1 | r""" |
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2 | Calculates the scattering from a **simple cubic lattice** with |
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3 | paracrystalline distortion. Thermal vibrations are considered to be |
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4 | negligible, and the size of the paracrystal is infinitely large. |
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5 | Paracrystalline distortion is assumed to be isotropic and characterized |
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6 | by a Gaussian distribution. |
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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 scattering intensity $I(q)$ is calculated as |
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12 | |
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13 | .. math:: |
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14 | |
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15 | I(q) = \text{scale}\frac{V_\text{lattice}P(q)Z(q)}{V_p} + \text{background} |
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16 | |
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17 | where scale is the volume fraction of spheres, $V_p$ is the volume of |
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18 | the primary particle, $V_\text{lattice}$ is a volume correction for the crystal |
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19 | structure, $P(q)$ is the form factor of the sphere (normalized), and |
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20 | $Z(q)$ is the paracrystalline structure factor for a simple cubic structure. |
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21 | |
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22 | Equation (16) of the 1987 reference is used to calculate $Z(q)$, using |
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23 | equations (13)-(15) from the 1987 paper for Z1, Z2, and Z3. |
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24 | |
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25 | The lattice correction (the occupied volume of the lattice) for a simple |
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26 | cubic structure of particles of radius *R* and nearest neighbor separation *D* is |
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27 | |
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28 | .. math:: |
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29 | |
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30 | V_\text{lattice}=\frac{4\pi}{3}\frac{R^3}{D^3} |
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31 | |
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32 | The distortion factor (one standard deviation) of the paracrystal is included |
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33 | in the calculation of $Z(q)$ |
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34 | |
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35 | .. math:: |
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36 | |
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37 | \Delta a = gD |
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38 | |
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39 | where *g* is a fractional distortion based on the nearest neighbor distance. |
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40 | |
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41 | The simple cubic lattice is |
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42 | |
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43 | .. figure:: img/sc_crystal_geometry.jpg |
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44 | |
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45 | For a crystal, diffraction peaks appear at reduced q-values given by |
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46 | |
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47 | .. math:: |
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48 | |
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49 | \frac{qD}{2\pi} = \sqrt{h^2+k^2+l^2} |
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50 | |
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51 | where for a simple cubic lattice any h, k, l are allowed and none are |
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52 | forbidden. Thus the peak positions correspond to (just the first 5) |
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53 | |
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54 | .. math:: |
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55 | :nowrap: |
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56 | |
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57 | \begin{align*} |
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58 | q/q_0 \quad & \quad 1 |
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59 | & \sqrt{2} \quad |
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60 | & \quad \sqrt{3} \quad |
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61 | & \sqrt{4} \quad |
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62 | & \quad \sqrt{5}\quad \\ |
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63 | Indices \quad & (100) |
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64 | & \quad (110) \quad |
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65 | & \quad (111) |
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66 | & (200) \quad |
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67 | & \quad (210) |
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68 | \end{align*} |
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69 | |
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70 | .. note:: |
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71 | |
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72 | The calculation of *Z(q)* is a double numerical integral that must be |
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73 | carried out with a high density of points to properly capture the sharp |
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74 | peaks of the paracrystalline scattering. |
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75 | So be warned that the calculation is SLOW. Go get some coffee. |
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76 | Fitting of any experimental data must be resolution smeared for any |
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77 | meaningful fit. This makes a triple integral. Very, very slow. |
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78 | Go get lunch! |
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79 | |
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80 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is |
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81 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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82 | be accurate. Note that we are not responsible for any incorrectness of the 2D |
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83 | model computation. |
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84 | |
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85 | .. figure:: img/sc_crystal_angle_definition.jpg |
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86 | |
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87 | Reference |
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88 | --------- |
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89 | Hideki Matsuoka et. al. *Physical Review B,* 36 (1987) 1754-1765 |
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90 | (Original Paper) |
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91 | |
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92 | Hideki Matsuoka et. al. *Physical Review B,* 41 (1990) 3854 -3856 |
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93 | (Corrections to FCC and BCC lattice structure calculation) |
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94 | |
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95 | """ |
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96 | |
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97 | from numpy import inf |
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98 | |
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99 | name = "sc_paracrystal" |
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100 | title = "Simple cubic lattice with paracrystalline distortion" |
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101 | description = """ |
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102 | P(q)=(scale/Vp)*V_lattice*P(q)*Z(q)+bkg where scale is the volume |
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103 | fraction of sphere, |
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104 | Vp = volume of the primary particle, |
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105 | V_lattice = volume correction for |
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106 | for the crystal structure, |
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107 | P(q)= form factor of the sphere (normalized), |
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108 | Z(q)= paracrystalline structure factor |
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109 | for a simple cubic structure. |
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110 | [Simple Cubic ParaCrystal Model] |
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111 | Parameters; |
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112 | scale: volume fraction of spheres |
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113 | bkg:background, R: radius of sphere |
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114 | dnn: Nearest neighbor distance |
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115 | d_factor: Paracrystal distortion factor |
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116 | radius: radius of the spheres |
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117 | sldSph: SLD of the sphere |
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118 | sldSolv: SLD of the solvent |
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119 | """ |
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120 | category = "shape:paracrystal" |
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121 | single = False |
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122 | # pylint: disable=bad-whitespace, line-too-long |
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123 | # ["name", "units", default, [lower, upper], "type","description"], |
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124 | parameters = [["dnn", "Ang", 220.0, [0.0, inf], "", "Nearest neighbor distance"], |
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125 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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126 | ["radius", "Ang", 40.0, [0.0, inf], "volume", "Radius of sphere"], |
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127 | ["sld", "1e-6/Ang^2", 3.0, [0.0, inf], "sld", "Sphere scattering length density"], |
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128 | ["sld_solvent", "1e-6/Ang^2", 6.3, [0.0, inf], "sld", "Solvent scattering length density"], |
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129 | ["theta", "degrees", 0.0, [-inf, inf], "orientation", "Orientation of the a1 axis w/respect incoming beam"], |
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130 | ["phi", "degrees", 0.0, [-inf, inf], "orientation", "Orientation of the a2 in the plane of the detector"], |
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131 | ["psi", "degrees", 0.0, [-inf, inf], "orientation", "Orientation of the a3 in the plane of the detector"], |
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132 | ] |
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133 | # pylint: enable=bad-whitespace, line-too-long |
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134 | |
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135 | source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c", "lib/gauss150.c", "sc_paracrystal.c"] |
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136 | |
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137 | demo = dict(scale=1, background=0, |
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138 | dnn=220.0, |
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139 | d_factor=0.06, |
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140 | radius=40.0, |
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141 | sld=3.0, |
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142 | sld_solvent=6.3, |
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143 | theta=0.0, |
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144 | phi=0.0, |
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145 | psi=0.0) |
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146 | |
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147 | tests = [ |
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148 | # Accuracy tests based on content in test/utest_extra_models.py |
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149 | [{}, 0.001, 10.3048], |
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150 | [{}, 0.215268, 0.00814889], |
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151 | [{}, (0.414467), 0.001313289] |
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152 | ] |
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153 | |
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154 | |
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