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)):

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:

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) = \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:

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} }

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:

${\begin{alignedat}{2}F(\mathbf {q} )&=\int \limits _{V}e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} \mathbf {r} \\\end{alignedat}}$

Form Factors

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

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