source: sasmodels/sasmodels/models/bcc_paracrystal.py @ b0c4271

core_shell_microgelscostrafo411magnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since b0c4271 was b0c4271, checked in by butler, 7 years ago

model documentation final format through core_shell_bicelle re: #646

  • Property mode set to 100644
File size: 5.2 KB
Line 
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/bcc_angle_definition.png
82
83    Orientation of the crystal with respect to the scattering plane.
84
85References
86----------
87
88.. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765
89   (Original Paper)
90.. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856
91   (Corrections to FCC and BCC lattice structure calculation)
92
93Authorship and Verification
94----------------------------
95
96* **Author:** NIST IGOR/DANSE **Date:** pre 2010
97* **Last Modified by:** Paul Butler **Date:** September 29, 2016
98* **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016
99"""
100
101from numpy import inf
102
103name = "bcc_paracrystal"
104title = "Body-centred cubic lattic with paracrystalline distortion"
105description = """
106    Calculates the scattering from a **body-centered cubic lattice** with
107    paracrystalline distortion. Thermal vibrations are considered to be
108    negligible, and the size of the paracrystal is infinitely large.
109    Paracrystalline distortion is assumed to be isotropic and characterized
110    by a Gaussian distribution.
111    """
112category = "shape:paracrystal"
113
114#note - calculation requires double precision
115single = False
116
117# pylint: disable=bad-whitespace, line-too-long
118#             ["name", "units", default, [lower, upper], "type","description" ],
119parameters = [["dnn",         "Ang",       220,    [-inf, inf], "",            "Nearest neighbour distance"],
120              ["d_factor",    "",            0.06, [-inf, inf], "",            "Paracrystal distortion factor"],
121              ["radius",      "Ang",        40,    [0, inf],    "volume",      "Particle radius"],
122              ["sld",         "1e-6/Ang^2",  4,    [-inf, inf], "sld",         "Particle scattering length density"],
123              ["sld_solvent", "1e-6/Ang^2",  1,    [-inf, inf], "sld",         "Solvent scattering length density"],
124              ["theta",       "degrees",    60,    [-inf, inf], "orientation", "In plane angle"],
125              ["phi",         "degrees",    60,    [-inf, inf], "orientation", "Out of plane angle"],
126              ["psi",         "degrees",    60,    [-inf, inf], "orientation", "Out of plane angle"]
127             ]
128# pylint: enable=bad-whitespace, line-too-long
129
130source = ["lib/sph_j1c.c", "lib/gauss150.c", "lib/sphere_form.c", "bcc_paracrystal.c"]
131
132# parameters for demo
133demo = dict(
134    scale=1, background=0,
135    dnn=220, d_factor=0.06, sld=4, sld_solvent=1,
136    radius=40,
137    theta=60, phi=60, psi=60,
138    radius_pd=.2, radius_pd_n=2,
139    theta_pd=15, theta_pd_n=0,
140    phi_pd=15, phi_pd_n=0,
141    psi_pd=15, psi_pd_n=0,
142    )
Note: See TracBrowser for help on using the repository browser.