Form Factor:Ellipsoid of revolution

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 $\phi$ angle in spherical coordinates), such that the size in the xy-plane is $R_{r}$ and along z is $R_{z}=\epsilon R_{r}$. A useful quantity is $R_{\theta }$, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle $\theta$ with respect to the z-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given $\theta$ angle, and provides the 'effective size' of the scattering object as seen by a q-vector pointing in that direction.

${\begin{alignedat}{2}R_{\theta }&={\sqrt {R_{z}^{2}\cos ^{2}\theta +R_{r}^{2}(1-\cos ^{2}\theta )}}\\&=R_{r}{\sqrt {1+(\epsilon ^{2}-1)\cos ^{2}\theta }}\\&=R_{r}{\sqrt {\sin ^{2}\theta +\epsilon ^{2}\cos ^{2}\theta }}\end{alignedat}}$

The ellipsoid is also characterized by:

$V_{ell}={\frac {4\pi }{3}}R_{z}R_{r}^{2}={\frac {4\pi }{3}}\epsilon R_{r}^{3}$

Form Factor Amplitude

$F_{ell}(\mathbf {q} )=\left\{{\begin{array}{c l}3\Delta \rho V_{ell}{\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}&\mathrm {when} \,\,q\neq 0\\\Delta \rho V_{ell}&\mathrm {when} \,\,q=0\\\end{array}}\right.$

Isotropic Form Factor Intensity

$P_{ell}(q)=\left\{{\begin{array}{c l}18\pi \Delta \rho ^{2}V_{ell}^{2}\int _{0}^{\pi }\left({\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}\right)^{2}\sin \theta \mathrm {d} \theta &\mathrm {when} \,\,q\neq 0\\4\pi \Delta \rho ^{2}V_{ell}^{2}&\mathrm {when} \,\,q=0\\\end{array}}\right.$

Sources

NCNR

From NCNR SANS Models documentation:

${\begin{alignedat}{2}P(q)&={\frac {\rm {scale}}{V_{ell}}}(\rho _{ell}-\rho _{solv})^{2}\int _{0}^{1}f^{2}[qr_{b}(1+x^{2}(v^{2}-1))^{1/2}]dx+bkg\\f(z)&=3V_{ell}{\frac {(\sin z-z\cos z)}{z^{3}}}\\V_{ell}&={\frac {4\pi }{3}}r_{a}r_{b}^{2}\\v&={\frac {r_{a}}{r_{b}}}\\\end{alignedat}}$

• Parameters:
  1. ${\rm {scale}}$ : Intensity scaling
  2. $r_{a}$ : rotation axis (Å)
  3. $r_{b}$ : orthogonal axis (Å)
  4. $\rho _{ell}-\rho _{solv}$ : scattering contrast (Å⁻²)
  5. ${\rm {background}}$ : incoherent background (cm⁻¹)

Pedersen

From Pedersen review, 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. doi: 10.1016/S0001-8686(97)00312-6

${\begin{alignedat}{2}&P(q,R,\epsilon )=\int _{0}^{\pi /2}F_{sphere}^{2}[q,r(R,\epsilon ,\alpha )]\sin \alpha d\alpha \\&r(R,\epsilon ,\alpha )=R\left(\sin ^{2}\alpha +\epsilon ^{2}\cos ^{2}\alpha \right)^{1/2}\end{alignedat}}$

Where:

$F_{sphere}={\frac {3\left[\sin(qr)-qr\cos(qr)\right]}{(qr)^{3}}}$

• Parameters:
  1. $R$ : radius (Å)
  2. $\epsilon R$ : orthogonal size (Å)

IsGISAXS

From IsGISAXS, Born form factors:

$F_{ell}(\mathbf {q} ,R,W,H,\alpha )=2\pi RWH{\frac {J_{1}(\gamma )}{\gamma }}\sin _{c}(q_{z}H/2)\exp(iq_{z}H/2)$

$\gamma ={\sqrt {(q_{x}R)^{2}+(q_{y}W)^{2}}}$

$V_{ell}=\pi RWH,\,S_{anpy}=\pi RW,\,R_{anpy}=Max(R,W)$

Where (presumably) J is a Bessel function:

$J_{1}(\gamma )={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\tau -x\sin \tau )\,\mathrm {d} \tau$

Sjoberg Monte Carlo Study

From 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. doi 10.1107/S0021889899006640

$F(\mathbf {q} )=3{\frac {\sin(qs)-qs\cos(qs)}{(qs)^{3}}}$

where:

$s=\left[a^{2}\cos ^{2}\gamma +b^{2}(1-\cos ^{2}\gamma )\right]^{1/2}$

where $\gamma$ is the angle between $\mathbf {q}$ and the a-axis vector of the ellipsoid of revolution (which also has axes b = c); $\cos \gamma$ is the inner product of unit vectors parallel to $\mathbf {q}$ and the a-axis. In some sense, s is the 'equivalent size' of a sphere that would lead to the scattering for a particular $\mathbf {q}$: it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the $\mathbf {q}$-vector.

Note that for $a=\epsilon b$:

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} 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} }

Derivations

Form Factor

For an ellipsoid oriented along the z-axis, we denote the size in-plane (in x and y) as $R_{r}$ and the size along z as $R_{z}=\epsilon R_{r}$. The parameter 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 \epsilon} denotes the shape of the ellipsoid: $\epsilon =1$ for a sphere, 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 \epsilon<1} for an oblate spheroid and $\epsilon >1$ for a prolate spheroid. The volume is thus:

$V_{ell}={\frac {4\pi }{3}}R_{z}R_{r}^{2}={\frac {4\pi }{3}}\epsilon R_{r}^{3}$

We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates $(r_{xy},z)$ (where 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 r_{xy}} is a distance in the xy-plane):

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} r_{xy} & = R_r \sin\theta \\ z & = R_z \cos\theta = \epsilon R_r \cos\theta \end{alignat} }

Where 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 \theta} is the angle with the z-axis. This lets us define a useful quantity, 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 R_{\theta}} , which is the distance to the point from the origin:

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} 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} }

The form factor is:

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_{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}}

Imagine instead that we compress/stretch the z dimension so that the ellipsoid becomes a sphere:

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} 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} }

This implies a coordinate transformation for the 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 \mathbf{q}} -vector of:

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} 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} }

Where 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 R_{\gamma}} is the 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 R_{\theta}} relation for a q-vector tilted at angle 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 \gamma} with respect to the z axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular 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 \mathbf{q}} vector sees a sphere-like scatterer with size (length-scale) given 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 R_{\gamma}} .

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_{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}}

We can then convert back:

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_{ell}(\mathbf{q}) & = 3 V_{ell} \frac{ \sin(q R_{\gamma}) - q R_{\gamma} \cos(q R_{\gamma}) }{ (q R_{\gamma})^3 } \end{alignat} }

Isotropic Form Factor Intensity

To average over all possible orientations, we use:

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_{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} }