core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4
5This model calculates an empirical functional form for SAS data characterized
6by a broad scattering peak. Many SAS spectra are characterized by a broad peak
7even though they are from amorphous soft materials. For example, soft systems
8that show a SAS peak include copolymers, polyelectrolytes, multiphase systems,
9layered structures, etc.
10
11The d-spacing corresponding to the broad peak is a characteristic distance
12between the scattering inhomogeneities (such as in lamellar, cylindrical, or
13spherical morphologies, or for bicontinuous structures).
14
15The scattering intensity $I(q)$ is calculated as
16
17.. math:: I(q) = \frac{A}{q^n} + \frac{C}{1 + (|q - q_0|\xi)^m} + B
18
19Here the peak position is related to the d-spacing as $q_0 = 2\pi / d_0$.
20
21$A$ is the Porod law scale factor, $n$ the Porod exponent, $C$ is the
22Lorentzian scale factor, $m$ the exponent of $q$, $\xi$ the screening length,
23and $B$ the flat background.
24
25For 2D data the scattering intensity is calculated in the same way as 1D,
26where the $q$ vector is defined as
27
28.. math:: q = \sqrt{q_x^2 + q_y^2}
29
30References
31----------
32
33None.
34
35Authorship and Verification
36----------------------------
37
38* **Author:** NIST IGOR/DANSE **Date:** pre 2010
40* **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016
41"""
42
43import numpy as np
44from numpy import inf, errstate
45
47title = "Broad Lorentzian type peak on top of a power law decay"
48description = """\
49      I(q) = scale_p/pow(q,exponent)+scale_l/
50      (1.0 + pow((fabs(q-q_peak)*length_l),exponent_l) )+ background
51
52      List of default parameters:
53      porod_scale = Porod term scaling
54      porod_exp = Porod exponent
55      lorentz_scale = Lorentzian term scaling
56      lorentz_length = Lorentzian screening length [A]
57      peak_pos = peak location [1/A]
58      lorentz_exp = Lorentzian exponent
59      background = Incoherent background"""
60category = "shape-independent"
61
63#             ["name", "units", default, [lower, upper], "type", "description"],
64parameters = [["porod_scale",    "",  1.0e-05, [-inf, inf], "", "Power law scale factor"],
65              ["porod_exp",      "",      3.0, [-inf, inf], "", "Exponent of power law"],
66              ["lorentz_scale",  "",     10.0, [-inf, inf], "", "Scale factor for broad Lorentzian peak"],
67              ["lorentz_length", "Ang",  50.0, [-inf, inf], "", "Lorentzian screening length"],
68              ["peak_pos",       "1/Ang", 0.1, [-inf, inf], "", "Peak position in q"],
69              ["lorentz_exp",    "",      2.0, [-inf, inf], "", "Exponent of Lorentz function"],
70             ]
72
73def Iq(q,
74       porod_scale=1.0e-5,
75       porod_exp=3.0,
76       lorentz_scale=10.0,
77       lorentz_length=50.0,
78       peak_pos=0.1,
79       lorentz_exp=2.0):
80    """
81    :param q:              Input q-value
82    :param porod_scale:    Power law scale factor
83    :param porod_exp:      Exponent of power law
84    :param lorentz_scale:  Scale factor for broad Lorentzian peak
85    :param lorentz_length: Lorentzian screening length
86    :param peak_pos:       Peak position in q
87    :param lorentz_exp:    Exponent of Lorentz function
88    :return:               Calculated intensity
89    """
90    z = abs(q - peak_pos) * lorentz_length
91    with errstate(divide='ignore'):
92        inten = (porod_scale / q ** porod_exp
93                 + lorentz_scale / (1 + z ** lorentz_exp))
94    return inten
95Iq.vectorized = True  # Iq accepts an array of q values
96
97def random():
98    """Return a random parameter set for the model."""
99    pars = dict(
100        scale=1,
101        porod_scale=10**np.random.uniform(-8, -5),
102        porod_exp=np.random.uniform(1, 6),
103        lorentz_scale=10**np.random.uniform(0.3, 6),
104        lorentz_length=10**np.random.uniform(0, 2),
105        peak_pos=10**np.random.uniform(-3, -1),
106        lorentz_exp=np.random.uniform(1, 4),
107    )
108    pars['lorentz_length'] /= pars['peak_pos']
109    pars['lorentz_scale'] *= pars['porod_scale'] / pars['peak_pos']**pars['porod_exp']
110    #pars['porod_scale'] = 0.
111    return pars
112
113demo = dict(scale=1, background=0,
114            porod_scale=1.0e-05, porod_exp=3,
115            lorentz_scale=10, lorentz_length=50, peak_pos=0.1, lorentz_exp=2)
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