1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | This model is a trivial extension of the core_shell_sphere function where the |
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6 | core is filled with solvent and is surrounded by $N$ shells of material |
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7 | (such as lipids) interleaved with $N - 1$ layers of solvent. For $N = 1$, this |
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8 | returns the same as the vesicle model, except for the normalisation, which here |
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9 | is to outermost volume. The shell thicknesses and SLD are constant for all |
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10 | shells as expected for a multilayer vesicle. |
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11 | |
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12 | .. figure:: img/multi_shell_geometry.jpg |
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13 | |
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14 | Geometry of the multilayer_vesicle model. |
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15 | |
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16 | See the :ref:`core-shell-sphere` model for more documentation. |
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17 | |
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18 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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19 | |
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20 | .. math:: |
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21 | P(q) = \text{scale} \cdot \frac{\phi}{V(R_N)} F^2(q) + \text{background} |
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22 | |
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23 | where |
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24 | |
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25 | .. math:: |
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26 | F(q) = (\rho_\text{shell}-\rho_\text{solv}) \sum_{i=1}^{N} \left[ |
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27 | 3V(r_i)\frac{\sin(qr_i) - qr_i\cos(qr_i)}{(qr_i)^3} |
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28 | - 3V(R_i)\frac{\sin(qR_i) - qR_i\cos(qR_i)}{(qR_i)^3} |
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29 | \right] |
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30 | |
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31 | for |
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32 | |
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33 | .. math:: |
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34 | |
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35 | r_i &= r_c + (i-1)(t_s + t_w) && \text{ solvent radius before shell } i \\ |
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36 | R_i &= r_i + t_s && \text{ shell radius for shell } i |
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37 | |
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38 | $\phi$ is the volume fraction of particles, $V(r)$ is the volume of a sphere |
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39 | of radius $r$, $r_c$ is the radius of the core, $t_s$ is the thickness of |
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40 | the shell, $t_w$ is the thickness of the solvent layer between the shells, |
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41 | $\rho_\text{shell}$ is the scattering length density of a shell, and |
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42 | $\rho_\text{solv}$ is the scattering length density of the solvent. |
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43 | |
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44 | USAGE NOTES |
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45 | |
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46 | * The outer-most shell radius $R_N$ is used as the effective radius |
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47 | for $P(Q)$ when $P(Q) * S(Q)$ is applied. |
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48 | calculations rather slow. |
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49 | * The number of shells is always rounded to an integer value as a non interger |
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50 | number of layers is not physical. |
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51 | * Thus Polydispersity should only be applied to number of shells **VERY |
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52 | CAREFULLY**. A possible legitimate use would be for mixed systems in which |
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53 | some vesicles have 1 shell, some have 2, etc. A polydispersity on $N$ can be |
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54 | used to model the data by using the "array distriubtion" feature. First |
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55 | create a file such as *shell_dist.txt* containing the relative portion |
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56 | of each vesicle size:: |
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57 | |
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58 | 1 20 |
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59 | 2 4 |
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60 | 3 1 |
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61 | |
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62 | Turn on polydispersity and select an array distribution for the *n_shells* |
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63 | parameter. Choose the above *shell_dist.txt* file, and the model will be |
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64 | computed with 80% 1-shell vesicles, 16% 2-shell vesicles and 4% |
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65 | 3-shell vesicles. |
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66 | * This is a highly non-linear, highly oscillatory (especially around the |
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67 | q-values that correspond to the repeat distance of the layers), model |
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68 | function complicated by the fact that the number of water/shell pairs must |
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69 | physically be an integer value, although the optimization treats it as a |
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70 | floating point value. Thus it may be that the resolution interpolation is not |
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71 | sufficiently fine grained in certain cases. Please report any such occurences |
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72 | to the SasView team. Generally, for the best possible experience: |
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73 | * Start with the best possible guess |
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74 | * Using a priori knowledge, hold as many parameters fixed as possible |
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75 | * if N=1, tw (water thickness) must by definition be zero. Both N and tw should |
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76 | be fixed during fitting. |
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77 | * If N>1, use constraints to keep N > 1 |
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78 | * Because N only really moves in integer steps, it may get "stuck" if the |
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79 | optimizer step size is too small so care should be taken |
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80 | If you experience problems with this please contact the SasView team and let |
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81 | them know the issue preferably with example data and model which fail to |
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82 | converge. |
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83 | |
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84 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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85 | of the q vector which is defined as: |
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86 | |
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87 | .. math:: |
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88 | |
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89 | q = \sqrt{q_x^2 + q_y^2} |
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90 | |
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91 | For information about polarised and magnetic scattering, see |
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92 | the :ref:`magnetism` documentation. |
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93 | |
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94 | References |
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95 | ---------- |
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96 | |
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97 | .. [#] B Cabane, *Small Angle Scattering Methods*, in *Surfactant Solutions: |
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98 | New Methods of Investigation*, Ch.2, Surfactant Science Series Vol. 22, Ed. |
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99 | R Zana and M Dekker, New York, (1987). |
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100 | |
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101 | Authorship and Verification |
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102 | ---------------------------- |
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103 | |
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104 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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105 | * **Converted to sasmodels by:** Piotr Rozyczko **Date:** Feb 24, 2016 |
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106 | * **Last Modified by:** Paul Kienzle **Date:** Feb 7, 2017 |
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107 | * **Last Reviewed by:** Paul Butler **Date:** March 12, 2017 |
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108 | |
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109 | """ |
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110 | |
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111 | from numpy import inf |
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112 | |
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113 | name = "multilayer_vesicle" |
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114 | title = "P(Q) for a Multi-lamellar vesicle" |
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115 | description = """ |
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116 | multilayer_vesicle model parameters; |
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117 | scale : scale factor for abs intensity if needed else 1.0 |
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118 | volfraction: volume fraction |
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119 | radius : Core radius of the multishell |
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120 | thick_shell: shell thickness |
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121 | thick_solvent: water thickness |
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122 | sld_solvent: solvent scattering length density |
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123 | sld: shell scattering length density |
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124 | n_shells:number of "shell plus solvent" layer pairs |
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125 | background: incoherent background |
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126 | """ |
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127 | category = "shape:sphere" |
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128 | |
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129 | # pylint: disable=bad-whitespace, line-too-long |
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130 | # ["name", "units", default, [lower, upper], "type","description"], |
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131 | parameters = [ |
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132 | ["volfraction", "", 0.05, [0.0, 1], "", "volume fraction of vesicles"], |
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133 | ["radius", "Ang", 60.0, [0.0, inf], "volume", "radius of solvent filled core"], |
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134 | ["thick_shell", "Ang", 10.0, [0.0, inf], "volume", "thickness of one shell"], |
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135 | ["thick_solvent", "Ang", 10.0, [0.0, inf], "volume", "solvent thickness between shells"], |
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136 | ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "sld", "solvent scattering length density"], |
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137 | ["sld", "1e-6/Ang^2", 0.4, [-inf, inf], "sld", "Shell scattering length density"], |
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138 | ["n_shells", "", 2.0, [1.0, inf], "volume", "Number of shell plus solvent layer pairs"], |
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139 | ] |
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140 | # pylint: enable=bad-whitespace, line-too-long |
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141 | |
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142 | # TODO: proposed syntax for specifying which parameters can be polydisperse |
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143 | #polydispersity = ["radius", "thick_shell"] |
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144 | |
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145 | source = ["lib/sas_3j1x_x.c", "multilayer_vesicle.c"] |
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146 | |
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147 | def ER(radius, thick_shell, thick_solvent, n_shells): |
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148 | n_shells = int(n_shells+0.5) |
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149 | return radius + n_shells * (thick_shell + thick_solvent) - thick_solvent |
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150 | |
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151 | demo = dict(scale=1, background=0, |
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152 | volfraction=0.05, |
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153 | radius=60.0, |
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154 | thick_shell=10.0, |
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155 | thick_solvent=10.0, |
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156 | sld_solvent=6.4, |
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157 | sld=0.4, |
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158 | n_shells=2.0) |
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159 | |
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160 | tests = [ |
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161 | # Accuracy tests based on content in test/utest_other_models.py |
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162 | [{'radius': 60.0, |
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163 | 'thick_shell': 10.0, |
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164 | 'thick_solvent': 10.0, |
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165 | 'sld_solvent': 6.4, |
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166 | 'sld': 0.4, |
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167 | 'n_shells': 2.0, |
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168 | 'scale': 1.0, |
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169 | 'background': 0.001, |
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170 | }, 0.001, 122.1405], |
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171 | |
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172 | [{'volfraction': 1.0, |
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173 | 'radius': 60.0, |
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174 | 'thick_shell': 10.0, |
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175 | 'thick_solvent': 10.0, |
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176 | 'sld_solvent': 6.4, |
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177 | 'sld': 0.4, |
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178 | 'n_shells': 2.0, |
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179 | 'scale': 1.0, |
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180 | 'background': 0.001, |
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181 | }, (0.001, 0.30903), 1.61873], |
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182 | ] |
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