[Ifeffit] Exafs distance resolution

[Matt Newville]

Eric,

> Is there a physical limitation determining exafs bond distance resolution?

There is a physical limit in determining bond distances from EXAFS.

> Very often the equation r = pi / 2 deltak is quoted as a measure for
> bond distance resolution.

The equation dr=pi/(2 *Delta k) gives the distance resolution: the
ability to separately see two distances (here Delta k is the data
range in k).  This is not the same thing as the precision with which
a single bond distance can be determined, which is generally
quite a bit better than the "resolution".

For Delta k ~= 15Ang^-1 (pretty good data), the resolution from the
equation above is about ~= 0.1Ang.  That is, one could expect to
reliably detect a splitting of distances by ~0.1Ang.

The precision in R from EXAFS experiments is typically 0.01Ang.
This is typically determined by a combination of noise in the data
and the accuracy of the phase shift calculations (say, from FEFF).
For certain cases, it's entirely feasible to detect *changes* in bond
distances with even better precision.  One paper not so long ago
(Pettifer, et al, Nature 435 pp78, 2005) claimed a precision of 10fm.

> But as i understand this equation is related
> to the fourier transform traditionally used for exafs analysis.
> If exafs fitting is done in k-space, on the raw exafs data without
> applying fourier or any other filtering transformation is there a
> physical limitation determining exafs bond distance resolution?

Whether or not Fourier transforms are used in the analysis is "mostly"
immaterial.   That is, EXAFS is an interference technique, and we
measure in k (or E) space to make statements about R space, so the
limits are fundamental, not an artifact of the analysis tools.  I say
mostly because practical use of Fourier transforms (Fast Fourier
Transforms with finite grids and extents) will impose additional
restrictions on resolution and precision -- but these are typically finer
than 0.1Ang for resolution and  0.01Ang for precision, and so are hardly
ever a concern.  As an example, FFTs in Ifeffit+Friends use a k-space
grid of 0.05Ang^-1 and kmax of 102.4Ang^-1, and a grid in R-space of
~0.03Ang.   This would limit resolution to about ~-0.03Ang, which
might be a limiting factor if you have data to k~=50Ang^-1.  It probably
limits precision too, though I do not know to what extent.

> This question comes down to the following practical problem. If one has a
> theoretical model developed using computational chemistry that predicts
> two different bond lengths within one shell,  e.g. an octahedral metal
> center surrounded by 6 oxygen atoms and this shell is predicted to be
> split in three subshells for wich the bond length differs only 0.05
> angstroms; and this model can be fit very well in k-space with the
> splitted shell, off course keeping the number of fit parameters below
> the nyquist criterion. Is there in such a case any physical reason not
> to fit the experimental data with the splitted shell , but with an
> averaged 6-atom shell with a larger Debye Waller factor?

My guess would be that the EXAFS could probably be fitted just as well with
one distance and a slightly larger sigma2 as with 3 separate distances.
But this would depend some on the data quality and it might be right at the
resolution limits, so I'd recommend trying both models.

Cheers,

--Matt