# Form Factor:Ellipsoid of revolution

An ellipsoid of revolution is a 'squashed' or 'stretched' sphere; technically an oblate or prolate spheroid, respectively.

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

{\displaystyle {\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:

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

### Form Factor Amplitude

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

### Isotropic Form Factor Intensity

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

## Sources

#### NCNR

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

{\displaystyle {\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:

${\displaystyle F_{sphere}={\frac {3\left[\sin(qr)-qr\cos(qr)\right]}{(qr)^{3}}}}$
• Parameters:
1. ${\displaystyle R}$ : radius (Å)
2. ${\displaystyle \epsilon R}$ : orthogonal size (Å)

#### IsGISAXS

${\displaystyle F_{ell}(\mathbf {q} ,R,W,H,\alpha )=2\pi RWH{\frac {J_{1}(\gamma )}{\gamma }}\sin _{c}(q_{z}H/2)\exp(iq_{z}H/2)}$
${\displaystyle \gamma ={\sqrt {(q_{x}R)^{2}+(q_{y}W)^{2}}}}$
${\displaystyle V_{ell}=\pi RWH,\,S_{anpy}=\pi RW,\,R_{anpy}=Max(R,W)}$

Where J is a Bessel function:

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

#### 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

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

where:

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

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

Note that for ${\displaystyle a=\epsilon b}$:

{\displaystyle {\begin{alignedat}{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{alignedat}}}

## Derivations

### Form Factor

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

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

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

{\displaystyle {\begin{alignedat}{2}r_{xy}&=R_{r}\sin \theta \\z&=R_{z}\cos \theta =\epsilon R_{r}\cos \theta \end{alignedat}}}

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

{\displaystyle {\begin{alignedat}{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{alignedat}}}

The form factor is:

{\displaystyle {\begin{alignedat}{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{alignedat}}}

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

{\displaystyle {\begin{alignedat}{2}x^{\prime }&=x\\y^{\prime }&=y\\z^{\prime }&=zR_{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{alignedat}}}

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

{\displaystyle {\begin{alignedat}{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{alignedat}}}

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

{\displaystyle {\begin{alignedat}{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{alignedat}}}

We can then convert back:

{\displaystyle {\begin{alignedat}{2}F_{ell}(\mathbf {q} )&=3V_{ell}{\frac {\sin(qR_{\gamma })-qR_{\gamma }\cos(qR_{\gamma })}{(qR_{\gamma })^{3}}}\end{alignedat}}}

### Isotropic Form Factor Intensity

To average over all possible orientations, we use:

{\displaystyle {\begin{alignedat}{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|3V_{ell}{\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}\right|^{2}\sin \theta \mathrm {d} \theta \mathrm {d} \phi \\&=9V_{ell}^{2}\int _{0}^{2\pi }\mathrm {d} \phi \int _{0}^{\pi }\left({\frac {\sin(qR_{\theta })-qR_{\theta }\cos(qR_{\theta })}{(qR_{\theta })^{3}}}\right)^{2}\sin \theta \mathrm {d} \theta \\&=18\pi 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 \end{alignedat}}}

## Approximating by a Sphere

One can approximate a spheroid using an isovolumic sphere of radius Reffective:

${\displaystyle V_{ell}={\frac {4\pi }{3}}R_{z}R_{r}^{2}}$
{\displaystyle {\begin{alignedat}{2}R_{\mathrm {effective} }&=\left({\frac {3V_{ell}}{4\pi }}\right)^{1/3}\\&=(R_{z}R_{r}^{2})^{1/3}\\\end{alignedat}}}