source: sasmodels/sasmodels/models/hollow_cylinder.py @ aea2e2a

core_shell_microgelscostrafo411magnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since aea2e2a was aea2e2a, checked in by richardh, 7 years ago

reparametrised hollow_cylinder to use thickness not outer radius

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1r"""
2This model provides the form factor, $P(q)$, for a monodisperse hollow right
3angle circular cylinder (rigid tube) where the form factor is normalized by the
4volume of the tube (i.e. not by the external volume).
5
6.. math::
7
8    P(q) = \text{scale} \left<F^2\right>/V_\text{shell} + \text{background}
9
10where the averaging $\left<\ldots\right>$ is applied only for the 1D calculation.
11
12The inside and outside of the hollow cylinder are assumed have the same SLD.
13
14Definition
15----------
16
17The 1D scattering intensity is calculated in the following way (Guinier, 1955)
18
19.. math::
20
21    P(q)           &= (\text{scale})V_\text{shell}\Delta\rho^2
22            \int_0^{1}\Psi^2
23            \left[q_z, R_\text{outer}(1-x^2)^{1/2},
24                       R_\text{core}(1-x^2)^{1/2}\right]
25            \left[\frac{\sin(qHx)}{qHx}\right]^2 dx \\
26    \Psi[q,y,z]    &= \frac{1}{1-\gamma^2}
27            \left[ \Lambda(qy) - \gamma^2\Lambda(qz) \right] \\
28    \Lambda(a)     &= 2 J_1(a) / a \\
29    \gamma         &= R_\text{core} / R_\text{outer} \\
30    V_\text{shell} &= \pi \left(R_\text{outer}^2 - R_\text{core}^2 \right)L \\
31    J_1(x)         &= (\sin(x)-x\cdot \cos(x)) / x^2
32
33where *scale* is a scale factor, $H = L/2$ and $J_1$ is the 1st order
34Bessel function.
35
36**NB**: The 2nd virial coefficient of the cylinder is calculated
37based on the outer radius and full length, which give an the effective radius
38for structure factor $S(q)$ when $P(q) \cdot S(q)$ is applied.
39
40In the parameters,the *radius* is $R_\text{core}$ while *thickness* is $R_\text{outer} - R_\text{core}$.
41
42To provide easy access to the orientation of the core-shell cylinder, we define
43the axis of the cylinder using two angles $\theta$ and $\phi$
44(see :ref:`cylinder model <cylinder-angle-definition>`).
45
46References
47----------
48
49L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and
50Neutron Scattering*, Plenum Press, New York, (1987)
51
52Authorship and Verification
53----------------------------
54
55* **Author:** NIST IGOR/DANSE **Date:** pre 2010
56* **Last Modified by:** Richard Heenan **Date:** October 06, 2016
57   (reparametrised to use thickness, not outer radius)
58* **Last Reviewed by:** Richard Heenan **Date:** October 06, 2016
59
60"""
61
62from numpy import pi, inf
63
64name = "hollow_cylinder"
65title = ""
66description = """
67P(q) = scale*<f*f>/Vol + background, where f is the scattering amplitude.
68radius = the radius of core
69thickness = the thickness of shell
70length = the total length of the cylinder
71sld = SLD of the shell
72sld_solvent = SLD of the solvent
73background = incoherent background
74"""
75category = "shape:cylinder"
76# pylint: disable=bad-whitespace, line-too-long
77#   ["name", "units", default, [lower, upper], "type","description"],
78parameters = [
79    ["radius",      "Ang",     20.0, [0, inf],    "volume",      "Cylinder core radius"],
80    ["thickness",   "Ang",     10.0, [0, inf],    "volume",      "Cylinder wall thickness"],
81    ["length",      "Ang",    400.0, [0, inf],    "volume",      "Cylinder total length"],
82    ["sld",         "1/Ang^2",  6.3, [-inf, inf], "sld",         "Cylinder sld"],
83    ["sld_solvent", "1/Ang^2",  1,   [-inf, inf], "sld",         "Solvent sld"],
84    ["theta",       "degrees", 90,   [-360, 360], "orientation", "Theta angle"],
85    ["phi",         "degrees",  0,   [-360, 360], "orientation", "Phi angle"],
86    ]
87# pylint: enable=bad-whitespace, line-too-long
88
89source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "hollow_cylinder.c"]
90
91# pylint: disable=W0613
92def ER(radius, thickness, length):
93    """
94    :param radius:      Cylinder core radius
95    :param thickness:   Cylinder wall thickness
96    :param length:      Cylinder length
97    :return:            Effective radius
98    """
99    router = radius + thickness
100    if router == 0 or length == 0:
101        return 0.0
102    len1 = router
103    len2 = length/2.0
104    term1 = len1*len1*2.0*len2/2.0
105    term2 = 1.0 + (len2/len1)*(1.0 + 1/len2/2.0)*(1.0 + pi*len1/len2/2.0)
106    ddd = 3.0*term1*term2
107    diam = pow(ddd, (1.0/3.0))
108    return diam
109
110def VR(radius, thickness, length):
111    """
112    :param radius:      Cylinder radius
113    :param thickness:   Cylinder wall thickness
114    :param length:      Cylinder length
115    :return:            Volf ratio for P(q)*S(q)
116    """
117    router = radius + thickness
118    vol_core = pi*radius*radius*length
119    vol_total = pi*router*router*length
120    vol_shell = vol_total - vol_core
121    return vol_shell, vol_total
122
123# parameters for demo
124demo = dict(scale=1.0, background=0.0, length=400.0, radius=20.0,
125            thickness=10, sld=6.3, sld_solvent=1, theta=90, phi=0,
126            thickness_pd=0.2, thickness_pd_n=9,
127            length_pd=.2, length_pd_n=10,
128            radius_pd=.2, radius_pd_n=9,
129            theta_pd=10, theta_pd_n=5,
130           )
131
132# Parameters for unit tests
133tests = [
134    [{}, 0.00005, 1764.926],
135    [{}, 'VR', 1.8],
136    [{}, 0.001, 1756.76]
137    ]
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