# 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 |
0.001 |

sld |
Cylinder scattering length density |
10 |
4 |

sld_solvent |
Solvent scattering length density |
10 |
1 |

radius |
Cylinder radius |
Å |
20 |

radius_cap |
Cap radius |
Å |
20 |

length |
Cylinder length |
Å |
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

where the amplitude \(A(q,\alpha)\) with the rod axis at angle \(\alpha\) to \(q\) is given as

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)

and its radius of gyration is

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**

H Kaya,

*J. Appl. Cryst.*, 37 (2004) 223-230H Kaya and N R deSouza,

*J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata)Onsager,

*Ann. New York Acad. Sci.*, 51 (1949) 627-659

**Authorship and Verification**

**Author:**NIST IGOR/DANSE**Date:**pre 2010**Last Modified by:**Paul Butler**Date:**September 30, 2016**Last Reviewed by:**Richard Heenan**Date:**January 4, 2017