source: sasmodels/sasmodels/models/two_power_law.py @ 6cefbc9

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Last change on this file since 6cefbc9 was 40a87fa, checked in by Paul Kienzle <pkienzle@…>, 8 years ago

lint and latex cleanup

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Line 
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
41**Last Modified by:** Wojciech Wpotrzebowski **on:** February 18, 2016
42
43**Last Reviewed by:** Paul Butler **on:** March 21, 2016
44"""
45
46from numpy import inf, power, empty, errstate
47
48name = "two_power_law"
49title = "This model calculates an empirical functional form for SAS data \
50characterized by two power laws."
51description = """
52            I(q) = coef_A*pow(qval,-1.0*power1) + background for q<=q_c
53            =C*pow(qval,-1.0*power2) + background for q>q_c
54            where C=coef_A*pow(q_c,-1.0*power1)/pow(q_c,-1.0*power2).
55
56            coef_A = scaling coefficent
57            q_c = crossover location [1/A]
58            power_1 (=m1) = power law exponent at low Q
59            power_2 (=m2) = power law exponent at high Q
60            background = Incoherent background [1/cm]
61        """
62category = "shape-independent"
63
64# pylint: disable=bad-whitespace, line-too-long
65#   ["name", "units", default, [lower, upper], "type", "description"],
66parameters = [
67    ["coefficent_1", "",       1.0, [-inf, inf], "", "coefficent A in low Q region"],
68    ["crossover",    "1/Ang",  0.04,[0, inf],    "", "crossover location"],
69    ["power_1",      "",       1.0, [0, inf],    "", "power law exponent at low Q"],
70    ["power_2",      "",       4.0, [0, inf],    "", "power law exponent at high Q"],
71    ]
72# pylint: enable=bad-whitespace, line-too-long
73
74
75def Iq(q,
76       coefficent_1=1.0,
77       crossover=0.04,
78       power_1=1.0,
79       power_2=4.0,
80      ):
81
82    """
83    :param q:                   Input q-value (float or [float, float])
84    :param coefficent_1:        Scaling coefficent at low Q
85    :param crossover:           Crossover location
86    :param power_1:             Exponent of power law function at low Q
87    :param power_2:             Exponent of power law function at high Q
88    :return:                    Calculated intensity
89    """
90    result= empty(q.shape, 'd')
91    index = (q <= crossover)
92    with errstate(divide='ignore'):
93        coefficent_2 = coefficent_1 * power(crossover, power_2 - power_1)
94        result[index] = coefficent_1 * power(q[index], -power_1)
95        result[~index] = coefficent_2 * power(q[~index], -power_2)
96    return result
97
98Iq.vectorized = True  # Iq accepts an array of q values
99
100demo = dict(scale=1, background=0.0,
101            coefficent_1=1.0,
102            crossover=0.04,
103            power_1=1.0,
104            power_2=4.0)
105
106tests = [
107    # Accuracy tests based on content in test/utest_extra_models.py
108    [{'coefficent_1':     1.0,
109      'crossover':  0.04,
110      'power_1':    1.0,
111      'power_2':    4.0,
112      'background': 0.0,
113     }, 0.001, 1000],
114
115    [{'coefficent_1':     1.0,
116      'crossover':  0.04,
117      'power_1':    1.0,
118      'power_2':    4.0,
119      'background': 0.0,
120     }, 0.150141, 0.125945],
121
122    [{'coeffcent_1':    1.0,
123      'crossover':  0.04,
124      'power_1':    1.0,
125      'power_2':    4.0,
126      'background': 0.0,
127     }, 0.442528, 0.00166884],
128
129    [{'coeffcent_1':    1.0,
130      'crossover':  0.04,
131      'power_1':    1.0,
132      'power_2':    4.0,
133      'background': 0.0,
134     }, (0.442528, 0.00166884), 0.00166884],
135
136]
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