triaxial_ellipsoid

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	10-6Å-2	4
sld_solvent	Solvent scattering length density	10-6Å-2	1
radius_equat_minor	Minor equatorial radius	Å	20
radius_equat_major	Major equatorial radius	Å	400
radius_polar	Polar radius	Å	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.

Definition

The form factor calculated is

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

where

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

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

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.

References

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