1 | r""" |
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2 | This model calculates an empirical functional form for SAS data using SpericalSLD profile |
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3 | |
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4 | Similarly to the OnionExpShellModel, this model provides the form factor, P(q), for a multi-shell sphere, |
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5 | where the interface between the each neighboring shells can be described by one of a number of functions |
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6 | including error, power-law, and exponential functions. This model is to calculate the scattering intensity |
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7 | by building a continuous custom SLD profile against the radius of the particle. The SLD profile is composed |
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8 | of a flat core, a flat solvent, a number (up to 9 ) flat shells, and the interfacial layers between |
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9 | the adjacent flat shells (or core, and solvent) (see below). |
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10 | |
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11 | .. figure:: img/spherical_sld_profile.gif |
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12 | |
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13 | Exemplary SLD profile |
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14 | |
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15 | Unlike the <onion> model (using an analytical integration), |
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16 | the interfacial layers here are sub-divided and numerically integrated assuming each of the sub-layers are described |
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17 | by a line function. The number of the sub-layer can be given by users by setting the integer values of npts_inter. |
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18 | The form factor is normalized by the total volume of the sphere. |
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19 | |
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20 | Definition |
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21 | ---------- |
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22 | |
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23 | The form factor $P(q)$ in 1D is calculated by: |
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24 | |
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25 | .. math:: |
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26 | |
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27 | P(q) = \frac{f^2}{V_\text{particle}} \text{ where } |
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28 | f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} + |
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29 | \sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent} |
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30 | |
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31 | For a spherically symmetric particle with a particle density $\rho_x(r)$ the sld function can be defined as: |
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32 | |
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33 | .. math:: |
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34 | |
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35 | f_x = 4 \pi \int_{0}^{\infty} \rho_x(r) \frac{\sin(qr)} {qr^2} r^2 dr |
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36 | |
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37 | |
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38 | so that individual terms can be calcualted as follows: |
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39 | |
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40 | .. math:: |
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41 | f_\text{core} = 4 \pi \int_{0}^{r_\text{core}} \rho_\text{core} \frac{\sin(qr)} {qr} r^2 dr = |
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42 | 3 \rho_\text{core} V(r_\text{core}) |
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43 | \Big[ \frac{\sin(qr_\text{core}) - qr_\text{core} \cos(qr_\text{core})} {qr_\text{core}^3} \Big] |
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44 | |
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45 | f_{\text{inter}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr |
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46 | |
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47 | f_{\text{shell}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{flat}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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48 | 3 \rho_{ \text{flat}_i } V ( r_{ \text{inter}_i } + \Delta t_{ \text{inter}_i } ) |
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49 | \Big[ \frac{\sin(qr_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) - q (r_{\text{inter}_i} + |
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50 | \Delta t_{ \text{inter}_i }) \cos(q( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) ) } |
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51 | {q ( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } )^3 } \Big] |
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52 | -3 \rho_{ \text{flat}_i } V(r_{ \text{inter}_i }) |
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53 | \Big[ \frac{\sin(qr_{\text{inter}_i}) - qr_{\text{flat}_i} \cos(qr_{\text{inter}_i}) } {qr_{\text{inter}_i}^3} \Big] |
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54 | |
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55 | f_\text{solvent} = 4 \pi \int_{r_N}^{\infty} \rho_\text{solvent} \frac{\sin(qr)} {qr} r^2 dr = |
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56 | 3 \rho_\text{solvent} V(r_N) |
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57 | \Big[ \frac{\sin(qr_N) - qr_N \cos(qr_N)} {qr_N^3} \Big] |
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58 | |
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59 | |
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60 | Here we assumed that the SLDs of the core and solvent are constant against $r$. |
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61 | The SLD at the interface between shells, $\rho_{\text {inter}_i}$ |
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62 | is calculated with a function chosen by an user, where the functions are |
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63 | |
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64 | Exp: |
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65 | |
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66 | .. math:: |
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67 | \rho_{{inter}_i} (r) = \begin{cases} |
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68 | B \exp\Big( \frac {\pm A(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A \neq 0 \\ |
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69 | B \Big( \frac {(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A = 0 \\ |
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70 | \end{cases} |
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71 | |
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72 | Power-Law |
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73 | |
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74 | .. math:: |
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75 | \rho_{{inter}_i} (r) = \begin{cases} |
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76 | \pm B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) ^A +C & \text{for} A \neq 0 \\ |
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77 | \rho_{\text{flat}_{i+1}} & \text{for} A = 0 \\ |
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78 | \end{cases} |
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79 | |
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80 | Erf: |
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81 | |
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82 | .. math:: |
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83 | \rho_{{inter}_i} (r) = \begin{cases} |
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84 | B \text{erf} \Big( \frac { A(r - r_{\text{flat}_i})} {\sqrt{2} \Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A \neq 0 \\ |
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85 | B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A = 0 \\ |
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86 | \end{cases} |
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87 | |
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88 | The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD |
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89 | is continuous at the boundaries of the interface as well as each sub-layers. Thus B and C are determined. |
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90 | |
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91 | Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-layer of the interface, we can find its contribution |
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92 | to the form factor $P(q)$ |
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93 | |
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94 | .. math:: |
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95 | f_{\text{inter}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr = |
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96 | 4 \pi \sum_{j=0}^{npts_{\text{inter}_i} -1 } |
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97 | \int_{r_j}^{r_{j+1}} \rho_{ \text{inter}_i } (r_j) \frac{\sin(qr)} {qr} r^2 dr \approx |
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98 | |
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99 | 4 \pi \sum_{j=0}^{npts_{\text{inter}_i} -1 } \Big[ |
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100 | 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } ( r_{j} ) V ( r_{ \text{sublayer}_j } ) |
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101 | \Big[ \frac {r_j^2 \beta_\text{out}^2 \sin(\beta_\text{out}) - (\beta_\text{out}^2-2) \cos(\beta_\text{out}) } |
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102 | {\beta_\text{out}^4 } \Big] |
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103 | |
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104 | - 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } ( r_{j} ) V ( r_{ \text{sublayer}_j-1 } ) |
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105 | \Big[ \frac {r_{j-1}^2 \sin(\beta_\text{in}) - (\beta_\text{in}^2-2) \cos(\beta_\text{in}) } |
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106 | {\beta_\text{in}^4 } \Big] |
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107 | |
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108 | + 3 \rho_{ \text{inter}_i } ( r_{j+1} ) V ( r_j ) |
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109 | \Big[ \frac {\sin(\beta_\text{out}) - \cos(\beta_\text{out}) } |
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110 | {\beta_\text{out}^4 } \Big] |
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111 | |
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112 | - 3 \rho_{ \text{inter}_i } ( r_{j} ) V ( r_j ) |
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113 | \Big[ \frac {\sin(\beta_\text{in}) - \cos(\beta_\text{in}) } |
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114 | {\beta_\text{in}^4 } \Big] |
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115 | \Big] |
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116 | |
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117 | where |
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118 | |
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119 | .. math:: |
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120 | V(a) = \frac {4\pi}{3}a^3 |
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121 | |
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122 | a_\text{in} ~ \frac{r_j}{r_{j+1} -r_j} \text{, } a_\text{out} ~ \frac{r_{j+1}}{r_{j+1} -r_j} |
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123 | |
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124 | \beta_\text{in} = qr_j \text{, } \beta_\text{out} = qr_{j+1} |
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125 | |
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126 | |
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127 | We assume the $\rho_{\text{inter}_i} (r)$ can be approximately linear within a sub-layer $j$ |
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128 | |
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129 | Finally form factor can be calculated by |
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130 | |
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131 | .. math:: |
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132 | |
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133 | P(q) = \frac{[f]^2} {V_\text{particle}} \text{where} V_\text{particle} = V(r_{\text{shell}_N}) |
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134 | |
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135 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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136 | where the $q$ vector is defined as |
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137 | |
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138 | .. math:: |
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139 | |
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140 | q = \sqrt{q_x^2 + q_y^2} |
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141 | |
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142 | |
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143 | .. figure:: img/spherical_sld_1d.jpg |
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144 | |
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145 | 1D plot using the default values (w/400 data point). |
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146 | |
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147 | .. figure:: img/spherical_sld_default_profile.jpg |
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148 | |
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149 | SLD profile from the default values. |
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150 | |
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151 | .. note:: |
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152 | The outer most radius is used as the effective radius for S(Q) when $P(Q) * S(Q)$ is applied. |
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153 | |
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154 | References |
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155 | ---------- |
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156 | L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, New York, (1987) |
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157 | |
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158 | """ |
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159 | |
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160 | from numpy import inf |
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161 | |
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162 | name = "spherical_sld" |
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163 | title = "Sperical SLD intensity calculation" |
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164 | description = """ |
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165 | I(q) = |
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166 | background = Incoherent background [1/cm] |
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167 | """ |
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168 | category = "sphere-based" |
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169 | |
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170 | # pylint: disable=bad-whitespace, line-too-long |
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171 | # ["name", "units", default, [lower, upper], "type", "description"], |
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172 | parameters = [["n_shells", "", 1, [0, 9], "", "number of shells"], |
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173 | ["thick_inter[n]", "Ang", 50, [-inf, inf], "", "intern layer thickness"], |
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174 | ["func_inter[n]", "", 0, [0, 4], "", "Erf:0, RPower:1, LPower:2, RExp:3, LExp:4"], |
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175 | ["sld_core", "1e-6/Ang^2", 2.07, [-inf, inf], "", "sld function flat"], |
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176 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "", "sld function solvent"], |
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177 | ["sld_flat[n]", "1e-6/Ang^2", 4.06, [-inf, inf], "", "sld function flat"], |
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178 | ["thick_inter[n]", "Ang", 50.0, [0, inf], "", "intern layer thickness"], |
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179 | ["thick_flat[n]", "Ang", 100.0, [0, inf], "", "flat layer_thickness"], |
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180 | ["inter_nu[n]", "", 2.5, [-inf, inf], "", "steepness parameter"], |
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181 | ["npts_inter", "", 35, [0, 35], "", "number of points in each sublayer Must be odd number"], |
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182 | ["core_rad", "Ang", 50.0, [0, inf], "", "intern layer thickness"], |
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183 | ] |
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184 | # pylint: enable=bad-whitespace, line-too-long |
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185 | #source = ["lib/librefl.c", "lib/sph_j1c.c", "spherical_sld.c"] |
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186 | |
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187 | def Iq(q, *args, **kw): |
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188 | return q |
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189 | |
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190 | def Iqxy(qx, *args, **kw): |
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191 | return qx |
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192 | |
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193 | |
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194 | demo = dict( |
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195 | n_shells=4, |
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196 | scale=1.0, |
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197 | solvent_sld=1.0, |
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198 | background=0.0, |
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199 | npts_inter=35.0, |
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200 | func_inter_0=0, |
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201 | nu_inter_0=2.5, |
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202 | rad_core_0=50.0, |
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203 | core0_sld=2.07, |
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204 | thick_inter_0=50.0, |
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205 | func_inter_1=0, |
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206 | nu_inter_1=2.5, |
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207 | thick_inter_1=50.0, |
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208 | flat1_sld=4.0, |
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209 | thick_flat_1=100.0, |
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210 | func_inter_2=0, |
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211 | nu_inter_2=2.5, |
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212 | thick_inter_2=50.0, |
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213 | flat2_sld=3.5, |
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214 | thick_flat_2=100.0, |
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215 | func_inter_3=0, |
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216 | nu_inter_3=2.5, |
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217 | thick_inter_3=50.0, |
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218 | flat3_sld=4.0, |
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219 | thick_flat_3=100.0, |
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220 | func_inter_4=0, |
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221 | nu_inter_4=2.5, |
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222 | thick_inter_4=50.0, |
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223 | flat4_sld=3.5, |
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224 | thick_flat_4=100.0, |
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225 | func_inter_5=0, |
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226 | nu_inter_5=2.5, |
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227 | thick_inter_5=50.0, |
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228 | flat5_sld=4.0, |
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229 | thick_flat_5=100.0, |
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230 | func_inter_6=0, |
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231 | nu_inter_6=2.5, |
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232 | thick_inter_6=50.0, |
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233 | flat6_sld=3.5, |
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234 | thick_flat_6=100.0, |
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235 | func_inter_7=0, |
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236 | nu_inter_7=2.5, |
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237 | thick_inter_7=50.0, |
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238 | flat7_sld=4.0, |
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239 | thick_flat_7=100.0, |
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240 | func_inter_8=0, |
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241 | nu_inter_8=2.5, |
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242 | thick_inter_8=50.0, |
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243 | flat8_sld=3.5, |
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244 | thick_flat_8=100.0, |
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245 | func_inter_9=0, |
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246 | nu_inter_9=2.5, |
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247 | thick_inter_9=50.0, |
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248 | flat9_sld=4.0, |
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249 | thick_flat_9=100.0, |
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250 | func_inter_10=0, |
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251 | nu_inter_10=2.5, |
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252 | thick_inter_10=50.0, |
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253 | flat10_sld=3.5, |
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254 | thick_flat_10=100.0 |
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255 | ) |
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256 | |
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257 | oldname = "SphereSLDModel" |
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258 | oldpars = dict( |
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259 | n_shells="n_shells", |
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260 | scale="scale", |
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261 | npts_inter='npts_inter', |
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262 | solvent_sld='sld_solv', |
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263 | func_inter_0='func_inter0', |
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264 | nu_inter_0='nu_inter0', |
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265 | background='background', |
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266 | rad_core_0='rad_core0', |
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267 | core0_sld='sld_core0', |
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268 | thick_inter_0='thick_inter0', |
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269 | func_inter_1='func_inter1', |
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270 | nu_inter_1='nu_inter1', |
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271 | thick_inter_1='thick_inter1', |
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272 | flat1_sld='sld_flat1', |
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273 | thick_flat_1='thick_flat1', |
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274 | func_inter_2='func_inter2', |
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275 | nu_inter_2='nu_inter2', |
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276 | thick_inter_2='thick_inter2', |
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277 | flat2_sld='sld_flat2', |
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278 | thick_flat_2='thick_flat2', |
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279 | func_inter_3='func_inter3', |
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280 | nu_inter_3='nu_inter3', |
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281 | thick_inter_3='thick_inter3', |
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282 | flat3_sld='sld_flat3', |
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283 | thick_flat_3='thick_flat3', |
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284 | func_inter_4='func_inter4', |
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285 | nu_inter_4='nu_inter4', |
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286 | thick_inter_4='thick_inter4', |
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287 | flat4_sld='sld_flat4', |
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288 | thick_flat_4='thick_flat4', |
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289 | func_inter_5='func_inter5', |
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290 | nu_inter_5='nu_inter5', |
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291 | thick_inter_5='thick_inter5', |
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292 | flat5_sld='sld_flat5', |
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293 | thick_flat_5='thick_flat5', |
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294 | func_inter_6='func_inter6', |
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295 | nu_inter_6='nu_inter6', |
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296 | thick_inter_6='thick_inter6', |
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297 | flat6_sld='sld_flat6', |
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298 | thick_flat_6='thick_flat6', |
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299 | func_inter_7='func_inter7', |
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300 | nu_inter_7='nu_inter7', |
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301 | thick_inter_7='thick_inter7', |
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302 | flat7_sld='sld_flat7', |
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303 | thick_flat_7='thick_flat7', |
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304 | func_inter_8='func_inter8', |
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305 | nu_inter_8='nu_inter8', |
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306 | thick_inter_8='thick_inter8', |
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307 | flat8_sld='sld_flat8', |
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308 | thick_flat_8='thick_flat8', |
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309 | func_inter_9='func_inter9', |
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310 | nu_inter_9='nu_inter9', |
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311 | thick_inter_9='thick_inter9', |
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312 | flat9_sld='sld_flat9', |
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313 | thick_flat_9='thick_flat9', |
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314 | func_inter_10='func_inter10', |
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315 | nu_inter_10='nu_inter10', |
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316 | thick_inter_10='thick_inter10', |
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317 | flat10_sld='sld_flat10', |
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318 | thick_flat_10='thick_flat10') |
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319 | |
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320 | #TODO: Not working yet |
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321 | tests = [ |
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322 | # Accuracy tests based on content in test/utest_extra_models.py |
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323 | [{'npts_iter':35, |
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324 | 'sld_solv':1, |
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325 | 'func_inter_1':0.0, |
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326 | 'nu_inter':2.5, |
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327 | 'thick_inter_1':50, |
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328 | 'sld_flat_1':4, |
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329 | 'thick_flat_1':100, |
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330 | 'func_inter_0':0.0, |
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331 | 'nu_inter_0':2.5, |
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332 | 'rad_core_0':50.0, |
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333 | 'sld_core_0':2.07, |
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334 | 'thick_inter_0':50, |
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335 | 'background': 0.0, |
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336 | }, 0.001, 1000], |
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337 | ] |
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