**FROM THE EXACT FORMULAE
TO THE PEAK SHAPES USED IN THE RIETVELD METHOD,**
**WHAT APPROXIMATIONS ARE
INTRODUCED ?**

**Let us see the meaning
of some analytical profile shapes regarding size and microstrain effects
as defined by Warren, in the hypothesis of symmetrical f profile
shapes :**

**The size distribution
function P(j), fraction of columns of length**

**Now some examples of profile
shapes as corresponding to selected size distribution functions :**

**Another one :**

**About the microstrain
:**

**A Gaussian peak shape
is expected if the distortion is Gaussian, and if the mean squared distortion
varies as the square of the distance :**

**< Z^{2}_{n}>
= n^{2} <Z^{2}_{1}>**

**A Lorentzian peak shape
is expected if the distortion is Gaussian, and if the mean squared distortion
varies as the distance :**

**< Z^{2}_{n}>
= n <Z^{2}_{1}>**

**Any hypothesis of this
order has chances to be more or less wrong, the best is to have parameters
sufficiently flexible...**

**Those equations are for
one series of harmonics in a reflection family ( 00l for instance).
If the broadening effects are different in each hkl reflection family
(anisotropy), then you will have a huge number of Fourier coefficients
to manage, a problem reputed impossible to solve because of reflection
overlapping.**

**Alternative is to try
to describe the anisotropy by global expressions depending on the hkl
indices**

**Let us scrutinize first
a typical very recent paper explaining how approximations are introduced
:**

**Fourier modelling of the
anisotropic line broadening of XRD profiles due to line and plane lattice
defects.**
**Scardi, P. & Leoni,
M. (1999). J. Appl. Cryst. 32, 671-682.**

**From the paper abstract
: Applications to face-centred cubic structure materials provided detailed
information on the defect structure : dislocation density and cut-off radius,
stacking- and twin-fault probabilities were refined together with the structural
parameters.**

**It is certainly interesting
to look accurately at the way the microstructure parameters were introduced.**

**Reading the paper, one
can note that the method was applied to two similar cases (fcc materials)
but no evidence was given about the reliability and accuracy of the estimated
physical parameters (size, dislocation density, stacking fault and twinning
probabilities...) which could be expected from this technique. No standard
material was studied, no alternative method was used for comparison. No
theoretical pattern was simulated by using the "true" (Warren) equations
(or better, the Debye scattering equation) which then could have been treated
by the approximate equations developed in the manuscript so that the effect
of those approximations could have been estimated.**

**The title is misleading.
"Fourier modelling" is hardly applicable here since pseudo-Voigt functions
are used : so that the profile shapes are not at all produced by inverting
size and strain non-analytical Fourier series.**

**The pseudo-Vogt functions
retained is :**

**Note that by doing this,
the authors have decided that the profiles will be pseudo-Voigt, without
any justification.**

**And it is said : "The
two components (Gaussian and Cauchy) have been assumed to have the same
half width at half maximum (HWHM)".**

**Why assuming that ?**

**This is a clear limitation
which cannot allow the authors to claim the proposal of a general method.
Moreover, what about the distribution of size effect in the Cauchy and
Gaussian parts, and what about of distortion effect repartition in the
Cauchy and Gaussian parts ? It will be shown later in these comments that
the assumption made here is the dubious key allowing to establish the final
equations. This assumption should be clearly justified on the point of
view of size and strain effects, but it is not.**

**Now, the Fourier transform
of the pseudo-Voigt is defined as :**

**And the size-strain model
has now to be introduced inside it.**

**The limit for L tending
toward 0 of the first derivative of the size-strain Fourier coefficient
is :**

**the negative inverse
mean size M_{e} of the coherently diffracting domains.**

**So that the authors assimilate
the first derivative limit of the pseudo-Voigt Fourier transform to the
same value. This seems logical after having arbitrarily decided that the
shapes will be pseudo-Voigtian :**

**This allows to relate
the widh parameter of the Gaussian and Lorentzian parts to the mean size
M_{e},
so that the Fourier transform of the pseudo-Voigt becomes :**

**The authors will use another
approximation in order to introduce the lattice-distortion terms :**

**And they will assume
that this expression is even true for L = M_{e}/2,
meaning that the size Fourier coefficients correspond to a Cauchy function
up to half the mean size, so that the size distribution corresponds also
to a Cauchy function up to half the mean size.**

**Afterthat, the size-strain
Fourier coefficient for L = M_{e}/2 becomes :**

**This equation is now made
equal to the pseudo-Voigt Fourier transform for L = M_{e}/2
and the final expression is obtained :**

**What is done here is to
make equal an expression containing only the mean size M_{e}
and an expression containing both the size and strain effects, for a particular
value of L = M_{e}/2. Both expressions being previously
highly arranged and approximated.**

**That final expression
connects the mean size Me and the mean square distortion <e ^{2}_{Me/2}>
, at L = M_{e}/2, with the HWHM and the mixing parameter
of a pseudo-Voigt function.**

**Note that this equation
established for L = M_{e}/2 is now implicitly declared
as valid for any L value.**

**Is that serious ?**
**A lot of equation manipulations,
hard to justify.**
**Well, it is published...
and there is no alternative.**

**So you will have to make
your own opinion about the credibility of such a demonstration, and the
meaning of the M_{e} and <e^{2}_{Me/2}>
values that you will extract from your data when applying all those approximations.**

**One has always in the
physical parameters the consequence of the approximations retained. Those
approximations remain highly disputable in this paper.**