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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 (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
----------
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
----------
#. A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
John Wiley and Sons, New York, (1955)
Authorship and Verification
----------------------------
* **Author:**
* **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 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", "sphere.c"]
have_Fq = True
radius_effective_modes = ["radius"]
#single = False
def random():
"""Return a random parameter set for the model."""
radius = 10**np.random.uniform(1.3, 4)
pars = dict(
radius=radius,
)
return pars
#2345678901234567890123456789012345678901234567890123456789012345678901234567890
tests = [
[{}, 0.2, 0.726362], # each test starts with default parameter values
# inside { }, unless modified. Then Q and expected value of I(Q)
# putting None for an expected result will pass the test if there are no
# errors from the routine, but without any check on the value of the result
[{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
"radius": 120.},
[0.01, 0.1, 0.2], [1.34836265e+04, 6.20114062e+00, 1.04733914e-01]],
[{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
# careful tests here R=120 Pd=.2, then with S(Q) at default Reff=50
# (but this gets changed to 120) phi=0,2
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45},
[0.01, 0.1, 0.2], [1.74395295e+04, 3.68016987e+00, 2.28843099e-01]],
# a list of Q values and list of expected results is also possible
[{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45},
0.01, 335839.88055473, 1.41045057e+11, 120.0, 8087664.122641933, 1.0],
# the longer list here checks F1, F2, R_eff, volume, volume_ratio
[{"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45},
0.1, 482.93824329, 29763977.79867414, 120.0, 8087664.122641933, 1.0],
[{"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45},
0.2, 1.23330406, 1850806.1197361, 120.0, 8087664.122641933, 1.0],
# But note P(Q) = F2/volume
# F and F^2 are "unscaled", with for n <F F*>S(q) or for beta approx
# I(q) = n [<F F*> + <F><F*> (S(q) - 1)]
# for n the number density and <.> the orientation average, and
# F = integral rho(r) exp(i q . r) dr.
# The number density is volume fraction divided by particle volume.
# Effectively, this leaves F = V drho form, where form is the usual
# 3 j1(qr)/(qr) or whatever depending on the shape.
# @S RESULTS using F1 and F2 from the longer test string above:
#
# I(Q) = (F2 + F1^2*(S(Q) -1))*volfraction*scale/Volume + background
#
# with by default scale=1.0, background=0.001
# NOTE currently S(Q) volfraction is also included in scaling
# structure_factor_mode 0 = normal decoupling approx,
# 1 = beta(Q) approx
# radius_effective_mode 0 is for free choice,
# 1 is use radius from F2(Q)
# (sphere only has two choices, other models may have more)
[{"@S": "hardsphere",
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45, "volfraction": 0.2,
#"radius_effective":50.0, # hard sphere structure factor
"structure_factor_mode": 1, # mode 0 = normal decoupling approx,
# 1 = beta(Q) approx
"radius_effective_mode": 0 # this used default hardsphere Reff=50
}, [0.01, 0.1, 0.2], [1.32473756e+03, 7.36633631e-01, 4.67686201e-02]],
[{"@S": "hardsphere",
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45,
"volfraction": 0.2,
"radius_effective": 45.0, # explicit Reff over rides either 50 or 120
"structure_factor_mode": 1, # beta approx
"radius_effective_mode": 0 #
}, 0.01, 1316.2990966463444],
[{"@S": "hardsphere",
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45,
"volfraction": 0.2,
"radius_effective": 120.0, # over ride Reff
"structure_factor_mode": 1, # beta approx
"radius_effective_mode": 0 # (mode=1 here also uses 120)
}, [0.01, 0.1, 0.2], [1.57928589e+03, 7.37067923e-01, 4.67686197e-02]],
[{"@S": "hardsphere",
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45,
"volfraction": 0.2,
#"radius_effective": 120.0, # hard sphere structure factor
"structure_factor_mode": 0, # normal decoupling approximation
"radius_effective_mode": 1 # this uses 120 from the form factor
}, [0.01, 0.1, 0.2], [1.10112335e+03, 7.41366536e-01, 4.66630207e-02]],
[{"@S": "hardsphere",
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45,
"volfraction": 0.2,
#"radius_effective": 50.0, # hard sphere structure factor
"structure_factor_mode": 0, # normal decoupling approximation
"radius_effective_mode": 0 # this used 50 the default for hardsphere
}, [0.01, 0.1, 0.2], [7.82803598e+02, 6.85943611e-01, 4.71586457e-02]],
# Check returned intermediate results.
# Note: Target values come from double precision dll calculation.
# TODO: Cross check results against other software.
[{"@S": "hardsphere",
"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45,
"volfraction": 0.2,
"radius_effective": 120.0, # hard sphere structure factor
"structure_factor_mode": 1, # normal decoupling approximation
"radius_effective_mode": 0, # mode 0 says ignore Reff from P
}, [0.01, 0.1, 0.2], [1.57928589e+03, 7.37067923e-01, 4.67686197e-02],
{"P(Q)": [3487.905895219423, 0.7360339734027279, 0.04576861975646704],
"S(Q)": [0.31569726516764035, 1.005886362143737, 0.9976927625183415],
#"beta(Q)": [0.7996623765645325, 0.007835960247334845, 8.218250904154009e-07],
"beta(Q)": None, # Single precision not good enough for 5 digits of beta
"S_eff(Q)": [0.4527888487743462, 1.0000461252997597, 0.9999999981038543],
"volume": 8087664.122641933,
"volume_ratio": 1.0,
"radius_effective": 0.0, # zero since mode is 0, and Reff isn't computed
}],
]