source: sasmodels/sasmodels/models/lamellar_stack_caille.py @ 7c57861

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Last change on this file since 7c57861 was 7c57861, checked in by butler, 7 years ago

normalize nearest neighbor spaccing dnn, vs bragg spacing d_spacing and
polydispersity in spacing sigma_d

  • Property mode set to 100644
File size: 3.9 KB
Line 
1r"""
2This model provides the scattering intensity, $I(q) = P(q) S(q)$, for a
3lamellar phase where a random distribution in solution are assumed.
4Here a Caille $S(q)$ is used for the lamellar stacks.
5
6Definition
7----------
8
9The scattering intensity $I(q)$ is
10
11.. math::
12
13    I(q) = 2\pi \frac{P(q)S(q)}{q^2\delta }
14
15The form factor is
16
17.. math::
18
19    P(q) = \frac{2\Delta\rho^2}{q^2}\left(1-\cos q\delta \right)
20
21and the structure factor is
22
23.. math::
24
25    S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right)
26           \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right)
27
28where
29
30.. math::
31    :nowrap:
32
33    \begin{align*}
34    \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right)
35              && \\
36    \gamma_E  &= 0.5772156649
37              && \text{Euler's constant} \\
38    \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}}
39              && \text{Caille constant}
40    \end{align*}
41
42Here $d$ = (repeat) d_spacing, $\delta$ = bilayer thickness,
43the contrast $\Delta\rho$ = SLD(headgroup) - SLD(solvent),
44$K$ = smectic bending elasticity, $B$ = compression modulus, and
45$N$ = number of lamellar plates (*n_plates*).
46
47NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the
48assumptions of the model are incorrect.** And due to a complication of the
49model function, users are responsible for making sure that all the assumptions
50are handled accurately (see the original reference below for more details).
51
52Non-integer numbers of stacks are calculated as a linear combination of
53results for the next lower and higher values.
54
55The 2D scattering intensity is calculated in the same way as 1D, where the
56$q$ vector is defined as
57
58.. math::
59
60    q = \sqrt{q_x^2 + q_y^2}
61
62
63References
64----------
65
66F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502
67
68also in J. Phys. Chem. B, 105, (2001) 11081-11088
69"""
70from numpy import inf
71
72name = "lamellar_stack_caille"
73title = "Random lamellar sheet with Caille structure factor"
74description = """\
75    [Random lamellar phase with Caille  structure factor]
76    randomly oriented stacks of infinite sheets
77    with Caille S(Q), having polydisperse spacing.
78    sld = sheet scattering length density
79    sld_solvent = solvent scattering length density
80    background = incoherent background
81    scale = scale factor
82"""
83category = "shape:lamellae"
84
85single = False  # TODO: check
86# pylint: disable=bad-whitespace, line-too-long
87#             ["name", "units", default, [lower, upper], "type","description"],
88parameters = [
89    ["thickness",        "Ang",      30.0,  [0, inf],   "volume", "sheet thickness"],
90    ["Nlayers",          "",          20,   [0, inf],   "",       "Number of layers"],
91    ["d_spacing",        "Ang",      400.,  [0.0,inf],  "volume", "lamellar d-spacing of Caille S(Q)"],
92    ["Caille_parameter", "1/Ang^2",    0.1, [0.0,0.8],  "",       "Caille parameter"],
93    ["sld",              "1e-6/Ang^2", 6.3, [-inf,inf], "sld",    "layer scattering length density"],
94    ["sld_solvent",      "1e-6/Ang^2", 1.0, [-inf,inf], "sld",    "Solvent scattering length density"],
95    ]
96# pylint: enable=bad-whitespace, line-too-long
97
98source = ["lamellar_stack_caille.c"]
99
100# No volume normalization despite having a volume parameter
101# This should perhaps be volume normalized?
102form_volume = """
103    return 1.0;
104    """
105
106# ER defaults to 0.0
107# VR defaults to 1.0
108
109demo = dict(scale=1, background=0,
110            thickness=67., Nlayers=3.75, d_spacing=200.,
111            Caille_parameter=0.268, sld=1.0, sld_solvent=6.34,
112            thickness_pd=0.1, thickness_pd_n=100,
113            d_spacing_pd=0.05, d_spacing_pd_n=40)
114
115#
116tests = [
117    [{'scale': 1.0, 'background': 0.0, 'thickness': 30., 'Nlayers': 20.0,
118      'd_spacing': 400., 'Caille_parameter': 0.1, 'sld': 6.3,
119      'sld_solvent': 1.0, 'thickness_pd': 0.0, 'd_spacing_pd': 0.0},
120     [0.001], [28895.13397]]
121    ]
122# ADDED by: RKH  ON: 18Mar2016  converted from sasview previously, now renaming everything & sorting the docs
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