Fuzzy Sphere

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

Definition

Scattering from spheres with a Gaussian interface.

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

$$ I(q) = \frac{\text{scale}}{V}(\Delta \rho)^2 A(q)^2 S(q) + \text{background}
$$
where the amplitude $A(q)$ describes the radial scattering length density profile of a homogeneous sphere convoluted with a Gaussian in order to give a function with a gradual drop-off in scattering length density (SLD) towards the interface (i.e. a sphere with a diffuse, or "fuzzy", interface):

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

**In this model,** $R$ **represents the radius at which the SLD has decreased to half of its value at the core, not the overall radius of a sphere**. This is a frequent source of confusion when applying this model. $\sigma_\text{fuzzy}$ is then the width of the fuzzy interface; strictly, the standard deviation from the average thickness of the interface.

From Reference [1]:

The inner regions ... that display a higher ... [SLD] are described by a radial box profile extending to a radius of approximately $R_\text{box} \sim R - 2 \sigma_\text{fuzzy}$. The profile approaches zero at $R_\text{SANS} \sim R + 2 \sigma_\text{fuzzy}$. Therefore, the overall size of the fuzzy sphere is approximated by ... $R_\text{SANS}$.

For this model to give meaningful results it is important $\sigma_\text{fuzzy} \ll R$. It is for the User to ensure that this condition is maintained, especially if applying polydispersity to one or both length scales. This model has been widely applied to the scattering from polymer microgel particles as illustrated below, where $R_\text{h}$ is the hydrodynamic radius.

Figure shows the fuzzy_sphere model applied to a microgel particle (adapted from [2], Fig 5).

Although the fuzzy sphere model often provides a good description of scattering data from such systems, advances in measurement techniques have highlighted that the *real-space* density profile can be far more complex than this model assumes [3].

This model is *not* suitable for describing spherical particles decorated with so-called polymer 'brushes' (where the SLD profile follows a parabolic decay) or spherical particles with terminally-attached polymer chains (where the SLD profile is expected to exhibit a maximum before the Gaussian decay).

To model more complex SLD profiles, see the `onion` and `spherical-sld` models.

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

#. E Ponomareva, B Tadgell, M Hildebrandt, M Krüsmann, S Prévost, P Mulvaney, M Karg, *Soft Matter*, 18 (2022) 807-825 #. F Scheffold, *Soft Matter*, 20 (2024) 8181-8184

Authorship and Verification

**Author:**
**Last Modified by: Steve King Date: April 16, 2026**
**Last Reviewed by: Paul Butler Date: May 15, 2026**