.. _binary-hard-sphere:

binary_hard_sphere
=======================================================

binary mixture of hard spheres with hard sphere interactions.

============== =========================================== ============ =============
Parameter      Description                                 Units        Default value
============== =========================================== ============ =============
scale          Scale factor or Volume fraction             None                     1
background     Source background                           |cm^-1|              0.001
radius_lg      radius of large particle                    |Ang|                  100
radius_sm      radius of small particle                    |Ang|                   25
volfraction_lg volume fraction of large particle           None                   0.1
volfraction_sm volume fraction of small particle           None                   0.2
sld_lg         scattering length density of large particle |1e-6Ang^-2|           3.5
sld_sm         scattering length density of small particle |1e-6Ang^-2|           0.5
sld_solvent    Solvent scattering length density           |1e-6Ang^-2|          6.36
============== =========================================== ============ =============

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


**Definition**

The binary hard sphere model provides the scattering intensity, for binary
mixture of hard spheres including hard sphere interaction between those
particles, using rhw Percus-Yevick closure. The calculation is an exact
multi-component solution that properly accounts for the 3 partial structure
factors as follows:

.. math::

    I(q) = (1-x)f_1^2(q) S_{11}(q) + 2[x(1-x)]^{1/2} f_1(q)f_2(q)S_{12}(q) +
    x\,f_2^2(q)S_{22}(q)

where $S_{ij}$ are the partial structure factors and $f_i$ are the scattering
amplitudes of the particles. The subscript 1 is for the smaller particle and 2
is for the larger. The number fraction of the larger particle,
($x = n2/(n1+n2)$, where $n$ = the number density) is internally calculated
based on the diameter ratio and the volume fractions.

.. math::
    :nowrap:

    \begin{align*}
    x &= \frac{(\phi_2 / \phi)\alpha^3}{(1-(\phi_2/\phi) + (\phi_2/\phi)
    \alpha^3)} \\
    \phi &= \phi_1 + \phi_2 = \text{total volume fraction} \\
    \alpha &= R_1/R_2 = \text{size ratio}
    \end{align*}

The 2D scattering intensity is the same as 1D, regardless of the orientation of
the *q* vector which is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}


**NOTE 1:** The volume fractions and the scattering contrasts are loosely
correlated, so holding as many parameters fixed to known values during fitting
will improve the robustness of the fit.

**NOTE 2:** Since the calculation uses the Percus-Yevick closure, all of the
limitations of that closure relation apply here. Specifically, one should be
wary of results for (total) volume fractions greater than approximately 40%.
Depending on the size ratios or number fractions, the limit on total volume
fraction may be lower.

**NOTE 3:** The heavy arithmetic operations also mean that at present the
function is poorly behaved at very low qr.  In some cases very large qr may
also be poorly behaved.  These should however be outside any useful region of
qr.

The code for this model is based originally on a c-library implementation by the
NIST Center for Neutron Research (Kline, 2006).

See the references for details.


.. figure:: img/binary_hard_sphere_autogenfig.png

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


**Source**

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

**References**

#. N W Ashcroft and D C Langreth, *Physical Review*, 156 (1967) 685-692
   [Errata found in *Phys. Rev.* 166 (1968) 934]

#. S R Kline, *J Appl. Cryst.*, 39 (2006) 895

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler **Date:** March 20, 2016
* **Last Reviewed by:** Paul Butler **Date:** March 20, 2016

