.. _fuzzy-sphere:

fuzzy_sphere
=======================================================

Scattering from spherical particles with a fuzzy surface.

=========== ======================================================================= ============ =============
Parameter   Description                                                             Units        Default value
=========== ======================================================================= ============ =============
scale       Scale factor or Volume fraction                                         None                     1
background  Source background                                                       |cm^-1|              0.001
sld         Particle scattering length density                                      |1e-6Ang^-2|             1
sld_solvent Solvent scattering length density                                       |1e-6Ang^-2|             3
radius      Sphere radius                                                           |Ang|                   60
fuzziness   std deviation of Gaussian convolution for interface (must be << radius) |Ang|                   10
=========== ======================================================================= ============ =============

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


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

**Definition**

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

.. math::

    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:

.. math::

    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

.. math:: q = \sqrt{{q_x}^2 + {q_y}^2}


.. figure:: img/fuzzy_sphere_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`fuzzy_sphere.py <src/fuzzy_sphere.py>`
$\ \star\ $ :download:`fuzzy_sphere.c <src/fuzzy_sphere.c>`
$\ \star\ $ :download:`sas_3j1x_x.c <src/sas_3j1x_x.c>`

**References**

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

**Authorship and Verification**

* **Author:**
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