source: sasmodels/sasmodels/models/flexible_cylinder.py @ 598a354

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1r"""
2This model provides the form factor, $P(q)$, for a flexible cylinder
3where the form factor is normalized by the volume of the cylinder.
4**Inter-cylinder interactions are NOT provided for.**
5
6.. math::
7
8    P(q) = \text{scale} \left<F^2\right>/V + \text{background}
9
10where the averaging $\left<\ldots\right>$ is applied only for the 1D
11calculation
12
13The 2D scattering intensity is the same as 1D, regardless of the orientation of
14the q vector which is defined as
15
16.. math::
17
18    q = \sqrt{q_x^2 + q_y^2}
19
20Definitions
21-----------
22
23.. figure:: img/flexible_cylinder_geometry.jpg
24
25
26The chain of contour length, $L$, (the total length) can be described as a
27chain of some number of locally stiff segments of length $l_p$, the persistence
28length (the length along the cylinder over which the flexible cylinder can be
29considered a rigid rod).
30The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain.
31
32In the parameters, the sld and sld\_solvent represent the SLD of the cylinder
33and solvent respectively.
34
35Our model uses the form factor calculations implemented in a c-library provided
36by the NIST Center for Neutron Research (Kline, 2006). This states:
37
38    'Method 3 With Excluded Volume' is used.
39    The model is a parametrization of simulations of a discrete representation
40    of the worm-like chain model of Kratky and Porod applied in the
41    pseudocontinuous limit.
42    See equations (13,26-27) in the original reference for the details.
43
44References
45----------
46
47.. [#] J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612
48
49Correction of the formula can be found in
50
51.. [#] W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles.* Langmuir, 22(15) 2006 6539-6548
52
53Source
54------
55
56`flexible_cylinder.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder.py>`_
57
58`flexible_cylinder.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder.c>`_
59
60Authorship and Verification
61----------------------------
62
63* **Author:**
64* **Last Modified by:**
65* **Last Reviewed by:**
66* **Source added by :** Steve King **Date:** March 25, 2019
67"""
68
69import numpy as np
70from numpy import inf
71
72name = "flexible_cylinder"
73title = "Flexible cylinder where the form factor is normalized by the volume " \
74        "of the cylinder."
75description = """Note : scale and contrast = (sld - sld_solvent) are both
76                multiplicative factors in the model and are perfectly
77                correlated. One or both of these parameters must be held fixed
78                during model fitting.
79              """
80
81category = "shape:cylinder"
82single = False  # double precision only!
83
84# pylint: disable=bad-whitespace, line-too-long
85#             ["name", "units", default, [lower, upper], "type", "description"],
86parameters = [
87    ["length",      "Ang",       1000.0, [0, inf],    "volume", "Length of the flexible cylinder"],
88    ["kuhn_length", "Ang",        100.0, [0, inf],    "volume", "Kuhn length of the flexible cylinder"],
89    ["radius",      "Ang",         20.0, [0, inf],    "volume", "Radius of the flexible cylinder"],
90    ["sld",         "1e-6/Ang^2",   1.0, [-inf, inf], "sld",    "Cylinder scattering length density"],
91    ["sld_solvent", "1e-6/Ang^2",   6.3, [-inf, inf], "sld",    "Solvent scattering length density"],
92    ]
93# pylint: enable=bad-whitespace, line-too-long
94source = ["lib/polevl.c", "lib/sas_J1.c", "lib/wrc_cyl.c", "flexible_cylinder.c"]
95
96def random():
97    """Return a random parameter set for the model."""
98    length = 10**np.random.uniform(2, 6)
99    radius = 10**np.random.uniform(1, 3)
100    kuhn_length = 10**np.random.uniform(-2, 0)*length
101    pars = dict(
102        length=length,
103        radius=radius,
104        kuhn_length=kuhn_length,
105    )
106    return pars
107
108tests = [
109    # Accuracy tests based on content in test/utest_other_models.py
110    [{'length':     1000.0,  # test T1
111      'kuhn_length': 100.0,
112      'radius':       20.0,
113      'sld':           1.0,
114      'sld_solvent':   6.3,
115      'background':    0.0001,
116     }, 0.001, 3509.2187],
117
118    # Additional tests with larger range of parameters
119    [{'length':    1000.0,  # test T2
120      'kuhn_length': 100.0,
121      'radius':       20.0,
122      'sld':           1.0,
123      'sld_solvent':   6.3,
124      'background':    0.0001,
125     }, 1.0, 0.000595345],
126    [{'length':        10.0,  # test T3
127      'kuhn_length': 800.0,
128      'radius':        2.0,
129      'sld':           6.0,
130      'sld_solvent':  12.3,
131      'background':    0.001,
132     }, 0.1, 1.55228],
133    [{'length':        100.0,  # test T4
134      'kuhn_length': 800.0,
135      'radius':       50.0,
136      'sld':           0.1,
137      'sld_solvent':   5.1,
138      'background':    0.0,
139     }, 1.0, 0.000938456]
140    ]
141
142# There are a few branches in the code that ought to have test values:
143#
144# For length > 4 * kuhn_length
145#        if length > 10 * kuhn_length then C is scaled by 3.06 (L/b)^(-0.44)
146#        q*kuhn_length <= 3.1  => Sexv_new
147#           dS/dQ < 0 has different behaviour from dS/dQ >= 0
148#  T2    q*kuhn_length > 3.1   => a_long
149#
150# For length <= 4 * kuhn_length
151#        q*kuhn_length <= max(1.9/Rg_short, 3.0)  => Sdebye((q*Rg)^2)
152#           q*Rg < 0.5 uses Pade approx, q*Rg > 1.0 uses math lib
153#  T3,T4 q*kuhn_length > max(1.9/Rg_short, 3.0)   => a_short
154#
155# Note that the transitions between branches may be abrupt.  You can see a
156# several percent change around length=10*kuhn_length and length=4*kuhn_length
157# using the following:
158#
159#    sascomp flexible_cylinder -calc=double -sets=10 length=10*kuhn_length,10.000001*kuhn_length
160#    sascomp flexible_cylinder -calc=double -sets=10 length=4*kuhn_length,4.000001*kuhn_length
161#
162# The transition between low q and high q around q*kuhn_length = 3 seems
163# to be good to 4 digits or better.  This was tested by computing the value
164# on each branches near the transition point and reporting the relative error
165# for kuhn lengths of 10, 100 and 1000 and a variety of length:kuhn_length
166# ratios.
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