KevinYager: /* Mathematics */

2014-10-29T13:58:29Z

‎Mathematics

KevinYager: Created page with "This is a summary/discussion of the results from: * S. Förster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney, R. Knott, J. [http://pubs.acs.org/do..."

2014-10-15T19:20:59Z

Created page with "This is a summary/discussion of the results from: * S. Förster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney, R. Knott, J. [http://pubs.acs.org/do..."

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This is a summary/discussion of the results from:
* S. Förster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney, R. Knott, J. [http://pubs.acs.org/doi/abs/10.1021/jp0467494 Scattering Curves of Ordered Mesoscopic Materials] ''Phys. Chem. B'' '''2005''', 109 (4), 1347–1360 [http://dx.doi.org/10.1021/jp0467494 doi: 10.1021/jp0467494]

==Mathematics==
Equation (1) describes the general scattered intensity from particles (phase ''1'') in a matrix (phase ''2''):

:<math>
I(q) = (b_1 - b_2)^2 \rho_N \left ( \left \langle F^2(q)\right \rangle + \left \langle F(q) \right\rangle ^2 [ \left \langle Z(q)\right\rangle - 1 ] \right )
</math>

The ''b1'' and ''b2'' are the scattering lengths, which basically describes how strongly each material "scatters" the x-rays. So the <math>(b_1 - b_2)</math> is the ''scattering contrast''. The ''F(q)'' is the Fourier transform of the particle form (related to the "Form Factor") and ''Z(q)'' is the [[lattice factor]] that describes the spatial distribution of the particles (related the "Structure Factor").

Equation (30) (with Equation (2)) recast this slightly:
:<math>
\begin{alignat}{2}
& I(q) = \left (b_1 - b_2 \right)^2 \rho_N P(q)S(q) \\
& P(q) = F^2(q) \\
& S(q) = 1 + \beta (q) \left ( Z_0(q) - 1 \right ) G(q)\\
\end{alignat}
</math>
Where ''P(q)'' is the '''[[form factor]]''' and ''S(q)'' is the '''[[structure factor]]'''. ''G(q)'' is a [[Debye-Waller factor]] for thermal disorder:
:<math>
G(q) = e^{-\sigma_a^2a^2q^2}
</math>
''Z0'' is the lattice factor computed from a sum over reciprocal space peaks ([[Miller indices]] {hkl}):
:<math>
Z_0(q) = \frac{(2\pi)^{d-1}c}{nv_d\Omega_dq^{d-1}}\sum_{\{hkl\}} m_{hkl}f_{hkl}^2L_{hkl}(q-q_{hkl})
</math>
where the pre-factor is affected by the dimensionality, ''d'', which also influences the projected volume <math>v_d</math>, the solid angle <math>\Omega_d</math>, and the lattice type, which influences the number of particles per [[unit cell]], ''n''. The sum over peaks {''hkl''} requires knowing the multiplicities (<math>m_{hkl}</math>), symmetry factors (<math>f_{hkl}</math>) and peak positions (<math>q_{hkl}</math>) for the given [[lattice]] type ([[Lattice:BCC|BCC]], [[Lattice:FCC|FCC]], etc.).

@@ Line 9: / Line 9: @@
 </math>
-The ''b<sub>1</sub>'' and ''b<sub>2</sub>'' are the scattering lengths, which basically describes how strongly each material "scatters" the x-rays. So the <math>(b_1 - b_2)</math> is the ''scattering contrast''. The ''F(q)'' is the Fourier transform of the particle form (related to the "Form Factor") and ''Z(q)'' is the [[lattice factor]] that describes the spatial distribution of the particles (related the "Structure Factor").
+The ''b<sub>1</sub>'' and ''b<sub>2</sub>'' are the scattering lengths, which basically describes how strongly each material "scatters" the x-rays. So the <math>(b_1 - b_2)</math> is the ''scattering contrast''. The ''F(q)'' is the [[Fourier transform]] of the particle form (related to the "[[Form Factor]]") and ''Z(q)'' is the [[lattice factor]] that describes the spatial distribution of the particles (related the "[[Structure Factor]]").
 Equation (30) (with Equation (2)) recast this slightly:

Paper:Scattering Curves of Ordered Mesoscopic Materials - Revision history

KevinYager: /* Mathematics */

KevinYager: Created page with "This is a summary/discussion of the results from: * S. Förster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney, R. Knott, J. [http://pubs.acs.org/do..."