Dear Mr Girgsdies,
Off hand I am not aware of any critical discussion. Let me add a few
thoughts though that may help to explain the issues at hand.
A Rietveld program calculates the diffraction pattern as a sum of all
Bragg reflections. Initially these Bragg peaks are treated as
infinitesimally sharp points at a fixed 2Theta position. This hold in
particular for the calculation of the structure factor, which is
calculated at the corresponding point in reciprocal space for the
integer values triplet hkl. In a second step these sharp peaks are
widened by a profile function to describe the experimentally observed
broad peaks. The broadening of the profile function has components due
to the instrumental resolution and sample contributions (size and strain).
The Rietveld treatment implicitly assumes perfect translational
periodicity, as all calculations in reciprocal space are limited to the
integer Bragg positions. The sample contribution to the broadening is
thus a bit of an artificial "trick" to get a good/reasonable agreement.
In the actual diffraction experiment the diffraction pattern arises form
the sum of all secondary waves emitted by all atoms. This sum of all the
secondary waves is a continuous function in reciprocal space. Only in
the limit of a perfect infinite crystal will the sum reduce to the Bragg
positions, and be zero at all other points in reciprocal space.
For a (very) small nanoparticle this sum of the secondary waves will
naturally "widen" the Bragg positions compared to those of a large
crystal. If one looks at a point slightly off the Bragg position, one
has to keep in mind that this diffraction angle differs from that of the
integer Bragg reflection. As a consequence, the individual atomic form
factors and the structure factor will differ from the values at the
integer Bragg position. This difference is not included in the Rietveld
algorithm. This difference can lead to an asymmetric profile function.
This profile may be asymmetric enough to have its maximum off the Bragg
position and one must be super careful not to mistake the location of
the maximum intensity of such an asymmetric profile with the actual
Bragg position. This is described nicely in Tchoubar & Drits X-ray
Diffraction by disordered lamellar structures. There are a bunch of
"lovely" papers that do misinterpret this.
The small nanoparticles below 3 nm diameter will add two more
complications to the situation.
A) The (irregular) surface will likely truncate the average bulk unit
cell at different positions around the nanoparticle. Thus the Rietveld
assumption that the crystal consists of identical unit cells is no
longer absolutely correct. This may change the relative intensities.
B) The surface is bound to be subject to
distortions/reconstructions/different surface chemistry compared to the
interior of the nanoparticle and will in many cases cause an appreciable
strain across the particle, which again is not part of the Rietveld
algorithm.
All in all I would recommend to calculate the diffraction pattern of
such small nanoparticles by use of the Debye-Scattering-Equation. This
algorithm adds up the diffraction pattern from the contribution of all
atom (pairs) and gives a direct diffraction pattern without the need of
a sample related profile function. The Debussy program by Antonio
Cervellino J.Appl.Cryst 48, 2026 (2015) and my own DISCUS program JAC
32, 838 (1999) "https://github.com/tproffen/DiffuseCode"; are two
examples of such programs.
The special issue of Acta Crystallographica A72 (2016) has several
papers related to the Deby-Scattering-Equation, Paolo Scardi and Matteo
Leoni have written several papers on the sample related profile function.
Sincerely
Reinhard Neder
Am 06.06.19 um 12:39 schrieb Frank Girgsdies:
Dear fellow Rietvelders,
Could anyone point me to some nice literature which critically
discusses the limitations of the Rietveld method when it comes to
nano-crystalline materials (specifically in the 1 to 3 nm range)?
As far as I'm aware, the core Rietveld literature seems to touch this
point only in the passing.
Background:
To the best of my knowledge, Rietveld-derived parameters (like lattice
constants or domain sizes) should not be trusted as being "physically
meaningful" anymore when you fit the powder pattern of a material in
the few nm range with standard Rietveld tools.
My naive understanding of this problem is that the physical principles
of diffraction (or rather the best way to model it) gradually change
when you go from long-range ordered to medium-/short-range ordered
materials.
Being a Rietveld practitioner rather than a theoretician, and having
no first-hand experience with WPPM and PDF methods, I am often
confronted with the problem to explain to my "customers" why I can't
extract trustworthy lattice constants or domain sizes from their
nano-crystalline samples, especially if it seems technically possible
to fit the pattern with a Rietveld program.
I think it would be nice if I could cite some critical discussion, or
overview article with further references, to put my finger on the
problem.
Especially in the catalysis community literature, my impression is
that the applicability of the Rietveld method is sometimes
overestimated, leading to overinterpretation of the results.
Any suggestions?
Best wishes,
Frank Girgsdies
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