# 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{alignedat}{2}G(q)&=e^{-\langle u^{2}\rangle q^{2}}\\&=e^{-\sigma _{\mathrm {rms} }^{2}q^{2}}\\&=e^{-\sigma _{a}^{2}a^{2}q^{2}}\end{alignedat}} 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.)