# core_shell_bicelle_elliptical

Elliptical cylinder with a core-shell scattering length density profile..

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
x_core axial ratio of core, X = r_polar/r_equatorial None 3
thick_rim Rim shell thickness 8
thick_face Cylinder face thickness 14
length Cylinder length 50
sld_core Cylinder core scattering length density 10-6-2 4
sld_face Cylinder face scattering length density 10-6-2 7
sld_rim Cylinder rim scattering length density 10-6-2 1
sld_solvent Solvent scattering length density 10-6-2 6
theta In plane angle degree 90
phi Out of plane angle degree 0
psi Major axis angle relative to Q degree 0

The returned value is scaled to units of cm-1 sr-1, absolute scale.

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.

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:

$\begin{split}\rho(r) = \begin{cases} &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) \\[1.5ex] &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) \end{cases}\end{split}$

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:

$I(Q,\alpha,\psi) = \frac{\text{scale}}{V_t} \cdot F(Q,\alpha, \psi)^2.sin(\alpha) + \text{background}$

where a numerical integration of $$F(Q,\alpha, \psi)^2.sin(\alpha)$$ is carried out over alpha and psi for:

\begin{split} \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}\end{split}

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 bicellles the theta, phi and psi orientation parameters only appear when fitting 2D data, see the elliptical_cylinder model for further information.

References

 [1]

Authorship and Verification

• Author: Richard Heenan Date: December 14, 2016