1.1.7. 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
radius_minor Ellipse minor radius 20
axis_ratio Ratio of major radius over minor radius None 1.5
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 cylinder axis to beam angle degree 90
phi rotation about beam degree 0
psi rotation about cylinder axis 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.

../_images/elliptical_cylinder_geometry.png

Fig. 1.20 Figure 1: Elliptical cylinder geometry \(a = r_\text{minor}\) and \(\nu = r_\text{major} / r_\text{minor}\) is the axis_ratio.

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.

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

../_images/elliptical_cylinder_angle_definition.png

Fig. 1.21 Figure 2: Definition of angles for oriented elliptical cylinder, where axis_ratio is drawn >1, and angle \(\Psi\) is now a rotation around the axis of the cylinder.

../_images/elliptical_cylinder_angle_projection.png

Fig. 1.22 Figure 3: Examples of the angles for oriented elliptical cylinders against the detector plane, with \(\Psi\) = 0.

The \(\theta\) and \(\phi\) parameters to orient the cylinder only appear in the model when fitting 2d data. On introducing “Orientational Distribution” in the angles, “distribution of theta” and “distribution of phi” parameters will appear. These are actually rotations about the axes \(\delta_1\) and \(\delta_2\) of the cylinder, the \(b\) and \(a\) axes of the cylinder cross section. (When \(\theta = \phi = 0\) these are parallel to the \(Y\) and \(X\) axes of the instrument.) The third orientation distribution, in \(\psi\), is about the \(c\) axis of the particle. Some experimentation may be required to understand the 2d patterns fully. (Earlier implementations had numerical integration issues in some circumstances when orientation distributions passed through 90 degrees, such situations, with very broad distributions, should still be approached with care.)

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.

../_images/elliptical_cylinder_averaging.png

Fig. 1.23 Figure 4: The intensities averaged from 2D over different numbers of bins and angles.

../_images/elliptical_cylinder_autogenfig.png

Fig. 1.24 Figure 5: 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) [see table 3.4]

Authorship and Verification