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Electronic structure of a σ-FeCr compound

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IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 20 (2008) 235234 (6pp)

Electronic structure of a

σ-FeCr compound

doi:10.1088/0953-8984/20/23/235234

J Cieslak1,3, J Tobola1, S M Dubiel1, S Kaprzyk1, W Steiner2and

M Reissner2

1Faculty of Physics and Applied Computer Science, AGH University of Science and

Technology, al. Mickiewicza 30, 30-059 Krakow, Poland

2Institute of Solid State Physics, Vienna University of Technology, A-1040 Wien, Austria

E-mail: cieslak@novell.ftj.agh.edu.pl

Received 28 January 2008, in final form 2 April 2008

Published 9 May 2008

Online at stacks.iop.org/JPhysCM/20/235234

Abstract

The electronic structure of a σ-FeCr compound in a paramagnetic state was calculated for the

first time in terms of isomer shifts and quadrupole splittings. The former were calculated using

the charge self-consistent Korringa–Kohn–Rostoker (KKR) Green’s function technique, while

the latter were estimated from an extended point charge model. The calculated quantities

combined with recently measured site occupancies were successfully used to analyze a

M¨ ossbauer spectrum recorded at room temperature using only five fitting parameters namely

background, total intensity, linewidth, IS0 (necessary to adjust the refined spectrum to the used

M¨ ossbauer source) and the QS proportionality factor. Theoretically determined changes of the

isomer shift for the σ-FeCr sample were found to be in line with the corresponding ones

measured on a α-FeCr sample.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The σ-phase belongs to the so-called Frank–Kasper phases,

whose characteristic feature is a complex crystallographic

structure, and, in particular, high coordination numbers.

Consequently, understanding of the physical properties of such

phases has been not straightforward and easy. In particular,

the time that elapsed between its discovery [1] and the

identification of its crystallographic structure [2] was about 30

years. The complexity of the structure results in difficulties

both in the interpretation of the experimental results as well

as in the theoretical calculations. Concerning the latter, only

a few papers have been published on the σ-phase in the Fe–

Cr system, none of them related to its electronic structure.

Nowadays, over 50 examples of binary alloy systems are

known in which the σ-phase is formed [3]. It occurs often in

materials that are technologically important and its presence

drastically deteriorates their mechanical properties.

the basic investigation of structural, electronic and magnetic

properties is of quite general interest.

Hence,

3Author to whom any correspondence should be addressed.

The Fe–Cr σ-phase precipitates from the α-phase

during an isothermal annealing in the temperature range of

∼530◦C < T < ∼830◦C mostly on the grain boundaries

and in the form of sub-micrometer sized needles and/or

lamellae.Its physical properties are substantially different

from those characteristic of the α-phase (bcc), from which it

precipitates [3].

M¨ ossbauer spectroscopy belongs to the most suitable

methods for the investigation of the structural properties

and magnetic behavior of the Fe–Cr σ-phase.

from its high sensitivity to the hyperfine parameters, which

are strongly influenced by the local configuration of nearest

neighbor (NN) atoms of the57Fe atom probe.

M¨ ossbauer spectrum of the Fe–Cr σ-phase must be composed

of at least five subspectra with various intensities related to

the iron occupation of five inequivalent crystallographic sites.

Although the corresponding hyperfine parameters (isomer

shift, IS and quadrupole splitting, QS) differ from each other,

the differences are comparable or smaller than the typical

experimental linewidths. Consequently, the spectrum is not

well resolved, even below the Curie temperature (Tc

30 K in σ-Fe53.8Cr46.2). Some of the parameters describing

the spectrum could be determined in other experiments

This stems

A typical

∼

0953-8984/08/235234+06$30.00

© 2008 IOP Publishing LtdPrinted in the UK

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J. Phys.: Condens. Matter 20 (2008) 235234J Cieslak et al

Figure 1.57Fe M¨ ossbauer spectrum of the σ-Fe53.8Cr46.2alloy

recorded at 300 K (dots) compared to the calculated one (solid line).

The subspectra related to the inequivalent sites are indicated by

colored lines. The spectra recorded at (a) 4.5 K and (b) 35 K are

shown in the insets for comparison.

(e.g. relative subspectra contributions should be proportional

to the Fe concentrations on particular sites determined in

the neutron diffraction experiment), but the rest (IS, QS and

linewidths ?) should be obtained from the fitting procedure.

Unfortunately, the number of known parameters is too small

and decompositionof the overall spectrum is not a unique task,

since one can make several mathematically correct fits with

different sets of fitting parameters. For that reason we decided

to calculate the electronic structure and the resulting hyperfine

parameters (IS, QS), in various atomic configurations of the σ

Fe–Cr system in order to interpret the experimental M¨ ossbauer

spectrum.

The earlier electronic structure computations of the

transition-element σ-phase focused mainly on crystal stability

and site occupancy preference of the constituent atoms on

five inequivalent sites [4].Later, it was reported [5] from

the full potential linearized augmented plane waves (FLAPW)

method that the calculated formation energy in Fe–Cr and Co–

Cr σ-phases remained in reasonable agreement with measured

enthalpies. Finally, the vibrational properties of the Fe–Cr

σ-phase were investigated by means of molecular dynamics

simulation [6], which showed some similarities to vibrational

behaviors of the glassy state.

The aim of our work is to calculate the electronic

structure and resulting hyperfine parameters for the most

plausible atomic configurations of the σ-FeCr system in order

to reconstruct the experimental M¨ ossbauer spectrum in the

paramagnetic state.

2. Experimental details

The procedure of σ-Fe53.8Cr46.2sample preparation is given in

detail elsewhere [7].

in transmission geometry using a standard spectrometer and a

57Co/Rh source for the 14.4 keV gamma rays at temperatures

4.5, 35 and 300 K. They are presented in figure 1. At first

glance, one notices that the spectra measured above the Curie

temperature (35 and 300 K) are quite similar, whereas a small

magnetic splitting can be observed at 4.5 K. In this work we

focus on the paramagnetic state only.

57Fe M¨ ossbauer spectra were recorded

Table 1. Atomic crystallographic positions and numbers of NN

atoms for the five lattice sites of the Fe–Cr σ-phase.

NN

Site

Crystallographic

positionsABCDETotal

A

B

C

D

E

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)

—

2

—

1

1

4

1

1

2

3

—

2

5

4

4

4

4

4

1

4

4

6

4

4

2

12

15

14

12

14

3. Computational details

The charge and spin self-consistent Korringa–Kohn–Rostoker

Green’s function method [8, 9, 11] was used to calculate the

electronic structure ofthe Fe–Cr σ-phase. The crystal potential

of muffin-tin (MT) form was constructed within the local

density approximation (LDA) framework using the Barth–

Hedin formula [10] for the exchange–correlation part. The

group symmetry of the unit cell of the σ-phase (P42/mnm,

#136) was lowered to allow for various configurations of

Fe/Cr atoms. The experimental values of lattice constants [7]

(a = 8.7891 ˚ A, c = 4.5559 ˚ A) and atomic positions

(table 1) were applied in all computations. For fully converged

crystal potentials electronic density of states (DOS), total, site-

composed and l-decomposed DOS (with lmax= 2 for Fe and

Cr atoms) were derived. Fully converged results were obtained

for ∼120 special k-point grids in the irreducible part of the

Brillouin zone but they were also checked for convergence

using a more dense k-mesh. Electronic DOSs were computed

using the tetrahedron k-space integration technique and ∼700

small tetrahedra [12].

3.1. Structural aspects

As aforementioned, the σ-phase has a complex close-packed

tetragonal structure with 30 atoms in the unit cell. Atoms are

distributed over five nonequivalent sites, called A, B, C, D and

E, the population of them is shown in table 1. Each position

can be characterized by:

(i) the total number of nearest neighbors (NN),

(ii) their belonging to one of five sublattices,

(iii) the distances to NN atoms,

(iv) the occupancy by Fe or Cr atoms.

The former three properties (i–iii) can be derived directly

from the space group information and are presented in tables 1

and 2. Each atom in the σ-phase structure is surrounded by

12–15 atoms belonging to various sites (all NN configurations

except for A–A and A–C are possible).

distances range from 2.265 ˚ A (E–E) to 2.922 ˚ A (B–E) (see

table 2 for average values) and can be slightly different even

within the same pairs of atoms (e.g. for C–D three different

valuescan be noticed). Moreover, the NNspatialdistributionis

notfar from spherical for each site. Site-occupation parameters

as found from the neutron diffraction data [7] are presented in

table 3. It clearly shows that all five sites are populated by

both alloy constituting elements. The distribution of Fe and

Their interatomic

2

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J. Phys.: Condens. Matter 20 (2008) 235234J Cieslak et al

Table 2. Distances between NN atoms for five lattice sites of the

Fe–Cr σ-phase. The average values with standard deviations

(in parentheses) are indicated, when more than one distance is

possible. The corresponding weighted mean values (dav) are given

in the last row.

Distances (˚ A)

SiteABCDE

A

B

C

D

E

dav

—

2.605 2.519 2.420

—2.420 2.749(0.196) 2.487(0.004) 2.766(0.006)

2.366 2.695 2.487(0.004) 2.422

2.547 2.864 2.766(0.006) 2.549(0.019) 2.280(0.018)

2.506 2.702 2.655

2.605 —2.366

2.695

2.547

2.864(0.005)

2.549(0.019)

2.5722.640

Cr atoms over these sites is neither random nor fully ordered,

but within each of the five crystallographic positions it remains

random [7]. The electronic structure and hyperfine interactions

of such complex and disordered systems might be investigated

by the well-established coherent potential approximation

(CPA) combined with the KKR method [11]. However, the

KKR-CPA computations of the binary σ-phase are (at the

moment)toocomplicatedand highlytime-consumingto obtain

acceptable results in a reasonable time. Additionally, the CPA

approach tends to average the parameters over all possible

Fe/Cr configurations for the given lattice site. On the other

hand, hyperfine interactions, which are the subject of the

M¨ ossbauer investigations, are mainly sensitive to the local

NN-configuration changes. For that reason we lowered the

symmetry of the unit cell (space group P42/mnm) to the

simple tetragonal one, and the calculations were carried out for

defined atom configurations using the KKR method adapted to

ordered systems. Inpractice, thetetragonalunitcellandatomic

positions were unchanged but variable occupancy made all 30

atomic positions crystallographically nonequivalent. In such a

specified unit cell each of the crystallographic positions was

occupied exclusively either by Fe or Cr atom. However, in our

numerical attempts we were constrained by the experimental

Fe/Cr concentrations on each of the five lattice sites and

the considered composition should be as close as possible

to the measured stoichiometry of the σ-Fe53.8Cr46.2.

concentrations used are given in table 3. For example, the

experimentally observed Fe occupancy of site C (8i) was

found to be P(Fe,C) = 0.413, whereas in our computations,

having eight atoms on the position previously defined as ‘C’,

we had to assume five Cr and three Fe atoms to reach the

closest concentration. It is worth noting that each lattice site

is surrounded by all other crystallographic sites occupied by

either Fe or Cr, which can be distributed in different ways. For

the above-mentionedexample of the C site, one can distinguish

56 possible nonequivalent Fe–Cr arrangements and over 105

possibilities for all 30 atoms in the unit cell.

there is no need to analyze all possible arrangements because

their influence on the hyperfine parameters can be accounted

for by the number of Fe/Cr atoms in the nearest shell of the

specified57Fe atom. Consequently, it was enough to restrict

our calculations to arbitrarily chosen atom configurations

(26 different arrangements), which covered most of all the

possible NN-values for each of the five lattice sites. Indeed,

The

Fortunately,

Table 3. Fe site-occupation parameters of the σ-FeCr alloy,

experimental and assumed for calculations. Ntstands for the

percentage of the total Fe atoms in the site (referred to the 30 atom

unit cell), whereas N describes the number of Fe atoms occupying

the site. Ncalrepresents the value used in the KKR calculations.

The corresponding relative occupancy (percentage) is given in

parentheses. The quadrupole splitting values QS for five lattice

sites as obtained from the fitting procedure are presented in the

last column.

Site

Nt

NNcal

QS (mm s−1)

A

B

C

D

E

11.3

6.4

20.5

44.9

17.0

1.826 (91.3)

1.040 (26.0)

3.304 (41.3)

7.208 (90.1)

2.744 (34.3)

2 (100.0)

1 (25.0)

3 (37.5)

7 (87.5)

3 (37.5)

0.34

0.24

0.18

0.21

0.45

the above-mentioned assumptions were later supported by

almost linear correlations between calculated charge densities

at the Fe nucleus and its Fe-NN numbers for all five lattice

sites, separately. These correlations were used in further

calculations.

3.2. Isomer shift, IS

The isomer shift is proportional to the difference between the

electron densities in the range of the nucleus for source and

absorber atoms:

IS =2π

5ZS(Z)e2R2?R

R

[ρA(0) − ρS(0)]

(1)

where Z is the nuclear charge of the M¨ ossbauer absorber,

S(Z) is a relativistic correction factor, e is the elementary

charge, R is the average radius of the M¨ ossbauer nucleus in the

ground and excited state and the ?R is the difference of these

two radii. ρi(0) is the non-relativistic electron density at the

nucleus for the M¨ ossbauer absorber (i = A) or source (i = S),

which is in practice derived from extrapolation of the electron

charge density to r = 0. Since all factors except the electron

density of the absorber are constant for a given spectrum, it is

sufficient for our purpose to consider a simplified equation for

the IS as follows:

IS = a[ρA(0) − b]

(2)

with a and b constants as determined in the calibration

procedure. According to [13], the value of a may vary from

0.367 to 0.403 (au3mm s−1) (au being atomic units). In our

work the average value of a = 0.385 (au3mm s−1) was used.

Since only differences between isomer shifts for particular

subspectra are important in the fitting procedure, the b-value

was assumed to be equal to 0.

3.3. Quadrupole splitting, QS

The shifts in the energy levels resulting from the interaction

between the nucleus with a quadrupole moment eQ and the

electric field gradient (EFG) are obtained from

?E =

eQVzz

4I(2I − 1)[3m2− I(I + 1)](1 + η2/3)1/2

(3)

3

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J. Phys.: Condens. Matter 20 (2008) 235234 J Cieslak et al

where Vzz= ∂2V/∂z2is the largest component of EFG along

a principal axis and η = (Vxx− Vyy)/Vzzis the asymmetry

parameter. The source of EFG at the nucleus is the character

of the surrounding electron charge distribution. There are two

principal classifications of this external charge:

(a) the electrons directly associated with the Fe nucleus, the

so-called valence contribution Vval

(b) the charges of other atoms in the lattice, the so-called

lattice contribution Vlat

zz,

zz.

Hence, the EFG tensor depends on both components [14].

In the most widely used computational approach, charges

external tothecentral atom are treated as pointcharges. Hence,

the Vij-values can beeasily calculated according totheformula

?

where x1 = x, x2 = y, x3 = z, ei is the electronic

charge on neighboring atom i and δij denotes the Kronecker

symbol. In thiswork the procedure for deriving the EFG tensor

was extended to the electron density distributed around each

atom, which goes beyond the point charge approximation. In

practice, the electron cloud inside each MT sphere was divided

into over 105small volume elements, which represented point

charges in equation (4). Radial charge density distributions for

Fe (ρFe

in the unit cell, were obtained from the KKR calculations

for each atom and for all considered atom configurations. In

the next step, the mean radial charge density distributions

were calculated using an averaging procedure, i.e. for each

of five lattice sites the ρL(r) (with L

was computed by averaging earlier calculated charge density

distributions weighted by the effective occupancy of Fe/Cr on

each site. Using these ρL(r) dependences, Vijwas evaluated

(equation (4)) by summation in the variable distance (in each

crystallographic direction) from the fixed atom. The procedure

was carried out with increasing summation distance until

resulting EFG-values became stable (the difference less than

0.1%). Finally, satisfactory results were obtained for a cube of

seven lattice constants around the57Fe atom, containing 3375

nearest unit cells (over 105atoms and electron densities inside

their MT spheres). The resulting EFG-tensors were then used

to calculate the energy shift values ?E for each lattice site.

The interaction between Vlat

of the57Fe-nucleus is complicated by the presence of the

M¨ ossbauer atom’s own electrons, especially in d-states. On

the other hand, since the charge density distribution around

each atom in the MT approximation is spherically symmetric,

the calculated Vzz-value related to the valence electron’s cloud

vanishes, which is a serious limitation of this approach. For

that reason we have to assume that the influence of Vlat

dominant and Vval

zz

is proportional to Vlat

the calculated energy shift values ?E are proportional to the

QS-values. The proportionality factor γ was assumed to be

identical for all lattice sites, and was calculated in the fitting

procedure of the experimental data. The resulting QS-values

are shown in table 3. It must be mentioned that the obtained

QS-values are only average values for each lattice site.

Vij=

ei(3xixj− r2δij)r−5

(4)

i(r)) and Cr (ρCr

i(r)), where i denotes each of 30 atoms

= A through E)

zzand the quadrupole moment

zzis

zz. Consequently,

Figure 2. Probability Pk(NN,c) for finding NN-Fe atoms in the first

coordination shell for the five inequivalent lattice sites in

σ-Fe53.8Cr46.2. k = A··· E, see equation (6). The solid lines are

shown as a guide for the eye only.

4. Results and discussion

4.1. Probability distribution calculation

Since hyperfine parameters depend mainly on the NN

configuration, it is worth calculating the probability of finding

a definite number of Fe atoms in the nearest shell for each

lattice site. In the case of a random distribution one can use

the Bernoulli formula describing the probability of finding n

atoms of type X in XcY1−calloy

?Ntot

where Ntotstands for the total number of NN atoms and c is the

atomic concentration of X. In the case of the σ-phase there are

five different NN-types with different Ntotand c values on each

site. The NN atoms are distributed on the specified lattice sites

(table 1), so the formula for the kth site should be modified:

PB(n,c) =

n

?

cn(1 − c)Ntot−n

(5)

Pk(NN,c) =

?

5 ?

i=1

PB(NNi,ci);

NN =

5

?

i=1

NNi

(6)

where index i denotes the crystallographic site, the summation

in equation (6) runs over all possible combinations of NNi

yielding NN, and c is the atomic concentration of Fe. This

procedure allows us to calculate the probability distribution for

each site separately. As can be clearly seen in figure 2, all

distributions are Gaussian-like curves with similar linewidths

but shifted relative to each other. The most probable numbers

of Fe-NN atoms are markedly different for each site and vary

from 5 (D site) to 9 (B site).

4.2. Charge density distributions

The electron densities at the nucleus for the M¨ ossbauer

absorber ρA(0) were calculated for each lattice site of the Fe–

Cr σ-phase. Figure 3 summarizes our results, depicting the

dependences of ρA(0) on the number of Fe-NN atoms. One

can notice correlations between these quantities, which agree

4

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J. Phys.: Condens. Matter 20 (2008) 235234J Cieslak et al

Figure 3. Electron densities at the nucleus of Fe calculated for each

lattice site of the Fe–Cr σ-phase (left scale) or, equivalently, isomer

shifts relative to the source, IS (right scale). Solid lines represent the

best linear fits between the ρAvalues and the number of Fe-NN

atoms. Different symbols denote the calculated ρAvalues

corresponding to inequivalent sites. The average values of the charge

density (or isomer shifts relative to the source, ISav) are indicated on

the right-hand side of the graph.

well with the anticipated relations. The slopes of the linear

regressions are quite similar and negative (−0.04 au−3per

one Fe-atom in the first coordination shell on average) for

all sites, which means that one Fe-NN atom decreases the

isomer shift value by ?IS ∼ 0.013 mm s−1. The particular

slopes calculated for the σ-phase can be compared to two

experimentally obtained values for the Fe–Cr α-phase. In this

case, ?IS equal to 0.020 mm s−1and 0.009 mm s−1for the

first and the second neighbor shell, respectively, when one Fe-

neighbor atom is added [15, 16]. A correlation between ?IS

for the Fe–Cr α- and σ-phases and the average interatomic

distances dav is presented in figure 4(a). We see that these

quantities markedly decrease in absolute value with increasing

distance, as can be expected. Also the average isomer shift

relative to the source ISavdecreases monotonically with dav-

values and becomes the lowest for the closest neighbors

(site A), as shown in figure 4(b). The determined ISavremain

significantly different for each crystallographic position.

4.3. RT M¨ ossbauer spectrum analysis

In our analysis we reasonably assumed the M¨ ossbauer

spectrum to be composed of five subspectra, each of them

represents the distribution of the IS. The relative area under

each subspectrum corresponding to the particular site should

be equal to the number of Fe atoms occupying this site.

Figure 4. (a) The slope of the isomer shift, ?IS, dependence versus

average NN distance, dav. Empty and filled circles stand for the

Fe–Cr σ- and α-phase, respectively. (b) The relative to the source

average isomer shift ISavversus dav.

The differences between the Lamb–M¨ ossbauer factors for

nonequivalent positions were considered to be very small, thus

these factors were identical in our analysis. Also, the ?-values

were assumed to be the same for all lines.

The RT M¨ ossbauer spectrum was successfully fitted using

the least square method. The relative subspectra intensities

were assumed to be known from recent neutron diffraction

experiments [7], whereas QS- and IS-values relative to the

source as well as the distinct linear dependence of IS on

the NN-Fe atoms were obtained from our KKR calculations.

Each subspectrum was constructed as a sum of double lines

with the same QS-values, IS-values linearly dependent on

Fe-NN numbers and probabilities determined according to

equation (6). It should be mentioned here that there are only

five fitting parameters in the model, namely: background, total

intensity,isomershiftfortheBsitesubspectrumIS0(necessary

to adjust the refined spectrum to the used M¨ ossbauer source),

linewidth ? and proportionality factor γ. The background,

total intensity and IS0 depend on the measurement conditions

only, whereas the two latter parameters are directly connected

with the intrinsic properties of the Fe–Cr σ-phase.

5. Conclusions

In summary, we have calculated the electronic structure in

terms of the hyperfine parameters such as the isomer shift and

the quadrupole splitting of the σ-phase in the FeCr system.

Theoretically calculated values of the spectral parameters

were successfully used in combination with experimentally

determined site-occupation probabilities to fully reconstruct a

57Fe site M¨ ossbauer spectrum recorded at room temperature

on the σ-FeCr sample, using only five adjustable parameters

(background, total intensity, ?, IS0 and γ factor). It was also

shown that theoretically determined ?IS-values and average

IS-values remain in line with the corresponding quantities

measured in the α-FeCr phase. The latter means that in both

phases the Fe site charge density scales linearly with NN and

NNN distances. Similar scaling behavior of the charge density

(isomer shift)has been shownto existfor the number of NN-Fe

atoms. It is, however, characteristic of a given site.

5

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J. Phys.: Condens. Matter 20 (2008) 235234 J Cieslak et al

Acknowledgments

This work was partly supported by the Polish Ministry of

Science and Higher Education and¨OAD Project 20/2003 as

well as grant No. N202210433.

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6