Changeset 67595af in sasmodels

Ignore:
Timestamp:
Mar 22, 2017 10:43:16 PM (3 years ago)
Branches:
master, core_shell_microgels, costrafo411, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
01ea374
Parents:
92dfe0c
git-author:
Paul Kienzle <pkienzle@…> (03/22/17 22:41:58)
git-committer:
Paul Kienzle <pkienzle@…> (03/22/17 22:43:16)
Message:

triaxial ellipsoid: update equations in the docs.

File:
1 edited

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Unmodified
 r925ad6e # Note: model title and parameter table are inserted automatically r""" All three axes are of different lengths with $R_a \leq R_b \leq R_c$ **Users should maintain this inequality for all calculations**. Definition ---------- .. figure:: img/triaxial_ellipsoid_geometry.jpg Ellipsoid with $R_a$ as *radius_equat_minor*, $R_b$ as *radius_equat_major* and $R_c$ as *radius_polar*.  For highest accuracy in the orientational average, prefer $R_c > R_b > R_a$. Given an ellipsoid .. math:: P(q) = \text{scale} V \left< F^2(q) \right> + \text{background} \frac{X^2}{R_a^2} + \frac{Y^2}{R_b^2} + \frac{Z^2}{R_c^2} = 1 where the volume $V = 4/3 \pi R_a R_b R_c$, and the averaging $\left<\ldots\right>$ is applied over all orientations for 1D. .. figure:: img/triaxial_ellipsoid_geometry.jpg Ellipsoid schematic. Definition ---------- The form factor calculated is the scattering is defined by the average over all orientations $\Omega$, .. math:: P(q) = \frac{\text{scale}}{V}\int_0^1\int_0^1 \Phi^2(qR_a^2\cos^2( \pi x/2) + qR_b^2\sin^2(\pi y/2)(1-y^2) + R_c^2y^2) dx dy P(q) = \text{scale}\frac{V}{4 \pi}\int_\Omega \Phi^2(qr) d\Omega + \text{background} where .. math:: \Phi(u) = 3 u^{-3} (\sin u - u \cos u) \Phi(qr) &= 3 j_1(qr)/qr = 3 (\sin qr - qr \cos qr)/(qr)^3 \\ r^2 &= R_a^2e^2 + R_b^2f^2 + R_c^2g^2 \\ V &= \tfrac{4}{3} \pi R_a R_b R_c The $e$, $f$ and $g$ terms are the projections of the orientation vector on $X$, $Y$ and $Z$ respectively.  Keeping the orientation fixed at the canonical axes, we can integrate over the incident direction using polar angle $-\pi/2 \le \gamma \le \pi/2$ and equatorial angle $0 \le \phi \le 2\pi$ (as defined in ref ), .. math:: \langle\Phi^2\rangle = \int_0^{2\pi} \int_{-\pi/2}^{\pi/2} \Phi^2(qr) \cos \gamma\,d\gamma d\phi with $e = \cos\gamma \sin\phi$, $f = \cos\gamma \cos\phi$ and $g = \sin\gamma$. A little algebra yields .. math:: r^2 = b^2(p_a \sin^2 \phi \cos^2 \gamma + 1 + p_c \sin^2 \gamma) for .. math:: p_a = \frac{a^2}{b^2} - 1 \text{ and } p_c = \frac{c^2}{b^2} - 1 Due to symmetry, the ranges can be restricted to a single quadrant $0 \le \gamma \le \pi/2$ and $0 \le \phi \le \pi/2$, scaling the resulting integral by 8. The computation is done using the substitution $u = \sin\gamma$, $du = \cos\gamma\,d\gamma$, giving .. math:: \langle\Phi^2\rangle &= 8 \int_0^{\pi/2} \int_0^1 \Phi^2(qr) du d\phi \\ r^2 &= b^2(p_a \sin^2(\phi)(1 - u^2) + 1 + p_c u^2) To provide easy access to the orientation of the triaxial ellipsoid, ---------- L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, New York, 1987.  Finnigan, J.A., Jacobs, D.J., 1971. *Light scattering by ellipsoidal particles in solution*, J. Phys. D: Appl. Phys. 4, 72-77. doi:10.1088/0022-3727/4/1/310 """ "Solvent scattering length density"], ["radius_equat_minor", "Ang", 20, [0, inf], "volume", "Minor equatorial radius"], "Minor equatorial radius, Ra"], ["radius_equat_major", "Ang", 400, [0, inf], "volume", "Major equatorial radius"], "Major equatorial radius, Rb"], ["radius_polar", "Ang", 10, [0, inf], "volume", "Polar radius"], "Polar radius, Rc"], ["theta", "degrees", 60, [-inf, inf], "orientation", "In plane angle"],