The Crystal Structure of As-quenched Fe–C Martensite

Abstract

Confusion exists in the literature as to whether the crystal structures are cubic or tetragonal in lath martensites of Fe–C alloys and low-alloy steels. Steels with a range of carbon contents have been quenched and examined by synchrotron x-ray diffraction. The presence of dislocations and residual local strains complicates the analysis since peak splitting of tetragonal lines is obscured by the broadening. Asymmetry of the 200,020/002 lines has been examined and synthesised using model peak functions. A new approach has been to study the 222 peaks which are unique (not split) for both cubic and tetragonal crystals. For low carbon steels (<~0.2%C) the structures are fully or almost completely cubic. Above about 0.7%C the martensite has tetragonal symmetry. Intermediate, medium carbon, steels consist of mixtures of cubic and tetragonal structures.

It is argued that martensite is always tetragonal at the moment when it forms from austenite but it can subsequently decompose into cubic by auto-tempering during the quench. Kinetics of decomposition have been measured using higher carbon steels and these are extrapolated to the situation of auto-tempering. A model is developed whereby the creation of tetragonal martensite according to the Koistinen-Marburger relationship is modified by its rate of decomposition. The final structures are predicted to depend on both the martensite start temperature (Ms) and the cooling rate.

1. Introduction

It may seem incredible that such an ubiquitous material as steel could still harbour doubts about its basic crystal structure but that is actually the case with many hardened martensitic steels. A vast literature exists on the subject of martensite and many excellent and relevant publications are from more than 50 years ago. We will refer selectively to these but useful reviews are found in.^1,2,3) In older literature, it was considered that the quenched steels all had a structure that was body centred tetragonal (BCT) and in which c/a varied linearly with the carbon content, a natural corollary of the transformation reaction as proposed by Bain.⁴⁾ The carbon atom positions inherited from the austenite are aligned on a single sub-lattice, extending the crystal along one of the cube axes. According to Roberts’ review,⁵⁾ the a and c parameters (in nm) vary with carbon content as:

  

$$\begin{array}{l}a=0.2867-0.0013\%\text{C}\\ c=0.2867+0.0116\%\text{C}\end{array}$$

For simplicity we describe such a behaviour as the ‘classical’ viewpoint which was widely held during the second half of the previous century. There is wide agreement that this is true for steels with carbon contents in excess of about 0.7%. However, the difficulties in solving diffraction data for lower carbon contents meant that no convincing results could be shown for steels containing less carbon than this. There are presently four different interpretations of the actual crystal structures in these lath martensites which are nowadays produced in large tonnages as commercial steel products. These cases are:

1. The classical behaviour applies with full tetragonality at all carbon contents. More recent analyses by Lu et al.⁶⁾ using Rietveld refinement of their x-ray diffraction data led to a conclusion of tetragonality for all carbon contents with lattice parameters quite close to the classical ones.

2. Several workers have described their low carbon martensites as tetragonal, although with lower c/a ratios than for the classical case.^e.g.7,8,9) This might be described as dilute tetragonality as not all the carbon atoms are concentrated on the specific sub-lattice. This conclusion has also been reached after Rietveld refinement but sometimes in surprising situations. For example, tetragonality has been claimed to exist even after tempering martensite to temperatures as high as 650°C⁷⁾ and also in steels containing no carbon.⁸⁾

3. Many more recent publications have described the structures as body centred cubic (BCC). Approximately 20 years ago, and then partly in connection with a concept proposed by Sherby et al.,¹⁰⁾ it became widely accepted that those lower carbon (<~0.6%C) steels have, in fact, body centred cubic structures and many subsequent observations on quenched and dual-phase steels have been interpreted in this way.^e.g.11,12) The model of Sherby et al. is quite complicated and is related to the belief that a sharp transition (‘H’ point) occurs at 0.6%C, see also a recent review.¹³⁾ In fact, this is unlikely to be true. Very fast cooling using ‘splat quenching’ resulted in tetragonal martensite in an iron-carbon material with as little as 0.21%C according to Cadeville et al.¹⁴⁾ and similar structures also occur in Fe–Ni–C alloys with less rapid cooling.¹⁵⁾ Thus, it seems likely that the lower limit for tetragonality may vary somewhat depending on the cooling rate.

An alternative interpretation is simply that the carbon atoms on the aligned sublattice can diffuse when the Ms temperature is sufficiently high, in a first stage of (auto-) tempering during the quench. Movement of the carbon atoms destroys the ordering and causes the structure to revert spontaneously to a cubic symmetry. This viewpoint can be found in various publications^16,17) including Honda¹⁸⁾ as long ago as 1927. The question is, however, still open to debate. There is some more recent support for this viewpoint from the in-situ neutron diffraction work of Wang et al.¹⁹⁾ on a 0.4%C-1%Cr steel and also in-situ synchrotron experiments of Kohne et al.²⁰⁾ The structure after initial cooling below the Ms temperature was identified as tetragonal but this rapidly changed to cubic on subsequent cooling or holding.

4. In an earlier review paper, Kurdjumov¹⁾ expressed the view that different crystal structures arise with different carbon contents in quenched steels. Thus, up to about 0.2%C the steels are cubic; between about 0.2%C and 0.6%C they consist of mixtures of cubic and tetragonal phases while above about 0.6%C they are purely tetragonal with the classical c/a relationship. However, this view does not appear to have been widely upheld.

Most literature concerns diffraction by x-rays or neutrons. However, other experimental methods have also been employed and the study by Tanaka et al.²¹⁾ is especially interesting. Very high-resolution electron back-scattering diffraction (EBSD) measurements were made on individual crystals of martensite with varying carbon contents. These workers reached a somewhat different conclusion whereby the martensite structure is truly cubic for carbon contents up to approximately 0.44%. For higher C-contents than this, the linear dependence of a and c was confirmed with the classic relationship. The c/a relationships for individual grains were also affected by local elastic strains which can be very large in martensite.²²⁾

The problem concerning interpretation of diffraction data arises because the spectral lines are greatly broadened by high dislocation densities in the martensite and additionally by large type II residual strains. This means that the splitting of lines that distinguishes tetragonal from cubic structures is largely obscured. It would appear that such a situation is well suited for clarification using the Rietveld refinement²³⁾ procedure but there are conditions that are very difficult, or almost impossible, to satisfy. These difficulties are explained more fully in Appendix 1 but, in principle, the influences of tetragonality and elastic strains on diffraction peaks are so similar that they cannot be distinguished from one another.

In addition to conventional peak analyses we have also paid attention to the high angle 222 reflections which are unique lines for both BCC and BCT crystal structures, see for example Cullity.²⁴⁾ These planes have a single lattice spacing with a unique diffraction line which is unaffected by any degree of tetragonality, so broadening is caused only by strain. It appears that this advantage has not been utilised in previous literature, probably because the 222 reflections are very weak. However, the high brightness of synchrotron sources means that this needs no longer be a limitation.

2. Experimental

A range of commercial steels was obtained and processed where necessary into sheets 0.5 mm thick. Their chemical compositions are given in Table 1. Samples were austenitised by holding for 5 minutes at temperatures that were about 70 degrees above their Ae₃ temperatures according to the iron-carbon phase diagram. Following this, they were quenched into saturated brine. Specimens for synchrotron XRD were prepared by chemically polishing these samples from both sides down to a thickness of 0.1 mm. The reduction of 0.2 mm from each side by chemical polishing ensured the removal of any decarburised layers that might have formed during the heat treatment. All the synchrotron specimens were examined metallographically which confirmed that no cementite remained undissolved and that the martensite structures were unifom in appearance. Pieces of the quenched steels were also annealed at 650°C for 30 minutes to fully temper the steel and precipitate all carbon as cementite. The purpose for this was to check whether the different contents of alloying elements had any significant effect on the steels’ lattice parameters.

Table 1. Chemical compositions of the steels in this investigation.

Name	%C	%Si	%Mn	P	S	N	Cr	%Ni	%Mo	%Al	%B	%Ti
0.056%C	0.056	0.21	1.79	0.009	0.003	0.0040	0.03	0.06	0.01	0.039	0.0002	0.004
0.103%C	0.103	0.21	1.60	0.012	0.002	0.0053	0.03	0.04	0.04	0.043	0.0003	0.04
0,202%C	0.202	0.19	1.06	0.009	0.002	0.0054	0.039	0.03	0.00	0.044	0.0022	0.034
0.263%C	0.263	0.19	0.50	0.008	0.003	0.0029	0.023	0.03	0.01	0.047	0.0023	0.036
0.348%C	0.348	0.01	0.38	0.011	–	0.0023	0.144	–	–	0.034	0.0047	0.032
0.489%C	0.489	0.251	0.776	0.016	0.027	–	0.09	0.13	0.02	0.00	0.0001	0.001
0.72%C	0.710	0.21	0.49	0.011	0.005	0.0023	0.088	0.130	0.022	0.005	–	0.001
0.76%C	0.756	0.239	0.695	0.018	0.003	–	0.212	0.052	0.023	0.007	–	0.002
1.02%C	1.020	0.23	0.39	0.009	0.002	0.006	0.21	0.06	–	0.014	–	0.001
1.00%C	1.004	0.26	0.44	0.008	0.001	–	0.20	0.09	0.03	0.006	–	–

Most of these specimens were examined in the powder diffraction line of the Australian Synchrotron using transmitted x-rays with a beam cross-section of 3×1.2 mm (H×W) and energy of 21 keV (refined wavelength λ= 0.05898 nm). Rapid acquisition of high-resolution diffraction data was achieved using a Mythen II micro-strip detector spanning an angular range of 50° in 2θ.

Length changes on tempering the 1.00%C steel were measured with a travelling microscope with a resolution of 0.001 mm over 50 mm lengths. Heat treatments were carried out successively for increasing times in an oil bath. The purpose of this was to quantify decomposition of the tetragonal structure in a similar manner to that used by Roberts et al.²⁵⁾

3. Results and Discussion

3.1. Synchrotron Measurements

The first analysis concerned the position of the 222 peaks and the resulting inter-planar spacings. As mentioned above, these are unique reflections for both cubic and tetragonal crystals. Results are presented in Fig. 1. The reproducibility of the values corresponds to the size of the symbols and it is seen that the tempered specimens have identical values within these limits. The small differences in other alloy element contents did not affect the measured spacing values for the different steels.

Fig. 1. Solid symbols show measured 222 planar spacings for the quenched steels as function of their carbon contents. Red diamonds are for the steels tempered at 650°C. Black crosses are modelled based on the 200,020/002 reflections. The black dashed line is from Roberts’ review.⁵⁾ (Online version in color.)

Figure 1 also shows measured values together with the expectation for 222 spacing using Roberts’ tetragonal relationship⁵⁾ as a dashed black line. The four steels with smallest carbon levels have lattices that are slightly expanded compared to pure iron which could arise from carbon atoms in interstitial cubic solid solution. The two steels with intermediate carbon levels are interesting as they do not appear to conform to either category and will be discussed further below. In principle these could represent an intermediate degree of tetragonality or an overlapping of peaks from cubic and tetragonal crystals.

Turning to the 200,020/002 peaks, it could be seen that steels with carbon contents in excess of 0.26%C showed visible asymmetry that was compatible with an effect of tetragonality. The degree of asymmetry was quantified using the same procedure as previously used in for example in refs.^26,27) The high angle side of the peak is mirrored and this doubled area is subtracted from the total intensity leaving a residue. The asymmetry or skewness factor is defined as the ratio of the residue to the doubled peak, expressed as a percentage in Fig. 2.

Fig. 2. Skewness factors for the 200 diffraction peaks as a function of the steels’ carbon contents. (Online version in color.)

For the four steels with the lowest carbon contents, the peaks are symmetrical within the accuracy of these measurements and these skewness factors are effectively zero. Although they appear to have no tetragonal contribution there is reason to suspect that a small amount may exist as discussed further below. The three steels with carbon levels above 0.7% had clearly split and skewed 200 peaks indicating tetragonal structures, as expected from published literature. The skewness factors in these cases corresponds to the ratios of diffracted intensities coming from the c and a planes in the crystals. On the basis of multiplicity, a value of 50% would be expected. However, it has been known since the work of Lipson et al.²⁸⁾ that the interstitial carbon atoms in tetragonal martensite displace the iron atoms to such an extent that the structure factor is reduced for 002 as compared to 200,020 so the diffracted intensity is lessened. Other publications confirm this behaviour,^e.g.29,30) with intensity ratios of typically 20–30%. The steels with intermediate carbon contents are notable in that their peaks are significantly skewed.

In addition to the martensite peaks, reflections from retained FCC austenite (RA) were seen in most of the samples. The content of this phase was calculated based on the {200} peak intensities for austenite compared to the {200} peaks for martensite. The counting statistics for the peaks are very good but some influence of texture on the resulting values cannot be ruled out.

The synchrotron 200,020/002 diffraction peaks have been analysed using the line profile fitting procedure in Topas software³¹⁾ with application of symmetric pseudo-Voigt functions. The number of free parameters was kept to a minimum by using standard d-spacings for the cubic (iron) and tetragonal phases (Roberts⁵⁾). The optimised parameters are listed in Table 2 while Fig. 3 compares the measured and simulated peak profiles for four of the steels. Agreement is very good. A single cubic structure applied for the three steels with lowest carbon contents and a pure tetragonal condition applied for the three highest carbon steels. The three intermediate carbon steels were interesting in that these required both cubic and tetragonal structures to achieve satisfactory fits. Since the absolute intensity in transmission geometry is very sensitive to specimen thickness, the intensity values in Table 2 are normalised to the same total intensity.

Table 2. Numerical values obtained by optimising the simulation of measured diffraction spectra with model peak functions and the percentage of retained austenite RA.

%C	Cubic 200			Tetragonal 200,020			Tetragonal 002
	d, nm	FWHM,°	Int.	d, nm	FWHM	Int. %	d, nm	FWHM,°	Int. %	%RA
0.056	0.14332	0.219	100%	0.14331	–	0	0.14367	–	0	0
0.103	0.14332	0.249	100%	0.14328	–	0	0.14395	–	0	0
0.212	0.14332	0.313	100%	0.14322	–	0	0.14452	–	0	2.13%
0.263	0.14332	0.350	88.0%	0.1432	0.670	9.5%	0.14488	0.670	2.5%	2.02%
0.348	0.14332	0.437	45.8%	0.1431	0.355	44.3%	0.14535	0.355	9.9%	1.2%
0.489	0.14332	0.700	27.6%	0.14305	0.400	57.9%	0.14620	0.400	14.5%	4.6%
0.72	0.14332	–	0	0.14288	0.420	83.8%	0.14753	0.450	16.2%	12.9%
0.76	0.14332	–	0	0.14285	0.430	85.6%	0.14775	0.300	14.4%	17.7%
1.02	0.14332	–	0	0.14269	0.358	89.9%	0.14927	0.300	10.1%	27.5%

Fig. 3. Measured and modelled 200 peaks for four quenched steels, (a) 0.103%C, (b) 0.348%C, (c) 0.489%C and (d) 0.72%C. (Online version in color.)

3.2. Phase Transformation Kinetics

The conversion from tetragonal to cubic structures during annealing has been studied in many previous works on high carbon steels,^{e.g.1,25,32,33)} often referred to as Stage 1 tempering. It has also been held as implicit in many cases of lower carbon steels that auto-tempering cannot readily be avoided during quenching when the Ms temperature is sufficiently high. We consider, therefore, that it is reasonable to hypothesise that martensite, at the moment of its formation, is always tetragonal but may not retain this during cooling to room temperature. In the following, we test this hypothesis and use it as the basis for a general model.

The 1.0%C steel was used as this is indubitably tetragonal after quenching, as was also confirmed by the split 200,020/002 peak in XRD. In this experiment, the quenched samples were cooled in liquid nitrogen to minimise the content of retained austenite before tempering. Following Roberts et al.³³⁾ measurements of length changes were made after sequential isothermal anneals at four temperatures as shown in Fig. 4. XRD measurements confirmed that these changes correspond to elimination of tetragonality. The kinetics were quantified by the time to reach a contraction in length of 0.1% which is near to 50% of the total change. These values are shown in Fig. 5 as an Arrhenius plot for a thermally activated process and for which the activation energy was determined to be 140 kJ/mol. The kinetic expression is given by the equation:

  

$$\mathrm{L}\mathrm{o}\mathrm{g}\left({\mathrm{t}}_{0.5}\right)=7\mathrm{\hspace{0.17em} }318/\mathrm{T}-15.224$$(1)

Fig. 4. Strains measured after isothermal annealing of the quenched 1.00%C steel at different temperatures. A photograph of one specimen is inset. (Online version in color.)

Fig. 5. Annealing times to give a strain of −0.1% at different temperatures as an Arrhenius plot. The length changes are a result of the transformation of martensite structure from tetragonal to cubic.

The Arrhenius relationship allows values to be estimated for other temperatures and some of these are shown in Fig. 6. Although the extrapolations are too large for complete reliability, they do permit some clear conclusions to be drawn. In particular, the time for decomposition at temperatures where martensite forms (~300°C) is only about 1 ms which implies that auto-tempering cannot be prevented by normal quenching such as in water even if the cooling rate is 1000°C/s. If the transformation takes place at room temperatures it would require around 100 years to happen, although some workers^e.g.34) consider that the tetragonal phase is actually a stable equilibrium phase at such low temperatures.

Fig. 6. Times for the transformation from tetragonal to cubic martensite at different temperatures calculated by extrapolation of the results in Fig. 5. (Online version in color.)

3.3. Modelling the Quenched Structures

The crystal structure(s) of quenched martensite have been modelled by combining functions for creation of tetragonal martensite during the transformation from austenite with its destruction by subsequent transformation to cubic where applicable.

The transformation from tetragonal to cubic martensite was calculated using equation 1 for conditions of constant cooling rate. Since this cannot be integrated analytically, the temperature range was digitized in 5 degree isothermal steps and the total transformation was obtained by summing these up to the value which was considered to imply transformation. When the Ms temperature is too low and/or the cooling rate is too high, there is insufficient thermal activation to cause the transformation. This is readily calculated and the result is shown in Fig. 7. Any situation where the martensite is formed below the line results in tetragonality while cubic structure results above this.

Fig. 7. Limiting conditions that permit/prevent the martensite to transform from tetragonal to cubic during cooling at different rates. (Online version in color.)

This picture is oversimplified in reality as the critical condition is not limited to the Ms temperature, since austenite continues to transform to martensite throughout the whole temperature range below Ms. It should be recalled that the martensite start temperature Ms tells where this begins to form during quenching but the content continues to rise as the temperature falls further. This behaviour is usually treated using the Koistinen-Marburger relationship³⁵⁾ which tells the fraction (f_m) of martensite as a function of temperature below Ms as:

  

$${\mathrm{f}}_{\mathrm{m}}=1–\mathrm{e}\mathrm{x}\mathrm{p}\left\{-{\alpha }_{\mathrm{m}}\left(\mathrm{M}\mathrm{s}-\mathrm{T}\right)\right\}$$(2)

where α_m is an empirical constant which has a value of about 0.012 (K⁻¹).

Results of the model calculation are mapped in Fig. 8 as a function of the Ms temperature and cooling rate. In Fig. 8, M^C and M^T are the fractions of cubic and tetragonal martensite respectively. Mixed structures with both cubic and tetragonal crystal structures are predicted to occur over a wide range of conditions, in agreement with the diffraction results. In view of the large extrapolation involved, these results should be considered as indicative rather than precise but they can be used to understand the behaviour of different steels. The Ms temperature of a steel depends principally on its chemical composition, especially the carbon content. Several empirical expressions have been deduced for this of which the one by Andrews³⁶⁾ is widely quoted.

  

$$\mathrm{M}\mathrm{s}=561–474\%\mathrm{C}-30\%\mathrm{M}\mathrm{n}$$(3)

Fig. 8. Map showing predicted crystal structures for martensite in steels cooled to room temperature as functions of the Ms temperature (or %C) and cooling rate; M^C and M^T are the volume fractions of cubic and tetragonal phases respectively. (Online version in color.)

The Ms temperatures of simple carbon steels containing typically 1%Mn are marked on the right-hand axis of Fig. 8 so the expected martensite structures can be estimated from this map. The structure can be fully tetragonal when the carbon content is sufficiently high and it may be almost entirely cubic when this is low and the cooling is slow, but in many cases the steels microstructure is a mixture of both phases.

3.4. General Discussion

Our present conclusions are in good agreement with the viewpoint expressed by Kurdjumov¹⁾ in 1960 although this has seldom, if ever, been reiterated in later literature. Quenched medium carbon steels generally have structures composed of mixtures of cubic and tetragonal crystals. Duplex or two-phase microstructures are, of course, very common in alloys but there is an important difference. Usually, the two phases exist in thermodynamic equilibrium but that is not the situation here. The mixed structures are a result of kinetics, not equilibrium conditions. A recurring theme in the literature is that a critical carbon content (H point at 0.6%C) exists which stabilises the tetragonal structure.^{e.g.1,10,34,37)} We believe now that this is not true. The limits of existence of tetragonality are a result of kinetic processes at high temperatures and not equilibrium conditions after cooling to room temperature. In fact, the splat cooling results of Cadeville et al.¹⁴⁾ in 1977 already showed that the cooling rate has to be taken into consideration and that tetragonality can result in steel with much lower carbon contents.

Further support for the present conclusions could be found when the 222 diffraction peaks were modelled using the data from the 200,020/220 peaks that are summarised in Table 2. The deduced fractions of the cubic and tetragonal phases were combined to create model 222 peaks which showed a high degree of overlapping. The maxima in these combined peaks were used to provide effective values of the 222 planar spacing which are also included in Fig. 1 as black crosses. These lie a small distance below the measured points which may be attributed to some interstitial carbon content in the BCC martensite since the lattice parameter for pure iron was used in the modelling. However, the variation with carbon content is very well described by the model values. Note also that no uncertainty arises from splitting of the tetragonal lines in this case.

Although direct evidence for tetragonality in the lower carbon steels was not seen, there is reason to believe that some tetragonal crystals existed there as well. A content of retained austenite was clearly present in the 0.202%C steel as shown in Table 2. The Koistinen-Marburger principle implies that some austenite should have been transformed to martensite above room temperature in the range where its decomposition would have been impossible. Accordingly, small, although unidentified, traces of tetragonal martensite must have been present also in these materials. Their presence could lead to misinterpretation of diffraction line profile analyses if these are assumed to arise from purely cubic structures.^e.g.38)

There are other developments that raise the relevance of this viewpoint. Additive manufacturing (AM) using laser or electron beam heating can results in cooling rates far higher than in conventional quenching. It can be expected that products made from low-medium carbon steels may consist of tetragonal martensite unless these are intentionally tempered subsequently. According to König et al.³⁹⁾ cooling rates may reach or exceed 10⁶ deg/s while solidification and subsequent cooling transformations could be monitored in real time synchrotron experiments during laser AM treatments. There would appear to be possibilities for affecting mechanical properties of steels in ways that have previously not been feasible.

4. Conclusions

Crystal structures have been determined in a range of steels with low, medium and high carbon contents. Based on a combination of synchrotron diffraction measurements and modelling it is concluded that both chemistry (Ms temperature) and cooling rate must be taken into account to explain their behaviour. The martensite may be purely tetragonal or mainly cubic but in most circumstances it consists of a mixture of these two phases. However, this is not a normal two-phase equilibrium structure but results from kinetic factors that govern the formation and decomposition of martensite during quenching.

Acknowledgements

Prof. Matthew Barnett is thanked for his support and provision of laboratory facilities at Deakin University. The authors also thank Prof. Maria Santofimia for helpful discussions. This research was undertaken on the powder diffraction beamline at the Australian synchrotron facility, Victoria, Australia.

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Appendix

Appendix 1. Relating to the Rietveld Refinement Method for Martensite

Rietveld refinement is an excellent and widely used method for finding optimal solutions when analysing diffraction data. At first sight it would appear to be an ideal tool for interpreting Fe–C martensitic structures, and has been applied in many cases for this purpose,^{(e.g.6,7,8,9)} but there are complications that render its applicability doubtful.

Two observations are worth making, to begin with. Firstly, ferrite is always tetragonal below the Curie temperature due to magnetostriction that extends the lattice along the cube axis parallel to the magnetisation direction. However, this effect is small, corresponding to a c/a ratio of only about 1.00003, which is significantly below the scale of the measurements under discussion here. Secondly, any statistical method such as Rietveld will always provide a better fit to experiment under the assumption of a tetragonal structure rather than a cubic one since there are more free parameters to be adjusted (two d-parameters and peak intensities instead of one, etc.). Since cubic symmetry is a limiting case of tetragonality, any unavoidable experimental error can then be better accommodated.

The main difficulty in the present situation arises because the diffraction peaks from Fe–C martensite are extremely broad. Broadening results from its high dislocation density⁴⁰⁾ as well as from the presence of severe Type II elastic strains²²⁾ and it varies with hkl because of elastic anisotropy. In the case of tetragonality, broadening is additionally caused by the overlapping of triplets of hkl lines which are closely spaced in the spectrum. It turns out that both of these effects vary with hkl in ways that are remarkably similar. An estimate of the relative broadening due to line splitting can be made as the standard error of the three hkl interplanar spacings, i.e. the standard deviation of these divided by their mean. The value of c/a chosen here was 1.02 although the magnitude of c/a is not of importance. The breadth of diffraction lines arising from dislocations and elastic strains scales as the inverse of Young’s moduli for iron in the corresponding normal directions. When these two estimates of broadening are plotted against one another for the first ten permissible reflections in Fig. 1A, the difficulty becomes apparent. It is notable that the majority of values for different hkl lie close to a linear relationship and are, indeed, almost in proportionality. Among these lower order reflections only 222 deviates significantly (having no line splitting) and this unfortunately has been omitted in all published experiments using Rietveld refinement.

Fig. 1A. Comparison of two possible sources of line broadening in martensite – splitting of the hkl lines in a tetragonal structure (with c/a = 1.02) and elastic anisotropy that controls internal strains normal to the diffracting planes. For most reflections the two possibilities are strongly correlated.

A result of this proportionality is that software has an almost insuperable task to distinguish the structure symmetry from the effect of strain unless it is provided with some additional information. Most commonly, isotropic strain is assumed and, in that case, the whole effect of anisotropic broadening becomes attributed to tetragonal distortion. Even if elastic anisotropy is incorporated in the Rietveld programme such as in the method of Thompson et al.,⁴¹⁾ this is done implicitly, based on crystal symmetry, without introducing independent elastic constant data, so the near proportionality shown in Fig. 1A will persist as a source of confusion.

Peak breadth is not the only information employed in the Rietveld refinement method; the whole spectrum is utilised. In the work of Lu et al.⁶⁾ it appears that only the 200 reflections were included and their asymmetry was interpreted in terms of the overlapping reflections from c and a peaks. Due to the crystallographic character of the austenite to martensite transformation, it may be expected that any resulting elastic strains will vary, depending on their directions in the martensite crystal lattice. Calculations based on the CPFEM transformation model in reference²²⁾ showed significant distortions of the predicted diffraction peaks in ways that depended on the assumed details of the transformation strain. Such information would be completely inaccessible to the Rietveld procedure and any contribution from it would necessarily be misinterpreted. Examination of martensite microstructures using EBSD has demonstrated that one third of the cube axes are associated with the Bain compression axes while the other two thirds are orthogonal to this.^e.g.42,43) The results of Lu et al. may, therefore, be understood on the basis that there exists residual (elastic) tensile strain in the Bain axis directions and compressive strain perpendicular to this, although these axes would be equal in a relaxed state. A recent report on a carbon-free Fe–Ni martensite by Fukui et al.⁴⁴⁾ demonstrated this effect and similar anisotropic strains have also been shown to arise during tempering of Fe–C martensite.²⁷⁾

To further aggregate these circumstances, the effect of different carbon contents also has a similar influence on strain and on peak splitting. The increase in dislocation density and Type II elastic strain leads to a nearly linear increase in peak breadth^27,40) with increasing carbon content for lath martensites and so does the c/a ratio for tetragonality. The confusion between elastic strains and the degree of tetragonality discussed above implies that any erroneously deduced c/a values will also increase linearly with the carbon content of the steel.