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For information about polarised and magnetic scattering, see
the :ref:`magnetism` documentation.
The 1D scattering intensity is calculated in the following way (Guinier, 1955)
.. math::
I(q) = \frac{\text{scale}}{V} \cdot \left[
3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3}
\right]^2 + \text{background}
where *scale* is a volume fraction, $V$ is the volume of the scatterer,
$r$ is the radius of the sphere and *background* is the background level.
*sld* and *sld_solvent* are the scattering length densities (SLDs) of the
scatterer and the solvent respectively, whose difference is $\Delta\rho$.
Note that if your data is in absolute scale, the *scale* should represent
the volume fraction (which is unitless) if you have a good fit. If not,
it should represent the volume fraction times a factor (by which your data
might need to be rescaled).
The 2D scattering intensity is the same as above, regardless of the
orientation of $\vec q$.
Validation of our code was done by comparing the output of the 1D model
to the output of the software provided by the NIST (Kline, 2006).
A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
John Wiley and Sons, New York, (1955)
* **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06
import numpy as np
from numpy import inf
name = "sphere"
title = "Spheres with uniform scattering length density"
description = """\
P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr))
/(qr)^3]^2 + background
r: radius of sphere
V: The volume of the scatter
sld: the SLD of the sphere
sld_solvent: the SLD of the solvent
category = "shape:sphere"
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld",
"Layer scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
"Solvent scattering length density"],
["radius", "Ang", 50, [0, inf], "volume",
"Sphere radius"],
source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c"]
# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
form_volume = """
return sphere_volume(radius);
Iq = """
return sphere_form(q, radius, sld, sld_solvent);
def ER(radius):
Return equivalent radius (ER)
return radius
# VR defaults to 1.0
def random():
radius = 10**np.random.uniform(1.3, 4)
pars = dict(
return pars
tests = [
[{}, 0.2, 0.726362],
[{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
"radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
0.2, 0.228843],
[{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "ER", 120.],
[{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "VR", 1.],