In [1]:
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
In [2]:
q = esc.var("Q")

SAXS. Form-factors. PolyCoreShell¶

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

Author: Denis Korolkov

Calculates the form factor for polydispersed spherical particles with a core-shell structure. The spherical particles have a polydisperse core with a constant shell thickness. The form factor is normalized by the average particle volume

Parameters¶

Parameter Variable Value
0 Scale 1.0
1 Core Radius (Å) 60.0
2 Shell Thickness (Å) 10.0
3 Core Polydispersity (0-1) 0.2
4 Core SLD ($Å^{-2}$) 1.0e-6
5 Shell SLD ($Å^{-2}$) 2.0e-6
6 Solvent SLD ($Å^{-2}$) 3.0e-6
7 incoherent Background(cm^{-1}) 0.000

Usage notes¶

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.

Parameter[3] (polydispersity) is constrained to keep it within its physical limits of (0,1).

The returned form factor is normalized by the average particle volume $\left<V\right>$:

image-6.png

where

image-7.png

and $z$ is the width parameter of the Schulz distribution,

image-8.png

The average particle diameter is 2*(Radius + shell thickness)

Reference¶

Bartlett, P.; Ottewill, R. H. J. Chem. Phys., 1992, 96, 3306.

In [3]:
# Definition of parameters
I0 = esc.par("Scale", 1, scale=1e8, fixed=True)
Rc = esc.par("Average Core Radius", 60, units=esc.angstr)
P = esc.par("Polydispersity", 0.2, units="", userlim=[0, 1])
S = esc.par("Shell thickness", 10, units=esc.angstr)
rho_c = esc.par("Core SLD", 1, scale=1e-6, units=f"{esc.angstr}⁻²")
rho_s = esc.par("Shell SLD", 2, scale=1e-6, units=f"{esc.angstr}⁻²")
rho_solv = esc.par("Solvent SLD", 3, scale=1e-6, units=f"{esc.angstr}⁻²")
bkgr = esc.par("Background", 0.0, userlim=[0, 0.03])

# Model equations
X = esc.var("X")


# integration parameter or actual particle radius
r = esc.par("Core radius", 60)

# distribution function
G = esc.schulz("G", r, Rc, P)


rs = r + S

Vc = 4 / 3 * np.pi * r**3
Vs = 4 / 3 * np.pi * rs**3

QRc = q * r
QRs = q * rs

jc = (esc.sin(QRc) - QRc * esc.cos(QRc)) / QRc**3
js = (esc.sin(QRs) - QRs * esc.cos(QRs)) / QRs**3

z = 1 / P**2 - 1
Va = 4 * np.pi / 3 * (z + 3) * (z + 2) / (z + 1) ** 2 * Rc**3

F = (3 * Vc * (rho_c - rho_s) * jc + 3 * Vs * (rho_s - rho_solv) * js) ** 2

std = Rc * P
# integral from 0 to 10*std
I = (
    I0
    / Va
    * esc.average(F, G, r, Rc, 0, 10 * std, maxiter=100, epsrel=1e-8, epsabs=1e-8)
    + bkgr
)


I.show(coordinates=np.linspace(0.001, 0.7, 256)).config(
    title="PolyCoreShell",
    xlog=True,
    ylog=True,
    xlabel=f"Q [{esc.angstr}⁻¹]",
    ylabel="P(q)[cm⁻¹]",
)
Out[3]:
In [4]:
# or using a special function to integrate with Schulz distribution
I = (
    I0
    / Va
    * esc.average_schulz(F, P, r, Rc, maxiter=100, epsrel=1e-8, epsabs=1e-8, numstd=10)
    + bkgr
)
I.show(coordinates=np.linspace(0.001, 0.7, 256)).config(
    title="PolyCoreShell",
    xlog=True,
    ylog=True,
    xlabel=f"Q [{esc.angstr}⁻¹]",
    ylabel="P(q)[cm⁻¹]",
)
Out[4]:
In [ ]: