Difference between revisions of "Debye-Waller factor"

From GISAXS
Jump to: navigation, search
(Mathematical form)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
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 '''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==
 
==Mathematical form==
Line 13: Line 15:
 
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 [[Tutorial:Qualitative_inspection#Higher_Orders|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]

Latest revision as of 10:01, 29 October 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.

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 , 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.

In the high-q limit, form factors frequently exhibit a scaling (c.f. sphere form factor), in which case one expects (since ):

(Which reproduces the scaling of the Porod law.)

See Also