1 | r""" |
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2 | This model calculates the scattering from a gel structure, |
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3 | but typically a physical rather than chemical network. |
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4 | It is modeled as a sum of a low-q exponential decay plus |
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5 | a lorentzian at higher-q values. |
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6 | |
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7 | Definition |
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8 | ---------- |
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9 | |
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10 | The scattering intensity I(q) is calculated as (Eqn. 5 from the reference) |
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11 | |
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12 | .. math:: |
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13 | |
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14 | I(q) = I_G(0)exp(-q^2\Xi ^2/2) + I_L(0)/(1+q^2\xi^2) |
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15 | |
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16 | $\Xi$ is the length scale of the static correlations in the gel, |
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17 | which can be attributed to the "frozen-in" crosslinks. |
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18 | $\xi is the dynamic correlation length, which can be attributed to the |
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19 | fluctuating polymer chains between crosslinks. |
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20 | $IG(0)$ and $IL(0)$ are the scaling factors for each of these structures. |
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21 | Think carefully about how these map to your particular system! |
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22 | |
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23 | .. note:: |
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24 | The peaked structure at higher $q$ values (Figure 2 from the reference) |
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25 | is not reproduced by the model. Peaks can be introduced into the model |
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26 | by summing this model with the PeakGaussModel function. |
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27 | |
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28 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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29 | where the $q$ vector is defined as |
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30 | |
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31 | .. math:: |
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32 | |
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33 | q = \sqrt{q_x^2 + q_y^2} |
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34 | |
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35 | |
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36 | .. figure:: img/gauss_lorentz_gel_1d.jpg |
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37 | |
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38 | 1D plot using the default values (w/500 data point). |
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39 | |
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40 | |
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41 | References |
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42 | ---------- |
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43 | |
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44 | G Evmenenko, E Theunissen, K Mortensen, H Reynaers, *Polymer*, |
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45 | 42 (2001) 2907-2913 |
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46 | |
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47 | """ |
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48 | |
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49 | from numpy import inf, sqrt, exp |
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50 | |
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51 | name = "gauss_lorentz_gel" |
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52 | title = "Gauss Lorentz Gel model of scattering from a gel structure" |
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53 | description = """ |
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54 | Class that evaluates a GaussLorentzGel model. |
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55 | |
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56 | I(q) = scale_g*exp(- q^2*Z^2 / 2)+scale_l/(1+q^2*z^2) |
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57 | + background |
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58 | List of default parameters: |
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59 | scale_g = Gauss scale factor |
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60 | Z = Static correlation length |
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61 | scale_l = Lorentzian scale factor |
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62 | z = Dynamic correlation length |
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63 | background = Incoherent background |
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64 | """ |
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65 | category = "shape-independent" |
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66 | # pylint: disable=bad-whitespace, line-too-long |
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67 | # ["name", "units", default, [lower, upper], "type", "description"], |
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68 | parameters = [["gauss_scale_factor", "", 100.0, [-inf, inf], "", "Gauss scale factor"], |
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69 | ["static_cor_length", "Ang", 100.0, [0, inf], "", "Static correlation length"], |
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70 | ["lorentz_scale_factor", "", 50.0, [-inf, inf], "", "Lorentzian scale factor"], |
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71 | ["dynamic_cor_length", "Ang", 20.0, [0, inf], "", "Dynamic correlation length"], |
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72 | ] |
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73 | # pylint: enable=bad-whitespace, line-too-long |
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74 | |
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75 | def Iq(q, |
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76 | gauss_scale_factor=100.0, |
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77 | static_cor_length=100.0, |
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78 | lorentz_scale_factor=50.0, |
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79 | dynamic_cor_length=20.0): |
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80 | """ |
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81 | |
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82 | :param q: Input q-value |
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83 | :param gauss_scale_factor: Gauss scale factor |
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84 | :param static_cor_length: Static correlation length |
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85 | :param lorentz_scale_factor: Lorentzian scale factor |
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86 | :param dynamic_cor_length: Dynamic correlation length |
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87 | :return: 1-D intensity |
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88 | """ |
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89 | |
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90 | term1 = gauss_scale_factor *\ |
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91 | exp(-1.0*q*q*static_cor_length*static_cor_length/2.0) |
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92 | term2 = lorentz_scale_factor /\ |
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93 | (1.0+(q*dynamic_cor_length)*(q*dynamic_cor_length)) |
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94 | |
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95 | return term1 + term2 |
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96 | |
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97 | Iq.vectorized = True # Iq accepts an array of q values |
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98 | |
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99 | |
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100 | def Iqxy(qx, qy, *args): |
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101 | """ |
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102 | :param qx: Input q_x-value |
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103 | :param qy: Input q_y-value |
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104 | :param args: Remaining aruments |
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105 | :return: 2-D intensity |
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106 | """ |
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107 | |
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108 | return Iq(sqrt(qx**2 + qy**2), *args) |
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109 | |
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110 | Iqxy.vectorized = True # Iqxy accepts an array of qx, qy values |
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111 | |
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112 | |
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113 | demo = dict(scale=1, background=0.1, |
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114 | gauss_scale_factor=100.0, |
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115 | static_cor_length=100.0, |
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116 | lorentz_scale_factor=50.0, |
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117 | dynamic_cor_length=20.0) |
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118 | |
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119 | oldname = "GaussLorentzGelModel" |
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120 | |
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121 | oldpars = dict(background='background', |
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122 | gauss_scale_factor='scale_g', |
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123 | static_cor_length='stat_colength', |
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124 | lorentz_scale_factor='scale_l', |
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125 | dynamic_cor_length='dyn_colength') |
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126 | |
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127 | tests = [ |
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128 | |
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129 | # Accuracy tests based on content in test/utest_extra_models.py |
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130 | [{'gauss_scale_factor': 100.0, |
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131 | 'static_cor_length': 100.0, |
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132 | 'lorentz_scale_factor': 50.0, |
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133 | 'dynamic_cor_length': 20.0, |
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134 | }, 0.001, 149.481], |
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135 | |
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136 | [{'gauss_scale_factor': 100.0, |
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137 | 'static_cor_length': 100.0, |
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138 | 'lorentz_scale_factor': 50.0, |
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139 | 'dynamic_cor_length': 20.0, |
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140 | }, 0.105363, 9.1903], |
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141 | |
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142 | [{'gauss_scale_factor': 100.0, |
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143 | 'static_cor_length': 100.0, |
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144 | 'lorentz_scale_factor': 50.0, |
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145 | 'dynamic_cor_length': 20.0, |
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146 | }, 0.441623, 0.632811], |
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147 | |
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148 | # Additional tests with larger range of parameters |
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149 | [{'gauss_scale_factor': 10.0, |
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150 | 'static_cor_length': 100.0, |
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151 | 'lorentz_scale_factor': 3.0, |
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152 | 'dynamic_cor_length': 1.0, |
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153 | }, 0.1, 2.9702970297], |
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154 | |
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155 | [{'gauss_scale_factor': 10.0, |
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156 | 'static_cor_length': 100.0, |
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157 | 'lorentz_scale_factor': 3.0, |
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158 | 'dynamic_cor_length': 1.0, |
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159 | 'background': 100.0 |
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160 | }, 5.0, 100.115384615], |
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161 | |
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162 | [{'gauss_scale_factor': 10.0, |
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163 | 'static_cor_length': 100.0, |
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164 | 'lorentz_scale_factor': 3.0, |
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165 | 'dynamic_cor_length': 1.0, |
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166 | }, 200., 7.49981250469e-05], |
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167 | ] |
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