1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
|
r"""
This model provides the form factor for $N$ spherical pearls of radius $R$
linearly joined by short strings (or segment length or edge separation)
$l$ $(= A - 2R)$. $A$ is the center-to-center pearl separation distance.
The thickness of each string is assumed to be negligible.
.. figure:: img/linear_pearls_geometry.jpg
Definition
----------
The output of the scattering intensity function for the linear_pearls model
is given by (Dobrynin, 1996)
.. math::
P(Q) = \frac{\text{scale}}{V}\left[ m_{p}^2
\left(N+2\sum_{n-1}^{N-1}(N-n)\frac{\sin(qnl)}{qnl}\right)
\left( 3\frac{\sin(qR)-qR\cos(qR)}{(qr)^3}\right)^2\right]
where the mass $m_p$ is $(SLD_{pearl}-SLD_{solvent})*(volume\ of\ N\ pearls)$.
V is the total volume.
The 2D scattering intensity is the same as P(q) above,
regardless of the orientation of the q vector.
References
----------
#. A V Dobrynin, M Rubinstein and S P Obukhov, *Macromol.*, 29 (1996) 2974-2979
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""
import numpy as np
from numpy import inf
name = "linear_pearls"
title = "Linear pearls model of scattering from spherical pearls."
description = """
Calculate form factor for Pearl Necklace Model
[Macromol. 1996, 29, 2974-2979]
Parameters:
sld_pearl: the SLD of the pearl spheres
sld_solv: the SLD of the solvent
num_pearls: number of the pearls
radius: the radius of a pearl
edge_separation: the length of string segment; surface to surface
"""
category = "shape:sphere"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
["radius", "Ang", 80.0, [0, inf], "", "Radius of the pearls"],
["edge_sep", "Ang", 350.0, [0, inf], "", "Length of the string segment - surface to surface"],
["num_pearls", "", 3.0, [1, inf], "", "Number of the pearls"],
["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "SLD of the pearl spheres"],
["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "SLD of the solvent"],
]
# pylint: enable=bad-whitespace, line-too-long
single = False
source = ["lib/sas_3j1x_x.c", "linear_pearls.c"]
def random():
"""Return a random parameter set for the model."""
radius = 10**np.random.uniform(1, 3) # 1 - 1000
edge_sep = 10**np.random.uniform(0, 3) # 1 - 1000
num_pearls = np.round(10**np.random.uniform(0.3, 3)) # 2 - 1000
pars = dict(
radius=radius,
edge_sep=edge_sep,
num_pearls=num_pearls,
)
return pars
_ = """
Tests temporarily disabled, until single-double precision accuracy issue solved.
tests = [
# Accuracy tests based on content in test/utest_model_pearlnecklace.py
[{'radius': 20.0,
'num_pearls': 2.0,
'sld': 1.0,
'sld_solvent': 6.3,
'edge_sep': 400.0,
}, 0.001, 185.135],
# Additional tests with larger range of parameters
[{'radius': 120.0,
'num_pearls': 5.0,
'sld': 2.0,
'sld_solvent': 2.3,
'edge_sep': 100.0,
}, 0.01, 45.4984],
[{'radius': 7.0,
'num_pearls': 2.0,
'sld': 10.0,
'sld_solvent': 99.3,
'edge_sep': 20.0,
}, 1.0, 0.632811],
]
"""
|
