Superball Model - Neither Sphere nor Cube -

    # superball model
# Note: model title and parameter table are inserted automatically

.. figure:: img/superball_realSpace.png

   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
[#WetterskogSuperball]_. The shape is described by

.. math::

    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

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

.. figure:: img/superball_geometry2d.png

    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

.. math::
    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}
            \mathrm{sin}(q_z a \zeta / 2),\\

.. math::
    \gamma =& \sqrt[2p]{1-x^{2p}}, \\
    \zeta =& \sqrt[2p]{1-x^{2p} -y^{2p}}.

The integral can be transformed to

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

which can be solved numerically.

The orientational average is then obtained by calculating

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


.. math::
    \vec{q} &= q \begin{pmatrix} \mathrm{cos} (\varphi) \mathrm{sin} (\theta)\\
    \mathrm{sin} (\varphi) \mathrm{sin} (\theta)\\
    \mathrm{cos} (\theta)\end{pmatrix}

The implemented orientationally averaged superball model is then fully given by

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



    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


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

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


` <>`_

`superball.c <>`_

Authorship and Verification

* **Author:** Dominique Dresen **Date:** March 27, 2019
* **Last Modified by:**  Dominique Dresen **Date:** March 27, 2019
* **Last Reviewed by:**  **Date:**
* **Source added by :** Dominique Dresen **Date:** March 27, 2019"""

import numpy as np
from numpy import inf

name = "superball"
title = "Superball with uniform scattering length density."
description = """
    I(q)= scale*V*(sld - sld_solvent)^2*P(q)+background
        P(q) = (2/pi) * double integral from 0 to pi/2 of ...
        AP = integral from -1 to 1 integral from -g to g of ...
        cos(R qx x) cos(R qy y]) sin(R qz Ze)
        g = (1 - x^(2p))^(1/(2p))
        Ze = (1 - x^(2p) - y^(2p))^(1/(2p))
category = "shape:superball"

#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld",
               "Superball scattering length density"],
              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
               "Solvent scattering length density"],
              ["length_a", "Ang", 50, [0, inf], "volume",
               "Cube edge length of the superball"],
              ["exponent_p", "", 2.5, [0, inf], "volume",
               "Exponent describing the roundness of the superball"],
              ["theta", "degrees", 0, [-360, 360], "orientation",
               "c axis to beam angle"],
              ["phi", "degrees", 0, [-360, 360], "orientation",
               "rotation about beam"],
              ["psi", "degrees", 0, [-360, 360], "orientation",
               "rotation about c axis"],
# lib/gauss76.c
# lib/gauss20.c
source = ["lib/gauss20.c", "lib/sas_gamma.c", "superball.c"]

have_Fq = True
effective_radius_type = [
    "radius of gyration",
    "equivalent volume sphere",
    "half length_a",

def random():
    """Return a random parameter set for the model."""
    length = np.random.uniform(10, 500)
    exponent = np.random.uniform(1.5, 5)
    pars = dict(
    return pars

# parameters for demo
demo = dict(scale=1, background=0,
            sld=6.3, sld_solvent=1.0,
            length_a=100, exponent_p=2.5,
            theta=45, phi=30, psi=15,
            length_a_pd=0.1, length_a_pd_n=10,
            theta_pd=10, theta_pd_n=1,
            phi_pd=10, phi_pd_n=1,
            psi_pd=10, psi_pd_n=1)

tests = [
    [{}, 0.2, 0.76833],
    [{"length_a": 100., "exponent_p": 1, "sld": 6., "sld_solvent": 1.},
     0.2, 0.7263],
    [{"length_a": 100., "exponent_p": 1000, "sld": 6., "sld_solvent": 1.},
     0.2, 0.2714],
    [{"length_a": 100., "exponent_p": 2.5, "sld": 6., "sld_solvent": 1.},
     0.2, 0.2810],
    [{"length_a": 100., "exponent_p": 2.5, "sld": 6., "sld_solvent": 1.,
      "length_a_pd": 0.1, "length_a_pd_n": 10},
     0.2, 0.49551865],
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