# 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 10-6-2 4
sld_solvent Solvent scattering length density 10-6-2 1
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.

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.

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.

References

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