# Debye-Waller factor

The **Debye-Waller factor** is a term (in scattering equations) which accounts for how thermal fluctuations extinguish scattering intensity (especially high-*q* peaks). This scattering intensity then appears as diffuse scattering. Conceptually, thermal fluctuations create disorder, because the atoms/particles oscillate about their equilibrium positions and thus the lattice is never (instantaneously) perfect.

The same conceptual framework can be used to describe static disorder. A defective lattice where many particles are displaced from their idealized positions will cause the structural scattering to be weakened, with diffuse scattering appearing instead.

## Mathematical form

For a lattice-size *a*, the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width , attenuating structural peaks like:

Where is the root-mean-square displacement of the lattice-spacing *a* (such that the spacing at time *t* is ), and is the relative displacement.

Thus, the intensity of the structural peaks is multiplied by , which attenuates the higher-order (high-*q*) peaks, and redistributes this intensity into a diffuse scattering term, which appears in the structure factor () as:

And thus appears in the overall intensity as:

where is the form factor.

In the high-*q* limit, form factors frequently exhibit a scaling (c.f. sphere form factor), in which case one expects (since ):

(Which reproduces the scaling of the Porod law.)