Difference between revisions of "Debye-Waller factor"

From GISAXS
Jump to: navigation, search
(Created page with "The '''Debye-Waller factor''' is a term (in scattering equations) which accounts for how thermal fluctuations extinguish scattering intensity (especially high-''q'' peaks). Th...")
 
Line 12: Line 12:
 
</math>
 
</math>
 
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:
 +
:<math>
 +
S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]
 +
</math>
 +
And thus appears in the overall intensity as:
 +
:<math>
 +
I_{\mathrm{diffuse}}(q) = P(q) \left[ 1- G(q) \right]
 +
</math>
 +
where <math>P(q)</math> is the [[form factor]].
  
  
 
==See Also==
 
==See Also==
 
* [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]
 
* [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]

Revision as of 21:38, 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 , attenuating structural peaks like:

Where is the root-mean-square displacement of the lattice-spacing a (such that the spacing at time t is ), and is the relative displacement.

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

And thus appears in the overall intensity as:

where is the form factor.


See Also