Yes, one can determine size distribution parameters by using Rietveld
refinement. In particular, the lognormal size distribution is defined by two
parameters (say, the average radius and the distribution dispersion, see,
for instance, (2) and (3) of JAC 37 (2004) 911, SSRR for short here, or
other references therein). It was first shown by Krill & Birringer that both
volume-weighted (Dv) and area-weighted (Da) domain size (that are normally
evaluated in a diffraction experiment) can be related to the average radius
and dispersion of the lognormal distribution; one obtains something like (5)
in the paper SSRR. Therefore, if one can evaluate both Dv and Da by Rietveld
refinement, it would be possible to determine the parameters of the size
distribution, as two independent parameters are required to define the
lognormal or similar types of bell-shaped distributions. Note here that a
different distribution can be used, which will change the relationship
between Dv & Da and the parameters of the distribution (for the gamma
distribution, see JAC 35 (2002) 338, for the equations equivalent to (5) in
SSRR). The value that is normally evaluated through the Rietveld refinement
is Dv, as the refinable parameters in the Thompson-Cox-Hastings (TCH) model
are based on the integral-breadth methods. This means that one would have to
use (9) and (15)-(18) in SSRR, to obtain Dv, which depends on both P and X
parameters. As the TCH model implicitly assumes Voigt functions for both
size and strain-broadened profiles ("double-Voigt" model), Da can be also
calculated, but from X only, as it depends only on the Lorentzian
size-broadened integral breadth, Da=1/(2betaL) (this and other consequences
of a "double-Voigt" model were shown/discussed in JAC 26 (1993) 97).
HOWEVER, as pointed out by others in previous messages, this assumes that
(i) Both observed and physically broadened profiles are Voigt functions,
which is implicit to the TCH model; (ii) Size distribution is lognormal,
gamma, or whatever we assume it to be. On the former, it is easy to see if
observed profiles can't be successfully fit ("super-Lorentzian" peak shapes,
for instance), which means that the TCH peak shape cannot be used. However,
an assumption that physically broadened profiles (size and strain) are also
Voigt function is more difficult to prove; if not and one uses the equations
described above, a systematic error will be introduced. On the latter, a
good fit in Rietveld means only that a lognormal or other assumed
distribution is one POSSIBLE approximation of the real size distribution in
the sample. However, this equally applies to all the other parameters
obtained through the Rietveld refinement and is not a special deficiency of
this model. Second, even if one obtains more information about the actual
size distribution via TEM, SEM, etc., sometimes it is very difficult to
discern between different bell-shaped size distributions, especially if the
size distribution is narrow.
Davor
************************************
Davor Balzar
Department of Physics & Astronomy
University of Denver
2112 E Wesley Ave
Denver, CO 80208
Phone: 303-871-2137
Fax: 303-871-4405
Web: www.du.edu/~balzar
************************************
************************************
National Institute of Standards and Technology (NIST)
Division 853
Boulder, CO 80305
Phone: 303-497-3006
Fax: 303-497-5030
Web: www.boulder.nist.gov/div853/balzar
************************************
> -----Original Message-----
> From: Leonid Solovyov [mailto:[EMAIL PROTECTED]
> Sent: Monday, November 22, 2004 3:12 AM
> To: [EMAIL PROTECTED]
> Subject: Size distribution from Rietveld refinement
>
> Dear Rietvelders,
>
> Despite the heated discussion of the problem, the initial question,
> which, actually, concerned the size distribution from Rietveld
> refinement, seems to be unsettled.
> Can we derive ANY information on the crystallite size distribution
> (based on sensible assumptions) from the Thompson-Cox-Hastings
> size-broadening parameters P and X normally obtained from Rietveld
> refinement?
> For the Ceria Size-Strain Round Robin sample the crystallite
> distribution dispersion was determined from the profile analysis
> assuming lognormal distribution. This suggests that the diffraction
> data contained this information. Why Rietveld refinement can not be
> used for this purpose?
> I realize that most simple questions may be most difficult to answer,
> but nevertheless...
>
> Regards,
> Leonid
>
>
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