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 $\sigma _{a}$, 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 (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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.

See Also

• Wikipedia: Debye-Waller factor