.. _porod:

porod
=======================================================

Porod function

========== =============================== ======= =============
Parameter  Description                     Units   Default value
========== =============================== ======= =============
scale      Scale factor or Volume fraction None                1
background Source background               |cm^-1|         0.001
========== =============================== ======= =============

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


This model is a special case of the power law model, deriving its special name
from the Porod Law which it models, where the power law exponent is fixed to -4.

.. math:: I(q) = C/q^4 + Background

Where the power law constant $C$ in this case is just the scale factor.

The Porod law, similar to the Guinier model, is a broadly applicable model to a
very restricted portion of the data. While the Guinier model applies to any
dilute particulate system regardless of shape or size but only to the very low
$q$ (being a taylor expansion around $q$ = 0), the Porod Law applies to any
scattering system with sharp scattering length density interfaces between the
phases but only at very high $q$ (technically in the limit as $q \to \infty$ ..
except of course that, practically, the continuum approach breaks down there).
It is based on the idea that at sufficiently high $q$ there is no shape
information left and all scattering is just reflections off the sharp
interfaces.

In the special case of a two phase system, the power law constant $C$ derived
from the appropriate $q$ limit portion of the data is known as the Porod
Constant and can be written as:

.. math:: C = 2\pi (\Delta\rho)^2 S_v

where $S_v$ is the specific surface area (ie, surface area / volume) of the
material under study, and $\Delta\rho$ is the contrast factor between the two
phases.

Thus, by extracting the Porod Constant from experimental data, and
knowing the contrast factor between the two phases, one can obtain the specific
surface area for the material. This can be very useful for example in
understanding porosity in materials, particularly when used in conjunction
with complementary techniques such as BET.

.. Note:: While it is straightforward to calculate the specific surface area,
    $S_v$ of the material from the Porod Constant returned here, the Invariant
    analysis panel will also compute the $S_v$ by entering the contrast term
    and the Porod Constant obtained here into the appropriate entry boxes.

    There are, however, several caveats regarding obtaining a good experimental
    value of the Porod Constant.

    * First, as it is the scale value, the data **must** be on an absolute scale.

    * Next of course, there must be a sufficiently large $q$ range that is in
      the Porod region to be able to be fit. But this may not always be
      possible: for example, polymer coils in solution will often not reach that
      limit within typical SAS ranges, nor even the $q$ ranges where the
      continuum approach of using the SLD is even valid.

    * For highly monodisperse systems with limited resolution smearing, the
      data will contain large oscillations which will make the estimate from
      this fit unreliable.  It will vary depending on the exact range of the fit
      that is chosen. This is because numerical integration using a finite
      number of points cannot properly capture the exact area under the data
      across these very sharp dips.

    * Ironically, large resolution smearing, and/or polydispersity smearing,
      will make the value obtained much more consistent and reliable. Thus the
      problem is less severe for typical real data than for simulated data that
      does not simulate any resolution smearing.

    * If in any doubt, using several values of $q_{min}$ in the fitting, and
      comparing with the value obtained in a different manner, is highly
      advisable. One such method uses the Porod plot ($I(q)* q^4$ vs. $q^4$).
      Fitting the highest $q$ (strictly speaking $q^4$) data to a straight line
      using as much of the high $q$ as fits well to a straight line, yields a
      slope which is the flat background and an intercept which is the Porod
      Constant.

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/porod_autogenfig.png

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


**Source**

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

**References**

#. G Porod. *Kolloid Zeit*. 124 (1951) 83
#. L A Feigin, D I Svergun, G W Taylor. *Structure Analysis by Small-Angle
   X-ray and Neutron Scattering*. Springer. (1987)

**Authorship and Verification**

* **Author:**
* **Last Modified by: Paul Butler, 28Mar2020**
* **Last Reviewed by:**

