elliptical_cylinder

Form factor for an elliptical cylinder.

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
length Length of the cylinder 400
sld Cylinder scattering length density 10-6-2 4
sld_solvent Solvent scattering length density 10-6-2 1
theta In plane angle degree 90
phi Out of plane angle degree 0
psi Major axis angle relative to Q degree 0

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

Definition for 2D (orientated system)

The angles $$\theta$$ and $$\phi$$ define the orientation of the axis of the cylinder. The angle $$\Psi$$ is defined as the orientation of the major axis of the ellipse with respect to the vector $$Q$$. A gaussian polydispersity can be added to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii.

The function calculated is

$I(\vec q)=\frac{1}{V_\text{cyl}}\int{d\psi}\int{d\phi}\int{ p(\theta,\phi,\psi)F^2(\vec q,\alpha,\psi)\sin(\alpha)d\alpha}$

with the functions

$F(q,\alpha,\psi) = 2\frac{J_1(a)\sin(b)}{ab}$

where

\begin{align}\begin{aligned}a = qr'\sin(\alpha)\\b = q\frac{L}{2}\cos(\alpha)\\r'=\frac{r_{minor}}{\sqrt{2}}\sqrt{(1+\nu^{2}) + (1-\nu^{2})cos(\psi)}\end{aligned}\end{align}

and the angle $$\psi$$ is defined as the orientation of the major axis of the ellipse with respect to the vector $$\vec q$$. The angle $$\alpha$$ is the angle between the axis of the cylinder and $$\vec q$$.

Definition for 1D (no preferred orientation)

The form factor is averaged over all possible orientation before normalized by the particle volume

$P(q) = \text{scale} <F^2> / V$

To provide easy access to the orientation of the elliptical cylinder, we define the axis of the cylinder using two angles $$\theta$$, $$\phi$$ and $$\Psi$$ (see cylinder orientation). The angle $$\Psi$$ is the rotational angle around its own long_c axis against the $$q$$ plane. For example, $$\Psi = 0$$ when the $$r_\text{minor}$$ axis is parallel to the $$x$$ axis of the detector.

All angle parameters are valid and given only for 2D calculation; ie, an oriented system.

NB: The 2nd virial coefficient of the cylinder is calculated based on the averaged radius $$(=\sqrt{r_\text{minor}^2 * \text{axis ratio}})$$ and length values, and used as the effective radius for $$S(Q)$$ when $$P(Q)*S(Q)$$ 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 the 2D calculation over all possible angles.

In the 2D average, more binning in the angle $$\phi$$ is necessary to get the proper result. The following figure shows the results of the averaging by varying the number of angular bins.

References

L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, (1987) [see table 3.4]

Authorship and Verification

• Author: