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r"""
For information about polarised and magnetic scattering, see
the :ref:`magnetism` documentation.
Definition
----------
The 1D scattering intensity is calculated in the following way [Guinier1955]_
.. 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, *background* is the background level and
*sld* and *sld_solvent* are the scattering length densities (SLDs) of the
scatterer and the solvent respectively.
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
----------
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).
References
----------
.. [Guinier1955] A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
John Wiley and Sons, New York, (1955)
Authorship and Verification
----------------------------
* **Author: P Kienzle**
* **Last Modified by:**
* **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06
"""
import numpy as np
from numpy import pi, inf, sin, cos, sqrt, log
name = " _sphere (python)"
title = "PAK testing ideas for 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], "",
"Layer scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "",
"Solvent scattering length density"],
["radius", "Ang", 50, [0, inf], "volume",
"Sphere radius"],
]
def form_volume(radius):
"""Calculate volume for sphere"""
return 1.333333333333333 * pi * radius ** 3
def radius_effective(mode, radius):
"""Calculate R_eff for sphere"""
return radius if mode else 0.
def Iq(q, sld, sld_solvent, radius):
"""Calculate I(q) for sphere"""
#print "q",q
#print "sld,r",sld,sld_solvent,radius
qr = q * radius
sn, cn = sin(qr), cos(qr)
## The natural expression for the Bessel function is the following:
## bes = 3 * (sn-qr*cn)/qr**3 if qr>0 else 1
## however, to support vector q values we need to handle the conditional
## as a vector, which we do by first evaluating the full expression
## everywhere, then fixing it up where it is broken. We should probably
## set numpy to ignore the 0/0 error before we do though...
bes = 3 * (sn - qr * cn) / qr ** 3 # may be 0/0 but we fix that next line
bes[qr == 0] = 1
fq = bes * (sld - sld_solvent) * form_volume(radius)
return 1.0e-4 * fq ** 2
Iq.vectorized = True # Iq accepts an array of q values
def sesans(z, sld, sld_solvent, radius):
"""
Calculate SESANS-correlation function for a solid sphere.
Wim Bouwman after formulae Timofei Kruglov J.Appl.Cryst. 2003 article
"""
d = z / radius
g = np.zeros_like(z)
g[d == 0] = 1.
low = ((d > 0) & (d < 2))
dlow = d[low]
dlow2 = dlow ** 2
g[low] = (sqrt(1 - dlow2/4.) * (1 + dlow2/8.)
+ dlow2/2.*(1 - dlow2/16.) * log(dlow / (2. + sqrt(4. - dlow2))))
return g
sesans.vectorized = True # sesans accepts an array of z values
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