# core_shell_sphere

Form factor for a monodisperse spherical particle with particle with a core-shell structure.

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
thickness Sphere shell thickness 10
sld_core core scattering length density 10-6-2 1
sld_shell shell scattering length density 10-6-2 2
sld_solvent Solvent scattering length density 10-6-2 3

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

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 Polarisation/Magnetic Scattering 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

$F^2(q) = \frac{3}{V_s}\left[ V_c(\rho_c-\rho_s)\frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3} + V_s(\rho_s-\rho_\text{solv})\frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3} \right]$

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.

References

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

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.