1 | # rectangular_prism model |
---|
2 | # Note: model title and parameter table are inserted automatically |
---|
3 | r""" |
---|
4 | |
---|
5 | This model provides the form factor, $P(q)$, for a rectangular prism. |
---|
6 | |
---|
7 | Note that this model is almost totally equivalent to the existing |
---|
8 | :ref:`parallelepiped` model. |
---|
9 | The only difference is that the way the relevant |
---|
10 | parameters are defined here ($a$, $b/a$, $c/a$ instead of $a$, $b$, $c$) |
---|
11 | which allows use of polydispersity with this model while keeping the shape of |
---|
12 | the prism (e.g. setting $b/a = 1$ and $c/a = 1$ and applying polydispersity |
---|
13 | to *a* will generate a distribution of cubes of different sizes). |
---|
14 | |
---|
15 | |
---|
16 | Definition |
---|
17 | ---------- |
---|
18 | |
---|
19 | The 1D scattering intensity for this model was calculated by Mittelbach and |
---|
20 | Porod (Mittelbach, 1961), but the implementation here is closer to the |
---|
21 | equations given by Nayuk and Huber (Nayuk, 2012). |
---|
22 | Note also that the angle definitions used in the code and the present |
---|
23 | documentation correspond to those used in (Nayuk, 2012) (see Fig. 1 of |
---|
24 | that reference), with $\theta$ corresponding to $\alpha$ in that paper, |
---|
25 | and not to the usual convention used for example in the |
---|
26 | :ref:`parallelepiped` model. |
---|
27 | |
---|
28 | In this model the scattering from a massive parallelepiped with an |
---|
29 | orientation with respect to the scattering vector given by $\theta$ |
---|
30 | and $\phi$ |
---|
31 | |
---|
32 | .. math:: |
---|
33 | |
---|
34 | A_P\,(q) = |
---|
35 | \frac{\sin \left( \tfrac{1}{2}qC \cos\theta \right) }{\tfrac{1}{2} qC \cos\theta} |
---|
36 | \,\times\, |
---|
37 | \frac{\sin \left( \tfrac{1}{2}qA \cos\theta \right) }{\tfrac{1}{2} qA \cos\theta} |
---|
38 | \,\times\ , |
---|
39 | \frac{\sin \left( \tfrac{1}{2}qB \cos\theta \right) }{\tfrac{1}{2} qB \cos\theta} |
---|
40 | |
---|
41 | where $A$, $B$ and $C$ are the sides of the parallelepiped and must fulfill |
---|
42 | $A \le B \le C$, $\theta$ is the angle between the $z$ axis and the |
---|
43 | longest axis of the parallelepiped $C$, and $\phi$ is the angle between the |
---|
44 | scattering vector (lying in the $xy$ plane) and the $y$ axis. |
---|
45 | |
---|
46 | The normalized form factor in 1D is obtained averaging over all possible |
---|
47 | orientations |
---|
48 | |
---|
49 | .. math:: |
---|
50 | P(q) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \, |
---|
51 | \int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi |
---|
52 | |
---|
53 | And the 1D scattering intensity is calculated as |
---|
54 | |
---|
55 | .. math:: |
---|
56 | I(q) = \text{scale} \times V \times (\rho_\text{p} - |
---|
57 | \rho_\text{solvent})^2 \times P(q) |
---|
58 | |
---|
59 | where $V$ is the volume of the rectangular prism, $\rho_\text{p}$ |
---|
60 | is the scattering length of the parallelepiped, $\rho_\text{solvent}$ |
---|
61 | is the scattering length of the solvent, and (if the data are in absolute |
---|
62 | units) *scale* represents the volume fraction (which is unitless). |
---|
63 | |
---|
64 | For 2d data the orientation of the particle is required, described using |
---|
65 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details |
---|
66 | of the calculation and angular dispersions see :ref:`orientation` . |
---|
67 | The angle $\Psi$ is the rotational angle around the long *C* axis. For example, |
---|
68 | $\Psi = 0$ when the *B* axis is parallel to the *x*-axis of the detector. |
---|
69 | |
---|
70 | For 2d, constraints must be applied during fitting to ensure that the inequality |
---|
71 | $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, |
---|
72 | but the results may be not correct. |
---|
73 | |
---|
74 | .. figure:: img/parallelepiped_angle_definition.png |
---|
75 | |
---|
76 | Definition of the angles for oriented core-shell parallelepipeds. |
---|
77 | Note that rotation $\theta$, initially in the $xz$ plane, is carried out first, then |
---|
78 | rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the cylinder. |
---|
79 | The neutron or X-ray beam is along the $z$ axis. |
---|
80 | |
---|
81 | .. figure:: img/parallelepiped_angle_projection.png |
---|
82 | |
---|
83 | Examples of the angles for oriented rectangular prisms against the |
---|
84 | detector plane. |
---|
85 | |
---|
86 | |
---|
87 | |
---|
88 | Validation |
---|
89 | ---------- |
---|
90 | |
---|
91 | Validation of the code was conducted by comparing the output of the 1D model |
---|
92 | to the output of the existing :ref:`parallelepiped` model. |
---|
93 | |
---|
94 | |
---|
95 | References |
---|
96 | ---------- |
---|
97 | |
---|
98 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
---|
99 | .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
---|
100 | .. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659 |
---|
101 | |
---|
102 | Authorship and Verification |
---|
103 | ---------------------------- |
---|
104 | |
---|
105 | * **Author:** |
---|
106 | * **Last Modified by:** |
---|
107 | * **Last Reviewed by:** |
---|
108 | """ |
---|
109 | |
---|
110 | import numpy as np |
---|
111 | from numpy import inf |
---|
112 | |
---|
113 | name = "rectangular_prism" |
---|
114 | title = "Rectangular parallelepiped with uniform scattering length density." |
---|
115 | description = """ |
---|
116 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,theta,phi)+background |
---|
117 | P(q,theta,phi) = (2/pi) * double integral from 0 to pi/2 of ... |
---|
118 | AP^2(q)*sin(theta)*dtheta*dphi |
---|
119 | AP = S(q*C*cos(theta)/2) * S(q*A*sin(theta)*sin(phi)/2) * S(q*B*sin(theta)*cos(phi)/2) |
---|
120 | S(x) = sin(x)/x |
---|
121 | """ |
---|
122 | category = "shape:parallelepiped" |
---|
123 | |
---|
124 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
125 | parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", |
---|
126 | "Parallelepiped scattering length density"], |
---|
127 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
---|
128 | "Solvent scattering length density"], |
---|
129 | ["length_a", "Ang", 35, [0, inf], "volume", |
---|
130 | "Shorter side of the parallelepiped"], |
---|
131 | ["b2a_ratio", "", 1, [0, inf], "volume", |
---|
132 | "Ratio sides b/a"], |
---|
133 | ["c2a_ratio", "", 1, [0, inf], "volume", |
---|
134 | "Ratio sides c/a"], |
---|
135 | ["theta", "degrees", 0, [-360, 360], "orientation", |
---|
136 | "c axis to beam angle"], |
---|
137 | ["phi", "degrees", 0, [-360, 360], "orientation", |
---|
138 | "rotation about beam"], |
---|
139 | ["psi", "degrees", 0, [-360, 360], "orientation", |
---|
140 | "rotation about c axis"], |
---|
141 | ] |
---|
142 | |
---|
143 | source = ["lib/gauss76.c", "rectangular_prism.c"] |
---|
144 | have_Fq = True |
---|
145 | radius_effective_modes = [ |
---|
146 | "equivalent cylinder excluded volume", "equivalent volume sphere", |
---|
147 | "half length_a", "half length_b", "half length_c", |
---|
148 | "equivalent circular cross-section", "half ab diagonal", "half diagonal", |
---|
149 | ] |
---|
150 | |
---|
151 | def random(): |
---|
152 | """Return a random parameter set for the model.""" |
---|
153 | a, b, c = 10**np.random.uniform(1, 4.7, size=3) |
---|
154 | pars = dict( |
---|
155 | length_a=a, |
---|
156 | b2a_ratio=b/a, |
---|
157 | c2a_ratio=c/a, |
---|
158 | ) |
---|
159 | return pars |
---|
160 | |
---|
161 | # parameters for demo |
---|
162 | demo = dict(scale=1, background=0, |
---|
163 | sld=6.3, sld_solvent=1.0, |
---|
164 | length_a=35, b2a_ratio=1, c2a_ratio=1, |
---|
165 | length_a_pd=0.1, length_a_pd_n=10, |
---|
166 | b2a_ratio_pd=0.1, b2a_ratio_pd_n=1, |
---|
167 | c2a_ratio_pd=0.1, c2a_ratio_pd_n=1) |
---|
168 | |
---|
169 | tests = [[{}, 0.2, 0.375248406825], |
---|
170 | [{}, [0.2], [0.375248406825]], |
---|
171 | ] |
---|