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.

Mathematical form

For a lattice-size a, the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_a} , attenuating structural peaks like:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(q \rightarrow \infty) = 1} ):

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

See Also

• Wikipedia: Debye-Waller factor