1 | r""" |
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2 | This model calculates the scattering from fractal-like aggregates based |
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3 | on 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 | :nowrap: |
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12 | |
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13 | \begin{align*} |
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14 | I(q) &= \text{scale} \times P(q)S(q) + \text{background} \\ |
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15 | P(q) &= F(qR)^2 \\ |
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16 | F(x) &= \frac{3\left[\sin(x)-x\cos(x)\right]}{x^3} \\ |
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17 | S(q) &= \Gamma(5-D_S)\xi^{\,5-D_S}\left[1+(q\xi)^2 \right]^{-(5-D_S)/2} |
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18 | \sin\left[-(5-D_S) \tan^{-1}(q\xi) \right] q^{-1} \\ |
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19 | \text{scale} &= \text{scale factor}\, N V^1(\rho_\text{particle} - \rho_\text{solvent})^2 \\ |
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20 | V &= \frac{4}{3}\pi R^3 |
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21 | \end{align*} |
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22 | |
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23 | where $R$ is the radius of the building block, $D_S$ is the **surface** fractal |
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24 | dimension, $\xi$ is the cut-off length, $\rho_\text{solvent}$ is the scattering |
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25 | length density of the solvent and $\rho_\text{particle}$ is the scattering |
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26 | length density of particles. |
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27 | |
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28 | .. note:: |
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29 | |
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30 | The surface fractal dimension is only valid if $1<D_S<3$. The result is |
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31 | only valid over a limited $q$ range, $\tfrac{5}{3-D_S}\xi^{\,-1} < q < R^{-1}$. |
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32 | See the reference for details. |
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33 | |
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34 | |
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35 | References |
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36 | ---------- |
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37 | |
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38 | .. [#] D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545 |
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39 | |
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40 | Source |
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41 | ------ |
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42 | |
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43 | `surface_fractal.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/surface_fractal.py>`_ |
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44 | |
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45 | `surface_fractal.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/surface_fractal.c>`_ |
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46 | |
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47 | Authorship and Verification |
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48 | ---------------------------- |
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49 | |
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50 | * **Author:** |
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51 | * **Last Modified by:** |
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52 | * **Last Reviewed by:** |
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53 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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54 | """ |
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55 | |
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56 | import numpy as np |
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57 | from numpy import inf |
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58 | |
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59 | name = "surface_fractal" |
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60 | title = "Fractal-like aggregates based on the Mildner reference" |
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61 | description = """\ |
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62 | [The scattering intensity I(x) = scale*P(x)*S(x) + background, where |
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63 | scale = scale_factor * V * delta^(2) |
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64 | p(x) = F(x*radius)^(2) |
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65 | F(x) = 3*[sin(x)-x cos(x)]/x**3 |
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66 | S(x) = [(gamma(5-Ds)*colength^(5-Ds)*[1+(x^2*colength^2)]^((Ds-5)/2) |
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67 | * sin[(Ds-5)*arctan(x*colength)])/x] |
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68 | where |
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69 | delta = sldParticle -sldSolv. |
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70 | radius = Particle radius |
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71 | fractal_dim_surf = Surface fractal dimension (Ds) |
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72 | co_length = Cut-off length |
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73 | background = background |
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74 | |
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75 | Ref. :Mildner, Hall,J Phys D Appl Phys(1986), 19, 1535-1545 |
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76 | Note I : This model is valid for 1<fractal_dim_surf<3 with limited q range. |
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77 | Note II: This model is not in absolute scale. |
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78 | """ |
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79 | category = "shape-independent" |
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80 | |
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81 | # pylint: disable=bad-whitespace, line-too-long |
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82 | # ["name", "units", default, [lower, upper], "type","description"], |
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83 | parameters = [["radius", "Ang", 10.0, [0, inf], "", |
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84 | "Particle radius"], |
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85 | ["fractal_dim_surf", "", 2.0, [1, 3], "", |
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86 | "Surface fractal dimension"], |
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87 | ["cutoff_length", "Ang", 500., [0.0, inf], "", |
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88 | "Cut-off Length"], |
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89 | ] |
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90 | # pylint: enable=bad-whitespace, line-too-long |
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91 | |
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92 | source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "surface_fractal.c"] |
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93 | |
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94 | def random(): |
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95 | """Return a random parameter set for the model.""" |
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96 | radius = 10**np.random.uniform(1, 4) |
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97 | fractal_dim_surf = np.random.uniform(1, 3-1e-6) |
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98 | cutoff_length = 1e6 # Sets the low q limit; keep it big for sim |
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99 | pars = dict( |
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100 | #background=0, |
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101 | scale=1, |
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102 | radius=radius, |
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103 | fractal_dim_surf=fractal_dim_surf, |
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104 | cutoff_length=cutoff_length, |
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105 | ) |
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106 | return pars |
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107 | |
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108 | tests = [ |
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109 | # Accuracy tests based on content in test/utest_other_models.py |
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110 | [{'radius': 10.0, |
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111 | 'fractal_dim_surf': 2.0, |
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112 | 'cutoff_length': 500.0, |
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113 | }, 0.05, 301428.66016], |
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114 | |
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115 | # Additional tests with larger range of parameters |
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116 | [{'radius': 1.0, |
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117 | 'fractal_dim_surf': 1.0, |
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118 | 'cutoff_length': 10.0, |
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119 | }, 0.332070182643, 1125.00421004], |
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120 | |
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121 | [{'radius': 3.5, |
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122 | 'fractal_dim_surf': 0.1, |
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123 | 'cutoff_length': 30.0, |
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124 | 'background': 0.01, |
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125 | }, 5.0, 0.00999998891322], |
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126 | |
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127 | [{'radius': 3.0, |
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128 | 'fractal_dim_surf': 1.0, |
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129 | 'cutoff_length': 33.0, |
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130 | 'scale': 0.1, |
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131 | }, 0.51, 2.50120147004], |
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132 | ] |
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