1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | This model provides the form factor for a circular cylinder with a |
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5 | core-shell scattering length density profile. Thus this is a variation |
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6 | of a core-shell cylinder or disc where the shell on the walls and ends |
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7 | may be of different thicknesses and scattering length densities. The form |
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8 | factor is normalized by the particle volume. |
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9 | |
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10 | .. _core-shell-bicelle-geometry: |
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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 cylindrical symmetry 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 | The output of the 1D scattering intensity function for randomly oriented |
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24 | cylinders is then given by the equation above. |
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25 | |
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26 | The *theta* and *phi* parameters are not used for the 1D output. |
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27 | Our implementation of the scattering kernel and the 1D scattering intensity |
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28 | use the c-library from NIST. |
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29 | |
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30 | .. figure:: img/cylinder_angle_definition.jpg |
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31 | |
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32 | Definition of the angles for the oriented core shell bicelle tmodel. |
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33 | |
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34 | .. figure:: img/cylinder_angle_projection.jpg |
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35 | :width: 600px |
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36 | |
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37 | Examples of the angles for oriented pp against the detector plane. |
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38 | |
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39 | References |
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40 | ---------- |
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41 | |
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42 | .. [#Matusmori] `N Matsumori and M Murata <http://dx.doi.org/10.1039/C0NP0000 |
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43 | 2G>`_, *Nat. Prod. Rep.* 27 (2010) 1480-1492 |
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44 | .. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and |
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45 | Neutron Scattering,* Plenum Press, New York, (1987) |
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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:** NIST IGOR/DANSE **Date:** pre 2010 |
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51 | * **Last Modified by:** Paul Butler **Date:** Septmber 30, 2016 |
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52 | * **Last Reviewed by:** Under Review **Date:** October 5, 2016 |
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53 | |
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54 | """ |
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55 | |
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56 | from numpy import inf, sin, cos |
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57 | |
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58 | name = "core_shell_bicelle" |
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59 | title = "Circular cylinder with a core-shell scattering length density profile.." |
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60 | description = """ |
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61 | P(q,alpha)= (scale/Vs)*f(q)^(2) + bkg, where: |
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62 | f(q)= Vt(sld_rim - sld_solvent)* sin[qLt.cos(alpha)/2] |
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63 | /[qLt.cos(alpha)/2]*J1(qRout.sin(alpha)) |
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64 | /[qRout.sin(alpha)]+ |
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65 | (sld_core-sld_face)*Vc*sin[qLcos(alpha)/2][[qL |
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66 | *cos(alpha)/2]*J1(qRc.sin(alpha)) |
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67 | /qRc.sin(alpha)]+ |
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68 | (sld_face-sld_rim)*(Vc+Vf)*sin[q(L+2.thick_face). |
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69 | cos(alpha)/2][[q(L+2.thick_face)*cos(alpha)/2]* |
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70 | J1(qRc.sin(alpha))/qRc.sin(alpha)] |
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71 | |
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72 | alpha:is the angle between the axis of |
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73 | the cylinder and the q-vector |
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74 | Vt = pi.(Rc + thick_rim)^2.Lt : total volume |
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75 | Vc = pi.Rc^2.L :the volume of the core |
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76 | Vf = 2.pi.Rc^2.thick_face |
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77 | Rc = radius: is the core radius |
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78 | L: the length of the core |
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79 | Lt = L + 2.thick_face: total length |
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80 | Rout = radius + thick_rim |
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81 | sld_core, sld_rim, sld_face:scattering length |
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82 | densities within the particle |
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83 | sld_solvent: the scattering length density |
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84 | of the solvent |
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85 | bkg: the background |
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86 | J1: the first order Bessel function |
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87 | theta: axis_theta of the cylinder |
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88 | phi: the axis_phi of the cylinder... |
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89 | """ |
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90 | category = "shape:cylinder" |
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91 | |
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92 | # pylint: disable=bad-whitespace, line-too-long |
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93 | # ["name", "units", default, [lower, upper], "type", "description"], |
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94 | parameters = [ |
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95 | ["radius", "Ang", 20, [0, inf], "volume", "Cylinder core radius"], |
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96 | ["thick_rim", "Ang", 10, [0, inf], "volume", "Rim shell thickness"], |
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97 | ["thick_face", "Ang", 10, [0, inf], "volume", "Cylinder face thickness"], |
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98 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder length"], |
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99 | ["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Cylinder core scattering length density"], |
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100 | ["sld_face", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder face scattering length density"], |
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101 | ["sld_rim", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder rim scattering length density"], |
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102 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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103 | ["theta", "degrees", 90, [-inf, inf], "orientation", "In plane angle"], |
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104 | ["phi", "degrees", 0, [-inf, inf], "orientation", "Out of plane angle"], |
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105 | ] |
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106 | |
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107 | # pylint: enable=bad-whitespace, line-too-long |
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108 | |
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109 | source = ["lib/Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", |
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110 | "core_shell_bicelle.c"] |
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111 | |
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112 | demo = dict(scale=1, background=0, |
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113 | radius=20.0, |
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114 | thick_rim=10.0, |
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115 | thick_face=10.0, |
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116 | length=400.0, |
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117 | sld_core=1.0, |
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118 | sld_face=4.0, |
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119 | sld_rim=4.0, |
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120 | sld_solvent=1.0, |
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121 | theta=90, |
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122 | phi=0) |
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123 | |
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124 | qx, qy = 0.4 * cos(90), 0.5 * sin(0) |
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125 | tests = [ |
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126 | # Accuracy tests based on content in test/utest_other_models.py |
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127 | [{'radius': 20.0, |
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128 | 'thick_rim': 10.0, |
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129 | 'thick_face': 10.0, |
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130 | 'length': 400.0, |
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131 | 'sld_core': 1.0, |
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132 | 'sld_face': 4.0, |
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133 | 'sld_rim': 4.0, |
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134 | 'sld_solvent': 1.0, |
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135 | 'background': 0.0, |
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136 | }, 0.001, 353.550], |
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137 | |
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138 | [{'radius': 20.0, |
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139 | 'thick_rim': 10.0, |
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140 | 'thick_face': 10.0, |
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141 | 'length': 400.0, |
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142 | 'sld_core': 1.0, |
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143 | 'sld_face': 4.0, |
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144 | 'sld_rim': 4.0, |
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145 | 'sld_solvent': 1.0, |
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146 | 'theta': 90.0, |
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147 | 'phi': 0.0, |
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148 | 'background': 0.00, |
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149 | }, (qx, qy), 24.9167], |
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150 | |
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151 | # Additional tests with larger range of parameters |
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152 | [{'radius': 3.0, |
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153 | 'thick_rim': 100.0, |
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154 | 'thick_face': 100.0, |
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155 | 'length': 1200.0, |
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156 | 'sld_core': 5.0, |
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157 | 'sld_face': 41.0, |
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158 | 'sld_rim': 42.0, |
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159 | 'sld_solvent': 21.0, |
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160 | }, 0.05, 1670.1828], |
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161 | ] |
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