--- "Von Dreele, Robert B." <[EMAIL PROTECTED]> wrote:
> Actually, I looked at Luca's little "show" & was
> sufficiently interested
> that GSAS will now plot sqrt(I) style plots. There
> is one "problem" -
> the value of "I" can be negative particularly after
> a background
> subtraction. These must be suppressed to zero for
> this plot to work -
> thus there is a small risk of something getting
> hidden.
I think that that is because although it is called a
SQRT plot, that is just because that is how it is
usually implemented. What is meant is a more general
principle, namely that of displaying the data points
divided by their ESDs (where, of course, the ESDs must
be calculated *from the raw number of counts*). I will
try to explain why.
As mentioned elsewhere in this thread, the main error
source in the raw number of counts is due to Poisson
statistics, which means that if I measure Iobs counts
for a given data point, then the ESD in that data
point is SQRT(Iobs). If I divide Iobs by its ESD I
get:
Iobs / SQRT(Iobs) = SQRT(Iobs)
It is the result, the SQRT(Iobs), that is generally
implemented, but in cases where this gives problems,
one should go back to the derivation, which allows the
SQRT to be generalised: instead of taking SQRT(Iobs),
one can divide by the ESD directly. For straigtforward
cases, the results are the same, but for the more
complicated cases, like background subtracted data,
dividing by the ESD is the only way to get the correct
result.
Example 1. Background subtraction.
Measured: Iobs = 100 counts, so ESD = SQRT(100) = 10.
The background at that point is determined to be, say,
102. The nett intensity at that point is then 100 -
102 = -2. However, the ESD of the point does not
change as a result of the background subtraction, and
is still 10. Plotting -2 / 10 gives the correct plot.
This suggests that simply plotting the SQRT is wrong
not only for background subtracted data, but for *all*
data where the raw number of counts has somehow been
modified. The two main examples are variable count
time (VCT) data where the separate ranges have been
rescaled before they can be recombined, and
synchrotron data that has been rescaled to correct for
beam decay. The generalisation to plot each point
divided by its ESD (calculated as the SQRT of the
observed, unmodified number of counts) again provides
the correct treatment:
Example 2. VCT data.
Measured: Iobs = 100 counts, so ESD = SQRT(100) = 10.
Rescaling to counts per second to combine the ranges
with different count times requires, say, rescaling by
a factor of 2, giving 100 * 2 = 200. For the relative
error to remain the same, the ESD must also be scaled
with the same factor, i.e. ESD = 10 * 2 = 20. Plotting
200 / 20 now gives the correct plot. Note that simply
plotting the SQRT would give a flattered plot, as the
data obviously has not somehow become more accurate by
multiplying everything by a constant (2 in this
example). If it were possible to male experimental
data more accurate that way, we could simply measure
powder patterns in a fraction of a second and just
multiply by a sufficiently big constant to achieve the
desired accuracy, which is clearly absurd.
Example 3. Synchrotron data, corrected for beam decay.
Measured: Iobs = 100 counts, so ESD = SQRT(100) = 10.
However, the intensity of the incoming beam is not
constant, and the individual data points must
therefore be rescaled before they can be combined.
Let's say that a factor of 2 is required for this data
point, then the rest of the argument is the same as
for Example 2.
This shows that it is very important to always keep
track of the (rescaled) original ESDs during
background subtraction etc.
I hope this made some sense.
Best wishes,
--
Dr Jacco van de Streek
Frankfurt University
Frankfurt am Main, Germany
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