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. Cylinder¶

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

Author: Denis Korolkov

Calculates the form factor for a monodisperse right circular cylinder with uniform scattering length density. The form factor is normalized by the particle volume such that $P(q) = I0*\left<f^2\right>/V + B$, where < > is an average over all possible orientations of the cylinder.

Parameters¶

Parameter Variable Value
0 Scale 1.0
1 Radius (Å) 20.0
2 Length (Å) 400.0
3 Contrast ($Å^{-2}$) 3.0e-6
4 Incoherent Background ($cm^{-1}$) 0.0

Usage notes¶

The function calculated is:

image.png

image-2.png

image-3.png and image-4.png

where $J_1(x)$ is the first order Bessel function. Alpha is defined as the angle between the cylinder axis and the scattering vector, $q$. The integral over alpha averages the form factor over all possible orientations of the cylinder with respect to $q$. The returned value is in units of $cm^{-1}$, on absolute scale.

image-5.png

Parameter[0] (scale) and Parameter[3] (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.

In [3]:
# Definition of parameters
I0 = esc.par("Scale", 1, scale=1e8, fixed=True)
R = esc.par("Radius", 20, units=esc.angstr)
L = esc.par("Length", 400, units=esc.angstr)
rho = esc.par("Contrast", 3, scale=1e-6, units=f"{esc.angstr}⁻²")
bkgr = esc.par("Background", 0.0, userlim=[0, 0.03])

# Model equations
V = np.pi * R**2 * L

H = L / 2

alpha = esc.var("alpha")

j0 = esc.sin(q * H * esc.cos(alpha)) / (q * H * esc.cos(alpha))
j1 = esc.cyl_bessel_j(1, q * R * esc.sin(alpha)) / (q * R * esc.sin(alpha))

F = 2 * rho * V * j0 * j1
P = (
    I0
    / V
    * esc.integral(
        F**2 * esc.sin(alpha),
        alpha,
        0,
        np.pi / 2.0,
        maxiter=50,
        epsabs=1e-8,
        epsrel=1e-8,
    )
)

P.show(coordinates=np.linspace(0.001, 0.7, 128)).config(
    title="Cylinder",
    xlog=True,
    ylog=True,
    xlabel=f"Q[{esc.angstr}⁻¹]",
    ylabel="P(q)[cm⁻¹]",
)
Out[3]:
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