# source:sasmodels/sasmodels/models/fcc.py@9aac25d

core_shell_microgelscostrafo411magnetic_modelrelease_v0.94release_v0.95ticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 9aac25d was 9aac25d, checked in by piotr, 7 years ago

Converted SCCrystalModel

• Property mode set to 100644
File size: 5.2 KB
Line
1#fcc paracrystal model
2#note model title and parameter table are automatically inserted
3#note - calculation requires double precision
4r"""
5Calculates the scattering from a **face-centered cubic lattice** with
6paracrystalline distortion. Thermal vibrations are considered to be
7negligible, and the size of the paracrystal is infinitely large.
8Paracrystalline distortion is assumed to be isotropic and characterized by
9a Gaussian distribution.
10
11Definition
12----------
13
14The scattering intensity $I(q)$ is calculated as
15
16.. math::
17
18    I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q)
19
20where *scale* is the volume fraction of spheres, $V_p$ is the volume of
21the primary particle, $V_\text{lattice}$ is a volume correction for the crystal
22structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$
23is the paracrystalline structure factor for a face-centered cubic structure.
24
25Equation (1) of the 1990 reference is used to calculate $Z(q)$, using
26equations (23)-(25) from the 1987 paper for $Z1$, $Z2$, and $Z3$.
27
28The lattice correction (the occupied volume of the lattice) for a
29face-centered cubic structure of particles of radius $R$ and nearest
30neighbor separation $D$ is
31
32.. math::
33
34   V_\text{lattice} = \frac{16\pi}{3}\frac{R^3}{\left(D\sqrt{2}\right)^3}
35
36The distortion factor (one standard deviation) of the paracrystal is
37included in the calculation of $Z(q)$
38
39.. math::
40
41    \Delta a = gD
42
43where $g$ is a fractional distortion based on the nearest neighbor distance.
44
45.. figure:: img/fcc_lattice.jpg
46
47    Face-centered cubic lattice.
48
49For a crystal, diffraction peaks appear at reduced q-values given by
50
51.. math::
52
53    \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2}
54
55where for a face-centered cubic lattice $h, k , l$ all odd or all
56even are allowed and reflections where $h, k, l$ are mixed odd/even
57are forbidden. Thus the peak positions correspond to (just the first 5)
58
59.. math::
60
61    \begin{array}{cccccc}
62    q/q_0 & 1 & \sqrt{4/3} & \sqrt{8/3} & \sqrt{11/3} & \sqrt{4} \\
63    \text{Indices} & (111)  & (200) & (220) & (311) & (222)
64    \end{array}
65
66**NB**: The calculation of $Z(q)$ is a double numerical integral that
67must be carried out with a high density of points to properly capture
68the sharp peaks of the paracrystalline scattering. So be warned that the
69calculation is SLOW. Go get some coffee. Fitting of any experimental data
70must be resolution smeared for any meaningful fit. This makes a triple
71integral. Very, very slow. Go get lunch!
72
73This example dataset is produced using 200 data points, *qmin* = 0.01 |Ang^-1|,
74*qmax* = 0.1 |Ang^-1| and the above default values.
75
76.. figure:: img/fcc_1d.jpg
77
78    1D plot in the linear scale using the default values (w/200 data point).
79
80The 2D (Anisotropic model) is based on the reference below where $I(q)$ is
81approximated for 1d scattering. Thus the scattering pattern for 2D may not
82be accurate. Note that we are not responsible for any incorrectness of the
832D model computation.
84
85.. figure:: img/crystal_orientation.png
86
87    Orientation of the crystal with respect to the scattering plane.
88
89.. figure:: img/fcc_2d.jpg
90
91    2D plot using the default values (w/200X200 pixels).
92
93References
94----------
95
96Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765
97(Original Paper)
98
99Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856
100(Corrections to FCC and BCC lattice structure calculation)
101"""
102
103from numpy import inf
104
105name = "fcc_paracrystal"
106title = "Face-centred cubic lattic with paracrystalline distortion"
107description = """
108    Calculates the scattering from a **face-centered cubic lattice** with paracrystalline distortion. Thermal vibrations
109    are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is
110    assumed to be isotropic and characterized by a Gaussian distribution.
111    """
112category = "shape:paracrystal"
113
114single = False
115
116#             ["name", "units", default, [lower, upper], "type","description"],
117parameters = [["dnn", "Ang", 220, [-inf, inf], "", "Nearest neighbour distance"],
118              ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"],
120              ["sld", "1e-6/Ang^2", 4, [-inf, inf], "", "Particle scattering length density"],
121              ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", "Solvent scattering length density"],
122              ["theta", "degrees", 60, [-inf, inf], "orientation", "In plane angle"],
123              ["phi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"],
124              ["psi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"]
125             ]
126
127source = ["lib/J1.c", "lib/gauss150.c", "lib/sphere_form.c", "fcc.c"]
128
129# parameters for demo
130demo = dict(scale=1, background=0,
131            dnn=220, d_factor=0.06, sld=4, solvent_sld=1,