes the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes, From the above power series indicates that this is a classical (zero-point motion).