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 provides the form factor for an elliptical cylinder with a |
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6 | core-shell scattering length density profile. Thus this is a variation |
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7 | of the core-shell bicelle model, but with an elliptical cylinder for the core. |
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8 | Outer shells on the rims and flat ends may be of different thicknesses and |
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9 | scattering length densities. The form factor is normalized by the total particle volume. |
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10 | |
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11 | |
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12 | .. figure:: img/core_shell_bicelle_geometry.png |
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13 | |
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14 | Schematic cross-section of bicelle. Note however that the model here |
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15 | calculates for rectangular, not curved, rims as shown below. |
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16 | |
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17 | .. figure:: img/core_shell_bicelle_parameters.png |
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18 | |
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19 | Cross section of model used here. Users will have |
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20 | to decide how to distribute "heads" and "tails" between the rim, face |
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21 | and core regions in order to estimate appropriate starting parameters. |
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22 | |
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23 | Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, |
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24 | the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the |
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25 | scattering length density variation along the bicelle axis is: |
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26 | |
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27 | .. math:: |
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28 | |
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29 | \rho(r) = |
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30 | \begin{cases} |
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31 | &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] |
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32 | &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; |
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33 | L \lt z\lt (L+2t) \\[1.5ex] |
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34 | &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) |
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35 | \end{cases} |
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36 | |
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37 | The form factor for the bicelle is calculated in cylindrical coordinates, where |
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38 | $\alpha$ is the angle between the $Q$ vector and the cylinder axis, and $\psi$ is the angle |
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39 | for the ellipsoidal cross section core, to give: |
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40 | |
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41 | .. math:: |
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42 | |
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43 | I(Q,\alpha,\psi) = \frac{\text{scale}}{V_t} \cdot |
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44 | F(Q,\alpha, \psi)^2.sin(\alpha) + \text{background} |
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45 | |
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46 | where a numerical integration of $F(Q,\alpha, \psi)^2.sin(\alpha)$ is carried out over \alpha and \psi for: |
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47 | |
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48 | .. math:: |
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49 | |
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50 | \begin{align} |
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51 | F(Q,\alpha,\psi) = &\bigg[ |
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52 | (\rho_c - \rho_f) V_c \frac{2J_1(QR'sin \alpha)}{QR'sin\alpha}\frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\ |
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53 | &+(\rho_f - \rho_r) V_{c+f} \frac{2J_1(QR'sin\alpha)}{QR'sin\alpha}\frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\ |
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54 | &+(\rho_r - \rho_s) V_t \frac{2J_1(Q(R'+t_r)sin\alpha)}{Q(R'+t_r)sin\alpha}\frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} |
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55 | \bigg] |
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56 | \end{align} |
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57 | |
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58 | where |
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59 | |
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60 | .. math:: |
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61 | |
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62 | R'=\frac{R}{\sqrt{2}}\sqrt{(1+X_{core}^{2}) + (1-X_{core}^{2})cos(\psi)} |
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63 | |
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64 | |
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65 | and $V_t = \pi.(R+t_r)(Xcore.R+t_r)^2.(L+2.t_f)$ is the total volume of the bicelle, |
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66 | $V_c = \pi.Xcore.R^2.L$ the volume of the core, $V_{c+f} = \pi.Xcore.R^2.(L+2.t_f)$ |
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67 | the volume of the core plus the volume of the faces, $R$ is the radius |
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68 | of the core, $Xcore$ is the axial ratio of the core, $L$ the length of the core, |
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69 | $t_f$ the thickness of the face, $t_r$ the thickness of the rim and $J_1$ the usual |
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70 | first order bessel function. The core has radii $R$ and $Xcore.R$ so is circular, |
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71 | as for the core_shell_bicelle model, for $Xcore$ =1. Note that you may need to |
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72 | limit the range of $Xcore$, especially if using the Monte-Carlo algorithm, as |
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73 | setting radius to $R/Xcore$ and axial ratio to $1/Xcore$ gives an equivalent solution! |
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74 | |
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75 | The output of the 1D scattering intensity function for randomly oriented |
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76 | bicelles is then given by integrating over all possible $\alpha$ and $\psi$. |
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77 | |
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78 | For oriented bicellles the *theta*, *phi* and *psi* orientation parameters only appear when fitting 2D data, |
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79 | see the :ref:`elliptical-cylinder` model for further information. |
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80 | |
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81 | |
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82 | .. figure:: img/elliptical_cylinder_angle_definition.jpg |
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83 | |
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84 | Definition of the angles for the oriented core_shell_bicelle_elliptical model. |
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85 | |
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86 | |
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87 | References |
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88 | ---------- |
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89 | |
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90 | .. [#] |
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91 | |
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92 | Authorship and Verification |
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93 | ---------------------------- |
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94 | |
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95 | * **Author:** Richard Heenan **Date:** December 14, 2016 |
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96 | * **Last Modified by:** Richard Heenan **Date:** December 14, 2016 |
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97 | * **Last Reviewed by:** Richard Heenan BEWARE 2d data yet to be checked **Date:** December 14, 2016 |
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98 | """ |
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99 | |
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100 | from numpy import inf, sin, cos |
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101 | |
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102 | name = "core_shell_bicelle_elliptical" |
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103 | title = "Elliptical cylinder with a core-shell scattering length density profile.." |
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104 | description = """ |
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105 | core_shell_bicelle_elliptical |
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106 | Elliptical cylinder core, optional shell on the two flat faces, and shell of |
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107 | uniform thickness on its rim (extending around the end faces). |
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108 | Please see full documentation for equations and further details. |
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109 | Involves a double numerical integral around the ellipsoid diameter |
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110 | and the angle of the cylinder axis to Q. |
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111 | Compare also the core_shell_bicelle and elliptical_cylinder models. |
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112 | """ |
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113 | category = "shape:cylinder" |
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114 | |
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115 | # pylint: disable=bad-whitespace, line-too-long |
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116 | # ["name", "units", default, [lower, upper], "type", "description"], |
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117 | parameters = [ |
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118 | ["radius", "Ang", 30, [0, inf], "volume", "Cylinder core radius"], |
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119 | ["x_core", "None", 3, [0, inf], "volume", "axial ratio of core, X = r_polar/r_equatorial"], |
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120 | ["thick_rim", "Ang", 8, [0, inf], "volume", "Rim shell thickness"], |
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121 | ["thick_face", "Ang", 14, [0, inf], "volume", "Cylinder face thickness"], |
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122 | ["length", "Ang", 50, [0, inf], "volume", "Cylinder length"], |
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123 | ["sld_core", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder core scattering length density"], |
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124 | ["sld_face", "1e-6/Ang^2", 7, [-inf, inf], "sld", "Cylinder face scattering length density"], |
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125 | ["sld_rim", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Cylinder rim scattering length density"], |
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126 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", "Solvent scattering length density"], |
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127 | ["theta", "degrees", 90, [-360, 360], "orientation", "In plane angle"], |
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128 | ["phi", "degrees", 0, [-360, 360], "orientation", "Out of plane angle"], |
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129 | ["psi", "degrees", 0, [-360, 360], "orientation", "Major axis angle relative to Q"], |
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130 | ] |
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131 | |
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132 | # pylint: enable=bad-whitespace, line-too-long |
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133 | |
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134 | source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", |
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135 | "core_shell_bicelle_elliptical.c"] |
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136 | |
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137 | demo = dict(scale=1, background=0, |
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138 | radius=30.0, |
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139 | x_core=3.0, |
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140 | thick_rim=8.0, |
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141 | thick_face=14.0, |
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142 | length=50.0, |
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143 | sld_core=4.0, |
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144 | sld_face=7.0, |
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145 | sld_rim=1.0, |
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146 | sld_solvent=6.0, |
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147 | theta=90, |
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148 | phi=0, |
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149 | psi=0) |
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150 | |
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151 | qx, qy = 0.4 * cos(90), 0.5 * sin(0) |
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152 | |
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153 | tests = [ |
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154 | [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0}, 'ER', 1], |
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155 | [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0}, 'VR', 1], |
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156 | |
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157 | [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0, |
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158 | 'sld_core':4.0, 'sld_face':7.0, 'sld_rim':1.0, 'sld_solvent':6.0, 'background':0.0}, |
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159 | 0.015, 286.540286], |
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160 | ] |
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