1 | # This is how a model would look if we were to stick it all into one file. |
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2 | r""" |
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3 | CylinderModel |
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4 | ============= |
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5 | |
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6 | This model provides the form factor for a right circular cylinder with uniform |
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7 | scattering length density. The form factor is normalized by the particle volume. |
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8 | |
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9 | For information about polarised and magnetic scattering, click here_. |
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10 | |
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11 | Definition |
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12 | ---------- |
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13 | |
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14 | The output of the 2D scattering intensity function for oriented cylinders is |
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15 | given by (Guinier, 1955) |
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16 | |
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17 | .. math:: |
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18 | |
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19 | P(q,\alpha) = \frac{\text{scale}}{V}f^2(q) + \text{bkg} |
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20 | |
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21 | where |
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22 | |
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23 | .. math:: |
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24 | |
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25 | f(q) = 2 (\Delta \rho) V |
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26 | \frac{\sin (q L/2 \cos \alpha)}{q L/2 \cos \alpha} |
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27 | \frac{J_1 (q r \sin \alpha)}{q r \sin \alpha} |
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28 | |
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29 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, $V$ |
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30 | is the volume of the cylinder, $L$ is the length of the cylinder, $r$ is the |
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31 | radius of the cylinder, and $d\rho$ (contrast) is the scattering length density |
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32 | difference between the scatterer and the solvent. $J_1$ is the first order |
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33 | Bessel function. |
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34 | |
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35 | To provide easy access to the orientation of the cylinder, we define the |
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36 | axis of the cylinder using two angles $\theta$ and $\phi$. Those angles |
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37 | are defined in Figure :num:`figure #CylinderModel-orientation`. |
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38 | |
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39 | .. _CylinderModel-orientation: |
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40 | |
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41 | .. figure:: img/image061.JPG |
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42 | |
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43 | Definition of the angles for oriented cylinders. |
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44 | |
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45 | .. figure:: img/image062.JPG |
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46 | |
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47 | Examples of the angles for oriented pp against the detector plane. |
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48 | |
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49 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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50 | radius and length values, and used as the effective radius for $S(Q)$ |
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51 | when $P(Q) \cdot S(Q)$ is applied. |
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52 | |
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53 | The returned value is scaled to units of |cm^-1| and the parameters of |
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54 | the CylinderModel are the following: |
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55 | |
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56 | %(parameters)s |
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57 | |
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58 | The output of the 1D scattering intensity function for randomly oriented |
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59 | cylinders is then given by |
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60 | |
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61 | .. math:: |
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62 | |
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63 | P(q) = \frac{\text{scale}}{V} |
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64 | \int_0^{\pi/2} f^2(q,\alpha) \sin \alpha d\alpha + \text{background} |
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65 | |
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66 | The *theta* and *phi* parameters are not used for the 1D output. Our |
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67 | implementation of the scattering kernel and the 1D scattering intensity |
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68 | use the c-library from NIST. |
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69 | |
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70 | Validation of the CylinderModel |
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71 | ------------------------------- |
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72 | |
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73 | Validation of our code was done by comparing the output of the 1D model |
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74 | to the output of the software provided by the NIST (Kline, 2006). |
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75 | Figure :num:`figure #CylinderModel-compare` shows a comparison of |
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76 | the 1D output of our model and the output of the NIST software. |
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77 | |
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78 | .. _CylinderModel-compare: |
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79 | |
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80 | .. figure:: img/image065.JPG |
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81 | |
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82 | Comparison of the SasView scattering intensity for a cylinder with the |
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83 | output of the NIST SANS analysis software. |
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84 | The parameters were set to: *Scale* = 1.0, *Radius* = 20 |Ang|, |
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85 | *Length* = 400 |Ang|, *Contrast* = 3e-6 |Ang^-2|, and |
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86 | *Background* = 0.01 |cm^-1|. |
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87 | |
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88 | In general, averaging over a distribution of orientations is done by |
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89 | evaluating the following |
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90 | |
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91 | .. math:: |
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92 | |
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93 | P(q) = \int_0^{\pi/2} d\phi |
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94 | \int_0^\pi p(\theta, \phi) P_0(q,\alpha) \sin \theta d\theta |
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95 | |
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96 | |
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97 | where $p(\theta,\phi)$ is the probability distribution for the orientation |
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98 | and $P_0(q,\alpha)$ is the scattering intensity for the fully oriented |
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99 | system. Since we have no other software to compare the implementation of |
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100 | the intensity for fully oriented cylinders, we can compare the result of |
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101 | averaging our 2D output using a uniform distribution $p(\theta, \phi) = 1.0$. |
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102 | Figure :num:`figure #CylinderModel-crosscheck` shows the result of |
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103 | such a cross-check. |
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104 | |
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105 | .. _CylinderModel-crosscheck: |
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106 | |
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107 | .. figure:: img/image066.JPG |
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108 | |
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109 | Comparison of the intensity for uniformly distributed cylinders |
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110 | calculated from our 2D model and the intensity from the NIST SANS |
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111 | analysis software. |
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112 | The parameters used were: *Scale* = 1.0, *Radius* = 20 |Ang|, |
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113 | *Length* = 400 |Ang|, *Contrast* = 3e-6 |Ang^-2|, and |
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114 | *Background* = 0.0 |cm^-1|. |
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115 | """ |
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116 | |
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117 | from numpy import pi, inf |
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118 | |
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119 | name = "cylinder" |
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120 | title = "Cylinder with uniform scattering length density" |
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121 | description = """ |
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122 | f(q)= 2*(sldCyl - sldSolv)*V*sin(qLcos(alpha/2)) |
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123 | /[qLcos(alpha/2)]*J1(qRsin(alpha/2))/[qRsin(alpha)] |
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124 | |
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125 | P(q,alpha)= scale/V*f(q)^(2)+background |
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126 | V: Volume of the cylinder |
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127 | R: Radius of the cylinder |
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128 | L: Length of the cylinder |
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129 | J1: The bessel function |
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130 | alpha: angle betweenthe axis of the |
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131 | cylinder and the q-vector for 1D |
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132 | :the ouput is P(q)=scale/V*integral |
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133 | from pi/2 to zero of... |
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134 | f(q)^(2)*sin(alpha)*dalpha+ bkg |
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135 | """ |
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136 | |
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137 | parameters = [ |
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138 | # [ "name", "units", default, [lower, upper], "type", |
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139 | # "description" ], |
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140 | [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "", |
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141 | "Cylinder scattering length density" ], |
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142 | [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "", |
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143 | "Solvent scattering length density" ], |
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144 | [ "radius", "Ang", 20, [0, inf], "volume", |
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145 | "Cylinder radius" ], |
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146 | [ "length", "Ang", 400, [0, inf], "volume", |
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147 | "Cylinder length" ], |
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148 | [ "theta", "degrees", 60, [-inf, inf], "orientation", |
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149 | "In plane angle" ], |
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150 | [ "phi", "degrees", 60, [-inf, inf], "orientation", |
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151 | "Out of plane angle" ], |
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152 | ] |
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153 | |
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154 | source = [ "lib/J1.c", "lib/gauss76.c", "lib/cylkernel.c" ] |
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155 | |
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156 | form_volume = """ |
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157 | return M_PI*radius*radius*length; |
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158 | """ |
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159 | |
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160 | Iq = """ |
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161 | const real h = REAL(0.5)*length; |
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162 | real summ = REAL(0.0); |
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163 | for (int i=0; i<76 ;i++) { |
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164 | //const real zi = ( Gauss76Z[i]*(uplim-lolim) + uplim + lolim )/2.0; |
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165 | const real zi = REAL(0.5)*(Gauss76Z[i]*M_PI_2 + M_PI_2); |
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166 | summ += Gauss76Wt[i] * CylKernel(q, radius, h, zi); |
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167 | } |
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168 | //const real form = (uplim-lolim)/2.0*summ; |
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169 | const real form = summ * M_PI_4; |
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170 | |
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171 | // Multiply by contrast^2, normalize by cylinder volume and convert to cm-1 |
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172 | // NOTE that for this (Fournet) definition of the integral, one must MULTIPLY by Vcyl |
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173 | // The additional volume factor is for polydisperse volume normalization. |
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174 | const real s = (sld - solvent_sld) * form_volume(radius, length); |
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175 | return REAL(1.0e-4) * form * s * s; |
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176 | """ |
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177 | |
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178 | Iqxy = """ |
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179 | real sn, cn; // slots to hold sincos function output |
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180 | |
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181 | // Compute angle alpha between q and the cylinder axis |
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182 | SINCOS(theta*M_PI_180, sn, cn); |
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183 | // # The following correction factor exists in sasview, but it can't be |
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184 | // # right, so we are leaving it out for now. |
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185 | // const real correction = fabs(cn)*M_PI_2; |
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186 | const real q = sqrt(qx*qx+qy*qy); |
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187 | const real cos_val = cn*cos(phi*M_PI_180)*(qx/q) + sn*(qy/q); |
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188 | const real alpha = acos(cos_val); |
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189 | |
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190 | // The following is CylKernel() / sin(alpha), but we are doing it in place |
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191 | // to avoid sin(alpha)/sin(alpha) for alpha = 0. It is also a teensy bit |
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192 | // faster since we don't mulitply and divide sin(alpha). |
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193 | SINCOS(alpha, sn, cn); |
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194 | const real besarg = q*radius*sn; |
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195 | const real siarg = REAL(0.5)*q*length*cn; |
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196 | // lim_{x->0} J1(x)/x = 1/2, lim_{x->0} sin(x)/x = 1 |
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197 | const real bj = (besarg == REAL(0.0) ? REAL(0.5) : J1(besarg)/besarg); |
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198 | const real si = (siarg == REAL(0.0) ? REAL(1.0) : sin(siarg)/siarg); |
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199 | const real form = REAL(4.0)*bj*bj*si*si; |
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200 | |
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201 | // Multiply by contrast^2, normalize by cylinder volume and convert to cm-1 |
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202 | // NOTE that for this (Fournet) definition of the integral, one must MULTIPLY by Vcyl |
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203 | // The additional volume factor is for polydisperse volume normalization. |
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204 | const real s = (sld - solvent_sld) * form_volume(radius, length); |
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205 | return REAL(1.0e-4) * form * s * s; // * correction; |
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206 | """ |
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207 | |
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208 | def ER(radius, length): |
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209 | ddd = 0.75*radius*(2*radius*length + (length+radius)*(length+pi*radius)) |
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210 | return 0.5 * (ddd)**(1./3.) |
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211 | |
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