2.1.1.1. Barbell¶

Cylinder with spherical end caps

Parameter	Description	Units	Default value
scale	Source intensity	None	1
background	Source background	cm^-1	0
sld	Barbell scattering length density	10^-6Å^-2	4
solvent_sld	Solvent scattering length density	10^-6Å^-2	1
bell_radius	Spherical bell radius	Å	40
radius	Cylindrical bar radius	Å	20
length	Cylinder bar length	Å	400
theta	In plane angle	degree	60
phi	Out of plane angle	degree	60

The returned value is scaled to units of cm^-1.

Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of the cylinder, L, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder and the center of the end cap radius lies outside of the cylinder. All dimensions of the BarBell are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values.

Definition¶

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

The barbell geometry is defined as

where r is the radius of the cylinder. All other parameters are as defined in the diagram.

Since the end cap radius R >= r and by definition for this geometry h < 0, h is then defined by r and R as

h = -1 * sqrt(R² - r²)

The scattered intensity I(q) is calculated as

\[\begin{split}I(Q) = \frac{(\Delta \rho)^2}{V} \left< A^2(Q)\right>\end{split}\]

where the amplitude A(q) is given as

\[\begin{split}A(Q) =&\ \pi r^2L {\sin\left(\tfrac12 QL\cos\theta\right) \over \tfrac12 QL\cos\theta} {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\ &\ + 4 \pi R^3 \int_{-h/R}^1 dt \cos\left[ Q\cos\theta \left(Rt + h + {\tfrac12} L\right)\right] \times (1-t^2) {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right] \over QR\sin\theta \left(1-t^2\right)^{1/2}}\end{split}\]

The < > brackets denote an average of the structure over all orientations. <A ²(q)> is then the form factor, P(q). The scale factor is equivalent to the volume fraction of cylinders, each of volume, V. Contrast 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 >= r is not enforced in the model! It is up to you to restrict this during analysis.

This example dataset is produced by running the Macro PlotBarbell(), using 200 data points, qmin = 0.001 Å^-1, qmax = 0.7 Å^-1, sld = 4e-6 Å^-2 and the default model values.

Figure. 1D plot using the default values (w/256 data point).

For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. For example, for θ = 45 deg and φ = 0 deg with default values for other parameters

Figure. 2D plot (w/(256X265) data points).

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

Figure. Examples of the angles for oriented pp against the detector plane.

REFERENCE¶

H Kaya, J. Appl. Cryst., 37 (2004) 37 223-230

H Kaya and N R deSouza, J. Appl. Cryst., 37 (2004) 508-509 (addenda and errata)

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2.1.1.1. Barbell

2.1.1.1. Barbell¶

Definition¶

REFERENCE¶