source: sasmodels/sasmodels/models/sc_paracrystal.py @ 48438f9

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1r"""
2Calculates the scattering from a **simple cubic lattice** with
3paracrystalline distortion. Thermal vibrations are considered to be
4negligible, and the size of the paracrystal is infinitely large.
5Paracrystalline distortion is assumed to be isotropic and characterized
6by a Gaussian distribution.
7
8Definition
9----------
10
11The scattering intensity $I(q)$ is calculated as
12
13.. math::
14
15    I(q) = \text{scale}\frac{V_\text{lattice}P(q)Z(q)}{V_p} + \text{background}
16
17where scale is the volume fraction of spheres, $V_p$ is the volume of
18the primary particle, $V_\text{lattice}$ is a volume correction for the crystal
19structure, $P(q)$ is the form factor of the sphere (normalized), and
20$Z(q)$ is the paracrystalline structure factor for a simple cubic structure.
21
22Equation (16) of the 1987 reference is used to calculate $Z(q)$, using
23equations (13)-(15) from the 1987 paper for Z1, Z2, and Z3.
24
25The lattice correction (the occupied volume of the lattice) for a simple
26cubic structure of particles of radius *R* and nearest neighbor separation *D* is
27
28.. math::
29
30    V_\text{lattice}=\frac{4\pi}{3}\frac{R^3}{D^3}
31
32The distortion factor (one standard deviation) of the paracrystal is included
33in the calculation of $Z(q)$
34
35.. math::
36
37    \Delta a = gD
38
39where *g* is a fractional distortion based on the nearest neighbor distance.
40
41The simple cubic lattice is
42
43.. figure:: img/sc_crystal_geometry.jpg
44
45For a crystal, diffraction peaks appear at reduced q-values given by
46
47.. math::
48
49    \frac{qD}{2\pi} = \sqrt{h^2+k^2+l^2}
50
51where for a simple cubic lattice any h, k, l are allowed and none are
52forbidden. Thus the peak positions correspond to (just the first 5)
53
54.. math::
55    :nowrap:
56
57    \begin{align*}
58    q/q_0 \quad & \quad 1
59                & \sqrt{2} \quad
60                & \quad  \sqrt{3} \quad
61                & \sqrt{4} \quad
62                & \quad \sqrt{5}\quad \\
63    Indices \quad & (100)
64                  & \quad (110) \quad
65                  & \quad (111)
66                  & (200) \quad
67                  & \quad (210)
68    \end{align*}
69
70.. note::
71
72    The calculation of *Z(q)* is a double numerical integral that must be
73    carried out with a high density of points to properly capture the sharp
74    peaks of the paracrystalline scattering.
75    So be warned that the calculation is SLOW. Go get some coffee.
76    Fitting of any experimental data must be resolution smeared for any
77    meaningful fit. This makes a triple integral. Very, very slow.
78    Go get lunch!
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 2D
83model computation.
84
85.. figure:: img/parallelepiped_angle_definition.png
86
87    Orientation of the crystal with respect to the scattering plane, when
88    $\theta = \phi = 0$ the $c$ axis is along the beam direction (the $z$ axis).
89
90Reference
91---------
92Hideki Matsuoka et. al. *Physical Review B,* 36 (1987) 1754-1765
93(Original Paper)
94
95Hideki Matsuoka et. al. *Physical Review B,* 41 (1990) 3854 -3856
96(Corrections to FCC and BCC lattice structure calculation)
97
98"""
99
100from numpy import inf
101
102name = "sc_paracrystal"
103title = "Simple cubic lattice with paracrystalline distortion"
104description = """
105        P(q)=(scale/Vp)*V_lattice*P(q)*Z(q)+bkg where scale is the volume
106        fraction of sphere,
107        Vp = volume of the primary particle,
108        V_lattice = volume correction for
109        for the crystal structure,
110        P(q)= form factor of the sphere (normalized),
111        Z(q)= paracrystalline structure factor
112        for a simple cubic structure.
113        [Simple Cubic ParaCrystal Model]
114        Parameters;
115        scale: volume fraction of spheres
116        bkg:background, R: radius of sphere
117        dnn: Nearest neighbor distance
118        d_factor: Paracrystal distortion factor
119        radius: radius of the spheres
120        sldSph: SLD of the sphere
121        sldSolv: SLD of the solvent
122        """
123category = "shape:paracrystal"
124single = False
125# pylint: disable=bad-whitespace, line-too-long
126#             ["name", "units", default, [lower, upper], "type","description"],
127parameters = [["dnn",         "Ang",       220.0, [0.0, inf],  "",            "Nearest neighbor distance"],
128              ["d_factor",    "",           0.06, [-inf, inf], "",            "Paracrystal distortion factor"],
129              ["radius",      "Ang",        40.0, [0.0, inf],  "volume",      "Radius of sphere"],
130              ["sld",  "1e-6/Ang^2",         3.0, [0.0, inf],  "sld",         "Sphere scattering length density"],
131              ["sld_solvent", "1e-6/Ang^2",  6.3, [0.0, inf],  "sld",         "Solvent scattering length density"],
132              ["theta",       "degrees",    0,    [-360, 360], "orientation", "c axis to beam angle"],
133              ["phi",         "degrees",    0,    [-360, 360], "orientation", "rotation about beam"],
134              ["psi",         "degrees",    0,    [-360, 360], "orientation", "rotation about c axis"]
135             ]
136# pylint: enable=bad-whitespace, line-too-long
137
138source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c", "lib/gauss150.c", "sc_paracrystal.c"]
139
140demo = dict(scale=1, background=0,
141            dnn=220.0,
142            d_factor=0.06,
143            radius=40.0,
144            sld=3.0,
145            sld_solvent=6.3,
146            theta=0.0,
147            phi=0.0,
148            psi=0.0)
149
150tests = [
151    # Accuracy tests based on content in test/utest_extra_models.py
152    [{}, 0.001, 10.3048],
153    [{}, 0.215268, 0.00814889],
154    [{}, (0.414467), 0.001313289]
155    ]
156
157
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