.. _stacked-disks:

stacked_disks
=======================================================

Form factor for a stacked set of non exfoliated core/shell disks

=========== ============================================================ ============ =============
Parameter   Description                                                  Units        Default value
=========== ============================================================ ============ =============
scale       Scale factor or Volume fraction                              None                     1
background  Source background                                            |cm^-1|              0.001
thick_core  Thickness of the core disk                                   |Ang|                   10
thick_layer Thickness of layer each side of core                         |Ang|                   10
radius      Radius of the stacked disk                                   |Ang|                   15
n_stacking  Number of stacked layer/core/layer disks                     None                     1
sigma_d     Sigma of nearest neighbor spacing                            |Ang|                    0
sld_core    Core scattering length density                               |1e-6Ang^-2|             4
sld_layer   Layer scattering length density                              |1e-6Ang^-2|             0
sld_solvent Solvent scattering length density                            |1e-6Ang^-2|             5
theta       Orientation of the stacked disk axis w/respect incoming beam degree                   0
phi         Rotation about beam                                          degree                   0
=========== ============================================================ ============ =============

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


**Definition**

This model provides the form factor, $P(q)$, for stacked discs (tactoids)
with a core/layer structure which is constructed itself as $P(q) S(Q)$
multiplying a $P(q)$ for individual core/layer disks by a structure factor
$S(q)$ proposed by Kratky and Porod in 1949\ [#Kratky1949]_ assuming the next
neighbor distance (d-spacing) in the stack of parallel discs obeys a Gaussian
distribution. As such the normalization of this "composite" form factor is
relative to the individual disk volume, not the volume of the stack of disks.
This model is appropriate for example for non non exfoliated clay particles
such as Laponite.

.. figure:: img/stacked_disks_geometry.png

   Geometry of a single core/layer disk

The scattered intensity $I(q)$ is calculated as

.. math::

    I(q) = N\int_{0}^{\pi /2}\left[ \Delta \rho_t
    \left( V_t f_t(q,\alpha) - V_c f_c(q,\alpha)\right) + \Delta
    \rho_c V_c f_c(q,\alpha)\right]^2 S(q,\alpha)\sin{\alpha}\ d\alpha
    + \text{background}

where the contrast

.. math::

    \Delta \rho_i = \rho_i - \rho_\text{solvent}

and $N$ is the number of individual (single) discs per unit volume, $\alpha$
is the angle between the axis of the disc and $q$, and $V_t$ and $V_c$ are the
total volume and the core volume of a single disc, respectively, and

.. math::

    f_t(q,\alpha) =
    \left(\frac{\sin(q(d+h)\cos{\alpha})}{q(d+h)\cos{\alpha}}\right)
    \left(\frac{2J_1(qR\sin{\alpha})}{qR\sin{\alpha}} \right)

    f_c(q,\alpha) =
    \left(\frac{\sin(qh)\cos{\alpha})}{qh\cos{\alpha}}\right)
    \left(\frac{2J_1(qR\sin{\alpha})}{qR\sin{\alpha}}\right)

where $d$ = thickness of the layer (*thick_layer*),
$2h$ = core thickness (*thick_core*), and $R$ = radius of the disc (*radius*).

.. math::

    S(q,\alpha) = 1 + \frac{1}{2}\sum_{k=1}^n(n-k)\cos{(kDq\cos{\alpha})}
    \exp\left[ -k(q)^2(D\cos{\alpha}~\sigma_d)^2/2\right]

where $n$ is the total number of the disc stacked (*n_stacking*),
$D = 2(d+h)$ is the next neighbor center-to-center distance (d-spacing),
and $\sigma_d$ = the Gaussian standard deviation of the d-spacing (*sigma_d*).
Note that $D\cos(\alpha)$ is the component of $D$ parallel to $q$ and the last
term in the equation above is effectively a Debye-Waller factor term.

.. note::

    1. Each assembly in the stack is layer/core/layer, so the spacing of the
    cores is core plus two layers. The 2nd virial coefficient of the cylinder
    is calculated based on the *radius* and *length*
    = *n_stacking* * (*thick_core* + 2 * *thick_layer*)
    values, and used as the effective radius for $S(Q)$ when $P(Q) * S(Q)$
    is applied.

    2. the resolution smearing calculation uses 76 Gaussian quadrature points
    to properly smear the model since the function is HIGHLY oscillatory,
    especially around the q-values that correspond to the repeat distance of
    the layers.

2d scattering from oriented stacks is calculated in the same way as for
cylinders, for further details of the calculation and angular dispersions
see :ref:`orientation`.

.. figure:: img/cylinder_angle_definition.png

    Angles $\theta$ and $\phi$ orient the stack of discs relative
    to the beam line coordinates, where the beam is along the $z$ axis.
    Rotation $\theta$, initially in the $xz$ plane, is carried out first,
    then rotation $\phi$ about the $z$ axis. Orientation distributions are
    described as rotations about two perpendicular axes $\delta_1$ and
    $\delta_2$ in the frame of the cylinder itself, which when
    $\theta = \phi = 0$ are parallel to the $Y$ and $X$ axes.


Our model is derived from the form factor calculations implemented in a
C-library provided by the NIST Center for Neutron Research\ [#Kline2006]_


.. figure:: img/stacked_disks_autogenfig.png

    1D and 2D plots corresponding to the default parameters of the model.


**Source**

:download:`stacked_disks.py <src/stacked_disks.py>`
$\ \star\ $ :download:`stacked_disks.c <src/stacked_disks.c>`
$\ \star\ $ :download:`gauss76.c <src/gauss76.c>`
$\ \star\ $ :download:`sas_J1.c <src/sas_J1.c>`
$\ \star\ $ :download:`polevl.c <src/polevl.c>`

**References**

See also Higgins and Benoit [#Higgins1994]_ and Guinier and
Fournet [#Guinier1955]_.

.. [#Kratky1949] O Kratky and G Porod, *J. Colloid Science*, 4, (1949) 35
.. [#Kline2006] S R Kline, *J Appl. Cryst.*, 39 (2006) 895
.. [#Higgins1994] J S Higgins and H C Benoit, *Polymers and Neutron Scattering*,
   Clarendon, Oxford, 1994
.. [#Guinier1955] A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*,
   John Wiley and Sons, New York, 1955

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler and Paul Kienzle **Date:** November 26, 2016
* **Last Reviewed by:** Paul Butler and Paul Kienzle **Date:** November 26, 2016

