# source:sasmodels/sasmodels/models/multilayer_vesicle.py@db1d9d5

ticket-1257-vesicle-productticket_1156ticket_822_more_unit_tests
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1r"""
2Definition
3----------
4
5This model is a trivial extension of the core_shell_sphere function where the
6core is filled with solvent and is surrounded by $N$ shells of material
7(such as lipids) interleaved with $N - 1$ layers of solvent. For $N = 1$, this
8returns the same as the vesicle model, except for the normalisation, which here
9is to outermost volume. The shell thicknesses and SLD are constant for all
10shells as expected for a multilayer vesicle.
11
12.. figure:: img/multi_shell_geometry.jpg
13
14    Geometry of the multilayer_vesicle model.
15
16See the :ref:core-shell-sphere model for more documentation.
17
18The 1D scattering intensity is calculated in the following way (Guinier, 1955)
19
20.. math::
21    P(q) = \text{scale} \cdot \frac{\phi}{V(R_N)} F^2(q) + \text{background}
22
23where
24
25.. math::
26     F(q) = (\rho_\text{shell}-\rho_\text{solv}) \sum_{i=1}^{N} \left[
27     3V(r_i)\frac{\sin(qr_i) - qr_i\cos(qr_i)}{(qr_i)^3}
28     - 3V(R_i)\frac{\sin(qR_i) - qR_i\cos(qR_i)}{(qR_i)^3}
29     \right]
30
31for
32
33.. math::
34
35     r_i &= r_c + (i-1)(t_s + t_w) \text{ solvent radius before shell } i \\
36     R_i &= r_i + t_s \text{ shell radius for shell } i
37
38$\phi$ is the volume fraction of particles, $V(r)$ is the volume of a sphere
39of radius $r$, $r_c$ is the radius of the core, $t_s$ is the thickness of
40the shell, $t_w$ is the thickness of the solvent layer between the shells,
41$\rho_\text{shell}$ is the scattering length density of a shell, and
42$\rho_\text{solv}$ is the scattering length density of the solvent.
43
44USAGE NOTES
45
46* The outer-most shell radius $R_N$ is used as the effective radius
47  for $P(Q)$ when $P(Q) * S(Q)$ is applied.
48  calculations rather slow.
49* The number of shells is always rounded to an integer value as a non interger
50  number of layers is not physical.
51* Thus Polydispersity should only be applied to number of shells **VERY
52  CAREFULLY**.  A possible legitimate use would be for mixed systems in which
53  some vesicles have 1 shell, some have 2, etc. A polydispersity on $N$ can be
54  used to model the data by using the "array distriubtion" feature. First
55  create a file such as *shell_dist.txt* containing the relative portion
56  of each vesicle size::
57
58    1 20
59    2  4
60    3  1
61
62  Turn on polydispersity and select an array distribution for the *n_shells*
63  parameter.  Choose the above *shell_dist.txt* file, and the model will be
64  computed with 80% 1-shell vesicles, 16% 2-shell vesicles and 4%
65  3-shell vesicles.
66* This is a highly non-linear, highly oscillatory (especially around the
67  q-values that correspond to the repeat distance of the layers), model
68  function complicated by the fact that the number of water/shell pairs must
69  physically be an integer value, although the optimization treats it as a
70  floating point value. Thus it may be that the resolution interpolation is not
71  sufficiently fine grained in certain cases. Please report any such occurences
72  to the SasView team. Generally, for the best possible experience:
73
75 - Using a priori knowledge, hold as many parameters fixed as possible
76 - if N=1, tw (water thickness) must by definition be zero. Both N and tw should
77   be fixed during fitting.
78 - If N>1, use constraints to keep N > 1
79 - Because N only really moves in integer steps, it may get "stuck" if the
80   optimizer step size is too small so care should be taken
82   them know the issue preferably with example data and model which fail to
83   converge.
84
85The 2D scattering intensity is the same as 1D, regardless of the orientation
86of the q vector which is defined as:
87
88.. math::
89
90    q = \sqrt{q_x^2 + q_y^2}
91
92For information about polarised and magnetic scattering, see
93the :ref:magnetism documentation.
94
95References
96----------
97
98.. [#] B Cabane, *Small Angle Scattering Methods*, in *Surfactant Solutions:
99   New Methods of Investigation*, Ch.2, Surfactant Science Series Vol. 22, Ed.
100   R Zana and M Dekker, New York, (1987).
101
102Source
103------
104
105multilayer_vesicle.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/multilayer_vesicle.py>_
106
107multilayer_vesicle.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/multilayer_vesicle.c>_
108
109Authorship and Verification
110----------------------------
111
112* **Author:** NIST IGOR/DANSE **Date:** pre 2010
113* **Converted to sasmodels by:** Piotr Rozyczko **Date:** Feb 24, 2016
115* **Last Reviewed by:** Steve King **Date:** March 28, 2019
116* **Source added by :** Steve King **Date:** March 25, 2019
117"""
118
119import numpy as np
120from numpy import inf
121
122name = "multilayer_vesicle"
123title = "Calculate form factor for a multi-lamellar vesicle"
124description = """
125    multilayer_vesicle model parameters;
126    scale : scale factor for abs intensity if needed else 1.0
127    volfraction: volume fraction
129    thick_shell: shell thickness
130    thick_solvent: water thickness
131    sld_solvent: solvent scattering length density
132    sld: shell scattering length density
133    n_shells:number of "shell plus solvent" layer pairs
134    background: incoherent background
135        """
136category = "shape:sphere"
137
139#   ["name", "units", default, [lower, upper], "type","description"],
140parameters = [
141    ["volfraction", "",  0.05, [0.0, 1],  "", "volume fraction of vesicles"],
142    ["radius", "Ang", 60.0, [0.0, inf],  "volume", "radius of solvent filled core"],
143    ["thick_shell", "Ang",        10.0, [0.0, inf],  "volume", "thickness of one shell"],
144    ["thick_solvent", "Ang",        10.0, [0.0, inf],  "volume", "solvent thickness between shells"],
145    ["sld_solvent",    "1e-6/Ang^2",  6.4, [-inf, inf], "sld", "solvent scattering length density"],
146    ["sld",   "1e-6/Ang^2",  0.4, [-inf, inf], "sld", "Shell scattering length density"],
147    ["n_shells",     "",            2.0, [1.0, inf],  "volume", "Number of shell plus solvent layer pairs (must be integer)"],
148    ]
150
151# TODO: proposed syntax for specifying which parameters can be polydisperse
153
154source = ["lib/sas_3j1x_x.c", "multilayer_vesicle.c"]
155have_Fq = True
157
158def random():
159    """Return a random parameter set for the model."""
160    volfraction = 10**np.random.uniform(-3, -0.5)  # scale from 0.1% to 30%
161    radius = 10**np.random.uniform(0, 2.5) # core less than 300 A
162    total_thick = 10**np.random.uniform(2, 4) # up to 10000 A of shells
163    # random number of shells, with shell+solvent thickness > 10 A
164    n_shells = int(10**np.random.uniform(0, np.log10(total_thick)-1)+0.5)
165    # split total shell thickness with preference for shell over solvent;
166    # make sure that shell thickness is at least 1 A
167    one_thick = total_thick/n_shells
168    thick_solvent = 10**np.random.uniform(-2, 0)*(one_thick - 1)
169    thick_shell = one_thick - thick_solvent
170    pars = dict(
171        scale=1,
172        volfraction=volfraction,
174        thick_shell=thick_shell,
175        thick_solvent=thick_solvent,
176        n_shells=n_shells,
177    )
178    return pars
179
180tests = [
181    # Accuracy tests based on content in test/utest_other_models.py
183      'thick_shell': 10.0,
184      'thick_solvent': 10.0,
185      'sld_solvent': 6.4,
186      'sld': 0.4,
187      'n_shells': 2.0,
188      'scale': 1.0,
189      'background': 0.001,
190     }, 0.001, 122.1405],
191
192    [{'volfraction': 1.0,