View source for 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 [http://www.nature.com/nmat/journal/v9/n11/full/nmat2870.html DNA-nanoparticle superlattices formed from anisotropic building blocks] ''Nature Materials'' '''2010''', ''9'', 913-917 [http://dx.doi.org/10.1038/nmat2870 doi: 10.1038/nmat2870]

This paper describes the formation of nanoparticle [[superlattices] from anisotropic nano-objects. In the [http://www.nature.com/nmat/journal/v9/n11/extref/nmat2870-s1.pdf Supplementary Information] information, the authors describe how to model x-ray scattering data from [[lattice]]s of anisotropic nanoparticles.

===Summary of Mathematics===
Randomly oriented crystals give scattering intensity:
:<math>
\begin{alignat}{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{alignat}
</math>

Where the '''[[structure factor]]''' is defined by an orientational average (randomly oriented crystal(s)):
:<math>
S(q) \equiv \left\langle \frac{|F(\mathbf{q})|^2}{P(q)} S(\mathbf{q}) \right\rangle 
</math>
and can be computed by:
:<math>
S(q) = \frac{c}{ q_{hkl}^2P(q_{hkl}) }
    \sum_{ \{hkl\} }^{m_{hkl} }
        \left|F(M \cdot \mathbf{q}_{hkl}) \sum_{i=1}^{n_c} e^{2\pi i(x_ih+y_ik+z_il)} \right|^2
        e^{-\sigma_D^2q_{hkl}^2a^2}
        L_{hkl}(q-q_{hkl})
</math>


Where ''c'' is a constant, and ''L'' is the [[peak shape|peak shape]]; such as:
:<math>
L = \frac{\sigma_L/(2\pi)}{(q-q_{hkl})^2 + (\sigma_L/2)^2}
</math>

The (isotropic) '''[[form factor]] intensity''' is an average over all possible particle orientations:
:<math>
\begin{alignat}{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{alignat}
</math>
The '''form factor amplitude''' is computed via:
:<math>
\begin{alignat}{2}

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

\end{alignat}
</math>

==Form Factors==
The SI also provides form factors for a variety of nano-object shapes:
* Pyramid
* Cube
* Cylinder
* Octahedron
* Rhombic dodecahedron (RD)
* Triangular prism