# Changeset 3b571ae in sasmodels

Ignore:
Timestamp:
Mar 22, 2017 6:32:33 PM (9 months ago)
Branches:
master, boltzmann, costrafo411, doc_update, generic_integration_loop, ticket-1043, ticket-786
Children:
61104c8
Parents:
b00a646
Message:

ellipsoid: fix docs so equations are consistent with code; rearrange code for speed and readability

Location:
sasmodels/models
Files:
2 edited

### Legend:

Unmodified
 r925ad6e .. math:: F(q,\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} F(q,\alpha) = \Delta \rho V \frac{3(\sin qr  - qr \cos qr)}{(qr)^3} and for .. math:: r(R_p,R_e,\alpha) = \left[ R_e^2 \sin^2 \alpha + R_p^2 \cos^2 \alpha \right]^{1/2} r = \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 between the scatterer and the solvent. $V = (4/3)\pi R_pR_e^2$ 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 between the scatterer and the solvent. For randomly oriented particles: For randomly oriented particles use the orientational average, .. math:: F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha} \langle F^2(q) \rangle = \int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)\,d\alpha} computed via substitution of $u=\sin(\alpha)$, $du=\cos(\alpha)\,d\alpha$ as .. math:: \langle F^2(q) \rangle = \int_0^1{F^2(q, u)\,du} with .. math:: r = R_e \left[ 1 + u^2\left(R_p^2/R_e^2 - 1\right)\right]^{1/2} To provide easy access to the orientation of the ellipsoid, we define :ref:cylinder orientation figure . For the ellipsoid, $\theta$ is the angle between the rotational axis and the $z$ -axis. and the $z$ -axis in the $xz$ plane followed by a rotation by $\phi$ in the $xy$ plane. NB: The 2nd virial coefficient of the solid ellipsoid is calculated based than 500. Model was also tested against the triaxial ellipsoid model with equal major and minor equatorial radii.  It is also consistent with the cyclinder model with polar radius equal to length and equatorial radius equal to radius. References ---------- *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press, New York, 1987. Authorship and Verification ---------------------------- * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Converted to sasmodels by:** Helen Park **Date:** July 9, 2014 * **Last Modified by:** Paul Kienzle **Date:** March 22, 2017 """