1 | r""" |
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2 | Calculates the scattering from fractal-like aggregates based on |
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3 | the Mildner reference. |
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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) = scale \times P(q)S(q) + background |
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13 | |
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14 | .. math:: |
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15 | |
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16 | P(q) = F(qR)^2 |
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17 | |
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18 | .. math:: |
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19 | |
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20 | F(x) = \frac{3\left[sin(x)-xcos(x)\right]}{x^3} |
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21 | |
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22 | .. math:: |
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23 | |
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24 | S(q) = \frac{\Gamma(D_m-1)\zeta^{D_m-1}}{\left[1+(q\zeta)^2 |
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25 | \right]^{(D_m-1)/2}} |
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26 | \frac{sin\left[(D_m - 1) tan^{-1}(q\zeta) \right]}{q} |
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27 | |
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28 | .. math:: |
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29 | |
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30 | scale = scale\_factor \times NV^2(\rho_\text{particle} - \rho_\text{solvent})^2 |
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31 | |
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32 | .. math:: |
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33 | |
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34 | V = \frac{4}{3}\pi R^3 |
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35 | |
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36 | where $R$ is the radius of the building block, $D_m$ is the **mass** fractal |
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37 | dimension, $\zeta$ is the cut-off length, $\rho_\text{solvent}$ is the scattering |
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38 | length density of the solvent, and $\rho_\text{particle}$ is the scattering |
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39 | length density of particles. |
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40 | |
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41 | .. note:: |
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42 | |
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43 | The mass fractal dimension ( $D_m$ ) is only |
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44 | valid if $1 < mass\_dim < 6$. It is also only valid over a limited |
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45 | $q$ range (see the reference for details). |
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46 | |
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47 | |
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48 | References |
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49 | ---------- |
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50 | |
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51 | .. [#] D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545 Equation(9) |
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52 | |
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53 | Source |
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54 | ------ |
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55 | |
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56 | `mass_fractal.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/mass_fractal.py>`_ |
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57 | |
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58 | `mass_fractal.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/mass_fractal.c>`_ |
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59 | |
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60 | Authorship and Verification |
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61 | ---------------------------- |
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62 | |
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63 | * **Author:** |
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64 | * **Last Modified by:** |
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65 | * **Last Reviewed by:** |
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66 | * **Source added by :** Steve King **Date:** March 25, 2019""" |
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67 | |
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68 | import numpy as np |
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69 | from numpy import inf |
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70 | |
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71 | name = "mass_fractal" |
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72 | title = "Mass Fractal model" |
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73 | description = """ |
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74 | The scattering intensity I(x) = scale*P(x)*S(x) + background, where |
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75 | scale = scale_factor * V * delta^(2) |
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76 | p(x)= F(x*radius)^(2) |
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77 | F(x) = 3*[sin(x)-x cos(x)]/x**3 |
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78 | S(x) = [(gamma(Dm-1)*colength^(Dm-1)*[1+(x^2*colength^2)]^((1-Dm)/2) |
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79 | * sin[(Dm-1)*arctan(x*colength)])/x] |
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80 | where delta = sldParticle -sldSolv. |
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81 | radius = Particle radius |
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82 | fractal_dim_mass = Mass fractal dimension |
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83 | cutoff_length = Cut-off length |
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84 | background = background |
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85 | Ref.:Mildner, Hall,J Phys D Appl Phys(1986), 9, 1535-1545 |
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86 | Note I: This model is valid for 1<fractal_dim_mass<6. |
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87 | Note II: This model is not in absolute scale. |
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88 | """ |
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89 | category = "shape-independent" |
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90 | |
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91 | # pylint: disable=bad-whitespace, line-too-long |
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92 | # ["name", "units", default, [lower, upper], "type","description"], |
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93 | parameters = [ |
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94 | ["radius", "Ang", 10.0, [0.0, inf], "", "Particle radius"], |
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95 | ["fractal_dim_mass", "", 1.9, [1.0, 6.0], "", "Mass fractal dimension"], |
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96 | ["cutoff_length", "Ang", 100.0, [0.0, inf], "", "Cut-off length"], |
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97 | ] |
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98 | # pylint: enable=bad-whitespace, line-too-long |
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99 | |
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100 | source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "mass_fractal.c"] |
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101 | |
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102 | def random(): |
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103 | """Return a random parameter set for the model.""" |
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104 | radius = 10**np.random.uniform(0.7, 4) |
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105 | cutoff_length = 10**np.random.uniform(0.7, 2)*radius |
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106 | # TODO: fractal dimension should range from 1 to 6 |
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107 | fractal_dim_mass = 2*np.random.beta(3, 4) + 1 |
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108 | #volfrac = 10**np.random.uniform(-4, -1) |
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109 | pars = dict( |
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110 | #background=0, |
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111 | scale=1, #1e5*volfrac/radius**(fractal_dim_mass), |
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112 | radius=radius, |
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113 | cutoff_length=cutoff_length, |
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114 | fractal_dim_mass=fractal_dim_mass, |
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115 | ) |
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116 | return pars |
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117 | |
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118 | demo = dict(scale=1, background=0, |
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119 | radius=10.0, |
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120 | fractal_dim_mass=1.9, |
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121 | cutoff_length=100.0) |
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122 | |
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123 | tests = [ |
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124 | |
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125 | # Accuracy tests based on content in test/utest_other_models.py |
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126 | [{'radius': 10.0, |
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127 | 'fractal_dim_mass': 1.9, |
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128 | 'cutoff_length': 100.0, |
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129 | }, 0.05, 279.59422], |
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130 | |
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131 | # Additional tests with larger range of parameters |
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132 | [{'radius': 2.0, |
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133 | 'fractal_dim_mass': 3.3, |
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134 | 'cutoff_length': 1.0, |
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135 | }, 0.5, 1.29116774904], |
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136 | |
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137 | [{'radius': 1.0, |
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138 | 'fractal_dim_mass': 1.3, |
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139 | 'cutoff_length': 1.0, |
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140 | 'background': 0.8, |
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141 | }, 0.001, 1.69747015932], |
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142 | |
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143 | [{'radius': 1.0, |
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144 | 'fractal_dim_mass': 2.3, |
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145 | 'cutoff_length': 1.0, |
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146 | 'scale': 10.0, |
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147 | }, 0.051, 11.6237966145], |
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148 | ] |
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