Ellipsoid

Steve Kline 20 JAN 1999

Alan Munter 08 JULY 1999, converted to Java

Calculates the form factor for a monodisperse ellipsoid (ellipsoid of revolution) with uniform scattering length density. The form factor is normalized by the particle volume such that P(q) = scale*<f*f>/Vol + bkg, where f is the scattering amplitude and the < > denote an average over all possible orientations of the ellipsoid.

Resolution smeared version is also provided.

Input Variables (default values):

Parameter	Variable	Value
0	Scale	1.0
1	Ra (rotation axis) (Å)	20.0
2	Rb (Å)	400.0
3	Contrast (Å-2)	3.0e-6
4	Incoherent Background (cm-1)	0.0

The function calculated is:

and

The returned value is scaled to units of [cm^-1], on absolute scale.

The ellipse is rotated about the axis of R_a to define the ellipsoid. If (as shown below) the R_b >R_a, the object is an oblate ellipsoid (disk-like). If R_b < R_a, then the object is a prolate ellipsoid (needle-like). If the two radii are equal, then the ellipsoid is a sphere.

If the object is an oblate ellipsoid (disk-like), it displays a limiting power-law slope of q^-2 at low q.

A prolate ellipsoid (needle-like), displays a limiting power-law slope of q^-1 at low q.

Parameter[0] (scale) and Parameter[3] (contrast) are both multiplicative factors in the model and are perfectly correlated. One or both of these parameters must be held fixed during model fitting.

If the scale factor Parameter[0] is set equal to the particle volume fraction, phi, the returned value is the scattered intensity per unit volume, I(q) = phi*P(q). However, no interparticle interference effects are included in this calculation.

Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).

This example dataset is produced by calculating the Ellipsoid using 128 data points, q_min = 0.001 Å^-1, q_max = 0.7 Å^-1 and the above default parameter values.