Cylinder with spherical end caps

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
sld Barbell scattering length density 10-6-2 4
sld_solvent Solvent scattering length density 10-6-2 1
radius_bell Spherical bell radius 40
radius Cylindrical bar radius 20
length Cylinder bar length 400
theta In plane angle degree 60
phi Out of plane angle degree 60

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


Calculates the scattering from a barbell-shaped cylinder. Like capped_cylinder, this is a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder, but with the center of the end cap radius lying outside of the cylinder. See the diagram for the details of the geometry and restrictions on parameter values.


Fig. 1 Barbell geometry, where \(r\) is radius, \(R\) is radius_bell 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

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

The \(\left<\ldots\right>\) brackets denote an average of the structure over all orientations. \(\left<A^2(q,\alpha)\right>\) 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 barbell is

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

and its radius of gyration is

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


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. 2 Definition of the angles for oriented 2D barbells.


Fig. 3 1D and 2D plots corresponding to the default parameters of the model.


[1]H Kaya, J. Appl. Cryst., 37 (2004) 37 223-230
[2]H Kaya and N R deSouza, J. Appl. Cryst., 37 (2004) 508-509 (addenda and errata)

Authorship and Verification

  • Author: NIST IGOR/DANSE Date: pre 2010
  • Last Modified by: Paul Butler Date: March 20, 2016
  • Last Reviewed by: Richard Heenan Date: January 4, 2017