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

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Line 
1r"""
2For information about polarised and magnetic scattering, see
3the :ref:`magnetism` documentation.
4
5Definition
6----------
7
8The 1D scattering intensity is calculated in the following way (Guinier, 1955)
9
10.. math::
11
12    I(q) = \frac{\text{scale}}{V} \cdot \left[
13        3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3}
14        \right]^2 + \text{background}
15
16where *scale* is a volume fraction, $V$ is the volume of the scatterer,
17$r$ is the radius of the sphere and *background* is the background level.
18*sld* and *sld_solvent* are the scattering length densities (SLDs) of the
19scatterer and the solvent respectively, whose difference is $\Delta\rho$.
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
36References
37----------
38
39.. [#] A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
40   John Wiley and Sons, New York, (1955)
41
42Authorship and Verification
43----------------------------
44
45* **Author:**
46* **Last Modified by:**
47* **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06
48"""
49
50import numpy as np
51from numpy import inf
52
53name = "sphere"
54title = "Spheres with uniform scattering length density"
55description = """\
56P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr))
57                /(qr)^3]^2 + background
58    r: radius of sphere
59    V: The volume of the scatter
60    sld: the SLD of the sphere
61    sld_solvent: the SLD of the solvent
62"""
63category = "shape:sphere"
64
65#             ["name", "units", default, [lower, upper], "type","description"],
66parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld",
67               "Layer scattering length density"],
68              ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
69               "Solvent scattering length density"],
70              ["radius", "Ang", 50, [0, inf], "volume",
71               "Sphere radius"],
72             ]
73
74source = ["lib/sas_3j1x_x.c", "sphere.c"]
75have_Fq = True
76radius_effective_modes = ["radius"]
77
78def random():
79    """Return a random parameter set for the model."""
80    radius = 10**np.random.uniform(1.3, 4)
81    pars = dict(
82        radius=radius,
83    )
84    return pars
85#2345678901234567890123456789012345678901234567890123456789012345678901234567890
86tests = [
87     [{}, 0.2, 0.726362], # each test starts with default parameter values
88     #            inside { }, unless modified. Then Q and expected value of I(Q)
89     # putting None for an expected result will pass the test if there are no
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.,
92       "radius": 120.}, [0.01,0.1,0.2],
93     [1.34836265e+04, 6.20114062e+00, 1.04733914e-01]],
94     [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
95     #  careful tests here R=120 Pd=.2, then with S(Q) at default Reff=50
96     #  (but this gets changed to 120) phi=0,2
97       "radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
98      [0.01,0.1,0.2], [1.74395295e+04, 3.68016987e+00, 2.28843099e-01]],
99     # a list of Q values and list of expected results is also possible
100    [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
101     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
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
104    [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
105      0.1, 482.93824329, 29763977.79867414, 120.0, 8087664.122641933, 1.0],
106    [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
107      0.2, 1.23330406, 1850806.1197361, 120.0, 8087664.122641933, 1.0],
108   #  But note P(Q) = F2/volume
109   #  F and F^2 are "unscaled", with for  n <F F*>S(q) or for beta approx
110   #          I(q) = n [<F F*> + <F><F*> (S(q) - 1)]
111   #  for n the number density and <.> the orientation average, and
112   #  F = integral rho(r) exp(i q . r) dr.
113   #  The number density is volume fraction divided by particle volume.
114   #  Effectively, this leaves F = V drho form, where form is the usual
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
122   #  structure_factor_mode 0 = normal decoupling approx,
123   #                        1 = beta(Q) approx
124   # radius_effective_mode  0 is for free choice,
125   #                        1 is use radius from F2(Q)
126   #    (sphere only has two choices, other models may have more)
127    [{"@S": "hardsphere",
128     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,"volfraction":0.2,
129     #"radius_effective":50.0,    # hard sphere structure factor
130     "structure_factor_mode": 1,  # mode 0 = normal decoupling approx,
131     #                                   1 = beta(Q) approx
132     "radius_effective_mode": 0   # this used default hardsphere Reff=50
133     }, [0.01,0.1,0.2], [1.32473756e+03, 7.36633631e-01, 4.67686201e-02]  ],
134    [{"@S": "hardsphere",
135     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,
136     "volfraction":0.2,
137     "radius_effective":45.0,     # explicit Reff over rides either 50 or 120
138     "structure_factor_mode": 1,  # beta approx
139     "radius_effective_mode": 0   #
140     }, 0.01, 1316.2990966463444 ],
141    [{"@S": "hardsphere",
142     "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,
143     "volfraction":0.2,
144     "radius_effective":120.0,    # over ride Reff
145     "structure_factor_mode": 1,  # beta approx
146     "radius_effective_mode": 0   # (mode=1 here also uses 120)
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,
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
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,
158     #"radius_effective":50.0,    # hard sphere structure factor
159     "structure_factor_mode": 0,  # normal decoupling approximation
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 ]]
162]
163#
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