Vibrational properties of alpha- and sigma-phase Fe-Cr alloy
S. M. Dubiel, J. Cieslak, W. Sturhahn, M. Sternik, P. Piekarz, S. Stankov, K. Parlinski
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Vibrational properties of alpha- and sigma-phase Fe-Cr alloy
S. M. Dubiel ∗ and J. Cieslak Faculty of Physics and Applied Computer Science,AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland
W. Sturhahn
Advanced Photon Source, Argonne National Laboratory,9700 South Cass Ave, Argonne Illinois 60439, USA
M. Sternik, P. Piekarz, and K. Parlinski
Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Krakow, Poland
S. Stankov
European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble,France. Present address: Karlsruher Institute of Technology, DE-76344 Eggenstein-Leopoldshafen, Germany (Dated: August 27, 2018)Experimental investigation as well as theoretical calculations, of the Fe-partial phonon density-of-states (DOS) for nominally Fe . Cr . alloy having (a) α - and (b) σ -phase structure were carriedout. The former at sector 3-ID of the Advanced Photon Source, using the method of nuclear resonantinelastic X-ray scattering, and the latter with the direct method [K. Parlinski et al., Phys. Rev.Lett. , 4063 (1997)]. The characteristic features of phonon DOS, which differentiate one phasefrom the other, were revealed and successfully reproduced by the theory. Various data pertinentto the dynamics such as Lamb-M¨ossbauer factor, f , kinetic energy per atom, E k , and the meanforce constant, D , were directly derived from the experiment and the theoretical calculations, whilevibrational specific heat at constant volume, C V , and vibrational entropy, S were calculated usingthe Fe-partial DOS. Using the values of f and C V , we determined values for Debye temperatures,Θ D . An excellent agreement for some quantities derived from experiment and first-principles theory,like C V and quite good one for others like D and S was obtained. For many years Fe-Cr alloy system has been of excep-tional scientific and technological interest. The formerstems on one hand from its interesting physical proper-ties such as magnetic ones, and on the other hand fromthe fact that it forms a solid solution within the wholeconcentration range preserving, at least metastably, thesame crystallographic structure (bcc). This, in turn,gives a unique chance for investigating the influence ofthe composition on various physical properties withinthe same structure as well as adequatly testing differ-ent theoretical models and theories. The latter followsfrom the fact that Fe-Cr alloys constitute the basic in-gredient of stainless steels (SS) that for a century havebeen one of the most important structural materials [1],and, consequently, some properties of SS are inheritedfrom the parent alloy. To the latter belongs a σ -phasethat can precipitate if a quasi-equiatomic Fe-Cr alloyundergoes an isothermal annealing in the temperaturerange ∼ ≤ T ≤∼ σ -phase has a tetrag-onal structure (type D h P4 /mnm) with 30 atoms dis-tributed over five different sites (Table 1). Its physicalproperties are, in general, quite different than those of the α -phase of similar composition. Some properties, like themagnetic ones, are even dramatically different [2], otherproperties, like the Debye temperature, seem to be verysimilar [3]. The latter is rather unexpected as the hard-ness of the σ -phase is by a factor of ∼ α -phase. To clarify this situation a more de-tailed knowledge of vibrational properties of the σ - and α -phases is essential. In addition, the σ -phase belongs toan important family of tetrahedrally close-packed Frank-Kasper phases and is one of the closest low-order crys-talline approximants for dodecagonal quasicrystals whichhave similar local structural properties with the icosahe-dral glass (ICG) [4]. The latter implies that the studyof the vibrational properties of the σ -phase should shedsome light on similar properties in ICGs. Challengedby this possibility and motivated by a lack of availableknowledge on dynamical properties of the real σ -phase wehave carried out both, an experimental investigation aswell as theoretical calculations, of the Fe-partial phonondensity-of-states (DOS) for nominally Fe . Cr . alloyhaving (a) α - and (b) σ -phase structure.The master alloy ( α -phase) was prepared by melting,in appropriate proportion, Fe- enriched ( ∼ µm from which two 5 × plates were cut out. They were next solution- treatedat 1273 K for 72 h. One of the samples was afterwardstransformed into the σ -phase by the isothermal anneal-ing at 973 K for 7 days. The verification of the transfor-mation into the σ -phase caused by such thermal proce-dure was done by recording a Fe M¨ossbauer spectrumat 295 K. Experiments were conducted at sector 3-ID of
TABLE I: Atomic crystallographic positions and numbers ofNN atoms for the five lattice sites of the Fe-Cr σ -phase.Site Crystallographic positions NNA B C D E TotalA 2i (0, 0, 0 ) - 4 - 4 4 12B 4f (0.4, 0.4, 0 ) 2 1 2 4 6 15C 8i (0.74, 0.66, 0 ) - 1 5 4 4 14D 8i (0.464, 0.131, 0 ) 1 2 4 1 4 12E 8j (0.183, 0.183, 0.252) 1 3 4 4 2 14 the Advanced Photon Source. The vibrational propertiessuch as the Fe-partial phonon density-of-states were stud-ied using the method of nuclear resonant inelastic X-rayscattering (NRIXS) [5, 6]. Synchrotron radiation X-rayswere monochromatized to a bandwidth of 1.2 meV andtuned in energy ranges of +/- 80 meV (room temperaturemeasurements) and -20 meV to +80 meV (measurementat 20 K) around the Fe nuclear transition energy of14.4125 keV. The X-ray flux and beam size at the sam-ple position were 4 × photons/s and 0.3 × , re-spectively. Data collection times were about 3 hours forthe room temperature measurement of each sample andabout 12 hours for the low temperature measurement ofthe σ -phase sample. We followed previously describedevaluation procedures [5, 6] using the publicly availablePHOENIX software [7]. The following quantities werederived directly from the data: Lamb-M¨ossbauer factor, f , kinetic energy per atom, E k , and the mean force con-stant, D . No specific assumptions about the characterof the vibrations had to be made to obtain these values.The Fe-partial DOS was derived by direct data inversionusing the Fourier-Log method under the assumption ofquasi- harmonic vibrations. The consistency of this pro-cedure was verified by independent calculation of Lamb-M¨ossbauer factor, kinetic energy per atom, and meanforce constant from the DOS and by agreement of thesevalues with same quantities obtained directly from thedata. Then the following quantities were calculated usingthe Fe-partial DOS: vibrational specific heat at constantvolume, C V , and vibrational entropy, S . The assignmentof Debye temperatures, Θ D , is based on the Debye model,i.e., the DOS is proportional to energy squared, and theyare widely used in the literature. With the determina-tion of the Fe-partial DOS, we have surpassed the Debyemodel but find it useful to provide Debye temperaturesfor comparison. Using the values of f and C V , we deter-mined commonly presented values for Θ D .In calculations, both phases of Fe-Cr alloy were mod-eled by the appropriate atomic configurations placed ina supercell with the periodic boundary conditions. Thedisordered α -Fe . Cr . alloy was approximated by the α -Fe Cr one, for which we used the 2 × × σ -Fe . Cr . sample wasapproximated by a σ -Fe . Cr . one. The latter wasstudied in the 1 × × d s and d s for the Cr and Fe atoms,respectively) are represented by plane wave expansions.The wave functions in the core region are evaluated us-ing the full-potential projector augmented-wave (PAW)method [10, 11]. The integrations in the reciprocal spacewere performed on the 8 × × × × α - and σ -phase, respectively. During the optimization,the Hellmann-Feynman (H-F) forces and the stress ten-sor were calculated and the structure optimization wasperformed in two steps. First the lattice constants weredetermined assuming the appropriate symmetry, then theatomic positions were found in a fixed unit cell. Thecrystal structure optimization was finished when resid-ual forces were less than 10 − eV/˚Aand stresses wereless than 0.1 kbar. The calculated lattice constants are a = 5 . a = 8 . c = 4 . µ B , inthe α and σ phase, respectively. On the Cr atoms, theantiparallel arrangement occurs with the negative meanmagnetic moments -0.17 and -0.28 µ B in the respectivephases.For the optimized structures the phonon dispersionsand density of states were calculated using the directmethod [12, 13]. The dynamical matrix of the crystalis constructed from the H-F forces generated while dis-placing atoms from their equilibrium positions. For con-sidered structures each atom must be displaced in threedirections. For the α and σ -phases, a complete set of H-F forces is obtained from 48 and 90 independent atomicdisplacements, respectively. The amplitude of the dis-placements equals 0.03 ˚A. To minimize systematic errorswe applied displacements in positive and negative direc-tions. Finally, the phonon frequencies are obtained by thediagonalization of the dynamical matrix for each wavevector. The phonon DOS is calculated by the randomsampling on the k-point grid in the reciprocal space, andthen the thermodynamic functions are obtained withinthe harmonic approximation.The phonon DOS’s measured on Fe for both phasesof the Fe-Cr alloy are presented in Figure 1. The dif-ferences in the energy range covered by the spectrumand the discrete structure of the spectrum are signifi-cant. The spectrum obtained for the α -phase is found tobe similar to that of pure Fe [14] exibiting a distinct peak FIG. 1: (Color online) Phonon DOS as measured on Featoms at 298 K for the α - (circles) and the σ -phase (triangles)on the Fe . Cr . samples. at 36 meV. The Fe-partial DOS spectrum of the σ -phasedemonstrates the additional high-frequency peak at 40meV, that is not observed neither in α -FeCr nor in purebcc-Fe. There is also the shift downward of the low-energy peak, which makes the entire spectrum broaderthan in the α -phase. In Figure 2 the measured and calcu-lated phonon DOS spectra of α -FeCr are compared. Theshape of both spectra is similar and two characteristicpeaks of measured spectrum are reproduced satisfacto-rily. As in the chosen supercell, there are eight Fe atoms,hence by taking into account five different atomic config-urations, one considers vibrations of forty independentFe atoms. Their partial DOS turned out to be different,but their shape was not correlated with the particularnearest-neighbor (NN) - next NN shell. Consequently,the final DOS was calculated using the partial contribu-tions with the same weights. The discrepencies betweenmeasured and calculated spectrum are likely caused byan incomplete representation of possible atomic configu-ration of FeCr disordered alloy in our model.Likewise, the calculations of σ -FeCr performed for onlyone configuration yield the phonon DOS exhibiting char-acteristic features of the experimental spectrum (Figure3). Observed discrepancies, like underestimation of theintensity of low-frequency peak or shift of high-frequencypeak, are not significant. Using the theoretical result, itis feasible to separate the individual contributions to thetotal DOS generated by the Fe atom placed at each par-ticular crystallographic position.Thus, we see that the Fe atoms on sites A and C arecausing the high-energy contributions to the DOS.TheNN sites of these Fe atoms are placed at distances shorterthan 2.48˚A. In pure bcc-Fe, all 8 NN atoms are situatedat the same distance of 2.485˚A. The shorter distancesbetween atoms result in the larger interatomic interation.Therefore, in σ -FeCr, the phonon frequencies higher than40 meV, which is a high-frequency limit of bcc-Fe DOS FIG. 2: (Color online) Phonon DOS as measured on Featoms at 298 K (circles) and as calculated (dotted line) forthe α -Fe . Cr . . spectrum, are observed. Also due to larger dispersion ofdistances in the σ -phase, its phonon spectrum is muchbroader than in the α -phase, where the atomic positionsare very close to those of the pure bcc-Fe. FIG. 3: (Color online) Phonon DOS as measured on Featoms at 298 K on the σ - Fe . Cr . sample (triangles) andas calculated (dotted line). DOS-curves for particular crys-tallographic sites are indicated, too. The data derived from the experiment and the theoret-ical calculations are displayed in Table I. One can see avery good agreement for some quantities like specific heatand quite good for other ones like force constant and en-tropy. The values of the Debye temperature derived fromthe specific heat are close to those calculated from theLamb-M¨ossbauer factor, and there is a small differencefor this quantity found for different phases. These find-ings disagree with experimentally found Θ D -values forthe two phases using second-order Doppler shifts fromM¨ossbauer spectroscopy [3]. In those studies, the Θ D -values for the α -phase were larger than the ones for the σ -phase. This apparent discrepancy is explained by the TABLE II: Physical quantities derived from the measured andcalculated Fe-partial DOS. The units are meV/atom for vi-brational kinetic energy, E k ; k B /atom for vibrational entropy, S , and specific heat C V ; N/m for mean force constant, D ; Kfor Debye temperatures, Θ D .experiment theoryquantity α σ α σf @298K 0.782 ± .
001 0.768 ± . E k @298K 42.4 ± . ± . D @298K 156 ± ± C V @298K 2.747 ± .
006 2.748 ± .
007 2.746 2.752 S @298K 3.252 ± .
006 3.347 ± .
007 3.354 3.412 f @20K 0.9150 ± . E k @20K 18.93 ± . D @20K 155.1 ± . f @0K 0.9194 ± . ± . E k @0K 19.32 ± .
07 18.95 ± . D ( f @298K) 417 398Θ D ( C V ) 399 398Θ D ( f @20K) 387Θ D ( f @0K) 403 385 fact that the center shift measured by M¨ossbauer spec-troscopy constitutes a sum of the chemical isomer shift,which is independent on the atomic motion, and thesecond-order Doppler shift, which is a relativistic cor-rection to the atomic energy levels purely due to mo-tion. The second-order Doppler shift is proportional tothe vibrational kinetic energy of the Fe atom [15], i.e.SOD[mm/s] = -0.00565 × E k [meV]. Only under the as-sumption that the chemical isomer shift is temperatureindependent, the M¨ossbauer measurement can providethe correct Debye temperature. This does not seem to bethe case here, and our NRIXS data present a notable im-provement in the understanding of the role of vibrationsin the Fe-Cr system. The value of the vibrational entropymeasured at 298 K for the α -phase agrees quite well withinelastic neutron scattering results on samples of similarcomposition [14]. Also the difference in the entropy val-ues, ∆ S = S σ − S α = 0 . ± . k B as determined inthe present experiment from the Fe-partial DOS agreeswell with the corresponding difference calculated fromthe equation ∆ S = 3 k B ln(Θ Dσ / Θ Dα ) = 0 . k B , where k B is the Boltzmann constant and Θ Di is the Debye tem-perature as determined from the second-order Dopplershift for the α ( i = α ) and the σ ( i = σ ) phase, respec-tively [3]. The corresponding theoretical value of ∆ S isequal to 0.058 k B . Taking into account the approxima-tions used in the calculations, the agreement seems to bequite satisfactory.In summary, we have revealed, both experimentallyand theoretically, significant differences in the partial-Fephonon DOS of the α and σ phases of a quasi-equiatomic Fe-Cr alloy. Although, the sigma phase is a very complexobject due to a high number of atoms per unit cell, fivedifferent sublattices with high coordination numbers (12-15), each showing chemical disorder, which altogetherresults in a huge number of possible atomic configura-tions, it was described reasonable well in terms of onlyone adequately chosen configuration. From the calcula-tions, it is also evident that the dynamics in particularsublattices is different. The method was also successfullyused to calculate the dynamics of the disordered alloy inthe alpha-phase. Although its crystallographic structureis much simpler, but, due to the chemical disorder, thenumber of possible atomic configurations is high, makingthe calculations not trivial. We have also obtained rele-vant thermodynamic quantities without necessity of us-ing empirical parameters. Such a complex alloy has beenstudied for the first time within the combined NRIXSand theoretical ab initio approach, and it may provideunderstanding of lattice dynamics in a wide variety ofdisordered systems.The results reported in this study were partly obtainedwithin the project supported by the Ministry of Sci-ence and Higher Education, Warsaw (grant No. N N202228837) and the Project no. 44/N-COST/2007/0. Useof the Advanced Photon Source was supported by theU. S. Department of Energy, Office of Science, Office ofBasic Energy Sciences, under Contract No. DE-AC02-06CH11357. ∗ Corresponding author: [email protected][1] K. H. Lo, C. H. Shek and J. K. L. Lai, Materials Sci.Eng. R: Reports , 39 (2009).[2] J. Cieslak, M. Reissner, W. Steiner and S. M. Dubiel,Phys. Stat. Sol. , 1794 (2008).[3] J. Cieslak, B. F. O. Costa and S. M. Dubiel, J. Phys.:Condens. Matter , 10899 (2006).[4] S. I. Simdyankin, S. N. Taraskin, M. Dzugutov, andS. R. Elliott, Phys. Rev. B , 3223 (2000).[5] W. Sturhahn, T. S. Toellner, E. E. Alp, X. Zhang,M. Ando, Y. Yoda, S. Kikuta, M. Seto, C. W. Kimballand B. Dabrowski, Phys. Rev. Lett. , 3832 (1995).[6] W. Sturhahn, J. Phys.: Condens. Matt. , S497 (2004).[7] W. Sturhahn, Hyperfine Interact. , 149 (2000).[8] G. Kresse, and J. Furtm¨uller, Phys. Rev. B , 11169(1996).[9] G. Kresse, and J. Furtm¨uller, Comput. Mater. Sci. , 15(1996).[10] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[11] G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999).[12] K. Parlinski, Z. Q. Li and Y. Kawazoe, Phys. Rev. Lett. , 4063 (1997).[13] K. Parlinski, Software PHONON, Krakow, 2007.[14] M. S. Lucas, M. Kresch, R. Stevens and B. Fultz, Phys.Rev. B , 184303 (2008).[15] W. Sturhahn, and A. I. Chumakov, Hyperfine Interact.123/124