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 | """ |
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5 | Calculates the scattering from a **body-centered cubic lattice** with paracrystalline distortion. Thermal vibrations |
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6 | are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is |
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7 | assumed to be isotropic and characterized by a Gaussian distribution. |
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8 | |
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9 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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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 | .. image:: img/image167.jpg |
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17 | |
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18 | where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume |
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19 | correction for the crystal structure, *P(q)* is the form factor of the sphere (normalized), and *Z(q)* is the |
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20 | 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 is used to calculate *Z(q)*, using equations (29)-(31) from the 1987 paper for |
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23 | *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 body-centered cubic structure of particles of radius |
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26 | *R* and nearest neighbor separation *D* is |
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27 | |
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28 | .. image:: img/image159.jpg |
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29 | |
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30 | The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* |
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31 | |
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32 | .. image:: img/image160.jpg |
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33 | |
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34 | where *g* is a fractional distortion based on the nearest neighbor distance. |
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35 | |
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36 | The body-centered cubic lattice is |
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37 | |
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38 | .. image:: img/image168.jpg |
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39 | |
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40 | For a crystal, diffraction peaks appear at reduced q-values given by |
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41 | |
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42 | .. image:: img/image162.jpg |
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43 | |
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44 | where for a body-centered cubic lattice, only reflections where (\ *h* + *k* + *l*\ ) = even are allowed and |
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45 | reflections where (\ *h* + *k* + *l*\ ) = odd are forbidden. Thus the peak positions correspond to (just the first 5) |
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46 | |
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47 | .. image:: img/image169.jpg |
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48 | |
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49 | **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** |
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50 | **points to properly capture the sharp peaks of the paracrystalline scattering.** So be warned that the calculation is |
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51 | SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This |
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52 | makes a triple integral. Very, very slow. Go get lunch! |
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53 | |
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54 | This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above |
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55 | default values. |
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56 | |
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57 | .. image:: img/image170.jpg |
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58 | |
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59 | *Figure. 1D plot in the linear scale using the default values (w/200 data point).* |
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60 | |
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61 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is approximated for 1d scattering. Thus the |
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62 | scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model |
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63 | computation. |
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64 | |
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65 | .. image:: img/image165.gif |
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66 | |
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67 | .. image:: img/image171.jpg |
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68 | |
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69 | *Figure. 2D plot using the default values (w/200X200 pixels).* |
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70 | |
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71 | REFERENCE |
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72 | |
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73 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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74 | (Original Paper) |
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75 | |
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76 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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77 | (Corrections to FCC and BCC lattice structure calculation) |
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78 | """ |
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79 | |
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80 | from numpy import pi, inf |
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81 | |
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82 | name = "BCCparacrystal" |
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83 | title = "Body-centred cubic lattic with paracrystalline distortion" |
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84 | description = """ |
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85 | Calculates the scattering from a **body-centered cubic lattice** with paracrystalline distortion. Thermal vibrations |
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86 | are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is |
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87 | assumed to be isotropic and characterized by a Gaussian distribution. |
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88 | """ |
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89 | |
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90 | parameters = [ |
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91 | # [ "name", "units", default, [lower, upper], "type","description" ], |
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92 | [ "dnn", "Ang", 220, [-inf,inf],"","Nearest neighbour distance"], |
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93 | [ "d_factor", "", 0.06,[-inf,inf],"","Paracrystal distortion factor" ], |
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94 | [ "radius", "Ang", 40, [0, inf], "volume","Particle radius" ], |
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95 | [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "", "Particle scattering length density" ], |
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96 | [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "","Solvent scattering length density" ], |
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97 | [ "theta", "degrees", 60, [-inf, inf], "orientation","In plane angle" ], |
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98 | [ "phi", "degrees", 60, [-inf, inf], "orientation","Out of plane angle" ], |
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99 | [ "psi", "degrees", 60, [-inf,inf], "orientation","Out of plane angle"] |
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100 | ] |
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101 | |
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102 | source = [ "lib/J1.c", "lib/gauss150.c", "bcc.c" ] |
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103 | |
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104 | def ER(radius, length): |
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105 | return 0 |
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106 | |
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107 | # parameters for demo |
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108 | demo = dict( |
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109 | scale=1, background=0, |
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110 | dnn=220, d_factor=0.06, sld=4, solvent_sld=1, |
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111 | radius=40, |
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112 | theta=60, phi=60, psi=60, |
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113 | radius_pd=.2, radius_pd_n=0.2, |
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114 | theta_pd=15, theta_pd_n=0, |
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115 | phi_pd=15, phi_pd_n=0, |
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116 | psi_pd=15, psi_pd_n=0, |
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117 | ) |
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118 | |
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119 | # For testing against the old sasview models, include the converted parameter |
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120 | # names and the target sasview model name. |
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121 | oldname='BCCrystalModel' |
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122 | oldpars=dict(sld='sldSph', |
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123 | solvent_sld='sldSolv') |
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