1 | # 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 V = \piR^2L. |
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5 | For information about polarised and magnetic scattering, see |
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6 | the :doc:`magnetic help <../sasgui/perspectives/fitting/mag_help>` documentation. |
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7 | |
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8 | Definition |
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9 | ---------- |
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10 | |
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11 | The output of the 2D scattering intensity function for oriented cylinders is |
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12 | given by (Guinier, 1955) |
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13 | |
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14 | .. math:: |
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15 | |
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16 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q) + \text{background} |
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17 | |
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18 | where |
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19 | |
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20 | .. math:: |
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21 | |
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22 | F(q) = 2 (\Delta \rho) V |
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23 | \frac{\sin \left(\tfrac12 qL\cos\alpha \right)} |
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24 | {\tfrac12 qL \cos \alpha} |
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25 | \frac{J_1 \left(q R \sin \alpha\right)}{q R \sin \alpha} |
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26 | |
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27 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, $V$ |
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28 | is the volume of the cylinder, $L$ is the length of the cylinder, $R$ is the |
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29 | radius of the cylinder, and $\Delta\rho$ (contrast) is the scattering length |
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30 | density difference between the scatterer and the solvent. $J_1$ is the |
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31 | first order Bessel function. |
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32 | |
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33 | To provide easy access to the orientation of the cylinder, we define the |
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34 | axis of the cylinder using two angles $\theta$ and $\phi$. Those angles |
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35 | are defined in :numref:`cylinder-angle-definition`. |
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36 | |
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37 | .. _cylinder-angle-definition: |
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38 | |
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39 | .. figure:: img/cylinder_angle_definition.jpg |
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40 | |
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41 | Definition of the angles for oriented cylinders. |
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42 | |
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43 | .. figure:: img/cylinder_angle_projection.jpg |
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44 | |
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45 | Examples of the angles for oriented cylinders against the detector plane. |
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46 | |
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47 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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48 | radius and length values, and used as the effective radius for $S(q)$ |
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49 | when $P(q) \cdot S(q)$ is applied. |
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50 | |
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51 | The output of the 1D scattering intensity function for randomly oriented |
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52 | cylinders is then given by |
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53 | |
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54 | .. math:: |
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55 | |
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56 | P(q) = \frac{\text{scale}}{V} |
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57 | \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background} |
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58 | |
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59 | The $\theta$ and $\phi$ parameters are not used for the 1D output. |
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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 the code was done by comparing the output of the 1D model |
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65 | to the output of the software provided by the NIST (Kline, 2006). |
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66 | The implementation of the intensity for fully oriented cylinders was done |
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67 | by averaging over a uniform distribution of orientations using |
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68 | |
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69 | .. math:: |
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70 | |
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71 | P(q) = \int_0^{\pi/2} d\phi |
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72 | \int_0^\pi p(\theta, \phi) P_0(q,\alpha) \sin \theta\ d\theta |
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73 | |
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74 | |
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75 | where $p(\theta,\phi)$ is the probability distribution for the orientation |
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76 | and $P_0(q,\alpha)$ is the scattering intensity for the fully oriented |
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77 | system, and then comparing to the 1D result. |
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78 | |
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79 | References |
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80 | ---------- |
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81 | |
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82 | None |
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83 | |
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84 | """ |
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85 | |
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86 | import numpy as np # type: ignore |
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87 | from numpy import pi, inf # type: ignore |
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88 | |
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89 | name = "cylinder" |
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90 | title = "Right circular cylinder with uniform scattering length density." |
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91 | description = """ |
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92 | f(q,alpha) = 2*(sld - sld_solvent)*V*sin(qLcos(alpha)/2)) |
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93 | /[qLcos(alpha)/2]*J1(qRsin(alpha))/[qRsin(alpha)] |
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94 | |
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95 | P(q,alpha)= scale/V*f(q,alpha)^(2)+background |
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96 | V: Volume of the cylinder |
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97 | R: Radius of the cylinder |
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98 | L: Length of the cylinder |
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99 | J1: The bessel function |
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100 | alpha: angle between the axis of the |
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101 | cylinder and the q-vector for 1D |
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102 | :the ouput is P(q)=scale/V*integral |
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103 | from pi/2 to zero of... |
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104 | f(q,alpha)^(2)*sin(alpha)*dalpha + background |
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105 | """ |
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106 | category = "shape:cylinder" |
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107 | |
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108 | # [ "name", "units", default, [lower, upper], "type", "description"], |
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109 | parameters = [["sld", "4e-6/Ang^2", 4, [-inf, inf], "sld", |
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110 | "Cylinder scattering length density"], |
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111 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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112 | "Solvent scattering length density"], |
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113 | ["radius", "Ang", 20, [0, inf], "volume", |
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114 | "Cylinder radius"], |
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115 | ["length", "Ang", 400, [0, inf], "volume", |
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116 | "Cylinder length"], |
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117 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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118 | "In plane angle"], |
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119 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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120 | "Out of plane angle"], |
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121 | ] |
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122 | |
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123 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"] |
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124 | |
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125 | def ER(radius, length): |
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126 | """ |
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127 | Return equivalent radius (ER) |
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128 | """ |
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129 | ddd = 0.75 * radius * (2 * radius * length + (length + radius) * (length + pi * radius)) |
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130 | return 0.5 * (ddd) ** (1. / 3.) |
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131 | |
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132 | # parameters for demo |
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133 | demo = dict(scale=1, background=0, |
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134 | sld=6, sld_solvent=1, |
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135 | radius=20, length=300, |
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136 | theta=60, phi=60, |
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137 | radius_pd=.2, radius_pd_n=9, |
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138 | length_pd=.2, length_pd_n=10, |
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139 | theta_pd=10, theta_pd_n=5, |
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140 | phi_pd=10, phi_pd_n=5) |
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141 | |
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142 | qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5) |
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143 | tests = [[{}, 0.2, 0.042761386790780453], |
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144 | [{}, [0.2], [0.042761386790780453]], |
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145 | [{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03514647218513852], |
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146 | [{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03514647218513852]], |
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147 | ] |
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148 | del qx, qy # not necessary to delete, but cleaner |
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149 | # ADDED by: RKH ON: 18Mar2016 renamed sld's etc |
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