# fuzzy_sphere

Scattering from spherical particles with a fuzzy surface.

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
sld Particle scattering length density 10-6-2 1
sld_solvent Solvent scattering length density 10-6-2 3
fuzziness std deviation of Gaussian convolution for interface (must be << radius) 10

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

For information about polarised and magnetic scattering, see the Polarisation/Magnetic Scattering 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