I think you already discussed the cause for X-ray peak broadening. But the first cause of grain size has the major effect and mostly refers to it. The real cause for such effect is the existance of differvent lattice value bigining from the smallest which belongs to the most pure and perfect single crystal of the material to those due to the effects of grain size, internal and surface defect, impurities and dislocations. all the diffracted X-ray due to the effects above will have a continuous range of theta beginning from minimum to the higher value which produces a peak broadening.

This very important question , so, I follow..................

@Artur: If you have a gradually changing mineral (in composition and also in lattice parameters, with a Gaussian distribution)...and you would grind it and measure with XRD...don't you think that you would observe simply broad peaks which could be explained by a high-defective phase?

And if this material really contains many defects (which prevents or limits investigations by CBED), additionally to the lattice parameter gradient, what you would expect as second signature?

The tetragonality of martensite is caused by the content of carbon. Extension of the lattice is caused by a non-relaxed internal stress. Consequently, the widening of the x-ray peaks is caused by both reasons.

Dear Tamaz Vasili Eterashvili, It is probable what you are saying but this is no proof. As Kurdjumov said: The grains are also very small, which also is one reason for broadened peaks. STOP: Correctly spoken: ..which MIGHT BE also one reason for broadened peaks. This is exactly why I am asking here: Do we really KNOW it, or do we only have some indications and no experimental proof and simply BELIEVE it since these models explain the up to now investigated experimental facts? We are quite flexible in belief, like the belief that atoms are small spheres which sometimes fit into holes between other spheres and sometimes not. We all know that this is not the case but we still use this argumentation in many models.

Until know I mainly read that the broadening of martensite reflection is a consequence of a high dislocation density...which can generate a lot of strain, and in case of bigger rotations it can be even interpreted as small incoherent volumes other people discriminate as separate grains. However, about different lattice parameters I never read anything serious.

Excellent question. In our XRD assays we observed variation of the widening of the peaks in the activated alkali materials due to grain size. Another variation of the amplification of the peaks was due to the technical factor of the preparation of the samples for reading in the XRD. In some cases the technical way of preparing the sample influences in a more significant way to mask the influence of lattice defects. My small contribution certify a same sample preparer to avoid stretching the peaks.

Thank you everyone for excellent discussions and sharing resources.

This is not the answer to your question but the method may be helpful. See J. Appl. Cryst. (1977). 10, 66-68 The approximation of symmetric X-ray peaks by Pearson type VII distributions. by M. M. HALL JR et al. The example given is for Fe-Ni- Ti martensite.

Dear Mr. Hall, as you mentioned already this might not be the answer to my question...but how is your impression about the expectation of Kurdjumov that a distribution of lattice parameters my also cause a broadening which does not display considerable differences to the common assumptions of grain size and strain within a material?

I have not worked in this area since finishing my PhD program so I am unable to answer your questions in general. However, attached is the Pearson type VII paper. This methodolgy has been used extensively to analyse diffraction peaks. Also attached is a paper that might be instructive relative to our use of the analysis method. The experimental austenitic/martensitic Fe-Ni-Ti alloys that were studied form Ni3Ti clusters during aging of the austenite that then prop open (via the Bain strain) the tetragonal lattice that forms on transformation to martensite. This is somewhat analogous to the martensitic Fe-C alloys where carbon in octahedral and tetrahedral lattice sites distorts the bbc lattice. In the Fe-Ni-Ti case, these tetragonal lattice strain centers then produce a periodic distortion of the bcc lattice, that is, a periodic distribution of the lattice parameter. In our case, "side peaks" appear on both sides of the main diffraction peak where the spacing of side peaks is a measure of the lattice strain.

I do not know the work of Kurdjumov but can certainly agree that "a distribution of lattice parameters may also cause a broadening". In our case the source of broadening is due to a PERIODIC distribution of lattice parameters that is distinctively different from broadening due to a random distribution of grain size and a uniform distribution (realtive to the wave length x-rays) of residual strain.

The broadening of XRD peaks during the martensitic phase transformation of solid solutions based on ZrO2 (T-ZrO2 ↔ M-ZrO2) are associated with the dispersion and the degree of crystallinity. Nanocrystalline powders have very wide peaks. When the size of the particles increases, as well as the degree of their crystallinity, the width of the peaks decreases.

@ E.V. Dudnik: I guess you did not read the contribution and the divergent arguments? Your argument describes only one correlation assuming the broadening is exclusively caused by the size of coherent volumes.

We are developing not new steel, but composites based on refractory oxides. Therefore, my answer is connected with the interpretation of the

broadening of XRD peaks concerning to the T-ZrO2 ↔ M-ZrO2

phase transformation of the martensitic type.

I'm not a specialist in steel, but may give some hints derived from theory (and experience with ceramic materials).

The different influences noted by Gert Nolze cause different types of broadening of reflections of XRD pattern.

a) grain size: Crystallite size (mean column length L) is the better name for evaluation of XRD scans. The angel dependecy of the broadening (integral line width b) is b = lambda /L /cos(theta).

d) lattice strain: lattice strain by point defects (micro strain, e) causes symmetric broadening (fwhm b) of the reflections b = 4 e tan(theta). The different angle dependencies of crystallite size and micro strain enable separation of both effects if enough peaks at low and high 2theta are available. This type of strain may cause symmetric broadening of the peaks only, but also anisotropic broadening.

b) concentration variations: Already Roberts (Journal of Metals 5 (1953) 203) showed a dependency of peak splitting of the tetragonal martensite from the contration of C. He also described an increasing volume of the unit cell with increasing contents (c) of C. For each value of c(C) there is a distinct peak position of the peaks, where the doublet of the (110) and (101) peaks is a proper example. If there is a distribution of c(C) instead of a fixed value for all crystallites, then there is a non-isotropic broadening of the peaks, which also follows b = 4 e tan(theta). But this broadening most probable is asymmetric for nearly all peaks. Asymmetric and anisotropic broadening involves high demands to the evaluation method (best: hgh level pattern decomposition). It is difficult to separate this effect from micro strain. Concentration variations is stress and causes strain, which may not be called micro strain, nevertheless.

c) different martensites: This is only a special case of b). If the function c(C) has two (or more) local maxima then this is like having different distinct martensites. If c(C) has only one maximum this is like having one martensite only with a high amount of anisotropic strain.

e) I don't know whether stacking faults may exist as an additional source of peak broadening and peak shift. The cubic aristo structure often shows stacking faults after cold working. The martensitic phase transition might also cause stress resulting in stacking faults.

Regards,

Roberd