1 | #bcc 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 **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 returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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11 | |
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12 | Definition |
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13 | ---------- |
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14 | |
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15 | The scattering intensity $I(q)$ is calculated as |
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16 | |
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17 | .. math: |
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18 | |
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19 | I(q) = \frac{\text{scale}}{V_P} V_\text{lattice} P(q) Z(q) |
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20 | |
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21 | |
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22 | where *scale* is the volume fraction of spheres, *Vp* is the volume of the |
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23 | primary particle, *V(lattice)* is a volume correction for the crystal |
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24 | structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$ |
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25 | is the paracrystalline structure factor for a body-centered cubic structure. |
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26 | |
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27 | Equation (1) of the 1990 reference is used to calculate $Z(q)$, using |
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28 | equations (29)-(31) from the 1987 paper for *Z1*\ , *Z2*\ , and *Z3*\ . |
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29 | |
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30 | The lattice correction (the occupied volume of the lattice) for a |
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31 | body-centered cubic structure of particles of radius $R$ and nearest neighbor |
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32 | separation $D$ is |
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33 | |
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34 | .. math: |
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35 | |
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36 | V_\text{lattice} = \frac{16\pi}{3} \frac{R^3}{\left(D\sqrt{2}\right)^3} |
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37 | |
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38 | |
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39 | The distortion factor (one standard deviation) of the paracrystal is included |
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40 | in the calculation of $Z(q)$ |
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41 | |
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42 | .. math: |
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43 | |
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44 | \Delta a = g D |
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45 | |
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46 | where $g$ is a fractional distortion based on the nearest neighbor distance. |
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47 | |
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48 | The body-centered cubic lattice is |
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49 | |
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50 | .. image:: img/bcc_lattice.jpg |
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51 | |
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52 | For a crystal, diffraction peaks appear at reduced q-values given by |
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53 | |
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54 | .. math: |
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55 | |
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56 | \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2} |
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57 | |
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58 | where for a body-centered cubic lattice, only reflections where |
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59 | $(h + k + l) = \text{even}$ are allowed and reflections where |
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60 | $(h + k + l) = \text{odd}$ are forbidden. Thus the peak positions |
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61 | correspond to (just the first 5) |
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62 | |
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63 | .. math: |
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64 | |
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65 | \begin{eqnarray} |
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66 | &q/q_o&&\quad 1&& \ \sqrt{2} && \ \sqrt{3} && \ \sqrt{4} && \ \sqrt{5} \\ |
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67 | &\text{Indices}&& (110) && (200) && (211) && (220) && (310) |
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68 | \end{eqnarray} |
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69 | |
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70 | **NB: The calculation of $Z(q)$ is a double numerical integral that must |
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71 | be carried out with a high density of points to properly capture the sharp |
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72 | peaks of the paracrystalline scattering.** So be warned that the calculation |
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73 | is SLOW. Go get some coffee. Fitting of any experimental data must be |
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74 | resolution smeared for any meaningful fit. This makes a triple integral. |
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75 | Very, very slow. Go get lunch! |
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76 | |
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77 | This example dataset is produced using 200 data points, |
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78 | *qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above default values. |
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79 | |
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80 | .. image:: img/bcc_1d.jpg |
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81 | |
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82 | *Figure. 1D plot in the linear scale using the default values |
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83 | (w/200 data point).* |
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84 | |
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85 | The 2D (Anisotropic model) is based on the reference below where $I(q)$ is |
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86 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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87 | be accurate. Note that we are not responsible for any incorrectness of the 2D |
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88 | model computation. |
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89 | |
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90 | .. image:: img/bcc_orientation.gif |
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91 | |
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92 | .. image:: img/bcc_2d.jpg |
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93 | |
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94 | *Figure. 2D plot using the default values (w/200X200 pixels).* |
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95 | |
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96 | REFERENCE |
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97 | --------- |
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98 | |
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99 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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100 | (Original Paper) |
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101 | |
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102 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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103 | (Corrections to FCC and BCC lattice structure calculation) |
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104 | """ |
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105 | |
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106 | from numpy import inf |
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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 paracrystalline distortion. Thermal vibrations |
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112 | are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is |
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113 | assumed to be isotropic and characterized by a Gaussian distribution. |
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114 | """ |
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115 | category = "shape:paracrystal" |
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116 | |
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117 | # ["name", "units", default, [lower, upper], "type","description" ], |
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118 | parameters = [["dnn", "Ang", 220, [-inf, inf], "", "Nearest neighbour distance"], |
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119 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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120 | ["radius", "Ang", 40, [0, inf], "volume", "Particle radius"], |
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121 | ["sld", "1e-6/Ang^2", 4, [-inf, inf], "", "Particle scattering length density"], |
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122 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", "Solvent scattering length density"], |
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123 | ["theta", "degrees", 60, [-inf, inf], "orientation", "In plane angle"], |
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124 | ["phi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"], |
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125 | ["psi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"] |
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126 | ] |
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127 | |
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128 | source = ["lib/J1.c", "lib/gauss150.c", "bcc.c"] |
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129 | |
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130 | # parameters for demo |
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131 | demo = dict( |
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132 | scale=1, background=0, |
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133 | dnn=220, d_factor=0.06, sld=4, solvent_sld=1, |
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134 | radius=40, |
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135 | theta=60, phi=60, psi=60, |
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136 | radius_pd=.2, radius_pd_n=2, |
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137 | theta_pd=15, theta_pd_n=0, |
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138 | phi_pd=15, phi_pd_n=0, |
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139 | psi_pd=15, psi_pd_n=0, |
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140 | ) |
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141 | |
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142 | # For testing against the old sasview models, include the converted parameter |
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143 | # names and the target sasview model name. |
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144 | oldname = 'BCCrystalModel' |
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145 | oldpars = dict(sld='sldSph', solvent_sld='sldSolv') |
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