# Fuzzy Sphere

## Description:

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

Definition

The scattering intensity $I(q)$ is calculated as:

$$I(q) = \frac{\text{scale}}{V}(\Delta \rho)^2 A^2(q) S(q) + \text{background}$$

where the amplitude $A(q)$ is given as the typical sphere scattering convoluted with a Gaussian to get a gradual drop-off in the scattering length density:

$$A(q) = \frac{3\left[\sin(qR) - qR \cos(qR)\right]}{(qR)^3} \exp\left(\frac{-(\sigma_\text{fuzzy}q)^2}{2}\right)$$
Here $A(q)^2$ is the form factor, $P(q)$. The scale is equivalent to the volume fraction of spheres, each of volume, $V$. Contrast $(\Delta \rho)$ is the difference of scattering length densities of the sphere and the surrounding solvent.

Poly-dispersion in radius and in fuzziness is provided for, though the fuzziness must be kept much smaller than the sphere radius for meaningful results.

From the reference:

The "fuzziness" of the interface is defined by the parameter $\sigma_\text{fuzzy}$. The particle radius $R$ represents the radius of the particle where the scattering length density profile decreased to 1/2 of the core density. $\sigma_\text{fuzzy}$ is the width of the smeared particle surface; i.e., the standard deviation from the average height of the fuzzy interface. The inner regions of the microgel that display a higher density are described by the radial box profile extending to a radius of approximately $R_\text{box} \sim R - 2 \sigma$. The profile approaches zero as $R_\text{sans} \sim R + 2\sigma$.

For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the $q$ vector is defined as

$$q = \sqrt{{q_x}^2 + {q_y}^2}$$
References

M Stieger, J. S Pedersen, P Lindner, W Richtering, *Langmuir*, 20 (2004) 7283-7292

Authorship and Verification

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