# cylinder model # Note: model title and parameter table are inserted automatically r""" For information about polarised and magnetic scattering, see the :ref:magnetism documentation. Definition ---------- The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955) .. math:: P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha).sin(\alpha) + \text{background} where .. math:: F(q,\alpha) = 2 (\Delta \rho) V \frac{\sin \left(\tfrac12 qL\cos\alpha \right)} {\tfrac12 qL \cos \alpha} \frac{J_1 \left(q R \sin \alpha\right)}{q R \sin \alpha} and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, $V =\pi R^2L$ is the volume of the cylinder, $L$ is the length of the cylinder, $R$ is the radius of the cylinder, and $\Delta\rho$ (contrast) is the scattering length density difference between the scatterer and the solvent. $J_1$ is the first order Bessel function. For randomly oriented particles: .. math:: F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}=\int_{0}^{1}{F^2(q,u)du} Numerical integration is simplified by a change of variable to $u = cos(\alpha)$ with $sin(\alpha)=\sqrt{1-u^2}$. The output of the 1D scattering intensity function for randomly oriented cylinders is thus given by .. math:: P(q) = \frac{\text{scale}}{V} \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background} NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and length values, and used as the effective radius for $S(q)$ when $P(q) \cdot S(q)$ is applied. For 2d scattering from oriented cylinders, we define the direction of the axis of the cylinder using two angles $\theta$ (note this is not the same as the scattering angle used in q) and $\phi$. Those angles are defined in :numref:cylinder-angle-definition , for further details see :ref:orientation. .. _cylinder-angle-definition: .. figure:: img/cylinder_angle_definition.png Angles $\theta$ and $\phi$ orient the cylinder relative to the beam line coordinates, where the beam is along the $z$ axis. Rotation $\theta$, initially in the $xz$ plane, is carried out first, then rotation $\phi$ about the $z$ axis. Orientation distributions are described as rotations about two perpendicular axes $\delta_1$ and $\delta_2$ in the frame of the cylinder itself, which when $\theta = \phi = 0$ are parallel to the $Y$ and $X$ axes. .. figure:: img/cylinder_angle_projection.png Examples for oriented cylinders. The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the model when fitting 2d data. Validation ---------- Validation of the code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). The implementation of the intensity for fully oriented cylinders was done by averaging over a uniform distribution of orientations using .. math:: P(q) = \int_0^{\pi/2} d\phi \int_0^\pi p(\theta) P_0(q,\theta) \sin \theta\ d\theta where $p(\theta,\phi) = 1$ is the probability distribution for the orientation and $P_0(q,\theta)$ is the scattering intensity for the fully oriented system, and then comparing to the 1D result. References ---------- #. J. Pedersen, *Adv. Colloid Interface Sci.*, 70 (1997) 171-210 #. G. Fournet, *Bull. Soc. Fr. Mineral. Cristallogr.*, 74 (1951) 39-113 #. L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659 Authorship and Verification ---------------------------- * **Author:** * **Last Modified by:** * **Last Reviewed by:** """ import numpy as np # type: ignore from numpy import pi, inf # type: ignore name = "cylinder" title = "Right circular cylinder with uniform scattering length density." description = """ f(q,alpha) = 2*(sld - sld_solvent)*V*sin(qLcos(alpha)/2)) /[qLcos(alpha)/2]*J1(qRsin(alpha))/[qRsin(alpha)] P(q,alpha)= scale/V*f(q,alpha)^(2)+background V: Volume of the cylinder R: Radius of the cylinder L: Length of the cylinder J1: The Bessel function alpha: angle between the axis of the cylinder and the q-vector for 1D :the ouput is P(q)=scale/V*integral from pi/2 to zero of... f(q,alpha)^(2)*sin(alpha)*dalpha + background """ category = "shape:cylinder" # pylint: disable=bad-whitespace, line-too-long # [ "name", "units", default, [lower, upper], "type", "description"], parameters = [ ["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder scattering length density"], ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], ["radius", "Ang", 20, [0, inf], "volume", "Cylinder radius"], ["length", "Ang", 400, [0, inf], "volume", "Cylinder length"], ["theta", "degrees", 60, [-360, 360], "orientation", "cylinder axis to beam angle"], ["phi", "degrees", 60, [-360, 360], "orientation", "rotation about beam"], ] # pylint: enable=bad-whitespace, line-too-long source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"] valid = "radius >= 0.0 && length >= 0.0" have_Fq = True radius_effective_modes = [ "excluded volume", "equivalent volume sphere", "radius", "half length", "half min dimension", "half max dimension", "half diagonal", ] def random(): """Return a random parameter set for the model.""" volume = 10**np.random.uniform(5, 12) length = 10**np.random.uniform(-2, 2)*volume**0.333 radius = np.sqrt(volume/length/np.pi) pars = dict( #scale=1, #background=0, length=length, radius=radius, ) return pars # Test 1-D and 2-D models qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5) theta, phi = 80.1534480601659, 10.1510817110481 # (10, 10) in sasview 3.x tests = [ [{}, 0.2, 0.042761386790780453], [{}, [0.2], [0.042761386790780453]], [{"scale": 1., "background": 0.}, [0.01, 0.05, 0.2], # these numerical results NOT independently verified [3.01823887e+02, 5.36661653e+01, 4.17613868e-02]], [{"scale": 1., "background": 0.}, # the longer list here checks F1, F2=I(Q)*V, R_eff, volume, volume_ratio 0.05, 2214.9614083, 26975556.88749548, 73.34013315261608, 502654.8245743669, 1.0], #2345678901234567890123456789012345678901234567890123456789012345678901234567890 [{"@S": "hardsphere", # MONODISPERSE "scale": 5., "background": 0., "volfraction": 0.2, "structure_factor_mode": 0, # normal decoupling approx "radius_effective_mode": 1, # Reff "excluded volume" }, [0.01, 0.05, 0.2], [7.35571916e+01, 5.78147797e+01, 4.15623248e-2] ], [{"@S": "hardsphere", "scale": 5., "background": 0., "volfraction": 0.2, "structure_factor_mode": 1, # beta approx "radius_effective_mode": 1, # Reff "excluded volume" }, [0.01, 0.05, 0.2], [8.29729770e+01, 5.44206752e+01, 4.17598382e-2] ], [{"@S": "hardsphere", # POLYDISPERSE "scale": 5., "background": 0., "volfraction": 0.2, "radius_pd": 0.2, "radius_pd_n": 15, "length_pd": 0.1, "structure_factor_mode": 0, # normal decoupling approx "radius_effective_mode": 1, # Reff "excluded volume" }, [0.01, 0.05, 0.2], [87.50350037, 63.95202427, 0.27889988] ], [{"@S": "hardsphere", "scale": 5., "background": 0., "volfraction": 0.2, "radius_pd": 0.2, "radius_pd_n": 15, "length_pd": 0.1, "structure_factor_mode": 1, # beta approx "radius_effective_mode": 1, # Reff "excluded volume" }, [0.01, 0.05, 0.2], [132.2101165, 59.89948174, 0.28048784] ], # [{'theta': theta, 'phi': phi}, (qx, qy), 0.03514647218513852], [{'theta': theta, 'phi': phi}, [(qx, qy)], [0.03514647218513852]], ] del qx, qy, theta, phi # not necessary to delete, but cleaner def _extend_with_reff_tests(radius, length): """Test R_eff and form volume calculations""" # V and Vr are the same for each R_eff mode V = pi*radius**2*length # shell volume = form volume for solid objects Vr = 1.0 # form:shell volume ratio # Use test value for I(0.2) from above to check Fsq value. Need to # remove scale and background before testing. q = 0.2 scale, background = V, 0.001 Fsq = (0.042761386790780453 - background)*scale F = None # Need target value for # Various values for R_eff, depending on mode r_effs = [ 0., 0.5*(0.75*radius*(2.0*radius*length + (radius + length)*(pi*radius + length)))**(1./3.), (0.75*radius**2*length)**(1./3.), radius, length/2., min(radius, length/2.), max(radius, length/2.), np.sqrt(4*radius**2 + length**2)/2., ] tests.extend([ ({'radius_effective_mode': 0}, q, F, Fsq, r_effs, V, Vr), ({'radius_effective_mode': 1}, q, F, Fsq, r_effs, V, Vr), ({'radius_effective_mode': 2}, q, F, Fsq, r_effs, V, Vr), ({'radius_effective_mode': 3}, q, F, Fsq, r_effs, V, Vr), ({'radius_effective_mode': 4}, q, F, Fsq, r_effs, V, Vr), ({'radius_effective_mode': 5}, q, F, Fsq, r_effs, V, Vr), ({'radius_effective_mode': 6}, q, F, Fsq, r_effs, V, Vr), ({'radius_effective_mode': 7}, q, F, Fsq, r_effs, V, Vr), ]) # Test Reff and volume with default model parameters _extend_with_reff_tests(parameters, parameters) del _extend_with_reff_tests # ADDED by: RKH ON: 18Mar2016 renamed sld's etc