# Sphere - sphere.py

```    ```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{	ext{scale}}{V} cdot left[
3V(Delta
ho) cdot frac{sin(qr) - qrcos(qr))}{(qr)^3}

ight]^2 + 	ext{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
ho\$.

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 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
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",
]

source = ["lib/sas_3j1x_x.c", "sphere.c"]
have_Fq = True

def random():
"""Return a random parameter set for the model."""
pars = dict(
)
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.,
[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
[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.,
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
0.1, 482.93824329, 29763977.79867414, 120.0, 8087664.122641933, 1.0],
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 strng 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
#                        1 is use radius from F2(Q)
#    (sphere only has two choices, other models may have more)
[{"@S": "hardsphere",
#"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",
"volfraction":0.2,
"radius_effective":45.0,     # explicit Reff over rides either 50 or 120
"structure_factor_mode": 1,  # beta approx
}, 0.01, 1316.2990966463444 ],
[{"@S": "hardsphere",
"volfraction":0.2,
"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",
"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",