1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | Calculates the scattering from a **body-centered cubic lattice** with |
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6 | paracrystalline distortion. Thermal vibrations are considered to be negligible, |
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7 | and the size of the paracrystal is infinitely large. Paracrystalline distortion |
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8 | is assumed to be isotropic and characterized by a Gaussian distribution. |
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9 | |
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10 | The scattering intensity $I(q)$ is calculated as |
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11 | |
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12 | .. math:: |
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13 | |
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14 | I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q) |
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15 | |
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16 | |
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17 | where *scale* is the volume fraction of spheres, $V_p$ is the volume of the |
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18 | 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 $Z(q)$ |
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20 | is the paracrystalline structure factor for a body-centered cubic structure. |
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21 | |
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22 | Equation (1) of the 1990 reference\ [#CIT1990]_ is used to calculate $Z(q)$, |
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23 | using equations (29)-(31) from the 1987 paper\ [#CIT1987]_ for $Z1$, $Z2$, and |
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24 | $Z3$. |
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25 | |
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26 | The lattice correction (the occupied volume of the lattice) for a |
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27 | body-centered cubic structure of particles of radius $R$ and nearest neighbor |
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28 | separation $D$ is |
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29 | |
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30 | .. math:: |
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31 | |
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32 | V_\text{lattice} = \frac{16\pi}{3} \frac{R^3}{\left(D\sqrt{2}\right)^3} |
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33 | |
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34 | |
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35 | The distortion factor (one standard deviation) of the paracrystal is included |
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36 | in the calculation of $Z(q)$ |
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37 | |
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38 | .. math:: |
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39 | |
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40 | \Delta a = g D |
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41 | |
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42 | where $g$ is a fractional distortion based on the nearest neighbor distance. |
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43 | |
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44 | |
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45 | .. figure:: img/bcc_geometry.jpg |
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46 | |
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47 | Body-centered cubic lattice. |
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48 | |
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49 | For a crystal, diffraction peaks appear at reduced q-values given by |
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50 | |
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51 | .. math:: |
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52 | |
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53 | \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2} |
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54 | |
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55 | where for a body-centered cubic lattice, only reflections where |
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56 | $(h + k + l) = \text{even}$ are allowed and reflections where |
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57 | $(h + k + l) = \text{odd}$ are forbidden. Thus the peak positions |
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58 | correspond to (just the first 5) |
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59 | |
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60 | .. math:: |
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61 | |
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62 | \begin{array}{lccccc} |
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63 | q/q_o & 1 & \sqrt{2} & \sqrt{3} & \sqrt{4} & \sqrt{5} \\ |
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64 | \text{Indices} & (110) & (200) & (211) & (220) & (310) \\ |
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65 | \end{array} |
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66 | |
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67 | .. note:: |
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68 | |
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69 | The calculation of $Z(q)$ is a double numerical integral that |
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70 | must be carried out with a high density of points to properly capture |
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71 | the sharp peaks of the paracrystalline scattering. |
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72 | So be warned that the calculation is slow. Fitting of any experimental data |
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73 | must be resolution smeared for any meaningful fit. This makes a triple integral |
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74 | which may be very slow. |
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75 | |
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76 | This example dataset is produced using 200 data points, |
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77 | *qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above default values. |
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78 | |
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79 | The 2D (Anisotropic model) is based on the reference below where $I(q)$ is |
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80 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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81 | be accurate, particularly at low $q$. For general details of the calculation and angular |
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82 | dispersions for oriented particles see :ref:`orientation` . |
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83 | Note that we are not responsible for any incorrectness of the 2D model computation. |
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84 | |
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85 | .. figure:: img/parallelepiped_angle_definition.png |
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86 | |
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87 | Orientation of the crystal with respect to the scattering plane, when |
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88 | $\theta = \phi = 0$ the $c$ axis is along the beam direction (the $z$ axis). |
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89 | |
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90 | References |
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91 | ---------- |
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92 | |
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93 | .. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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94 | (Original Paper) |
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95 | .. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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96 | (Corrections to FCC and BCC lattice structure calculation) |
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97 | |
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98 | Authorship and Verification |
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99 | ---------------------------- |
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100 | |
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101 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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102 | * **Last Modified by:** Paul Butler **Date:** September 29, 2016 |
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103 | * **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016 |
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104 | """ |
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105 | |
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106 | from numpy import inf, pi |
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107 | |
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108 | name = "bcc_paracrystal" |
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109 | title = "Body-centred cubic lattic with paracrystalline distortion" |
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110 | description = """ |
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111 | Calculates the scattering from a **body-centered cubic lattice** with |
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112 | paracrystalline distortion. Thermal vibrations are considered to be |
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113 | negligible, and the size of the paracrystal is infinitely large. |
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114 | Paracrystalline distortion is assumed to be isotropic and characterized |
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115 | by a Gaussian distribution. |
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116 | """ |
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117 | category = "shape:paracrystal" |
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118 | |
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119 | #note - calculation requires double precision |
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120 | single = False |
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121 | |
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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, [-inf, inf], "", "Nearest neighbour distance"], |
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125 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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126 | ["radius", "Ang", 40, [0, inf], "volume", "Particle radius"], |
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127 | ["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Particle scattering length density"], |
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128 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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129 | ["theta", "degrees", 60, [-360, 360], "orientation", "c axis to beam angle"], |
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130 | ["phi", "degrees", 60, [-360, 360], "orientation", "rotation about beam"], |
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131 | ["psi", "degrees", 60, [-360, 360], "orientation", "rotation about c axis"] |
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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/gauss150.c", "lib/sphere_form.c", "bcc_paracrystal.c"] |
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136 | |
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137 | def random(): |
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138 | import numpy as np |
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139 | # Define lattice spacing as a multiple of the particle radius |
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140 | # using the formulat a = 4 r/sqrt(3). Systems which are ordered |
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141 | # are probably mostly filled, so use a distribution which goes from |
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142 | # zero to one, but leaving 90% of them within 80% of the |
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143 | # maximum bcc packing. Lattice distortion values are empirically |
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144 | # useful between 0.01 and 0.7. Use an exponential distribution |
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145 | # in this range 'cuz its easy. |
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146 | radius = 10**np.random.uniform(1.3, 4) |
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147 | d_factor = 10**np.random.uniform(-2, -0.7) # sigma_d in 0.01-0.7 |
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148 | dnn_fraction = np.random.beta(a=10, b=1) |
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149 | dnn = radius*4/np.sqrt(3)/dnn_fraction |
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150 | pars = dict( |
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151 | #sld=1, sld_solvent=0, scale=1, background=1e-32, |
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152 | dnn=dnn, |
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153 | d_factor=d_factor, |
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154 | radius=radius, |
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155 | ) |
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156 | return pars |
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157 | |
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158 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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159 | # add 2d test later |
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160 | q = 4.*pi/220. |
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161 | tests = [ |
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162 | [{}, [0.001, q, 0.215268], [1.46601394721, 2.85851284174, 0.00866710287078]], |
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163 | [{'theta': 20.0, 'phi': 30, 'psi': 40.0}, (-0.017, 0.035), 2082.20264399], |
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164 | [{'theta': 20.0, 'phi': 30, 'psi': 40.0}, (-0.081, 0.011), 0.436323144781], |
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165 | ] |
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