source: sasmodels/sasmodels/models/binary_hard_sphere.py @ b0c4271

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model documentation final format through core_shell_bicelle re: #646

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1r"""
2Definition
3----------
4
5The binary hard sphere model provides the scattering intensity, for binary
6mixture of hard spheres including hard sphere interaction between those
7particles, using rhw Percus-Yevick closure. The calculation is an exact
8multi-component solution that properly accounts for the 3 partial structure
9factors as follows:
10
11.. math::
12
13    I(q) = (1-x)f_1^2(q) S_{11}(q) + 2[x(1-x)]^{1/2} f_1(q)f_2(q)S_{12}(q) +
14    x\,f_2^2(q)S_{22}(q)
15
16where $S_{ij}$ are the partial structure factors and $f_i$ are the scattering
17amplitudes of the particles. The subscript 1 is for the smaller particle and 2
18is for the larger. The number fraction of the larger particle,
19($x = n2/(n1+n2)$, where $n$ = the number density) is internally calculated
20based on the diameter ratio and the volume fractions.
21
22.. math::
23    :nowrap:
24
25    \begin{align}
26    x &= \frac{(\phi_2 / \phi)\alpha^3}{(1-(\phi_2/\phi) + (\phi_2/\phi)
27    \alpha^3)} \\
28    \phi &= \phi_1 + \phi_2 = \text{total volume fraction} \\
29    \alpha &= R_1/R_2 = \text{size ratio}
30    \end{align}
31
32The 2D scattering intensity is the same as 1D, regardless of the orientation of
33the *q* vector which is defined as
34
35.. math::
36
37    q = \sqrt{q_x^2 + q_y^2}
38
39
40**NOTE 1:** The volume fractions and the scattering contrasts are loosely
41correlated, so holding as many parameters fixed to known values during fitting
42will improve the robustness of the fit.
43
44**NOTE 2:** Since the calculation uses the Percus-Yevick closure, all of the
45limitations of that closure relation apply here. Specifically, one should be
46wary of results for (total) volume fractions greater than approximately 40%.
47Depending on the size ratios or number fractions, the limit on total volume
48fraction may be lower.
49
50**NOTE 3:** The heavy arithmatic operations also mean that at present the
51function is poorly behaved at very low qr.  In some cases very large qr may
52also be poorly behaved.  These should however be outside any useful region of
53qr.
54
55The code for this model is based originally on a c-library implementation by the
56NIST Center for Neutron Research (Kline, 2006).
57
58See the references for details.
59
60References
61----------
62
63.. [#] N W Ashcroft and D C Langreth, *Physical Review*, 156 (1967) 685-692
64   [Errata found in *Phys. Rev.* 166 (1968) 934]
65.. [#] S R Kline, *J Appl. Cryst.*, 39 (2006) 895
66
67Authorship and Verification
68----------------------------
69
70* **Author:** NIST IGOR/DANSE **Date:** pre 2010
71* **Last Modified by:** Paul Butler **Date:** March 20, 2016
72* **Last Reviewed by:** Paul Butler **Date:** March 20, 2016
73"""
74
75from numpy import inf
76
77category = "shape:sphere"
78single = False  # double precision only!
79
80name = "binary_hard_sphere"
81title = "binary mixture of hard spheres with hard sphere interactions."
82description = """Describes the scattering from a mixture of two distinct
83monodisperse, hard sphere particles.
84        [Parameters];
85        radius_lg: large radius of binary hard sphere,
86        radius_sm: small radius of binary hard sphere,
87        volfraction_lg: volume fraction of large spheres,
88        volfraction_sm: volume fraction of small spheres,
89        sld_lg: large sphere  scattering length density,
90        sld_sm: small sphere scattering length density,
91        sld_solvent: solvent scattering length density.
92"""
93#             ["name", "units", default, [lower, upper], "type", "description"],
94parameters = [["radius_lg", "Ang", 100, [0, inf], "",
95               "radius of large particle"],
96              ["radius_sm", "Ang", 25, [0, inf], "",
97               "radius of small particle"],
98              ["volfraction_lg", "", 0.1, [0, 1], "",
99               "volume fraction of large particle"],
100              ["volfraction_sm", "", 0.2, [0, 1], "",
101               "volume fraction of small particle"],
102              ["sld_lg", "1e-6/Ang^2", 3.5, [-inf, inf], "sld",
103               "scattering length density of large particle"],
104              ["sld_sm", "1e-6/Ang^2", 0.5, [-inf, inf], "sld",
105               "scattering length density of small particle"],
106              ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld",
107               "Solvent scattering length density"],
108             ]
109
110source = ["lib/sph_j1c.c", "binary_hard_sphere.c"]
111
112# parameters for demo and documentation
113demo = dict(sld_lg=3.5, sld_sm=0.5, sld_solvent=6.36,
114            radius_lg=100, radius_sm=20,
115            volfraction_lg=0.1, volfraction_sm=0.2)
116
117# NOTE: test results taken from values returned by SasView 3.1.2
118tests = [[{}, 0.001, 25.8927262013]]
119
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