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## superball

Superball with uniform scattering length density.

Parameter | Description | Units | Default value
--- | --- | --- | ---
scale | Scale factor or Volume fraction | None | 1
background | Source background | cm-1 | 0.001
sld | Superball scattering length density | 10-6Å-2 | 4
sld_solvent | Solvent scattering length density | 10-6Å-2 | 1
length_a | Cube edge length of the superball | Å | 50
exponent_p | Exponent describing the roundness of the superball | None | 2.5
theta | c axis to beam angle | degree | 0
phi | rotation about beam | degree | 0
psi | rotation about c axis | degree | 0

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

Definition

Fig. 107 Figure 1: Superball visualisation for varied values of the parameter p.

This model calculates the scattering of a superball, which represents a cube with rounded edges. It can be used to describe nanoparticles that deviate from the perfect cube shape as it is often observed experimentally [1]. The shape is described by

\[x^{2p} + y^{2p} + z^{2p} \leq \biggl( \frac{a}{2} \biggr)^{2p}\]

with \(a\) the cube edge length of the superball and \(p\) a parameter that describes the roundness of the edges. In the limiting cases \(p=1\) the superball corresponds to a sphere with radius \(R = a/2\) and for \(p = \infty\) to a cube with edge length \(a\). The exponent \(p\) is related to \(a\) and the face diagonal \(d\) via

\[p = \frac{1}{1 + 2 \mathrm{log}_2 (a/d)}.\]

Fig. 108 Figure 2: Cross-sectional view of a superball showing the principal axis length \(a\), the face-diagonal \(d\) and the superball radius \(R\).

The oriented form factor is determined by solving

\[\begin{split}p_o(\vec{q}) =& \int_{V} \mathrm{d} \vec{r} e^{i \vec{q} \cdot \vec{r}}\\ =& \frac{a^3}{8} \int_{-1}^{1} \mathrm{d} x \int_{-\gamma}^{\gamma} \mathrm{d} y \int_{-\zeta}^{\zeta} \mathrm{d} z e^{i a (q_x x + q_y y + q_z z) / 2}\\ =& \frac{a^2}{2 q_z} \int_{-1}^{1} \mathrm{d} x \int_{-\gamma}^{\gamma} \mathrm{d} y e^{i a(q_x x + q_y y)/2} \sin(q_z a \zeta / 2),\end{split}\]

with

\[\begin{split}\gamma =& \sqrt[2p]{1-x^{2p}}, \\ \zeta =& \sqrt[2p]{1-x^{2p} -y^{2p}}.\end{split}\]

The integral can be transformed to

\[p_o(\vec{q}) = \frac{2 a^2}{q_z} \int_{0}^{1} \mathrm{d} x \, \cos \biggl(\frac{a q_x x}{2} \biggr) \int_{0}^{\gamma} \mathrm{d} y \, \cos \biggl( \frac{a q_y y}{2} \biggr) \sin \biggl( \frac{a q_z \zeta}{2} \biggr),\]

which can be solved numerically.

The orientational average is then obtained by calculating

\[P(q) = \int_0^{\tfrac{\pi}{2}} \mathrm{d} \varphi \int_0^{\tfrac{\pi}{2}} \mathrm{d} \theta \, \sin (\theta) | p_o(\vec{q}) |^2\]

with

\[\begin{split}\vec{q} &= q \begin{pmatrix} \cos (\varphi) \sin (\theta)\\ \sin (\varphi) \sin(\theta)\\ \cos (\theta)\end{pmatrix}\end{split}\]

The implemented orientationally averaged superball model is then fully given by [2]

\[I(q) = \mathrm{scale} (\Delta \rho)^2 P(q) + \mathrm{background}.\]

FITTING NOTES

Validation

> The code is validated by reproducing the spherical form factor implemented in SasView for \(p = 1\) and the parallelepiped form factor with \(a = b = c\) for \(p = 1000\). The form factors match in the first order oscillation with a precision in the order of \(10^{-4}\). The agreement worsens for higher order oscillations and beyond the third order oscillations a higher order Gauss quadrature rule needs to be used to keep the agreement below \(10^{-3}\). This is however avoided in this implementation to keep the computation time fast.

Fig. 109 Figure 3: 1D and 2D plots corresponding to the default parameters of the model.

Source

`superball.py` \(\ \star\ \) `superball.c` \(\ \star\ \) `sas_gamma.c` \(\ \star\ \) `gauss20.c`

References

[1]

5. Wetterskog, A. Klapper, S. Disch, E. Josten, R. P. Hermann, U. Rücker, T. Brückel, L. Bergström and G. Salazar-Alvarez, Nanoscale, 8 (2016) 15571

[2]

4. Dresen, A. Qdemat, S. Ulusoy, F. Mees, D. Zakutna, E. Wetterskog, E. Kentzinger, G. Salazar-Alvarez, S. Disch, J. Phys. Chem. C (2021), doi: 10.1021/acs.jpcc.1c06082

Source

superball.py

superball.c

Authorship and Verification

- Author: Dominique Dresen Date: March 27, 2019

- Last Modified by: Dominique Dresen Date: March 27, 2019

- Last Reviewed by: Dirk Honecker Date: November 05, 2021

- Source added by : Dominique Dresen Date: March 27, 2019