Paper:DNA-nanoparticle superlattices formed from anisotropic building blocks

This is a summary/discussion of the results from:

• Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin DNA-nanoparticle superlattices formed from anisotropic building blocks Nature Materials 2010, 9, 913-917 doi: 10.1038/nmat2870

This paper describes the formation of nanoparticle [[superlattices] from anisotropic nano-objects. In the Supplementary Information information, the authors describe how to model x-ray scattering data from lattices of anisotropic nanoparticles.

Summary of Mathematics

Randomly oriented crystals give scattering intensity:

${\begin{alignedat}{2}I(q)&=\langle |F(\mathbf {q} )|^{2}S(\mathbf {q} )\rangle \\&=P(q)\left\langle {\frac {|F(\mathbf {q} )|^{2}}{P(q)}}S(\mathbf {q} )\right\rangle \\&=P(q)S(q)\end{alignedat}}$

Where the structure factor is defined by an orientational average (randomly oriented crystal(s)):

$S(q)\equiv \left\langle {\frac {|F(\mathbf {q} )|^{2}}{P(q)}}S(\mathbf {q} )\right\rangle$

and can be computed by:

$S(q)={\frac {c}{q_{hkl}^{2}P(q_{hkl})}}\sum _{\{hkl\}}^{m_{hkl}}\left|F(M\cdot \mathbf {q} _{hkl})\sum _{i=1}^{n_{c}}e^{2\pi i(x_{i}h+y_{i}k+z_{i}l)}\right|^{2}e^{-\sigma _{D}^{2}q_{hkl}^{2}a^{2}}L_{hkl}(q-q_{hkl})$

Where c is a constant, and L is the peak shape; such as:

$L={\frac {\sigma _{L}/(2\pi )}{(q-q_{hkl})^{2}+(\sigma _{L}/2)^{2}}}$

The (isotropic) form factor intensity is an average over all possible particle orientations:

${\begin{alignedat}{2}P(q)&=\left\langle |F(\mathbf {q} )|^{2}\right\rangle \\&=\int \limits _{S}|F(\mathbf {q} )|^{2}\mathrm {d} \mathbf {s} \\&=\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi }|F(-q\sin \theta \cos \phi ,q\sin \theta \sin \phi ,q\cos \theta )|^{2}\sin \theta \mathrm {d} \theta \mathrm {d} \phi \end{alignedat}}$

The form factor amplitude is computed via:

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} F(\mathbf{q}) & = \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\ \end{alignat} }