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
radius Cylinder radius 20
radius_cap Cap radius 20
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.


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.


Fig. 4 Capped cylinder geometry, where \(r\) is radius, \(R\) is radius_cap and \(L\) is length. Since the end cap radius \(R \geq r\) and by definition for this geometry \(h \le 0\), \(h\) 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

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

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

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


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.


Fig. 5 Definition of the angles for oriented 2D cylinders.


Fig. 6 1D and 2D plots corresponding to the default parameters of the model.

Source \(\ \star\ \) capped_cylinder.c \(\ \star\ \) gauss76.c \(\ \star\ \) sas_J1.c \(\ \star\ \) polevl.c


  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
  • Last Modified by: Paul Butler Date: September 30, 2016
  • Last Reviewed by: Richard Heenan Date: January 4, 2017