1 | r""" |
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2 | This model calculates an empirical functional form for SAS data characterized |
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3 | by two Lorentzian-type functions. |
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4 | |
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5 | Definition |
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6 | ---------- |
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7 | |
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8 | The scattering intensity $I(q)$ is calculated as |
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9 | |
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10 | .. math:: |
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11 | |
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12 | I(q) = \frac{A}{1 +(Q\xi_1)^n} + \frac{C}{1 +(Q\xi_2)^m} + \text{B} |
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13 | |
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14 | where $A$ = Lorentzian scale factor #1, $C$ = Lorentzian scale #2, |
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15 | $\xi_1$ and $\xi_2$ are the corresponding correlation lengths, and $n$ and |
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16 | $m$ are the respective power law exponents (set $n = m = 2$ for |
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17 | Ornstein-Zernicke behaviour). |
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18 | |
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19 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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20 | where the $q$ vector is defined as |
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21 | |
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22 | .. math:: |
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23 | |
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24 | q = \sqrt{q_x^2 + q_y^2} |
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25 | |
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26 | |
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27 | .. figure:: img/two_lorentzian.jpg |
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28 | |
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29 | 1D plot using the default values (w/500 data point). |
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30 | |
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31 | References |
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32 | ---------- |
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33 | |
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34 | None. |
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35 | |
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36 | """ |
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37 | |
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38 | from math import sqrt |
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39 | from numpy import inf, power |
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40 | |
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41 | name = "two_lorentzian" |
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42 | title = "Two Lorentzian type peak" |
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43 | description = """I(q) = scale_1/(1.0 + pow((q*length_1),exponent_1)) |
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44 | + scale_2/(1.0 + pow((q*length_2),exponent_2) )+ background |
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45 | |
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46 | scale_1 = Lorentzian term scaling #1 |
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47 | length_1 = Lorentzian screening length #1 [A] |
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48 | exponent_1 = Lorentzian exponent #1 |
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49 | scale_2 = Lorentzian term scaling #2 |
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50 | length_2 = Lorentzian screening length #2 [A] |
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51 | exponent_2 = Lorentzian exponent #2 |
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52 | background = Incoherent background |
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53 | """ |
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54 | category = "shape-independent" |
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55 | |
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56 | # pylint: disable=bad-whitespace, line-too-long |
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57 | # ["name", "units", default, [lower, upper], "type", "description"], |
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58 | parameters = [["lorentz_scale_1", "", 10.0, [-inf, inf], "", "First power law scale factor"], |
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59 | ["lorentz_length_1", "Ang", 100.0, [-inf, inf], "", "First Lorentzian screening length"], |
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60 | ["lorentz_exp_1", "", 3.0, [-inf, inf], "", "First exponent of power law"], |
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61 | ["lorentz_scale_2", "", 1.0, [-inf, inf], "", "Second scale factor for broad Lorentzian peak"], |
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62 | ["lorentz_length_2", "Ang", 10.0, [-inf, inf], "", "Second Lorentzian screening length"], |
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63 | ["lorentz_exp_2", "", 2.0, [-inf, inf], "", "Second exponent of power law"], |
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64 | ] |
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65 | # pylint: enable=bad-whitespace, line-too-long |
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66 | |
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67 | |
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68 | def Iq(q, |
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69 | lorentz_scale_1=10.0, |
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70 | lorentz_length_1=100.0, |
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71 | lorentz_exp_1=3.0, |
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72 | lorentz_scale_2=1.0, |
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73 | lorentz_length_2=10.0, |
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74 | lorentz_exp_2=2.0): |
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75 | |
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76 | """ |
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77 | :param q: Input q-value (float or [float, float]) |
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78 | :param lorentz_scale_1: Second scale factor for broad Lorentzian peak |
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79 | :param lorentz_length_1: First Lorentzian screening length |
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80 | :param lorentz_exp_1: Exponent of the second Lorentz function |
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81 | :param lorentz_scale_2: Second scale factor for broad Lorentzian peak |
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82 | :param lorentz_length_2: Second Lorentzian screening length |
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83 | :param lorentz_exp_2: Exponent of the second Lorentz function |
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84 | :return: Calculated intensity |
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85 | """ |
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86 | # pylint: disable=bad-whitespace |
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87 | intensity = lorentz_scale_1/(1.0 + |
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88 | power(q*lorentz_length_1, lorentz_exp_1)) |
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89 | intensity += lorentz_scale_2/(1.0 + |
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90 | power(q*lorentz_length_2, lorentz_exp_2)) |
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91 | # pylint: enable=bad-whitespace |
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92 | |
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93 | return intensity |
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94 | |
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95 | Iq.vectorized = True # Iq accepts an array of q values |
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96 | |
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97 | |
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98 | def Iqxy(qx, qy, *args): |
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99 | """ |
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100 | :param qx: Input q_x-value |
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101 | :param qy: Input q_y-value |
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102 | :param args: Remaining arguments |
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103 | :return: 2D-Intensity |
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104 | """ |
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105 | |
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106 | return Iq(sqrt(qx**2 + qy**2), *args) |
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107 | |
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108 | Iqxy.vectorized = True # Iqxy accepts an array of qx, qy values |
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109 | |
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110 | |
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111 | demo = dict(scale=1, background=0.1, |
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112 | lorentz_scale_1=10, |
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113 | lorentz_length_1=100.0, |
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114 | lorentz_exp_1=3.0, |
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115 | lorentz_scale_2=1, |
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116 | lorentz_length_2=10, |
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117 | lorentz_exp_2=2.0) |
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118 | |
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119 | oldname = "TwoLorentzianModel" |
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120 | oldpars = dict(background='background', |
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121 | lorentz_scale_1='scale_1', |
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122 | lorentz_scale_2='scale_2', |
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123 | lorentz_length_1='length_1', |
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124 | lorentz_length_2='length_2', |
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125 | lorentz_exp_1='exponent_1', |
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126 | lorentz_exp_2='exponent_2') |
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127 | |
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128 | tests = [ |
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129 | |
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130 | # Accuracy tests based on content in test/utest_extra_models.py |
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131 | [{'lorentz_scale_1': 10.0, |
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132 | 'lorentz_length_1': 100.0, |
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133 | 'lorentz_exp_1': 3.0, |
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134 | 'lorentz_scale_2': 1.0, |
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135 | 'lorentz_length_2': 10.0, |
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136 | 'lorentz_exp_2': 2.0, |
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137 | 'background': 0.1, |
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138 | }, 0.001, 11.08991], |
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139 | |
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140 | [{'lorentz_scale_1': 10.0, |
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141 | 'lorentz_length_1': 100.0, |
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142 | 'lorentz_exp_1': 3.0, |
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143 | 'lorentz_scale_2': 1.0, |
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144 | 'lorentz_length_2': 10.0, |
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145 | 'lorentz_exp_2': 2.0, |
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146 | 'background': 0.1, |
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147 | }, 0.150141, 0.410245], |
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148 | |
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149 | [{'lorentz_scale_1': 10.0, |
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150 | 'lorentz_length_1': 100.0, |
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151 | 'lorentz_exp_1': 3.0, |
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152 | 'lorentz_scale_2': 1.0, |
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153 | 'lorentz_length_2': 10.0, |
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154 | 'lorentz_exp_2': 2.0, |
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155 | 'background': 0.1, |
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156 | }, 0.442528, 0.148699], |
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157 | |
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158 | # Additional tests with larger range of parameters |
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159 | [{'lorentz_scale_1': 10.0, |
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160 | 'lorentz_length_1': 100.0, |
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161 | 'lorentz_exp_1': 3.0, |
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162 | 'lorentz_scale_2': 1.0, |
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163 | 'lorentz_length_2': 10.0, |
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164 | 'lorentz_exp_2': 2.0, |
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165 | }, 0.000332070182643, 10.9996228107], |
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166 | |
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167 | [{'lorentz_scale_1': 0.0, |
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168 | 'lorentz_length_1': 0.0, |
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169 | 'lorentz_exp_1': 0.0, |
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170 | 'lorentz_scale_2': 0.0, |
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171 | 'lorentz_length_2': 0.0, |
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172 | 'lorentz_exp_2': 0.0, |
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173 | 'background': 100.0 |
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174 | }, 5.0, 100.0], |
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175 | |
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176 | [{'lorentz_scale_1': 200.0, |
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177 | 'lorentz_length_1': 10.0, |
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178 | 'lorentz_exp_1': 0.1, |
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179 | 'lorentz_scale_2': 0.1, |
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180 | 'lorentz_length_2': 5.0, |
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181 | 'lorentz_exp_2': 2.0 |
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182 | }, 20000., 45.5659201896], |
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183 | ] |
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