capped_cylinder

Right circular cylinder with spherical end caps and uniform SLD

Parameter	Description	Units	Default value
scale	Source intensity	None	1
background	Source background	cm-1	0.001
sld	Cylinder scattering length density	10-6Å-2	4
sld_solvent	Solvent scattering length density	10-6Å-2	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.

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.

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

capped_cylinder.py \(\ \star\ \) capped_cylinder.c \(\ \star\ \) lib/gauss76.c \(\ \star\ \) lib/sas_J1.c \(\ \star\ \) lib/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)
3. 12. 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