Paper:DNA-nanoparticle superlattices formed from anisotropic building blocks - Revision history

KevinYager: /* Summary of Mathematics */

2015-01-14T21:49:19Z

‎Summary of Mathematics

KevinYager: /* Form Factors */

2014-12-03T19:22:48Z

‎Form Factors

KevinYager at 19:21, 3 December 2014

2014-12-03T19:21:21Z

KevinYager: /* Summary of Mathematics */

2014-10-15T19:09:51Z

‎Summary of Mathematics

KevinYager: /* Summary of Mathematics */

2014-10-15T19:05:51Z

‎Summary of Mathematics

KevinYager at 19:04, 15 October 2014

2014-10-15T19:04:42Z

KevinYager: Created page with "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 [ht..."

2014-10-15T19:02:23Z

Created page with "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 [ht..."

New page

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 [[X-ray peak shape|peak shape]].

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>

@@ Line 33: / Line 33: @@
 </math>
-The (isotropic) '''[[form factor]] intensity''' is an average over all possible particle orientations:
+Note that the presented form of <math>\scriptstyle S(q)</math> is closely-related to the [[lattice factor]]. The (isotropic) '''[[form factor]] intensity''' is an average over all possible particle orientations:
 :<math>
 \begin{alignat}{2}

@@ Line 52: / Line 52: @@
 ==Form Factors==
 The SI also provides form factors for a variety of nano-object shapes:
-* Pyramid
+* [[Form Factor:Pyramid|Pyramid]]
-* Cube
+* [[Form Factor:Cube|Cube]]
-* Cylinder
+* [[Form Factor:Cylinder|Cylinder]]
-* Octahedron
+* [[Form Factor:Octahedron|Octahedron]]
 * Rhombic dodecahedron (RD)
 * Triangular prism

← Older revision		Revision as of 19:09, 15 October 2014
Line 49:		Line 49:
	\end{alignat}		\end{alignat}
	</math>		</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

← Older revision		Revision as of 19:04, 15 October 2014
Line 28:		Line 28:


−	Where ''c'' is a constant, and ''L'' is the [[~~X-ray~~ peak shape\|peak shape]].	+	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:		The (isotropic) '''form factor intensity''' is an average over all possible particle orientations:

@@ Line 2: / Line 2: @@
 * 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.
+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===

@@ Line 14: / Line 14: @@
 </math>
-Where the '''structure factor''' is defined by an orientational average (randomly oriented crystal(s)):
+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
@@ Line 33: / Line 33: @@
 </math>
-The (isotropic) '''form factor intensity''' is an average over all possible particle orientations:
+The (isotropic) '''[[form factor]] intensity''' is an average over all possible particle orientations:
 :<math>
 \begin{alignat}{2}