source: sasmodels/sasmodels/models/lamellar_hg_stack_caille.py @ 99658f6

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Last change on this file since 99658f6 was 304c775, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

provide method for testing Fq results. Refs #1202.

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1# Note: model title and parameter table are inserted automatically
2r"""
3This model provides the scattering intensity, $I(q) = P(q)S(q)$, for a lamellar
4phase where a random distribution in solution are assumed. Here a Caille $S(q)$
5is used for the lamellar stacks.
6
7The scattering intensity $I(q)$ is
8
9.. math::
10
11    I(q) = 2 \pi \frac{P(q)S(q)}{q^2\delta }
12
13
14The form factor $P(q)$ is
15
16.. math::
17
18        P(q) = \frac{4}{q^2}\big\{
19        \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right]
20            + \Delta\rho_T\sin(q\delta_T)\big\}^2
21
22and the structure factor $S(q)$ is
23
24.. math::
25
26    S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right)
27        \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right)
28
29where
30
31.. math::
32    :nowrap:
33
34    \begin{align*}
35    \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right)
36              &&  \\
37    \gamma_E  &= 0.5772156649
38              && \text{Euler's constant} \\
39    \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}}
40              && \text{Caille constant}
41    \end{align*}
42
43
44$\delta_T$ is the tail length (or *length_tail*), $\delta_H$ is the head
45thickness (or *length_head*), $\Delta\rho_H$ is SLD(headgroup) - SLD(solvent),
46and $\Delta\rho_T$ is SLD(tail) - SLD(headgroup). Here $d$ is (repeat) spacing,
47$K$ is smectic bending elasticity, $B$ is compression modulus, and $N$ is the
48number of lamellar plates (*Nlayers*).
49
50NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the
51assumptions of the model are incorrect.**  And due to a complication of the
52model function, users are responsible for making sure that all the assumptions
53are handled accurately (see the original reference below for more details).
54
55Non-integer numbers of stacks are calculated as a linear combination of
56results for the next lower and higher values.
57
58Be aware that the computations may be very slow.
59
60The 2D scattering intensity is calculated in the same way as 1D, where
61the $q$ vector is defined as
62
63.. math::
64
65    q = \sqrt{q_x^2 + q_y^2}
66
67
68References
69----------
70
71F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502
72
73also in J. Phys. Chem. B, 105, (2001) 11081-11088
74"""
75
76import numpy as np
77from numpy import inf
78
79name = "lamellar_hg_stack_caille"
80title = "Random lamellar head/tail/tail/head sheet with Caille structure factor"
81description = """\
82    [Random lamellar phase with Caille  structure factor]
83        randomly oriented stacks of infinite sheets
84        with Caille S(Q), having polydisperse spacing.
85        layer thickness =(H+T+T+H) = 2(Head+Tail)
86        sld = Tail scattering length density
87        sld_head = Head scattering length density
88        sld_solvent = solvent scattering length density
89        background = incoherent background
90        scale = scale factor
91"""
92category = "shape:lamellae"
93
94single = False  # TODO: check
95parameters = [
96    #   [ "name", "units", default, [lower, upper], "type",
97    #     "description" ],
98    ["length_tail", "Ang", 10, [0, inf], "volume",
99     "Tail thickness"],
100    ["length_head", "Ang", 2, [0, inf], "volume",
101     "head thickness"],
102    ["Nlayers", "", 30, [1, inf], "",
103     "Number of layers"],
104    ["d_spacing", "Ang", 40., [0.0, inf], "volume",
105     "lamellar d-spacing of Caille S(Q)"],
106    ["Caille_parameter", "", 0.001, [0.0, 0.8], "",
107     "Caille parameter"],
108    ["sld", "1e-6/Ang^2", 0.4, [-inf, inf], "sld",
109     "Tail scattering length density"],
110    ["sld_head", "1e-6/Ang^2", 2.0, [-inf, inf], "sld",
111     "Head scattering length density"],
112    ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
113     "Solvent scattering length density"],
114    ]
115
116source = ["lamellar_hg_stack_caille.c"]
117
118# No volume normalization despite having a volume parameter
119# This should perhaps be volume normalized?
120form_volume = """
121    return 1.0;
122    """
123
124def random():
125    total_thickness = 10**np.random.uniform(2, 4.7)
126    Nlayers = np.random.randint(2, 200)
127    d_spacing = total_thickness / Nlayers
128    thickness = d_spacing * np.random.uniform(0, 1)
129    length_head = thickness * np.random.uniform(0, 1)
130    length_tail = thickness - length_head
131    Caille_parameter = np.random.uniform(0, 0.8)
132    pars = dict(
133        length_head=length_head,
134        length_tail=length_tail,
135        Nlayers=Nlayers,
136        d_spacing=d_spacing,
137        Caille_parameter=Caille_parameter,
138    )
139    return pars
140
141demo = dict(
142    scale=1, background=0,
143    Nlayers=20, d_spacing=200., Caille_parameter=0.05,
144    length_tail=15, length_head=10,
145    #sld=-1, sld_head=4.0, sld_solvent=6.0,
146    sld=-1, sld_head=4.1, sld_solvent=6.0,
147    length_tail_pd=0.1, length_tail_pd_n=20,
148    length_head_pd=0.05, length_head_pd_n=30,
149    d_spacing_pd=0.2, d_spacing_pd_n=40,
150    )
151
152#
153tests = [[{'scale': 1.0, 'background': 0.0, 'length_tail': 10.0, 'length_head': 2.0,
154           'Nlayers': 30.0, 'd_spacing': 40., 'Caille_parameter': 0.001, 'sld': 0.4,
155           'sld_head': 2.0, 'sld_solvent': 6.0, 'length_tail_pd': 0.0,
156           'length_head_pd': 0.0, 'd_spacing_pd': 0.0}, [0.001], [6838238.571488]]]
157# ADDED by: RKH  ON: 18Mar2016  converted from sasview previously, now renaming everything & sorting the docs
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