[f94d8a2] | 1 | r""" |
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[168052c] | 2 | This model provides the form factor, $P(q)$, for a flexible cylinder |
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| 3 | where the form factor is normalized by the volume of the cylinder. |
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[f94d8a2] | 4 | **Inter-cylinder interactions are NOT provided for.** |
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| 5 | |
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| 6 | .. math:: |
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| 7 | |
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| 8 | P(q) = \text{scale} \left<F^2\right>/V + \text{background} |
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| 9 | |
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[168052c] | 10 | where the averaging $\left<\ldots\right>$ is applied only for the 1D |
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| 11 | calculation |
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[f94d8a2] | 12 | |
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[168052c] | 13 | The 2D scattering intensity is the same as 1D, regardless of the orientation of |
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| 14 | the q vector which is defined as |
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[f94d8a2] | 15 | |
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| 16 | .. math:: |
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| 17 | |
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| 18 | q = \sqrt{q_x^2 + q_y^2} |
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| 19 | |
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| 20 | Definitions |
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| 21 | ----------- |
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| 22 | |
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| 23 | .. figure:: img/flexible_cylinder_geometry.jpg |
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| 24 | |
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| 25 | |
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[168052c] | 26 | The chain of contour length, $L$, (the total length) can be described as a |
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| 27 | chain of some number of locally stiff segments of length $l_p$, the persistence |
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| 28 | length (the length along the cylinder over which the flexible cylinder can be |
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| 29 | considered a rigid rod). |
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[f94d8a2] | 30 | The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain. |
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| 31 | |
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[ce8bed9] | 32 | In the parameters, the sld and sld\_solvent represent the SLD of the cylinder |
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[168052c] | 33 | and solvent respectively. |
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[f94d8a2] | 34 | |
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[168052c] | 35 | Our model uses the form factor calculations implemented in a c-library provided |
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[598a354] | 36 | by the NIST Center for Neutron Research (Kline, 2006). This states: |
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[f94d8a2] | 37 | |
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| 38 | 'Method 3 With Excluded Volume' is used. |
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| 39 | The model is a parametrization of simulations of a discrete representation |
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[168052c] | 40 | of the worm-like chain model of Kratky and Porod applied in the |
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| 41 | pseudocontinuous limit. |
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[f94d8a2] | 42 | See equations (13,26-27) in the original reference for the details. |
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| 43 | |
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| 44 | References |
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| 45 | ---------- |
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| 46 | |
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[0507e09] | 47 | .. [#] 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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[f94d8a2] | 48 | |
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| 49 | Correction of the formula can be found in |
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| 50 | |
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[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 |
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| 52 | |
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| 53 | Source |
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| 54 | ------ |
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| 55 | |
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| 56 | `flexible_cylinder.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder.py>`_ |
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| 57 | |
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| 58 | `flexible_cylinder.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder.c>`_ |
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| 59 | |
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| 60 | Authorship and Verification |
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| 61 | ---------------------------- |
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| 62 | |
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| 63 | * **Author:** |
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| 64 | * **Last Modified by:** |
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| 65 | * **Last Reviewed by:** |
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| 66 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[f94d8a2] | 67 | """ |
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[2d81cfe] | 68 | |
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| 69 | import numpy as np |
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[f94d8a2] | 70 | from numpy import inf |
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| 71 | |
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| 72 | name = "flexible_cylinder" |
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[598a354] | 73 | title = "Flexible cylinder where the form factor is normalized by the volume " \ |
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[168052c] | 74 | "of the cylinder." |
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[e65a3e7] | 75 | description = """Note : scale and contrast = (sld - sld_solvent) are both |
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[168052c] | 76 | multiplicative factors in the model and are perfectly |
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| 77 | correlated. One or both of these parameters must be held fixed |
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[f94d8a2] | 78 | during model fitting. |
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| 79 | """ |
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| 80 | |
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| 81 | category = "shape:cylinder" |
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[e65a3e7] | 82 | single = False # double precision only! |
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[f94d8a2] | 83 | |
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[168052c] | 84 | # pylint: disable=bad-whitespace, line-too-long |
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[f94d8a2] | 85 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 86 | parameters = [ |
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[168052c] | 87 | ["length", "Ang", 1000.0, [0, inf], "volume", "Length of the flexible cylinder"], |
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| 88 | ["kuhn_length", "Ang", 100.0, [0, inf], "volume", "Kuhn length of the flexible cylinder"], |
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| 89 | ["radius", "Ang", 20.0, [0, inf], "volume", "Radius of the flexible cylinder"], |
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[42356c8] | 90 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Cylinder scattering length density"], |
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| 91 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], |
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[168052c] | 92 | ] |
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| 93 | # pylint: enable=bad-whitespace, line-too-long |
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[26141cb] | 94 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/wrc_cyl.c", "flexible_cylinder.c"] |
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[f94d8a2] | 95 | |
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[31df0c9] | 96 | def random(): |
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[b297ba9] | 97 | """Return a random parameter set for the model.""" |
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[31df0c9] | 98 | length = 10**np.random.uniform(2, 6) |
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| 99 | radius = 10**np.random.uniform(1, 3) |
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[a8631ca] | 100 | kuhn_length = 10**np.random.uniform(-2, 0)*length |
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[31df0c9] | 101 | pars = dict( |
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| 102 | length=length, |
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| 103 | radius=radius, |
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| 104 | kuhn_length=kuhn_length, |
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| 105 | ) |
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| 106 | return pars |
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[f94d8a2] | 107 | |
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| 108 | tests = [ |
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[168052c] | 109 | # Accuracy tests based on content in test/utest_other_models.py |
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[2573fa1] | 110 | [{'length': 1000.0, # test T1 |
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| 111 | 'kuhn_length': 100.0, |
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| 112 | 'radius': 20.0, |
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| 113 | 'sld': 1.0, |
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| 114 | 'sld_solvent': 6.3, |
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| 115 | 'background': 0.0001, |
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| 116 | }, 0.001, 3509.2187], |
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[168052c] | 117 | |
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| 118 | # Additional tests with larger range of parameters |
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[18a2bfc] | 119 | [{'length': 1000.0, # test T2 |
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[168052c] | 120 | 'kuhn_length': 100.0, |
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| 121 | 'radius': 20.0, |
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| 122 | 'sld': 1.0, |
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[e65a3e7] | 123 | 'sld_solvent': 6.3, |
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[168052c] | 124 | 'background': 0.0001, |
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| 125 | }, 1.0, 0.000595345], |
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[18a2bfc] | 126 | [{'length': 10.0, # test T3 |
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[168052c] | 127 | 'kuhn_length': 800.0, |
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| 128 | 'radius': 2.0, |
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| 129 | 'sld': 6.0, |
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[e65a3e7] | 130 | 'sld_solvent': 12.3, |
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[168052c] | 131 | 'background': 0.001, |
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| 132 | }, 0.1, 1.55228], |
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[18a2bfc] | 133 | [{'length': 100.0, # test T4 |
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[168052c] | 134 | 'kuhn_length': 800.0, |
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| 135 | 'radius': 50.0, |
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| 136 | 'sld': 0.1, |
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[e65a3e7] | 137 | 'sld_solvent': 5.1, |
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[168052c] | 138 | 'background': 0.0, |
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| 139 | }, 1.0, 0.000938456] |
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| 140 | ] |
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[18a2bfc] | 141 | |
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| 142 | # There are a few branches in the code that ought to have test values: |
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| 143 | # |
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| 144 | # For length > 4 * kuhn_length |
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| 145 | # if length > 10 * kuhn_length then C is scaled by 3.06 (L/b)^(-0.44) |
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| 146 | # q*kuhn_length <= 3.1 => Sexv_new |
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| 147 | # dS/dQ < 0 has different behaviour from dS/dQ >= 0 |
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| 148 | # T2 q*kuhn_length > 3.1 => a_long |
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| 149 | # |
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| 150 | # For length <= 4 * kuhn_length |
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| 151 | # q*kuhn_length <= max(1.9/Rg_short, 3.0) => Sdebye((q*Rg)^2) |
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| 152 | # q*Rg < 0.5 uses Pade approx, q*Rg > 1.0 uses math lib |
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| 153 | # T3,T4 q*kuhn_length > max(1.9/Rg_short, 3.0) => a_short |
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| 154 | # |
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| 155 | # Note that the transitions between branches may be abrupt. You can see a |
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| 156 | # several percent change around length=10*kuhn_length and length=4*kuhn_length |
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| 157 | # using the following: |
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| 158 | # |
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| 159 | # sascomp flexible_cylinder -calc=double -sets=10 length=10*kuhn_length,10.000001*kuhn_length |
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| 160 | # sascomp flexible_cylinder -calc=double -sets=10 length=4*kuhn_length,4.000001*kuhn_length |
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| 161 | # |
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| 162 | # The transition between low q and high q around q*kuhn_length = 3 seems |
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| 163 | # to be good to 4 digits or better. This was tested by computing the value |
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| 164 | # on each branches near the transition point and reporting the relative error |
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| 165 | # for kuhn lengths of 10, 100 and 1000 and a variety of length:kuhn_length |
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| 166 | # ratios. |
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