# barbell

Cylinder with spherical end caps

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

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

40

20

length

Cylinder bar length

400

theta

Barbell axis to beam angle

degree

60

phi

degree

60

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

Definition

Calculates the scattering from a barbell-shaped cylinder. Like capped_cylinder, this is a spherocylinder 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.

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

$\begin{split}R_g^2 =&\ \left[ \tfrac{12}{5}R^4 + R^3\left(3L + \tfrac{18}{5} h\right) + R^2\left(L^2 + Lh + \tfrac25 h^2\right) + R\left(\tfrac14 L^3 + \tfrac12 L^2h - Lh^2\right) \right. \\ &\ \left. + Lh^4 - \tfrac12 L^2h^3 - \tfrac14 L^3h + \tfrac25 h^4\right] \left( 4R^2 + 3LR + 2Rh - 3Lh - 2h^2\right)^{-1}\end{split}$

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.

Source

barbell.py $$\ \star\$$ barbell.c $$\ \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