1 | # core shell cylinder model |
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2 | # Note: model title and parameter table are inserted automatically |
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3 | r""" |
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4 | The form factor is normalized by the particle volume. |
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5 | |
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6 | Definition |
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7 | ---------- |
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8 | |
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9 | The output of the 2D scattering intensity function for oriented core-shell |
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10 | cylinders is given by (Kline, 2006) |
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11 | |
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12 | .. math:: |
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13 | |
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14 | P(q,\alpha) = \frac{\text{scale}}{V_s} f^2(q) + \text{background} |
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15 | |
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16 | where |
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17 | |
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18 | .. math:: |
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19 | |
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20 | f(q) = (\rho_c - \rho_s) V_c \ |
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21 | \frac{\sin( Q L/2 \cos \alpha)}{Q L/2 \cos \alpha} \ |
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22 | \frac{2 J_1 (Q R \sin \alpha)}{Q R \sin \alpha} |
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23 | + (\rho_s - \rho_\text{solv}) V_s \ |
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24 | \frac{\sin( Q (L/2+T) \cos \alpha)}{Q (L/2+T) \cos \alpha} |
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25 | \frac{2 J_1 (Q (R+T) \sin \alpha)}{Q (R+T) \sin \alpha} |
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26 | |
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27 | and |
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28 | |
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29 | .. math:: |
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30 | |
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31 | V_s = \pi (R + T)^2 \dot (L + 2T) |
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32 | |
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33 | |
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34 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, |
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35 | $V_s$ is the volume of the outer shell (i.e. the total volume, including |
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36 | the shell), $V_c$ is the volume of the core, $L$ is the length of the core, |
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37 | $R$ is the radius of the core, $T$ is the thickness of the shell, $\rho_c$ |
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38 | is the scattering length density of the core, $\rho_s$ is the scattering |
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39 | length density of the shell, $\rho_\text{solv}$ is the scattering length |
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40 | density of the solvent, and *background* is the background level. The outer |
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41 | radius of the shell is given by $R+T$ and the total length of the outer |
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42 | shell is given by $L+2T$. $J1$ is the first order Bessel function. |
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43 | |
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44 | .. figure:: img/image069.JPG |
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45 | |
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46 | Core shell cylinder schematic. |
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47 | |
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48 | To provide easy access to the orientation of the core-shell cylinder, we |
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49 | define the axis of the cylinder using two angles $\theta$ and $\phi$. As |
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50 | for the case of the cylinder, those angles are defined in |
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51 | Figure :num:`figure #cylinder-orientation`. |
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52 | |
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53 | NB: The 2nd virial coefficient of the cylinder is calculated based on |
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54 | the radius and 2 length values, and used as the effective radius for |
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55 | $S(Q)$ when $P(Q) \dot S(Q)$ is applied. |
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56 | |
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57 | The $\theta$ and $\phi$ parameters are not used for the 1D output. Our |
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58 | implementation of the scattering kernel and the 1D scattering intensity |
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59 | use the c-library from NIST. |
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60 | |
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61 | Validation |
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62 | ---------- |
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63 | |
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64 | Validation of our code was done by comparing the output of the 1D model to |
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65 | the output of the software provided by the NIST (Kline, 2006). |
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66 | Figure :num:`figure #core-shell-cylinder-comparison-1d` shows a comparison |
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67 | of the 1D output of our model and the output of the NIST software. |
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68 | |
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69 | .. _core-shell-cylinder-comparison-1d: |
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70 | |
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71 | .. figure:: img/image070.JPG |
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72 | |
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73 | Comparison of the SasView scattering intensity for a core-shell cylinder |
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74 | with the output of the NIST SANS analysis software. The parameters were |
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75 | set to: *scale=1.0 |Ang|*, *radius=20 |Ang|*, *thickness=10 |Ang|*, |
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76 | *length=400 |Ang|*, *core_sld=1e-6 |Ang^-2|*, *shell_sld=4e-6 |Ang^-2|*, |
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77 | *solvent_sld=1e-6 |Ang^-2|*, and *background=0.01 |cm^-1|*. |
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78 | |
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79 | Averaging over a distribution of orientation is done by evaluating the |
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80 | equation above. Since we have no other software to compare the |
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81 | implementation of the intensity for fully oriented cylinders, we can |
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82 | compare the result of averaging our 2D output using a uniform |
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83 | distribution $p(\theta,\phi)* = 1.0$. |
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84 | Figure :num:`figure #core-shell-cylinder-comparison-2d` shows the result |
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85 | of such a cross-check. |
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86 | |
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87 | .. _core-shell-cylinder-comparison-2d: |
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88 | |
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89 | .. figure:: img/image071.JPG |
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90 | |
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91 | Comparison of the intensity for uniformly distributed core-shell |
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92 | cylinders calculated from our 2D model and the intensity from the |
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93 | NIST SANS analysis software. The parameters used were: *scale*=1.0*, |
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94 | *radius=20 |Ang|*, *thickness=10 |Ang|*, *length*=400 |Ang|*, |
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95 | *core_sld=1e-6 |Ang^-2|*, *shell_sld=4e-6 |Ang^-2|*, |
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96 | *solvent_sld=1e-6 |Ang^-2|*, and *background=0.0 |cm^-1|*. |
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97 | |
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98 | 2013/11/26 - Description reviewed by Heenan, R. |
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99 | """ |
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100 | |
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101 | from numpy import pi, inf |
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102 | |
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103 | name = "cylinder" |
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104 | title = "Right circular cylinder with a core-shell scattering length density profile." |
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105 | description = """ |
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106 | P(q,alpha)= scale/Vs*f(q)^(2) + background, |
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107 | where: f(q)= 2(core_sld - solvant_sld) |
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108 | * Vc*sin[qLcos(alpha/2)] |
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109 | /[qLcos(alpha/2)]*J1(qRsin(alpha)) |
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110 | /[qRsin(alpha)]+2(shell_sld-solvent_sld) |
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111 | *Vs*sin[q(L+T)cos(alpha/2)][[q(L+T) |
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112 | *cos(alpha/2)]*J1(q(R+T)sin(alpha)) |
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113 | /q(R+T)sin(alpha)] |
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114 | |
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115 | alpha:is the angle between the axis of |
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116 | the cylinder and the q-vector |
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117 | Vs: the volume of the outer shell |
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118 | Vc: the volume of the core |
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119 | L: the length of the core |
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120 | shell_sld: the scattering length density of the shell |
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121 | solvent_sld: the scattering length density of the solvent |
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122 | background: the background |
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123 | T: the thickness |
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124 | R+T: is the outer radius |
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125 | L+2T: The total length of the outershell |
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126 | J1: the first order Bessel function |
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127 | theta: axis_theta of the cylinder |
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128 | phi: the axis_phi of the cylinder |
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129 | """ |
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130 | |
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131 | parameters = [ |
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132 | # [ "name", "units", default, [lower, upper], "type", |
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133 | # "description" ], |
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134 | [ "core_sld", "1e-6/Ang^2", 4, [-inf,inf], "", |
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135 | "Cylinder core scattering length density" ], |
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136 | [ "shell_sld", "1e-6/Ang^2", 4, [-inf,inf], "", |
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137 | "Cylinder shell scattering length density" ], |
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138 | [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "", |
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139 | "Solvent scattering length density" ], |
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140 | [ "radius", "Ang", 20, [0, inf], "volume", |
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141 | "Cylinder core radius" ], |
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142 | [ "thickness", "Ang", 20, [0, inf], "volume", |
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143 | "Cylinder shell thickness" ], |
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144 | [ "length", "Ang", 400, [0, inf], "volume", |
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145 | "Cylinder length" ], |
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146 | [ "theta", "degrees", 60, [-inf, inf], "orientation", |
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147 | "In plane angle" ], |
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148 | [ "phi", "degrees", 60, [-inf, inf], "orientation", |
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149 | "Out of plane angle" ], |
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150 | ] |
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151 | |
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152 | source = [ "lib/J1.c", "lib/gauss76.c", "core_shell_cylinder.c"] |
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153 | |
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154 | def ER(radius, thickness, length): |
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155 | radius = radius + thickness |
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156 | length = length + 2*thickness |
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157 | ddd = 0.75*radius*(2*radius*length + (length+radius)*(length+pi*radius)) |
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158 | return 0.5 * (ddd)**(1./3.) |
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159 | |
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160 | def VR(radius, thickness, length): |
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161 | whole = pi * (radius+thickness)**2 * (length+2*thickness) |
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162 | core = pi * radius**2 * length |
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163 | return whole, whole-core |
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164 | |
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