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

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[f94d8a2]1r"""
[168052c]2This model provides the form factor, $P(q)$, for a flexible cylinder
3where the form factor is normalized by the volume of the cylinder.
[f94d8a2]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
[168052c]10where the averaging $\left<\ldots\right>$ is applied only for the 1D
11calculation
[f94d8a2]12
[168052c]13The 2D scattering intensity is the same as 1D, regardless of the orientation of
14the q vector which is defined as
[f94d8a2]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
[168052c]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).
[f94d8a2]30The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain.
31
[ce8bed9]32In the parameters, the sld and sld\_solvent represent the SLD of the cylinder
[168052c]33and solvent respectively.
[f94d8a2]34
[168052c]35Our model uses the form factor calculations implemented in a c-library provided
[598a354]36by the NIST Center for Neutron Research (Kline, 2006). This states:
[f94d8a2]37
38    'Method 3 With Excluded Volume' is used.
39    The model is a parametrization of simulations of a discrete representation
[168052c]40    of the worm-like chain model of Kratky and Porod applied in the
41    pseudocontinuous limit.
[f94d8a2]42    See equations (13,26-27) in the original reference for the details.
43
44References
45----------
46
[0507e09]47.. [#] J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612
[f94d8a2]48
49Correction of the formula can be found in
50
[0507e09]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
[f94d8a2]67"""
[2d81cfe]68
69import numpy as np
[f94d8a2]70from numpy import inf
71
72name = "flexible_cylinder"
[598a354]73title = "Flexible cylinder where the form factor is normalized by the volume " \
[168052c]74        "of the cylinder."
[e65a3e7]75description = """Note : scale and contrast = (sld - sld_solvent) are both
[168052c]76                multiplicative factors in the model and are perfectly
77                correlated. One or both of these parameters must be held fixed
[f94d8a2]78                during model fitting.
79              """
80
81category = "shape:cylinder"
[e65a3e7]82single = False  # double precision only!
[f94d8a2]83
[168052c]84# pylint: disable=bad-whitespace, line-too-long
[f94d8a2]85#             ["name", "units", default, [lower, upper], "type", "description"],
86parameters = [
[168052c]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"],
[42356c8]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"],
[168052c]92    ]
93# pylint: enable=bad-whitespace, line-too-long
[26141cb]94source = ["lib/polevl.c", "lib/sas_J1.c", "lib/wrc_cyl.c", "flexible_cylinder.c"]
[f94d8a2]95
[31df0c9]96def random():
[b297ba9]97    """Return a random parameter set for the model."""
[31df0c9]98    length = 10**np.random.uniform(2, 6)
99    radius = 10**np.random.uniform(1, 3)
[a8631ca]100    kuhn_length = 10**np.random.uniform(-2, 0)*length
[31df0c9]101    pars = dict(
102        length=length,
103        radius=radius,
104        kuhn_length=kuhn_length,
105    )
106    return pars
[f94d8a2]107
108tests = [
[168052c]109    # Accuracy tests based on content in test/utest_other_models.py
[2573fa1]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],
[168052c]117
118    # Additional tests with larger range of parameters
[18a2bfc]119    [{'length':    1000.0,  # test T2
[168052c]120      'kuhn_length': 100.0,
121      'radius':       20.0,
122      'sld':           1.0,
[e65a3e7]123      'sld_solvent':   6.3,
[168052c]124      'background':    0.0001,
125     }, 1.0, 0.000595345],
[18a2bfc]126    [{'length':        10.0,  # test T3
[168052c]127      'kuhn_length': 800.0,
128      'radius':        2.0,
129      'sld':           6.0,
[e65a3e7]130      'sld_solvent':  12.3,
[168052c]131      'background':    0.001,
132     }, 0.1, 1.55228],
[18a2bfc]133    [{'length':        100.0,  # test T4
[168052c]134      'kuhn_length': 800.0,
135      'radius':       50.0,
136      'sld':           0.1,
[e65a3e7]137      'sld_solvent':   5.1,
[168052c]138      'background':    0.0,
139     }, 1.0, 0.000938456]
140    ]
[18a2bfc]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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