Difference between revisions of "Debye-Waller factor"
| Line 13: | Line 13: | ||
Where <math>\sigma_{\mathrm{rms}} \equiv \sqrt{ \langle u^2 \rangle }</math> is the [http://en.wikipedia.org/wiki/Root_mean_square root-mean-square] displacement of the lattice-spacing ''a'' (such that the spacing at time ''t'' is <math>a+u(t)</math>), and <math>\sigma_a \equiv \sigma_{\mathrm{rms}}/a</math> is the relative displacement. | Where <math>\sigma_{\mathrm{rms}} \equiv \sqrt{ \langle u^2 \rangle }</math> is the [http://en.wikipedia.org/wiki/Root_mean_square root-mean-square] displacement of the lattice-spacing ''a'' (such that the spacing at time ''t'' is <math>a+u(t)</math>), and <math>\sigma_a \equiv \sigma_{\mathrm{rms}}/a</math> is the relative displacement. | ||
| − | Thus, the intensity of the structural peaks is multiplied by <math>G(q)</math>, which attenuates the higher-order (high-''q'') peaks, and redistributes this intensity into a diffuse scattering term, which appears in the [[structure factor]] (<math>S(q)</math>) as: | + | Thus, the intensity of the structural peaks is multiplied by <math>G(q)</math>, which attenuates the higher-order (high-''q'') peaks, and redistributes this intensity into a [[diffuse scattering]] term, which appears in the [[structure factor]] (<math>S(q)</math>) as: |
:<math> | :<math> | ||
S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right] | S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right] | ||
Revision as of 21:52, 3 June 2014
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 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_{\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 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 a+u(t)} ), and 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 \equiv \sigma_{\mathrm{rms}}/a} is the relative displacement.
Thus, the intensity of the structural peaks is multiplied by 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)} , 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 (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 S(q)} ) as:
- 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 S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right] }
And thus appears in the overall intensity as:
- 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 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.
