1 | r""" |
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2 | .. note:: Please read the Validation section below. |
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3 | |
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4 | Definition |
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5 | ---------- |
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6 | This model calculates the structure factor of a polyelectrolyte solution with |
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7 | the RPA expression derived by Borue and Erukhimovich\ [#Borue]_. Note however |
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8 | that the fitting procedure here does not follow the notation in that reference |
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9 | as 's' and 't' are **not** decoupled. Instead the scattering intensity $I(q)$ |
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10 | is calculated as |
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11 | |
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12 | .. math:: |
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13 | |
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14 | I(q) = K\frac{q^2+k^2}{4\pi L_b\alpha ^2} |
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15 | \frac{1}{1+r_{0}^4(q^2+k^2)(q^2-12hC_a/b^2)} + background |
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16 | |
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17 | k^2 = 4\pi L_b(2C_s + \alpha C_a) |
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18 | |
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19 | r_{0}^2 = \frac{b}{\alpha \sqrt{C_a 48\pi L_b}} |
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20 | |
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21 | where |
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22 | |
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23 | $K$ is the contrast factor for the polymer which is defined differently than in |
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24 | other models and is given in barns where 1 $barn = 10^{-24}$ $cm^2$. $K$ is |
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25 | defined as: |
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26 | |
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27 | .. math:: |
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28 | |
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29 | K = a^2 |
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30 | |
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31 | a = b_p - (v_p/v_s) b_s |
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32 | |
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33 | where: |
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34 | |
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35 | - $b_p$ and $b_s$ are **sum of the scattering lengths of the atoms** |
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36 | constituting the polymer monomer and the solvent molecules, respectively. |
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37 | |
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38 | - $v_p$ and $v_s$ are the partial molar volume of the polymer and the |
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39 | solvent, respectively. |
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40 | |
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41 | - $L_b$ is the Bjerrum length (|Ang|) - **Note:** This parameter needs to be |
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42 | kept constant for a given solvent and temperature! |
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43 | |
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44 | - $h$ is the virial parameter (|Ang^3|) - **Note:** See [#Borue]_ for the |
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45 | correct interpretation of this parameter. It incorporates second and third |
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46 | virial coefficients and can be *negative*. |
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47 | |
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48 | - $b$ is the monomer length (|Ang|). |
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49 | |
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50 | - $C_s$ is the concentration of monovalent salt(1/|Ang^3| - internally converted from mol/L). |
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51 | |
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52 | - $\alpha$ is the degree of ionization (the ratio of charged monomers to the total |
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53 | number of monomers) |
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54 | |
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55 | - $C_a$ is the polymer molar concentration (1/|Ang^3| - internally converted from mol/L) |
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56 | |
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57 | - $background$ is the incoherent background. |
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58 | |
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59 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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60 | where the $\vec q$ vector is defined as |
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61 | |
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62 | .. math:: |
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63 | |
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64 | q = \sqrt{q_x^2 + q_y^2} |
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65 | |
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66 | Validation |
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67 | ---------- |
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68 | |
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69 | As of the last revision, this code is believed to be correct. However it |
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70 | needs further validation and should be used with caution at this time. The |
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71 | history of this code goes back to a 1998 implementation. It was recently noted |
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72 | that in that implementation, while both the polymer concentration and salt |
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73 | concentration were converted from experimental units of mol/L to more |
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74 | dimensionally useful units of 1/|Ang^3|, only the converted version of the |
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75 | polymer concentration was actually being used in the calculation while the |
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76 | unconverted salt concentration (still in apparent units of mol/L) was being |
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77 | used. This was carried through to Sasmodels as used for SasView 4.1 (though |
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78 | the line of code converting the salt concentration to the new units was removed |
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79 | somewhere along the line). Simple dimensional analysis of the calculation shows |
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80 | that the converted salt concentration should be used, which the original code |
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81 | suggests was the intention, so this has now been corrected (for SasView 4.2). |
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82 | Once better validation has been performed this note will be removed. |
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83 | |
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84 | References |
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85 | ---------- |
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86 | |
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87 | .. [#Borue] V Y Borue, I Y Erukhimovich, *Macromolecules*, 21 (1988) 3240 |
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88 | .. [#] J F Joanny, L Leibler, *Journal de Physique*, 51 (1990) 545 |
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89 | .. [#] A Moussaid, F Schosseler, J P Munch, S Candau, *J. Journal de Physique |
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90 | II France*, 3 (1993) 573 |
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91 | .. [#] E Raphael, J F Joanny, *Europhysics Letters*, 11 (1990) 179 |
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92 | |
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93 | Authorship and Verification |
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94 | ---------------------------- |
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95 | |
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96 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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97 | * **Last Modified by:** Paul Butler **Date:** September 25, 2018 |
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98 | * **Last Reviewed by:** Paul Butler **Date:** September 25, 2018 |
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99 | """ |
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100 | |
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101 | import numpy as np |
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102 | from numpy import inf, pi, sqrt |
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103 | |
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104 | name = "be_polyelectrolyte" |
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105 | title = "Polyelectrolyte with the RPA expression derived by Borue and Erukhimovich" |
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106 | description = """ |
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107 | Evaluate |
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108 | F(x) = K 1/(4 pi Lb (alpha)^(2)) (q^(2)+k2)/(1+(r02)^(2)) |
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109 | (q^(2)+k2) (q^(2)-(12 h C/b^(2))) |
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110 | |
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111 | has 3 internal parameters : |
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112 | The inverse Debye Length: K2 = 4 pi Lb (2 Cs+alpha C) |
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113 | r02 =1/alpha/Ca^(0.5) (B/(48 pi Lb)^(0.5)) |
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114 | Ca = 6.022136e-4 C |
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115 | """ |
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116 | category = "shape-independent" |
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117 | |
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118 | # pylint: disable=bad-whitespace, line-too-long |
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119 | # ["name", "units", default, [lower, upper], "type", "description"], |
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120 | parameters = [ |
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121 | ["contrast_factor", "barns", 10.0, [-inf, inf], "", "Contrast factor of the polymer"], |
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122 | ["bjerrum_length", "Ang", 7.1, [0, inf], "", "Bjerrum length"], |
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123 | ["virial_param", "Ang^3", 12.0, [-inf, inf], "", "Virial parameter"], |
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124 | ["monomer_length", "Ang", 10.0, [0, inf], "", "Monomer length"], |
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125 | ["salt_concentration", "mol/L", 0.0, [-inf, inf], "", "Concentration of monovalent salt"], |
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126 | ["ionization_degree", "", 0.05, [0, inf], "", "Degree of ionization"], |
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127 | ["polymer_concentration", "mol/L", 0.7, [0, inf], "", "Polymer molar concentration"], |
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128 | ] |
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129 | # pylint: enable=bad-whitespace, line-too-long |
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130 | |
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131 | |
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132 | def Iq(q, |
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133 | contrast_factor, |
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134 | bjerrum_length, |
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135 | virial_param, |
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136 | monomer_length, |
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137 | salt_concentration, |
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138 | ionization_degree, |
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139 | polymer_concentration): |
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140 | """ |
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141 | :params: see parameter table |
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142 | :return: 1-D form factor for polyelectrolytes in low salt |
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143 | |
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144 | parameter names, units, default values, and behavior (volume, sld etc) are |
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145 | defined in the parameter table. The concentrations are converted from |
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146 | experimental mol/L to dimensionaly useful 1/A3 in first two lines |
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147 | """ |
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148 | |
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149 | concentration_pol = polymer_concentration * 6.022136e-4 |
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150 | concentration_salt = salt_concentration * 6.022136e-4 |
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151 | |
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152 | k_square = 4.0 * pi * bjerrum_length * (2*concentration_salt + |
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153 | ionization_degree * concentration_pol) |
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154 | |
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155 | r0_square = 1.0/ionization_degree/sqrt(concentration_pol) * \ |
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156 | (monomer_length/sqrt((48.0*pi*bjerrum_length))) |
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157 | |
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158 | term1 = contrast_factor/(4.0 * pi * bjerrum_length * |
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159 | ionization_degree**2) * (q**2 + k_square) |
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160 | |
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161 | term2 = 1.0 + r0_square**2 * (q**2 + k_square) * \ |
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162 | (q**2 - (12.0 * virial_param * concentration_pol/(monomer_length**2))) |
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163 | |
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164 | return term1/term2 |
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165 | |
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166 | Iq.vectorized = True # Iq accepts an array of q values |
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167 | |
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168 | def random(): |
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169 | """Return a random parameter set for the model.""" |
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170 | # TODO: review random be_polyelectrolyte model generation |
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171 | pars = dict( |
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172 | scale=10000, #background=0, |
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173 | #polymer_concentration=0.7, |
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174 | polymer_concentration=np.random.beta(5, 3), # around 70% |
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175 | #salt_concentration=0.0, |
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176 | # keep salt concentration extremely low |
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177 | # and use explicit molar to match polymer concentration |
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178 | salt_concentration=np.random.beta(1, 100)*6.022136e-4, |
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179 | #contrast_factor=10.0, |
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180 | contrast_fact=np.random.uniform(1, 100), |
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181 | #bjerrum_length=7.1, |
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182 | bjerrum_length=np.random.uniform(1, 10), |
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183 | #virial_param=12.0, |
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184 | virial_param=np.random.uniform(-1000, 30), |
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185 | #monomer_length=10.0, |
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186 | monomer_length=10.0**(4*np.random.beta(1.5, 3)), |
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187 | #ionization_degree=0.05, |
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188 | ionization_degree=np.random.beta(1.5, 4), |
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189 | ) |
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190 | return pars |
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191 | |
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192 | demo = dict(scale=1, background=0.1, |
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193 | contrast_factor=10.0, |
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194 | bjerrum_length=7.1, |
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195 | virial_param=12.0, |
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196 | monomer_length=10.0, |
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197 | salt_concentration=0.0, |
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198 | ionization_degree=0.05, |
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199 | polymer_concentration=0.7) |
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200 | |
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201 | tests = [ |
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202 | |
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203 | # Accuracy tests based on content in test/utest_other_models.py |
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204 | # Note that these should some day be validated beyond this self validation |
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205 | # (circular reasoning). -- i.e. the "good value," at least for those with |
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206 | # non zero salt concentrations, were obtained by running the current |
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207 | # model in SasView and copying the appropriate result here. |
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208 | # PDB -- Sep 26, 2018 |
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209 | [{'contrast_factor': 10.0, |
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210 | 'bjerrum_length': 7.1, |
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211 | 'virial_param': 12.0, |
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212 | 'monomer_length': 10.0, |
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213 | 'salt_concentration': 0.0, |
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214 | 'ionization_degree': 0.05, |
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215 | 'polymer_concentration': 0.7, |
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216 | 'background': 0.001, |
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217 | }, 0.001, 0.0948379], |
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218 | |
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219 | [{'contrast_factor': 10.0, |
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220 | 'bjerrum_length': 100.0, |
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221 | 'virial_param': 3.0, |
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222 | 'monomer_length': 5.0, |
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223 | 'salt_concentration': 1.0, |
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224 | 'ionization_degree': 0.1, |
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225 | 'polymer_concentration': 1.0, |
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226 | 'background': 0.0, |
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227 | }, 0.1, 0.253469484], |
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228 | |
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229 | [{'contrast_factor': 10.0, |
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230 | 'bjerrum_length': 100.0, |
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231 | 'virial_param': 3.0, |
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232 | 'monomer_length': 5.0, |
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233 | 'salt_concentration': 1.0, |
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234 | 'ionization_degree': 0.1, |
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235 | 'polymer_concentration': 1.0, |
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236 | 'background': 1.0, |
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237 | }, 0.05, 1.738358122], |
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238 | |
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239 | [{'contrast_factor': 100.0, |
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240 | 'bjerrum_length': 10.0, |
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241 | 'virial_param': 12.0, |
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242 | 'monomer_length': 10.0, |
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243 | 'salt_concentration': 0.1, |
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244 | 'ionization_degree': 0.5, |
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245 | 'polymer_concentration': 0.1, |
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246 | 'background': 0.01, |
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247 | }, 0.5, 0.012881893], |
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248 | ] |
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