1 | # pylint: disable=line-too-long |
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2 | r""" |
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3 | Definition for 2D (orientated system) |
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4 | ------------------------------------- |
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5 | |
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6 | The angles $\theta$ and $\phi$ define the orientation of the axis of the |
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7 | cylinder. The angle $\Psi$ is defined as the orientation of the major |
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8 | axis of the ellipse with respect to the vector $Q$. A gaussian polydispersity |
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9 | can be added to any of the orientation angles, and also for the minor |
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10 | radius and the ratio of the ellipse radii. |
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11 | |
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12 | .. figure:: img/elliptical_cylinder_geometry.png |
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13 | |
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14 | Elliptical cylinder geometry $a = r_\text{minor}$ |
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15 | and $\nu = r_\text{major} / r_\text{minor}$ is the *axis_ratio*. |
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16 | |
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17 | The function calculated is |
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18 | |
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19 | .. math:: |
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20 | |
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21 | I(\vec q)=\frac{1}{V_\text{cyl}}\int{d\psi}\int{d\phi}\int{ |
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22 | p(\theta,\phi,\psi)F^2(\vec q,\alpha,\psi)\sin(\theta)d\theta} |
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23 | |
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24 | with the functions |
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25 | |
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26 | .. math:: |
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27 | |
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28 | F(\vec q,\alpha,\psi) = 2\frac{J_1(a)\sin(b)}{ab} |
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29 | |
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30 | where |
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31 | |
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32 | .. math:: |
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33 | |
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34 | a &= \vec q\sin(\alpha)\left[ |
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35 | r^2_\text{major}\sin^2(\psi)+r^2_\text{minor}\cos(\psi) \right]^{1/2} |
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36 | |
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37 | b &= \vec q\frac{L}{2}\cos(\alpha) |
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38 | |
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39 | and the angle $\Psi$ is defined as the orientation of the major axis of the |
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40 | ellipse with respect to the vector $\vec q$. The angle $\alpha$ is the angle |
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41 | between the axis of the cylinder and $\vec q$. |
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42 | |
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43 | |
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44 | Definition for 1D (no preferred orientation) |
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45 | -------------------------------------------- |
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46 | |
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47 | The form factor is averaged over all possible orientation before normalized |
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48 | by the particle volume |
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49 | |
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50 | .. math:: |
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51 | |
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52 | P(q) = \text{scale} <F^2> / V |
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53 | |
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54 | To provide easy access to the orientation of the elliptical cylinder, we |
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55 | define the axis of the cylinder using two angles $\theta$, $\phi$ and $\Psi$ |
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56 | (see :ref:`cylinder orientation <cylinder-angle-definition>`). The angle |
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57 | $\Psi$ is the rotational angle around its own long_c axis against the $q$ plane. |
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58 | For example, $\Psi = 0$ when the $r_\text{minor}$ axis is parallel to the |
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59 | $x$ axis of the detector. |
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60 | |
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61 | All angle parameters are valid and given only for 2D calculation; ie, an |
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62 | oriented system. |
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63 | |
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64 | .. figure:: img/elliptical_cylinder_angle_definition.jpg |
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65 | |
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66 | Definition of angles for 2D |
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67 | |
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68 | .. figure:: img/cylinder_angle_projection.jpg |
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69 | |
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70 | Examples of the angles for oriented elliptical cylinders against the |
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71 | detector plane. |
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72 | |
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73 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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74 | averaged radius $(=\sqrt{r_\text{minor}^2 * \text{axis ratio}})$ and length |
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75 | values, and used as the effective radius for $S(Q)$ when $P(Q)*S(Q)$ is applied. |
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76 | |
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77 | |
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78 | Validation |
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79 | ---------- |
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80 | |
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81 | Validation of our code was done by comparing the output of the 1D calculation |
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82 | to the angular average of the output of the 2D calculation over all possible |
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83 | angles. |
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84 | |
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85 | In the 2D average, more binning in the angle $\phi$ is necessary to get the |
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86 | proper result. The following figure shows the results of the averaging by |
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87 | varying the number of angular bins. |
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88 | |
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89 | .. figure:: img/elliptical_cylinder_averaging.png |
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90 | |
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91 | The intensities averaged from 2D over different numbers of bins and angles. |
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92 | |
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93 | References |
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94 | ---------- |
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95 | |
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96 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and |
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97 | Neutron Scattering*, Plenum, New York, (1987) |
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98 | """ |
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99 | |
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100 | from numpy import pi, inf, sqrt |
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101 | |
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102 | name = "elliptical_cylinder" |
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103 | title = "Form factor for an elliptical cylinder." |
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104 | description = """ |
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105 | Form factor for an elliptical cylinder. |
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106 | 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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107 | """ |
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108 | category = "shape:cylinder" |
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109 | |
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110 | # pylint: disable=bad-whitespace, line-too-long |
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111 | # ["name", "units", default, [lower, upper], "type","description"], |
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112 | parameters = [["r_minor", "Ang", 20.0, [0, inf], "volume", "Ellipse minor radius"], |
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113 | ["axis_ratio", "", 1.5, [1, inf], "volume", "Ratio of major radius over minor radius"], |
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114 | ["length", "Ang", 400.0, [1, inf], "volume", "Length of the cylinder"], |
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115 | ["sld", "1e-6/Ang^2", 4.0, [-inf, inf], "sld", "Cylinder scattering length density"], |
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116 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Solvent scattering length density"], |
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117 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "In plane angle"], |
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118 | ["phi", "degrees", 0, [-360, 360], "orientation", "Out of plane angle"], |
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119 | ["psi", "degrees", 0, [-360, 360], "orientation", "Major axis angle relative to Q"]] |
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120 | |
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121 | # pylint: enable=bad-whitespace, line-too-long |
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122 | |
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123 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "lib/gauss20.c", |
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124 | "elliptical_cylinder.c"] |
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125 | |
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126 | demo = dict(scale=1, background=0, r_minor=100, axis_ratio=1.5, length=400.0, |
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127 | sld=4.0, sld_solvent=1.0, theta=10.0, phi=20, psi=30, |
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128 | theta_pd=10, phi_pd=2, psi_pd=3) |
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129 | |
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130 | def ER(r_minor, axis_ratio, length): |
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131 | """ |
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132 | Equivalent radius |
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133 | @param r_minor: Ellipse minor radius |
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134 | @param axis_ratio: Ratio of major radius over minor radius |
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135 | @param length: Length of the cylinder |
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136 | """ |
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137 | radius = sqrt(r_minor * r_minor * axis_ratio) |
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138 | ddd = 0.75 * radius * (2 * radius * length |
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139 | + (length + radius) * (length + pi * radius)) |
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140 | return 0.5 * (ddd) ** (1. / 3.) |
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141 | |
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142 | tests = [ |
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143 | [{'r_minor': 20.0, 'axis_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024], |
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144 | [{'r_minor': 20.0, 'axis_ratio': 1.2, 'length':300.0}, 'VR', 1], |
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145 | |
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146 | # The SasView test result was 0.00169, with a background of 0.001 |
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147 | [{'r_minor': 20.0, 'axis_ratio': 1.5, 'sld': 4.0, 'length':400.0, |
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148 | 'sld_solvent':1.0, 'background':0.0}, |
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149 | 0.001, 675.504402], |
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150 | ] |
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