source: sasmodels/sasmodels/models/bcc_paracrystal.py @ 1f65db5

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docs edits for bcc_ fc_ sc_paracrystal

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1r"""
2Definition
3----------
4
5Calculates the scattering from a **body-centered cubic lattice** with
6paracrystalline distortion. Thermal vibrations are considered to be negligible,
7and the size of the paracrystal is infinitely large. Paracrystalline distortion
8is assumed to be isotropic and characterized by a Gaussian distribution.
9
10The scattering intensity $I(q)$ is calculated as
11
12.. math::
13
14    I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q)
15
16
17where *scale* is the volume fraction of spheres, $V_p$ is the volume of the
18primary particle, $V_\text{lattice}$ is a volume correction for the crystal
19structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$
20is the paracrystalline structure factor for a body-centered cubic structure.
21
22Equation (1) of the 1990 reference\ [#CIT1990]_ is used to calculate $Z(q)$,
23using equations (29)-(31) from the 1987 paper\ [#CIT1987]_ for $Z1$, $Z2$, and
24$Z3$.
25
26The lattice correction (the occupied volume of the lattice) for a
27body-centered cubic structure of particles of radius $R$ and nearest neighbor
28separation $D$ is
29
30.. math::
31
32    V_\text{lattice} = \frac{16\pi}{3} \frac{R^3}{\left(D\sqrt{2}\right)^3}
33
34
35The distortion factor (one standard deviation) of the paracrystal is included
36in the calculation of $Z(q)$
37
38.. math::
39
40    \Delta a = g D
41
42where $g$ is a fractional distortion based on the nearest neighbor distance.
43
44
45.. figure:: img/bcc_geometry.jpg
46
47    Body-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 body-centered cubic lattice, only reflections where
56$(h + k + l) = \text{even}$ are allowed and reflections where
57$(h + k + l) = \text{odd}$ are forbidden. Thus the peak positions
58correspond to (just the first 5)
59
60.. math::
61
62    \begin{array}{lccccc}
63    q/q_o          &   1   & \sqrt{2} & \sqrt{3} & \sqrt{4} & \sqrt{5} \\
64    \text{Indices} & (110) &    (200) & (211)    & (220)    & (310)    \\
65    \end{array}
66
67**NB**: The calculation of $Z(q)$ is a double numerical integral that must
68be carried out with a high density of points to properly capture the sharp
69peaks of the paracrystalline scattering. So be warned that the calculation
70is SLOW. Go get some coffee. Fitting of any experimental data must be
71resolution smeared for any meaningful fit. This makes a triple integral.
72Very, very slow. Go get lunch!
73
74This example dataset is produced using 200 data points,
75*qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above default values.
76
77The 2D (Anisotropic model) is based on the reference below where $I(q)$ is
78approximated for 1d scattering. Thus the scattering pattern for 2D may not
79be accurate.
80
81.. figure:: img/parallelepiped_angle_definition.png
82
83    Orientation of the crystal with respect to the scattering plane, when
84    $\theta = \phi = 0$ the $c$ axis is along the beam direction (the $z$ axis).
85
86References
87----------
88
89.. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765
90   (Original Paper)
91.. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856
92   (Corrections to FCC and BCC lattice structure calculation)
93
94Authorship and Verification
95----------------------------
96
97* **Author:** NIST IGOR/DANSE **Date:** pre 2010
98* **Last Modified by:** Paul Butler **Date:** September 29, 2016
99* **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016
100"""
101
102from numpy import inf, pi
103
104name = "bcc_paracrystal"
105title = "Body-centred cubic lattic with paracrystalline distortion"
106description = """
107    Calculates the scattering from a **body-centered cubic lattice** with
108    paracrystalline distortion. Thermal vibrations are considered to be
109    negligible, and the size of the paracrystal is infinitely large.
110    Paracrystalline distortion is assumed to be isotropic and characterized
111    by a Gaussian distribution.
112    """
113category = "shape:paracrystal"
114
115#note - calculation requires double precision
116single = False
117
118# pylint: disable=bad-whitespace, line-too-long
119#             ["name", "units", default, [lower, upper], "type","description" ],
120parameters = [["dnn",         "Ang",       220,    [-inf, inf], "",            "Nearest neighbour distance"],
121              ["d_factor",    "",            0.06, [-inf, inf], "",            "Paracrystal distortion factor"],
122              ["radius",      "Ang",        40,    [0, inf],    "volume",      "Particle radius"],
123              ["sld",         "1e-6/Ang^2",  4,    [-inf, inf], "sld",         "Particle scattering length density"],
124              ["sld_solvent", "1e-6/Ang^2",  1,    [-inf, inf], "sld",         "Solvent scattering length density"],
125              ["theta",       "degrees",    60,    [-inf, inf], "orientation", "In plane angle"],
126              ["phi",         "degrees",    60,    [-inf, inf], "orientation", "Out of plane angle"],
127              ["psi",         "degrees",    60,    [-inf, inf], "orientation", "Out of plane angle"]
128             ]
129# pylint: enable=bad-whitespace, line-too-long
130
131source = ["lib/sas_3j1x_x.c", "lib/gauss150.c", "lib/sphere_form.c", "bcc_paracrystal.c"]
132
133# parameters for demo
134demo = dict(
135    scale=1, background=0,
136    dnn=220, d_factor=0.06, sld=4, sld_solvent=1,
137    radius=40,
138    theta=60, phi=60, psi=60,
139    radius_pd=.2, radius_pd_n=2,
140    theta_pd=15, theta_pd_n=0,
141    phi_pd=15, phi_pd_n=0,
142    psi_pd=15, psi_pd_n=0,
143    )
144# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
145# add 2d test later
146q =4.*pi/220.
147tests = [
148    [{ },
149     [0.001, q, 0.215268], [1.46601394721, 2.85851284174, 0.00866710287078]],
150]
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