An ellipsoid of revolution is a 'squashed' or 'stretched' sphere; technically an oblate or prolate spheroid, respectively.
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 angle in spherical coordinates), such that the size in the xy-plane is and along z is . A useful quantity is , which is the distance from the origin to the surface of the ellipsoid for a line titled at angle with respect to the z-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given angle, and provides the 'effective size' of the scattering object as seen by a q-vector pointing in that direction.
The ellipsoid is also characterized by:
Form Factor Amplitude
Isotropic Form Factor Intensity
From NCNR SANS Models documentation:
- : Intensity scaling
- : rotation axis (Å)
- : orthogonal axis (Å)
- : scattering contrast (Å−2)
- : incoherent background (cm−1)
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
- : radius (Å)
- : orthogonal size (Å)
From IsGISAXS, Born form factors:
Where J is a Bessel function:
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
where is the angle between and the a-axis vector of the ellipsoid of revolution (which also has axes b = c); is the inner product of unit vectors parallel to and the a-axis. In some sense, s is the 'equivalent size' of a sphere that would lead to the scattering for a particular : it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the -vector.
Note that for :
For an ellipsoid oriented along the z-axis, we denote the size in-plane (in x and y) as and the size along z as . The parameter denotes the shape of the ellipsoid: for a sphere, for an oblate spheroid and for a prolate spheroid. The volume is thus:
We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates (where is a distance in the xy-plane):
Where is the angle with the z-axis. This lets us define a useful quantity, , which is the distance to the point from the origin:
The form factor is:
Imagine instead that we compress/stretch the z dimension so that the ellipsoid becomes a sphere:
This implies a coordinate transformation for the -vector of:
Where is the relation for a q-vector tilted at angle with respect to the z axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular vector sees a sphere-like scatterer with size (length-scale) given by .
We can then convert back:
Isotropic Form Factor Intensity
To average over all possible orientations, we use:
Approximating by a Sphere
One can approximate a spheroid using an isovolumic sphere of radius Reffective: