Vibrational properties of alpha- and sigma-phase Fe-Cr alloy.
ABSTRACT Experimental and theoretical studies, of the Fe-partial phonon density of states (PDOS) for Fe52.5Cr47.5 alloy having alpha and sigma phases were carried out. The former using the nuclear resonant inelastic x-ray scattering method, and the latter with the direct one. Characteristic features of PDOS, which distinguish one phase from the other, were revealed and successfully reproduced by the theory. Data pertinent to the dynamics such as the Lamb-Mössbauer factor, f, the kinetic energy per atom, E(k), and the mean force constant, D, were directly derived, while vibrational specific heat at constant volume, C(V), and vibrational entropy, S were calculated using the Fe partial PDOS. Based on the values of f and C(V), we determined Debye temperatures, Theta(D). An excellent agreement for some quantities derived from experiment and first-principles theory, like C(V) and quite good ones for others like D and S were obtained.
- Journal of Materials Science 11/2012; 47(21):7385-7398. · 2.31 Impact Factor
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ABSTRACT: In steels and single-crystal superalloys the control of the formation of topologically close-packed (TCP) phases is critical for the performance of the material. The structural stability of TCP phases in multi-component transition-metal alloys may be rationalized in terms of the average valence-electron count N and the composition-dependent relative volume-difference dV/V. We elucidate the interplay of these factors by comparing density-functional theory calculations to an empirical structure map based on experimental data. In particular, we calculate the heat of formation for the TCP phases A15, C14, C15, C36, χ, μ and σ for all possible binary occupations of the Wyckoff positions. We discuss the isovalent systems V/Nb–Ta to highlight the role of atomic-size difference and observe the expected stabilization of C14/C15/C36/μ by dV/V at N=0 in V–Ta. In the systems V/Nb–Re, we focus on the well-known trend of A15→σ→χ stability with increasing N and show that the influence of dV/V is too weak to stabilize C14/C15/C36/μ in Nb–Re. As an example for a significant influence of both N and dV/V, we also consider the systems Cr/Mo–Co. Here the sequence A15→σ→χ is observed in both systems but in Mo–Co the large size-mismatch stabilizes C14/C15/C36/μ. We also include V/Nb–Co that cover the entire valence range of TCP stability and also show the stabilization of C14/C15/C36/μ. Moreover, the combination of a large volume-difference with a large mismatch in valence-electron count reduces the stability of the A15/σ/χ phases in Nb–Co as compared to V–Co. By comparison to non-magnetic calculations we also find that magnetism is of minor importance for the structural stability of TCP phases in Cr/Mo–Co and in V/Nb–Co.New Journal of Physics 11/2013; 15:115016. · 4.06 Impact Factor
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ABSTRACT: Using density-functional theory in combination with the direct force method and molecular dynamics we investigate the vibrational properties of a binary Cr-Re σ-phase. In the harmonic approximation, we have computed phonon dispersion curves and density of states, evidencing structural and chemical effects. We found that the σ-phase is mechanically unstable in some configurations, for example, when all crystallographic sites are occupied by Re atoms. By using a molecular-dynamics-based method, we have analysed the anharmonicity in the system and found negligible effects (∼0.5 kJ/mol) on the Helmholtz energy of the binary Cr-Re σ-phase up to 2000 K (∼0.8Tm). Finally, we show that the vibrational contribution has significant consequences on the disordering of the σ-phase at high temperature.The Journal of Chemical Physics 04/2014; 140(14):144502. · 3.12 Impact Factor
Vibrational Properties of ?- and ?-Phase Fe-Cr Alloy
S.M. Dubiel,1,*J. Cieslak,1W. Sturhahn,2M. Sternik,3P. Piekarz,3S. Stankov,4and K. Parlinski3
1Faculty of Physics and Applied Computer Science, AGH University of Science and Technology,
aleja Mickiewicza 30, 30-059 Krakow, Poland
2Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne Illinois 60439, USA
3Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Krakow, Poland
4Institute for Synchrotron Radiation, Karlsruher Institute of Technology, DE-76344 Eggenstein-Leopoldshafen, Germany
(Received 18 December 2009; published 16 April 2010)
Experimental and theoretical studies, of the Fe-partial phonon density of states (PDOS) for Fe52:5Cr47:5
alloy having ? and ? phases were carried out. The former using the nuclear resonant inelastic x-ray
scattering method, and the latter with the direct one. Characteristic features of PDOS, which distinguish
one phase from the other, were revealed and successfully reproduced by the theory. Data pertinent to the
dynamics such as the Lamb-Mo ¨ssbauer factor, f, the kinetic energy per atom, Ek, and the mean force
constant, D, were directly derived, while vibrational specific heat at constant volume, CV, and vibrational
entropy, S were calculated using the Fe partial PDOS. Based on the values of f and CV, we determined
Debye temperatures, ?D. An excellent agreement for some quantities derived from experiment and first-
principles theory, like CVand quite good ones for others like D and S were obtained.
DOI: 10.1103/PhysRevLett.104.155503PACS numbers: 63.20.dd, 63.20.dh, 63.20.dk, 63.50.Gh
For many years, the Fe-Cr alloy system has been of
exceptional scientific and technological interest. The alloy
shows interesting physical properties such as magnetism
and forms a solid solution within the whole concentration
range preserving, at least metastable, the same crystallo-
graphic structure (bcc). This, in turn, gives a unique chance
to investigate the effect of the composition on various
physical properties within the same structure as well as
to adequately test different theoretical models and theories.
In fact, Fe-Cr alloys constitute the basic ingredient of
stainless steels that for a century have been one of the
most important structural materials , and, hence, some
properties of stainless steels are inherited from the parent
alloy. A ? phase can precipitate if a quasiequiatomic Fe-Cr
alloy undergoes an isothermal annealingin the temperature
range of ?800 K ? T ? ?1100 K. It has a tetragonal
structure (type D14
4hP42=mnm) with 30 atoms distributed
over five different sites (Table I). Its physical properties
are, in general, quite different than those of the ? phase of
similar composition. Some experimentally found behavior,
such as magnetic properties, is even dramatically different
. Other properties, such as the Debye temperature, ?D,
seem to be very similar . The similarity of ?Dvalues is
rather unexpected because the hardness of the ? phase is
larger by a factor of ?3 than that of the ? phase. To clarify
this point, a more detailed knowledge of vibrational prop-
erties of the ? and ? phases is essential. In addition, the ?
phase belongs to an important family of tetrahedrally
close-packed Frank-Kasper phases and is one of the closest
low-order crystalline approximants for dodecagonal qua-
sicrystals which have similar local structural properties
with the icosahedral glass . The study of the vibrational
properties of the ? phase therefore should shed some light
on similar properties in icosahedral glass. Challenged by
this possibility and motivated by a lack of available knowl-
edge on dynamical properties of the real ? phase we have
carried out both an experimental investigation as well as
theoretical calculations of the Fe partial phonon density of
The master alloy (? phase) was prepared by melting, in
appropriate proportion,57Fe-enriched (?95%) iron with
chromium. The ingot was then cold rolled down to a
thickness of about 30 ?m from which two 5 ? 5 mm2
plates were cut out. The plates were then solution treated
at 1273 K for 72 h. Afterwards one of the samples was
transformed into the ? phase by the isothermal annealing
at 973 K for 7 days. The change into the ? phase was
verified by recording a57Fe Mo ¨ssbauer spectrum at 295 K.
Experiments were conducted at sector 3-ID of the
Advanced Photon Source [5,6]. The vibrational properties
such as the Fe partial phonon DOS were studied using the
method of nuclear resonant inelastic x-ray scattering
(NRIXS) [6,7]. Synchrotron radiation x-rays were mono-
chromatized to a bandwidth of 1.2 meV and tuned in
sublatticies, NFe, and numbers of NN atoms for the five lattice
sites of the Fe-Cr ? phase.
Atomic crystallographic positions, Fe occupancy of
Site Crystallographic positions NFe
ABC D E Total
2i (0, 0, 0 )
4f (0.4, 0.4, 0 )
8i (0.74, 0.66, 0 )
8i (0.464, 0.131, 0 )
8j (0.183, 0.183, 0.252)
??? 4 ??? 4 4
PRL 104, 155503 (2010)
16 APRIL 2010
? 2010 The American Physical Society
energy ranges of ?80 meV (measurements at 298 K) and
?20 to þ80 meV (measurement at 20 K) around the57Fe
nuclear transition energy of 14.4125 keV. The x-ray flux
and beam size at the sample position were 4 ?
109photons=s and 0:3 ? 2 mm2, respectively. Data col-
lection times were about 3 h for the 298 K measurement of
each sample and about 12 h for the 20 K measurement of
the ?-phase sample. We followed previously described
evaluation procedures [6,7] using the publicly available
PHOENIX software . The following quantities were de-
rived directly from the data: Lamb-Mo ¨ssbauer factor, f,
kinetic energy per atom, Ek, and the mean force constant,
D. No specific assumptions about the character of the
vibrations had to be made to get these values. The Fe
partial DOS was derived by direct data inversion using
the Fourier-Log method under the assumption of quasihar-
monic vibrations. The consistency of this procedure was
verified by independent calculation of f, Ek, and D from
the DOS and by agreement of these values with the same
quantities obtained directly from the data. The calculation
of the three quantities from moments or DOS utilize differ-
ent energy regions of the data. Hence the agreement pro-
vides sufficient confidence in our choice of energy range
and in the reliability of the values within the statistical
errors of the data. The following quantities were calculated
using the Fe partial DOS: vibrational specific heat at
constant volume, CV, vibrational entropy, S, and Debye
sound velocity, vD. The Debye sound velocity was derived
from the low-energy part of the DOS, which approximates
parabolic shape , using a refined extrapolation proce-
dure . The assignment of Debye temperatures is based
on the Debye model, i.e., the DOS is proportionaltoenergy
squared, and theyarewidely used in the literature. With the
determination of the Fe partial DOS, we have surpassed the
Debye model but find it useful to provide ?Dfor compari-
son. Using the values of f and CV, we determined com-
monly presented values for ?D.
In the calculations, both phases of Fe-Cr alloy were
modeled by the appropriate atomic configurations placed
in a supercell with periodic boundary conditions. The
disordered ?-Fe52:5Cr47:5alloy was approximated by the
?-Fe50Cr50composition, for which we used a 2 ? 2 ? 2
bcc cell with 16 atoms. For random distribution of atoms,
about 500 different atomic configurations should be con-
sidered inprinciple. However,limitedcomputingresources
resulted in the inclusion of just five configurations that
were chosen randomly. The real ?-Fe52:5Cr47:5sample
was approximated by a ?-Fe53:3Cr46:7composition which
was studied using a 1 ? 1 ? 1 tetragonal supercell with 30
atoms (16 Fe and 14 Cr atoms, the distribution of Fe over
the sites is given in Table I). Structure optimization was
achieved by spin-polarized density functional total energy
calculations performed within the generalized gradient
approximation using the VASP package . The valence
electrons for each atom (electron configuration: d5s1and
d6s2for the Cr and Fe atoms, respectively) are represented
by plane wave expansions. The wave functions in the core
region are evaluated using the full-potential projector
augmented-wave method [12,13]. Integrations in recipro-
cal space were performed on a 8 ? 8 ? 8 and 4 ? 4 ? 4
grid for the ? and ? phase, respectively. Structure optimi-
zation was performed as follows. The lattice constants
were determined assuming the appropriate symmetry,
then the atomic positions were found in the fixed-size
unit cell. This procedure was repeated until residual forces
were less than 10?5eV=? A and stresses were less than
0.1 kbar. The calculated lattice constants are a ¼ 5:64?A
and a ¼ 8:60?A, c ¼ 4:74?A for the cubic and tetragonal
symmetry, respectively. The optimized magnetic moments
of the Fe atoms are ordered ferromagnetically with average
values of 1.88 and 0:96?Bin the ? and ? phase, respec-
tively. The magnetic moments of the Cr atoms are in
antiparallel arrangement with average values of ?0:013
and ?0:28?B, respectively. From these values, average
magnetic moments of 0:38?B for the ? phase and
0:95?Bfor the ? phase were derived. The ?-phase value
compares rather well with results of others for similar
compositions [14,15]. The differences likely follow from
unlike atomic configurations taken into account in these
For the optimized structures, the phonon dispersions and
DOS were calculated with the direct method [16,17]. The
dynamical matrix of the crystal was constructed from the
Hellmann-Feynman forces generated while displacing
atoms from their equilibrium positions. Each atom must
be displaced in three directions, and a complete set of
Hellmann-Feynman forces was obtained from 48 and 90
independent atomic displacements for the ? and ? phase,
respectively. The amplitude of the displacements equals
0.03 A˚. Tominimize systematic errorswe applieddisplace-
ments in positive and negative directions. Finally, the
phonon frequencies were obtained by the diagonalization
of the dynamical matrix for each wave vector. The phonon
DOS was calculated by the random sampling of the k-point
grid in reciprocal space, and then the thermodynamic func-
tions were obtained within the harmonic approximation.
The phonon DOS measured on57Fe for both phases of
the Fe-Cr alloy and are presented in Fig. 1(a). The differ-
ences in the energy range covered by the spectra and their
discrete structure are significant. The DOS obtained for the
? phase is found to be similar to that of pure Fe 
showing a distinct peak at 36 meV. The Fe partial DOS
spectrum of the ? phase has an additional high-frequency
peak at 40 meV that is observed neither in ? Fe-Cr nor in
pure bcc Fe. In addition, the downward shift of the low-
energy peak broadens the entire spectrum compared to the
? phase. In Fig. 1(b) the measured and calculated phonon
DOSof ?Fe-Crare compared. Theshape ofboth spectra is
similar and two characteristic peaks of measured spectrum
are reproduced adequately. The eight Fe atoms of the
chosen supercell in combination with five various atomic
configurations produce 40 independent contributions to the
PRL 104, 155503 (2010)
16 APRIL 2010
DOS which turned out to be different, but their shape was
not correlated with the particular nearest-neighbor (NN) or
next NN shell. Consequently, the final DOS was calculated
using the partial contributions with the same weights. The
discrepancies between measured and calculated spectra are
likely caused by an incomplete representation of possible
atomic configurations of Fe-Cr disorder in our model.
Likewise, the calculations of ? Fe-Cr performed for
only one configuration yield the phonon DOS exhibiting
characteristic features of the experimental spectrum
[Fig. 1(c)]. Observed disaccord, such as underestimation
of the intensity of the low-frequency peak or the shift of the
high-frequency peak, is not significant. The theoretical
result permits to separate individual contributions to the
DOS generated by Fe atoms placed at each particular
crystallographic position. For example, we see that the
Fe atoms on sites A and C are causing the high-energy
contributions to the DOS. The NN sites of these Fe atoms
are placed at distances shorter than 2.48 A˚, whereas in pure
bcc Fe, all 8 NN atoms are situated at the same distance of
2.485 A˚. The shorter distances result in larger interatomic
iteration, and phonon frequencies higher than 40 meVare
observed in ? Fe-Cr but not in bcc Fe. Also the larger
dispersion of distances in the ? phase cause the DOS to
broaden in comparison to the ? phase where the atomic
positions are very close to those of bcc Fe.
The data derived from the experiment and the theoretical
calculations are displayed in Table II. The values of ?D
derived from the specific heat are close to those calculated
from the Lamb-Mo ¨ssbauer factor, and they differ little for
different phases. These findings disagree with ?Dvalues
derived from second-order Doppler shifts of Mo ¨ssbauer
spectroscopy measurements . In those studies, the ?D
values for the ? phase were larger. This apparent discrep-
ancy is explained by the fact that the center shift measured
cal isomer shift, which is independent of the atomic mo-
tion, and the second-order Doppler (SOD) shift, which is a
relativistic correction to the atomic energy levels purely
due to motion. The SOD shift is proportional to the vibra-
tional kinetic energy of the57Fe atom, i.e., SODðmm=sÞ ¼
?0:00565 ? ½EkðmeVÞ? . Only under the assumption
that the chemical isomer shift is temperature independent,
the Mo ¨ssbauer measurement can provide the correct value
of ?D. This does not seem to be the case here, and our
NRIXS data present a notable improvement in the under-
standing of the role of vibrations in the Fe-Cr system.
The Debye sound velocity of the ? phase exceeds the
value for the ? phase by about 2.5% which is mostly due to
the used densities of ??¼ 8 g=cm3and ??¼ 7:68 g=cm3
from the calculations. An average elasticity defined by
Dis virtually identical for the different crystal structures
but about 8% higher than the value for bcc Fe with vD¼
3:49ð5Þ km=s and ? ¼ 8 g=cm3(including enrichment).
This result is consistent with the almost identical ?D
values derived from the specific heat. It also suggests that
for Fe-Cr alloys average elasticity or Debye temperatures
are poor indicators for hardness.
The value of the vibrational entropy measured at 298 K
for the ? phase agrees quite well with inelastic neutron
scattering results on samples of similar composition .
Also the difference in the entropy values, ?S ¼
S?? S?¼ ð0:095 ? 0:009ÞkBas found in the present ex-
periment from the Fe-partial DOS agrees well with the
corresponding difference calculated from the equation
?S ¼ 3kBlnð?D?=?D?Þ ¼ 0:07kB,
Boltzmann constant and ?Diis the Debye temperature as
determined from the SOD shift for the ? (i ¼ ?) and the ?
(i ¼ ?) phase, respectively . The corresponding theo-
retical value of ?Sis equal to 0:058kB. Taking into account
the approximations used in the calculations, the agreement
seems to be quite satisfactory, all the more so the presently
found values of ?S are remarkably close to thevalue of the
difference in entropy for elemental Fe and Cr as found by a
combination of ab initio and CALPHAD approach .
FIG. 1 (color online).
atoms at 298 K for the ? (circles) and the ? phase (triangles) of
the Fe52:5Cr47:5 samples. Maximum error bars are shown;
(b) measured (circles) and calculated (dots) DOS for the
?-Fe52:5Cr47:5; (c) measured (triangles) and calculated (dots)
DOS for the ?-Fe52:5Cr47:5. DOS-curves for particular crystallo-
graphic sites are indicated, too.
(a) Phonon DOS as measured on57Fe
PRL 104, 155503 (2010)
16 APRIL 2010
In summary, we have revealed, both experimentally
and theoretically, significant differences in the partial
Fe phonon DOS of the ? and ? phases of a quasiequi-
atomic Fe-Cr alloy. The ? phase is a very complex struc-
ture due to a high number of atoms per unit cell and five
different sublattices with high coordination numbers (12–
15), each showing chemical disorder, which altogether
results in a huge number of possible atomic configurations.
Nevertheless the DOS was described reasonably well in
terms of only one adequately chosen configuration. From
the calculations, it is also evident that the dynamics, in
particular, sublattices are different. The method was also
successfully used to calculate the dynamics of the disor-
dered alloy in the ? phase. Even though its crystallo-
graphic structure is much simpler, the number of possible
atomic configurations is still high due to the chemical
disorder rendering the calculations nontrivial. We have
also obtained relevant thermodynamic quantities without
necessity of using empirical parameters. Such a complex
alloy has been studied for the first time within the com-
bined NRIXS and theoretical ab initio approach, and it
may provide understanding of lattice dynamics in a wide
variety of disordered systems.
The results reported in this study were partly obtained
within the project supported by the Ministry of Science and
Higher Education, Warsaw (Grant No. N N202 228837)
and Project No. 44/N-COST/2007/0. Use of the Advanced
Photon Source was supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy
Sciences, under Contract No. DE-AC02-06CH11357.
*Corresponding author: email@example.com
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units are meV/atom for vibrational kinetic energy Ek, kB=atom for vibrational entropy S and
specific heat CV, N=m for mean force constant D, km=s for Debye sound velocity vD, K for
Debye temperature ?D.
Physical quantities derived from the measured and calculated Fe partial DOS. The
f at 298 K
Ekat 298 K
D at 298 K
CVat 298 K
S at 298 K
vDat 298 K
f at 20 K
Ekat 20 K
D at 20 K
f at 0 K
Ekat 0 K
0:782 ? 0:001
42:4 ? 0:1
156 ? 1
2:747 ? 0:006
3:252 ? 0:006
3:63 ? 0:02
0:768 ? 0:002
41:4 ? 0:2
157 ? 2
2:748 ? 0:007
3:347 ? 0:007
3:72 ? 0:02
0:9150 ? 0:0002
18:93 ? 0:06
155:1 ? 0:7
3:87 ? 0:04
0:9149 ? 0:0003
18:95 ? 0:06
0:9194 ? 0:0001
19:32 ? 0:07
PRL 104, 155503 (2010)
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