1 | r""" |
---|
2 | Definition |
---|
3 | ---------- |
---|
4 | |
---|
5 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
---|
6 | |
---|
7 | .. math:: |
---|
8 | |
---|
9 | P(q) = \frac{\phi}{V_\text{shell}} \left[ |
---|
10 | \frac{3V_{\text{core}}({\rho_{\text{solvent}} |
---|
11 | - \rho_{\text{shell}})j_1(qR_{\text{core}})}}{qR_{\text{core}}} |
---|
12 | + \frac{3V_{\text{tot}}(\rho_{\text{shell}} |
---|
13 | - \rho_{\text{solvent}}) j_1(qR_{\text{tot}})}{qR_{\text{tot}}} |
---|
14 | \right]^2 + \text{background} |
---|
15 | |
---|
16 | |
---|
17 | where $\phi$ is the volume fraction of shell material, $V_{shell}$ is the volume |
---|
18 | of the shell, $V_{\text{cor}}$ is the volume of the core, $V_{\text{tot}}$ is |
---|
19 | the total volume, $R_{\text{core}}$ is the radius of the core, $R_{\text{tot}}$ |
---|
20 | is the outer radius of the shell, $\rho_{\text{solvent}}$ is the scattering |
---|
21 | length density of the solvent (which is the same as for the core in this case), |
---|
22 | $\rho_{\text{scale}}$ is the scattering length density of the shell, background |
---|
23 | is a flat background level (due for example to incoherent scattering in the |
---|
24 | case of neutrons), and $j_1$ is the spherical bessel function |
---|
25 | $j_1 = (\sin(x) - x \cos(x))/ x^2$. |
---|
26 | |
---|
27 | The functional form is identical to a "typical" core-shell structure, except |
---|
28 | that the scattering is normalized by the volume that is contributing to the |
---|
29 | scattering, namely the volume of the shell alone, the scattering length density |
---|
30 | of the core is fixed the same as that of the solvent, the scale factor when the |
---|
31 | data are on an absolute scale is equivalent to the volume fraction of material |
---|
32 | in the shell rather than the entire core+shell sphere, and the parameterization |
---|
33 | is done in terms of the core radius = $R_{\text{core}}$ and the shell |
---|
34 | thickness = $R_{\text{tot}} - R_{\text{core}}$. |
---|
35 | |
---|
36 | .. figure:: img/vesicle_geometry.jpg |
---|
37 | |
---|
38 | Vesicle geometry. |
---|
39 | |
---|
40 | The 2D scattering intensity is the same as *P(q)* above, regardless of the |
---|
41 | orientation of the *q* vector which is defined as |
---|
42 | |
---|
43 | .. math:: |
---|
44 | |
---|
45 | q = \sqrt{q_x^2 + q_y^2} |
---|
46 | |
---|
47 | |
---|
48 | NB: The outer most radius (= *radius* + *thickness*) is used as the effective |
---|
49 | radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
50 | |
---|
51 | |
---|
52 | References |
---|
53 | ---------- |
---|
54 | |
---|
55 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and |
---|
56 | Sons, New York, (1955) |
---|
57 | |
---|
58 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
---|
59 | * **Last Modified by:** Paul Butler **Date:** March 20, 2016 |
---|
60 | * **Last Reviewed by:** Paul Butler **Date:** March 20, 2016 |
---|
61 | """ |
---|
62 | |
---|
63 | import numpy as np |
---|
64 | from numpy import pi, inf |
---|
65 | |
---|
66 | name = "vesicle" |
---|
67 | title = "This model provides the form factor, *P(q)*, for an unilamellar \ |
---|
68 | vesicle. This is model is effectively identical to the hollow sphere \ |
---|
69 | reparameterized to be more intuitive for a vesicle and normalizing the \ |
---|
70 | form factor by the volume of the shell." |
---|
71 | description = """ |
---|
72 | Model parameters: |
---|
73 | radius : the core radius of the vesicle |
---|
74 | thickness: the shell thickness |
---|
75 | sld: the shell SLD |
---|
76 | sld_solvent: the solvent (and core) SLD |
---|
77 | background: incoherent background |
---|
78 | volfraction: shell volume fraction |
---|
79 | scale : scale factor = 1 if on absolute scale""" |
---|
80 | category = "shape:sphere" |
---|
81 | |
---|
82 | # [ "name", "units", default, [lower, upper], "type", "description"], |
---|
83 | parameters = [["sld", "1e-6/Ang^2", 0.5, [-inf, inf], "sld", |
---|
84 | "vesicle shell scattering length density"], |
---|
85 | ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld", |
---|
86 | "solvent scattering length density"], |
---|
87 | ["volfraction", "", 0.05, [0, 1.0], "", |
---|
88 | "volume fraction of shell"], |
---|
89 | ["radius", "Ang", 100, [0, inf], "volume", |
---|
90 | "vesicle core radius"], |
---|
91 | ["thickness", "Ang", 30, [0, inf], "volume", |
---|
92 | "vesicle shell thickness"], |
---|
93 | ] |
---|
94 | |
---|
95 | source = ["lib/sas_3j1x_x.c", "vesicle.c"] |
---|
96 | |
---|
97 | def ER(radius, thickness): |
---|
98 | ''' |
---|
99 | returns the effective radius used in the S*P calculation |
---|
100 | |
---|
101 | :param radius: core radius |
---|
102 | :param thickness: shell thickness |
---|
103 | ''' |
---|
104 | return radius + thickness |
---|
105 | |
---|
106 | def VR(radius, thickness): |
---|
107 | ''' |
---|
108 | returns the volumes of the shell and of the whole sphere including the |
---|
109 | core plus shell - is used to normalize when including polydispersity. |
---|
110 | |
---|
111 | :param radius: core radius |
---|
112 | :param thickness: shell thickness |
---|
113 | :return whole: volume of core and shell |
---|
114 | :return whole-core: volume of the shell |
---|
115 | ''' |
---|
116 | |
---|
117 | whole = 4./3. * pi * (radius + thickness)**3 |
---|
118 | core = 4./3. * pi * radius**3 |
---|
119 | return whole, whole - core |
---|
120 | |
---|
121 | def random(): |
---|
122 | total_radius = 10**np.random.uniform(1.3, 5) |
---|
123 | radius = total_radius * np.random.uniform(0, 1) |
---|
124 | thickness = total_radius - radius |
---|
125 | volfraction = 10**np.random.uniform(-3, -1) |
---|
126 | pars = dict( |
---|
127 | #background=0, |
---|
128 | scale=1, # volfraction is part of the model, so scale=1 |
---|
129 | radius=radius, |
---|
130 | thickness=thickness, |
---|
131 | volfraction=volfraction, |
---|
132 | ) |
---|
133 | return pars |
---|
134 | |
---|
135 | # parameters for demo |
---|
136 | demo = dict(sld=0.5, sld_solvent=6.36, |
---|
137 | volfraction=0.05, |
---|
138 | radius=100, thickness=30, |
---|
139 | radius_pd=.2, radius_pd_n=10, |
---|
140 | thickness_pd=.2, thickness_pd_n=10) |
---|
141 | |
---|
142 | # NOTE: test results taken from values returned by SasView 3.1.2, with |
---|
143 | # 0.001 added for a non-zero default background. |
---|
144 | tests = [[{}, 0.0005, 859.916526646], |
---|
145 | [{}, 0.100600200401, 1.77063682331], |
---|
146 | [{}, 0.5, 0.00355351388906], |
---|
147 | [{}, 'ER', 130.], |
---|
148 | [{}, 'VR', 0.54483386436], |
---|
149 | ] |
---|