# Paper:Scattering Curves of Ordered Mesoscopic Materials

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## Mathematics

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

${\displaystyle I(q)=(b_{1}-b_{2})^{2}\rho _{N}\left(\left\langle F^{2}(q)\right\rangle +\left\langle F(q)\right\rangle ^{2}[\left\langle Z(q)\right\rangle -1]\right)}$

The b1 and b2 are the scattering lengths, which basically describes how strongly each material "scatters" the x-rays. So the ${\displaystyle (b_{1}-b_{2})}$ 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:

{\displaystyle {\begin{alignedat}{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{alignedat}}}

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

${\displaystyle G(q)=e^{-\sigma _{a}^{2}a^{2}q^{2}}}$

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

${\displaystyle Z_{0}(q)={\frac {(2\pi )^{d-1}c}{nv_{d}\Omega _{d}q^{d-1}}}\sum _{\{hkl\}}m_{hkl}f_{hkl}^{2}L_{hkl}(q-q_{hkl})}$

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