1 | # ellipsoid model |
---|
2 | # Note: model title and parameter table are inserted automatically |
---|
3 | r""" |
---|
4 | The form factor is normalized by the particle volume. |
---|
5 | |
---|
6 | Definition |
---|
7 | ---------- |
---|
8 | |
---|
9 | The output of the 2D scattering intensity function for oriented ellipsoids |
---|
10 | is given by (Feigin, 1987) |
---|
11 | |
---|
12 | .. math:: |
---|
13 | |
---|
14 | P(Q,\alpha) = {\text{scale} \over V} F^2(Q) + \text{background} |
---|
15 | |
---|
16 | where |
---|
17 | |
---|
18 | .. math:: |
---|
19 | |
---|
20 | F(Q) = {3 (\Delta rho)) V (\sin[Qr(R_p,R_e,\alpha)] |
---|
21 | - \cos[Qr(R_p,R_e,\alpha)]) |
---|
22 | \over [Qr(R_p,R_e,\alpha)]^3 } |
---|
23 | |
---|
24 | and |
---|
25 | |
---|
26 | .. math:: |
---|
27 | |
---|
28 | r(R_p,R_e,\alpha) = \left[ R_e^2 \sin^2 \alpha |
---|
29 | + R_p^2 \cos^2 \alpha \right]^{1/2} |
---|
30 | |
---|
31 | |
---|
32 | $\alpha$ is the angle between the axis of the ellipsoid and $\vec q$, |
---|
33 | $V$ is the volume of the ellipsoid, $R_p$ is the polar radius along the |
---|
34 | rotational axis of the ellipsoid, $R_e$ is the equatorial radius perpendicular |
---|
35 | to the rotational axis of the ellipsoid and $\Delta \rho$ (contrast) is the |
---|
36 | scattering length density difference between the scatterer and the solvent. |
---|
37 | |
---|
38 | To provide easy access to the orientation of the ellipsoid, we define |
---|
39 | the rotation axis of the ellipsoid using two angles $\theta$ and $\phi$. |
---|
40 | These angles are defined in the |
---|
41 | :ref:`cylinder orientation figure <cylinder-orientation>`. |
---|
42 | For the ellipsoid, $\theta$ is the angle between the rotational axis |
---|
43 | and the $z$-axis. |
---|
44 | |
---|
45 | NB: The 2nd virial coefficient of the solid ellipsoid is calculated based |
---|
46 | on the $R_p$ and $R_e$ values, and used as the effective radius for |
---|
47 | $S(Q)$ when $P(Q) \cdot S(Q)$ is applied. |
---|
48 | |
---|
49 | .. _ellipsoid-1d: |
---|
50 | |
---|
51 | .. figure:: img/ellipsoid_1d.JPG |
---|
52 | |
---|
53 | The output of the 1D scattering intensity function for randomly oriented |
---|
54 | ellipsoids given by the equation above. |
---|
55 | |
---|
56 | |
---|
57 | The $\theta$ and $\phi$ parameters are not used for the 1D output. Our |
---|
58 | implementation of the scattering kernel and the 1D scattering intensity |
---|
59 | use the c-library from NIST. |
---|
60 | |
---|
61 | .. _ellipsoid-geometry: |
---|
62 | |
---|
63 | .. figure:: img/ellipsoid_geometry.JPG |
---|
64 | |
---|
65 | The angles for oriented ellipsoid. |
---|
66 | |
---|
67 | Validation |
---|
68 | ---------- |
---|
69 | |
---|
70 | Validation of our code was done by comparing the output of the 1D model |
---|
71 | to the output of the software provided by the NIST (Kline, 2006). |
---|
72 | :num:`Figure ellipsoid-comparison-1d` below shows a comparison of |
---|
73 | the 1D output of our model and the output of the NIST software. |
---|
74 | |
---|
75 | .. _ellipsoid-comparison-1d: |
---|
76 | |
---|
77 | .. figure:: img/ellipsoid_comparison_1d.jpg |
---|
78 | |
---|
79 | Comparison of the SasView scattering intensity for an ellipsoid |
---|
80 | with the output of the NIST SANS analysis software. The parameters |
---|
81 | were set to: *scale* = 1.0, *rpolar* = 20 |Ang|, |
---|
82 | *requatorial* =400 |Ang|, *contrast* = 3e-6 |Ang^-2|, |
---|
83 | and *background* = 0.01 |cm^-1|. |
---|
84 | |
---|
85 | Averaging over a distribution of orientation is done by evaluating the |
---|
86 | equation above. Since we have no other software to compare the |
---|
87 | implementation of the intensity for fully oriented ellipsoids, we can |
---|
88 | compare the result of averaging our 2D output using a uniform distribution |
---|
89 | $p(\theta,\phi) = 1.0$. :num:`Figure #ellipsoid-comparison-2d` |
---|
90 | shows the result of such a cross-check. |
---|
91 | |
---|
92 | .. _ellipsoid-comparison-2d: |
---|
93 | |
---|
94 | .. figure:: img/ellipsoid_comparison_2d.jpg |
---|
95 | |
---|
96 | Comparison of the intensity for uniformly distributed ellipsoids |
---|
97 | calculated from our 2D model and the intensity from the NIST SANS |
---|
98 | analysis software. The parameters used were: *scale* = 1.0, |
---|
99 | *rpolar* = 20 |Ang|, *requatorial* = 400 |Ang|, |
---|
100 | *contrast* = 3e-6 |Ang^-2|, and *background* = 0.0 |cm^-1|. |
---|
101 | |
---|
102 | The discrepancy above *q* = 0.3 |cm^-1| is due to the way the form factors |
---|
103 | are calculated in the c-library provided by NIST. A numerical integration |
---|
104 | has to be performed to obtain $P(Q)$ for randomly oriented particles. |
---|
105 | The NIST software performs that integration with a 76-point Gaussian |
---|
106 | quadrature rule, which will become imprecise at high $Q$ where the amplitude |
---|
107 | varies quickly as a function of $Q$. The SasView result shown has been |
---|
108 | obtained by summing over 501 equidistant points. Our result was found |
---|
109 | to be stable over the range of $Q$ shown for a number of points higher |
---|
110 | than 500. |
---|
111 | |
---|
112 | REFERENCE |
---|
113 | |
---|
114 | L A Feigin and D I Svergun. *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
---|
115 | New York, 1987. |
---|
116 | """ |
---|
117 | |
---|
118 | from numpy import inf |
---|
119 | |
---|
120 | name = "ellipsoid" |
---|
121 | title = "Ellipsoid of revolution with uniform scattering length density." |
---|
122 | |
---|
123 | description = """\ |
---|
124 | P(q.alpha)= scale*f(q)^2 + background, where f(q)= 3*(sld |
---|
125 | - solvent_sld)*V*[sin(q*r(Rp,Re,alpha)) |
---|
126 | -q*r*cos(qr(Rp,Re,alpha))] |
---|
127 | /[qr(Rp,Re,alpha)]^3" |
---|
128 | |
---|
129 | r(Rp,Re,alpha)= [Re^(2)*(sin(alpha))^2 |
---|
130 | + Rp^(2)*(cos(alpha))^2]^(1/2) |
---|
131 | |
---|
132 | sld: SLD of the ellipsoid |
---|
133 | solvent_sld: SLD of the solvent |
---|
134 | V: volume of the ellipsoid |
---|
135 | Rp: polar radius of the ellipsoid |
---|
136 | Re: equatorial radius of the ellipsoid |
---|
137 | """ |
---|
138 | category = "shape:ellipsoid" |
---|
139 | |
---|
140 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
141 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "", |
---|
142 | "Ellipsoid scattering length density"], |
---|
143 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
---|
144 | "Solvent scattering length density"], |
---|
145 | ["rpolar", "Ang", 20, [0, inf], "volume", |
---|
146 | "Polar radius"], |
---|
147 | ["requatorial", "Ang", 400, [0, inf], "volume", |
---|
148 | "Equatorial radius"], |
---|
149 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
---|
150 | "In plane angle"], |
---|
151 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
---|
152 | "Out of plane angle"], |
---|
153 | ] |
---|
154 | |
---|
155 | source = ["lib/J1.c", "lib/gauss76.c", "ellipsoid.c"] |
---|
156 | |
---|
157 | def ER(rpolar, requatorial): |
---|
158 | import numpy as np |
---|
159 | |
---|
160 | ee = np.empty_like(rpolar) |
---|
161 | idx = rpolar > requatorial |
---|
162 | ee[idx] = (rpolar[idx] ** 2 - requatorial[idx] ** 2) / rpolar[idx] ** 2 |
---|
163 | idx = rpolar < requatorial |
---|
164 | ee[idx] = (requatorial[idx] ** 2 - rpolar[idx] ** 2) / requatorial[idx] ** 2 |
---|
165 | idx = rpolar == requatorial |
---|
166 | ee[idx] = 2 * rpolar[idx] |
---|
167 | valid = (rpolar * requatorial != 0) |
---|
168 | bd = 1.0 - ee[valid] |
---|
169 | e1 = np.sqrt(ee[valid]) |
---|
170 | b1 = 1.0 + np.arcsin(e1) / (e1 * np.sqrt(bd)) |
---|
171 | bL = (1.0 + e1) / (1.0 - e1) |
---|
172 | b2 = 1.0 + bd / 2 / e1 * np.log(bL) |
---|
173 | delta = 0.75 * b1 * b2 |
---|
174 | |
---|
175 | ddd = np.zeros_like(rpolar) |
---|
176 | ddd[valid] = 2.0 * (delta + 1.0) * rpolar * requatorial ** 2 |
---|
177 | return 0.5 * ddd ** (1.0 / 3.0) |
---|
178 | |
---|
179 | |
---|
180 | demo = dict(scale=1, background=0, |
---|
181 | sld=6, solvent_sld=1, |
---|
182 | rpolar=50, requatorial=30, |
---|
183 | theta=30, phi=15, |
---|
184 | rpolar_pd=.2, rpolar_pd_n=15, |
---|
185 | requatorial_pd=.2, requatorial_pd_n=15, |
---|
186 | theta_pd=15, theta_pd_n=45, |
---|
187 | phi_pd=15, phi_pd_n=1) |
---|
188 | oldname = 'EllipsoidModel' |
---|
189 | oldpars = dict(theta='axis_theta', phi='axis_phi', |
---|
190 | sld='sldEll', solvent_sld='sldSolv', |
---|
191 | rpolar='radius_a', requatorial='radius_b') |
---|