# source:sasmodels/sasmodels/models/two_power_law.py@2d81cfe

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Last change on this file since 2d81cfe was 2d81cfe, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

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1r"""
2Definition
3----------
4
5The scattering intensity $I(q)$ is calculated as
6
7.. math::
8
9    I(q) = \begin{cases}
10    A q^{-m1} + \text{background} & q <= q_c \\
11    C q^{-m2} + \text{background} & q > q_c
12    \end{cases}
13
14where $q_c$ = the location of the crossover from one slope to the other,
15$A$ = the scaling coefficent that sets the overall intensity of the lower Q
16power law region, $m1$ = power law exponent at low Q, and $m2$ = power law
17exponent at high Q.  The scaling of the second power law region (coefficent C)
18is then automatically scaled to match the first by following formula:
19
20.. math::
21    C = \frac{A q_c^{m2}}{q_c^{m1}}
22
23.. note::
24    Be sure to enter the power law exponents as positive values!
25
26For 2D data the scattering intensity is calculated in the same way as 1D,
27where the $q$ vector is defined as
28
29.. math::
30
31    q = \sqrt{q_x^2 + q_y^2}
32
33
34References
35----------
36
37None.
38
39**Author:** NIST IGOR/DANSE **on:** pre 2010
40
42
43**Last Reviewed by:** Paul Butler **on:** March 21, 2016
44"""
45
46import numpy as np
47from numpy import inf, power, empty, errstate
48
49name = "two_power_law"
50title = "This model calculates an empirical functional form for SAS data \
51characterized by two power laws."
52description = """
53            I(q) = coef_A*pow(qval,-1.0*power1) + background for q<=q_c
54            =C*pow(qval,-1.0*power2) + background for q>q_c
55            where C=coef_A*pow(q_c,-1.0*power1)/pow(q_c,-1.0*power2).
56
57            coef_A = scaling coefficent
58            q_c = crossover location [1/A]
59            power_1 (=m1) = power law exponent at low Q
60            power_2 (=m2) = power law exponent at high Q
61            background = Incoherent background [1/cm]
62        """
63category = "shape-independent"
64
66#   ["name", "units", default, [lower, upper], "type", "description"],
67parameters = [
68    ["coefficent_1", "",       1.0, [-inf, inf], "", "coefficent A in low Q region"],
69    ["crossover",    "1/Ang",  0.04,[0, inf],    "", "crossover location"],
70    ["power_1",      "",       1.0, [0, inf],    "", "power law exponent at low Q"],
71    ["power_2",      "",       4.0, [0, inf],    "", "power law exponent at high Q"],
72    ]
74
75
76def Iq(q,
77       coefficent_1=1.0,
78       crossover=0.04,
79       power_1=1.0,
80       power_2=4.0,
81      ):
82
83    """
84    :param q:                   Input q-value (float or [float, float])
85    :param coefficent_1:        Scaling coefficent at low Q
86    :param crossover:           Crossover location
87    :param power_1:             Exponent of power law function at low Q
88    :param power_2:             Exponent of power law function at high Q
89    :return:                    Calculated intensity
90    """
91    result = empty(q.shape, 'd')
92    index = (q <= crossover)
93    with errstate(divide='ignore'):
94        coefficent_2 = coefficent_1 * power(crossover, power_2 - power_1)
95        result[index] = coefficent_1 * power(q[index], -power_1)
96        result[~index] = coefficent_2 * power(q[~index], -power_2)
97    return result
98
99Iq.vectorized = True  # Iq accepts an array of q values
100
101def random():
102    coefficient_1 = 1
103    crossover = 10**np.random.uniform(-3, -1)
104    power_1 = np.random.uniform(1, 6)
105    power_2 = np.random.uniform(1, 6)
106    pars = dict(
107        scale=1, #background=0,
108        coefficient_1=coefficient_1,
109        crossover=crossover,
110        power_1=power_1,
111        power_2=power_2,
112    )
113    return pars
114
115demo = dict(scale=1, background=0.0,
116            coefficent_1=1.0,
117            crossover=0.04,
118            power_1=1.0,
119            power_2=4.0)
120
121tests = [
122    # Accuracy tests based on content in test/utest_extra_models.py
123    [{'coefficent_1':     1.0,
124      'crossover':  0.04,
125      'power_1':    1.0,
126      'power_2':    4.0,
127      'background': 0.0,
128     }, 0.001, 1000],
129
130    [{'coefficent_1':     1.0,
131      'crossover':  0.04,
132      'power_1':    1.0,
133      'power_2':    4.0,
134      'background': 0.0,
135     }, 0.150141, 0.125945],
136
137    [{'coefficent_1':    1.0,
138      'crossover':  0.04,
139      'power_1':    1.0,
140      'power_2':    4.0,
141      'background': 0.0,
142     }, 0.442528, 0.00166884],
143
144    [{'coefficent_1':    1.0,
145      'crossover':  0.04,
146      'power_1':    1.0,
147      'power_2':    4.0,
148      'background': 0.0,
149     }, (0.442528, 0.00166884), 0.00166884],
150
151]
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