.. _teubner-strey:

teubner_strey
=======================================================

Teubner-Strey model of microemulsions

============= =============================== ============ =============
Parameter     Description                     Units        Default value
============= =============================== ============ =============
scale         Scale factor or Volume fraction None                     1
background    Source background               |cm^-1|              0.001
volfraction_a Volume fraction of phase a      None                   0.5
sld_a         SLD of phase a                  |1e-6Ang^-2|           0.3
sld_b         SLD of phase b                  |1e-6Ang^-2|           6.3
d             Domain size (periodicity)       |Ang|                  100
xi            Correlation length              |Ang|                   30
============= =============================== ============ =============

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


**Definition**

This model calculates the scattered intensity of a two-component system
using the Teubner-Strey model. Unlike :ref:`dab` this function generates
a peak. A two-phase material can be characterised by two length scales -
a correlation length and a domain size (periodicity).

The original paper by Teubner and Strey defined the function as:

.. math::

    I(q) \propto \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background}

where the parameters $a_2$, $c_1$ and $c_2$ are defined in terms of the
periodicity, $d$, and correlation length $\xi$ as:

.. math::

    a_2 &= \biggl[1+\bigl(\frac{2\pi\xi}{d}\bigr)^2\biggr]^2\\
    c_1 &= -2\xi^2\bigl(\frac{2\pi\xi}{d}\bigr)^2+2\xi^2\\
    c_2 &= \xi^4

and thus, the periodicity, $d$ is given by

.. math::

    d = 2\pi\left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2}
                  - \frac14\frac{c_1}{c_2}\right]^{-1/2}

and the correlation length, $\xi$, is given by

.. math::

    \xi = \left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2}
                  + \frac14\frac{c_1}{c_2}\right]^{-1/2}

Here the model is parameterised in terms of  $d$ and $\xi$ and with an explicit
volume fraction for one phase, $\phi_a$, and contrast,
$\delta\rho^2 = (\rho_a - \rho_b)^2$ :

.. math::

    I(q) = \frac{8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi}
        {a_2 + c_1q^2 + c_2q^4}

where :math:`8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi` is the constant of
proportionality from the first equation above.

In the case of a microemulsion, $a_2 > 0$, $c_1 < 0$, and $c_2 >0$.

For 2D data, 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/teubner_strey_autogenfig.png

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


**Source**

:download:`teubner_strey.py <src/teubner_strey.py>`

**References**

#. M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195
#. K V Schubert, R Strey, S R Kline and E W Kaler,
   *J. Chem. Phys.*, 101 (1994) 5343
#. H Endo, M Mihailescu, M. Monkenbusch, J Allgaier, G Gompper, D Richter,
   B Jakobs, T Sottmann, R Strey, and I Grillo,
   *J. Chem. Phys.*, 115 (2001), 580

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