[aea2e2a] | 1 | r""" |
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[15a7577] | 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[aea2e2a] | 5 | This model provides the form factor, $P(q)$, for a monodisperse hollow right |
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[15a7577] | 6 | angle circular cylinder (rigid tube) where the The inside and outside of the |
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| 7 | hollow cylinder are assumed to have the same SLD and the form factor is thus |
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| 8 | normalized by the volume of the tube (i.e. not by the total cylinder volume). |
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[aea2e2a] | 9 | |
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| 10 | .. math:: |
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| 11 | |
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| 12 | P(q) = \text{scale} \left<F^2\right>/V_\text{shell} + \text{background} |
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| 13 | |
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[15a7577] | 14 | where the averaging $\left<\ldots\right>$ is applied only for the 1D |
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| 15 | calculation. If Intensity is given on an absolute scale, the scale factor here |
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| 16 | is the volume fraction of the shell. This differs from |
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| 17 | the :ref:`core-shell-cylinder` in that, in that case, scale is the volume |
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| 18 | fraction of the entire cylinder (core+shell). The application might be for a |
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| 19 | bilayer which wraps into a hollow tube and the volume fraction of material is |
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| 20 | all in the shell, whereas the :ref:`core-shell-cylinder` model might be used for |
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| 21 | a cylindrical micelle where the tails in the core have a different SLD than the |
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| 22 | headgroups (in the shell) and the volume fraction of material comes fromm the |
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| 23 | whole cyclinder. NOTE: the hollow_cylinder represents a tube whereas the |
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| 24 | core_shell_cylinder includes a shell layer covering the ends (end caps) as well. |
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[aea2e2a] | 25 | |
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| 26 | |
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| 27 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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| 28 | |
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| 29 | .. math:: |
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| 30 | |
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| 31 | P(q) &= (\text{scale})V_\text{shell}\Delta\rho^2 |
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| 32 | \int_0^{1}\Psi^2 |
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| 33 | \left[q_z, R_\text{outer}(1-x^2)^{1/2}, |
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| 34 | R_\text{core}(1-x^2)^{1/2}\right] |
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| 35 | \left[\frac{\sin(qHx)}{qHx}\right]^2 dx \\ |
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| 36 | \Psi[q,y,z] &= \frac{1}{1-\gamma^2} |
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| 37 | \left[ \Lambda(qy) - \gamma^2\Lambda(qz) \right] \\ |
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| 38 | \Lambda(a) &= 2 J_1(a) / a \\ |
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| 39 | \gamma &= R_\text{core} / R_\text{outer} \\ |
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| 40 | V_\text{shell} &= \pi \left(R_\text{outer}^2 - R_\text{core}^2 \right)L \\ |
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| 41 | J_1(x) &= (\sin(x)-x\cdot \cos(x)) / x^2 |
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| 42 | |
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| 43 | where *scale* is a scale factor, $H = L/2$ and $J_1$ is the 1st order |
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| 44 | Bessel function. |
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| 45 | |
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| 46 | **NB**: The 2nd virial coefficient of the cylinder is calculated |
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| 47 | based on the outer radius and full length, which give an the effective radius |
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| 48 | for structure factor $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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| 49 | |
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[2d81cfe] | 50 | In the parameters,the *radius* is $R_\text{core}$ while *thickness* |
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| 51 | is $R_\text{outer} - R_\text{core}$. |
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[aea2e2a] | 52 | |
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| 53 | To provide easy access to the orientation of the core-shell cylinder, we define |
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| 54 | the axis of the cylinder using two angles $\theta$ and $\phi$ |
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| 55 | (see :ref:`cylinder model <cylinder-angle-definition>`). |
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| 56 | |
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| 57 | References |
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| 58 | ---------- |
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| 59 | |
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[455aaa1] | 60 | .. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and |
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| 61 | Neutron Scattering*, Plenum Press, New York, (1987) |
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[aea2e2a] | 62 | |
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| 63 | Authorship and Verification |
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| 64 | ---------------------------- |
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| 65 | |
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| 66 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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[15a7577] | 67 | * **Last Modified by:** Paul Butler **Date:** September 06, 2018 |
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| 68 | (corrected VR calculation) |
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| 69 | * **Last Reviewed by:** Paul Butler **Date:** September 06, 2018 |
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[aea2e2a] | 70 | """ |
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[304c775] | 71 | from __future__ import division |
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[aea2e2a] | 72 | |
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[2d81cfe] | 73 | import numpy as np |
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[0b56f38] | 74 | from numpy import pi, inf, sin, cos |
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[aea2e2a] | 75 | |
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| 76 | name = "hollow_cylinder" |
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| 77 | title = "" |
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| 78 | description = """ |
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| 79 | P(q) = scale*<f*f>/Vol + background, where f is the scattering amplitude. |
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| 80 | radius = the radius of core |
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| 81 | thickness = the thickness of shell |
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| 82 | length = the total length of the cylinder |
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| 83 | sld = SLD of the shell |
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| 84 | sld_solvent = SLD of the solvent |
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| 85 | background = incoherent background |
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| 86 | """ |
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| 87 | category = "shape:cylinder" |
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| 88 | # pylint: disable=bad-whitespace, line-too-long |
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| 89 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 90 | parameters = [ |
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| 91 | ["radius", "Ang", 20.0, [0, inf], "volume", "Cylinder core radius"], |
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| 92 | ["thickness", "Ang", 10.0, [0, inf], "volume", "Cylinder wall thickness"], |
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| 93 | ["length", "Ang", 400.0, [0, inf], "volume", "Cylinder total length"], |
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[f102a96] | 94 | ["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Cylinder sld"], |
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| 95 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent sld"], |
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[9b79f29] | 96 | ["theta", "degrees", 90, [-360, 360], "orientation", "Cylinder axis to beam angle"], |
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| 97 | ["phi", "degrees", 0, [-360, 360], "orientation", "Rotation about beam"], |
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[aea2e2a] | 98 | ] |
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| 99 | # pylint: enable=bad-whitespace, line-too-long |
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| 100 | |
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| 101 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "hollow_cylinder.c"] |
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[71b751d] | 102 | have_Fq = True |
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[ee60aa7] | 103 | effective_radius_type = [ |
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| 104 | "equivalent sphere", "outer radius", "half length", |
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| 105 | "half outer min dimension", "half outer max dimension", |
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| 106 | "half outer diagonal", |
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| 107 | ] |
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[aea2e2a] | 108 | |
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[31df0c9] | 109 | def random(): |
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[8f04da4] | 110 | length = 10**np.random.uniform(1, 4.7) |
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| 111 | outer_radius = 10**np.random.uniform(1, 4.7) |
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| 112 | # Use a distribution with a preference for thin shell or thin core |
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| 113 | # Avoid core,shell radii < 1 |
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| 114 | thickness = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1 |
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| 115 | radius = outer_radius - thickness |
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[31df0c9] | 116 | pars = dict( |
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| 117 | length=length, |
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| 118 | radius=radius, |
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[8f04da4] | 119 | thickness=thickness, |
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[31df0c9] | 120 | ) |
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| 121 | return pars |
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| 122 | |
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[8f04da4] | 123 | # parameters for demo |
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| 124 | demo = dict(scale=1.0, background=0.0, length=400.0, radius=20.0, |
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| 125 | thickness=10, sld=6.3, sld_solvent=1, theta=90, phi=0, |
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| 126 | thickness_pd=0.2, thickness_pd_n=9, |
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| 127 | length_pd=.2, length_pd_n=10, |
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| 128 | radius_pd=.2, radius_pd_n=9, |
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| 129 | theta_pd=10, theta_pd_n=5, |
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| 130 | ) |
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[0b56f38] | 131 | q = 0.1 |
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| 132 | # april 6 2017, rkh added a 2d unit test, assume correct! |
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| 133 | qx = q*cos(pi/6.0) |
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| 134 | qy = q*sin(pi/6.0) |
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[304c775] | 135 | radius = parameters[0][2] |
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| 136 | thickness = parameters[1][2] |
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| 137 | length = parameters[2][2] |
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[aea2e2a] | 138 | # Parameters for unit tests |
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| 139 | tests = [ |
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| 140 | [{}, 0.00005, 1764.926], |
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[304c775] | 141 | [{}, 0.1, None, None, |
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| 142 | (3./4*(radius+thickness)**2*length)**(1./3), # R_eff from volume |
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| 143 | pi*((radius+thickness)**2-radius**2)*length, # shell volume |
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| 144 | (radius+thickness)**2/((radius+thickness)**2 - radius**2), # form:shell ratio |
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| 145 | ], |
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[0b56f38] | 146 | [{}, 0.001, 1756.76], |
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[2d81cfe] | 147 | [{}, (qx, qy), 2.36885476192], |
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| 148 | ] |
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[0b56f38] | 149 | del qx, qy # not necessary to delete, but cleaner |
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