# [Ifeffit] can sigma square ever be less than zero?

Bruce Ravel bravel at bnl.gov
Mon Jun 28 15:41:40 CDT 2010

```On Monday 28 June 2010 04:10:12 pm Chris Patridge wrote:
> Hello all,
>
> I am working on W L3 edge data.  W is acting as a substitution dopant in
> vanadium dioxide at rather low concentration.  In a past mailing
> conversation discussing Feff6 overestimation of E0 for heavier elements
> it was mentioned that the E0 could be past the rising edge due to the
> white line from W data.  Well using this comment I aligned data using
> the theory method well explained by Shelly Kelly SnO2 example.
> Literature suggests W approximates WO2 cubic structure locally instead
> of the VO2 unit structure.  Then fitting the first oxygen coordination
> shell paths which are well isolated from the other paths, it gives
> reasonable values for amplitude and enot of 0.77 (0.17) and -1.14 (3.06)
> respectively.  delr is -0.067 (0.028) and then ss comes out to -0.00036
> (0.00414).  Can ss be negative if the uncertainty brings it above 0?

In a numerical sense -- the sense in which the covarience matrix of
the fit is evaluated -- there is no problem at all with a negative
sigma squared.  In that case, a negative value is what was required to
find the best fit, in the numerical sense.

Of course, this just emphasizes a point often repeated on this mailing
list.  A fit requires a human to make an interpretation.  In a
physical sense, sigma^2 cannot be negative.  Just look at it -- it is
the square of soemthing!  sigma is a real-valued quantity, the square
of a real number cannot be negative.

So, how should we interpret a fit that gives a negative sigma^2?
Well, you have to think about what sigma^2 is correlated with.  It is
certainly correlated with amplitude.  If the amplitude is much too
small for some reason (say, because you, for some reason, fixed N*S02
to some value that is too small), then the fit will have to make
sigma^2 smaller to compensate.

But the negative sigma^2 might also be telling you something about the
amount of disroder in the strutucral model you used in your Feff
calculation.  You might use some crystallography as the starting point
for the Feff calculation and that crystallography might result in a
very disrodered shell.  Well, if the local structure is much less
disordered than that but you retain that amount of structural disorder
by using all those slightly different paths, then the fit has to
somehow compensate for the excess structural disorder in the fitting
model.  It may do so by making sigma^2 really small, even negative.

So, in general a negative sigma^2 is probably trying to tell you
sigma^2 is unphysical, so a defensible interpretation of the fitting
result cannot include a negative sigma^2.  However, obtaining such a
thing at some point during the analysis might be a useful hint about
how to modify your fitting model.

HTH,
B

--

Bruce Ravel  ------------------------------------ bravel at bnl.gov

National Institute of Standards and Technology
Synchrotron Methods Group at NSLS --- Beamlines U7A, X24A, X23A2
Building 535A
Upton NY, 11973

My homepage:    http://xafs.org/BruceRavel
EXAFS software: http://cars9.uchicago.edu/~ravel/software/exafs/

```