torus_elliptical_shell

core-shell torus with elliptical cross-section

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

radius

Torus major radius R

Å

100

radius_core

Elliptical core minor radius a

Å

5

thickness

Shell thickness t

Å

2

core_nu

Aspect ratio of core cross-section

None

1

shell_nu

Aspect ratio of shell cross-section

None

1

sld_core

Core SLD

10-6Å-2

0

sld_shell

Shell SLD

10-6Å-2

1

sld_solvent

Solvent SLD

10-6Å-2

0

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

Definition

This model computes the scattering from a torus with an elliptical tube cross-section and a concentric shell.

The geometric parameters are:

  • \(R\): major (ring) radius of the torus centerline

  • \(a\): core minor radius in the radial direction

  • \(\nu_{core}\): aspect ratio of the elliptical core cross-section

  • \(\nu_{shell}\): aspect ratio of the elliptical shell cross-section

  • \(t\): shell thickness

../../_images/torus_elliptical_shell_geometry.png

Fig. 41 Schematic geometry of the torus with elliptical tube cross-section.

The core and shell have independent aspect ratios, so the core semi-axes are \(a\) and \(\nu_{core} \cdot a\), and the outer semi-axes are \(a+t\) and \(\nu_{core} \cdot a + \nu_{shell} \cdot t\).

For a given orientation angle \(\theta\) between the torus symmetry axis and \(\vec q\), the kernel evaluates the amplitude using a numerical integral over the tube section:

\[F(q,\theta; x, \Delta\rho, \nu) = 4\pi\,\Delta\rho\int_{R-x}^{R+x} r\,J_0\!\left(qr\sin\theta\right) \frac{\sin\!\left(q\,\gamma\cos\theta\right)}{q\cos\theta}\,dr\]

with

\[\gamma = \nu\sqrt{x^2-(r-R)^2}\]

The core-shell amplitude is formed from outer and inner contributions:

\[F_{cs}(q,\theta) = F\!\left(q,\theta;a + t,\rho_{shell}-\rho_{solvent},\nu_{outer}\right) -F\!\left(q,\theta;a,\rho_{shell}-\rho_{core},\nu_{core}\right)\]

where \(\nu_{outer} = \frac{\nu_{shell}t + \nu_{core}a}{a+t}\) is the aspect ratio of the outer ellipse.

and the orientationally averaged intensity is

\[I(q) = \int_0^{\pi/2}\!\left|F_{cs}(q,\theta)\right|^2\sin\theta\,d\theta\]
../../_images/torus_elliptical_shell_autogenfig.png

Fig. 42 1D plot corresponding to the default parameters of the model.

Source

torus_elliptical_shell.py \(\ \star\ \) torus_elliptical_shell.c \(\ \star\ \) gauss76.c \(\ \star\ \) sas_J0.c \(\ \star\ \) polevl.c

References #. T. Kawaguchi, J. Appl. Crystallogr, 34(2001) 580-584 #. S. Förster, J. Phys. Chem., 103(1999) 6657-6668

Authorship and Verification