# source:sasmodels/sasmodels/models/pearl_necklace.py@c1e44e5

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1r"""
2This model provides the form factor for a pearl necklace composed of two
3elements: *N* pearls (homogeneous spheres of radius *R*) freely jointed by *M*
4rods (like strings - with a total mass *Mw* = *M* \* *m*\ :sub:r + *N* \* *m*\
5:sub:s, and the string segment length (or edge separation) *l*
6(= *A* - 2\ *R*)). *A* is the center-to-center pearl separation distance.
7
8.. figure:: img/pearl_necklace_geometry.jpg
9
10    Pearl Necklace schematic
11
12Definition
13----------
14
15The output of the scattering intensity function for the pearl_necklace is
16given by (Schweins, 2004)
17
18.. math::
19
20    I(q)=\frac{ \text{scale} }{V} \cdot \frac{(S_{ss}(q)+S_{ff}(q)+S_{fs}(q))}
21        {(M \cdot m_f + N \cdot m_s)^2} + \text{bkg}
22
23where
24
25.. math::
26
27    S_{ss}(q) &= 2m_s^2\psi^2(q)\left[\frac{N}{1-sin(qA)/qA}-\frac{N}{2}-
28        \frac{1-(sin(qA)/qA)^N}{(1-sin(qA)/qA)^2}\cdot\frac{sin(qA)}{qA}\right] \\
29    S_{ff}(q) &= m_r^2\left[M\left\{2\Lambda(q)-\left(\frac{sin(ql/2)}{ql/2}\right)\right\}+
30        \frac{2M\beta^2(q)}{1-sin(qA)/qA}-2\beta^2(q)\cdot
31        \frac{1-(sin(qA)/qA)^M}{(1-sin(qA)/qA)^2}\right] \\
32    S_{fs}(q) &= m_r \beta (q) \cdot m_s \psi (q) \cdot 4\left[
33        \frac{N-1}{1-sin(qA)/qA}-\frac{1-(sin(qA)/qA)^{N-1}}{(1-sin(qA)/qA)^2}
34        \cdot \frac{sin(qA)}{qA}\right] \\
35    \psi(q) &= 3 \cdot \frac{sin(qR)-(qR)\cdot cos(qR)}{(qR)^3} \\
36    \Lambda(q) &= \frac{\int_0^{ql}\frac{sin(t)}{t}dt}{ql} \\
37    \beta(q) &= \frac{\int_{qR}^{q(A-R)}\frac{sin(t)}{t}dt}{ql}
38
39where the mass *m*\ :sub:i is (SLD\ :sub:i - SLD\ :sub:solvent) \*
40(volume of the *N* pearls/rods). *V* is the total volume of the necklace.
41
42.. note::
43
44   *num_pearls* must be an integer.
45
46The 2D scattering intensity is the same as $P(q)$ above, regardless of the
47orientation of the *q* vector.
48
49References
50----------
51
52.. [#] R Schweins and K Huber, *Particle Scattering Factor of Pearl Necklace Chains*,
53       *Macromol. Symp.* 211 (2004) 25-42 2004
54
55.. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
56
57Authorship and Verification
58----------------------------
59
60* **Author:**
62* **Last Reviewed by:** Steve King **Date:** March 28, 2019
63"""
64
65import numpy as np
66from numpy import inf
67
68name = "pearl_necklace"
69title = "Colloidal spheres chained together with no preferential orientation"
70description = """
71Calculate form factor for Pearl Necklace Model
72[Macromol. Symp. 2004, 211, 25-42]
73Parameters:
74background:background
75scale: scale factor
76sld: the SLD of the pearl spheres
77sld_string: the SLD of the strings
78sld_solvent: the SLD of the solvent
79num_pearls: number of the pearls
81edge_sep: the length of string segment; surface to surface
82thick_string: thickness (ie, diameter) of the string
83"""
84category = "shape:cylinder"
85
86#             ["name", "units", default, [lower, upper], "type","description"],
87parameters = [["radius", "Ang", 80.0, [0, inf], "volume",
88               "Mean radius of the chained spheres"],
89              ["edge_sep", "Ang", 350.0, [0, inf], "volume",
90               "Mean separation of chained particles"],
91              ["thick_string", "Ang", 2.5, [0, inf], "volume",
92               "Thickness of the chain linkage"],
93              ["num_pearls", "none", 3, [1, inf], "volume",
94               "Number of pearls in the necklace (must be integer)"],
95              ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld",
96               "Scattering length density of the chained spheres"],
97              ["sld_string", "1e-6/Ang^2", 1.0, [-inf, inf], "sld",
98               "Scattering length density of the chain linkage"],
99              ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld",
100               "Scattering length density of the solvent"],
101             ]
102
103source = ["lib/sas_Si.c", "lib/sas_3j1x_x.c", "pearl_necklace.c"]
104single = False  # use double precision unless told otherwise
106
107def random():
108    """Return a random parameter set for the model."""
109    radius = 10**np.random.uniform(1, 3) # 1 - 1000
111    edge_sep = 10**np.random.uniform(0, 3)  # 1 - 1000
112    num_pearls = np.round(10**np.random.uniform(0.3, 3)) # 2 - 1000
113    pars = dict(
115        edge_sep=edge_sep,
116        thick_string=thick_string,
117        num_pearls=num_pearls,
118    )
119    return pars
120
121# parameters for demo
122demo = dict(scale=1, background=0, radius=80.0, edge_sep=350.0,
123            num_pearls=3, sld=1, sld_solvent=6.3, sld_string=1,
124            thick_string=2.5,