1 | # parallelepiped 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 | For information about polarised and magnetic scattering, click here_. |
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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 | This model provides the form factor, *P(q)*, for a rectangular parallelepiped |
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12 | (below) where the form factor is normalized by the volume of the |
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13 | parallelepiped. If you need to apply polydispersity, see also the |
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14 | RectangularPrismModel_. |
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15 | |
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16 | The calculated form factor is: |
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17 | |
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18 | .. math:: |
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19 | |
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20 | P(Q) = {\text{scale} \over V} F^2(Q) + \text{background} |
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21 | |
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22 | where the volume *V* = *A B C* and the averaging < > is applied over all |
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23 | orientations for 1D. |
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24 | |
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25 | .. image:: img/parallelepiped.jpg |
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26 | |
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27 | *Figure. Parallelepiped with the corresponding Definition of sides. |
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28 | |
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29 | The edge of the solid must satisfy the condition that** *A* < *B* < *C*. |
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30 | Then, assuming *a* = *A* / *B* < 1, *b* = *B* / *B* = 1, and |
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31 | *c* = *C* / *B* > 1, the form factor is |
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32 | |
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33 | .. math:: |
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34 | |
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35 | P(q) = \frac{\textstyle{scale}}{V}\int_0^1 \phi(\mu \sqrt{1-\sigma^2},a) |
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36 | [S(\mu c \sigma/2)]^2 d\sigma |
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37 | |
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38 | with |
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39 | |
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40 | .. math:: |
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41 | |
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42 | \phi(\mu,a) = \int_0^1 \{S[\frac{\mu}{2}\cos(\frac{\pi}{2}u)] |
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43 | S[\frac{\mu a}{2}\sin(\frac{\pi}{2}u)]\}^2 du |
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44 | |
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45 | S(x) = \frac{\sin x}{x} |
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46 | |
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47 | \mu = qB |
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48 | |
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49 | and the contrast is defined as |
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50 | |
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51 | .. math:: |
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52 | |
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53 | \Delta\rho = \rho_{\textstyle p} - \rho_{\textstyle solvent} |
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54 | |
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55 | The scattering intensity per unit volume is returned in units of |cm^-1|; |
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56 | ie, *I(q)* = |phi| *P(q)*\ . |
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57 | |
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58 | NB: The 2nd virial coefficient of the parallelpiped is calculated based on |
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59 | the averaged effective radius (= sqrt(*short_a* \* *short_b* / |pi|)) and |
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60 | length(= *long_c*) values, and used as the effective radius for |
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61 | *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
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62 | |
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63 | To provide easy access to the orientation of the parallelepiped, we define |
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64 | three angles |theta|, |phi| and |bigpsi|. The definition of |theta| and |phi| |
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65 | is the same as for the cylinder model (see also figures below). |
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66 | The angle |bigpsi| is the rotational angle around the *long_c* axis against |
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67 | the *q* plane. For example, |bigpsi| = 0 when the *short_b* axis is parallel |
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68 | to the *x*-axis of the detector. |
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69 | |
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70 | |
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71 | .. _parallelepiped-orientation: |
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72 | |
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73 | .. figure:: img/orientation.jpg |
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74 | |
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75 | Definition of the angles for oriented parallelepipeds. |
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76 | |
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77 | .. figure:: img/orientation2.jpg |
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78 | |
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79 | Examples of the angles for oriented parallelepipeds against the detector plane. |
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80 | |
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81 | |
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82 | Validation |
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83 | ---------- |
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84 | |
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85 | Validation of the code was done by comparing the output of the 1D calculation |
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86 | to the angular average of the output of a 2D calculation over all possible |
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87 | angles. The Figure below shows the comparison where the solid dot refers to |
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88 | averaged 2D while the line represents the result of the 1D calculation (for |
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89 | the averaging, 76, 180, 76 points are taken for the angles of |theta|, |phi|, |
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90 | and |psi| respectively). |
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91 | |
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92 | .. _parallelepiped-compare: |
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93 | |
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94 | .. figure:: img/parallelepiped_compare.jpg |
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95 | |
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96 | *Figure. Comparison between 1D and averaged 2D.* |
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97 | |
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98 | This model reimplements the form factor calculations implemented in a c-library |
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99 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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100 | |
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101 | """ |
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102 | |
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103 | from numpy import pi, inf, sqrt |
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104 | |
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105 | name = "parallelepiped" |
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106 | title = "Rectangular parallelepiped with uniform scattering length density." |
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107 | description = """ |
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108 | P(q)= scale/V*integral from 0 to 1 of ... |
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109 | phi(mu*sqrt(1-sigma^2),a) * S(mu*c*sigma/2)^2 * dsigma |
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110 | |
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111 | phi(mu,a) = integral from 0 to 1 of .. |
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112 | (S((mu/2)*cos(pi*u/2))*S((mu*a/2)*sin(pi*u/2)))^2 * du |
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113 | S(x) = sin(x)/x |
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114 | mu = q*B |
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115 | """ |
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116 | category = "shape:parallelpiped" |
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117 | |
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118 | # ["name", "units", default, [lower, upper], "type","description"], |
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119 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "", |
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120 | "Parallelepiped scattering length density"], |
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121 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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122 | "Solvent scattering length density"], |
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123 | ["a_side", "Ang", 35, [0, inf], "volume", |
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124 | "Shorter side of the parallelepiped"], |
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125 | ["b_side", "Ang", 75, [0, inf], "volume", |
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126 | "Second side of the parallelepiped"], |
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127 | ["c_side", "Ang", 400, [0, inf], "volume", |
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128 | "Larger side of the parallelepiped"], |
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129 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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130 | "In plane angle"], |
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131 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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132 | "Out of plane angle"], |
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133 | ["psi", "degrees", 60, [-inf, inf], "orientation", |
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134 | "Rotation angle around its own c axis against q plane"], |
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135 | ] |
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136 | |
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137 | source = ["lib/J1.c", "lib/gauss76.c", "parallelepiped.c"] |
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138 | |
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139 | def ER(a_side, b_side, c_side): |
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140 | |
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141 | # surface average radius (rough approximation) |
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142 | surf_rad = sqrt(a_side * b_side / pi) |
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143 | |
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144 | # DiamCyl recoded here (to check and possibly put in a library?) |
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145 | a = surf_rad |
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146 | b = 0.5 * c_side |
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147 | t1 = a * a * b |
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148 | t2 = 1.0 + (b / a) * (1.0 + a / b / 2.0) * (1.0 + pi * a / b / 2.0) |
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149 | ddd = 3.0 * t1 * t2 |
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150 | |
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151 | return 0.5 * (ddd) ** (1. / 3.) |
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152 | |
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153 | # parameters for demo |
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154 | demo = dict(scale=1, background=0, |
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155 | sld=6.3e-6, solvent_sld=1.0e-6, |
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156 | a_side=35, b_side=75, c_side=400, |
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157 | theta=45, phi=30, psi=15, |
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158 | a_side_pd=0.1, a_side_pd_n=10, |
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159 | b_side_pd=0.1, b_side_pd_n=1, |
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160 | c_side_pd=0.1, c_side_pd_n=10, |
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161 | theta_pd=10, theta_pd_n=1, |
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162 | phi_pd=10, phi_pd_n=1, |
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163 | psi_pd=10, psi_pd_n=10) |
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164 | |
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165 | # For testing against the old sasview models, include the converted parameter |
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166 | # names and the target sasview model name. |
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167 | oldname = 'ParallelepipedModel' |
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168 | oldpars = dict(theta='parallel_theta', phi='parallel_phi', psi='parallel_psi', |
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169 | a_side='short_a', b_side='short_b', c_side='long_c', |
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170 | sld='sldPipe', solvent_sld='sldSolv') |
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171 | |
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