ellipsoid

Ellipsoid of revolution with uniform scattering length density.

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_polar	Polar radius	Å	20
radius_equatorial	Equatorial radius	Å	400
theta	In plane angle	degree	60
phi	Out of plane angle	degree	60

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

The form factor is normalized by the particle volume

Definition

The output of the 2D scattering intensity function for oriented ellipsoids is given by (Feigin, 1987)

\[P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background}\]

where

\[F(q,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)] - \cos[qr(R_p,R_e,\alpha)])} {[qr(R_p,R_e,\alpha)]^3}\]

and

\[r(R_p,R_e,\alpha) = \left[ R_e^2 \sin^2 \alpha + R_p^2 \cos^2 \alpha \right]^{1/2}\]

\(\alpha\) is the angle between the axis of the ellipsoid and \(\vec q\), \(V = (4/3)\pi R_pR_e^2\) is the volume of the ellipsoid , \(R_p\) is the polar radius along the rotational axis of the ellipsoid, \(R_e\) is the equatorial radius perpendicular to the rotational axis of the ellipsoid and \(\Delta \rho\) (contrast) is the scattering length density difference between the scatterer and the solvent.

For randomly oriented particles:

\[F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}\]

To provide easy access to the orientation of the ellipsoid, we define the rotation axis of the ellipsoid using two angles \(\theta\) and \(\phi\). These angles are defined in the cylinder orientation figure. For the ellipsoid, \(\theta\) is the angle between the rotational axis and the \(z\) -axis.

NB: The 2nd virial coefficient of the solid ellipsoid is calculated based on the \(R_p\) and \(R_e\) values, and used as the effective radius for \(S(q)\) when \(P(q) \cdot S(q)\) is applied.

The \(\theta\) and \(\phi\) parameters are not used for the 1D output.

Validation

Validation of the code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006).

The implementation of the intensity for fully oriented ellipsoids was validated by averaging the 2D output using a uniform distribution \(p(\theta,\phi) = 1.0\) and comparing with the output of the 1D calculation.

Fig. 35 Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: scale = 1.0, radius_polar = 20 Å, radius_equatorial = 400 Å, contrast = 3e-6 Å^-2, and background = 0.0 cm^-1.

The discrepancy above \(q\) = 0.3 cm^-1 is due to the way the form factors are calculated in the c-library provided by NIST. A numerical integration has to be performed to obtain \(P(q)\) for randomly oriented particles. The NIST software performs that integration with a 76-point Gaussian quadrature rule, which will become imprecise at high \(q\) where the amplitude varies quickly as a function of \(q\). The SasView result shown has been obtained by summing over 501 equidistant points. Our result was found to be stable over the range of \(q\) shown for a number of points higher than 500.

Fig. 36 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 Press, New York, 1987.