Paper:Scattering Curves of Ordered Mesoscopic Materials

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. Scattering Curves of Ordered Mesoscopic Materials Phys. Chem. B 2005, 109 (4), 1347–1360 doi: 10.1021/jp0467494

Mathematics

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

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(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 ) }

The b₁ and b₂ are the scattering lengths, which basically describes how strongly each material "scatters" the x-rays. So the 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 (b_1 - b_2)} 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:

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} & 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} }

Where P(q) is the form factor and S(q) is the structure factor. G(q) is a Debye-Waller factor for thermal disorder:

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) = e^{-\sigma_a^2a^2q^2} }

Z₀ is the lattice factor computed from a sum over reciprocal space peaks (Miller indices {hkl}):

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 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}) }

where the pre-factor is affected by the dimensionality, d, which also influences the projected volume 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 v_d} , the solid angle 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 \Omega_d} , and the lattice type, which influences the number of particles per unit cell, n. The sum over peaks {hkl} requires knowing the multiplicities (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 m_{hkl}} ), symmetry factors (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 f_{hkl}} ) and peak positions (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 q_{hkl}} ) for the given lattice type (BCC, FCC, etc.).