[c612162] | 1 | # pylint: disable=line-too-long |
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[a8b3cdb] | 2 | r""" |
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| 3 | This function calculates the scattering from an elliptical cylinder. |
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| 4 | |
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| 5 | Definition for 2D (orientated system) |
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| 6 | ------------------------------------- |
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| 7 | |
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| 8 | The angles |theta| and |phi| define the orientation of the axis of the cylinder. The angle |bigpsi| is defined as the |
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| 9 | orientation of the major axis of the ellipse with respect to the vector *Q*\ . A gaussian polydispersity can be added |
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| 10 | to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii. |
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| 11 | |
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| 12 | .. image:: img/elliptical_cylinder_geometry.gif |
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| 13 | |
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| 14 | *Figure.* *a* = *r_minor* and |nu|\ :sub:`n` = $r_ratio$ (i.e., $r_major / r_minor$). |
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| 15 | |
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| 16 | The function calculated is |
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| 17 | |
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| 18 | .. math:: |
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| 19 | I(\mathbf{q})=\frac{1}{V_{cyl}}\int{d\psi}\int{d\phi}\int{p(\theta,\phi,\psi)F^2(\mathbf{q},\alpha,\psi)\sin(\theta)d\theta} |
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| 20 | |
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| 21 | with the functions |
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| 22 | |
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| 23 | .. math:: |
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| 24 | F(\mathbf{q},\alpha,\psi)=2\frac{J_1(a)\sin(b)}{ab} |
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| 25 | \\ |
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| 26 | a = \mathbf{q}\sin(\alpha)\left[ r^2_{major}\sin^2(\psi)+r^2_{minor}\cos(\psi) \right]^{1/2} |
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| 27 | \\ |
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| 28 | b=\mathbf{q}\frac{L}{2}\cos(\alpha) |
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| 29 | |
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| 30 | and the angle |bigpsi| is defined as the orientation of the major axis of the ellipse with respect to the vector *q*\ . |
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| 31 | |
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| 32 | |
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| 33 | Definition for 1D (no preferred orientation) |
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| 34 | -------------------------------------------- |
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| 35 | |
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| 36 | The form factor is averaged over all possible orientation before normalized by the particle volume |
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| 37 | |
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| 38 | .. math:: |
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| 39 | P(q) = scale <F^2> / V |
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| 40 | |
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| 41 | The returned value is scaled to units of |cm^-1|. |
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| 42 | |
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| 43 | To provide easy access to the orientation of the elliptical cylinder, we define the axis of the cylinder using two |
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| 44 | angles |theta|, |phi| and |bigpsi|. As for the case of the cylinder, the angles |theta| and |phi| are defined on |
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| 45 | Figure 2 of CylinderModel. The angle |bigpsi| is the rotational angle around its own long_c axis against the *q* plane. |
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| 46 | For example, |bigpsi| = 0 when the *r_minor* axis is parallel to the *x*\ -axis of the detector. |
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| 47 | |
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| 48 | All angle parameters are valid and given only for 2D calculation; ie, an oriented system. |
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| 49 | |
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| 50 | .. image:: img/elliptical_cylinder_geometry_2d.jpg |
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| 51 | |
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| 52 | *Figure. Definition of angles for 2D* |
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| 53 | |
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| 54 | .. image:: img/core_shell_bicelle_fig2.jpg |
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| 55 | |
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| 56 | *Figure. Examples of the angles for oriented elliptical cylinders against the detector plane.* |
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| 57 | |
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| 58 | NB: The 2nd virial coefficient of the cylinder is calculated based on the averaged radius (= sqrt(*r_minor*\ :sup:`2` \* *r_ratio*)) |
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| 59 | and length values, and used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
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| 60 | |
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| 61 | |
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| 62 | .. image:: img/elliptical_cylinder_comparison_1d.jpg |
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| 63 | |
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| 64 | *Figure. 1D plot using the default values (w/1000 data point).* |
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| 65 | |
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| 66 | Validation |
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| 67 | ---------- |
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| 68 | |
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| 69 | Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of |
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| 70 | the 2D calculation over all possible angles. The figure below shows the comparison where the solid dot refers to |
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| 71 | averaged 2D values while the line represents the result of the 1D calculation (for the 2D averaging, values of 76, 180, |
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| 72 | and 76 degrees are taken for the angles of |theta|, |phi|, and |bigpsi| respectively). |
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| 73 | |
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| 74 | .. image:: img/elliptical_cylinder_validation_1d.gif |
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| 75 | |
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| 76 | *Figure. Comparison between 1D and averaged 2D.* |
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| 77 | |
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| 78 | In the 2D average, more binning in the angle |phi| is necessary to get the proper result. The following figure shows |
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| 79 | the results of the averaging by varying the number of angular bins. |
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| 80 | |
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| 81 | .. image:: img/elliptical_cylinder_averaging.gif |
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| 82 | |
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| 83 | *Figure. The intensities averaged from 2D over different numbers of bins and angles.* |
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| 84 | |
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| 85 | Reference |
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| 86 | --------- |
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| 87 | |
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| 88 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
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| 89 | New York, (1987) |
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| 90 | """ |
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| 91 | |
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| 92 | import math |
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| 93 | from numpy import pi, inf |
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| 94 | |
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| 95 | name = "elliptical_cylinder" |
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| 96 | title = "Form factor for an elliptical cylinder." |
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| 97 | description = """ |
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[c612162] | 98 | Form factor for an elliptical cylinder. |
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| 99 | See L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, (1987). |
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[a8b3cdb] | 100 | """ |
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| 101 | category = "shape:cylinder" |
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| 102 | |
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| 103 | # pylint: disable=bad-whitespace, line-too-long |
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| 104 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 105 | parameters = [["r_minor", "Ang", 20.0, [0, inf], "volume", "Ellipse minor radius"], |
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| 106 | ["r_ratio", "", 1.5, [1, inf], "volume", "Ratio of major radius over minor radius"], |
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| 107 | ["length", "Ang", 400.0, [1, inf], "volume", "Length of the cylinder"], |
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| 108 | ["sld", "1e-6/Ang^2", 4.0, [-inf, inf], "", "Cylinder scattering length density"], |
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| 109 | ["solvent_sld", "1e-6/Ang^2", 1.0, [-inf, inf], "", "Solvent scattering length density"], |
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| 110 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "In plane angle"], |
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| 111 | ["phi", "degrees", 0, [-360, 360], "orientation", "Out of plane angle"], |
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| 112 | ["psi", "degrees", 0, [-360, 360], "orientation", "Major axis angle relative to Q"]] |
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| 113 | |
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| 114 | # pylint: enable=bad-whitespace, line-too-long |
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| 115 | |
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| 116 | source = ["lib/nr_bess_j1.c", "lib/gauss76.c", "lib/gauss20.c", "elliptical_cylinder.c"] |
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| 117 | |
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| 118 | demo = dict(scale=1, background=0, r_minor=100, r_ratio=1.5, length=400.0, |
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| 119 | sld=4.0, solvent_sld=1.0, theta=10.0, phi=20, psi=30, theta_pd=10, phi_pd=2, psi_pd=3) |
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| 120 | |
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| 121 | oldname = 'EllipticalCylinderModel' |
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| 122 | oldpars = dict(theta='cyl_theta', phi='cyl_phi', psi='cyl_psi', sld='sldCyl', solvent_sld='sldSolv') |
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| 123 | |
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| 124 | def ER(r_minor, r_ratio, length): |
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| 125 | """ |
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| 126 | Equivalent radius |
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| 127 | @param r_minor: Ellipse minor radius |
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| 128 | @param r_ratio: Ratio of major radius over minor radius |
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| 129 | @param length: Length of the cylinder |
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| 130 | """ |
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[c612162] | 131 | r = math.sqrt(r_minor * r_minor * r_ratio) |
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[a8b3cdb] | 132 | ddd = 0.75 * r * (2 * r * length + (length + r) * (length + pi * r)) |
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| 133 | return 0.5 * (ddd) ** (1. / 3.) |
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| 134 | |
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| 135 | tests = [[{'r_minor': 20.0, 'r_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024], |
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| 136 | [{'r_minor': 20.0, 'r_ratio': 1.2, 'length':300.0}, 'VR', 1], |
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| 137 | |
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| 138 | # The SasView test result was 0.00169, with a background of 0.001 |
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| 139 | [{'r_minor': 20.0, |
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| 140 | 'r_ratio': 1.5, |
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| 141 | 'sld': 4.0, |
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| 142 | 'length':400.0, |
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| 143 | 'solvent_sld':1.0, |
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| 144 | 'background':0.0 |
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| 145 | }, 0.001, 675.504402]] |
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