Changeset 40a87fa in sasmodels for sasmodels/models/elliptical_cylinder.py
- Timestamp:
- Aug 8, 2016 9:24:11 AM (8 years ago)
- Branches:
- master, core_shell_microgels, costrafo411, magnetic_model, release_v0.94, release_v0.95, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
- Children:
- 2472141
- Parents:
- 2d65d51
- File:
-
- 1 edited
-
sasmodels/models/elliptical_cylinder.py (modified) (9 diffs)
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sasmodels/models/elliptical_cylinder.py
r42356c8 r40a87fa 1 1 # pylint: disable=line-too-long 2 2 r""" 3 This function calculates the scattering from an elliptical cylinder.4 5 3 Definition for 2D (orientated system) 6 4 ------------------------------------- 7 5 8 The angles |theta| and |phi| define the orientation of the axis of the cylinder. The angle |bigpsi| is defined as the 9 orientation of the major axis of the ellipse with respect to the vector *Q*\ . A gaussian polydispersity can be added 10 to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii. 6 The angles $\theta$ and $\phi$ define the orientation of the axis of the 7 cylinder. The angle $\Psi$ is defined as the orientation of the major 8 axis of the ellipse with respect to the vector $Q$. A gaussian polydispersity 9 can be added to any of the orientation angles, and also for the minor 10 radius and the ratio of the ellipse radii. 11 11 12 12 .. figure:: img/elliptical_cylinder_geometry.png 13 13 14 Elliptical cylinder geometry $a$ = $r_{minor}$ and \ |nu|\ = $axis\_ratio$ = $r_{major} / r_{minor}$ 14 Elliptical cylinder geometry $a = r_\text{minor}$ 15 and $\nu = r_\text{major} / r_\text{minor}$ is the *axis_ratio*. 15 16 16 17 The function calculated is … … 18 19 .. math:: 19 20 20 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} 21 I(\vec q)=\frac{1}{V_\text{cyl}}\int{d\psi}\int{d\phi}\int{ 22 p(\theta,\phi,\psi)F^2(\vec q,\alpha,\psi)\sin(\theta)d\theta} 21 23 22 24 with the functions … … 24 26 .. math:: 25 27 26 F(\mathbf{q},\alpha,\psi)=2\frac{J_1(a)\sin(b)}{ab} 27 \\ 28 where a = \mathbf{q}\sin(\alpha)\left[ r^2_{major}\sin^2(\psi)+r^2_{minor}\cos(\psi) \right]^{1/2} 29 \\ 30 b=\mathbf{q}\frac{L}{2}\cos(\alpha) 28 F(\vec q,\alpha,\psi) = 2\frac{J_1(a)\sin(b)}{ab} 31 29 32 and the angle |bigpsi| is defined as the orientation of the major axis of the ellipse with respect to the vector $\vec q$ . 33 The angle $\alpha$ is the angle between the axis of the cylinder and $\vec q$. 30 where 31 32 .. math:: 33 34 a &= \vec q\sin(\alpha)\left[ 35 r^2_\text{major}\sin^2(\psi)+r^2_\text{minor}\cos(\psi) \right]^{1/2} 36 37 b &= \vec q\frac{L}{2}\cos(\alpha) 38 39 and the angle $\Psi$ is defined as the orientation of the major axis of the 40 ellipse with respect to the vector $\vec q$. The angle $\alpha$ is the angle 41 between the axis of the cylinder and $\vec q$. 34 42 35 43 … … 37 45 -------------------------------------------- 38 46 39 The form factor is averaged over all possible orientation before normalized by the particle volume 47 The form factor is averaged over all possible orientation before normalized 48 by the particle volume 40 49 41 50 .. math:: 42 P(q) = scale <F^2> / V43 51 44 To provide easy access to the orientation of the elliptical cylinder, we define the axis of the cylinder using two 45 angles |theta|, |phi| and |bigpsi| (see :ref:`cylinder orientation <cylinder-angle-definition>`). 46 The angle |bigpsi| is the rotational angle around its own long_c axis against the *q* plane. 47 For example, |bigpsi| = 0 when the *r_minor* axis is parallel to the *x*\ -axis of the detector. 52 P(q) = \text{scale} <F^2> / V 48 53 49 All angle parameters are valid and given only for 2D calculation; ie, an oriented system. 54 To provide easy access to the orientation of the elliptical cylinder, we 55 define the axis of the cylinder using two angles $\theta$, $\phi$ and $\Psi$ 56 (see :ref:`cylinder orientation <cylinder-angle-definition>`). The angle 57 $\Psi$ is the rotational angle around its own long_c axis against the $q$ plane. 58 For example, $\Psi = 0$ when the $r_\text{minor}$ axis is parallel to the 59 $x$ axis of the detector. 60 61 All angle parameters are valid and given only for 2D calculation; ie, an 62 oriented system. 50 63 51 64 .. figure:: img/elliptical_cylinder_angle_definition.jpg … … 55 68 .. figure:: img/cylinder_angle_projection.jpg 56 69 57 Examples of the angles for oriented elliptical cylinders against the detector plane. 70 Examples of the angles for oriented elliptical cylinders against the 71 detector plane. 58 72 59 NB: The 2nd virial coefficient of the cylinder is calculated based on the averaged radius (= sqrt(*r_minor*\ :sup:`2` \* *axis_ratio*)) 60 and length values, and used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 73 NB: The 2nd virial coefficient of the cylinder is calculated based on the 74 averaged radius $(=\sqrt{r_\text{minor}^2 * \text{axis ratio}})$ and length 75 values, and used as the effective radius for $S(Q)$ when $P(Q)*S(Q)$ is applied. 61 76 62 77 … … 64 79 ---------- 65 80 66 Validation of our code was done by comparing the output of the 1D calculation to the 67 angular average of the output of the 2D calculation over all possible angles. 81 Validation of our code was done by comparing the output of the 1D calculation 82 to the angular average of the output of the 2D calculation over all possible 83 angles. 68 84 69 In the 2D average, more binning in the angle |phi| is necessary to get the proper result. 70 The following figure shows the results of the averaging by varying the number of angular bins. 85 In the 2D average, more binning in the angle $\phi$ is necessary to get the 86 proper result. The following figure shows the results of the averaging by 87 varying the number of angular bins. 71 88 72 89 .. figure:: img/elliptical_cylinder_averaging.png … … 77 94 ---------- 78 95 79 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum,80 Ne w York, (1987)96 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and 97 Neutron Scattering*, Plenum, New York, (1987) 81 98 """ 82 99 … … 104 121 # pylint: enable=bad-whitespace, line-too-long 105 122 106 source = ["lib/polevl.c","lib/sas_J1.c", "lib/gauss76.c", "lib/gauss20.c", "elliptical_cylinder.c"] 123 source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "lib/gauss20.c", 124 "elliptical_cylinder.c"] 107 125 108 126 demo = dict(scale=1, background=0, r_minor=100, axis_ratio=1.5, length=400.0, 109 sld=4.0, sld_solvent=1.0, theta=10.0, phi=20, psi=30, theta_pd=10, phi_pd=2, psi_pd=3) 127 sld=4.0, sld_solvent=1.0, theta=10.0, phi=20, psi=30, 128 theta_pd=10, phi_pd=2, psi_pd=3) 110 129 111 130 def ER(r_minor, axis_ratio, length): … … 117 136 """ 118 137 radius = sqrt(r_minor * r_minor * axis_ratio) 119 ddd = 0.75 * radius * (2 * radius * length + (length + radius) * (length + pi * radius)) 138 ddd = 0.75 * radius * (2 * radius * length 139 + (length + radius) * (length + pi * radius)) 120 140 return 0.5 * (ddd) ** (1. / 3.) 121 141 122 tests = [[{'r_minor': 20.0, 'axis_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024], 123 [{'r_minor': 20.0, 'axis_ratio': 1.2, 'length':300.0}, 'VR', 1], 142 tests = [ 143 [{'r_minor': 20.0, 'axis_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024], 144 [{'r_minor': 20.0, 'axis_ratio': 1.2, 'length':300.0}, 'VR', 1], 124 145 125 # The SasView test result was 0.00169, with a background of 0.001 126 [{'r_minor': 20.0, 127 'axis_ratio': 1.5, 128 'sld': 4.0, 129 'length':400.0, 130 'sld_solvent':1.0, 131 'background':0.0 132 }, 0.001, 675.504402]] 146 # The SasView test result was 0.00169, with a background of 0.001 147 [{'r_minor': 20.0, 'axis_ratio': 1.5, 'sld': 4.0, 'length':400.0, 148 'sld_solvent':1.0, 'background':0.0}, 149 0.001, 675.504402], 150 ]
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