Date: Fri, 11 May 2001 13:11:47 -0400
From: "Brian H. Toby" [[email protected]]
Organization: NIST Center for Neutron Research
To: [email protected]

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. 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.

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

-- 
********************************************************************
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
********************************************************************

Date: Mon, 14 May 2001 11:57:28 +0200
To: [email protected]
From: Jonathan WRIGHT ([email protected])
Subject: Re: Resolution in powder diffraction

Brian H. Toby wrote:

> ... <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>

D-spacings are distances between lattice planes and provide me at least
with a nice physical picture of Bragg's law. In this way I guess I am
correlating real space distances with reciprocal space distances, but I
think I prefer remain a novice than give up what little understanding I
have :) 

>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. 

High resolution at low Q is handy for indexing, and is essential for
tackling moderately complex magnetic structures with powder neutron data.
ToF diffractometers which use slow neutrons (eg OSIRIS at ISIS) appear to
be doing particularly well for resolution in the d=3-10 Angstrom region (I
hope we eventually see such machines at SNS and ESS). Peak broadening
effects and splittings can be easier to interpret where there is less peak
overlap, so high resolution in this range can again be useful. It's a
question of applying the right instrument or setup to the particular
problem in mind.

>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. 

There are a lot of different trade offs to take into account when choosing
a wavelength, with x-rays at least. If low angle asymmetry is a problem for
the experiment in question then you can use a small beam on sample and/or
narrower receiving slits. My recent experience at ESRF is that a lot of the
asymmetry can be removed without destroying the count rate even with a
large unit cell and fairly hard x-rays. (nb. removed by slits, not
deconvolution)

For ToF neutrons the moderator asymmetry is less dominant as ToF increases,
and this is where the resolution is maximised anyway. It can be fitted very
well with the right programs and the advantages of going to very low
d-spacings (high Q) also has to be taken into account. 

Comparing resolution functions only tells part of the story, the real
question is how much information did you gain from the experiment. This can
depend on a lot of other factors, not least background, Q-range and count
rate which are rarely compared in quite the same way. Remember the sample
can still ruin the background and the resolution regardless of the
instrument :)

Cheers,

Jon