# capped_cylinder

Right circular cylinder with spherical end caps and uniform SLD

Parameter Description Units Default value
scale Scale factor or Volume fraction None 1
background Source background cm-1 0.001
sld Cylinder scattering length density 10-6-2 4
sld_solvent Solvent scattering length density 10-6-2 1
length Cylinder length 400
theta cylinder axis to beam angle degree 60
phi rotation about beam degree 60

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

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.

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

$\begin{split}A(q) =&\ \pi r^2L \frac{\sin\left(\tfrac12 qL\cos\alpha\right)} {\tfrac12 qL\cos\alpha} \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\ &\ + 4 \pi R^3 \int_{-h/R}^1 dt \cos\left[ q\cos\alpha \left(Rt + h + {\tfrac12} L\right)\right] \times (1-t^2) \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]} {qR\sin\alpha \left(1-t^2\right)^{1/2}}\end{split}$

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)

$V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h - \tfrac13h^3\right)$

and its radius of gyration is

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

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.

Source

capped_cylinder.py $$\ \star\$$ capped_cylinder.c $$\ \star\$$ gauss76.c $$\ \star\$$ sas_J1.c $$\ \star\$$ polevl.c

References

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

Authorship and Verification

• Author: NIST IGOR/DANSE Date: pre 2010