source: sasmodels/sasmodels/models/core_shell_parallelepiped.py @ 5bc6d21

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 5bc6d21 was 5bc6d21, checked in by butler, 14 months ago

more standardization of text

moved validation to its own section (is that the standard?) and moved
parallelipiped frong stuff around so start with Definition as per
standard.

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Line 
1r"""
2Definition
3----------
4
5Calculates the form factor for a rectangular solid with a core-shell structure.
6The thickness and the scattering length density of the shell or
7"rim" can be different on each (pair) of faces.
8
9The form factor is normalized by the particle volume $V$ such that
10
11.. math::
12
13    I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle
14    + \text{background}
15
16where $\langle \ldots \rangle$ is an average over all possible orientations
17of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be
18pulled out of the form factor term due to the multiple slds in the model.
19
20The core of the solid is defined by the dimensions $A$, $B$, $C$ such that
21$A < B < C$.
22
23.. figure:: img/parallelepiped_geometry.jpg
24
25   Core of the core shell parallelepiped with the corresponding definition
26   of sides.
27
28
29There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension
30(on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$
31$(=t_C)$ faces. The projection in the $AB$ plane is
32
33.. figure:: img/core_shell_parallelepiped_projection.jpg
34
35   AB cut through the core-shell parallelipiped showing the cross secion of
36   four of the six shell slabs. As can be seen, this model leaves **"gaps"**
37   at the corners of the solid.
38
39
40The total volume of the solid is thus given as
41
42.. math::
43
44    V = ABC + 2t_ABC + 2t_BAC + 2t_CAB
45
46The intensity calculated follows the :ref:`parallelepiped` model, with the
47core-shell intensity being calculated as the square of the sum of the
48amplitudes of the core and the slabs on the edges. The scattering amplitude is
49computed for a particular orientation of the core-shell parallelepiped with
50respect to the scattering vector and then averaged over all possible
51orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis
52of the parallelepiped, and $\beta$ is the angle between the projection of the
53particle in the $xy$ detector plane and the $y$ axis.
54
55.. math::
56
57    P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha
58    \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta}
59
60and
61
62.. math::
63
64    F(q,\alpha,\beta)
65    &= (\rho_\text{core}-\rho_\text{solvent})
66       S(Q_A, A) S(Q_B, B) S(Q_C, C) \\
67    &+ (\rho_\text{A}-\rho_\text{solvent})
68        \left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\
69    &+ (\rho_\text{B}-\rho_\text{solvent})
70        S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\
71    &+ (\rho_\text{C}-\rho_\text{solvent})
72        S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right]
73
74with
75
76.. math::
77
78    S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L}
79
80and
81
82.. math::
83
84    Q_A &= q \sin\alpha \sin\beta \\
85    Q_B &= q \sin\alpha \cos\beta \\
86    Q_C &= q \cos\alpha
87
88
89where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$
90are the scattering lengths of the parallelepiped core, and the rectangular
91slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$
92is the scattering length of the solvent.
93
94.. note::
95
96   the code actually implements two substitutions: $d(cos\alpha)$ is
97   substituted for -$sin\alpha \ d\alpha$ (note that in the
98   :ref:`parallelepiped` code this is explicitly implemented with
99   $\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that
100   $du = \pi/2 \ d\beta$.  Thus both integrals go from 0 to 1 rather than 0
101   to $\pi/2$.
102
103FITTING NOTES
104~~~~~~~~~~~~~
105
106If the scale is set equal to the particle volume fraction, $\phi$, the returned
107value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. However,
108**no interparticle interference effects are included in this calculation.**
109
110There are many parameters in this model. Hold as many fixed as possible with
111known values, or you will certainly end up at a solution that is unphysical.
112
113The returned value is in units of |cm^-1|, on absolute scale.
114
115NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated
116based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$
117and length $(C+2t_C)$ values, after appropriately sorting the three dimensions
118to give an oblate or prolate particle, to give an effective radius
119for $S(q)$ when $P(q) * S(q)$ is applied.
120
121For 2d data the orientation of the particle is required, described using
122angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below. For further
123details of the calculation and angular dispersions see :ref:`orientation`.
124The angle $\Psi$ is the rotational angle around the *long_c* axis. For example,
125$\Psi = 0$ when the *short_b* axis is parallel to the *x*-axis of the detector.
126
127.. note:: For 2d, constraints must be applied during fitting to ensure that the
128   inequality $A < B < C$ is not violated, and hence the correct definition
129   of angles is preserved. The calculation will not report an error,
130   but the results may be not correct.
131
132.. figure:: img/parallelepiped_angle_definition.png
133
134    Definition of the angles for oriented core-shell parallelepipeds.
135    Note that rotation $\theta$, initially in the $xz$ plane, is carried
136    out first, then rotation $\phi$ about the $z$ axis, finally rotation
137    $\Psi$ is now around the axis of the particle. The neutron or X-ray
138    beam is along the $z$ axis.
139
140.. figure:: img/parallelepiped_angle_projection.png
141
142    Examples of the angles for oriented core-shell parallelepipeds against the
143    detector plane.
144
145
146Validation
147----------
148
149Cross-checked against hollow rectangular prism and rectangular prism for equal
150thickness overlapping sides, and by Monte Carlo sampling of points within the
151shape for non-uniform, non-overlapping sides.
152
153
154References
155----------
156
157.. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211
158    Equations (1), (13-14). (in German)
159.. [#] D Singh (2009). *Small angle scattering studies of self assembly in
160   lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available
161   from Proquest <http://search.proquest.com/docview/304915826?accountid
162   =26379>`_
163
164Authorship and Verification
165----------------------------
166
167* **Author:** NIST IGOR/DANSE **Date:** pre 2010
168* **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016
169* **Last Modified by:** Paul Kienzle **Date:** October 17, 2017
170"""
171
172import numpy as np
173from numpy import pi, inf, sqrt, cos, sin
174
175name = "core_shell_parallelepiped"
176title = "Rectangular solid with a core-shell structure."
177description = """
178     P(q)=
179"""
180category = "shape:parallelepiped"
181
182#             ["name", "units", default, [lower, upper], "type","description"],
183parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld",
184               "Parallelepiped core scattering length density"],
185              ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld",
186               "Parallelepiped A rim scattering length density"],
187              ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld",
188               "Parallelepiped B rim scattering length density"],
189              ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld",
190               "Parallelepiped C rim scattering length density"],
191              ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
192               "Solvent scattering length density"],
193              ["length_a", "Ang", 35, [0, inf], "volume",
194               "Shorter side of the parallelepiped"],
195              ["length_b", "Ang", 75, [0, inf], "volume",
196               "Second side of the parallelepiped"],
197              ["length_c", "Ang", 400, [0, inf], "volume",
198               "Larger side of the parallelepiped"],
199              ["thick_rim_a", "Ang", 10, [0, inf], "volume",
200               "Thickness of A rim"],
201              ["thick_rim_b", "Ang", 10, [0, inf], "volume",
202               "Thickness of B rim"],
203              ["thick_rim_c", "Ang", 10, [0, inf], "volume",
204               "Thickness of C rim"],
205              ["theta", "degrees", 0, [-360, 360], "orientation",
206               "c axis to beam angle"],
207              ["phi", "degrees", 0, [-360, 360], "orientation",
208               "rotation about beam"],
209              ["psi", "degrees", 0, [-360, 360], "orientation",
210               "rotation about c axis"],
211             ]
212
213source = ["lib/gauss76.c", "core_shell_parallelepiped.c"]
214
215
216def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c):
217    """
218        Return equivalent radius (ER)
219    """
220    from .parallelepiped import ER as ER_p
221
222    a = length_a + 2*thick_rim_a
223    b = length_b + 2*thick_rim_b
224    c = length_c + 2*thick_rim_c
225    return ER_p(a, b, c)
226
227# VR defaults to 1.0
228
229def random():
230    outer = 10**np.random.uniform(1, 4.7, size=3)
231    thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1
232    length = outer - thick
233    pars = dict(
234        length_a=length[0],
235        length_b=length[1],
236        length_c=length[2],
237        thick_rim_a=thick[0],
238        thick_rim_b=thick[1],
239        thick_rim_c=thick[2],
240    )
241    return pars
242
243# parameters for demo
244demo = dict(scale=1, background=0.0,
245            sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6,
246            length_a=35, length_b=75, length_c=400,
247            thick_rim_a=10, thick_rim_b=10, thick_rim_c=10,
248            theta=0, phi=0, psi=0,
249            length_a_pd=0.1, length_a_pd_n=1,
250            length_b_pd=0.1, length_b_pd_n=1,
251            length_c_pd=0.1, length_c_pd_n=1,
252            thick_rim_a_pd=0.1, thick_rim_a_pd_n=1,
253            thick_rim_b_pd=0.1, thick_rim_b_pd_n=1,
254            thick_rim_c_pd=0.1, thick_rim_c_pd_n=1,
255            theta_pd=10, theta_pd_n=1,
256            phi_pd=10, phi_pd_n=1,
257            psi_pd=10, psi_pd_n=1)
258
259# rkh 7/4/17 add random unit test for 2d, note make all params different,
260# 2d values not tested against other codes or models
261if 0:  # pak: model rewrite; need to update tests
262    qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.)
263    tests = [[{}, 0.2, 0.533149288477],
264             [{}, [0.2], [0.533149288477]],
265             [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222],
266             [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]],
267            ]
268    del qx, qy  # not necessary to delete, but cleaner
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