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

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.

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.

../../_images/capped_cylinder_geometry.jpg

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

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

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.

../../_images/cylinder_angle_definition.png

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

../../_images/capped_cylinder_autogenfig.png

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

  • Last Modified by: Paul Butler Date: September 30, 2016

  • Last Reviewed by: Richard Heenan Date: January 4, 2017