In mathematics, the family of Debye functions is defined by
Contents
- Relation to other functions
- Series Expansion
- Limiting values
- The Debye model
- Internal energy and heat capacity
- Mean squared displacement
- Implementations
- References
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.
Relation to other functions
The Debye functions are closely related to the Polylogarithm.
Series Expansion
According to,
Limiting values
For
For
The Debye model
The Debye model has a density of vibrational states
with the Debye frequency ωD.
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