source: sasmodels/sasmodels/models/core_shell_parallelepiped.py @ 10ee838

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 10ee838 was 10ee838, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

core_shell_parallelepiped: use ER from parallelepiped, with a, b, c dimensions including the rim

  • Property mode set to 100644
File size: 8.9 KB
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) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background}
14
15where $\langle \ldots \rangle$ is an average over all possible orientations
16of the rectangular solid.
17
18The function calculated is the form factor of the rectangular solid below.
19The core of the solid is defined by the dimensions $A$, $B$, $C$ such that
20$A < B < C$.
21
22.. image:: img/core_shell_parallelepiped_geometry.jpg
23
24There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension
25(on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$
26$(=t_C)$ faces. The projection in the $AB$ plane is then
27
28.. image:: img/core_shell_parallelepiped_projection.jpg
29
30The volume of the solid is
31
32.. math::
33
34    V = ABC + 2t_ABC + 2t_BAC + 2t_CAB
35
36**meaning that there are "gaps" at the corners of the solid.**
37
38The intensity calculated follows the :ref:`parallelepiped` model, with the
39core-shell intensity being calculated as the square of the sum of the
40amplitudes of the core and the slabs on the edges.
41
42the scattering amplitude is computed for a particular orientation of the
43core-shell parallelepiped with respect to the scattering vector and then
44averaged over all possible orientations, where $\alpha$ is the angle between
45the $z$ axis and the $C$ axis of the parallelepiped, $\beta$ is
46the angle between projection of the particle in the $xy$ detector plane
47and the $y$ axis.
48
49.. math::
50
51    F(Q)
52    &= (\rho_\text{core}-\rho_\text{solvent})
53       S(Q_A, A) S(Q_B, B) S(Q_C, C) \\
54    &+ (\rho_\text{A}-\rho_\text{solvent})
55        \left[S(Q_A, A+2t_A) - S(Q_A, Q)\right] S(Q_B, B) S(Q_C, C) \\
56    &+ (\rho_\text{B}-\rho_\text{solvent})
57        S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\
58    &+ (\rho_\text{C}-\rho_\text{solvent})
59        S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right]
60
61with
62
63.. math::
64
65    S(Q, L) = L \frac{\sin \tfrac{1}{2} Q L}{\tfrac{1}{2} Q L}
66
67and
68
69.. math::
70
71    Q_A &= \sin\alpha \sin\beta \\
72    Q_B &= \sin\alpha \cos\beta \\
73    Q_C &= \cos\alpha
74
75
76where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$
77are the scattering length of the parallelepiped core, and the rectangular
78slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$
79is the scattering length of the solvent.
80
81FITTING NOTES
82~~~~~~~~~~~~~
83
84If the scale is set equal to the particle volume fraction, $\phi$, the returned
85value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. However,
86**no interparticle interference effects are included in this calculation.**
87
88There are many parameters in this model. Hold as many fixed as possible with
89known values, or you will certainly end up at a solution that is unphysical.
90
91The returned value is in units of |cm^-1|, on absolute scale.
92
93NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated
94based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$
95and length $(C+2t_C)$ values, after appropriately sorting the three dimensions
96to give an oblate or prolate particle, to give an effective radius,
97for $S(Q)$ when $P(Q) * S(Q)$ is applied.
98
99For 2d data the orientation of the particle is required, described using
100angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further
101details of the calculation and angular dispersions see :ref:`orientation`.
102The angle $\Psi$ is the rotational angle around the *long_c* axis. For example,
103$\Psi = 0$ when the *short_b* axis is parallel to the *x*-axis of the detector.
104
105For 2d, constraints must be applied during fitting to ensure that the
106inequality $A < B < C$ is not violated, and hence the correct definition
107of angles is preserved. The calculation will not report an error,
108but the results may be not correct.
109
110.. figure:: img/parallelepiped_angle_definition.png
111
112    Definition of the angles for oriented core-shell parallelepipeds.
113    Note that rotation $\theta$, initially in the $xz$ plane, is carried
114    out first, then rotation $\phi$ about the $z$ axis, finally rotation
115    $\Psi$ is now around the axis of the cylinder. The neutron or X-ray
116    beam is along the $z$ axis.
117
118.. figure:: img/parallelepiped_angle_projection.png
119
120    Examples of the angles for oriented core-shell parallelepipeds against the
121    detector plane.
122
123References
124----------
125
126.. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211
127    Equations (1), (13-14). (in German)
128.. [#] D Singh (2009). *Small angle scattering studies of self assembly in
129   lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available
130   from Proquest <http://search.proquest.com/docview/304915826?accountid
131   =26379>`_
132
133Authorship and Verification
134----------------------------
135
136* **Author:** NIST IGOR/DANSE **Date:** pre 2010
137* **Converted to sasmodels by:** Miguel Gonzales **Date:** February 26, 2016
138* **Last Modified by:** Wojciech Potrzebowski **Date:** January 11, 2017
139* **Currently Under review by:** Paul Butler
140"""
141
142import numpy as np
143from numpy import pi, inf, sqrt, cos, sin
144
145name = "core_shell_parallelepiped"
146title = "Rectangular solid with a core-shell structure."
147description = """
148     P(q)=
149"""
150category = "shape:parallelepiped"
151
152#             ["name", "units", default, [lower, upper], "type","description"],
153parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld",
154               "Parallelepiped core scattering length density"],
155              ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld",
156               "Parallelepiped A rim scattering length density"],
157              ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld",
158               "Parallelepiped B rim scattering length density"],
159              ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld",
160               "Parallelepiped C rim scattering length density"],
161              ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
162               "Solvent scattering length density"],
163              ["length_a", "Ang", 35, [0, inf], "volume",
164               "Shorter side of the parallelepiped"],
165              ["length_b", "Ang", 75, [0, inf], "volume",
166               "Second side of the parallelepiped"],
167              ["length_c", "Ang", 400, [0, inf], "volume",
168               "Larger side of the parallelepiped"],
169              ["thick_rim_a", "Ang", 10, [0, inf], "volume",
170               "Thickness of A rim"],
171              ["thick_rim_b", "Ang", 10, [0, inf], "volume",
172               "Thickness of B rim"],
173              ["thick_rim_c", "Ang", 10, [0, inf], "volume",
174               "Thickness of C rim"],
175              ["theta", "degrees", 0, [-360, 360], "orientation",
176               "c axis to beam angle"],
177              ["phi", "degrees", 0, [-360, 360], "orientation",
178               "rotation about beam"],
179              ["psi", "degrees", 0, [-360, 360], "orientation",
180               "rotation about c axis"],
181             ]
182
183source = ["lib/gauss76.c", "core_shell_parallelepiped.c"]
184
185
186def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c):
187    """
188        Return equivalent radius (ER)
189    """
190    from .parallelepiped import ER as ER_p
191
192    a = length_a + 2*thick_rim_a
193    b = length_b + 2*thick_rim_b
194    c = length_c + 2*thick_rim_c
195    return ER_p(a, b, c)
196
197# VR defaults to 1.0
198
199def random():
200    outer = 10**np.random.uniform(1, 4.7, size=3)
201    thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1
202    length = outer - thick
203    pars = dict(
204        length_a=length[0],
205        length_b=length[1],
206        length_c=length[2],
207        thick_rim_a=thick[0],
208        thick_rim_b=thick[1],
209        thick_rim_c=thick[2],
210    )
211    return pars
212
213# parameters for demo
214demo = dict(scale=1, background=0.0,
215            sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6,
216            length_a=35, length_b=75, length_c=400,
217            thick_rim_a=10, thick_rim_b=10, thick_rim_c=10,
218            theta=0, phi=0, psi=0,
219            length_a_pd=0.1, length_a_pd_n=1,
220            length_b_pd=0.1, length_b_pd_n=1,
221            length_c_pd=0.1, length_c_pd_n=1,
222            thick_rim_a_pd=0.1, thick_rim_a_pd_n=1,
223            thick_rim_b_pd=0.1, thick_rim_b_pd_n=1,
224            thick_rim_c_pd=0.1, thick_rim_c_pd_n=1,
225            theta_pd=10, theta_pd_n=1,
226            phi_pd=10, phi_pd_n=1,
227            psi_pd=10, psi_pd_n=1)
228
229# rkh 7/4/17 add random unit test for 2d, note make all params different,
230# 2d values not tested against other codes or models
231if 0:  # pak: model rewrite; need to update tests
232    qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.)
233    tests = [[{}, 0.2, 0.533149288477],
234             [{}, [0.2], [0.533149288477]],
235             [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222],
236             [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]],
237            ]
238    del qx, qy  # not necessary to delete, but cleaner
Note: See TracBrowser for help on using the repository browser.