1 | # pylint: disable=line-too-long |
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2 | r""" |
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3 | This function calculates the scattering from an elliptical cylinder. |
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4 | |
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5 | Definition for 2D (orientated system) |
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6 | ------------------------------------- |
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7 | |
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8 | The angles |theta| and |phi| define the orientation of the axis of the cylinder. The angle |bigpsi| is defined as the |
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9 | orientation of the major axis of the ellipse with respect to the vector *Q*\ . A gaussian polydispersity can be added |
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10 | to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii. |
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11 | |
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12 | .. image:: img/elliptical_cylinder_geometry.gif |
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13 | |
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14 | *Figure.* *a* = *r_minor* and |nu|\ :sub:`n` = $r_ratio$ (i.e., $r_major / r_minor$). |
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15 | |
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16 | The function calculated is |
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17 | |
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18 | .. math:: |
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19 | I(\mathbf{q})=\frac{1}{V_{cyl}}\int{d\psi}\int{d\phi}\int{p(\theta,\phi,\psi)F^2(\mathbf{q},\alpha,\psi)\sin(\theta)d\theta} |
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20 | |
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21 | with the functions |
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22 | |
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23 | .. math:: |
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24 | F(\mathbf{q},\alpha,\psi)=2\frac{J_1(a)\sin(b)}{ab} |
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25 | \\ |
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26 | a = \mathbf{q}\sin(\alpha)\left[ r^2_{major}\sin^2(\psi)+r^2_{minor}\cos(\psi) \right]^{1/2} |
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27 | \\ |
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28 | b=\mathbf{q}\frac{L}{2}\cos(\alpha) |
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29 | |
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30 | and the angle |bigpsi| is defined as the orientation of the major axis of the ellipse with respect to the vector *q*\ . |
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31 | |
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32 | |
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33 | Definition for 1D (no preferred orientation) |
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34 | -------------------------------------------- |
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35 | |
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36 | The form factor is averaged over all possible orientation before normalized by the particle volume |
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37 | |
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38 | .. math:: |
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39 | P(q) = scale <F^2> / V |
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40 | |
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41 | The returned value is scaled to units of |cm^-1|. |
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42 | |
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43 | To provide easy access to the orientation of the elliptical cylinder, we define the axis of the cylinder using two |
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44 | angles |theta|, |phi| and |bigpsi|. As for the case of the cylinder, the angles |theta| and |phi| are defined on |
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45 | Figure 2 of CylinderModel. The angle |bigpsi| is the rotational angle around its own long_c axis against the *q* plane. |
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46 | For example, |bigpsi| = 0 when the *r_minor* axis is parallel to the *x*\ -axis of the detector. |
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47 | |
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48 | All angle parameters are valid and given only for 2D calculation; ie, an oriented system. |
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49 | |
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50 | .. image:: img/elliptical_cylinder_geometry_2d.jpg |
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51 | |
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52 | *Figure. Definition of angles for 2D* |
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53 | |
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54 | .. image:: img/core_shell_bicelle_fig2.jpg |
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55 | |
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56 | *Figure. Examples of the angles for oriented elliptical cylinders against the detector plane.* |
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57 | |
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58 | NB: The 2nd virial coefficient of the cylinder is calculated based on the averaged radius (= sqrt(*r_minor*\ :sup:`2` \* *r_ratio*)) |
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59 | and length values, and used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
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60 | |
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61 | |
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62 | .. image:: img/elliptical_cylinder_comparison_1d.jpg |
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63 | |
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64 | *Figure. 1D plot using the default values (w/1000 data point).* |
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65 | |
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66 | Validation |
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67 | ---------- |
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68 | |
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69 | Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of |
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70 | the 2D calculation over all possible angles. The figure below shows the comparison where the solid dot refers to |
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71 | averaged 2D values while the line represents the result of the 1D calculation (for the 2D averaging, values of 76, 180, |
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72 | and 76 degrees are taken for the angles of |theta|, |phi|, and |bigpsi| respectively). |
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73 | |
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74 | .. image:: img/elliptical_cylinder_validation_1d.gif |
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75 | |
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76 | *Figure. Comparison between 1D and averaged 2D.* |
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77 | |
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78 | In the 2D average, more binning in the angle |phi| is necessary to get the proper result. The following figure shows |
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79 | the results of the averaging by varying the number of angular bins. |
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80 | |
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81 | .. image:: img/elliptical_cylinder_averaging.gif |
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82 | |
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83 | *Figure. The intensities averaged from 2D over different numbers of bins and angles.* |
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84 | |
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85 | Reference |
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86 | --------- |
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87 | |
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88 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
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89 | New York, (1987) |
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90 | """ |
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91 | |
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92 | import math |
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93 | from numpy import pi, inf |
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94 | |
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95 | name = "elliptical_cylinder" |
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96 | title = "Form factor for an elliptical cylinder." |
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97 | description = """ |
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98 | Form factor for an elliptical cylinder. |
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99 | See L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, (1987). |
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100 | """ |
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101 | category = "shape:cylinder" |
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102 | |
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103 | # pylint: disable=bad-whitespace, line-too-long |
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104 | # ["name", "units", default, [lower, upper], "type","description"], |
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105 | parameters = [["r_minor", "Ang", 20.0, [0, inf], "volume", "Ellipse minor radius"], |
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106 | ["r_ratio", "", 1.5, [1, inf], "volume", "Ratio of major radius over minor radius"], |
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107 | ["length", "Ang", 400.0, [1, inf], "volume", "Length of the cylinder"], |
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108 | ["sld", "1e-6/Ang^2", 4.0, [-inf, inf], "", "Cylinder scattering length density"], |
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109 | ["solvent_sld", "1e-6/Ang^2", 1.0, [-inf, inf], "", "Solvent scattering length density"], |
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110 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "In plane angle"], |
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111 | ["phi", "degrees", 0, [-360, 360], "orientation", "Out of plane angle"], |
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112 | ["psi", "degrees", 0, [-360, 360], "orientation", "Major axis angle relative to Q"]] |
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113 | |
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114 | # pylint: enable=bad-whitespace, line-too-long |
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115 | |
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116 | source = ["lib/nr_bess_j1.c", "lib/gauss76.c", "lib/gauss20.c", "elliptical_cylinder.c"] |
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117 | |
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118 | demo = dict(scale=1, background=0, r_minor=100, r_ratio=1.5, length=400.0, |
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119 | sld=4.0, solvent_sld=1.0, theta=10.0, phi=20, psi=30, theta_pd=10, phi_pd=2, psi_pd=3) |
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120 | |
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121 | oldname = 'EllipticalCylinderModel' |
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122 | oldpars = dict(theta='cyl_theta', phi='cyl_phi', psi='cyl_psi', sld='sldCyl', solvent_sld='sldSolv') |
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123 | |
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124 | def ER(r_minor, r_ratio, length): |
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125 | """ |
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126 | Equivalent radius |
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127 | @param r_minor: Ellipse minor radius |
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128 | @param r_ratio: Ratio of major radius over minor radius |
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129 | @param length: Length of the cylinder |
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130 | """ |
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131 | r = math.sqrt(r_minor * r_minor * r_ratio) |
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132 | ddd = 0.75 * r * (2 * r * length + (length + r) * (length + pi * r)) |
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133 | return 0.5 * (ddd) ** (1. / 3.) |
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134 | |
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135 | tests = [[{'r_minor': 20.0, 'r_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024], |
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136 | [{'r_minor': 20.0, 'r_ratio': 1.2, 'length':300.0}, 'VR', 1], |
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137 | |
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138 | # The SasView test result was 0.00169, with a background of 0.001 |
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139 | [{'r_minor': 20.0, |
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140 | 'r_ratio': 1.5, |
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141 | 'sld': 4.0, |
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142 | 'length':400.0, |
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143 | 'solvent_sld':1.0, |
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144 | 'background':0.0 |
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145 | }, 0.001, 675.504402]] |
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