..
   Athena document is copyright 2016 Bruce Ravel and released under
   The Creative Commons Attribution-ShareAlike License
   http://creativecommons.org/licenses/by-sa/3.0/

.. _lr_sec:

Log-ratio/phase-difference analysis
===================================


Log-ratio/phase-difference analysis is a purely empirical technique
that can be used to determine changes in the distribution parameters
of the first coordination shell for certain kinds of data. This
technique is most useful for data with a well-separated first
coordination shell, i.e. one for which data from higher coordination
shells is spectrally isolated from the first shell. An excellent
example of a situation where this analysis would be useful would be a
temperature series. In that case, log-ratio analysis can be used to
measure the variation in |sigma|\ :sup:`2` as a function of
temperature and phase difference analysis can be used to measure
variation in bond length.

The approach of log-ratio/phase-difference analysis is to compare
unknown data to a well known standard. The standard might be a room
temperature measurement in the case of a temperature series. Or it might
be a well ordered oxide for comparison with more disordered oxide forms.
In any case, the analysis returns the differences between the standard
and the unknown of the various cumulants of the distribution of atoms in
the first shell.

The analysis is performed by first by Fourier filtering both the data
and unknown to spectrally isolate the signal from the first coordination
shell. Then polynomials are fitted to the log of the ratio of the
amplitudes of the χ(q) functions and to the difference of the phases of
the χ(q) function.

::
   
      Log-Ratio(q)  = ln( Amp_unknown(q) / Amp_standard(q) )
      Phase-Diff(q) = Phase_unknown(q) - Phase_standard(q)

These are fit using the cumulant expansion:


.. code-block:: text
   
      Log-Ratio(q)  = c_0 + 2*c_2 * q^2 + (2/3)*c_4 * q^4
      Phase-diff(q) = 2 * c_1 * q - (4/3) * c_3 * q^3

The standard is selected from the menu at the top of this tool. The
unknown is the current group, i.e. the one highlighted in the group
list.

There are some other caveats along with the fact that this technique
requires a spectrally isolated first shell.

#. A perfectly ordered coordination shell is not necessary, but the
   distribution of atoms in the first shell must be well described by an
   expansion in a small number of cumulants. Thus, this technique may be
   inappropriate for highly disordered materials.

#. Although this analysis is often referred to in the literature as
   :quoted:`model independent analysis`, that is as much a shortcoming as an
   advantage. The technique can be used to measure changes in structural
   cumulants, but cannot be used to measure absolute values of those
   cumulants.

#. This analysis cannot be performed on higher coordination shells
   except in the vary rare case of a second shell which is spectrally
   isolated both from the first shell and from all higher shells.

#. The presence of spectral weight from any higher shell or from any
   multiple scattering paths will add considerable systematic
   uncertainty to this analysis.

When you select :guilabel:`Log-Ratio` from the main menu, the main window is
replaced with the log-ratio/phase-difference tool, shown below.

.. _fig-lr:

.. figure:: ../../_images/lr.png
   :target: ../_images/lr.png
   :align: center

   The log-ratio/phase-difference tool.

Once you select Fourier transform parameters and ranges for use in the
analysis, you can click the :button:`Fit,light` button to perform the
fit. The fit to the log-ratio of the amplitudes is plotted when the
fit is over. The buttons can be used to plot either the fit to the
log-ratio of amplitudes or the fit to the phase difference. The fits
use all cumulants up to the fourth. Depending on your data quality,
choice of Fourier transform ranges can be important and quite
sensitive. You can save the log-ratio and phase-difference data along
with the fits to a file by clicking the :button:`Save ratio data and
fit,light` button.


.. subfigstart::

.. _lr_fit:
   
.. figure:: ../../_images/lr_fit.png
   :target: ../_images/lr_fit.png
   :width: 100%

   The results of the log-ratio fit to the Cu metal.


.. _lr_pd_fit:
   
.. figure:: ../../_images/lr_pd_fit.png
   :target: ../_images/lr_pd_fit.png
   :width: 100%

   The results of the phase-difference fit to the Cu metal.

.. subfigend::
   :width: 0.4
   :label: _fig-lrplots

The three plot buttons behave just like the purple plot buttons would
behave were only the standard and unknown marked.

My favorite explanation of this analysis method is

    .. bibliography:: ../athena.bib
       :filter: author % "Bunker"
       :list: bullet
