In [1]:
import numpy as np
import escape as esc

esc.require("0.9.8")

Loading material database from /home/dkor/Data/Development/workspace_escape/escape-core/python/src/escape/scattering/../data/mdb/materials.db

SAXS. Form-factors. Superball¶

Author: Denis Korolkov

Calculates the form factor, $P(q)$ for particles with superball geometry which describes a smooth transition between cubic and spherical morphologies with rounded corners and edges.

Parameters¶

Parameter Variable Value
0 Scale 1.0
1 Average Radius (nm) 10.0
2 Power, p 1.0
3 Polydispersity (0-1) 0.05
4 SLD ($nm^{-2}$) 1.0e-4
5 incoherent Background(cm^{-1}) 0.000

Parameter[0] (scale) is correlated with the SLD's that describe the particle. No more than one of these parameters can be free during model fitting.

Usage notes¶

image.png

A superball is a geometric body that is defined by the shape parameter $p$.

Any point of a superball geometry satisfies the following relation

image.png

where $a$ is the edge length of the superball. The case $p = 1$ is equivalent to the sphere with the radius $r = a/2$, and the case $p = ∞$ corresponds to a cube.

The form factor amplitude has no analytic expression and needs to be solved numerically. The amplitude of the oriented superball form factor $p_{orient}(\vec q)$ for an arbitrary $\vec q$ direction is given by

image.png

where

image.png

The integral splits onto real and imaginary part wherte the latter vanishes and the former can be transformed to

image.png

The particle size distribution of choice $\Gamma$ leads to the integral

image.png

and the orientation distribution is performed by integrating over all possible directions

image.png

The particle volume

image.png

Reference¶

D. Dresen, A. Qdemat, S. Ulusoy, F. Mees, D. Zákutná, E. Wetterskog, E. Kentzinger, G. Salazar-Alvarez, S. Disch, J. Phys. Chem. C 2021, 125, 23356−23363

In [2]:
# Definition of parameters

p = esc.par("P", 1, userlim=[0.01, 50])
r = esc.par("R", 10, units="nm", userlim=[-100000, 100000])
r0 = esc.par("R0", 10, units="nm", userlim=[-100000, 100000])
s = esc.par("Polydispersity", 0.05)

# Definition of variables
x = esc.var("x")
y = esc.var("y")
z = esc.var("z")

Q = esc.var("Q")
phi = esc.var("Phi")
theta = esc.var("Theta")

Qx = Q * esc.cos(phi) * esc.sin(theta)
Qy = Q * esc.sin(phi) * esc.sin(theta)
Qz = Q * esc.cos(theta)

# \ksi and \gamma
Ksi = (1 - x ** (2 * p) - y ** (2 * p)) ** (1 / (2 * p))
G = (1 - x ** (2 * p)) ** (1 / (2 * p))

# form-factor integrals
Fxy = esc.cos(r * Qy * y) * esc.sin(r * Qz * Ksi)
Fy = esc.integral(Fxy, y, 0, G, 21, epsabs=1e-8, epsrel=1e-8)
F = esc.integral(esc.cos(r * Qx * x) * Fy, x, 0, 1, 21, epsabs=1e-8, epsrel=1e-8)

# form-factor for certain particle orientation
Pori = (F * 8 * r**2 / Qz) ** 2

# integral over size distribution
PoAv = esc.average_lognorm(Pori, s, r, r0, 21, epsrel=1e-8, epsabs=1e-8)

# average volume, integral has analytical solution
V_c = 8 / (12 * p**2) * esc.tgamma(1 / (2 * p)) ** 3 / esc.tgamma(3 / (2 * p))
V_av = V_c * r0**3 * esc.exp(4.5 * s**2)

# integral over phi angles
PoAv_phi = esc.integral(PoAv, phi, 0, np.pi / 2, 21)
# untegral over theta angles
PoAv_theta = 1 / V_av * esc.integral(PoAv_phi, theta, 0, np.pi / 2, 21)

K_Q = esc.kernel("P(Q)", PoAv_theta)
In [3]:
coords = np.linspace(0.001, 3, 100)
K_Q.show(coordinates=coords).config(xlog=True, ylog=True, xlabel="Q", ylabel="I")
Out[3]:
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