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

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1# Note: model title and parameter table are inserted automatically
2r"""
3This model calculates the scattering from a stack of repeating lamellar
4structures. The stacks of lamellae (infinite in lateral dimension) are
5treated as a paracrystal to account for the repeating spacing. The repeat
6distance is further characterized by a Gaussian polydispersity. **This model
7can be used for large multilamellar vesicles.**
8
9Definition
10----------
11
12In the equations below,
13
14- *scale* is used instead of the mass per area of the bilayer $\Gamma_m$
15  (this corresponds to the volume fraction of the material in the bilayer,
16  *not* the total excluded volume of the paracrystal),
17
18- *sld* $-$ *sld_solvent* is the contrast $\Delta \rho$,
19
20- *thickness* is the layer thickness $t$,
21
22- *Nlayers* is the number of layers $N$,
23
24- *d_spacing* is the average distance between adjacent layers
25  $\langle D \rangle$, and
26
27- *sigma_d* is the relative standard deviation of the Gaussian
28  layer distance distribution $\sigma_D / \langle D \rangle$.
29
30
31The scattering intensity $I(q)$ is calculated as
32
33.. math::
34
35    I(q) = 2\pi\Delta\rho^2\Gamma_m\frac{P_\text{bil}(q)}{q^2} Z_N(q)
36
37The form factor of the bilayer is approximated as the cross section of an
38infinite, planar bilayer of thickness $t$ (compare the equations for the
39lamellar model).
40
41.. math::
42
43    P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2
44
45$Z_N(q)$ describes the interference effects for aggregates
46consisting of more than one bilayer. The equations used are (3-5)
47from the Bergstrom reference:
48
49.. math::
50
51
52    Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)}
53        + x_N S_N + (1 - x_N) S_{N+1}
54
55where
56
57.. math::
58
59    S_N(q) = \frac{a_N}{N}[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2
60
61and
62
63.. math::
64
65    a_N &= 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) \\
66        &\quad - 4w^{N+2}\cos(Nq \langle D \rangle)
67        + 2 w^{N+3}\cos[(N-1)q \langle D \rangle]
68        + 2w^{N+1}\cos[(N+1)q \langle D \rangle]
69
70for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$.
71
72Non-integer numbers of stacks are calculated as a linear combination of
73the lower and higher values
74
75.. math::
76
77    N_L = x_N N + (1 - x_N)(N+1)
78
79The 2D scattering intensity is the same as 1D, regardless of the orientation
80of the $q$ vector which is defined as
81
82.. math::
83
84    q = \sqrt{q_x^2 + q_y^2}
85
86
87Reference
88---------
89
90M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf,
91*J. Phys. Chem. B*, 103 (1999) 9888-9897
92"""
93
94import numpy as np
95from numpy import inf
96
97name = "lamellar_stack_paracrystal"
98title = "Random lamellar sheet with paracrystal structure factor"
99description = """\
100    [Random lamellar phase with paracrystal structure factor]
101        randomly oriented stacks of infinite sheets
102        with paracrytal S(Q), having polydisperse spacing.
103        sld = sheet scattering length density
104        sld_solvent = solvent scattering length density
105        background = incoherent background
106        scale = scale factor
107"""
108category = "shape:lamellae"
109
110single = False
111
112#             ["name", "units", default, [lower, upper], "type","description"],
113parameters = [["thickness", "Ang", 33.0, [0, inf], "volume",
114               "sheet thickness"],
115              ["Nlayers", "", 20, [1, inf], "",
116               "Number of layers"],
117              ["d_spacing", "Ang", 250., [0.0, inf], "",
118               "lamellar spacing of paracrystal stack"],
119              ["sigma_d", "Ang", 0.0, [0.0, inf], "",
120               "Sigma (polydispersity) of the lamellar spacing"],
121              ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld",
122               "layer scattering length density"],
123              ["sld_solvent", "1e-6/Ang^2", 6.34, [-inf, inf], "sld",
124               "Solvent scattering length density"],
125             ]
126
127
128source = ["lamellar_stack_paracrystal.c"]
129
130form_volume = """
131    return 1.0;
132"""
133
134def random():
135    total_thickness = 10**np.random.uniform(2, 4.7)
136    Nlayers = np.random.randint(2, 200)
137    d_spacing = total_thickness / Nlayers
138    thickness = d_spacing * np.random.uniform(0, 1)
139    # Let polydispersity peak around 15%; 95% < 0.4; max=100%
140    sigma_d = np.random.beta(1.5, 7)
141    pars = dict(
142        thickness=thickness,
143        Nlayers=Nlayers,
144        d_spacing=d_spacing,
145        sigma_d=sigma_d,
146    )
147    return pars
148
149# ER defaults to 0.0
150# VR defaults to 1.0
151
152demo = dict(scale=1, background=0,
153            thickness=33, Nlayers=20, d_spacing=250, sigma_d=0.2,
154            sld=1.0, sld_solvent=6.34,
155            thickness_pd=0.2, thickness_pd_n=40)
156
157#
158tests = [
159    [{'scale': 1.0, 'background': 0.0, 'thickness': 33., 'Nlayers': 20.0,
160      'd_spacing': 250., 'sigma_d': 0.2, 'sld': 1.0,
161      'sld_solvent': 6.34, 'thickness_pd': 0.0, 'thickness_pd_n': 40},
162     [0.001, 0.215268], [21829.3, 0.00487686]],
163]
164# ADDED by: RKH  ON: 18Mar2016  converted from sasview previously,
165# now renaming everything & sorting the docs
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