r""" Definition ---------- Parameters for this model are the core axial ratio X and a shell thickness, which are more often what we would like to determine and makes the model better behaved, particularly when polydispersity is applied than the four independent radii used in the original parameterization of this model. .. figure:: img/core_shell_ellipsoid_geometry.png The geometric parameters of this model are shown in the diagram above, which shows (a) a cut through at the circular equator and (b) a cross section through the poles, of a prolate ellipsoid. When *X_core < 1* the core is oblate; when *X_core > 1* it is prolate. *X_core = 1* is a spherical core. For a fixed shell thickness *XpolarShell = 1*, to scale the shell thickness pro-rata with the radius set or constrain *XpolarShell = X_core*. When including an $S(q)$, the radius in $S(q)$ is calculated to be that of a sphere with the same 2nd virial coefficient of the outer surface of the ellipsoid. This may have some undesirable effects if the aspect ratio of the ellipsoid is large (ie, if $X << 1$ or $X >> 1$ ), when the $S(q)$ - which assumes spheres - will not in any case be valid. Generating a custom product model will enable separate effective volume fraction and effective radius in the $S(q)$. If SAS data are in absolute units, and the SLDs are correct, then scale should be the total volume fraction of the "outer particle". When $S(q)$ is introduced this moves to the $S(q)$ volume fraction, and scale should then be 1.0, or contain some other units conversion factor (for example, if you have SAXS data). The calculation of intensity follows that for the solid ellipsoid, but with separate terms for the core-shell and shell-solvent boundaries. .. math:: P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background} where .. math:: :nowrap: \begin{align*} F(q,\alpha) = &f(q,radius\_equat\_core,radius\_equat\_core.x\_core,\alpha) \\ &+ f(q,radius\_equat\_core + thick\_shell, radius\_equat\_core.x\_core + thick\_shell.x\_polar\_shell,\alpha) \end{align*} where .. math:: f(q,R_e,R_p,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)] - \cos[qr(R_p,R_e,\alpha)])} {[qr(R_p,R_e,\alpha)]^3} and .. math:: r(R_e,R_p,\alpha) = \left[ R_e^2 \sin^2 \alpha + R_p^2 \cos^2 \alpha \right]^{1/2} $\alpha$ is the angle between the axis of the ellipsoid and $\vec q$, $V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the polar radius along the rotational axis of the ellipsoid, $R_e$ is the equatorial radius perpendicular to the rotational axis of the ellipsoid and $\Delta \rho$ (contrast) is the scattering length density difference, either $(sld\_core - sld\_shell)$ or $(sld\_shell - sld\_solvent)$. For randomly oriented particles: .. math:: F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha} For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters will appear when fitting 2D data, see the :ref:`elliptical-cylinder` model for further information. References ---------- see for example: Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461. Berr, S. J. Phys. Chem., 1987, 91, 4760. Authorship and Verification ---------------------------- * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Last Modified by:** Richard Heenan (reparametrised model) **Date:** 2015 * **Last Reviewed by:** Richard Heenan **Date:** October 6, 2016 """ import numpy as np from numpy import inf, sin, cos, pi name = "core_shell_ellipsoid" title = "Form factor for an spheroid ellipsoid particle with a core shell structure." description = """ [core_shell_ellipsoid] Calculates the form factor for an spheroid ellipsoid particle with a core_shell structure. The form factor is averaged over all possible orientations of the ellipsoid such that P(q) = scale*/Vol + bkg, where f is the single particle scattering amplitude. [Parameters]: radius_equat_core = equatorial radius of core, x_core = ratio of core polar/equatorial radii, thick_shell = equatorial radius of outer surface, x_polar_shell = ratio of polar shell thickness to equatorial shell thickness, sld_core = SLD_core sld_shell = SLD_shell sld_solvent = SLD_solvent background = Incoherent bkg scale =scale Note:It is the users' responsibility to ensure that shell radii are larger than core radii. oblate: polar radius < equatorial radius prolate : polar radius > equatorial radius - this new model will make this easier and polydispersity integrals more logical (as previously the shell could disappear). """ category = "shape:ellipsoid" # pylint: disable=bad-whitespace, line-too-long # ["name", "units", default, [lower, upper], "type", "description"], parameters = [ ["radius_equat_core","Ang", 20, [0, inf], "volume", "Equatorial radius of core"], ["x_core", "None", 3, [0, inf], "volume", "axial ratio of core, X = r_polar/r_equatorial"], ["thick_shell", "Ang", 30, [0, inf], "volume", "thickness of shell at equator"], ["x_polar_shell", "", 1, [0, inf], "volume", "ratio of thickness of shell at pole to that at equator"], ["sld_core", "1e-6/Ang^2", 2, [-inf, inf], "sld", "Core scattering length density"], ["sld_shell", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Shell scattering length density"], ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], ["theta", "degrees", 0, [-360, 360], "orientation", "elipsoid axis to beam angle"], ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], ] # pylint: enable=bad-whitespace, line-too-long source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "core_shell_ellipsoid.c"] have_Fq = True def ER(radius_equat_core, x_core, thick_shell, x_polar_shell): """ Returns the effective radius used in the S*P calculation """ from .ellipsoid import ER as ellipsoid_ER polar_outer = radius_equat_core*x_core + thick_shell*x_polar_shell equat_outer = radius_equat_core + thick_shell return ellipsoid_ER(polar_outer, equat_outer) def random(): volume = 10**np.random.uniform(5, 12) outer_polar = 10**np.random.uniform(1.3, 4) outer_equatorial = np.sqrt(volume/outer_polar) # ignore 4/3 pi # Use a distribution with a preference for thin shell or thin core # Avoid core,shell radii < 1 thickness_polar = np.random.beta(0.5, 0.5)*(outer_polar-2) + 1 thickness_equatorial = np.random.beta(0.5, 0.5)*(outer_equatorial-2) + 1 radius_polar = outer_polar - thickness_polar radius_equatorial = outer_equatorial - thickness_equatorial x_core = radius_polar/radius_equatorial x_polar_shell = thickness_polar/thickness_equatorial pars = dict( #background=0, sld=0, sld_solvent=1, radius_equat_core=radius_equatorial, x_core=x_core, thick_shell=thickness_equatorial, x_polar_shell=x_polar_shell, ) return pars q = 0.1 # tests had in old coords theta=0, phi=0; new coords theta=90, phi=0 qx = q*cos(pi/6.0) qy = q*sin(pi/6.0) # 11Jan2017 RKH sorted tests after redefinition of angles tests = [ # Accuracy tests based on content in test/utest_coreshellellipsoidXTmodel.py [{'radius_equat_core': 200.0, 'x_core': 0.1, 'thick_shell': 50.0, 'x_polar_shell': 0.2, 'sld_core': 2.0, 'sld_shell': 1.0, 'sld_solvent': 6.3, 'background': 0.001, 'scale': 1.0, }, 1.0, 0.00189402], # Additional tests with larger range of parameters [{'background': 0.01}, 0.1, 11.6915], [{'radius_equat_core': 20.0, 'x_core': 200.0, 'thick_shell': 54.0, 'x_polar_shell': 3.0, 'sld_core': 20.0, 'sld_shell': 10.0, 'sld_solvent': 6.0, 'background': 0.0, 'scale': 1.0, }, 0.01, 8688.53], # 2D tests [{'background': 0.001, 'theta': 90.0, 'phi': 0.0, }, (0.4, 0.5), 0.00690673], [{'radius_equat_core': 20.0, 'x_core': 200.0, 'thick_shell': 54.0, 'x_polar_shell': 3.0, 'sld_core': 20.0, 'sld_shell': 10.0, 'sld_solvent': 6.0, 'background': 0.01, 'scale': 0.01, 'theta': 90.0, 'phi': 0.0, }, (qx, qy), 0.01000025], ]