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:

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} 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)):

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 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:

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 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:

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} 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:

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} }

Form Factors

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

• Pyramid
• Cube
• Cylinder
• Octahedron
• Rhombic dodecahedron (RD)
• Triangular prism