Can you please give the source of the P42/n description of martensite? I think I have seen it before but I cannot currently find it. Isn't it an attempt to describe the Carbon ordering (with respect to I4/mmm) in terms of a supercell? The presence of spacial modulation of the C atom occupation in Zener ordered martensite is known from quite diffuse additional scattering in SAED (Nagakura and others) and there have been other attempts to set up crystal structures to describe (or approximate) that ordering (the Pnma case from Transactions of the Japan Institute of Metals (1979) 20, p100-p110 or J. Appl. Crystallogr. 14 (1981) 329., P2/m symmetry, see also Acta Mat 140 (2017) 433). Very likely the P42/n description is only one of several possible approximations, where even the size of the supercell is variable (like that of an approximant of an incommensurate structure).

I referred to Ohba, T. Space group consideration of martensitic transformation in ferrous alloy. Inter. Conf. Mart. Transf. (ICOMAT), 2017, however, I think that I found this description in another paper as well.

I also found some references where lower symmetries are indicated, however, I assumed that a general assumption is that martensite is tetragonal. My impression is only that a Zener ordering and P 4_2/n (using no supercell) is impossible.

Ok. I have found the abstract on the net, which is telling us little. Are the proceedings already published? It remains unclear, on which basis P42/n symmetry is adopted.

The number of domain states is that which is expected from the ratio of the space group orders (with respect to I4/mmm the space group order reduces by a factor of 24), if the volume of the conventional unit cell remains constant.

I think you can, indeed, describe the normal Zener ordered structure of tetragonal martensite in the first setting of P42/n with exactly the same 000 and 001/2 sites for Fe and C as commonly used in the I4/mmm description. That P42/n description, however, misses the additional symmetry elements present in I4/mmm. Since two atomic sites lack free fractional coordinates, it also cannot be periodic displacements of Fe and C leading to a symmetry break to P42/n. In case of use of the second setting of P42/n, the fractional coordinates change by 1/4 1/4 1/4. Hence I can see no reason why a P42/n description of tetragaonal martensite using no supercell should make sense. Maybe originally a supercell description is behind the P42/n space group symmetry.

If p 42/n is used, the atomic positions of all three possible unit cell definitions generate exactly the same positions again as in austenite (only differently described by a selection of other basis vectors), see first image where the basis centering carbon atom is shown in blue. Taking the translation symmetry into account (blue vectors) the same atom positions result for each of the shown three structure definitions. This means (from my point of view) that this reduction of symmetry enables a tetragonal distortion but no ordering of carbon atoms. In case of an ordering I would assume that some carbon positions will disappear for the new structure description. In other words, during symmetry reduction the asymmetric unit has to contain two independent positions which enables to delete one of them: here (0.5 , 0.5 , 0), isn't it?

In the second image another symmetry is used and display the ordering generating (in sum) exactly the same positions as in austenite, however, display now an ordering of carbon within layers. Comparing all three unit cell definitions, they are now really different to each other and describing the so-called carbon sublattices correctly. If there is a mistake, where is it?

I think I see you problem. It appears that you emenate too much from the fcc structures. Zener ordering is not the ordering you show in your second image. Zener ordering refers to the bcc/Im-3m description of the alpha phase, which contains 3 octahedral sites per Fe atoms (6b Wyckoff site of Im-3m), which are strongly distorted. Zener ordering implies that that one of these sites (which can be grouped as x, y and z, occording to the orientation of the short axis of the octahedron) is occupyied preferentially as compared to the other two ones, leading to symmetry reduction to I4/mmm. There the occupied ones are 2b sites of I4/mmm and the unoccupies are the 4c sites (e.g. x + y). If you keep c/a close to 1-1.1 (as in tetragonal martensite) you will still recognise these sites as octahedral sites. Interestingly the Zener ordering starting from Im-3m does not change in the translational symmetry, i.e. no superstructure reflections will evolve.

If you start from Fm-3m, you will recongnise only the z octahedral sites, as c/a = 1.41. The others are some of the 1/4 1/4 0 position in your fcc struccture.

One might argue, that in nature one starts from austenite, i.e. from Fm-3m symmetry, containing only 1 Octahedral site per Fe atom. Taking into account the usually adopted Bain-like lattice correspondence (I did not say orientation relationship!), the martensite transformation towards alpha' prime will atomatically lead to Zener ordering. There is some discussion in the literature, whether the Zener ordering is simply inherited from the fcc->martensite transformation or whether it is a true equilibrium phenomenon of bcc-Fe containing a corresponding C content.But Zener ordering refers to preferential ordering of one of the octahedral sites in bcc-Fe.

I see. Thanks!

I was busy with some other issue for a while so that I could not spend the time I would like to.

I realized that in a very pragmatic way Zener ordering can be easily described, and in fact, nothing speaks against this kind of drawings. What still puzzles me is the connection to the phase transformation from austenite. As far as I understood it, the carbon is assumed to occupy the octahedral interstices, and finally, after transformation everything indicates that the carbon is still sitting in (quite heavily distorted) octahedral interstices, i.e., despite the now actually bigger tetrahedral interstices in fcc they are not used. Carbon obviously does not change its coordination. From my point of view, this means that the relative positions during transformation should not change. If I now compare the common sketch of Zener ordering based on a bcc unit cell, there are too many (6 instead of 2) which are not used in fcc. I mean, even close to these interstices no carbon position is described in fcc.

If I select another c-axis for martensite, another subset of the 6 interstices are used, however, these are exactly the same carbon atoms as before, i.e., referred to the former fcc cell, all carbon atoms (for the sake of simplicity let us assume the positions are all occupied) are used so that in fact now ordering occurs. To me it seems, that there is only a possibly strain-controlled switch of the c-axes but de-facto no ordering at all.

The attached figure should express this. In order to get an impression about the unused octahedral interstices ( [1/2,0,0], i.e., all together 4 ), in the right of the three bcc unit cells red circles are inserted to mark them.

To my opinion, the former sketches suggest some kind of choice, carbon atoms have between three different layerings, what sees to be me not really the case. What do you think?

We agree that there are 3 sets of octahedral sites and 6 tetrahedral sites per Fe atom in the bcc structure. In fact the tetrahedral sites are “larger” than the octahedral sites but experimental evidence and DFT calculations indicate that C or N are quite unhappy on the tetrahedral sites. These are even only saddle points in the energy landscape at 0 K. Insertion of C/N on the octahedral sites is more favourable since the space is accommodated by displacement of the closest Fe atoms. These displacements are quite large and easy, because this is the direction in which displacement of the Fe atoms is most easy and this sis the direction directly away from C/N.

The thermodynamic theory developed by Zener consider the interactions by these displacement fields and finds that there can be a phase transition of 1st order: Below an order disorder transition temperature, which is proportional to the C content, occupation of one of the three octahedral sites becomes preferred over the other two, thus reducing configurational entropy and decreasing the interaction (“elastic”) energy. Thereby, the structure becomes tetragonally distorted. However, this order-disorder transition cannot be observed under equilibrium condition, since it is not possible introduce sufficient C into bcc-Fe below the eutectoid temperature without that cementite forms. The only way is via C-containing austenite. That gets instable at lower temperatures, but you have to quench to retain the C content in the alpha-Fe formed. However, as you have indicated, performing the Bain distortion already gives the correct ordering of C. Hence some people argue that the thermodynamic framework by Zener is not needed to explain the tetragonality. However, below a critical C content no Zener order is observed (actually close to the threshold composition it is difficult to decide because the reflections are broad due to the high dislocation content of the corresponding lath martensite).

Concerning your question: Do the C atoms have a chance to occupy some of the unfavourable sites in tetragonal martensite if the Zener ordering is already inherited from the austenite. I would say yes. If you are below the critical C content, the martensite ends up cubic. That means, the C atoms redistribute over the sites starting from Zener ordering inherited from the austenite. Moreover there is some, mainly Soviet literature which hint at some changes of the c/a after quenching, hinting at some equilibration of the C distribution between the favourable and unfavourable C sites within tetragonal martensites. You have to keep in mind that C can diffuse in alpha-Fe relatively easy even at ambient temperature. Of course the situation is complicated by the high dislocation density, i.e. C is also happy to end up in the strain fields of the dislocations.

Thanks, Andreas. I agree that this entire problem is definitely extremely complex and not easy to interpret unequivocally because of the numerous other accommodating parameters like improper quenching, diffusion at low temperature, the high defect density, or finally the very low concentration of C.

The graphic I've added was only a further attempt to show that an occupation of all possible octahedral sites in austenite can equally explain each different Zener ordering. Thus, it looks to me that there is only an artificial ordering which results from the transition from an fcc to a bcc (or bct) unit cell. If they are statistically really equal, another driving force exists for the selection of the c-axis, i.e., the generated strain caused by the collective movement. The figure indicates that in all orderings C atoms obviously do not change their coordination to the neighborhood, i.e., the adjacent Fe atoms do not change for all three orderings. Only the distortion direction referred to the ideal octahedral coordination changes. Two Fe-C distances becomes shorter whereas four become longer. In other words, caused by the collective movement, Fe atoms - let us call them 1 to 6 - describe in all three orderings the end points of the same distorted octahedron. Only in one case 1 and 4 have the shortest distance to the central C atom, or 2 and 5, or 3 and 6. And if a statistical distribution of C in austenite is assumed for a much smaller concentration, this should not change the character of ordering as well. Or am I totally wrong somewhere?