Ellipsoid of uniform scattering length density with three independent axes.

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
sld Ellipsoid scattering length density 10-6-2 4
sld_solvent Solvent scattering length density 10-6-2 1
radius_equat_minor Minor equatorial radius 20
radius_equat_major Major equatorial radius 400
radius_polar Polar radius 10
theta In plane angle degree 60
phi Out of plane angle degree 60
psi Out of plane angle degree 60

The returned value is scaled to units of cm-1 sr-1, absolute scale.

All three axes are of different lengths with \(R_a \leq R_b \leq R_c\) Users should maintain this inequality for all calculations.

\[P(q) = \text{scale} V \left< F^2(q) \right> + \text{background}\]

where the volume \(V = 4/3 \pi R_a R_b R_c\), and the averaging \(\left<\ldots\right>\) is applied over all orientations for 1D.


Fig. 37 Ellipsoid schematic.


The form factor calculated is

\[P(q) = \frac{\text{scale}}{V}\int_0^1\int_0^1 \Phi^2(qR_a^2\cos^2( \pi x/2) + qR_b^2\sin^2(\pi y/2)(1-y^2) + R_c^2y^2) dx dy\]


\[\Phi(u) = 3 u^{-3} (\sin u - u \cos u)\]

To provide easy access to the orientation of the triaxial ellipsoid, we define the axis of the cylinder using the angles \(\theta\), \(\phi\) and \(\psi\). These angles are defined on Fig. 38 . The angle \(\psi\) is the rotational angle around its own \(c\) axis against the \(q\) plane. For example, \(\psi = 0\) when the \(a\) axis is parallel to the \(x\) axis of the detector.


Fig. 38 The angles for oriented ellipsoid.

The radius-of-gyration for this system is \(R_g^2 = (R_a R_b R_c)^2/5\).

The contrast is defined as SLD(ellipsoid) - SLD(solvent). In the parameters, \(R_a\) is the minor equatorial radius, \(R_b\) is the major equatorial radius, and \(R_c\) is the polar radius of the ellipsoid.

NB: The 2nd virial coefficient of the triaxial solid ellipsoid is calculated based on the polar radius \(R_p = R_c\) and equatorial radius \(R_e = \sqrt{R_a R_b}\), and used as the effective radius for \(S(q)\) when \(P(q) \cdot S(q)\) is applied.


Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of 2D calculation over all possible angles.


Fig. 39 1D and 2D plots corresponding to the default parameters of the model.


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