.. _linear-pearls:

linear_pearls
=======================================================

Linear pearls model of scattering from spherical pearls.

=========== ================================================= ============ =============
Parameter   Description                                       Units        Default value
=========== ================================================= ============ =============
scale       Scale factor or Volume fraction                   None                     1
background  Source background                                 |cm^-1|              0.001
radius      Radius of the pearls                              |Ang|                   80
edge_sep    Length of the string segment - surface to surface |Ang|                  350
num_pearls  Number of the pearls                              None                     3
sld         SLD of the pearl spheres                          |1e-6Ang^-2|             1
sld_solvent SLD of the solvent                                |1e-6Ang^-2|           6.3
=========== ================================================= ============ =============

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


This model provides the form factor for $N$ spherical pearls of radius $R$
linearly joined by short strings (or segment length or edge separation)
$l$ $(= A - 2R)$. $A$ is the center-to-center pearl separation distance.
The thickness of each string is assumed to be negligible.

.. figure:: img/linear_pearls_geometry.jpg


**Definition**

The output of the scattering intensity function for the linear_pearls model
is given by (Dobrynin, 1996)

.. math::

    P(Q) = \frac{\text{scale}}{V}\left[ m_{p}^2
    \left(N+2\sum_{n-1}^{N-1}(N-n)\frac{\sin(qnl)}{qnl}\right)
    \left( 3\frac{\sin(qR)-qR\cos(qR)}{(qr)^3}\right)^2\right]

where the mass $m_p$ is $(SLD_{pearl}-SLD_{solvent})*(volume\ of\ N\ pearls)$.
V is the total volume.

The 2D scattering intensity is the same as P(q) above,
regardless of the orientation of the q vector.


.. figure:: img/linear_pearls_autogenfig.png

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


**Source**

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

**References**

#.  A V Dobrynin, M Rubinstein and S P Obukhov, *Macromol.*, 29 (1996) 2974-2979

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