# Stacking Fault Energy Determination in Fe-Mn-Al-C Austenitic Steels by X-ray Diffraction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}and 4.4 mJ/m

^{2}for alloys of 0% Al and 3% Al, respectively. This would lead them to be within the following plastic deformation mechanism, while for 8% Al the uncertainty is negligible.

## 1. Introduction

^{3}) is required to obtain electron diffractions and the sample does not represent the generalities of the microstructure or of the bulk [19], (ii) dislocations can only be observed as thin lines at the nanoscale [20] and special attention is required to not confuse them with contrast phenomena, (iii) deviations in measurements may exceed the average value [20], (iv) the probability of finding dislocations with the required geometries is low, (v) the precision depends largely on the models with which the data are interpreted and the skill of the person who performs and interprets the studies, and (vi) this technique is generally limited to steels with low values of SFE and no previous deformation since these two conditions are required in order to observe and measure the radius of the dislocation node [21] or clearly distinguish dissociated dislocations.

## 2. About the Stacking Fault and Stacking Fault Energy

## 3. About the X-ray Diffraction Technique for Determining the SFE

- $SFE$ = stacking fault energy (mJ/m
^{2}) - ${\mathsf{{\rm K}}}_{111}{\omega}_{0}$ = 6.6 (constant value)
- $A=2{C}_{44}/\left({C}_{11}-{C}_{12}\right)$, A is the Zener elastic anisotropy and ${C}_{ij}$ are elastic stiffness coefficients
- ${G}_{111}=1/3\left({C}_{44}+{C}_{11}-{C}_{12}\right)$ is the shear modulus in <111> direction (GPa.)
- ${a}_{0}$ = lattice constant ($\u212b$)
- ${\u03f5}_{50}^{2}{}_{111}$ = root mean square microstrain in the <111> direction averaged over the distance of 50 $\u212b$
- $\alpha $ = stacking fault probability

#### 3.1. XRD Background Setting

^{®}, (Origin lab corporation, Northampton, MA, USA) X’Pert HighScore

^{®}, (Malvern Panalytical, Marlvern, UK) and FullProf

^{®}, (ILL, Genobre, France) among others.

#### 3.2. XRD Determination of the Mean Square Microstrain $\langle {\epsilon}^{2}\left(L\right)\rangle $

^{2}(L)> as a function of 1/L. It should be noted that the Breadth program is found within the Shadow package.

#### 3.3. Determination of Peak Positions

#### 3.4. Stacking Fault Probability

- $\Delta \left(2{\theta}_{hkl}\right)$ = change in the position of the diffraction lines
- ${\theta}_{hkl}$ = the diffraction angle for each peak
- $\xb0{{\displaystyle \sum}}_{b}\left(\text{}\pm \text{}\right){L}_{0}/{h}_{0}^{2}\left(u+b\right)$ = constant specific to each h k l reflection (Table 1)

_{hkl}and α), and thereby allowing for the computation of the variables shown in the Equation (14) using less squares. This method has been used by multiple authors to calculate the SFP in austenitic steels, with results that are close to 3.2% variation, compared to the other models [68,69,70,71].

#### 3.5. Elastic Constants

_{11}, C

_{12}and C

_{44}by up to 22%. Moreover, increasing the Mn content for rates of Fe/Mn of 4.00 and 2.33, resulted in the reduction of the C

_{11}and C

_{12}constants by 6%, but the value of C

_{44}is independent of the Mn content. For the case of Fe-Cr ferromagnetic alloys (b.c.c. structures), Zhang, et al. [74] found that the elastic parameters exhibit an anomalous composition dependence around 5% of Cr attributable to volume expansion at low concentrations. This is represented to a greater extent by the constant ${C}_{11}$, which represents approximately 50% of the value reported for Fe-Mn-based alloys. The use of these constants would result in the overestimation of the SFE value.

## 4. Experimental Procedure

#### 4.1. Specimen Preparation

^{3}. To carry out the XRD tests, the surfaces of the specimens were brought to a mirror-like finish, starting with # 400 sandpaper and working up to # 1200. Afterwards, the specimens were passed through a polishing cloth using 1 and 0.3 µm alumina suspension.

#### 4.2. X-ray Diffraction

^{2}.

^{−4}and a lattice parameter of 3.614 Å. The program BREADTH outputted an MSM of 50 Å with a value of 10.07 × 10

^{−6}.

#### 4.3. Determination of the SFE

^{3}depending on the units of the established variables.

## 5. Results and Discussions

_{111}in Equation (1) proportionally affect the calculation of the SFE and their values are a function of the elastic constants; these in turn were obtained from other alloy systems that do not necessarily contain the same alloys or in the same proportions. In the absence of experimental data, theoretical values have been used to calculate the SFE in manganese steels by XRD. Based on the considerations above, an analysis was performed with the values reported in the literature for Fe-Mn base alloys. The analysis consisted of using the different elastic constants reported in the literature for other alloy systems in order to calculate the SFE of the austenitic Hadfield steel in the present work (control or reference sample). The aim was to compute the percentage error in the determination of the SFE when taking values of the elastic constants of different alloy systems, as displayed in Table 3. The MSM was calculated by the program BREATH using the Voigt convolution model, which outputted the SFE value in the expected range. The mean SFE value was 24.32 mJ/m

^{2}, which was taken as a basis for the different studies of the SFE and was within the range established in the literature of 23 ± 2, as stated above.

^{2}. This corresponds to the TWIP deformation mechanism, with a small part of the surface in the TRIP range where the SFE is below 20 mJ/m

^{2}.

_{11}and C

_{12}within the range of possible values places this alloy in the TWIP category (Figure 8a). Similar behavior occurs with the Fe-22Mn-0.9C-3Al alloy for the TWIP and MBIP mechanisms (Figure 8b). In contrast, the most likely mechanism is MBIP for the 22Mn-0.9C-8Al alloy (Figure 8c). Therefore, the selection of the elastic constants plays a very important role in determining the SFE and the predominant mechanism of the alloy.

## 6. Conclusions

- The flow diagram presents the calculation of the SFE using data obtained by XRD in addition to values of the elastic constants. The procedure was verified with a widely used commercial Hadfield-type alloy, where the values obtained were within the range established by previous investigations.
- Average SFE reference values can be obtained using elastic constants of alloys with similar compositions, which serve an alternative when it is not possible to retrieve the values from experimental tests or computational calculations. However, for Hadfield steel, the variation of the elastic constants in the range in which they have been reported generates a variation in the calculated SFE of 30%.
- ${C}_{11}$ and ${C}_{12}$ are within the ranges reported for austenitic steels generates variations of 36.6%, 28%, and 28.4% in the value of the SFE for the Fe-22Mn-XAl-0.9C alloys studied with 0%, 3%, and 8% Al, respectively; representing the possibility that these alloys present TRIP or TWIP deformation mechanisms for the case of 0% and TWIP or MBIP for 3% Al content. In the case of the alloy with 8% Al, the probable deformation mechanism is MBIP even with the variation in SFE.
- The SFE variation is 11.6%, 12.3%, and 11.5% for alloys with 0%, 3%, and 8% Al, respectively. When changing ${C}_{44}$ between the extreme values reported for this constant reflected in a smaller effect concerning the variations of ${C}_{11}$ and ${C}_{12}$.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

SFE | Stacking fault energy, mJ/m^{2} |

SFP | Stacking fault probability |

MSM | Mean square microstrain |

A | Zener elastic anisotropy |

G | Shear modulus |

β | Integral Breadth o FWHM |

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**Figure 1.**Representation of the stacking fault sequence in a f.c.c. structure. (

**a**) represent the sequence for a f.c.c. structure, (

**b**) h.c.p. structure, and (

**c**) f.c.c. to h.c.p. and twin.

**Figure 2.**Schematic representation of the plastic deformation mechanisms in austenitic manganese steels.

**Figure 5.**XRD for Hadfield commercial alloy. 2θc is the diffraction angle with maximum intensity. wL and wG are the Lorentzian and Gaussian breadth with respective errors. XRD extract from [79].

**Figure 6.**Effect of the variation in the elastic constants ${C}_{11}$ and ${C}_{12}$ for limit values of ${C}_{44}$ on the SFE for Hadfield steel.

**Figure 8.**Variation in the elastic constants ${C}_{11}$ and ${C}_{12}$ for limit values of ${C}_{44}$ and the effect on the SFE for (

**a**) Fe-22Mn-0.9C-0Al, (

**b**) Fe-22Mn-0.9C-3Al and (

**c**) Fe-22Mn-0.9C-8Al.

Indices of Reflection [H K L] | ${{\displaystyle \sum}}_{\mathit{b}}\left(\text{}\pm \text{}\right){\mathit{L}}_{0}/{\mathit{h}}_{0}^{2}\left(\mathit{u}+\mathit{b}\right)$ |
---|---|

1 1 0 | $1/4$ |

2 0 0 | $-1/2$ |

2 2 0 | $1/4$ |

3 1 1 | $-1/11$ |

2 2 2 | $-1/8$ |

4 0 0 | $1/4$ |

Alloy | Fe (% wt) | Mn (% wt) | Al (% wt) | C (% wt) |
---|---|---|---|---|

Fe-22Mn-0.9C-0Al | Balance | 20.5 | 0 | 0.87 |

Fe-22Mn-0.9C-3Al | Balance | 22.2 | 3.5 | 0.84 |

Fe-22Mn-0.9C-8Al | Balance | 22.1 | 8.3 | 0.89 |

Reference | Composition of Alloys (wt. pc) | Methodology | C_{11} [GPa] | C_{12} [GPa] | C_{44} [GPa] | Determined SFE of the Hadfield Using These Elastic Constants (mJ/m^{2}) |
---|---|---|---|---|---|---|

Music, et al. [83] | Fe-10Mn | ab initio | 210 | 153 | 135 | 20.53 |

Bampton, et al. [84] | Fe-18Cr-12N-3Mo | Crystal Grown | 235 | 138.5 | 117 | 29.2 |

Endoh, et al. [85] | Fe-30Mn | Atomic Force | 200 ± 9 | 127 ± 6 | 130 ± 3 | 24.1 ± 0.9 |

Gebhardt, Music, Kossmann, Ekholm, Abrikosov, Vitos and Schneider [73] | Fe-25Mn-2Al | ab initio | 153.6 | 105 | 135.5 | 18.5 |

Pierce, Nowag, Montagne, Jiménez, Wittig and Ghisleni [24] | Fe-18Mn-1.5Al-0.6C | Nanoindentation | 169 ± 6 | 82 ± 3 | 96 ± 4 | 26.9 ± 1 |

Lenkkeri [86] | Fe-38.5Mn | Ultrasound | 169.2 | 97.7 | 140.1 | 25.9 |

Cankurtaran, Saunders, Ray, Wang, Kawald, Pelzl and Bach [77] | Fe-40Mn | Ultrasound | 170 | 98 | 141 | 24.27 |

Stinville, et al. [87] | 316L | Nanoindentation | 196 | 129 | 116 | 21.9 |

Pierce, Nowag, Montagne, Jiménez, Wittig and Ghisleni [24] | Fe-22Mn-3Al-3Si | Nanoindentation | 175 ± 7 | 83 ± 3 | 97 ± 4 | 27.3 ± 1.1 |

**Table 4.**Values of the Rietveld refinement parameters where a is the lattice parameter, Vol is the crystal volume, X

^{2}is the chi square, and F

^{2}(R) is the difference between the theoretical and experimental intensities.

Alloy | Phase | $\mathbf{a}\text{}\left[\mathbf{\u212b}\right]$ ± 0.005 | $\mathbf{Vol}\text{}\left[{\mathbf{\u212b}}^{3}\right]\text{}\pm \text{}0.6$ | X^{2} | F^{2}(R) |
---|---|---|---|---|---|

Fe-22Mn-0.9C-0Al | $\gamma $ | 3.627 | 47.713 | 5.8 | 0.0431 |

Fe-22Mn-0.9C-3Al | $\gamma $ | 3.634 | 47.990 | 3.9 | 0.0383 |

Fe-22Mn-0.9C-8Al | $\gamma $ | 3.671 | 49.471 | 5.2 | 0.0523 |

Alloy | SFPx104 | ${\mathit{\epsilon}}^{2}\left(\mathit{L}\right)$ | SFE * (mJ/m ^{2}) | SFE ** (mJ/m ^{2}) |
---|---|---|---|---|

Fe-22Mn-0.9C-0Al | 9.62 ± 2.68 | 8.92 | 17.53 ± 2.47 | 10.99 |

Fe-22Mn-0.9C-3Al | 6.52 ± 2.96 | 13.56 | 35.61 ± 4.76 | 33.42 |

Fe-22Mn-0.9C-8Al | 7.48 ± 3.24 | 21.86 | 50.76 ± 6.73 | 53.35 |

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**MDPI and ACS Style**

Castañeda, J.A.; Zambrano, O.A.; Alcázar, G.A.; Rodríguez, S.A.; Coronado, J.J. Stacking Fault Energy Determination in Fe-Mn-Al-C Austenitic Steels by X-ray Diffraction. *Metals* **2021**, *11*, 1701.
https://doi.org/10.3390/met11111701

**AMA Style**

Castañeda JA, Zambrano OA, Alcázar GA, Rodríguez SA, Coronado JJ. Stacking Fault Energy Determination in Fe-Mn-Al-C Austenitic Steels by X-ray Diffraction. *Metals*. 2021; 11(11):1701.
https://doi.org/10.3390/met11111701

**Chicago/Turabian Style**

Castañeda, Jaime A., Oscar A. Zambrano, Germán A. Alcázar, Sara A. Rodríguez, and John J. Coronado. 2021. "Stacking Fault Energy Determination in Fe-Mn-Al-C Austenitic Steels by X-ray Diffraction" *Metals* 11, no. 11: 1701.
https://doi.org/10.3390/met11111701