[421e55c] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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| 5 | The scattering intensity $I(q)$ is calculated as |
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| 6 | |
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| 7 | .. math:: |
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| 8 | |
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| 9 | I(q) = \frac{A}{1 +(Q\xi_1)^n} + \frac{C}{1 +(Q\xi_2)^m} + \text{B} |
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| 10 | |
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| 11 | where $A$ = Lorentzian scale factor #1, $C$ = Lorentzian scale #2, |
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[513efc5] | 12 | $\xi_1$ and $\xi_2$ are the corresponding correlation lengths, and $n$ and |
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| 13 | $m$ are the respective power law exponents (set $n = m = 2$ for |
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| 14 | Ornstein-Zernicke behaviour). |
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[421e55c] | 15 | |
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| 16 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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| 17 | where the $q$ vector is defined as |
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| 18 | |
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| 19 | .. math:: |
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| 20 | |
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| 21 | q = \sqrt{q_x^2 + q_y^2} |
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| 22 | |
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| 23 | |
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| 24 | References |
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| 25 | ---------- |
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| 26 | |
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| 27 | None. |
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| 28 | |
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[ef07e95] | 29 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 30 | * **Last Modified by:** Piotr rozyczko **Date:** January 29, 2016 |
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| 31 | * **Last Reviewed by:** Paul Butler **Date:** March 21, 2016 |
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[421e55c] | 32 | """ |
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| 33 | |
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[2d81cfe] | 34 | import numpy as np |
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[2c74c11] | 35 | from numpy import inf, power |
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[421e55c] | 36 | |
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| 37 | name = "two_lorentzian" |
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[caa7b4a] | 38 | title = "This model calculates an empirical functional form for SAS data \ |
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| 39 | characterized by two Lorentzian-type functions." |
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[421e55c] | 40 | description = """I(q) = scale_1/(1.0 + pow((q*length_1),exponent_1)) |
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| 41 | + scale_2/(1.0 + pow((q*length_2),exponent_2) )+ background |
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| 42 | |
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| 43 | scale_1 = Lorentzian term scaling #1 |
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| 44 | length_1 = Lorentzian screening length #1 [A] |
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| 45 | exponent_1 = Lorentzian exponent #1 |
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| 46 | scale_2 = Lorentzian term scaling #2 |
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| 47 | length_2 = Lorentzian screening length #2 [A] |
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| 48 | exponent_2 = Lorentzian exponent #2 |
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| 49 | background = Incoherent background |
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| 50 | """ |
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| 51 | category = "shape-independent" |
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| 52 | |
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[168052c] | 53 | # pylint: disable=bad-whitespace, line-too-long |
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[513efc5] | 54 | # ["name", "units", default, [lower, upper], "type", "description"], |
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[168052c] | 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 | # pylint: enable=bad-whitespace, line-too-long |
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[421e55c] | 63 | |
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| 64 | |
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[513efc5] | 65 | def Iq(q, |
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[168052c] | 66 | lorentz_scale_1=10.0, |
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| 67 | lorentz_length_1=100.0, |
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| 68 | lorentz_exp_1=3.0, |
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| 69 | lorentz_scale_2=1.0, |
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| 70 | lorentz_length_2=10.0, |
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| 71 | lorentz_exp_2=2.0): |
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[421e55c] | 72 | |
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| 73 | """ |
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| 74 | :param q: Input q-value (float or [float, float]) |
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| 75 | :param lorentz_scale_1: Second scale factor for broad Lorentzian peak |
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| 76 | :param lorentz_length_1: First Lorentzian screening length |
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| 77 | :param lorentz_exp_1: Exponent of the second Lorentz function |
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| 78 | :param lorentz_scale_2: Second scale factor for broad Lorentzian peak |
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| 79 | :param lorentz_length_2: Second Lorentzian screening length |
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| 80 | :param lorentz_exp_2: Exponent of the second Lorentz function |
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| 81 | :return: Calculated intensity |
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| 82 | """ |
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[168052c] | 83 | # pylint: disable=bad-whitespace |
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[513efc5] | 84 | intensity = lorentz_scale_1/(1.0 + |
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| 85 | power(q*lorentz_length_1, lorentz_exp_1)) |
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| 86 | intensity += lorentz_scale_2/(1.0 + |
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| 87 | power(q*lorentz_length_2, lorentz_exp_2)) |
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[168052c] | 88 | # pylint: enable=bad-whitespace |
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[421e55c] | 89 | |
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| 90 | return intensity |
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| 91 | |
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| 92 | Iq.vectorized = True # Iq accepts an array of q values |
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| 93 | |
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[48462b0] | 94 | def random(): |
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| 95 | scale = 10**np.random.uniform(0, 4, 2) |
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| 96 | length = 10**np.random.uniform(1, 4, 2) |
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| 97 | expon = np.random.uniform(1, 6, 2) |
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| 98 | |
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| 99 | pars = dict( |
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| 100 | #background=0, |
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| 101 | scale=1, # scale provided in model |
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| 102 | lorentz_scale_1=scale[0], |
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| 103 | lorentz_length_1=length[0], |
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| 104 | lorentz_exp_1=expon[0], |
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| 105 | lorentz_scale_2=scale[1], |
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| 106 | lorentz_length_2=length[1], |
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| 107 | lorentz_exp_2=expon[1], |
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| 108 | ) |
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| 109 | return pars |
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| 110 | |
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[421e55c] | 111 | |
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| 112 | demo = dict(scale=1, background=0.1, |
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[168052c] | 113 | lorentz_scale_1=10, |
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| 114 | lorentz_length_1=100.0, |
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| 115 | lorentz_exp_1=3.0, |
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| 116 | lorentz_scale_2=1, |
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| 117 | lorentz_length_2=10, |
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| 118 | lorentz_exp_2=2.0) |
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[421e55c] | 119 | |
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[07a6700] | 120 | tests = [ |
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[168052c] | 121 | |
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| 122 | # Accuracy tests based on content in test/utest_extra_models.py |
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| 123 | [{'lorentz_scale_1': 10.0, |
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| 124 | 'lorentz_length_1': 100.0, |
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| 125 | 'lorentz_exp_1': 3.0, |
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| 126 | 'lorentz_scale_2': 1.0, |
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| 127 | 'lorentz_length_2': 10.0, |
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| 128 | 'lorentz_exp_2': 2.0, |
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| 129 | 'background': 0.1, |
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| 130 | }, 0.001, 11.08991], |
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| 131 | |
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| 132 | [{'lorentz_scale_1': 10.0, |
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| 133 | 'lorentz_length_1': 100.0, |
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| 134 | 'lorentz_exp_1': 3.0, |
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| 135 | 'lorentz_scale_2': 1.0, |
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| 136 | 'lorentz_length_2': 10.0, |
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| 137 | 'lorentz_exp_2': 2.0, |
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| 138 | 'background': 0.1, |
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| 139 | }, 0.150141, 0.410245], |
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| 140 | |
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| 141 | [{'lorentz_scale_1': 10.0, |
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| 142 | 'lorentz_length_1': 100.0, |
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| 143 | 'lorentz_exp_1': 3.0, |
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| 144 | 'lorentz_scale_2': 1.0, |
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| 145 | 'lorentz_length_2': 10.0, |
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| 146 | 'lorentz_exp_2': 2.0, |
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| 147 | 'background': 0.1, |
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| 148 | }, 0.442528, 0.148699], |
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| 149 | |
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| 150 | # Additional tests with larger range of parameters |
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| 151 | [{'lorentz_scale_1': 10.0, |
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| 152 | 'lorentz_length_1': 100.0, |
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| 153 | 'lorentz_exp_1': 3.0, |
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| 154 | 'lorentz_scale_2': 1.0, |
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| 155 | 'lorentz_length_2': 10.0, |
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| 156 | 'lorentz_exp_2': 2.0, |
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| 157 | }, 0.000332070182643, 10.9996228107], |
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| 158 | |
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| 159 | [{'lorentz_scale_1': 0.0, |
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| 160 | 'lorentz_length_1': 0.0, |
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| 161 | 'lorentz_exp_1': 0.0, |
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| 162 | 'lorentz_scale_2': 0.0, |
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| 163 | 'lorentz_length_2': 0.0, |
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| 164 | 'lorentz_exp_2': 0.0, |
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| 165 | 'background': 100.0 |
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| 166 | }, 5.0, 100.0], |
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| 167 | |
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| 168 | [{'lorentz_scale_1': 200.0, |
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| 169 | 'lorentz_length_1': 10.0, |
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| 170 | 'lorentz_exp_1': 0.1, |
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| 171 | 'lorentz_scale_2': 0.1, |
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| 172 | 'lorentz_length_2': 5.0, |
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| 173 | 'lorentz_exp_2': 2.0 |
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| 174 | }, 20000., 45.5659201896], |
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| 175 | ] |
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