# 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*):

The *b _{1}* and

*b*are the scattering lengths, which basically describes how strongly each material "scatters" the x-rays. So the is the

_{2}*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:

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

*Z _{0}* is the lattice factor computed from a sum over reciprocal space peaks (Miller indices {hkl}):

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