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. Sphere

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

Author: Denis Korolkov

Calculates the form factor, $P(q)$, for a monodisperse spherical particle with uniform scattering length density. The form factor is normalized by the particle volume as described below.

Parameters

Parameter	Variable	Value
0	Scale	1.0
1	Radius (Å)	60.0
2	Contrast ($Å^{-2}$)	1.0e-6
3	Incoherent Background ($cm^{-1}$)	0.0

Usage notes

The function calculated is:

The returned value is scaled to units of $cm^{-1}$

Parameter[0] (scale) and Parameter[2] (contrast) are both multiplicative factors in the model and are perfectly correlated. One or both of these parameters must be held fixed during model fitting.

Reference

Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York, (1955).

I0 = esc.par("Scale", 1, scale=1e8, fixed=True)
R = esc.par("Radius", 60, units=esc.angstr)
rho = esc.par("Contrast", 1, scale=1e-6, units=f"{esc.angstr}⁻²")
bkgr = esc.par("Background", 0.0, userlim=[0, 0.03])

V = 4 / 3 * np.pi * esc.pow(R, 3)
QR = q * R
P1 = (
    I0 / V * esc.pow(3 * V * rho * (esc.sin(QR) - QR * esc.cos(QR)) / esc.pow(QR, 3), 2)
    + bkgr
)
P1.show(coordinates=np.linspace(0.001, 0.3, 128)).config(
    title="Sphere", xlog=True, ylog=True, xlabel=f"Q{esc.angstr}⁻¹", ylabel="P(q)[cm⁻¹]"
)