Capped Cylinder

Description:

Definitions

Calculates the scattering from a cylinder with spherical section end-caps. Like `barbell`, this is a sphereocylinder with end caps that have a radius larger than that of the cylinder, but with the center of the end cap radius lying within the cylinder. This model simply becomes a convex lens when the length of the cylinder $L=0$. See the diagram for the details of the geometry and restrictions on parameter values.

Capped cylinder geometry, where $r$ is *radius*, $R$ is *bell_radius* and $L$ is *length*. Since the end cap radius $R \geq r$ and by definition for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as $h = - \sqrt{R^2 - r^2}$

The scattered intensity $I(q)$ is calculated as

$$I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right>$$
where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as

$$A(q) = \pi r^2L \frac{\sin\left(\tfrac12 qL\cos\alpha\right)} {\tfrac12 qL\cos\alpha} \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\ + 4 \pi R^3 \int_{-h/R}^1 dt \cos\left[ q\cos\alpha \left(Rt + h + {\tfrac12} L\right)\right] \times (1-t^2) \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]} {qR\sin\alpha \left(1-t^2\right)^{1/2}}$$
The $\left<\ldots\right>$ brackets denote an average of the structure over all orientations. $\left< A^2(q)\right>$ is then the form factor, $P(q)$. The scale factor is equivalent to the volume fraction of cylinders, each of volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length densities of the cylinder and the surrounding solvent.

The volume of the capped cylinder is (with $h$ as a positive value here)

$$V = \pi r_c^2 L + \tfrac{2\pi}{3}(R-h)^2(2R + h)$$

and its radius of gyration is

$$R_g^2 = \left[ \tfrac{12}{5}R^5 + R^4\left(6h+\tfrac32 L\right) + R^2\left(4h^2 + L^2 + 4Lh\right) + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1}$$

.. note::

The requirement that $R \geq r$ is not enforced in the model! It is up to you to restrict this during analysis.

The 2D scattering intensity is calculated similar to the 2D cylinder model.

Definition of the angles for oriented 2D cylinders.

References

H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230
H Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda
L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
and errata)

Authorship and Verification

**Author:** NIST IGOR/DANSE **Date:** pre 2010