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 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 is the -th Bernoulli number.

Limiting values

If is the gamma function and is the Riemann zeta function, then, for,

Derivative

The derivative obeys the relation
where 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 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 of the vector 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 follows that the mean square displacement at high temperatures is linear in temperature
The absence of indicates that this is a classical result. Because goes to zero for it follows that for
.

Implementations