# source:sasmodels/sasmodels/models/lamellarCailleHG.py@652a78a

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Last change on this file since 652a78a was 652a78a, checked in by mathieu, 8 years ago

Fix pylint

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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)}{\delta q^2}
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 *tail_length*), $\delta_H$ is the head
45thickness (or *head_length*), $\Delta\rho_H$ 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
58The 2D scattering intensity is calculated in the same way as 1D, where
59the $q$ vector is defined as
60
61.. math::
62
63    q = \sqrt{q_x^2 + q_y^2}
64
65.. figure:: img/lamellarCailleHG_1d.jpg
66
67    1D plot using the default values (w/6000 data point).
68
69References
70----------
71
72F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502
73
74also in J. Phys. Chem. B, 105, (2001) 11081-11088
75"""
76from numpy import inf
77
78name = "lamellarCailleHG"
79title = "Random lamellar sheet with Caille structure factor"
80description = """\
81    [Random lamellar phase with Caille  structure factor]
82        randomly oriented stacks of infinite sheets
83        with Caille S(Q), having polydisperse spacing.
84        layer thickness =(H+T+T+H) = 2(Head+Tail)
85        sld = Tail scattering length density
86        sld_head = Head scattering length density
87        sld_solvent = solvent scattering length density
88        background = incoherent background
89        scale = scale factor
90"""
91category = "shape:lamellae"
92
93single = False
94parameters = [
95    #   [ "name", "units", default, [lower, upper], "type",
96    #     "description" ],
97    ["tail_length", "Ang", 10, [0, inf], "volume",
98     "Tail thickness"],
99    ["head_length", "Ang", 2, [0, inf], "volume",
101    ["Nlayers", "", 30, [0, inf], "",
102     "Number of layers"],
103    ["spacing", "Ang", 40., [0.0, inf], "volume",
104     "d-spacing of Caille S(Q)"],
105    ["Caille_parameter", "", 0.001, [0.0, 0.8], "",
106     "Caille parameter"],
107    ["sld", "1e-6/Ang^2", 0.4, [-inf, inf], "",
108     "Tail scattering length density"],
109    ["head_sld", "1e-6/Ang^2", 2.0, [-inf, inf], "",
110     "Head scattering length density"],
111    ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "",
112     "Solvent scattering length density"],
113    ]
114
115source = ["lamellarCailleHG_kernel.c"]
116
117# No volume normalization despite having a volume parameter
118# This should perhaps be volume normalized?
119form_volume = """
120    return 1.0;
121    """
122
123Iqxy = """
124    return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS);
125    """
126
127# ER defaults to 0.0
128# VR defaults to 1.0
129
130demo = dict(
131    scale=1, background=0,
132    Nlayers=20, spacing=200., Caille_parameter=0.05,
134    #sld=-1, head_sld=4.0, solvent_sld=6.0,
135    sld=-1, head_sld=4.1, solvent_sld=6.0,
136    tail_length_pd=0.1, tail_length_pd_n=20,
138    spacing_pd=0.2, spacing_pd_n=40,
139    )
140
141oldname = 'LamellarPSHGModel'
142
143oldpars = dict(
144    tail_length='deltaT', head_length='deltaH', Nlayers='n_plates',