1 | r""" |
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2 | This model calculates an empirical functional form for SAS data characterized by two Lorentzian-type functions. |
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3 | |
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4 | Definition |
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5 | ---------- |
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6 | |
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7 | The scattering intensity $I(q)$ is calculated as |
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8 | |
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9 | .. math:: |
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10 | |
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11 | I(q) = \frac{A}{1 +(Q\xi_1)^n} + \frac{C}{1 +(Q\xi_2)^m} + \text{B} |
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12 | |
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13 | where $A$ = Lorentzian scale factor #1, $C$ = Lorentzian scale #2, |
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14 | $\xi_1$ and $\xi_2$ are the corresponding correlation lengths, and $n$ and $m$ are the respective |
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15 | power law exponents (set $n = m = 2$ for Ornstein-Zernicke behaviour). |
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16 | |
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17 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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18 | where the $q$ vector is defined as |
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19 | |
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20 | .. math:: |
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21 | |
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22 | q = \sqrt{q_x^2 + q_y^2} |
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23 | |
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24 | |
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25 | .. figure:: img/two_lorentzian.jpg |
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26 | |
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27 | 1D plot using the default values (w/500 data point). |
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28 | |
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29 | References |
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30 | ---------- |
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31 | |
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32 | None. |
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33 | |
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34 | """ |
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35 | |
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36 | from math import sqrt |
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37 | from numpy import inf, power |
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38 | |
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39 | name = "two_lorentzian" |
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40 | title = "Two Lorentzian type peak" |
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41 | description = """I(q) = scale_1/(1.0 + pow((q*length_1),exponent_1)) |
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42 | + scale_2/(1.0 + pow((q*length_2),exponent_2) )+ background |
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43 | |
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44 | scale_1 = Lorentzian term scaling #1 |
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45 | length_1 = Lorentzian screening length #1 [A] |
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46 | exponent_1 = Lorentzian exponent #1 |
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47 | scale_2 = Lorentzian term scaling #2 |
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48 | length_2 = Lorentzian screening length #2 [A] |
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49 | exponent_2 = Lorentzian exponent #2 |
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50 | background = Incoherent background |
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51 | """ |
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52 | category = "shape-independent" |
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53 | |
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54 | # ["name", "units", default, [lower, upper], "type", "description"], |
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55 | parameters = [["lorentz_scale_1", "", 10.0, [-inf, inf], "", "First power law scale factor"], |
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56 | ["lorentz_length_1", "Ang", 100.0, [-inf, inf], "", "First Lorentzian screening length"], |
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57 | ["lorentz_exp_1", "", 3.0, [-inf, inf], "", "First exponent of power law"], |
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58 | ["lorentz_scale_2", "", 1.0, [-inf, inf], "", "Second scale factor for broad Lorentzian peak"], |
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59 | ["lorentz_length_2", "Ang", 10.0, [-inf, inf], "", "Second Lorentzian screening length"], |
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60 | ["lorentz_exp_2", "", 2.0, [-inf, inf], "", "Second exponent of power law"], |
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61 | ] |
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62 | |
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63 | |
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64 | def Iq(q, lorentz_scale_1, lorentz_length_1, lorentz_exp_1, lorentz_scale_2, lorentz_length_2, lorentz_exp_2): |
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65 | |
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66 | """ |
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67 | :param q: Input q-value (float or [float, float]) |
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68 | :param lorentz_scale_1: Second scale factor for broad Lorentzian peak |
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69 | :param lorentz_length_1: First Lorentzian screening length |
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70 | :param lorentz_exp_1: Exponent of the second Lorentz function |
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71 | :param lorentz_scale_2: Second scale factor for broad Lorentzian peak |
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72 | :param lorentz_length_2: Second Lorentzian screening length |
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73 | :param lorentz_exp_2: Exponent of the second Lorentz function |
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74 | :return: Calculated intensity |
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75 | """ |
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76 | |
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77 | intensity = lorentz_scale_1/(1.0 + power(q*lorentz_length_1, lorentz_exp_1)) |
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78 | intensity += lorentz_scale_2/(1.0 + power(q*lorentz_length_2, lorentz_exp_2)) |
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79 | |
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80 | return intensity |
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81 | |
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82 | Iq.vectorized = True # Iq accepts an array of q values |
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83 | |
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84 | |
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85 | def Iqxy(qx, qy, *args): |
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86 | iq = Iq(sqrt(qx**2 + qy**2), *args) |
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87 | |
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88 | return iq |
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89 | |
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90 | Iqxy.vectorized = True # Iqxy accepts an array of qx, qy values |
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91 | |
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92 | |
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93 | demo = dict(scale=1, background=0.1, |
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94 | lorentz_scale_1=10, lorentz_length_1=100.0, lorentz_exp_1=3.0, |
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95 | lorentz_scale_2=1, lorentz_length_2=10, lorentz_exp_2=2.0) |
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96 | |
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97 | oldname = "TwoLorentzianModel" |
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98 | oldpars = dict(background='background', |
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99 | lorentz_scale_1='scale_1', lorentz_scale_2='scale_2', |
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100 | lorentz_length_1='length_1', lorentz_length_2='length_2', |
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101 | lorentz_exp_1='exponent_1', lorentz_exp_2='exponent_2') |
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102 | |
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103 | tests = [[{'lorentz_scale_1': 10, 'lorentz_length_1': 100.0, 'lorentz_exp_1' : 3.0, |
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104 | 'lorentz_scale_2': 1, 'lorentz_length_2': 10.0, 'lorentz_exp_2' : 2.0, |
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105 | }, 0.000332070182643, 10.9996228107], |
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106 | |
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107 | [{'lorentz_scale_1': 0, 'lorentz_length_1': 0.0, 'lorentz_exp_1' : 0.0, |
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108 | 'lorentz_scale_2': 0, 'lorentz_length_2': 0.0, 'lorentz_exp_2' : 0.0, |
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109 | 'background':100. |
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110 | }, 5.0, 100.0], |
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111 | |
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112 | [{'lorentz_scale_1': 200, 'lorentz_length_1': 10.0, 'lorentz_exp_1' : 0.1, |
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113 | 'lorentz_scale_2': 0.1, 'lorentz_length_2': 5.0, 'lorentz_exp_2' : 2.0 |
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114 | }, 20000., 45.5659201896], |
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115 | ] |
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