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

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 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 R_effective:

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

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