KevinYager: /* Equations */

2014-11-14T16:57:00Z

‎Equations

KevinYager: /* NCNR */

2014-06-18T14:34:53Z

‎NCNR

KevinYager at 22:17, 3 June 2014

2014-06-03T22:17:29Z

KevinYager at 22:15, 3 June 2014

2014-06-03T22:15:51Z

KevinYager at 22:14, 3 June 2014

2014-06-03T22:14:55Z

KevinYager: Created page with "200px ==Equations== For spheres of radius ''R'' (volume $V_{sphere}=4\pi R^3/3$ ): ===Form Factor Amplitude=== :: $F_{sphere}(q) = \left\{..."$

2014-06-03T22:13:27Z

Created page with "200px ==Equations== For spheres of radius ''R'' (volume <math>V_{sphere}=4\pi R^3/3</math>): ===Form Factor Amplitude=== ::<math> F_{sphere}(q) = \left\{..."

New page

[[Image:Sphere.png|200px]]
==Equations==
For spheres of radius ''R'' (volume <math>V_{sphere}=4\pi R^3/3</math>):
===Form Factor Amplitude===
::<math>
F_{sphere}(q) = \left\{

\begin{array}{c l}

3 \Delta\rho V_{sphere} \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 }
& \mathrm{when} \,\, q\neq0\\
\Delta\rho V_{sphere}
& \mathrm{when} \,\, q=0 \\
\end{array}

\right.
</math>
===Isotropic Form Factor Intensity===

::<math>
P_{sphere}(q) = \left\{

\begin{array}{c l}

36 \pi \Delta\rho^2 V_{sphere}^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ (qR)^6 }

& \mathrm{when} \,\, q\neq0\\
4\pi \Delta\rho^2 V_{sphere}^2
& \mathrm{when} \,\, q=0\\
\end{array}

\right.
</math>

==Sources==
====NCNR====
From [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Sphere.html NCNR SANS Models documentation]:
:<math>P(q)=\frac{ \rm{scale} }{ V }\left[ \frac{ 3V(\Delta\rho)( \sin(qr)-qr \cos(qr)) }{ (qr)^3 } \right]^2 + \rm{background}</math>
* ''Parameters:''
*# <math>\rm{scale}</math> : Intensity scaling
*# <math>r</math> : sphere radius (Å)
*# <math>\Delta\rho</math> : scattering contrast (Å<sup>−2</sup>), <math>\Delta\rho = SLD_{core} - SLD_{solvent}</math>
*# <math>\rm{background}</math> : incoherent background (cm<sup>−1</sup>)

====Pedersen====
From Pedersen review, [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]
:<math> F(q, r)= \frac{ 3 \left[ \sin(qr)-qr \cos(qr) \right ] }{ (qr)^3 } </math>
* ''Parameters:''
*# <math>r</math> : sphere radius (Å)

====IsGISAXS====
From [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:
:<math> F(\mathbf{q}, r)=4\pi r^3 \frac{ \sin(qr)-qr \cos(qr) }{ (qr)^3 } \exp{(i q_z r)} </math>
:<math> V = \frac{4}{3} \pi r^3 , S = \pi r^2 </math>
* ''Parameters:''
*# <math>r</math> : sphere radius (Å)

==Code==
<source lang="python" line>
def sphere(self, q, r, scale=1.0, contrast=0.1, background=0.0):

V = (4/3)*numpy.pi*(r**3)

return (scale/V)*(( 3*V*contrast*(sin(q*r)-q*r*cos(q*r) )/( (q*r)**3 ) )**2) + background

</source>

==Derivations==
===Form Factor===
For a sphere of radius ''R'', the volume is:
::<math> V_{sphere} = \frac{4}{3} \pi R^3</math>
We can use a [http://en.wikipedia.org/wiki/Spherical_coordinate_system spherical coordinates], where <math>\theta</math> denotes the angle with respect to the <math>+q_z</math> axis, and <math>\phi</math> is the in-plane angle (i.e. with respect to the <math>+x</math> axis):
::<math>
\begin{alignat}{2}
& \mathbf{r} = (x,y,z) = (r\sin\theta\cos\phi , r\sin\theta\sin\phi , r\cos\theta) \\
& \mathbf{q} = (q_x,q_y,q_z) \\
& q = |\mathbf{q}|^2 = \sqrt{ q_x^2 + q_y^2 + q_z^2 }
\end{alignat}
</math>
Where the form factor is:
:<math>
\begin{alignat}{2}

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

& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r=0}^{R} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta \mathrm{d}\phi \\
\end{alignat}
</math>

We take advantage of spherical symmetry. E.g. we can rotate any '''q''' onto a particular axis, such as <math>q_z</math>. So that:
::<math> \mathbf{q}=(0,0,q_z) </math>
::<math>
\begin{alignat}{2}
\mathbf{q}\cdot\mathbf{r} & =q_x x + q_y y + q_z z \\
& = q_z z \\
& = q r \cos\theta
\end{alignat}
</math>
And so:
:<math>
\begin{alignat}{2}
F_{sphere}(q) & = \int_{0}^{2\pi} \mathrm{d}\phi \int_{0}^{\pi}\int_{0}^{R} e^{i q r \cos\theta } r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta \\
& = [2\pi] \int_{0}^{\pi}\int_{0}^{R} ( \cos(q r \cos\theta) + i \sin(q r \cos\theta) ) r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta
\end{alignat}
</math>
A simple variable substitution:
::<math>u = q r \cos\theta</math>
::<math>\mathrm{d}u = - q r \sin\theta \mathrm{d}\theta</math>
Yields:
:<math>
\begin{alignat}{2}
F_{sphere}(q) & = 2\pi \int_{0}^{R} r^2 \left[\int_{0}^{\pi}( \cos(u) + i \sin(u) )\frac{-\mathrm{d}u}{q r} \right] \mathrm{d}r \\
& = 2\pi \int_{0}^{R} r^2 \frac{-1}{qr} \left[ \sin(u) - i \cos(u) \right]_{\theta=0}^{\pi} \mathrm{d}r \\
& = 2\pi \int_{0}^{R} \frac{-r}{q} \left[ \sin(q r \cos\theta) - i \cos(q r \cos\theta) \right]_{\theta=0}^{\pi} \mathrm{d}r \\
& = 2\pi \int_{0}^{R} \frac{-r}{q} \left[ \sin(- q r ) - i \cos(- q r) - \sin(q r) + i \cos(q r) \right] \mathrm{d}r \\
& = 2\pi \int_{0}^{R} \frac{r}{q} \left[ 2 \sin(q r ) \right] \mathrm{d}r \\
& = \frac{4\pi}{q} \int_{0}^{R} r \sin(q r ) \mathrm{d}r \\
\end{alignat}
</math>
Using [http://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions#Integrands_involving_only_sine the fact that]:
::<math>\int x\sin ax\;dx = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C\,\!</math>
We integrate:
:<math>
\begin{alignat}{2}
F_{sphere}(q) & = \frac{4\pi}{q} \left[ \frac{\sin(qr)}{q^2} - \frac{r\cos(qr)}{q} \right]_{r=0}^R \\
& = \frac{4\pi}{q} \left[ \frac{\sin(qR)}{q^2} - \frac{R\cos(qR)}{q} - \frac{\sin(0)}{q^2} + \frac{0\cos(q0)}{q} \right] \\
& = \frac{4\pi}{1} \left[ \frac{\sin(qR)}{q^3} - \frac{R\cos(qR)}{q^2} - 0 + 0 \right] \\
& = 4\pi \left[ \frac{\sin(qR)}{q^3} - \frac{R\cos(qR)}{q^2} \right] \\
& = 4\pi R^3 \left[ \frac{\sin(qR)}{q^3R^3} - \frac{qR\cos(qR)}{q^3R^3} \right] \\
& = \frac{3\times4\pi R^3}{3} \left[ \frac{\sin(qR)-qRcos(qR)}{q^3R^3} \right] \\
& = 3 V_{sphere} \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 }

\end{alignat}
</math>

===Form Factor at ''q''=0===
At very small q:
:<math>
\begin{alignat}{2}

\lim_{q\to0}F_{sphere}(q)
& = \frac{3 V_{sphere}}{R^3} \lim_{q\to0} \frac{ \sin(qR)- qR \cos(qR) }{ q^3 } \\
& = 4\pi \lim_{q\to0} \frac{ \sin(qR)- qR \cos(qR) }{ q^3 } \\

& = 4\pi \lim_{q\to0} \frac{ 1 }{q^3}\left[
qR-\frac{(qR)^3}{3!}+...
\right]- \frac{R}{q^2} \left[
1-\frac{(qR)^2}{2!}+...
\right] \\

& = 4\pi \lim_{q\to0}
\frac{R}{q^2}-\frac{R^3}{3!}+\frac{O((qR)^5)}{q^3}
-\frac{R}{q^2}+\frac{R^3}{2!}+\frac{R\times O((qR)^4)}{q^2}
\\

& = 4\pi \lim_{q\to0}
R^3\left( \frac{1}{2}-\frac{1}{6}\right)
+ O(q^2)
\\

& = 4\pi \lim_{q\to0}
\frac{R^3}{3}
+ O(q^2)
\\

& = \frac{4\pi R^3}{3} \\
& = V_{sphere} \\

\end{alignat}
</math>

===Isotropic Form Factor Intensity===
To average over all possible orientations, we use:
:<math>
\begin{alignat}{2}
P(q) & = \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>
For a sphere:
:<math>
\begin{alignat}{2}
P_{sphere}(q) & = \int_{0}^{2\pi}\int_{0}^{\pi} | F_{sphere}(q) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\
& = \int_{0}^{2\pi}\int_{0}^{\pi} \left| 3 V_{sphere} \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 } \right|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi
\end{alignat}
</math>
Note that the spherical symmetry guarantees that the integrand does not depend on <math>\phi</math> or <math>\theta</math>:
:<math>
\begin{alignat}{2}
P_{sphere}(q) & = \left( 3 V_{sphere} \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 } \right)^2 \int_{0}^{2\pi}\int_{0}^{\pi} \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\
& = 3^2 V_{sphere}^2 \left( \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 } \right)^2 \left[\int_{0}^{2\pi}\mathrm{d}\phi\right]\left[\int_{0}^{\pi} \sin\theta\mathrm{d}\theta\right] \\
& = 9 V_{sphere}^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ (qR)^6 } \left[ 2\pi \right]\left[ 2 \right] \\
& = 36 \pi V_{sphere}^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ (qR)^6 } \\
\end{alignat}
</math>
===Isotropic Form Factor Intensity at ''q''=0===
At ''q''=0, we expect:
:<math>
P_{sphere}\left(0\right) = 4 \pi V_{sphere}^2
</math>

===Isotropic Form Factor Intensity at large ''q''===
Note that:
:<math>
\begin{alignat}{2}
P_{sphere}(q)
& = 36 \pi V_{sphere}^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ (qR)^6 } \\
& = 36 \pi \left( \frac{4 \pi R^3}{3} \right)^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ q^6 R^6 } \\
& = 64 \pi^3 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ q^6 } \\
\end{alignat}
</math>
For large ''q'', the <math>-q R</math> term dominates the numerator:
:<math>
\begin{alignat}{2}
\lim_{q \rightarrow \infty} P_{sphere}(q)
& = \lim_{q \rightarrow \infty} 64 \pi^3 \frac{ (\sin(qR) - qR \cos(qR))^2 }{ q^6 } \\
& = \lim_{q \rightarrow \infty} 64 \pi^3 \frac{ q^2 R^2 \cos^2(qR) }{ q^6 } \\
& = 64 \pi^3 R^2 \lim_{q \rightarrow \infty} \frac{ \cos^2(qR) }{ q^4 } \\
\end{alignat}
</math>
The oscillation of the numerator is overwhelmed by the decay of the denominator:
:<math>
\begin{alignat}{2}
\lim_{q \rightarrow \infty} P_{sphere}(q)
& \approx \frac{ 64 \pi^3 R^2 }{ q^4 } \\
\end{alignat}
</math>

@@ Line 1: / Line 1: @@
 [[Image:Sphere.png|200px|right]]
-This page provides the equations for calculating the form factor of a sphere (including derivations).
+This page provides the equations for calculating the [[form factor]] of a sphere (including derivations).
 ==Equations==
 For spheres of radius ''R'' (volume <math>V_{sphere}=4\pi R^3/3</math>):

@@ Line 1: / Line 1: @@
-[[Image:Sphere.png|200px]]
+[[Image:Sphere.png|200px|right]]
 This page provides the equations for calculating the form factor of a sphere (including derivations).
 ==Equations==

@@ Line 34: / Line 34: @@
 \right.
 </math>
 ==Sources==

@@ Line 43: / Line 43: @@
 *# <math>r</math> : sphere radius (Å)
 *# <math>\Delta\rho</math> : scattering contrast (Å<sup>−2</sup>), <math>\Delta\rho = SLD_{core} - SLD_{solvent}</math>
-*# <math>\rm{background}</math> : incoherent background (cm<sup>−1</sup>)
+*# <math>\rm{background}</math> : incoherent [[background]] (cm<sup>−1</sup>)
 ====Pedersen====