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 10-6Å-2 4
sld_solvent Solvent scattering length density 10-6Å-2 1
radius_bell Spherical bell radius Å 40
radius Cylindrical bar radius Å 20
length Cylinder bar length Å 400
theta Barbell 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.

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.

../../_images/barbell_geometry.jpg

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 \(h \ge 0\), \(h\) 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

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

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.

../../_images/cylinder_angle_definition.png

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

../../_images/barbell_autogenfig.png

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

Source

barbell.py \(\ \star\ \) barbell.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)
    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: March 20, 2016
  • Last Reviewed by: Richard Heenan Date: January 4, 2017