1 | # Note: model title and parameter table are inserted automatically |
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2 | r""" |
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3 | This model calculates the scattering from a stack of repeating lamellar |
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4 | structures. The stacks of lamellae (infinite in lateral dimension) are |
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5 | treated as a paracrystal to account for the repeating spacing. The repeat |
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6 | distance is further characterized by a Gaussian polydispersity. **This model |
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7 | can be used for large multilamellar vesicles.** |
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8 | |
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9 | Definition |
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10 | ---------- |
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11 | |
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12 | In the equations below, |
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13 | |
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14 | - *scale* is used instead of the mass per area of the bilayer $\Gamma_m$ |
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15 | (this corresponds to the volume fraction of the material in the bilayer, |
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16 | *not* the total excluded volume of the paracrystal), |
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17 | |
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18 | - *sld* $-$ *sld_solvent* is the contrast $\Delta \rho$, |
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19 | |
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20 | - *thickness* is the layer thickness $t$, |
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21 | |
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22 | - *Nlayers* is the number of layers $N$, |
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23 | |
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24 | - *d_spacing* is the average distance between adjacent layers |
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25 | $\langle D \rangle$, and |
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26 | |
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27 | - *sigma_d* is the relative standard deviation of the Gaussian |
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28 | layer distance distribution $\sigma_D / \langle D \rangle$. |
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29 | |
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30 | |
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31 | The scattering intensity $I(q)$ is calculated as |
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32 | |
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33 | .. math:: |
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34 | |
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35 | I(q) = 2\pi\Delta\rho^2\Gamma_m\frac{P_\text{bil}(q)}{q^2} Z_N(q) |
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36 | |
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37 | The form factor of the bilayer is approximated as the cross section of an |
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38 | infinite, planar bilayer of thickness $t$ (compare the equations for the |
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39 | lamellar model). |
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40 | |
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41 | .. math:: |
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42 | |
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43 | P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2 |
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44 | |
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45 | $Z_N(q)$ describes the interference effects for aggregates |
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46 | consisting of more than one bilayer. The equations used are (3-5) |
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47 | from the Bergstrom reference: |
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48 | |
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49 | .. math:: |
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50 | |
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51 | |
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52 | Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)} |
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53 | + x_N S_N + (1 - x_N) S_{N+1} |
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54 | |
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55 | where |
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56 | |
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57 | .. math:: |
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58 | |
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59 | S_N(q) = \frac{a_N}{N}[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2 |
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60 | |
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61 | and |
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62 | |
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63 | .. math:: |
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64 | |
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65 | a_N &= 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) \\ |
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66 | &\quad - 4w^{N+2}\cos(Nq \langle D \rangle) |
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67 | + 2 w^{N+3}\cos[(N-1)q \langle D \rangle] |
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68 | + 2w^{N+1}\cos[(N+1)q \langle D \rangle] |
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69 | |
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70 | for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$. |
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71 | |
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72 | Non-integer numbers of stacks are calculated as a linear combination of |
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73 | the lower and higher values |
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74 | |
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75 | .. math:: |
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76 | |
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77 | N_L = x_N N + (1 - x_N)(N+1) |
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78 | |
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79 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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80 | of the $q$ vector which is defined as |
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81 | |
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82 | .. math:: |
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83 | |
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84 | q = \sqrt{q_x^2 + q_y^2} |
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85 | |
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86 | |
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87 | Reference |
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88 | --------- |
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89 | |
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90 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, |
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91 | *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
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92 | |
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93 | """ |
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94 | |
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95 | from numpy import inf |
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96 | |
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97 | name = "lamellar_stack_paracrystal" |
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98 | title = "Random lamellar sheet with paracrystal structure factor" |
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99 | description = """\ |
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100 | [Random lamellar phase with paracrystal structure factor] |
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101 | randomly oriented stacks of infinite sheets |
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102 | with paracrytal S(Q), having polydisperse spacing. |
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103 | sld = sheet scattering length density |
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104 | sld_solvent = solvent scattering length density |
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105 | background = incoherent background |
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106 | scale = scale factor |
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107 | """ |
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108 | category = "shape:lamellae" |
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109 | |
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110 | single = False |
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111 | |
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112 | # ["name", "units", default, [lower, upper], "type","description"], |
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113 | parameters = [["thickness", "Ang", 33.0, [0, inf], "volume", |
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114 | "sheet thickness"], |
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115 | ["Nlayers", "", 20, [1, inf], "", |
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116 | "Number of layers"], |
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117 | ["d_spacing", "Ang", 250., [0.0, inf], "", |
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118 | "lamellar spacing of paracrystal stack"], |
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119 | ["sigma_d", "Ang", 0.0, [0.0, inf], "", |
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120 | "Sigma (polydispersity) of the lamellar spacing"], |
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121 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", |
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122 | "layer scattering length density"], |
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123 | ["sld_solvent", "1e-6/Ang^2", 6.34, [-inf, inf], "sld", |
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124 | "Solvent scattering length density"], |
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125 | ] |
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126 | |
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127 | |
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128 | source = ["lamellar_stack_paracrystal.c"] |
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129 | |
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130 | form_volume = """ |
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131 | return 1.0; |
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132 | """ |
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133 | |
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134 | def random(): |
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135 | import numpy as np |
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136 | total_thickness = 10**np.random.uniform(2, 4.7) |
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137 | Nlayers = np.random.randint(2, 200) |
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138 | d_spacing = total_thickness / Nlayers |
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139 | thickness = d_spacing * np.random.uniform(0, 1) |
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140 | # Let polydispersity peak around 15%; 95% < 0.4; max=100% |
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141 | sigma_d = np.random.beta(1.5, 7) |
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142 | scale = thickness * 10**np.random.uniform(-7, -4) |
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143 | pars = dict( |
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144 | scale=1, |
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145 | thickness=thickness, |
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146 | Nlayers=Nlayers, |
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147 | d_spacing=d_spacing, |
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148 | sigma_d=sigma_d, |
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149 | ) |
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150 | return pars |
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151 | |
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152 | # ER defaults to 0.0 |
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153 | # VR defaults to 1.0 |
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154 | |
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155 | demo = dict(scale=1, background=0, |
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156 | thickness=33, Nlayers=20, d_spacing=250, sigma_d=0.2, |
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157 | sld=1.0, sld_solvent=6.34, |
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158 | thickness_pd=0.2, thickness_pd_n=40) |
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159 | |
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160 | # |
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161 | tests = [ |
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162 | [{'scale': 1.0, 'background': 0.0, 'thickness': 33.,'Nlayers': 20.0, |
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163 | 'd_spacing': 250., 'sigma_d': 0.2, 'sld': 1.0, |
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164 | 'sld_solvent': 6.34, 'thickness_pd': 0.0, 'thickness_pd_n': 40 }, |
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165 | [0.001, 0.215268], [21829.3, 0.00487686]], |
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166 | ] |
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167 | # ADDED by: RKH ON: 18Mar2016 converted from sasview previously, now renaming everything & sorting the docs |
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