1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | TThis model provides the form factor, *P(q)*, for an unilamellar vesicle. This |
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6 | s model is effectively identical to the hollow sphere reparameterized to be |
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7 | more intuitive for a vesicle and normalizing the form factor by the volume of |
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8 | the shell. The 1D scattering intensity is calculated in the following way |
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9 | (Guinier,1955\ [#Guinier1955]_) |
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10 | |
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11 | .. math:: |
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12 | |
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13 | P(q) = \frac{\phi}{V_\text{shell}} \left[ |
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14 | \frac{3V_{\text{core}}({\rho_{\text{solvent}} |
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15 | - \rho_{\text{shell}})j_1(qR_{\text{core}})}}{qR_{\text{core}}} |
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16 | + \frac{3V_{\text{tot}}(\rho_{\text{shell}} |
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17 | - \rho_{\text{solvent}}) j_1(qR_{\text{tot}})}{qR_{\text{tot}}} |
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18 | \right]^2 + \text{background} |
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19 | |
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20 | |
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21 | where $\phi$ is the volume fraction of shell material, $V_{shell}$ is the volume |
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22 | of the shell, $V_{\text{cor}}$ is the volume of the core, $V_{\text{tot}}$ is |
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23 | the total volume, $R_{\text{core}}$ is the radius of the core, $R_{\text{tot}}$ |
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24 | is the outer radius of the shell, $\rho_{\text{solvent}}$ is the scattering |
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25 | length density of the solvent (which is the same as for the core in this case), |
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26 | $\rho_{\text{scale}}$ is the scattering length density of the shell, background |
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27 | is a flat background level (due for example to incoherent scattering in the |
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28 | case of neutrons), and $j_1$ is the spherical bessel function |
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29 | $j_1 = (\sin(x) - x \cos(x))/ x^2$. |
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30 | |
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31 | The functional form is identical to a "typical" core-shell structure, except |
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32 | that the scattering is normalized by the volume that is contributing to the |
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33 | scattering, namely the volume of the shell alone, the scattering length density |
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34 | of the core is fixed the same as that of the solvent, the scale factor when the |
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35 | data are on an absolute scale is equivalent to the volume fraction of material |
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36 | in the shell rather than the entire core+shell sphere, and the parameterization |
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37 | is done in terms of the core radius = $R_{\text{core}}$ and the shell |
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38 | thickness = $R_{\text{tot}} - R_{\text{core}}$. |
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39 | |
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40 | .. figure:: img/vesicle_geometry.jpg |
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41 | |
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42 | Vesicle geometry. |
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43 | |
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44 | The 2D scattering intensity is the same as *P(q)* above, regardless of the |
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45 | orientation of the *q* vector which is defined as |
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46 | |
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47 | .. math:: |
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48 | |
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49 | q = \sqrt{q_x^2 + q_y^2} |
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50 | |
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51 | |
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52 | NB: The outer most radius (= *radius* + *thickness*) is used as the effective |
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53 | radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
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54 | |
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55 | |
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56 | References |
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57 | ---------- |
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58 | |
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59 | .. [#Guinier1955] A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and |
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60 | Sons, New York, (1955) |
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61 | |
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62 | |
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63 | Authorship and Verification |
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64 | ---------------------------- |
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65 | |
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66 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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67 | * **Last Modified by:** Paul Butler **Date:** March 20, 2016 |
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68 | * **Last Reviewed by:** Paul Butler **Date:** September 7, 2018 |
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69 | """ |
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70 | |
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71 | import numpy as np |
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72 | from numpy import pi, inf |
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73 | |
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74 | name = "vesicle" |
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75 | title = "Vesicle model representing a hollow sphere" |
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76 | description = """ |
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77 | Model parameters: |
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78 | radius : the core radius of the vesicle |
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79 | thickness: the shell thickness |
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80 | sld: the shell SLD |
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81 | sld_solvent: the solvent (and core) SLD |
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82 | background: incoherent background |
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83 | volfraction: shell volume fraction |
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84 | scale : scale factor = 1 if on absolute scale""" |
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85 | category = "shape:sphere" |
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86 | |
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87 | # [ "name", "units", default, [lower, upper], "type", "description"], |
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88 | parameters = [["sld", "1e-6/Ang^2", 0.5, [-inf, inf], "sld", |
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89 | "vesicle shell scattering length density"], |
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90 | ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld", |
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91 | "solvent scattering length density"], |
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92 | ["volfraction", "", 0.05, [0, 1.0], "", |
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93 | "volume fraction of shell"], |
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94 | ["radius", "Ang", 100, [0, inf], "volume", |
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95 | "vesicle core radius"], |
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96 | ["thickness", "Ang", 30, [0, inf], "volume", |
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97 | "vesicle shell thickness"], |
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98 | ] |
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99 | |
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100 | source = ["lib/sas_3j1x_x.c", "vesicle.c"] |
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101 | |
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102 | def ER(radius, thickness): |
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103 | ''' |
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104 | returns the effective radius used in the S*P calculation |
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105 | |
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106 | :param radius: core radius |
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107 | :param thickness: shell thickness |
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108 | ''' |
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109 | return radius + thickness |
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110 | |
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111 | def VR(radius, thickness): |
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112 | ''' |
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113 | returns the volumes of the shell and of the whole sphere including the |
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114 | core plus shell - is used to normalize when including polydispersity. |
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115 | |
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116 | :param radius: core radius |
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117 | :param thickness: shell thickness |
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118 | :return whole: volume of core and shell |
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119 | :return whole-core: volume of the shell |
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120 | ''' |
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121 | |
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122 | whole = 4./3. * pi * (radius + thickness)**3 |
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123 | core = 4./3. * pi * radius**3 |
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124 | return whole, whole - core |
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125 | |
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126 | def random(): |
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127 | total_radius = 10**np.random.uniform(1.3, 5) |
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128 | radius = total_radius * np.random.uniform(0, 1) |
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129 | thickness = total_radius - radius |
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130 | volfraction = 10**np.random.uniform(-3, -1) |
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131 | pars = dict( |
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132 | #background=0, |
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133 | scale=1, # volfraction is part of the model, so scale=1 |
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134 | radius=radius, |
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135 | thickness=thickness, |
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136 | volfraction=volfraction, |
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137 | ) |
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138 | return pars |
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139 | |
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140 | # parameters for demo |
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141 | demo = dict(sld=0.5, sld_solvent=6.36, |
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142 | volfraction=0.05, |
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143 | radius=100, thickness=30, |
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144 | radius_pd=.2, radius_pd_n=10, |
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145 | thickness_pd=.2, thickness_pd_n=10) |
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146 | |
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147 | # NOTE: test results taken from values returned by SasView 3.1.2, with |
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148 | # 0.001 added for a non-zero default background. |
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149 | tests = [[{}, 0.0005, 859.916526646], |
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150 | [{}, 0.100600200401, 1.77063682331], |
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151 | [{}, 0.5, 0.00355351388906], |
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152 | [{}, 'ER', 130.], |
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153 | [{}, 'VR', 0.54483386436], |
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154 | ] |
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