source: sasmodels/sasmodels/models/sphere.py @ d57b06c

Last change on this file since d57b06c was d57b06c, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

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[5d4777d]1r"""
[40a87fa]2For information about polarised and magnetic scattering, see
[9a4811a]3the :ref:`magnetism` documentation.
[19dcb933]4
5Definition
6----------
7
8The 1D scattering intensity is calculated in the following way (Guinier, 1955)
9
10.. math::
11
[eb69cce]12    I(q) = \frac{\text{scale}}{V} \cdot \left[
13        3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3}
[19dcb933]14        \right]^2 + \text{background}
15
16where *scale* is a volume fraction, $V$ is the volume of the scatterer,
[7e6bea81]17$r$ is the radius of the sphere and *background* is the background level.
[49da079]18*sld* and *sld_solvent* are the scattering length densities (SLDs) of the
[7e6bea81]19scatterer and the solvent respectively, whose difference is $\Delta\rho$.
[19dcb933]20
21Note that if your data is in absolute scale, the *scale* should represent
22the volume fraction (which is unitless) if you have a good fit. If not,
23it should represent the volume fraction times a factor (by which your data
24might need to be rescaled).
25
26The 2D scattering intensity is the same as above, regardless of the
27orientation of $\vec q$.
28
29Validation
30----------
31
32Validation of our code was done by comparing the output of the 1D model
33to the output of the software provided by the NIST (Kline, 2006).
34
35
[eb69cce]36References
37----------
[19dcb933]38
[934a001]39.. [#] A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
40   John Wiley and Sons, New York, (1955)
[19dcb933]41
[0507e09]42Authorship and Verification
43----------------------------
44
[c1e44e5]45* **Author:**
46* **Last Modified by:**
[ef07e95]47* **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06
[5d4777d]48"""
49
[2d81cfe]50import numpy as np
[3c56da87]51from numpy import inf
[5d4777d]52
53name = "sphere"
[19dcb933]54title = "Spheres with uniform scattering length density"
[5d4777d]55description = """\
[49da079]56P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr))
[eb69cce]57                /(qr)^3]^2 + background
58    r: radius of sphere
[19dcb933]59    V: The volume of the scatter
60    sld: the SLD of the sphere
[49da079]61    sld_solvent: the SLD of the solvent
[5d4777d]62"""
[a5d0d00]63category = "shape:sphere"
[5d4777d]64
[3e428ec]65#             ["name", "units", default, [lower, upper], "type","description"],
[42356c8]66parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld",
[3e428ec]67               "Layer scattering length density"],
[42356c8]68              ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
[3e428ec]69               "Solvent scattering length density"],
70              ["radius", "Ang", 50, [0, inf], "volume",
71               "Sphere radius"],
72             ]
[5d4777d]73
[b297ba9]74source = ["lib/sas_3j1x_x.c", "sphere.c"]
[71b751d]75have_Fq = True
[a34b811]76radius_effective_modes = ["radius"]
[c036ddb]77
[404ebbd]78def random():
[b297ba9]79    """Return a random parameter set for the model."""
[404ebbd]80    radius = 10**np.random.uniform(1.3, 4)
81    pars = dict(
82        radius=radius,
83    )
84    return pars
[934a001]85#2345678901234567890123456789012345678901234567890123456789012345678901234567890
[7e6bea81]86tests = [
[d57b06c]87     [{}, 0.2, 0.726362], # each test starts with default parameter values
[934a001]88     #            inside { }, unless modified. Then Q and expected value of I(Q)
[d57b06c]89     # putting None for an expected result will pass the test if there are no
[934a001]90     # errors from the routine, but without any check on the value of the result
91    [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
[d57b06c]92       "radius": 120.}, [0.01,0.1,0.2],
[934a001]93     [1.34836265e+04, 6.20114062e+00, 1.04733914e-01]],
[6140894]94     [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
[d57b06c]95     #  careful tests here R=120 Pd=.2, then with S(Q) at default Reff=50
[934a001]96     #  (but this gets changed to 120) phi=0,2
[6140894]97       "radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
[d57b06c]98      [0.01,0.1,0.2], [1.74395295e+04, 3.68016987e+00, 2.28843099e-01]],
[934a001]99     # a list of Q values and list of expected results is also possible
[7e6bea81]100    [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
[934a001]101     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
[d57b06c]102      0.01, 335839.88055473, 1.41045057e+11, 120.0, 8087664.122641933, 1.0],
103     # the longer list here checks  F1, F2, R_eff, volume, volume_ratio
[db3947c]104    [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
[d57b06c]105      0.1, 482.93824329, 29763977.79867414, 120.0, 8087664.122641933, 1.0],
[304c775]106    [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
[934a001]107      0.2, 1.23330406, 1850806.1197361, 120.0, 8087664.122641933, 1.0],
[6140894]108   #  But note P(Q) = F2/volume
[d57b06c]109   #  F and F^2 are "unscaled", with for  n <F F*>S(q) or for beta approx
[934a001]110   #          I(q) = n [<F F*> + <F><F*> (S(q) - 1)]
[d57b06c]111   #  for n the number density and <.> the orientation average, and
[934a001]112   #  F = integral rho(r) exp(i q . r) dr.
[a34b811]113   #  The number density is volume fraction divided by particle volume.
[d57b06c]114   #  Effectively, this leaves F = V drho form, where form is the usual
[934a001]115   #  3 j1(qr)/(qr) or whatever depending on the shape.
116   # @S RESULTS using F1 and F2 from the longer test strng above:
117   #
118   # I(Q) = (F2 + F1^2*(S(Q) -1))*volfraction*scale/Volume  + background
119   #
120   # with by default scale=1.0, background=0.001
121   # NOTE currently S(Q) volfraction is also included in scaling
[d57b06c]122   #  structure_factor_mode 0 = normal decoupling approx,
[934a001]123   #                        1 = beta(Q) approx
[d57b06c]124   # radius_effective_mode  0 is for free choice,
[934a001]125   #                        1 is use radius from F2(Q)
126   #    (sphere only has two choices, other models may have more)
[b39bf3b]127    [{"@S": "hardsphere",
[934a001]128     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,"volfraction":0.2,
129     #"radius_effective":50.0,    # hard sphere structure factor
[d57b06c]130     "structure_factor_mode": 1,  # mode 0 = normal decoupling approx,
[934a001]131     #                                   1 = beta(Q) approx
[d57b06c]132     "radius_effective_mode": 0   # this used default hardsphere Reff=50
[934a001]133     }, [0.01,0.1,0.2], [1.32473756e+03, 7.36633631e-01, 4.67686201e-02]  ],
[6140894]134    [{"@S": "hardsphere",
135     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,
136     "volfraction":0.2,
[934a001]137     "radius_effective":45.0,     # explicit Reff over rides either 50 or 120
138     "structure_factor_mode": 1,  # beta approx
[d57b06c]139     "radius_effective_mode": 0   #
[934a001]140     }, 0.01, 1316.2990966463444 ],
[6140894]141    [{"@S": "hardsphere",
142     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,
143     "volfraction":0.2,
[934a001]144     "radius_effective":120.0,    # over ride Reff
145     "structure_factor_mode": 1,  # beta approx
[d57b06c]146     "radius_effective_mode": 0   # (mode=1 here also uses 120)
[db3947c]147     }, [0.01,0.1,0.2], [1.57928589e+03, 7.37067923e-01, 4.67686197e-02  ]],
148    [{"@S": "hardsphere",
149     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,
150     "volfraction":0.2,
[934a001]151     #"radius_effective":120.0,   # hard sphere structure factor
152     "structure_factor_mode": 0,  # normal decoupling approximation
153     "radius_effective_mode": 1   # this uses 120 from the form factor
[db3947c]154     }, [0.01,0.1,0.2], [1.10112335e+03, 7.41366536e-01, 4.66630207e-02]],
155    [{"@S": "hardsphere",
156     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,
157     "volfraction":0.2,
[934a001]158     #"radius_effective":50.0,    # hard sphere structure factor
159     "structure_factor_mode": 0,  # normal decoupling approximation
[db3947c]160     "radius_effective_mode": 0   # this used 50 the default for hardsphere
161     }, [0.01,0.1,0.2], [7.82803598e+02, 6.85943611e-01, 4.71586457e-02 ]]
[7e6bea81]162]
[934a001]163#
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