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. 95 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. 96 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. 97 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