source: sasmodels/sasmodels/models/lamellar_hg_stack_caille.py @ 71b751d

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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
124# ER defaults to 0.0
125# VR defaults to 1.0
126
127def random():
128    total_thickness = 10**np.random.uniform(2, 4.7)
129    Nlayers = np.random.randint(2, 200)
130    d_spacing = total_thickness / Nlayers
131    thickness = d_spacing * np.random.uniform(0, 1)
132    length_head = thickness * np.random.uniform(0, 1)
133    length_tail = thickness - length_head
134    Caille_parameter = np.random.uniform(0, 0.8)
135    pars = dict(
136        length_head=length_head,
137        length_tail=length_tail,
138        Nlayers=Nlayers,
139        d_spacing=d_spacing,
140        Caille_parameter=Caille_parameter,
141    )
142    return pars
143
144demo = dict(
145    scale=1, background=0,
146    Nlayers=20, d_spacing=200., Caille_parameter=0.05,
147    length_tail=15, length_head=10,
148    #sld=-1, sld_head=4.0, sld_solvent=6.0,
149    sld=-1, sld_head=4.1, sld_solvent=6.0,
150    length_tail_pd=0.1, length_tail_pd_n=20,
151    length_head_pd=0.05, length_head_pd_n=30,
152    d_spacing_pd=0.2, d_spacing_pd_n=40,
153    )
154
155#
156tests = [[{'scale': 1.0, 'background': 0.0, 'length_tail': 10.0, 'length_head': 2.0,
157           'Nlayers': 30.0, 'd_spacing': 40., 'Caille_parameter': 0.001, 'sld': 0.4,
158           'sld_head': 2.0, 'sld_solvent': 6.0, 'length_tail_pd': 0.0,
159           'length_head_pd': 0.0, 'd_spacing_pd': 0.0}, [0.001], [6838238.571488]]]
160# ADDED by: RKH  ON: 18Mar2016  converted from sasview previously, now renaming everything & sorting the docs
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