Debye–Waller factor

The Debye -Waller factor ( DWF, by Peter Debye and Ivar Waller ) describes the temperature dependence of the intensity of the coherently elastically scattered radiation in a crystal lattice. Only these elastic scattering is subject to the Laue conditions; the complementary, inelastic scattering is referred to as thermal- diffuse.

In neutron scattering the term Debye -Waller factor is partially applied indiscriminately to coherent and incoherent scattering; part is for the latter but also used the more accurate term Lamb - Mössbauer factor.

Here, I0 the intensity of the incident wave, which is reduced by the factor of the exponential function to I. G is a reciprocal lattice vector, and u is the temperature-dependent oscillation of the atoms.

The Bragg reflections are even more attenuated due to the lattice vibrations, the higher the temperature and the higher is its order.

When considering a harmonic oscillator with the energy:

Can the temperature-dependent Debye- Waller factor written as follows:

Derivation

The structure factor is a measure for the relative intensity of the Miller indices, specific diffraction reflex.

The sum runs over all atoms of the base. This is a position vector pointing from a fixed reference point within the unit cell of the core -th atom, a reciprocal lattice vector and the atomic scattering factor of the - th atom:

The volume and the scattering power (eg electron density in X-ray diffraction, the charge density in electron diffraction ) of the - th atom.

Considering the thermal motion of atoms, it is time dependent. Now it decomposed into a mean residence ( equilibrium position, resting ) and the deflection ( time-dependent). The latter leads to the Debye-Waller factor.

The oscillation periods are very short ( s ) over the observation period, so there is always a time average is measured. The time average of the structure factor

For small deflections in developing the exponential to second order

The first order vanishes, since the displacements take place randomly in all directions in space ( temporal average of zero ) and not with the direction of correlated. The second order

This is the angle between and. Is averaged over all directions in three-dimensional space, so integration over the unit sphere:

Used in the exponential function, this gives:

The structure factor writes now:

For similar atoms is approximately equal for all. Thus, one can draw the second exponential factor before the sum:

Is the structure factor of the static case ( rigid lattice, no movement of atoms). The intensity is proportional to the squared magnitude of the structure factor. The time-averaged intensity is thus

The average intensity is lowered with respect to the static case of the Debye-Waller factor.

The Debye -Waller factor is at most 1, then when the atoms do not oscillate ( corresponds to the static case, approximately at K). At higher temperatures the exponential factor becomes larger, thus smaller. Due to thermal motion of the atoms, the reflexes are not broadened, but its intensity decreased. However, it appears a diffuse ground between the reflexes as a result of energy conservation.

The Debye -Waller factor and thus the intensity is also the smaller, is bigger, so the higher the Miller indices of the lattice planes, which takes place at the Bragg reflection are.