# 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 $\sigma_a$, attenuating structural peaks like:

\begin{alignat}{2} G(q) & = e^{-\langle u^2 \rangle q^2} \\ & = e^{-\sigma_{\mathrm{rms}}^2q^2} \\ & = e^{-\sigma_a^2a^2q^2} \end{alignat}

Where $\sigma_{\mathrm{rms}} \equiv \sqrt{ \langle u^2 \rangle }$ is the root-mean-square displacement of the lattice-spacing a (such that the spacing at time t is $a+u(t)$), and $\sigma_a \equiv \sigma_{\mathrm{rms}}/a$ is the relative displacement.

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

$S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]$

And thus appears in the overall intensity as:

$I_{\mathrm{diffuse}}(q) = P(q) \left[ 1- G(q) \right]$

where $P(q)$ is the form factor.

In the high-q limit, form factors frequently exhibit a $q^{-4}$ scaling (c.f. sphere form factor), in which case one expects (since $G(q \rightarrow \infty) = 0$):

$I_{\mathrm{diffuse}}(q \rightarrow \infty) \propto q^{-4}$

(Which reproduces the scaling of the Porod law.)