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.
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,[3]
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 (
zero-point motion).
Guseinov, I. I.; Mamedov, B. A. (2007). "Calculation of Integer and noninteger n-Dimensional Debye Functions using Binomial Coefficients and Incomplete Gamma Functions". Int. J. Thermophys. 28 (4): 1420–1426.
Bibcode:
2007IJT....28.1420G.
doi:
10.1007/s10765-007-0256-1.
S2CID120284032.
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.
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,[3]
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 (
zero-point motion).
Guseinov, I. I.; Mamedov, B. A. (2007). "Calculation of Integer and noninteger n-Dimensional Debye Functions using Binomial Coefficients and Incomplete Gamma Functions". Int. J. Thermophys. 28 (4): 1420–1426.
Bibcode:
2007IJT....28.1420G.
doi:
10.1007/s10765-007-0256-1.
S2CID120284032.