1 | r""" |
---|
2 | This model provides the form factor, $P(q)$, for a multi-shell sphere where |
---|
3 | the scattering length density (SLD) of each shell is described by an |
---|
4 | exponential, linear, or constant function. The form factor is normalized by |
---|
5 | the volume of the sphere where the SLD is not identical to the SLD of the |
---|
6 | solvent. We currently provide up to 9 shells with this model. |
---|
7 | |
---|
8 | NB: *radius* represents the core radius $r_0$ and |
---|
9 | *thickness[k]* represents the thickness of the shell, $r_{k+1} - r_k$. |
---|
10 | |
---|
11 | Definition |
---|
12 | ---------- |
---|
13 | |
---|
14 | The 1D scattering intensity is calculated in the following way |
---|
15 | |
---|
16 | .. math:: |
---|
17 | |
---|
18 | P(q) = [f]^2 / V_\text{particle} |
---|
19 | |
---|
20 | where |
---|
21 | |
---|
22 | .. math:: |
---|
23 | :nowrap: |
---|
24 | |
---|
25 | \begin{align*} |
---|
26 | f &= f_\text{core} |
---|
27 | + \left(\sum_{\text{shell}=1}^N f_\text{shell}\right) |
---|
28 | + f_\text{solvent} |
---|
29 | \end{align*} |
---|
30 | |
---|
31 | The shells are spherically symmetric with particle density $\rho(r)$ and |
---|
32 | constant SLD within the core and solvent, so |
---|
33 | |
---|
34 | .. math:: |
---|
35 | :nowrap: |
---|
36 | |
---|
37 | \begin{align*} |
---|
38 | f_\text{core} |
---|
39 | &= 4\pi\int_0^{r_\text{core}} \rho_\text{core} |
---|
40 | \frac{\sin(qr)}{qr}\, r^2\,\mathrm{d}r |
---|
41 | &= 3\rho_\text{core} V(r_\text{core}) |
---|
42 | \frac{j_1(qr_\text{core})}{qr_\text{core}} \\ |
---|
43 | f_\text{shell} |
---|
44 | &= 4\pi\int_{r_{\text{shell}-1}}^{r_\text{shell}} |
---|
45 | \rho_\text{shell}(r)\frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r \\ |
---|
46 | f_\text{solvent} |
---|
47 | &= 4\pi\int_{r_N}^\infty |
---|
48 | \rho_\text{solvent}\frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r |
---|
49 | &= -3\rho_\text{solvent}V(r_N)\frac{j_1(q r_N)}{q r_N} |
---|
50 | \end{align*} |
---|
51 | |
---|
52 | where the spherical bessel function $j_1$ is |
---|
53 | |
---|
54 | .. math:: |
---|
55 | |
---|
56 | j_1(x) = \frac{\sin(x)}{x^2} - \frac{\cos(x)}{x} |
---|
57 | |
---|
58 | and the volume is $V(r) = \frac{4\pi}{3}r^3$. The volume of the particle |
---|
59 | is determined by the radius of the outer shell, so $V_\text{particle} = V(r_N)$. |
---|
60 | |
---|
61 | Now lets consider the SLD of a shell defined by |
---|
62 | |
---|
63 | .. math:: |
---|
64 | |
---|
65 | \rho_\text{shell}(r) = \begin{cases} |
---|
66 | B\exp\left(A(r-r_{\text{shell}-1})/\Delta t_\text{shell}\right) |
---|
67 | + C & \mbox{for } A \neq 0 \\ |
---|
68 | \rho_\text{in} = \text{constant} & \mbox{for } A = 0 |
---|
69 | \end{cases} |
---|
70 | |
---|
71 | An example of a possible SLD profile is shown below where |
---|
72 | $\rho_\text{in}$ and $\Delta t_\text{shell}$ stand for the |
---|
73 | SLD of the inner side of the $k^\text{th}$ shell and the |
---|
74 | thickness of the $k^\text{th}$ shell in the equation above, respectively. |
---|
75 | |
---|
76 | For $A > 0$, |
---|
77 | |
---|
78 | .. math:: |
---|
79 | |
---|
80 | f_\text{shell} &= 4 \pi \int_{r_{\text{shell}-1}}^{r_\text{shell}} |
---|
81 | \left[ B\exp |
---|
82 | \left(A (r - r_{\text{shell}-1}) / \Delta t_\text{shell} \right) + C |
---|
83 | \right] \frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r \\ |
---|
84 | &= 3BV(r_\text{shell}) e^A h(\alpha_\text{out},\beta_\text{out}) |
---|
85 | - 3BV(r_{\text{shell}-1}) h(\alpha_\text{in},\beta_\text{in}) |
---|
86 | + 3CV(r_{\text{shell}}) \frac{j_1(\beta_\text{out})}{\beta_\text{out}} |
---|
87 | - 3CV(r_{\text{shell}-1}) \frac{j_1(\beta_\text{in})}{\beta_\text{in}} |
---|
88 | |
---|
89 | for |
---|
90 | |
---|
91 | .. math:: |
---|
92 | :nowrap: |
---|
93 | |
---|
94 | \begin{align*} |
---|
95 | B&=\frac{\rho_\text{out} - \rho_\text{in}}{e^A-1} |
---|
96 | & C &= \frac{\rho_\text{in}e^A - \rho_\text{out}}{e^A-1} \\ |
---|
97 | \alpha_\text{in} &= A\frac{r_{\text{shell}-1}}{\Delta t_\text{shell}} |
---|
98 | & \alpha_\text{out} &= A\frac{r_\text{shell}}{\Delta t_\text{shell}} \\ |
---|
99 | \beta_\text{in} &= qr_{\text{shell}-1} |
---|
100 | & \beta_\text{out} &= qr_\text{shell} \\ |
---|
101 | \end{align*} |
---|
102 | |
---|
103 | where $h$ is |
---|
104 | |
---|
105 | .. math:: |
---|
106 | |
---|
107 | h(x,y) = \frac{x \sin(y) - y\cos(y)}{(x^2+y^2)y} |
---|
108 | - \frac{(x^2-y^2)\sin(y) - 2xy\cos(y)}{(x^2+y^2)^2y} |
---|
109 | |
---|
110 | |
---|
111 | For $A \sim 0$, e.g., $A = -0.0001$, this function converges to that of the |
---|
112 | linear SLD profile with |
---|
113 | $\rho_\text{shell}(r) \approx A(r-r_{\text{shell}-1})/\Delta t_\text{shell})+B$, |
---|
114 | so this case is equivalent to |
---|
115 | |
---|
116 | .. math:: |
---|
117 | :nowrap: |
---|
118 | |
---|
119 | \begin{align*} |
---|
120 | f_\text{shell} |
---|
121 | &= |
---|
122 | 3 V(r_\text{shell}) \frac{\Delta\rho_\text{shell}}{\Delta t_\text{shell}} |
---|
123 | \left[\frac{ |
---|
124 | 2 \cos(qr_\text{out}) |
---|
125 | + qr_\text{out} \sin(qr_\text{out}) |
---|
126 | }{ |
---|
127 | (qr_\text{out})^4 |
---|
128 | }\right] \\ |
---|
129 | &{} |
---|
130 | -3 V(r_\text{shell}) \frac{\Delta\rho_\text{shell}}{\Delta t_\text{shell}} |
---|
131 | \left[\frac{ |
---|
132 | 2\cos(qr_\text{in}) |
---|
133 | +qr_\text{in}\sin(qr_\text{in}) |
---|
134 | }{ |
---|
135 | (qr_\text{in})^4 |
---|
136 | }\right] \\ |
---|
137 | &{} |
---|
138 | +3\rho_\text{out}V(r_\text{shell}) \frac{j_1(qr_\text{out})}{qr_\text{out}} |
---|
139 | -3\rho_\text{in}V(r_{\text{shell}-1}) \frac{j_1(qr_\text{in})}{qr_\text{in}} |
---|
140 | \end{align*} |
---|
141 | |
---|
142 | For $A = 0$, the exponential function has no dependence on the radius (so that |
---|
143 | $\rho_\text{out}$ is ignored in this case) and becomes flat. We set the constant |
---|
144 | to $\rho_\text{in}$ for convenience, and thus the form factor contributed by |
---|
145 | the shells is |
---|
146 | |
---|
147 | .. math:: |
---|
148 | |
---|
149 | f_\text{shell} = |
---|
150 | 3\rho_\text{in}V(r_\text{shell}) |
---|
151 | \frac{j_1(qr_\text{out})}{qr_\text{out}} |
---|
152 | - 3\rho_\text{in}V(r_{\text{shell}-1}) |
---|
153 | \frac{j_1(qr_\text{in})}{qr_\text{in}} |
---|
154 | |
---|
155 | .. figure:: img/onion_geometry.png |
---|
156 | |
---|
157 | Example of an onion model profile. |
---|
158 | |
---|
159 | The 2D scattering intensity is the same as $P(q)$ above, regardless of the |
---|
160 | orientation of the $q$ vector which is defined as |
---|
161 | |
---|
162 | .. math:: |
---|
163 | |
---|
164 | q = \sqrt{q_x^2 + q_y^2} |
---|
165 | |
---|
166 | NB: The outer most radius is used as the effective radius for $S(q)$ |
---|
167 | when $P(q) S(q)$ is applied. |
---|
168 | |
---|
169 | References |
---|
170 | ---------- |
---|
171 | |
---|
172 | .. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press, New York, 1987. |
---|
173 | |
---|
174 | Source |
---|
175 | ------ |
---|
176 | |
---|
177 | `onion.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/onion.py>`_ |
---|
178 | |
---|
179 | `onion.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/onion.c>`_ |
---|
180 | |
---|
181 | Authorship and Verification |
---|
182 | ---------------------------- |
---|
183 | |
---|
184 | * **Author:** |
---|
185 | * **Last Modified by:** |
---|
186 | * **Last Reviewed by:** |
---|
187 | * **Source added by :** Steve King **Date:** March 25, 2019 |
---|
188 | """ |
---|
189 | |
---|
190 | # |
---|
191 | # Give a polynomial $\rho(r) = Ar^3 + Br^2 + Cr + D$ for density, |
---|
192 | # |
---|
193 | # .. math:: |
---|
194 | # |
---|
195 | # f = 4 \pi \int_a^b \rho(r) \sin(qr)/(qr) \mathrm{d}r = h(b) - h(a) |
---|
196 | # |
---|
197 | # where |
---|
198 | # |
---|
199 | # .. math:: |
---|
200 | # |
---|
201 | # h(r) = \frac{4 \pi}{q^6}\left[ |
---|
202 | # (q^3(4Ar^3 + 3Br^2 + 2Cr + D) - q(24Ar + 6B)) \sin(qr) |
---|
203 | # - (q^4(Ar^4 + Br^3 + Cr^2 + Dr) - q^2(12Ar^2 + 6Br + 2C) + 24A) \cos(qr) |
---|
204 | # \right] |
---|
205 | # |
---|
206 | # Use the monotonic spline to get the polynomial coefficients for each shell. |
---|
207 | # |
---|
208 | # Order 0 |
---|
209 | # |
---|
210 | # .. math:: |
---|
211 | # |
---|
212 | # h(r) = \frac{4 \pi}{q^3} \left[ |
---|
213 | # - \cos(qr) (Ar) q |
---|
214 | # + \sin(qr) (A) |
---|
215 | # \right] |
---|
216 | # |
---|
217 | # Order 1 |
---|
218 | # |
---|
219 | # .. math:: |
---|
220 | # |
---|
221 | # h(r) = \frac{4 \pi}{q^4} \left[ |
---|
222 | # - \cos(qr) ( Ar^2 + Br) q^2 |
---|
223 | # + \sin(qr) ( Ar + B ) q |
---|
224 | # + \cos(qr) (2A ) |
---|
225 | # \right] |
---|
226 | # |
---|
227 | # Order 2 |
---|
228 | # |
---|
229 | # .. math:: |
---|
230 | # h(r) = \frac{4 \pi}{q^5} \left[ |
---|
231 | # - \cos(qr) ( Ar^3 + Br^2 + Cr) q^3 |
---|
232 | # + \sin(qr) (3Ar^2 + 2Br + C ) q^2 |
---|
233 | # + \cos(qr) (6Ar + 2B ) q |
---|
234 | # - \sin(qr) (6A ) |
---|
235 | # |
---|
236 | # Order 3 |
---|
237 | # |
---|
238 | # h(r) = \frac{4 \pi}{q^6}\left[ |
---|
239 | # - \cos(qr) ( Ar^4 + Br^3 + Cr^2 + Dr) q^4 |
---|
240 | # + \sin(qr) ( 4Ar^3 + 3Br^2 + 2Cr + D ) q^3 |
---|
241 | # + \cos(qr) (12Ar^2 + 6Br + 2C ) q^2 |
---|
242 | # - \sin(qr) (24Ar + 6B ) q |
---|
243 | # - \cos(qr) (24A ) |
---|
244 | # \right] |
---|
245 | # |
---|
246 | # Order p |
---|
247 | # |
---|
248 | # h(r) = \frac{4 \pi}{q^{2}} |
---|
249 | # \sum_{k=0}^p -\frac{d^k\cos(qr)}{dr^k} \frac{d^k r\rho(r)}{dr^k} (qr)^{-k} |
---|
250 | # |
---|
251 | # Given the equation |
---|
252 | # |
---|
253 | # f = sum_(k=0)^(n-1) h_k(r_(k+1)) - h_k(r_k) |
---|
254 | # |
---|
255 | # we can rearrange the terms so that |
---|
256 | # |
---|
257 | # f = sum_0^(n-1) h_k(r_(k+1)) - sum_0^(n-1) h_k(r_k) |
---|
258 | # = sum_1^n h_(k-1)(r_k) - sum_0^(n-1) h_k(r_k) |
---|
259 | # = h_(n-1)(r_n) - h_0(r_0) + sum_1^(n-1) [h_(k-1)(r_k) - h_k(r_k)] |
---|
260 | # = h_(n-1)(r_n) - h_0(r_0) - sum_1^(n-1) h_(Delta k)(r_k) |
---|
261 | # |
---|
262 | # where |
---|
263 | # |
---|
264 | # h_(Delta k)(r) = h(Delta rho_k, r) |
---|
265 | # |
---|
266 | # for |
---|
267 | # |
---|
268 | # Delta rho_k = (A_k-A_(k-1)) r^p + (B_k-B_(k-1)) r^(p-1) + ... |
---|
269 | # |
---|
270 | # Using l'H\^opital's Rule 6 times on the order 3 polynomial, |
---|
271 | # |
---|
272 | # lim_(q->0) h(r) = (140D r^3 + 180C r^4 + 144B r^5 + 120A r^6)/720 |
---|
273 | # |
---|
274 | |
---|
275 | from __future__ import division |
---|
276 | |
---|
277 | from math import fabs, exp, expm1 |
---|
278 | |
---|
279 | import numpy as np |
---|
280 | from numpy import inf, nan |
---|
281 | |
---|
282 | name = "onion" |
---|
283 | title = "Onion shell model with constant, linear or exponential density" |
---|
284 | |
---|
285 | description = """\ |
---|
286 | Form factor of mutishells normalized by the volume. Here each shell is |
---|
287 | described by an exponential function; |
---|
288 | |
---|
289 | I) For A_shell != 0, |
---|
290 | f(r) = B*exp(A_shell*(r-r_in)/thick_shell)+C |
---|
291 | where |
---|
292 | B=(sld_out-sld_in)/(exp(A_shell)-1) |
---|
293 | C=sld_in-B. |
---|
294 | Note that in the above case, the function becomes a linear function |
---|
295 | as A_shell --> 0+ or 0-. |
---|
296 | |
---|
297 | II) For the exact point of A_shell == 0, |
---|
298 | f(r) = sld_in ,i.e., it crosses over flat function |
---|
299 | Note that the 'sld_out' becaomes NULL in this case. |
---|
300 | |
---|
301 | background:background, |
---|
302 | rad_core0: radius of sphere(core) |
---|
303 | thick_shell#:the thickness of the shell# |
---|
304 | sld_core0: the SLD of the sphere |
---|
305 | sld_solv: the SLD of the solvent |
---|
306 | sld_shell: the SLD of the shell# |
---|
307 | A_shell#: the coefficient in the exponential function |
---|
308 | """ |
---|
309 | |
---|
310 | category = "shape:sphere" |
---|
311 | |
---|
312 | # TODO: n is a volume parameter that is not polydisperse |
---|
313 | |
---|
314 | # pylint: disable=bad-whitespace, line-too-long |
---|
315 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
316 | parameters = [ |
---|
317 | ["sld_core", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Core scattering length density"], |
---|
318 | ["radius_core", "Ang", 200., [0, inf], "volume", "Radius of the core"], |
---|
319 | ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "sld", "Solvent scattering length density"], |
---|
320 | ["n_shells", "", 1, [0, 10], "volume", "number of shells"], |
---|
321 | ["sld_in[n_shells]", "1e-6/Ang^2", 1.7, [-inf, inf], "sld", "scattering length density at the inner radius of shell k"], |
---|
322 | ["sld_out[n_shells]", "1e-6/Ang^2", 2.0, [-inf, inf], "sld", "scattering length density at the outer radius of shell k"], |
---|
323 | ["thickness[n_shells]", "Ang", 40., [0, inf], "volume", "Thickness of shell k"], |
---|
324 | ["A[n_shells]", "", 1.0, [-inf, inf], "", "Decay rate of shell k"], |
---|
325 | ] |
---|
326 | # pylint: enable=bad-whitespace, line-too-long |
---|
327 | |
---|
328 | source = ["lib/sas_3j1x_x.c", "onion.c"] |
---|
329 | single = False |
---|
330 | have_Fq = True |
---|
331 | effective_radius_type = ["outer radius"] |
---|
332 | |
---|
333 | profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)'] |
---|
334 | def profile(sld_core, radius_core, sld_solvent, n_shells, |
---|
335 | sld_in, sld_out, thickness, A): |
---|
336 | """ |
---|
337 | Returns shape profile with x=radius, y=SLD. |
---|
338 | """ |
---|
339 | n_shells = int(n_shells+0.5) |
---|
340 | total_radius = 1.25*(sum(thickness[:n_shells]) + radius_core + 1) |
---|
341 | dz = total_radius/400 # 400 points for a smooth plot |
---|
342 | |
---|
343 | z = [] |
---|
344 | rho = [] |
---|
345 | |
---|
346 | # add in the core |
---|
347 | z.append(0) |
---|
348 | rho.append(sld_core) |
---|
349 | z.append(radius_core) |
---|
350 | rho.append(sld_core) |
---|
351 | |
---|
352 | # add in the shells |
---|
353 | for k in range(int(n_shells)): |
---|
354 | # Left side of each shells |
---|
355 | z_current = z[-1] |
---|
356 | z.append(z_current) |
---|
357 | rho.append(sld_in[k]) |
---|
358 | |
---|
359 | if fabs(A[k]) < 1.0e-16: |
---|
360 | # flat shell |
---|
361 | z.append(z_current + thickness[k]) |
---|
362 | rho.append(sld_in[k]) |
---|
363 | else: |
---|
364 | # exponential shell |
---|
365 | # num_steps must be at least 1, so use floor()+1 rather than ceil |
---|
366 | # to protect against a thickness0. |
---|
367 | num_steps = np.floor(thickness[k]/dz) + 1 |
---|
368 | slope = (sld_out[k] - sld_in[k]) / expm1(A[k]) |
---|
369 | const = (sld_in[k] - slope) |
---|
370 | for z_shell in np.linspace(0, thickness[k], num_steps+1): |
---|
371 | z.append(z_current+z_shell) |
---|
372 | rho.append(slope*exp(A[k]*z_shell/thickness[k]) + const) |
---|
373 | |
---|
374 | # add in the solvent |
---|
375 | z.append(z[-1]) |
---|
376 | rho.append(sld_solvent) |
---|
377 | z.append(total_radius) |
---|
378 | rho.append(sld_solvent) |
---|
379 | |
---|
380 | return np.asarray(z), np.asarray(rho) |
---|
381 | |
---|
382 | # TODO: no random parameter function for onion model |
---|
383 | |
---|
384 | demo = { |
---|
385 | "sld_solvent": 2.2, |
---|
386 | "sld_core": 1.0, |
---|
387 | "radius_core": 100, |
---|
388 | "n_shells": 4, |
---|
389 | "sld_in": [0.5, 1.5, 0.9, 2.0], |
---|
390 | "sld_out": [nan, 0.9, 1.2, 1.6], |
---|
391 | "thickness": [50, 75, 150, 75], |
---|
392 | "A": [0, -1, 1e-4, 1], |
---|
393 | # Could also specify them individually as |
---|
394 | # "A1": 0, "A2": -1, "A3": 1e-4, "A4": 1, |
---|
395 | #"radius_core_pd_n": 10, |
---|
396 | #"radius_core_pd": 0.4, |
---|
397 | #"thickness4_pd_n": 10, |
---|
398 | #"thickness4_pd": 0.4, |
---|
399 | } |
---|