1 | r""" |
---|
2 | Definition |
---|
3 | ---------- |
---|
4 | |
---|
5 | Calculates the form factor for a rectangular solid with a core-shell structure. |
---|
6 | The thickness and the scattering length density of the shell or "rim" can be |
---|
7 | different on each (pair) of faces. The three dimensions of the core of the |
---|
8 | parallelepiped (strictly here a cuboid) may be given in *any* size order as |
---|
9 | long as the particles are randomly oriented (i.e. take on all possible |
---|
10 | orientations see notes on 2D below). To avoid multiple fit solutions, |
---|
11 | especially with Monte-Carlo fit methods, it may be advisable to restrict their |
---|
12 | ranges. There may be a number of closely similar "best fits", so some trial and |
---|
13 | error, or fixing of some dimensions at expected values, may help. |
---|
14 | |
---|
15 | The form factor is normalized by the particle volume $V$ such that |
---|
16 | |
---|
17 | .. math:: |
---|
18 | |
---|
19 | I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle |
---|
20 | + \text{background} |
---|
21 | |
---|
22 | where $\langle \ldots \rangle$ is an average over all possible orientations |
---|
23 | of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be |
---|
24 | pulled out of the form factor term due to the multiple slds in the model. |
---|
25 | |
---|
26 | The core of the solid is defined by the dimensions $A$, $B$, $C$ here shown |
---|
27 | such that $A < B < C$. |
---|
28 | |
---|
29 | .. figure:: img/parallelepiped_geometry.jpg |
---|
30 | |
---|
31 | Core of the core shell parallelepiped with the corresponding definition |
---|
32 | of sides. |
---|
33 | |
---|
34 | |
---|
35 | There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension |
---|
36 | (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ |
---|
37 | $(=t_C)$ faces. The projection in the $AB$ plane is |
---|
38 | |
---|
39 | .. figure:: img/core_shell_parallelepiped_projection.jpg |
---|
40 | |
---|
41 | AB cut through the core-shell parallelipiped showing the cross secion of |
---|
42 | four of the six shell slabs. As can be seen, this model leaves **"gaps"** |
---|
43 | at the corners of the solid. |
---|
44 | |
---|
45 | |
---|
46 | The total volume of the solid is thus given as |
---|
47 | |
---|
48 | .. math:: |
---|
49 | |
---|
50 | V = ABC + 2t_ABC + 2t_BAC + 2t_CAB |
---|
51 | |
---|
52 | The intensity calculated follows the :ref:`parallelepiped` model, with the |
---|
53 | core-shell intensity being calculated as the square of the sum of the |
---|
54 | amplitudes of the core and the slabs on the edges. The scattering amplitude is |
---|
55 | computed for a particular orientation of the core-shell parallelepiped with |
---|
56 | respect to the scattering vector and then averaged over all possible |
---|
57 | orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis |
---|
58 | of the parallelepiped, and $\beta$ is the angle between the projection of the |
---|
59 | particle in the $xy$ detector plane and the $y$ axis. |
---|
60 | |
---|
61 | .. math:: |
---|
62 | |
---|
63 | P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha |
---|
64 | \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta} |
---|
65 | |
---|
66 | and |
---|
67 | |
---|
68 | .. math:: |
---|
69 | |
---|
70 | F(q,\alpha,\beta) |
---|
71 | &= (\rho_\text{core}-\rho_\text{solvent}) |
---|
72 | S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ |
---|
73 | &+ (\rho_\text{A}-\rho_\text{solvent}) |
---|
74 | \left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\ |
---|
75 | &+ (\rho_\text{B}-\rho_\text{solvent}) |
---|
76 | S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ |
---|
77 | &+ (\rho_\text{C}-\rho_\text{solvent}) |
---|
78 | S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right] |
---|
79 | |
---|
80 | with |
---|
81 | |
---|
82 | .. math:: |
---|
83 | |
---|
84 | S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L} |
---|
85 | |
---|
86 | and |
---|
87 | |
---|
88 | .. math:: |
---|
89 | |
---|
90 | Q_A &= q \sin\alpha \sin\beta \\ |
---|
91 | Q_B &= q \sin\alpha \cos\beta \\ |
---|
92 | Q_C &= q \cos\alpha |
---|
93 | |
---|
94 | |
---|
95 | where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$ |
---|
96 | are the scattering lengths of the parallelepiped core, and the rectangular |
---|
97 | slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$ |
---|
98 | is the scattering length of the solvent. |
---|
99 | |
---|
100 | .. note:: |
---|
101 | |
---|
102 | the code actually implements two substitutions: $d(cos\alpha)$ is |
---|
103 | substituted for -$sin\alpha \ d\alpha$ (note that in the |
---|
104 | :ref:`parallelepiped` code this is explicitly implemented with |
---|
105 | $\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that |
---|
106 | $du = \pi/2 \ d\beta$. Thus both integrals go from 0 to 1 rather than 0 |
---|
107 | to $\pi/2$. |
---|
108 | |
---|
109 | FITTING NOTES |
---|
110 | ~~~~~~~~~~~~~ |
---|
111 | |
---|
112 | #. There are many parameters in this model. Hold as many fixed as possible with |
---|
113 | known values, or you will certainly end up at a solution that is unphysical. |
---|
114 | |
---|
115 | #. The 2nd virial coefficient of the core_shell_parallelepiped is calculated |
---|
116 | based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ |
---|
117 | and length $(C+2t_C)$ values, after appropriately sorting the three |
---|
118 | dimensions to give an oblate or prolate particle, to give an effective radius |
---|
119 | for $S(q)$ when $P(q) * S(q)$ is applied. |
---|
120 | |
---|
121 | #. For 2d data the orientation of the particle is required, described using |
---|
122 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ |
---|
123 | and $\phi$ define the orientation of the director in the laboratry reference |
---|
124 | frame of the beam direction ($z$) and detector plane ($x-y$ plane), while |
---|
125 | the angle $\Psi$ is effectively the rotational angle around the particle |
---|
126 | $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the |
---|
127 | $B$ axis oriented parallel to the y-axis of the detector with $A$ along |
---|
128 | the x-axis. For other $\theta$, $\phi$ values, the order of rotations |
---|
129 | matters. In particular, the parallelepiped must first be rotated $\theta$ |
---|
130 | degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ |
---|
131 | axis (in the $x-y$ plane). Applying orientational distribution to the |
---|
132 | particle orientation (i.e `jitter` to one or more of these angles) can get |
---|
133 | more confusing as `jitter` is defined **NOT** with respect to the laboratory |
---|
134 | frame but the particle reference frame. It is thus highly recmmended to |
---|
135 | read :ref:`orientation` for further details of the calculation and angular |
---|
136 | dispersions. |
---|
137 | |
---|
138 | .. note:: For 2d, constraints must be applied during fitting to ensure that the |
---|
139 | order of sides chosen is not altered, and hence that the correct definition |
---|
140 | of angles is preserved. For the default choice shown here, that means |
---|
141 | ensuring that the inequality $A < B < C$ is not violated, The calculation |
---|
142 | will not report an error, but the results may be not correct. |
---|
143 | |
---|
144 | .. figure:: img/parallelepiped_angle_definition.png |
---|
145 | |
---|
146 | Definition of the angles for oriented core-shell parallelepipeds. |
---|
147 | Note that rotation $\theta$, initially in the $x-z$ plane, is carried |
---|
148 | out first, then rotation $\phi$ about the $z$ axis, finally rotation |
---|
149 | $\Psi$ is now around the $C$ axis of the particle. The neutron or X-ray |
---|
150 | beam is along the $z$ axis and the detecotr defines the $x-y$ plane. |
---|
151 | |
---|
152 | .. figure:: img/parallelepiped_angle_projection.png |
---|
153 | |
---|
154 | Examples of the angles for oriented core-shell parallelepipeds against the |
---|
155 | detector plane. |
---|
156 | |
---|
157 | |
---|
158 | Validation |
---|
159 | ---------- |
---|
160 | |
---|
161 | Cross-checked against hollow rectangular prism and rectangular prism for equal |
---|
162 | thickness overlapping sides, and by Monte Carlo sampling of points within the |
---|
163 | shape for non-uniform, non-overlapping sides. |
---|
164 | |
---|
165 | |
---|
166 | References |
---|
167 | ---------- |
---|
168 | |
---|
169 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
---|
170 | Equations (1), (13-14). (in German) |
---|
171 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
---|
172 | lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available |
---|
173 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
---|
174 | =26379>`_ |
---|
175 | .. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659 |
---|
176 | |
---|
177 | Source |
---|
178 | ------ |
---|
179 | |
---|
180 | `core_shell_parallelepiped.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/core_shell_parallelepiped.py>`_ |
---|
181 | |
---|
182 | `core_shell_parallelepiped.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/core_shell_parallelepiped.c>`_ |
---|
183 | |
---|
184 | Authorship and Verification |
---|
185 | ---------------------------- |
---|
186 | |
---|
187 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
---|
188 | * **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016 |
---|
189 | * **Last Modified by:** Paul Kienzle **Date:** October 17, 2017 |
---|
190 | * **Last Reviewed by:** Paul Butler **Date:** May 24, 2018 - documentation |
---|
191 | updated |
---|
192 | * **Source added by :** Steve King **Date:** March 25, 2019 |
---|
193 | """ |
---|
194 | |
---|
195 | import numpy as np |
---|
196 | from numpy import inf |
---|
197 | |
---|
198 | name = "core_shell_parallelepiped" |
---|
199 | title = "Rectangular solid with a core-shell structure." |
---|
200 | description = """ |
---|
201 | P(q)= |
---|
202 | """ |
---|
203 | category = "shape:parallelepiped" |
---|
204 | |
---|
205 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
206 | parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
---|
207 | "Parallelepiped core scattering length density"], |
---|
208 | ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
---|
209 | "Parallelepiped A rim scattering length density"], |
---|
210 | ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
---|
211 | "Parallelepiped B rim scattering length density"], |
---|
212 | ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
---|
213 | "Parallelepiped C rim scattering length density"], |
---|
214 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
---|
215 | "Solvent scattering length density"], |
---|
216 | ["length_a", "Ang", 35, [0, inf], "volume", |
---|
217 | "Shorter side of the parallelepiped"], |
---|
218 | ["length_b", "Ang", 75, [0, inf], "volume", |
---|
219 | "Second side of the parallelepiped"], |
---|
220 | ["length_c", "Ang", 400, [0, inf], "volume", |
---|
221 | "Larger side of the parallelepiped"], |
---|
222 | ["thick_rim_a", "Ang", 10, [0, inf], "volume", |
---|
223 | "Thickness of A rim"], |
---|
224 | ["thick_rim_b", "Ang", 10, [0, inf], "volume", |
---|
225 | "Thickness of B rim"], |
---|
226 | ["thick_rim_c", "Ang", 10, [0, inf], "volume", |
---|
227 | "Thickness of C rim"], |
---|
228 | ["theta", "degrees", 0, [-360, 360], "orientation", |
---|
229 | "c axis to beam angle"], |
---|
230 | ["phi", "degrees", 0, [-360, 360], "orientation", |
---|
231 | "rotation about beam"], |
---|
232 | ["psi", "degrees", 0, [-360, 360], "orientation", |
---|
233 | "rotation about c axis"], |
---|
234 | ] |
---|
235 | |
---|
236 | source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] |
---|
237 | have_Fq = True |
---|
238 | radius_effective_modes = [ |
---|
239 | "equivalent cylinder excluded volume", |
---|
240 | "equivalent volume sphere", |
---|
241 | "half outer length_a", "half outer length_b", "half outer length_c", |
---|
242 | "equivalent circular cross-section", |
---|
243 | "half outer ab diagonal", "half outer diagonal", |
---|
244 | ] |
---|
245 | |
---|
246 | def random(): |
---|
247 | """Return a random parameter set for the model.""" |
---|
248 | outer = 10**np.random.uniform(1, 4.7, size=3) |
---|
249 | thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1 |
---|
250 | length = outer - thick |
---|
251 | pars = dict( |
---|
252 | length_a=length[0], |
---|
253 | length_b=length[1], |
---|
254 | length_c=length[2], |
---|
255 | thick_rim_a=thick[0], |
---|
256 | thick_rim_b=thick[1], |
---|
257 | thick_rim_c=thick[2], |
---|
258 | ) |
---|
259 | return pars |
---|
260 | |
---|
261 | # parameters for demo |
---|
262 | demo = dict(scale=1, background=0.0, |
---|
263 | sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, |
---|
264 | length_a=35, length_b=75, length_c=400, |
---|
265 | thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, |
---|
266 | theta=0, phi=0, psi=0, |
---|
267 | length_a_pd=0.1, length_a_pd_n=1, |
---|
268 | length_b_pd=0.1, length_b_pd_n=1, |
---|
269 | length_c_pd=0.1, length_c_pd_n=1, |
---|
270 | thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, |
---|
271 | thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, |
---|
272 | thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, |
---|
273 | theta_pd=10, theta_pd_n=1, |
---|
274 | phi_pd=10, phi_pd_n=1, |
---|
275 | psi_pd=10, psi_pd_n=1) |
---|
276 | |
---|
277 | # rkh 7/4/17 add random unit test for 2d, note make all params different, |
---|
278 | # 2d values not tested against other codes or models |
---|
279 | if 0: # pak: model rewrite; need to update tests |
---|
280 | qx, qy = 0.2 * np.cos(np.pi/6.), 0.2 * np.sin(np.pi/6.) |
---|
281 | tests = [[{}, 0.2, 0.533149288477], |
---|
282 | [{}, [0.2], [0.533149288477]], |
---|
283 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222], |
---|
284 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]], |
---|
285 | ] |
---|
286 | del qx, qy # not necessary to delete, but cleaner |
---|