# Debye-Waller factor

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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 ${\displaystyle \sigma _{a}}$, attenuating structural peaks like:

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

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

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

And thus appears in the overall intensity as:

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

where ${\displaystyle P(q)}$ is the form factor.

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

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

(Which reproduces the scaling of the Porod law.)