source:sasmodels/sasmodels/models/poly_gauss_coil.py@0507e09

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1#poly_gauss_coil model
2#conversion of Poly_GaussCoil.py
3#converted by Steve King, Mar 2016
4r"""
5This empirical model describes the scattering from *polydisperse* polymer
6chains in theta solvents or polymer melts, assuming a Schulz-Zimm type
7molecular weight distribution.
8
9To describe the scattering from *monodisperse* polymer chains, see the
10:ref:mono-gauss-coil model.
11
12Definition
13----------
14
15.. math::
16
17     I(q) = \text{scale} \cdot I_0 \cdot P(q) + \text{background}
18
19where
20
21.. math::
22
23     I_0 &= \phi_\text{poly} \cdot V \cdot (\rho_\text{poly}-\rho_\text{solv})^2 \\
24     P(q) &= 2 [(1 + UZ)^{-1/U} + Z - 1] / [(1 + U) Z^2] \\
25     Z &= [(q R_g)^2] / (1 + 2U) \\
26     U &= (Mw / Mn) - 1 = \text{polydispersity ratio} - 1 \\
27     V &= M / (N_A \delta)
28
29Here, $\phi_\text{poly}$, is the volume fraction of polymer, $V$ is the
30volume of a polymer coil, $M$ is the molecular weight of the polymer,
31$N_A$ is Avogadro's Number, $\delta$ is the bulk density of the polymer,
32$\rho_\text{poly}$ is the sld of the polymer, $\rho_\text{solv}$ is the
33sld of the solvent, and $R_g$ is the radius of gyration of the polymer coil.
34
35The 2D scattering intensity is calculated in the same way as the 1D,
36but where the $q$ vector is redefined as
37
38.. math::
39
40    q = \sqrt{q_x^2 + q_y^2}
41
42References
43----------
44
45.. [#] O Glatter and O Kratky (editors), *Small Angle X-ray Scattering*, Academic Press, (1982) Page 404
46.. [#] J S Higgins, H C Benoit, *Polymers and Neutron Scattering*, Oxford Science Publications, (1996)
47.. [#] S M King, *Small Angle Neutron Scattering* in *Modern Techniques for Polymer Characterisation*, Wiley, (1999)
48.. [#] http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf
49
50Source
51------
52
53poly_gauss_coil.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/poly_gauss_coil.py>_
54
55poly_gauss_coil.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/poly_gauss_coil.c>_
56
57Authorship and Verification
58----------------------------
59
60* **Author:**
62* **Last Reviewed by:**
63* **Source added by :** Steve King **Date:** March 25, 2019
64"""
65
66import numpy as np
67from numpy import inf, expm1, power
68
69name = "poly_gauss_coil"
70title = "Scattering from polydisperse polymer coils"
71
72description = """
73    Evaluates the scattering from
74    polydisperse polymer chains.
75    """
76category = "shape-independent"
77
79#   ["name", "units", default, [lower, upper], "type", "description"],
80parameters = [
81    ["i_zero",          "1/cm", 70.0, [0.0, inf], "", "Intensity at q=0"],
82    ["rg",  "Ang", 75.0, [0.0, inf], "", "Radius of gyration"],
83    ["polydispersity",  "None",  2.0, [1.0, inf], "", "Polymer Mw/Mn"],
84    ]
86
87# NB: Scale and Background are implicit parameters on every model
88def Iq(q, i_zero, rg, polydispersity):
89    # pylint: disable = missing-docstring
90    u = polydispersity - 1.0
91    z = q**2 * (rg**2 / (1.0 + 2.0*u))
92
93    # need to trap the case of the polydispersity being 1 (ie, monodisperse!)
94    if polydispersity == 1.0:
95        result = 2.0 * (expm1(-z) + z)
96        index = q != 0.
97        result[index] /= z[index]**2
98        result[~index] = 1.0
99    else:
100        # Taylor series around z=0 of (2*(1+uz)^(-1/u) + z - 1) / (z^2(u+1))
101        p = [
102            #(-1 - 20*u - 155*u**2 - 580*u**3 - 1044*u**4 - 720*u**5) / 2520.,
103            #(+1 + 14*u + 71*u**2 + 154*u**3 + 120*u**4) / 360.,
104            #(-1 - 9*u - 26*u**2 - 24*u**3) / 60.,
105            (+1 + 5*u + 6*u**2) / 12.,
106            (-1 - 2*u) / 3.,
107            (+1),
108            ]
109        result = 2.0 * (power(1.0 + u*z, -1.0/u) + z - 1.0) / (1.0 + u)
110        index = z > 1e-4
111        result[index] /= z[index]**2
112        result[~index] = np.polyval(p, z[~index])
113    return i_zero * result
114Iq.vectorized = True  # Iq accepts an array of q values
115
116def random():
117    """Return a random parameter set for the model."""
118    rg = 10**np.random.uniform(0, 4)
119    #rg = 1e3
120    polydispersity = 10**np.random.uniform(0, 3)
121    pars = dict(
122        #scale=1, background=0,
123        i_zero=1e7, # i_zero is a simple scale
124        rg=rg,
125        polydispersity=polydispersity,
126    )
127    return pars
128
129demo = dict(scale=1.0,
130            i_zero=70.0,
131            rg=75.0,
132            polydispersity=2.0,
133            background=0.0)
134
135# these unit test values taken from SasView 3.1.2
136tests = [
137    [{'scale': 1.0, 'i_zero': 70.0, 'rg': 75.0,
138      'polydispersity': 2.0, 'background': 0.0},
139     [0.0106939, 0.469418], [57.6405, 0.169016]],
140    ]
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