Dear Neil & others,
Instrumental resolution is best expressed (IMHO) as FWHM(Q) vs Q (see
for example http://www.ncnr.nist.gov/xtal/bt1_for_physicists.html),
where Q=2pi/d; not in 2theta because 2theta is dependent on wavelength
and thus cannot be compared between techniques and not d-space because
that overemphasizes the low Q (low angle) portion of the pattern which
has minor leverage on most results and further is where one rarely needs
to worry about optimizing instrumental resolution. <SOAPBOX>Besides, Q
is the Fourier conjugate of x (the coordinates); d-spaces have no
physical significance. D-spaces can mislead a novice into correlating
reciprocal space distances with real space distances.</SOAPBOX>
What does delta-d/d mean? Assuming we are talking about FWHM as
delta-d/d (that is not the only choice) then if delta-d/d = 0.2%, for a
d=0.5 A peak,
the FWHM is .002*0.5 = .001 A. You can make a change of variables
directly with a bit of calculus, or you can also compute the FWHM in any
other units by converting d=.5+.001/2 and d=.5-.001/2.
Why is resolution often reported as delta-d/d? Because delta-d/d is to
first order a constant for TOF instruments (which is equivalent to
constant delta-Q/Q). BTW, constant delta-Q/Q is not really what one
wants. One would [usually] prefer that delta-Q decrease with Q, so that
resolution increases as peaks get more dense and because high Q data
typically has more leverage on lattice constants and structural
parameters, but constant delta-Q/Q is what nature gives us for TOF and
TOF (now at ISIS and someday the SNS) is the best we are likely to get
for neutrons. For CW instruments, the resolution functions are more
complex, but not necessarily better.
I would argue that the ability we have at NIST to "tune" the resolution
is a very good thing. This is because better resolution is not always
better. There is always a tradeoff between resolution and signal to
noise. Besides, not all samples even have high Q peaks. Different
experiments demand different measurements.
One final comment. People tend to sweep imperfections under the rug. In
the plot on my web page, low angle asymmetry is subtracted out -- but it
should not be. For very short wavelength machines, the asymmetry can be
severe for the bulk of the pattern. Same is true for other types of
aberrations. I do not know how people treat the very asymmetric peaks
obtained in TOF measurements when they quote "delta-d/d." Of course, the
FWHM values should be obtained from a real sample (as ours were) and one
that has negligible sample broadening effects.
Brian
"Hyatt, Neil" wrote:
>
> Dear All,
>
> The resolution of a diffractometer is often quoted in terms of "delta d / d"
> or as full-width-at-half-height (FWHM, on the two-theta scale). The concept
> of FWHM is fairly easy to understand and visualise - but I'm having trouble
> grasping the concept of what "delta d / d" means in physical terms. What I
> would like to know is:
>
> 1. How is "delta d / d" formally defined (an equation would help here!) - I
> guess it arsies somehow from differentiating Braggs Law?
>
> 2. When a value of "delta d / d" is quoted in the literature, is this
> conventionally taken to be the best achievable resolution for a given
> diffractometer set-up?
>
> 3. Is there an "accepted" way of measuring resolution? I guess the answer
> here is to use a suitable "standard" specimen with a known or "zero"
> reflection broadening component arising from crystallite stress / strain
> etc.
>
> Finally, am I right in thinking that talking about resolution in terms of
> "delta d / d" rather than FWHM is advantageous since it allows a relative
> comparison of the resolution afforded by different techniques, e.g.
> time-of-flight neutron vs. constant wavelength X-ray?
>
> I'd be really grateful to hear from anyone with thoughts on this issue.
>
> All the best,
>
> Neil Hyatt.
--
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Brian H. Toby, Ph.D. Leader, Crystallography Team
[EMAIL PROTECTED] NIST Center for Neutron Research, Stop 8562
voice: 301-975-4297 National Institute of Standards & Technology
FAX: 301-921-9847 Gaithersburg, MD 20899-8562
http://www.ncnr.nist.gov/xtal
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