Form Factor:Ellipsoid of revolution - Revision history

JoachimWuttke: /* IsGISAXS */ I confirm that J_1 is a Bessel function

2020-05-11T14:17:59Z

‎IsGISAXS: I confirm that J_1 is a Bessel function

KevinYager: /* IsGISAXS */

2015-12-21T14:03:59Z

‎IsGISAXS

KevinYager: /* NCNR */

2014-06-18T14:35:43Z

‎NCNR

68.194.136.6 at 01:57, 5 June 2014

2014-06-05T01:57:26Z

68.194.136.6: /* Approximating by a Sphere */

2014-06-05T01:57:11Z

‎Approximating by a Sphere

68.194.136.6 at 01:56, 5 June 2014

2014-06-05T01:56:31Z

68.194.136.6: Created page with "==Equations== For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid..."

2014-06-05T01:53:49Z

Created page with "==Equations== For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid..."

New page

==Equations==
For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid aligned along the ''z''-direction (rotation about ''z''-axis, i.e. sweeping the <math>\phi</math> angle in spherical coordinates), such that the size in the ''xy''-plane is <math>R_r</math> and along ''z'' is <math>R_z = \epsilon R_r</math>. A useful quantity is <math>R_{\theta}</math>, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle <math>\theta</math> with respect to the ''z''-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given <math>\theta</math> angle, and provides the 'effective size' of the scattering object as seen by a ''q''-vector pointing in that direction.
::<math>
\begin{alignat}{2}
R_{\theta} & = \sqrt{ R_z^2 \cos^2 \theta + R_r^2(1- \cos^2\theta) } \\
& = R_r \sqrt{ 1 + (\epsilon^2-1) \cos^2 \theta } \\
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta }
\end{alignat}
</math>
The ellipsoid is also characterized by:
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3
</math>
===Form Factor Amplitude===
::<math>
F_{ell}(\mathbf{q}) = \left\{

\begin{array}{c l}

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

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

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

\begin{array}{c l}

18 \pi \Delta\rho^2 V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta
& \mathrm{when} \,\, q\neq0\\
4\pi \Delta\rho^2 V_{ell}^2
& \mathrm{when} \,\, q=0\\
\end{array}

\right.
</math>

==Sources==
====NCNR====
From [http://www.ncnr.nist.gov/resources/sansmodels/Ellipsoid.html NCNR SANS Models documentation]:
:<math>
\begin{alignat}{2}
P(q) & =\frac{ \rm{scale} }{ V_{ell} }(\rho_{ell}-\rho_{solv})^2 \int_0^1 f^2 [ qr_b(1+x^2(v^2-1))^{1/2} ] dx + bkg \\

f(z) & = 3 V_{ell} \frac{(\sin z - z \cos z)}{z^3} \\

V_{ell} & = \frac{4 \pi}{3} r_a r_b^2 \\
v & = \frac{r_a}{r_b} \\

\end{alignat}
</math>
* ''Parameters:''
*# <math>\rm{scale}</math> : Intensity scaling
*# <math>r_a</math> : rotation axis (Å)
*# <math>r_b</math> : orthogonal axis (Å)
*# <math>\rho_{ell}-\rho_{solv}</math> : scattering contrast (Å<sup>−2</sup>)
*# <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>
\begin{alignat}{2}
& P(q, R, \epsilon)= \int_{0}^{\pi / 2} F_{sphere}^2[q,r(R,\epsilon,\alpha)] \sin \alpha d\alpha \\
& r(R,\epsilon,\alpha) = R \left( \sin^2\alpha + \epsilon^2 \cos^2 \alpha \right)^{1/2}
\end{alignat}
</math>
Where:
:<math>F_{sphere} = \frac{ 3 \left[ \sin(qr)-qr \cos(qr) \right ] }{ (qr)^3 }
</math>
* ''Parameters:''
*# <math>R</math> : radius (Å)
*# <math>\epsilon R</math> : orthogonal size (Å)

====IsGISAXS====
From [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:
:<math> F_{ell}(\mathbf{q}, R, W, H, \alpha) = 2 \pi RWH \frac{ J_1 (\gamma) }{ \gamma } \sin_c(q_z H/2) \exp( i q_z H/2 )</math>
:<math>\gamma = \sqrt{ (q_x R)^2 + (q_y W)^2 } </math>
:<math> V_{ell} = \pi RWH, \, S_{anpy} = \pi R W , \, R_{anpy} = Max(R,W) </math>
Where (presumably) ''J'' is a [http://en.wikipedia.org/wiki/Bessel_function Bessel function]:
::<math> J_1(\gamma) = \frac{1}{\pi} \int_0^\pi \cos (\tau - x \sin \tau) \,\mathrm{d}\tau
</math>

====Sjoberg Monte Carlo Study====
From [http://scripts.iucr.org/cgi-bin/paper?S0021889899006640 Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics], Bo Sjöberg, J.Appl. Cryst. (1999), 32, 917-923. [http://dx.doi.org/10.1107/S0021889899006640 doi 10.1107/S0021889899006640]

:<math>F(\mathbf{q}) = 3 \frac{ \sin(qs) - qs \cos(qs) }{(qs)^3}
</math>
where:
::<math>s=\left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2}
</math>
where <math>\gamma</math> is the angle between <math>\mathbf{q}</math> and the ''a''-axis vector of the ellipsoid of revolution (which also has axes ''b'' = ''c''); <math>\cos\gamma</math> is the inner product of unit vectors parallel to <math>\mathbf{q}</math> and the ''a''-axis. In some sense, ''s'' is the 'equivalent size' of a sphere that would lead to the scattering for a particular <math>\mathbf{q}</math>: it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the <math>\mathbf{q}</math>-vector.

Note that for <math> a = \epsilon b</math>:

::<math>
\begin{alignat}{2}
s & = \left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\
& = \left[ b^2\epsilon^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\
& = b \left[ \epsilon^2\cos^2\gamma + (1-\cos^2\gamma) \right]^{1/2} \\
& = b \left[ \epsilon^2\cos^2\gamma + \sin^2\gamma \right]^{1/2} \\
& = b \left[ 1 + (\epsilon^2-1)\cos^2\gamma \right]^{1/2}
\end{alignat}
</math>

==Derivations==
===Form Factor===
For an ellipsoid oriented along the ''z''-axis, we denote the size in-plane (in ''x'' and ''y'') as <math>R_r</math> and the size along ''z'' as <math>R_z=\epsilon R_r</math>. The parameter <math>\epsilon</math> denotes the shape of the ellipsoid: <math>\epsilon=1</math> for a sphere, <math>\epsilon<1</math> for an oblate spheroid and <math>\epsilon>1</math> for a prolate spheroid. The volume is thus:

::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3
</math>
We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates <math>(r_{xy},z)</math> (where <math>r_{xy}</math> is a distance in the ''xy''-plane):
::<math>
\begin{alignat}{2}
r_{xy} & = R_r \sin\theta \\
z & = R_z \cos\theta = \epsilon R_r \cos\theta
\end{alignat}
</math>
Where <math>\theta</math> is the angle with the ''z''-axis. This lets us define a useful quantity, <math>R_{\theta}</math>, which is the distance to the point from the origin:
::<math>
\begin{alignat}{2}
R_{\theta}
& = \sqrt{ (R_r \sin\theta)^2 + (R_z \cos \theta)^2 } \\
& = \sqrt{ R_r^2 \sin^2\theta + \epsilon^2 R_r^2 \cos^2 \theta } \\
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta } \\
\end{alignat}
</math>

The form factor is:
:<math>
\begin{alignat}{2}

F_{ell}(\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_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta \mathrm{d}\phi \\

& = 2 \pi \int_{0}^{\pi} \left [ \int_{0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \right ] \sin\theta \mathrm{d}\theta \\

\end{alignat}</math>

Imagine instead that we compress/stretch the ''z'' dimension so that the ellipsoid becomes a sphere:
::<math>
\begin{alignat}{2}
x^{\prime} & = x \\
y^{\prime} & = y \\
z^{\prime} & = z R_r/R_z=z/\epsilon \\
r^{\prime} & = \left| \mathbf{r}^{\prime} \right| = r \frac{R_r}{R_{\gamma}} \\
\mathrm{d}V & = \mathrm{d}x\mathrm{d}y\mathrm{d}z = \mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\epsilon\mathrm{d}z^{\prime} = \epsilon \mathrm{d}V^{\prime}
\end{alignat}
</math>

This implies a coordinate transformation for the <math>\mathbf{q}</math>-vector of:
::<math>
\begin{alignat}{2}
q_x^{\prime} & = q_x \\
q_y^{\prime} & = q_y \\
q_z^{\prime} & = q_z R_z/R_r = q_z \epsilon \\
q^{\prime} & = \left| \mathbf{q}^{\prime} \right| = q \frac{R_{\gamma}}{R_r}
\end{alignat}
</math>
Where <math>R_{\gamma}</math> is the <math>R_{\theta}</math> relation for a ''q''-vector tilted at angle <math>\gamma</math> with respect to the ''z'' axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular <math>\mathbf{q}</math> vector sees a sphere-like scatterer with size (length-scale) given by <math>R_{\gamma}</math>.
:<math>
\begin{alignat}{2}
F_{ell}(\mathbf{q})
& = \epsilon \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r^{\prime}=0}^{R_r}
e^{i \mathbf{q}^{\prime} \cdot \mathbf{r}^{\prime} } r^{\prime 2} \mathrm{d}r^{\prime}
\sin\theta \mathrm{d}\theta \mathrm{d}\phi \\

& = 3 \left( \frac{4 \pi}{3} \epsilon R_r^3 \right) \frac{ \sin(q^{\prime} R_r) - q^{\prime} R_r \cos(q^{\prime} R_r) }{ (q^{\prime} R_r)^3 }

\end{alignat}</math>

We can then convert back:

:<math>
\begin{alignat}{2}
F_{ell}(\mathbf{q}) & = 3 V_{ell} \frac{ \sin(q R_{\gamma}) - q R_{\gamma} \cos(q R_{\gamma}) }{ (q R_{\gamma})^3 }
\end{alignat}
</math>

===Isotropic Form Factor Intensity===
To average over all possible orientations, we use:
:<math>
\begin{alignat}{2}
P_{ell}(q)
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F_{ell}(\mathbf{q}) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\
& = \int_{0}^{2\pi}\int_{0}^{\pi} \left| 3 V_{ell} \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\
& = 9 V_{ell}^2 \int_{0}^{2\pi}\mathrm{d}\phi \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta \\
& = 18 \pi V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta
\end{alignat}
</math>

← Older revision		Revision as of 01:57, 5 June 2014
Line 1:		Line 1:
−	An '''ellipsoid of revolution''' is a 'squashed' or 'stretched' sphere; technically an oblate or prolate spheroid, respectively.	+	An '''ellipsoid of revolution''' is a 'squashed' or 'stretched' [[Form Factor:Sphere\|sphere]]; technically an oblate or prolate spheroid, respectively.

	==Equations==		==Equations==

@@ Line 201: / Line 201: @@
 ==Approximating by a Sphere==
-One can approximate a spheroid using an isovolumic sphere of radius ''R''<sub>effective</sub>:
+One can approximate a spheroid using an isovolumic [[Form Factor:Sphere|sphere]] of radius ''R''<sub>effective</sub>:
 :<math>V_{ell}
    = \frac{ 4\pi }{ 3 } R_z R_r^2 </math>

@@ Line 85: / Line 85: @@
 :<math>\gamma = \sqrt{ (q_x R)^2 + (q_y W)^2 } </math>
 :<math> V_{ell} = \pi RWH, \, S_{anpy} = \pi R W , \, R_{anpy} = Max(R,W) </math>
-Where (presumably) ''J'' is a [http://en.wikipedia.org/wiki/Bessel_function Bessel function]:
+Where ''J'' is a [http://en.wikipedia.org/wiki/Bessel_function Bessel function]:
 ::<math> J_1(\gamma) = \frac{1}{\pi} \int_0^\pi \cos (\tau - x \sin \tau) \,\mathrm{d}\tau
 </math>

@@ Line 81: / Line 81: @@
 ====IsGISAXS====
-From [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:
+From [http://www.insp.jussieu.fr/oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:
 :<math> F_{ell}(\mathbf{q}, R, W, H, \alpha) = 2 \pi RWH \frac{ J_1 (\gamma) }{ \gamma } \sin_c(q_z H/2) \exp( i q_z H/2 )</math>
 :<math>\gamma = \sqrt{ (q_x R)^2 + (q_y W)^2 } </math>

@@ Line 63: / Line 63: @@
 *# <math>r_b</math> : orthogonal axis (Å)
 *# <math>\rho_{ell}-\rho_{solv}</math> : scattering contrast (Å<sup>−2</sup>)
-*# <math>\rm{background}</math> : incoherent background (cm<sup>−1</sup>)
+*# <math>\rm{background}</math> : incoherent [[background]] (cm<sup>−1</sup>)
 ====Pedersen====

@@ Line 1: / Line 1: @@
 ==Equations==
 For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid aligned along the ''z''-direction (rotation about ''z''-axis, i.e. sweeping the <math>\phi</math> angle in spherical coordinates), such that the size in the ''xy''-plane is <math>R_r</math> and along ''z'' is <math>R_z = \epsilon R_r</math>. A useful quantity is <math>R_{\theta}</math>, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle <math>\theta</math> with respect to the ''z''-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given <math>\theta</math> angle, and provides the 'effective size' of the scattering object as seen by a ''q''-vector pointing in that direction.
@@ Line 194: / Line 196: @@
    & = 9 V_{ell}^2 \int_{0}^{2\pi}\mathrm{d}\phi \int_{0}^{\pi} \left( \frac{  \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta \\
    & = 18 \pi V_{ell}^2 \int_{0}^{\pi} \left( \frac{  \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta
 \end{alignat}
 </math>