Re: U, V, W vs Fundamental Parameters: Re: A few questions

Paolo G. Radaelli ( (no email) )
Mon, 25 May 1998 10:01:53 +0100

Dear Lachlan,

the parameters U,V and W are not some kind of "fudge" factors, but reflect a
well-defined (albeit incomplete) physical understanding of the resolution
process, at least in the context for which they were developed, i.e.,
monochromatisation of a white (neutron) beam by a mosaic crystal. They are
the parameters of the so-called "Cagliotti" function:

D2th=sqrt(U*tan(th)^2 + V*tan(th) + W),

which was developed in 1958 by my fellow countrymen Cagliotti, Paoletti and
Ricci (Nucl. Intrum. V3, (1958) pp. 223-228). Interestingly, I was told
that Cagliotti is a particle physicist by training.

This function comes about when you consider a system with white beam
divergence alpha1, sample divergence alpha2, detector divergence alpha3 and
"take-off" angle 2thM (the take-off angle is the angle between the white
beam and the monochromatic beam). All the divergences are in the scattering
plane. It is a simple matter of geometry to demonstrate that, assuming
Gaussian profiles, the resolution is given by the formula:

D2th=sqrt((alpha1^2+alpha2^2)*(tan(th)/tan(thM))^2-2*alpha2^2*(tan(th)/tan(t
hM))+(alpha2^2+alpha3^2)))

from this follows:

U=(alpha1^2+alpha2^2)/(tan(thM))^2
V=-2*alpha2^2/tan(thM)
W=alpha2^2+alpha3^2

What determines the three divergences depends on the actual instrument
configuration. In an open geometry (no collimators) alpha1=alpha2=2betaM,
where betaM is the mosaic spread of the monochromator, and alpha3 is a
convolution of sample and detector sizes divided by the sample-detector
distance. In the presence of Soller collimators, the these angles become
equal to the collimation angle.

This formula works very well for CW neutron diffractometers, both high-res
and high-flux. It is an approximation only in the sense that it ignores the
effects out of the scattering plane (sample height, axial divergence,
monochromatic beam divergence etc.), but these things usually have a
relatively small effect on the resolution (they affect the profiles in a
dramatic way, though). Also, the approximation is made that all
resolution-limited peaks are Gaussians. This is quite true, in general, but
there are some exceptions, notably when the monochromator mosaic dictates
the peak shape and when the aforementioned off-plane effects are important.
However, even in more sophisticated treatments of the peak profiles, the
Cagliotti function is always used as a kernel which is then convoluted with
an intrinsically asymmetric (or otherwise non-Gaussian) profile.

As far as other intruments are concerned, the Cagliotti function also works
well for synchrotron x-ray diffractometers (with double-bounce
monochromators). I have not analysed the geometry of it myself, but the
take-off angles are obviously very small. Modern Lab diffractometers have
considerably more complex geometries, so it is very well possible that the
Cagliotti function may not be completely adequate.

Paolo
Dr. Paolo G. Radaelli
ISIS Facility
Rutherford Appleton Laboratory, Bldg. R3
Chilton, Didcot
Oxon. OX11 0QX
United Kingdom

Phone : (+44) 1235-44 6290
FAX : (+44) 1235-44 5642
e-mail: P.G.Radaelli@rl.ac.uk
pgr@isise.rl.ac.uk