[504abee] | 1 | r""" |
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| 2 | This model calculates the form factor for a flexible cylinder with an |
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| 3 | elliptical cross section and a uniform scattering length density. |
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| 4 | The non-negligible diameter of the cylinder is included by accounting |
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| 5 | for excluded volume interactions within the walk of a single cylinder. |
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[e53b881] | 6 | **Inter-cylinder interactions are NOT provided for.** |
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| 7 | |
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[504abee] | 8 | The form factor is normalized by the particle volume such that |
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| 9 | |
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| 10 | .. math:: |
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| 11 | |
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| 12 | P(q) = \text{scale} \left<F^2\right>/V + \text{background} |
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| 13 | |
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| 14 | where the averaging $\left<\ldots\right>$ is over all possible orientations |
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| 15 | of the flexible cylinder. |
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| 16 | |
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| 17 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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| 18 | of the q vector which is defined as |
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| 19 | |
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| 20 | .. math:: |
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| 21 | |
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| 22 | q = \sqrt{q_x^2 + q_y^2} |
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| 23 | |
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| 24 | |
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| 25 | Definitions |
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| 26 | ----------- |
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| 27 | |
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[e53b881] | 28 | The function is calculated in a similar way to that for the |
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| 29 | :ref:`flexible-cylinder` model in reference [1] below using the author's |
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| 30 | "Method 3 With Excluded Volume". |
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| 31 | |
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[504abee] | 32 | The model is a parameterization of simulations of a discrete representation of |
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| 33 | the worm-like chain model of Kratky and Porod applied in the pseudo-continuous |
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| 34 | limit. See equations (13, 26-27) in the original reference for the details. |
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| 35 | |
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| 36 | .. note:: |
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| 37 | |
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| 38 | There are several typos in the original reference that have been corrected |
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| 39 | by WRC. Details of the corrections are in the reference below. Most notably |
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| 40 | |
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| 41 | - Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$ |
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| 42 | |
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| 43 | - Equations (23) and (24) are incorrect; WRC has entered these into |
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| 44 | Mathematica and solved analytically. The results were then converted to |
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| 45 | code. |
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| 46 | |
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| 47 | - Equation (27) should be $q0 = max(a3/sqrt(RgSquare),3)$ instead of |
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| 48 | $max(a3*b/sqrt(RgSquare),3)$ |
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| 49 | |
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| 50 | - The scattering function is negative for a range of parameter values and |
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| 51 | q-values that are experimentally accessible. A correction function has been |
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| 52 | added to give the proper behavior. |
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| 53 | |
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| 54 | .. figure:: img/flexible_cylinder_ex_geometry.jpg |
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| 55 | |
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| 56 | |
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| 57 | The chain of contour length, $L$, (the total length) can be described as a chain |
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| 58 | of some number of locally stiff segments of length $l_p$, the persistence length |
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| 59 | (the length along the cylinder over which the flexible cylinder can be considered |
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| 60 | a rigid rod). |
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| 61 | The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain. |
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| 62 | |
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| 63 | The cross section of the cylinder is elliptical, with minor radius $a$ . |
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[e53b881] | 64 | The major radius is larger, so of course, **the axis_ratio must be |
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[504abee] | 65 | greater than one.** Simple constraints should be applied during curve fitting to |
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| 66 | maintain this inequality. |
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| 67 | |
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[ce8bed9] | 68 | In the parameters, the $sld$ and $sld\_solvent$ represent the SLD of the |
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[504abee] | 69 | chain/cylinder and solvent respectively. The *scale*, and the contrast are both |
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| 70 | multiplicative factors in the model and are perfectly correlated. One or both of |
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| 71 | these parameters must be held fixed during model fitting. |
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| 72 | |
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[e53b881] | 73 | **This is a model with complex behaviour depending on the ratio of** $L/b$ **and the |
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| 74 | reader is strongly encouraged to read reference [1] before use.** |
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[504abee] | 75 | |
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| 76 | References |
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| 77 | ---------- |
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| 78 | |
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[0507e09] | 79 | .. [#] J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612 |
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[504abee] | 80 | |
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| 81 | Correction of the formula can be found in |
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| 82 | |
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[0507e09] | 83 | .. [#] 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 |
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| 84 | |
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| 85 | Source |
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| 86 | ------ |
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| 87 | |
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| 88 | `flexible_cylinder_elliptical.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder_elliptical.py>`_ |
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| 89 | |
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| 90 | `flexible_cylinder_elliptical.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder_elliptical.c>`_ |
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| 91 | |
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| 92 | Authorship and Verification |
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| 93 | ---------------------------- |
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| 94 | |
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| 95 | * **Author:** |
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| 96 | * **Last Modified by:** |
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| 97 | * **Last Reviewed by:** |
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| 98 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[504abee] | 99 | """ |
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[2d81cfe] | 100 | |
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| 101 | import numpy as np |
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[504abee] | 102 | from numpy import inf |
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| 103 | |
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[65bf704] | 104 | name = "flexible_cylinder_elliptical" |
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[504abee] | 105 | title = "Flexible cylinder wth an elliptical cross section and a uniform " \ |
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| 106 | "scattering length density." |
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| 107 | description = """Note : scale and contrast=sldCyl-sldSolv are both multiplicative |
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| 108 | factors in the |
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| 109 | model and are perfectly correlated. One or |
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| 110 | both of these parameters must be held fixed |
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| 111 | during model fitting. |
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| 112 | """ |
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[13ed84c] | 113 | single = False |
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[504abee] | 114 | |
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| 115 | category = "shape:cylinder" |
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| 116 | # pylint: disable=bad-whitespace, line-too-long |
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| 117 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 118 | parameters = [ |
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| 119 | ["length", "Ang", 1000.0, [0, inf], "volume", "Length of the flexible cylinder"], |
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| 120 | ["kuhn_length", "Ang", 100.0, [0, inf], "volume", "Kuhn length of the flexible cylinder"], |
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[74fd96f] | 121 | ["radius", "Ang", 20.0, [1, inf], "volume", "Radius of the flexible cylinder"], |
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[ce8bed9] | 122 | ["axis_ratio", "", 1.5, [0, inf], "", "Axis_ratio (major_radius/minor_radius"], |
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[42356c8] | 123 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Cylinder scattering length density"], |
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| 124 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], |
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[504abee] | 125 | ] |
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| 126 | # pylint: enable=bad-whitespace, line-too-long |
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| 127 | |
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[40a87fa] | 128 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "lib/wrc_cyl.c", |
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| 129 | "flexible_cylinder_elliptical.c"] |
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[504abee] | 130 | |
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[31df0c9] | 131 | def random(): |
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[b297ba9] | 132 | """Return a random parameter set for the model.""" |
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[31df0c9] | 133 | length = 10**np.random.uniform(2, 6) |
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| 134 | radius = 10**np.random.uniform(1, 3) |
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| 135 | axis_ratio = 10**np.random.uniform(-1, 1) |
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| 136 | kuhn_length = 10**np.random.uniform(-2, -0.7)*length # at least 10 segments |
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| 137 | pars = dict( |
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| 138 | length=length, |
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| 139 | radius=radius, |
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| 140 | axis_ratio=axis_ratio, |
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| 141 | kuhn_length=kuhn_length, |
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| 142 | ) |
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| 143 | return pars |
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[504abee] | 144 | |
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| 145 | tests = [ |
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| 146 | # Accuracy tests based on content in test/utest_other_models.py |
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| 147 | # Currently fails in OCL |
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| 148 | # [{'length': 1000.0, |
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| 149 | # 'kuhn_length': 100.0, |
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| 150 | # 'radius': 20.0, |
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| 151 | # 'axis_ratio': 1.5, |
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| 152 | # 'sld': 1.0, |
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[ce8bed9] | 153 | # 'sld_solvent': 6.3, |
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[504abee] | 154 | # 'background': 0.0001, |
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| 155 | # }, 0.001, 3509.2187], |
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| 156 | |
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| 157 | # Additional tests with larger range of parameters |
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| 158 | [{'length': 1000.0, |
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| 159 | 'kuhn_length': 100.0, |
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| 160 | 'radius': 20.0, |
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| 161 | 'axis_ratio': 1.5, |
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| 162 | 'sld': 1.0, |
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[ce8bed9] | 163 | 'sld_solvent': 6.3, |
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[504abee] | 164 | 'background': 0.0001, |
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| 165 | }, 1.0, 0.00223819], |
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| 166 | [{'length': 10.0, |
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| 167 | 'kuhn_length': 800.0, |
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| 168 | 'radius': 2.0, |
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| 169 | 'axis_ratio': 0.5, |
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| 170 | 'sld': 6.0, |
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[ce8bed9] | 171 | 'sld_solvent': 12.3, |
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[504abee] | 172 | 'background': 0.001, |
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| 173 | }, 0.1, 0.390281], |
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| 174 | [{'length': 100.0, |
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| 175 | 'kuhn_length': 800.0, |
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| 176 | 'radius': 50.0, |
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| 177 | 'axis_ratio': 4.5, |
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| 178 | 'sld': 0.1, |
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[ce8bed9] | 179 | 'sld_solvent': 5.1, |
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[504abee] | 180 | 'background': 0.0, |
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| 181 | }, 1.0, 0.0016338264790] |
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| 182 | ] |
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