source: sasmodels/sasmodels/models/polymer_excl_volume.py @ 58c3367

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Last change on this file since 58c3367 was 40a87fa, checked in by Paul Kienzle <pkienzle@…>, 8 years ago

lint and latex cleanup

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1r"""
2This model describes the scattering from polymer chains subject to excluded
3volume effects and has been used as a template for describing mass fractals.
4
5Definition
6----------
7
8The form factor was originally presented in the following integral form
9(Benoit, 1957)
10
11.. math::
12
13    P(Q)=2\int_0^{1}dx(1-x)exp\left[-\frac{Q^2a^2}{6}n^{2v}x^{2v}\right]
14
15where $\nu$ is the excluded volume parameter
16(which is related to the Porod exponent $m$ as $\nu=1/m$ ),
17$a$ is the statistical segment length of the polymer chain,
18and $n$ is the degree of polymerization.
19This integral was later put into an almost analytical form as follows
20(Hammouda, 1993)
21
22.. math::
23
24    P(Q)=\frac{1}{\nu U^{1/2\nu}}\gamma\left(\frac{1}{2\nu},U\right) -
25    \frac{1}{\nu U^{1/\nu}}\gamma\left(\frac{1}{\nu},U\right)
26
27where $\gamma(x,U)$ is the incomplete gamma function
28
29.. math::
30
31    \gamma(x,U)=\int_0^{U}dt\ exp(-t)t^{x-1}
32
33and the variable $U$ is given in terms of the scattering vector $Q$ as
34
35.. math::
36
37    U=\frac{Q^2a^2n^{2\nu}}{6} = \frac{Q^2R_{g}^2(2\nu+1)(2\nu+2)}{6}
38
39The square of the radius-of-gyration is defined as
40
41.. math::
42
43    R_{g}^2 = \frac{a^2n^{2\nu}}{(2\nu+1)(2\nu+2)}
44
45Note that this model applies only in the mass fractal range (ie, $5/3<=m<=3$ )
46and **does not apply** to surface fractals ( $3<m<=4$ ).
47It also does not reproduce the rigid rod limit (m=1) because it assumes chain
48flexibility from the outset. It may cover a portion of the semi-flexible chain
49range ( $1<m<5/3$ ).
50
51A low-Q expansion yields the Guinier form and a high-Q expansion yields the
52Porod form which is given by
53
54.. math::
55
56    P(Q\rightarrow \infty) = \frac{1}{\nu U^{1/2\nu}}\Gamma\left(
57    \frac{1}{2\nu}\right) - \frac{1}{\nu U^{1/\nu}}\Gamma\left(
58    \frac{1}{\nu}\right)
59
60Here $\Gamma(x) = \gamma(x,\infty)$ is the gamma function.
61
62The asymptotic limit is dominated by the first term
63
64.. math::
65
66    P(Q\rightarrow \infty) \sim \frac{1}{\nu U^{1/2\nu}}\Gamma\left(\frac{1}{2\nu}\right) =
67    \frac{m}{\left(QR_{g}\right)^m}\left[\frac{6}{(2\nu +1)(2\nu +2)} \right]^{m/2}
68    \Gamma (m/2)
69
70The special case when $\nu=0.5$ (or $m=1/\nu=2$ ) corresponds to Gaussian chains for
71which the form factor is given by the familiar Debye function.
72
73.. math::
74
75    P(Q) = \frac{2}{Q^4R_{g}^4} \left[exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right]
76
77For 2D data: The 2D scattering intensity is calculated in the same way as 1D,
78where the $q$ vector is defined as
79
80.. math::
81
82    q = \sqrt{q_x^2 + q_y^2}
83
84
85References
86----------
87
88H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247
89
90B Hammouda, *SANS from Homogeneous Polymer Mixtures - A Unified Overview,
91Advances in Polym. Sci.* 106(1993) 87-133
92
93"""
94
95from numpy import inf, power, errstate
96from scipy.special import gammainc, gamma
97
98name = "polymer_excl_volume"
99title = "Polymer Excluded Volume model"
100description = """Compute the scattering intensity from polymers with excluded
101                volume effects.
102                rg:         radius of gyration
103                porod_exp:  Porod exponent
104              """
105category = "shape-independent"
106
107# pylint: disable=bad-whitespace, line-too-long
108#   ["name", "units", default, [lower, upper], "type", "description"],
109parameters = [
110    ["rg",        "Ang", 60.0, [0, inf],    "", "Radius of Gyration"],
111    ["porod_exp", "",     3.0, [-inf, inf], "", "Porod exponent"],
112]
113# pylint: enable=bad-whitespace, line-too-long
114
115
116def Iq(q, rg=60.0, porod_exp=3.0):
117    """
118    :param q:         Input q-value (float or [float, float])
119    :param rg:        Radius of gyration
120    :param porod_exp: Porod exponent
121    :return:          Calculated intensity
122    """
123    usub = (q*rg)**2 * (2.0/porod_exp + 1.0) * (2.0/porod_exp + 2.0)/6.0
124    with errstate(divide='ignore', invalid='ignore'):
125        upow = power(usub, -0.5*porod_exp)
126        result= (porod_exp*upow *
127                 (gamma(0.5*porod_exp)*gammainc(0.5*porod_exp, usub) -
128                  upow*gamma(porod_exp)*gammainc(porod_exp, usub)))
129    result[q <= 0] = 1.0
130
131    return result
132
133Iq.vectorized = True  # Iq accepts an array of q values
134
135tests = [
136    # Accuracy tests based on content in test/polyexclvol_default_igor.txt
137    [{'rg': 60, 'porod_exp': 3.0}, 0.001, 0.999801],
138    [{'rg': 60, 'porod_exp': 3.0}, 0.105363, 0.0172751],
139    [{'rg': 60, 'porod_exp': 3.0, 'background': 0.0}, 0.665075, 6.56261e-05],
140
141    # Additional tests with larger range of parameters
142    [{'rg': 10, 'porod_exp': 4.0}, 0.1, 0.724436675809],
143    [{'rg': 2.2, 'porod_exp': 22.0, 'background': 100.0}, 5.0, 100.0],
144    [{'rg': 1.1, 'porod_exp': 1, 'background': 10.0, 'scale': 1.25},
145     20000., 10.0000712097]
146    ]
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