# Core Shell Sphere

## Description:

.. _core_shell_sphere:

This model provides the form factor, $P(q)$, for a spherical particle with a core-shell structure. The form factor is normalized by the particle volume.

For information about polarised and magnetic scattering, see the `magnetism` documentation.

Definition

The 1D scattering intensity is calculated in the following way (Guinier, 1955)

$$P(q) = \frac{\text{scale}}{V} F^2(q) + \text{background}$$
where

$$F(q) = \frac{3}{V_s}\left[ V_c(\rho_c-\rho_s)\frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3} + V_s(\rho_s-\rho_\text{solv})\frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3} \right]$$
where $V_s$ is the volume of the whole particle, $V_c$ is the volume of the core, $r_s$ = $radius$ + $thickness$ is the radius of the particle, $r_c$ is the radius of the core, $\rho_c$ is the scattering length density of the core, $\rho_s$ is the scattering length density of the shell, $\rho_\text{solv}$, is the scattering length density of the solvent.

The 2D scattering intensity is the same as $P(q)$ above, regardless of the orientation of the $q$ vector.

NB: The outer most radius (ie, = radius + thickness) is used as the effective radius for $S(Q)$ when $P(Q) \cdot S(Q)$ is applied.

Validation

Validation of our code was done by comparing the output of the 1D model to the output of the software provided by NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software.

References

A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955)

Authorship and Verification

**Author:**