1 | r""" |
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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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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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10 | where the averaging $\left<\ldots\right>$ is applied only for the 1D |
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11 | calculation |
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12 | |
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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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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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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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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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32 | In the parameters, the sld and sld\_solvent represent the SLD of the cylinder |
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33 | and solvent respectively. |
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34 | |
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35 | Our model uses the form factor calculations in reference [1] as implemented in a |
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36 | c-library provided by the NIST Center for Neutron Research (Kline, 2006). This states: |
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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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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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42 | See equations (13,26-27) in the original reference for the details. |
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43 | |
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44 | .. note:: |
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45 | |
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46 | There are several typos in the original reference that have been corrected |
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47 | by WRC [2]. Details of the corrections are in the reference below. Most notably |
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48 | |
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49 | - Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$ |
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50 | |
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51 | - Equations (23) and (24) are incorrect; WRC has entered these into |
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52 | Mathematica and solved analytically. The results were then converted to |
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53 | code. |
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54 | |
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55 | - Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of |
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56 | $max(a3*b(Rg^2)^{1/2},3)$ |
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57 | |
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58 | - The scattering function is negative for a range of parameter values and |
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59 | q-values that are experimentally accessible. A correction function has been |
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60 | added to give the proper behavior. |
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61 | |
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62 | |
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63 | **This is a model with complex behaviour depending on the ratio of** $L/b$ **and the |
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64 | reader is strongly encouraged to read reference [1] before use.** |
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65 | |
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66 | .. note:: |
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67 | |
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68 | There are several typos in the original reference that have been corrected |
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69 | by WRC [2]. Details of the corrections are in the reference below. Most notably |
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70 | |
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71 | - Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$ |
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72 | |
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73 | - Equations (23) and (24) are incorrect; WRC has entered these into |
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74 | Mathematica and solved analytically. The results were then converted to |
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75 | code. |
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76 | |
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77 | - Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of |
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78 | $max(a3*b(Rg^2)^{1/2},3)$ |
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79 | |
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80 | - The scattering function is negative for a range of parameter values and |
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81 | q-values that are experimentally accessible. A correction function has been |
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82 | added to give the proper behavior. |
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83 | |
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84 | |
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85 | **This is a model with complex behaviour depending on the ratio of** $L/b$ **and the |
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86 | reader is strongly encouraged to read reference [1] before use.** |
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87 | |
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88 | References |
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89 | ---------- |
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90 | |
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91 | .. [#] 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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92 | |
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93 | Correction of the formula can be found in |
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94 | |
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95 | .. [#] 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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96 | |
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97 | Source |
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98 | ------ |
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99 | |
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100 | `flexible_cylinder.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder.py>`_ |
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101 | |
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102 | `flexible_cylinder.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/flexible_cylinder.c>`_ |
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103 | |
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104 | `wrc_cyl.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/lib/wrc_cyl.c>`_ |
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105 | |
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106 | Authorship and Verification |
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107 | ---------------------------- |
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108 | |
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109 | * **Author:** |
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110 | * **Last Modified by:** |
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111 | * **Last Reviewed by:** Steve King **Date:** March 26, 2019 |
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112 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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113 | """ |
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114 | |
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115 | import numpy as np |
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116 | from numpy import inf |
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117 | |
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118 | name = "flexible_cylinder" |
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119 | title = "Flexible cylinder where the form factor is normalized by the volume " \ |
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120 | "of the cylinder." |
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121 | description = """Note : scale and contrast = (sld - sld_solvent) are both |
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122 | multiplicative factors in the model and are perfectly |
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123 | correlated. One or both of these parameters must be held fixed |
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124 | during model fitting. |
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125 | """ |
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126 | |
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127 | category = "shape:cylinder" |
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128 | single = False # double precision only! |
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129 | |
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130 | # pylint: disable=bad-whitespace, line-too-long |
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131 | # ["name", "units", default, [lower, upper], "type", "description"], |
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132 | parameters = [ |
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133 | ["length", "Ang", 1000.0, [0, inf], "volume", "Length of the flexible cylinder"], |
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134 | ["kuhn_length", "Ang", 100.0, [0, inf], "volume", "Kuhn length of the flexible cylinder"], |
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135 | ["radius", "Ang", 20.0, [0, inf], "volume", "Radius of the flexible cylinder"], |
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136 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Cylinder scattering length density"], |
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137 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], |
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138 | ] |
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139 | # pylint: enable=bad-whitespace, line-too-long |
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140 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/wrc_cyl.c", "flexible_cylinder.c"] |
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141 | |
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142 | def random(): |
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143 | """Return a random parameter set for the model.""" |
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144 | length = 10**np.random.uniform(2, 6) |
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145 | radius = 10**np.random.uniform(1, 3) |
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146 | kuhn_length = 10**np.random.uniform(-2, 0)*length |
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147 | pars = dict( |
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148 | length=length, |
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149 | radius=radius, |
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150 | kuhn_length=kuhn_length, |
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151 | ) |
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152 | return pars |
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153 | |
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154 | tests = [ |
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155 | # Accuracy tests based on content in test/utest_other_models.py |
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156 | [{'length': 1000.0, # test T1 |
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157 | 'kuhn_length': 100.0, |
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158 | 'radius': 20.0, |
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159 | 'sld': 1.0, |
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160 | 'sld_solvent': 6.3, |
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161 | 'background': 0.0001, |
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162 | }, 0.001, 3509.2187], |
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163 | |
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164 | # Additional tests with larger range of parameters |
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165 | [{'length': 1000.0, # test T2 |
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166 | 'kuhn_length': 100.0, |
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167 | 'radius': 20.0, |
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168 | 'sld': 1.0, |
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169 | 'sld_solvent': 6.3, |
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170 | 'background': 0.0001, |
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171 | }, 1.0, 0.000595345], |
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172 | [{'length': 10.0, # test T3 |
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173 | 'kuhn_length': 800.0, |
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174 | 'radius': 2.0, |
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175 | 'sld': 6.0, |
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176 | 'sld_solvent': 12.3, |
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177 | 'background': 0.001, |
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178 | }, 0.1, 1.55228], |
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179 | [{'length': 100.0, # test T4 |
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180 | 'kuhn_length': 800.0, |
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181 | 'radius': 50.0, |
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182 | 'sld': 0.1, |
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183 | 'sld_solvent': 5.1, |
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184 | 'background': 0.0, |
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185 | }, 1.0, 0.000938456] |
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186 | ] |
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187 | |
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188 | # There are a few branches in the code that ought to have test values: |
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189 | # |
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190 | # For length > 4 * kuhn_length |
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191 | # if length > 10 * kuhn_length then C is scaled by 3.06 (L/b)^(-0.44) |
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192 | # q*kuhn_length <= 3.1 => Sexv_new |
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193 | # dS/dQ < 0 has different behaviour from dS/dQ >= 0 |
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194 | # T2 q*kuhn_length > 3.1 => a_long |
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195 | # |
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196 | # For length <= 4 * kuhn_length |
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197 | # q*kuhn_length <= max(1.9/Rg_short, 3.0) => Sdebye((q*Rg)^2) |
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198 | # q*Rg < 0.5 uses Pade approx, q*Rg > 1.0 uses math lib |
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199 | # T3,T4 q*kuhn_length > max(1.9/Rg_short, 3.0) => a_short |
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200 | # |
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201 | # Note that the transitions between branches may be abrupt. You can see a |
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202 | # several percent change around length=10*kuhn_length and length=4*kuhn_length |
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203 | # using the following: |
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204 | # |
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205 | # sascomp flexible_cylinder -calc=double -sets=10 length=10*kuhn_length,10.000001*kuhn_length |
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206 | # sascomp flexible_cylinder -calc=double -sets=10 length=4*kuhn_length,4.000001*kuhn_length |
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207 | # |
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208 | # The transition between low q and high q around q*kuhn_length = 3 seems |
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209 | # to be good to 4 digits or better. This was tested by computing the value |
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210 | # on each branches near the transition point and reporting the relative error |
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211 | # for kuhn lengths of 10, 100 and 1000 and a variety of length:kuhn_length |
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212 | # ratios. |
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