Debye-Waller factor - Revision history

KevinYager at 14:01, 29 October 2014

2014-10-29T14:01:21Z

KevinYager at 15:25, 14 June 2014

2014-06-14T15:25:34Z

66.249.83.102: /* Mathematical form */

2014-06-09T21:21:41Z

‎Mathematical form

KevinYager: /* Mathematical form */

2014-06-04T21:26:25Z

‎Mathematical form

68.194.136.6: /* Mathematical form */

2014-06-04T02:01:59Z

‎Mathematical form

68.194.136.6: /* Mathematical form */

2014-06-04T01:58:42Z

‎Mathematical form

68.194.136.6 at 01:52, 4 June 2014

2014-06-04T01:52:19Z

68.194.136.6 at 01:38, 4 June 2014

2014-06-04T01:38:11Z

68.194.136.6: 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..."

2014-06-04T00:32:34Z

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..."

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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 <math>\sigma_a</math>, attenuating structural peaks like:

:<math>
\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}
</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.

==See Also==
* [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]

← Older revision		Revision as of 14:01, 29 October 2014
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.		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.

@@ 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.
 ==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.
-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>
 S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]

@@ Line 23: / Line 23: @@
 where <math>P(q)</math> is the [[form factor]].
-In the high-''q'' limit, form factors frequently exhibit a <math>q^{-4}</math> scaling (c.f. [[Form_Factor:Sphere#Isotropic_Form_Factor_Intensity_at_large_q|sphere form factor]]), in which case one expects (since <math>G(q \rightarrow \infty) = 1</math>):
+In the high-''q'' limit, form factors frequently exhibit a <math>q^{-4}</math> scaling (c.f. [[Form_Factor:Sphere#Isotropic_Form_Factor_Intensity_at_large_q|sphere form factor]]), in which case one expects (since <math>G(q \rightarrow \infty) = 0</math>):
 :<math>
 I_{\mathrm{diffuse}}(q \rightarrow \infty) \propto q^{-4}

@@ 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.
-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>
 S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]

@@ Line 2: / Line 2: @@
 ==Mathematical form==
-For a lattice-size ''a'', the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width <math>\sigma_a</math>, attenuating structural peaks like:
+For a [[lattice]]-size ''a'', the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width <math>\sigma_a</math>, attenuating structural peaks like:
 :<math>

@@ Line 27: / Line 27: @@
 I_{\mathrm{diffuse}}(q \rightarrow \infty) \propto q^{-4}
 </math>
 ==See Also==
 * [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]