import escape as esc
import numpy as np

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

q = esc.var("Q")

SAXS. Form-factors. OblateCoreShell

Original: https://www.ncnr.nist.gov/resources/sansmodels/OblateCoreShell.html

Author: Denis Korolkov

Calculates the form factor for an oblate ellipsoid particle with a core/shell structure. The form factor is averaged over all possible orientations of the ellipsoid.

Parameters

Parameter	Variable	Value
0	Scale	1.0
1	Major Core Radius (Å)	200.0
2	Minor Core Radius (Å)	20.0
3	Major Shell Radius (Å)	250.0
4	Minor Shell Radius (Å)	30.0
5	Contrast (core-shell) ($Å^{-2}$)	1.0e-6
6	Contrast (shell-solvent) (Å^{-2})	1.0e-6
7	Incoherent Background (cm-1)	0.0

Usage notes

The function calculated is:

where

and

Reference

Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.

Berr, S. J. Phys. Chem., 1987, 91, 4760.

# Definition of parameters
I0 = esc.par("Scale", 1, scale=1e8, fixed=True)

RCmaj = esc.par("Major Core Radius", 200, units=esc.angstr)
RCmin = esc.par("Minor Core Radius", 20, units=esc.angstr)
RSmaj = esc.par("Major Shell Radius", 250, units=esc.angstr)
RSmin = esc.par("Minor Shell Radius", 30, units=esc.angstr)

rho_cs = esc.par("Core-Shell SLD", 1, scale=1e-6, units=f"{esc.angstr}⁻²")
rho_ss = esc.par("Shell-Solvent SLD", 1, scale=1e-6, units=f"{esc.angstr}⁻²")
bkgr = esc.par("Background", 0.0, userlim=[0, 0.03])

# Model equations

# rotation angle
alpha = esc.var("alpha")


Vc = 4 / 3 * np.pi * RCmin * RCmaj**2
Vs = 4 / 3 * np.pi * RSmin * RSmaj**2

Uc = q * esc.sqrt((RCmaj**2 * (1 - alpha**2) + RCmin**2 * alpha**2))
Us = q * esc.sqrt((RSmaj**2 * (1 - alpha**2) + RSmin**2 * alpha**2))

Jc = (esc.sin(Uc) - Uc * esc.cos(Uc)) / Uc**2
Js = (esc.sin(Us) - Us * esc.cos(Us)) / Us**2

F = 3 * Vc * rho_cs * Jc / Uc + 3 * Vs * rho_ss * Js / Us

P = (
    I0 / Vs * esc.integral(F * F, alpha, 0, 1, maxiter=100, epsrel=1e-8, epsabs=1e-8)
    + bkgr
)

P.show(coordinates=np.linspace(0.001, 0.7, 128)).config(
    title="CoreShell",
    xlog=True,
    ylog=True,
    xlabel=f"Q [{esc.angstr}⁻¹]",
    ylabel="P(q)[cm⁻¹]",
)