1 | #fcc paracrystal model |
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2 | #note model title and parameter table are automatically inserted |
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3 | #note - calculation requires double precision |
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4 | r""" |
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5 | Calculates the scattering from a **face-centered cubic lattice** with |
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6 | paracrystalline distortion. Thermal vibrations are considered to be |
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7 | negligible, and the size of the paracrystal is infinitely large. |
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8 | Paracrystalline distortion is assumed to be isotropic and characterized by |
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9 | a Gaussian distribution. |
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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 scattering intensity $I(q)$ is calculated as |
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15 | |
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16 | .. math:: |
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17 | |
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18 | I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q) |
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19 | |
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20 | where *scale* is the volume fraction of spheres, $V_p$ is the volume of |
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21 | the primary particle, $V_\text{lattice}$ is a volume correction for the crystal |
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22 | structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$ |
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23 | is the paracrystalline structure factor for a face-centered cubic structure. |
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24 | |
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25 | Equation (1) of the 1990 reference is used to calculate $Z(q)$, using |
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26 | equations (23)-(25) from the 1987 paper for $Z1$, $Z2$, and $Z3$. |
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27 | |
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28 | The lattice correction (the occupied volume of the lattice) for a |
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29 | face-centered cubic structure of particles of radius $R$ and nearest |
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30 | neighbor separation $D$ is |
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31 | |
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32 | .. math:: |
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33 | |
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34 | V_\text{lattice} = \frac{16\pi}{3}\frac{R^3}{\left(D\sqrt{2}\right)^3} |
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35 | |
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36 | The distortion factor (one standard deviation) of the paracrystal is |
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37 | included in the calculation of $Z(q)$ |
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38 | |
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39 | .. math:: |
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40 | |
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41 | \Delta a = gD |
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42 | |
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43 | where $g$ is a fractional distortion based on the nearest neighbor distance. |
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44 | |
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45 | .. figure:: img/fcc_geometry.jpg |
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46 | |
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47 | Face-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 face-centered cubic lattice $h, k , l$ all odd or all |
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56 | even are allowed and reflections where $h, k, l$ are mixed odd/even |
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57 | are forbidden. Thus the peak positions correspond to (just the first 5) |
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58 | |
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59 | .. math:: |
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60 | |
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61 | \begin{array}{cccccc} |
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62 | q/q_0 & 1 & \sqrt{4/3} & \sqrt{8/3} & \sqrt{11/3} & \sqrt{4} \\ |
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63 | \text{Indices} & (111) & (200) & (220) & (311) & (222) |
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64 | \end{array} |
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65 | |
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66 | .. note:: |
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67 | |
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68 | The calculation of $Z(q)$ is a double numerical integral that |
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69 | must be carried out with a high density of points to properly capture |
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70 | the sharp peaks of the paracrystalline scattering. |
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71 | So be warned that the calculation is slow. Fitting of any experimental data |
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72 | must be resolution smeared for any meaningful fit. This makes a triple integral |
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73 | which may be very slow. |
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74 | |
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75 | The 2D (Anisotropic model) is based on the reference below where $I(q)$ is |
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76 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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77 | be accurate particularly at low $q$. For general details of the calculation |
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78 | and angular dispersions for oriented particles see :ref:`orientation` . |
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79 | Note that we are not responsible for any incorrectness of the |
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80 | 2D model computation. |
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81 | |
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82 | .. figure:: img/parallelepiped_angle_definition.png |
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83 | |
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84 | Orientation of the crystal with respect to the scattering plane, when |
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85 | $\theta = \phi = 0$ the $c$ axis is along the beam direction (the $z$ axis). |
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86 | |
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87 | References |
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88 | ---------- |
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89 | |
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90 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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91 | (Original Paper) |
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92 | |
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93 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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94 | (Corrections to FCC and BCC lattice structure calculation) |
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95 | """ |
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96 | |
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97 | from numpy import inf, pi |
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98 | |
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99 | name = "fcc_paracrystal" |
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100 | title = "Face-centred cubic lattic with paracrystalline distortion" |
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101 | description = """ |
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102 | Calculates the scattering from a **face-centered cubic lattice** with paracrystalline distortion. Thermal vibrations |
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103 | are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is |
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104 | assumed to be isotropic and characterized by a Gaussian distribution. |
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105 | """ |
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106 | category = "shape:paracrystal" |
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107 | |
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108 | single = False |
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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 = [["dnn", "Ang", 220, [-inf, inf], "", "Nearest neighbour distance"], |
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113 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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114 | ["radius", "Ang", 40, [0, inf], "volume", "Particle radius"], |
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115 | ["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Particle scattering length density"], |
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116 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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117 | ["theta", "degrees", 60, [-360, 360], "orientation", "c axis to beam angle"], |
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118 | ["phi", "degrees", 60, [-360, 360], "orientation", "rotation about beam"], |
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119 | ["psi", "degrees", 60, [-360, 360], "orientation", "rotation about c axis"] |
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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/sas_3j1x_x.c", "lib/gauss150.c", "lib/sphere_form.c", "fcc_paracrystal.c"] |
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124 | |
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125 | def random(): |
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126 | import numpy as np |
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127 | # copied from bcc_paracrystal |
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128 | radius = 10**np.random.uniform(1.3, 4) |
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129 | d_factor = 10**np.random.uniform(-2, -0.7) # sigma_d in 0.01-0.7 |
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130 | dnn_fraction = np.random.beta(a=10, b=1) |
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131 | dnn = radius*4/np.sqrt(2)/dnn_fraction |
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132 | pars = dict( |
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133 | #sld=1, sld_solvent=0, scale=1, background=1e-32, |
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134 | dnn=dnn, |
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135 | d_factor=d_factor, |
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136 | radius=radius, |
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137 | ) |
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138 | return pars |
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139 | |
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140 | # april 10 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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141 | # TODO: fix the 2d tests |
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142 | q = 4.*pi/220. |
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143 | tests = [ |
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144 | [{}, [0.001, q, 0.215268], [0.275164706668, 5.7776842567, 0.00958167119232]], |
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145 | #[{}, (-0.047, -0.007), 238.103096286], |
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146 | #[{}, (0.053, 0.063), 0.863609587796], |
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147 | ] |
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