# source:sasmodels/sasmodels/models/hollow_rectangular_prism.py@8de1477

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 8de1477 was 8de1477, checked in by Paul Kienzle <pkienzle@…>, 6 years ago

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1# rectangular_prism model
2# Note: model title and parameter table are inserted automatically
3r"""
4
5This model provides the form factor, $P(q)$, for a hollow rectangular
6parallelepiped with a wall of thickness $\Delta$.
7It computes only the 1D scattering, not the 2D.
8
9Definition
10----------
11
12The 1D scattering intensity for this model is calculated by forming
13the difference of the amplitudes of two massive parallelepipeds
14differing in their outermost dimensions in each direction by the
15same length increment $2\Delta$ (Nayuk, 2012).
16
17As in the case of the massive parallelepiped model (:ref:rectangular-prism),
18the scattering amplitude is computed for a particular orientation of the
19parallelepiped with respect to the scattering vector and then averaged over all
20possible orientations, giving
21
22.. math::
23  P(q) =  \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \,
24  \int_0^{\frac{\pi}{2}} A_{P\Delta}^2(q) \, \sin\theta \, d\theta \, d\phi
25
26where $\theta$ is the angle between the $z$ axis and the longest axis
27of the parallelepiped, $\phi$ is the angle between the scattering vector
28(lying in the $xy$ plane) and the $y$ axis, and
29
30.. math::
31  :nowrap:
32
33  \begin{align*}
34  A_{P\Delta}(q) & =  A B C
35    \left[\frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)}
36    {\left( q \frac{C}{2} \cos\theta \right)} \right]
37    \left[\frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)}
38    {\left( q \frac{A}{2} \sin\theta \sin\phi \right)}\right]
39    \left[\frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)}
40    {\left( q \frac{B}{2} \sin\theta \cos\phi \right)}\right] \\
41    & - 8
42    \left(\frac{A}{2}-\Delta\right) \left(\frac{B}{2}-\Delta\right) \left(\frac{C}{2}-\Delta\right)
43    \left[ \frac{\sin \bigl[ q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta \bigr]}
44    {q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta} \right]
45    \left[ \frac{\sin \bigl[ q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi \bigr]}
46    {q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi} \right]
47    \left[ \frac{\sin \bigl[ q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi \bigr]}
48    {q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi} \right]
49  \end{align*}
50
51where $A$, $B$ and $C$ are the external sides of the parallelepiped fulfilling
52$A \le B \le C$, and the volume $V$ of the parallelepiped is
53
54.. math::
55  V = A B C \, - \, (A - 2\Delta) (B - 2\Delta) (C - 2\Delta)
56
57The 1D scattering intensity is then calculated as
58
59.. math::
60  I(q) = \text{scale} \times V \times (\rho_\text{p} -
61  \rho_\text{solvent})^2 \times P(q) + \text{background}
62
63where $\rho_\text{p}$ is the scattering length of the parallelepiped,
64$\rho_\text{solvent}$ is the scattering length of the solvent,
65and (if the data are in absolute units) *scale* represents the volume fraction
66(which is unitless).
67
68**The 2D scattering intensity is not computed by this model.**
69
70
71Validation
72----------
73
74Validation of the code was conducted by qualitatively comparing the output
75of the 1D model to the curves shown in (Nayuk, 2012).
76
77
78References
79----------
80
81R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854
82
83"""
84
85from numpy import pi, inf, sqrt
86
87name = "hollow_rectangular_prism"
88title = "Hollow rectangular parallelepiped with uniform scattering length density."
89description = """
90    I(q)= scale*V*(sld - sld_solvent)^2*P(q,theta,phi)+background
91        P(q,theta,phi) = (2/pi/V^2) * double integral from 0 to pi/2 of ...
92           (AP1-AP2)^2(q)*sin(theta)*dtheta*dphi
93        AP1 = S(q*C*cos(theta)/2) * S(q*A*sin(theta)*sin(phi)/2) * S(q*B*sin(theta)*cos(phi)/2)
94        AP2 = S(q*C'*cos(theta)) * S(q*A'*sin(theta)*sin(phi)) * S(q*B'*sin(theta)*cos(phi))
95        C' = (C/2-thickness)
96        B' = (B/2-thickness)
97        A' = (A/2-thickness)
98        S(x) = sin(x)/x
99"""
100category = "shape:parallelepiped"
101
102#             ["name", "units", default, [lower, upper], "type","description"],
103parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld",
104               "Parallelepiped scattering length density"],
105              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
106               "Solvent scattering length density"],
107              ["length_a", "Ang", 35, [0, inf], "volume",
108               "Shorter side of the parallelepiped"],
109              ["b2a_ratio", "Ang", 1, [0, inf], "volume",
110               "Ratio sides b/a"],
111              ["c2a_ratio", "Ang", 1, [0, inf], "volume",
112               "Ratio sides c/a"],
113              ["thickness", "Ang", 1, [0, inf], "volume",
114               "Thickness of parallelepiped"],
115              ["theta", "degrees", 0, [-360, 360], "orientation",
116               "c axis to beam angle"],
117              ["phi", "degrees", 0, [-360, 360], "orientation",
119              ["psi", "degrees", 0, [-360, 360], "orientation",
121             ]
122
123source = ["lib/gauss76.c", "hollow_rectangular_prism.c"]
124
125def ER(length_a, b2a_ratio, c2a_ratio, thickness):
126    """
128    thickness parameter not used
129    """
130    b_side = length_a * b2a_ratio
131    c_side = length_a * c2a_ratio
132
133    # surface average radius (rough approximation)
134    surf_rad = sqrt(length_a * b_side / pi)
135
137    return 0.5 * (ddd) ** (1. / 3.)
138
139def VR(length_a, b2a_ratio, c2a_ratio, thickness):
140    """
141    Return shell volume and total volume
142    """
143    b_side = length_a * b2a_ratio
144    c_side = length_a * c2a_ratio
145    a_core = length_a - 2.0*thickness
146    b_core = b_side - 2.0*thickness
147    c_core = c_side - 2.0*thickness
148    vol_core = a_core * b_core * c_core
149    vol_total = length_a * b_side * c_side
150    vol_shell = vol_total - vol_core
151    return vol_total, vol_shell
152
153
154def random():
155    import numpy as np
156    a, b, c = 10**np.random.uniform(1, 4.7, size=3)
157    # Thickness is limited to 1/2 the smallest dimension
158    # Use a distribution with a preference for thin shell or thin core
159    # Avoid core,shell radii < 1
160    min_dim = 0.5*min(a, b, c)
161    thickness = np.random.beta(0.5, 0.5)*(min_dim-2) + 1
162    #print(a, b, c, thickness, thickness/min_dim)
163    pars = dict(
164        length_a=a,
165        b2a_ratio=b/a,
166        c2a_ratio=c/a,
167        thickness=thickness,
168    )
169    return pars
170
171
172# parameters for demo
173demo = dict(scale=1, background=0,
174            sld=6.3, sld_solvent=1.0,
175            length_a=35, b2a_ratio=1, c2a_ratio=1, thickness=1,
176            length_a_pd=0.1, length_a_pd_n=10,
177            b2a_ratio_pd=0.1, b2a_ratio_pd_n=1,
178            c2a_ratio_pd=0.1, c2a_ratio_pd_n=1)
179
180tests = [[{}, 0.2, 0.76687283098],
181         [{}, [0.2], [0.76687283098]],
182        ]
183
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