# 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 $\phi$ 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

{\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

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

{\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 $R_{r}$ and the size along z as $R_{z}=\epsilon R_{r}$ . The parameter $\epsilon$ denotes the shape of the ellipsoid: $\epsilon =1$ for a sphere, $\epsilon <1$ 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 $r_{xy}$ is a distance in the xy-plane):

{\begin{alignedat}{2}r_{xy}&=R_{r}\sin \theta \\z&=R_{z}\cos \theta =\epsilon R_{r}\cos \theta \end{alignedat}} Where $\theta$ is the angle with the z-axis. This lets us define a useful quantity, $R_{\theta }$ , which is the distance to the point from the origin:

{\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:

{\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:

{\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 $\mathbf {q}$ -vector of:

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

{\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:

{\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:

{\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:

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