# Core Shell Bicelle Elliptical

## Description:

Definition

This model provides the form factor for an elliptical cylinder with a core-shell scattering length density profile. Thus this is a variation of the core-shell bicelle model, but with an elliptical cylinder for the core. Outer shells on the rims and flat ends may be of different thicknesses and scattering length densities. The form factor is normalized by the total particle volume.

Schematic cross-section of bicelle. Note however that the model here calculates for rectangular, not curved, rims as shown below.

Cross section of model used here. Users will have to decide how to distribute "heads" and "tails" between the rim, face and core regions in order to estimate appropriate starting parameters.

Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the scattering length density variation along the bicelle axis is:

$$\rho(r) = \begin{cases} rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) \\[1.5ex] rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) \end{cases}$$
The form factor for the bicelle is calculated in cylindrical coordinates, where $\alpha$ is the angle between the $Q$ vector and the cylinder axis, and $\psi$ is the angle for the ellipsoidal cross section core, to give:

$$I(Q,\alpha,\psi) = \frac{\text{scale}}{V_t} \cdot F(Q,\alpha, \psi)^2 \cdot sin(\alpha) + \text{background}$$
where a numerical integration of $F(Q,\alpha, \psi)^2 \cdot sin(\alpha)$ is carried out over \alpha and \psi for:

\begin{align*} F(Q,\alpha,\psi) = bigg[ (\rho_c - \rho_f) V_c \frac{2J_1(QR'sin \alpha)}{QR'sin\alpha} \frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\ &+(\rho_f - \rho_r) V_{c+f} \frac{2J_1(QR'sin\alpha)}{QR'sin\alpha} \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\ &+(\rho_r - \rho_s) V_t \frac{2J_1(Q(R'+t_r)sin\alpha)}{Q(R'+t_r)sin\alpha} \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \bigg] \end{align*}
where

$$R'=\frac{R}{\sqrt{2}}\sqrt{(1+X_{core}^{2}) + (1-X_{core}^{2})cos(\psi)}$$

and $V_t = \pi.(R+t_r)(Xcore.R+t_r)^2.(L+2.t_f)$ is the total volume of the bicelle, $V_c = \pi.Xcore.R^2.L$ the volume of the core, $V_{c+f} = \pi.Xcore.R^2.(L+2.t_f)$ the volume of the core plus the volume of the faces, $R$ is the radius of the core, $Xcore$ is the axial ratio of the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$ the thickness of the rim and $J_1$ the usual first order bessel function. The core has radii $R$ and $Xcore.R$ so is circular, as for the core_shell_bicelle model, for $Xcore$ =1. Note that you may need to limit the range of $Xcore$, especially if using the Monte-Carlo algorithm, as setting radius to $R/Xcore$ and axial ratio to $1/Xcore$ gives an equivalent solution!

The output of the 1D scattering intensity function for randomly oriented bicelles is then given by integrating over all possible $\alpha$ and $\psi$.

For oriented bicelles the *theta*, *phi* and *psi* orientation parameters will appear when fitting 2D data, see the `elliptical-cylinder` model for further information.

Definition of the angles for the oriented core_shell_bicelle_elliptical particles.

Model verified using Monte Carlo simulation for 1D and 2D scattering.

References

L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659

Authorship and Verification

**Author:** Richard Heenan **Date:** December 14, 2016