# Paper:DNA-nanoparticle superlattices formed from anisotropic building blocks

This is a summary/discussion of the results from:

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{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}

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:

$S(q) = \frac{c}{ q_{hkl}^2P(q_{hkl}) } \sum_{ \{hkl\} }^{m_{hkl} } \left|F(M \cdot \mathbf{q}_{hkl}) \sum_{i=1}^{n_c} e^{2\pi i(x_ih+y_ik+z_il)} \right|^2 e^{-\sigma_D^2q_{hkl}^2a^2} L_{hkl}(q-q_{hkl})$

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

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

Note that the presented form of $\scriptstyle S(q)$ is closely-related to the lattice factor. The (isotropic) form factor intensity is an average over all possible particle orientations:

\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}

The form factor amplitude is computed via:

\begin{alignat}{2} F(\mathbf{q}) & = \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\ \end{alignat}

## Form Factors

The SI also provides form factors for a variety of nano-object shapes: