# core_shell_bicelle

Circular 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
thick_rim Rim shell thickness 10
thick_face Cylinder face thickness 10
length Cylinder length 50
sld_core Cylinder core scattering length density 10-6-2 1
sld_face Cylinder face scattering length density 10-6-2 4
sld_rim Cylinder rim scattering length density 10-6-2 4
sld_solvent Solvent scattering length density 10-6-2 1
theta In plane angle degree 90
phi Out of plane angle degree 0

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

Definition

This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. Thus this is a variation of a core-shell cylinder or disc where the shell on the walls and ends may be of different thicknesses and scattering length densities. The form factor is normalized by the 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 cylinder 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, to give:

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

where

\begin{split} \begin{align} F(Q,\alpha) = &\bigg[ (\rho_c - \rho_f) V_c \frac{2J_1(QRsin \alpha)}{QRsin\alpha}\frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\ &+(\rho_f - \rho_r) V_{c+f} \frac{2J_1(QRsin\alpha)}{QRsin\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 $$V_t$$ is the total volume of the bicelle, $$V_c$$ the volume of the core, $$V_{c+f}$$ the volume of the core plus the volume of the faces, $$R$$ is the radius 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 output of the 1D scattering intensity function for randomly oriented cylinders is then given by integrating over all possible $$\theta$$ and $$\phi$$.

The theta and phi parameters are not used for the 1D output. Our implementation of the scattering kernel and the 1D scattering intensity use the c-library from NIST.

References

 [1] D Singh (2009). Small angle scattering studies of self assembly in lipid mixtures, John’s Hopkins University Thesis (2009) 223-225. Available from Proquest

Authorship and Verification

• Author: NIST IGOR/DANSE Date: pre 2010