Acta Materialia 53 (2005) 255–264

www.actamat-journals.com

A combined cluster variation method and ab initio approach to thec-Fe[N]/c 0-Fe4N1�x phase equilibrium

S. Shang, A.J. Bottger *

Department of Materials Science and Engineering, Delft University of Technology, Rotterdamseweg 137, 2628 AL Delft, The Netherlands

Received 21 July 2004; received in revised form 3 September 2004; accepted 12 September 2004

Abstract

A combination of ab initio and statistical thermodynamics, i.e., the cluster variation method (CVM), is applied to describe the

FCC-based c-Fe[N]/c 0-Fe4N1� x phase equilibrium. The ab initio calculated internal energies of ordered compounds, using the full-

potential (linearized) augmented plane-wave plus the local orbital method including the magnetic contributions, are used to obtain a

set of volume-dependent effective cluster interactions which parametrize the internal energy of FCC-based Fe–N alloys. The

c-Fe[N]/c 0-Fe4N1� x phase boundaries, the lattice parameters and the distribution of N atoms over the octahedral interstitial sites

are calculated by applying the tetrahedron approximation. The vibrational contributions are also considered by using the Debye–

Gruneisen model. The current calculations of the phase boundaries, the lattice parameters and the interstitial nitrogen distribution

agree well with experimental data available in the literature.

� 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Iron–nitrogen system; Phase diagram; Ab initio calculations; Cluster variation method; Cluster expansion; Mossbauer spectroscopy

1. Introduction

Nitriding is the thermochemical treatment often ap-

plied to iron-based alloys (steels) to improve the corro-sion and wear resistance of surfaces. The ability to

predict relative phase stabilities and thermodynamic

properties of the iron–nitrogen (Fe–N) system is a pre-

requisite in controlling the process of nitriding. Most

of the thermodynamic properties that enable the calcula-

tion of absorption isotherms or other process parameters

can only be obtained through calculations since many of

the phases that develop are metastable precipitates [1,2].This work aims to develop a method to calculate thermo-

dynamic properties of nitrides starting from first princi-

ples. To this end the well-studied c and c 0 phases have

been chosen to explore such an approach.

1359-6454/$30.00 � 2004 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2004.09.009

* Corresponding author. Tel.: +3115 2782243; fax: +3115 2786730.

E-mail address: A.B.Bottger@tnw.tudelft.nl (A.J. Bottger).

The c-Fe[N] and c 0-Fe4N1�x phases are both face

centered cubic (FCC) structures in which the N atoms

reside predominantly in the octahedral interstices

formed by FCC Fe matrix atoms, whereas the octahe-dral interstitial sites also form a FCC sublattice. The

structures can be described using two interpenetrating

FCC sublattices: the M (matrix) sublattice fully occu-

pied by Fe atoms, and the octahedral I (interstitial) sub-

lattice occupied by nitrogen (N) atoms and vacancies

(Va). The ordering of interstitials N on the I-sublattice

can be described as the ordering of a binary nitrogen-va-

cancy (N, Va) substitutional system [3–5]. In our previ-ous investigation [3], the c-Fe[N]/c 0-Fe4N1�x phase

equilibrium was studied based on the phenomenological

Lennard–Jones pair potential determined through fitting

the experimental data of the phase boundary. With that

approach, the calculated phase equilibria and degree

and type of ordering of the N atoms agree well with

the available experimental data; this showed that

describing interstitial systems on the basis of the

ll rights reserved.

256 S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264

I-sublattice only is a successful approach. In this work

thermodynamic quantities of the Fe–N system are ob-

tained via a first principle approach. Ab initio calcula-

tion of the ground state properties of a set of ordered

FCC based compounds is performed to obtain the effec-

tive cluster interactions (ECIs) [6] used to calculate theinternal energy at 0 K. The Debye–Gruneisen model

[7] is applied to account for vibrational contributions

and the cluster variation method (CVM) [6] is used to

calculate the configurational entropy. The resulting c-Fe[N]/c 0-Fe4N1�x phase boundaries, the lattice parame-

ters and the distribution of N atoms over the octahedral

interstitial sites are discussed and compared to the avail-

able experimental data.

2. Theory

The thermodynamic calculations in the present work

are based on the Gibbs free energy G (per atom) given

by

GðV ;T Þ ¼ EabðV Þ þEvibðV ;T Þ � T ½Sconf þ SvibðV ;T Þ� þ pV ;

ð1Þwhere Eab(V) is the internal energy at 0 K estimated

via ab initio calculations and the cluster expansion

method; details are given in Sections 2.1 and 2.2.

Evib(V,T) and Svib(V,T) are, respectively, the vibra-

tional contributions to the internal energy and entro-py, which are obtained on the basis of the Debye–

Gruneisen model [7], see Section 2.3. Sconf is the

CVM configurational entropy described in Section

2.4. Other parameters, T, p and V are the tempera-

ture, external pressure and volume per atom respec-

tively. In Section 2.5, the calculation of the Gibbs

energy will be discussed, and the determination of

the phase equilibria and lattice parameters in theframework of CVM are described.

2.1. Details of the ab initio calculations

The ab initio calculations have been performed using

the relativistic full-potential (linearized) augmented

Table 1

The space group, the assigned k-mesh (and k points in the irreducible Brill

functions nm (see [6] for definition) within the tetrahedron cluster for FCC-b

Structure Space group k-Mesh and k-point in IBZ Correlation f

m = 0 Empty

Fe (A1, HS) Fm�3m 20 · 20 · 20 (256) 1

Fe4N (L12) Pm�3m 14 · 14 · 14 (84) 1

Fe4N (D022) I4/mmm 14 · 14 · 14 (240) 1

Fe4N (D023) I4/mmm 11 · 11 · 11 (126) 1

Fe4N2 (L10) P4/mmm 16 · 16 · 11 (216) 1

Fe4N3 (L12) Pm�3m 14 · 14 · 14 (84) 1

FeN (A1) Fm�3m 17 · 17 · 17 (165) 1

plane-wave plus local orbitals (L/APW + lo) method to-

gether with the generalized gradient approximation

(GGA) [8] as implemented in the WIEN2K code [9].

In the calculation the core and valence states are distin-

guished, inside the muffin-tin (MT) sphere there are 10

charges for Fe and 2 for N. Here, a mixed LAPW/APW + lo basis set is employed, the LAPW is used for

the global basis set, the APW + lo basis is used for the

s, p, d valence states of Fe and the s, p valence states

of N; additional local orbitals (LOs) are added to de-

scribed the Fe 3s, 3p (semicore) states and N 2s state.

In order to describe the unoccupied state better, an extra

LO for Fe with a linearized energy above the Fermi en-

ergy is also added. This extra flexibility turns out to benecessary in order to obtain consistent results for the to-

tal energy for Fe (energy vs. volume curve with the extra

LO is smooth) [10]. In the present work, the lineariza-

tion energies for each state used in the L/APW + lo cal-

culations are carefully chosen according to the method

described in [10].

For all the Fe–N compounds, the MT sphere radii R

are chosen as 1.8 a.u. for Fe and 1.5 for N. The plane-wave cutoff Kmax is defined by RKmax = 8.5, where R is

minimum sphere radius. Inside the atomic sphere, the

maximum angular momentum (lmax) is chosen as 12,

and the maximum angular momentum (lns max) of 6 is

used in the computation of the non-spherical matrix ele-

ments. The parameter Gmax to control the Fourier

expansion in the interstitial region is assigned to 18

Ry1/2. These relatively large values for RKmax, lmax,lns max and Gmax are chosen in order to obtain a suffi-

ciently accurate total energy, especially for (FCC) Fe.

The number of sampling k points used in the calcula-

tions for Fe–N compounds are given in Table 1. The

structures of the interstitial sublattices of these com-

pounds are shown in Fig. 1. Ordered compounds are

chosen only because then the calculations can be accom-

plished in a reasonable time. The convergence criterionfor the calculations of the internal energy (of the unit

cell) is taken as 0.02 mRy, except for Fe (A1) and FeN

(A1), where a smaller value of 0.001 mRy could be

accomplished because there are less independent atoms

in the unit cell. It should be noted that the structure des-

ouin zone (IBZ)) used in the ab initio calculation and the correlation

ased Fe–N compounds.

unctions

m = 1 Point m = 2 Pair m = 3 Triangle m = 4 Tetrahedron

1 1 1 1

1/2 0 �1/2 �1

1/2 0 �1/2 �1

1/2 0 �1/2 �1

0 �1/3 0 1

�1/2 0 1/2 �1

�1 1 �1 1

Fig. 1. The FCC structure showing the two sublattices: the matrix

sublattice (sites occupied by Fe) and the interstitial sublattice (sites

occupied by N or Va) together with the schematics of FCC based Fe–

N compounds (within the interstitial sublattice) used in the ab initio

calculations. The structure designations are referring to the structure

of the interstitial sublattice. The tetrahedron cluster is also indicated

within the interstitial sublattice. The symbols a, b, c and d represent thefour interpenetrating (interstitial) sublattices. The site occupations of

the tetrahedron cluster are denoted by i, j, k and l take the values 1 or

2, when occupied by N or vacancy, respectively.

S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264 257

ignations given above (and also in Table 1) are referring

to the structure of the FCC interstitial sublattice and not

to that of the overall structure. This implies for instance

that the FeN structure is represented here by A1 insteadof the usual B1. Due to the ferromagnetic nature of Fe–

N compounds, all the calculations are performed within

the spin-polarized approximation, for FCC Fe the high

spin (HS) state is chosen as will be discussed next.

In the performance of the ab initio calculations for

FCC based Fe–N compounds, the complex ferromag-

netic nature of the FCC Fe and c 0-Fe4N needs to be con-

sidered. Experimentally FCC Fe shows differentmagnetic states depending on the lattice parameter (epit-

axially grown FCC pure Fe) [11] and, temperature and

composition (bulk solid solutions) [12]. It has been

shown that although the ground state of FCC Fe is

AFM (anti-ferromagnetic), at high temperatures (above

the Neel temperature �50 K [12]) FCC Fe shows a

moment–volume instability, i.e., a transition from the

anti-ferromagnetic to a ferromagnetic state that involvesvolume and magnetic moment changes. The observation

of an excess thermal expansion known as the anti-Invar

effect indicates the presence of two kinds of FM (ferro-

magnetic) states: the low spin LS FM and the high

spin HS FM states of FCC Fe [12,13], where the low

spin state is of lower volume (and magnetic moment)

than the high spin state. In the c-Fe[N] phase the addi-

tion of N atoms increases the volume of the FCC Fe ma-trix; therefore the HS FM state of FCC Fe matrix is

expected be the favorable one and is chosen for the cur-

rent calculations. For the c 0-Fe4N phase, a small Invar

effect is also reported in the literature [12,14], in this case

the volume difference is small: the estimated volume dif-

ference caused by the Invar effect at 900 K, as estimated

according to the method given in [14], is less than 0.02%,

which is negligible in the context of the presentcalculations.

2.2. Cluster expansion and effective cluster interaction

energy

In this section, the cluster expansion method used to

calculate the internal energy Eab used in Eq. (1) from

the above discussed ab initio calculations is described.The cluster expansion method (CEM) can serve as a

bridge between the ab initio calculated internal energies

of ordered structures (Section 2.1) and the internal energy

for any structure [6]. In this approach the internal energy

is expressed in terms of a generalised Isingmodel in which

the energy is expanded in a series of basic functions Jm(withm = 0,. . .,mmax), the so-called effective cluster inter-

action energies [6]. The ECIs are averages of the clusterenergy of many atomic configurations [15]. Within the

framework of the cluster expansion method, the energy

per atom for any phase can then be described as [6],

EðV Þ ¼Xmmax

m¼0

nmJmðV Þ; ð2Þ

where in the case of the tetrahedron approximation

mmax = 4, the coefficients nm are the correlation func-

tions defined in [6,16], and V is the volume per atom.

The values of nm pertaining to the clusters of the set of

FCC-based compounds used in present work, are shownin Table 1. The ECIs (Jm) can be extracted from the

internal energies of a series of ordered compounds. In

the present work the method to obtain Jm is described

in matrix form as,

J ¼ n�1 � Eab; ð3Þ

where the n�1 is the inverse (or pseudo-inverse, in the

case that the number of ordered structures is different

from the number of ECIs to be determined) of the cor-relation function matrix n. Eab is the energy matrix for a

series of ordered structures (as shown in Table 1 for the

present work) obtained from ab initio calculations.

The influence of the volume on the internal energy

of each ordered structure h is obtained by calculating

258 S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264

the internal energy for several volumes (P6 in the pre-

sent calculation), and by subsequently fitting the fol-

lowing equation of state (EOS) [17] to the resulting

energies,

EhabðV Þ ¼ aþ bV �1=3 þ cV �2=3 þ dV �1; ð4Þ

where V is the volume and a, b, c, d are the fit parame-

ters. This EOS is used to estimate the equilibrium energy

(Ee), volume (Ve), bulk modulus (Be) and the pressure

derivative of the bulk modulus ðB0eÞ. See Appendix A

for details. The most important reasons to choose the

expression given in Eq. (4) are that: (i) this EOS was

found to describe the data better than the commonly

used Murnaghan EOS [17]; (ii) it can be written in ma-

trix form enabling the fit parameters to be easily solved

by (pseudo)-inversion; (iii) Eq. (4) can be substituted

into Eq. (3) directly.

Previous CVM results [3] and also the present results(see Section 3.5) show that the main energy contributions

to c-Fe[N] and c 0-Fe4N1�x phases are the clusters with

zero and one N atom. Therefore, in the current work

somewhat more ordered Fe4N compounds namely

Fe4N (L12, D022 and D023), are chosen; in that way the

energy contributions of clusters containing a small

amount of N atoms are more accurately represented

within the ECIs. Since the compounds Fe4N with L12,D022 and D023 structures have the same correlation func-

tions within the tetrahedron approximation as shown in

Table 1, the following weights [18] are employed to rep-

resent their contributions to Fe4N, i.e., w(Fe4N,

L12):w(Fe4N, D022):w(Fe4N, D023) = 0.1:0.3:0.6. Note

that the investigated c and c 0 are cubic phases and the lat-

tice parameter ratio c/a for the relaxed tetragonal struc-

tures is near to the ideal value for the stacked cubic cells,e.g., the relaxed c/a of Fe4N (D022) is 2.005 [19]. In this

work, the possible (tetragonal) distortion of the lattice

is not explicitly considered because no tetragonality has

been observed experimentally so far for any of the

iron–nitrides in the temperature and composition range

considered. During the ab initio calculations the c/a for

the tetragonal compounds Fe4N2 (L10) and Fe4N (D022and D023) are fixed at 1, 2 and 4, respectively. In the pre-sent work, the compounds Fe (A1, HS) and FeN (A1) are

taken as the reference states. This implies that Fe4Nk

(k = 0, 1, 2, 3 or 4) is described as Fe4� k(FeN)k, and

the relative energy per atom for Fe4Nk, DEFeNk=4

ab is ob-

tained by ½EFe4Nkab � ð4� kÞEFe

ab � kEFeNab �=4.

2.3. The vibrational contributions

The vibrational internal energy Evib(V,T) and vibra-

tional entropy Svib(V,T) are estimated using the De-

bye–Gruneisen model, according to [7],

EvibðV ; T Þ ¼9

8kBHD þ 3kBTD

HD

T

� �; ð5Þ

SvibðV ; T Þ ¼ 3kB4

3D

HD

T

� �� ln 1� exp �HD

T

� �� �� �;

ð6Þwhere kB is Boltzmann�s constant, T is the temperature,and D(HD/T) the Debye function is DðxÞ ¼ 3=x3R x0t3=ðet � 1Þdt. The Debye temperature, HD, is writ-

ten as follows, as deduced from the expression given in

[7].

HD ¼ cV 1=6e;0

Be;0

M

� �1=2 V e;0

V

� �c

; ð7Þ

where c ¼ ð1þ B0e;0Þ=2� 2=3 is the Gruneisen constant

for the case of high temperature [13], B0e;0 is the pressure

derivative of the bulk modulus Be,0, V is the volume. The

subscript ‘‘0’’ indicates the value at 0 K. M is the aver-

age molecular weight per atom (the definition of �peratom� is given in Section 2.2), c is a constant. The value

of c is calculated as 155.124 if Be,0 is given in GPa, M in

g and Ve,0 in cm3/mol. The properties required to esti-mate the Debye temperature, such as the

B0e;0; Be;0 and V e;0 are obtained from the ab initio calcu-

lations and Eq. (4).

2.4. The tetrahedron approximation and the

configurational entropy

For both phases c and c 0 the tetrahedron approxima-tion of the CVM is chosen to calculate the distribution

of N atoms and vacancies Va over the FCC I-sublattice,

which is subdivided into four interpenetrating FCC sub-

lattices, denoted by a, b, c and d. For each of these four

interpenetrating FCC lattices the site occupation is de-

scribed by i, j, k and l, respectively, which can take the

values of 1 (if the site is occupied by N) and 2 (if the site

is occupied by Va). The thus constructed tetrahedroncluster within the I-sublattice is shown in Fig. 1. The

general expression for the configurational entropy per

lattice point for a FCC ordered state using tetrahedron

cluster approximation is given by

Sconf=kB ¼�2Xijkl

Lðqabcdijkl Þþ

Xij

Lðqabij Þþ

Xik

Lðqacik Þ

"

þXil

Lðqadil Þþ

Xjk

Lðqbcjk Þþ

Xjl

Lðqbdjl Þþ

Xkl

Lðqcdkl Þ#

�5

4

Xi

Lðqai Þþ

Xj

Lðqbj Þþ

Xk

LðqckÞþ

Xl

Lðqdl Þ

" #;

ð8Þwhere kB is Boltzmann�s constant, the symbol q is the

cluster probability for a point ðqai Þ, pair ðq

abij Þ and tetra-

hedron cluster ðqabcdijkl Þ, respectively. The function

L(x) = xlnx. For a given ordered phase or a disor-dered phase, Eq. (8) can be simplified further, see for

example [20].

S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264 259

2.5. The calculation of the c-Fe[N]/ c 0-Fe4N1�x phase

equilibrium and lattice parameters

To calculate the phase boundaries in the CVM the

grand potential function is used, for each phase the ther-

modynamic function X, the grand potential function, isgiven by

XðV ; T ; l�1; l

�2Þ ¼ ðEab þ EvibÞ � T ðSconf þ SvibÞ þ pV

�X2n¼1

l�nxn þ k 1�

Xijkl

qabcdijkl

!; ð9Þ

where xn is the mole fraction of component n (n = 1, 2,

i.e., N and Va, respectively), in the phase considered,

for the tetrahedron approximation xn ¼ ðqan þ qb

nþqcn þ qd

nÞ=4. l* is the effective chemical potential defined

as l�n ¼ ln � ðl1 þ l2Þ=2, where ln is the chemical poten-

tial of component n. k is the Lagrange multiplier to the

constraintP

ijklqabcdijlk ¼ 1.

In the present calculation the (Eab + Evib � TSvib) vs.

volume curves are fitted to the EOS of Eq. (4) for the

temperatures of interest, meanwhile the high tempera-

ture equilibrium properties ðEe; B0e; Be and V eÞ consid-

ering the vibrational contributions are estimated

accordingly. The values for X and the corresponding

configurations of clusters are obtained by minimizing

X with respect to qabcdijkl (see Eq. (10)). The correlation

function nm in the energy term (Eq. (2)) is transferred

into the cluster probability according to the method gi-

ven in [21]. The volume per cluster site is obtained by

minimizing X with respect to the volume (seeEq. (11)), where atmospheric pressure is taken as the ref-

erence pressure. The aforementioned optimization pro-

cedure is performed by the natural iteration method

[22]. The thermodynamic equilibrium between the cand c 0 phases is determined by the T, l* for which Xare the same for both phases

oX

oqabcdijkl

�����T ;l�;V

¼ 0; ð10Þ

oXoV

����T ;l�;qabcdijkl

¼ �p: ð11Þ

3. Results and discussion

In this part the results of the ab initio calculations at

0 K, the vibrational contributions to the high tempera-

ture equilibrium properties (an intermediate tempera-

ture in the c/c 0 phase region, i.e., 900 K is taken as an

example, see Section 3.2) are presented at first. Nextthe c-Fe[N]/c 0-Fe4N1�x phase boundaries with and

without vibrational effects are calculated in the frame-

work of the CVM. Based on the calculated phase equi-

librium the lattice parameters and the distribution of

N atoms in both c and c 0 phases are calculated and com-

pared with the available experimental data.

3.1. Ab initio results

From the ab initio calculations the volume dependent

energy per atom (at 0 K) for each of the chosen ordered

compounds (see Section 2) is obtained. The influence of

composition, temperature and magnetic state on the vol-ume will be discussed.

The estimated equilibrium properties, i.e., the relative

energy per atom DEe as defined in Section 2.2, the vol-

ume per atom Ve, the bulk modulus Be and the pressure

derivative of the bulk modulus B0e both at 0 and 900 K

(including the vibrational contributions) are given in Ta-

ble 2 for each ordered compound, along with the mag-

netic moment per atom at 0 K and the Debyetemperature at 0 and 900 K. Among the four estimated

properties, DEe and Ve are not very sensitive to the qual-

ity of the EOS fitting (i.e., the scatter of the energy vs.

volume curve), whereas Be and especially B0e are sensitive

to the quality of EOS fitting. As an indication for the

quality of the obtained results the average fitting error

per atom, i.e., the average difference |Eab � Efit| at 0 K,

is also given in Table 2.The results in Table 2 show that the volume is gener-

ally an increasing function of the N content. The calcu-

lated largest bulk modulus for FCC based compounds

appeared at yN = 0.5 (N atom fraction in I-sublattice).

The Debye temperature (see Eq. (7)) shows the same

trend as the bulk modulus, which is expected since the

Debye temperature is in (direct) proportion to the

square of the bulk modulus. The Debye temperature de-creases with the increasing of temperature through the

temperature dependence of the volume. For the com-

pounds with yN = 0.25, the Fe4N (L12) (i.e., the stoichi-

ometric c 0 phase) has the lowest energy which agrees

well with the observation of that phase experimentally

[23], Fe4N (D022) has the highest energy, and the energy

of Fe4N (D023) is in between that of the Fe4N (L12) and

the Fe4N (D022). This is consistent with the structuralfeatures of D023 that can be described as a set of alter-

nating L12 and D022 lattices. For what concerns the vol-

ume and the average magnetic moment per atom for

these three compounds, they show the same trends:

i.e., Fe4N (L12) > Fe4N (D023) > Fe4N (D022). For the

Fe–N compounds with the same N content, the volume

differences are associated with the differences of the

average magnetic moment per atom MMe,

MM e ¼ ðMMuc �MMNÞ=nFe; ð12Þwhere MMuc is the MM per unit cell, MMN is the MM

of N atoms in the unit cell, nFe is the number of Fe

atoms in the unit cell. Table 2 shows that the larger

the magnetic moment the larger the volume per atom

Fig. 2. Calculated (lines) and experimental (symbols) thermal expan-

sion coefficients for c-Fe (r) and c 0-Fe4N (h, evaluated from lattice

parameters at 293, 523 and 773 K [26]; n, evaluated from volumes at

4.2 and 293 K [27]).

Table 2

Calculated equilibrium properties at 0 and 900 K (including the vibrational contributions) together with the available experimental data: the relative

energy per atom DEe, atomic volume per atom Ve, bulk modulus Be, pressure derivative of the bulk modulus B0e, average magnetic moment per atom

MMe, Debye temperature (HD)e. The average error to the fit of the EOS of Eq. (4) is also given

Compounds DEe (mRy/atom) Ve (a.u.3/atom) Be (GPa) B0

e (HD)e (K) MMe (lB/atom) av_error (mRy/atom)

Fe (A1, HS) 0 K 0 81.082 181.261 5.439 388.7 2.57 0.016

900 K �21.003 85.056 145.738 5.776 344.0

Exp. 81.7a

Fe4N (D022) 0 K �22.602 89.757 158.645 5.100 358.7 2.17 0.010

900 K �44.875 93.937 131.157 5.336 321.9

Fe4N (D023) 0 K �23.403 90.517 154.606 2.690 354.8 2.32 0.046

900 K �45.171 92.790 149.111 2.702 345.9

Fe4N (L12) 0 K �24.626 92.185 191.133 3.176 395.5 2.49 0.002

900 K �44.557 94.116 182.789 3.183 384.0

Exp. 91.8b 2.21c

Fe2N (L10) 0 K �25.418 98.426 216.701 6.352 413.7 2.12 0.011

900 K �45.302 102.343 174.546 6.869 367.9

Fe4N3 (L12) 0 K �15.635 106.577 209.491 2.914 401.2 2.17 0.025

900 K �35.230 108.164 203.717 2.918 393.6

FeN (A1) 0 K 0 113.158 132.852 3.063 314.5 2.19 0.056

900 K �23.899 115.821 126.668 3.069 304.7

a Estimated value at 4 K [24].b a = 0.379 nm at room temperature [23].c Extrapolated value at 0 K [25].

260 S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264

for Fe4N (L12, D023, D022). The larger volume has more

space to hold the interstitial N atom, which leads to the

lowest energy, i.e., the stable phase Fe4N (L12). The

influence of temperature on the equilibrium properties

estimated by the Debye–Gruneisen model, also shown

in Table 2, indicates that with the increase of tempera-

ture, the DEe; V e; B0e and ðHDÞe increase, whereas Be

decreases.On the basis of the thus obtained changes in equi-

librium volume as a function of temperature the ther-

mal expansion coefficient, a, is calculated for c-Fe (A1,

HS) and c 0-Fe4N (L12). The following expression is

used,

aðT Þ ¼ ½3V eðT Þ��1 dV eðT ÞdT

: ð13Þ

In Fig. 2 the thermal expansion coefficients obtained

on the basis of experimental data [24,26,27] and those

calculated using the Debye–Gruneisen model are given

as a function of temperature. The calculated thermal

expansion coefficient agrees well with the available

experimental data for the c 0 phase. For the c-Fe phase

the calculated values are somewhat lower than the

experimental ones. This discrepancy is assigned to: (i)the anti-Invar effect that has not been considered in

the present work but has been experimentally observed

for FCC Fe [12,24], and; (ii) the quality of the fit to

the EOS, i.e., the average fit error (mRy/atom) is a little

larger for c-Fe (A1, HS) because of the approximations

used for electron energies applied in the ab initio calcu-

lation, which will influence the value of B0e.

3.2. Calculation of phase diagram

On the basis of the equilibrium properties (see

Table 2), the internal energy of Eq. (4) can be obtained

as indicated in Appendix A. According to Eqs. (2) and

(4), the volume dependent Jm (ECIs) are calculated

and shown in Fig. 3 for 0 and 900 K, the latter includes

the vibrational effects. Fig. 3 shows that the main contri-

butions are J0, J1 and J2, the value of J4 is very small,

Fig. 3. Calculated effective cluster interactions (ECIs) for FCC based

Fe–N system both at 0 K (solid lines) and 900 K (dotted lines), the

latter includes the vibrational contributions.

S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264 261

which suggests that clusters larger than the tetrahedron

cluster can be ignored in the present case. With increas-

ing temperature, the values of the ECIs change: in par-

ticular the value of J0 decreases observably, indicating

that the disordered phase becomes more stable at high

temperature.

The calculated c-Fe[N]/c 0-Fe4N1�x phase bounda-ries with and without vibrational contributions are

shown in Fig. 4 together with the available experimen-

tal data [28,29]. Obviously the calculated (c + c 0)/c 0

phase boundary agrees well with the experimental data

regardless of the consideration of vibrational effects.

The c/(c + c 0) phase boundary on the other hand is

strongly affected by the vibrational contributions. A

good agreement between calculations and experimental

Fig. 4. Calculated c-Fe[N]/c 0-Fe4N1�x phase boundaries with (solid

lines) and without (dotted lines) vibrational influences together with

experimental data (symbols) [28,29], where h indicates the c/(c + c 0)

phase boundary, s indicates the (c + c 0)/c 0 phase boundary.

data is obtained when vibrational effects are included

in the calculations. Further investigation shows that

the calculated c/(c + c 0) phase boundary is very sensi-

tive to the equilibrium volumes of Fe (A1, HS) and

Fe4N (L12, D022, D023). The changes in the volume

result in large variations of the energy which obvi-ously influences the phase boundaries. This is also

the main reason why the accurate (but time consum-

ing) full potential method (L/APW + lo) is used to ob-

tain the equilibrium properties for the ordered

compounds.

The influences of the equilibrium volume on the posi-

tion of c/(c + c 0) phase boundary can also be under-

stood from the viewpoint of the geometry of thestructure. An increase of V0 for Fe (A1, HS), will in-

crease the size of the octahedral interstitial site and

therefore more space is available to hold an interstitial

N atom, and thus the solubility of N in the c phase will

increase, i.e., the c/(c + c 0) phase boundary will shift to

high N content. A decrease of V0 for Fe4N, implies that

the expansion of FCC Fe caused by the occupation of N

atom in octahedral interstitial site is smaller, then theFCC Fe is capable of accommodating a higher intersti-

tial N atom content, the solubility of N in c phase will

also increase.

Although the correspondence between the calcula-

tions and experimental data of the phase boundaries is

very good, some differences occur (see Fig. 4). These dif-

ferences between the calculated and experimental phase

boundaries can be caused by the approximations madein the calculations. The most important approximations

are: (1) the limitation in the number of basic compounds

considered to calculate the ECIs; (2) the accuracy of the

ab initio simulations because of the approximations

made in electron energies, and the consequences of that

on the volume dependencies (Gruneisen constant); (3)

the way the magnetic contributions are treated, i.e., at

high temperature Fe–N phases are paramagnetic,whereas the low temperature magnetic state has been

used in the calculations.

3.3. The lattice parameter

The CVM calculated lattice parameters at 900 K

for the c-Fe[N] and c 0-Fe4N1�x phases together with

the experimental data [23,24] are shown in Fig. 5.The calculated lattice parameter of the c 0 phase agrees

well with the experimental one, while the calculated

lattice parameter for the c phase is about 2% larger

than the experimental one. The difference that occurs

for the c phase could be caused by the fact that:

(1) only the high spin state of FCC Fe is considered;

(2) the Gruneisen constant for FCC Fe could not be

estimated accurately, i.e., the average error to the fitof the EOS of Eq. (4) is somewhat larger for c-Fe(A1, HS).

Fig. 5. Calculated lattice parameters at 900 K (lines) together with the

experimental ones (symbols) for c-Fe[N] and c 0-Fe4N1�x phases. r is

the experimental datum for c-Fe at 1150 K [24]. s and h are the

experimental data for c 0-Fe4N at room temperature [23] and 900 K

(evaluated value according to the thermal expansion coefficients, see

Fig. 2), respectively.

262 S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264

3.4. Distribution of N atoms in the c 0-Fe4N1�x phase

The calculated tetrahedron cluster distribution varia-

bles qabcdijkl of the c 0-Fe4N1�x phase pertaining to a tem-

perature of 900 K are given in Fig. 6 as a function of

the atomic percentage of N atoms, the qabcdijkl with very

small possibilities (less than 0.01) are not included.

Fig. 6. Calculated tetrahedron cluster distribution variables for the

c 0-Fe4N1�x phase at 900 K.

Fig. 6 shows that only tetrahedron clusters with zero

and one N atom appear in the composition range from

about 18–20 at.% N. The occurrence of clusters contain-

ing one N atom increases upon increasing nitrogen con-

tent, whereas the occurrence of clusters without N atom

decreases. The dominantly appearing tetrahedron clus-ter with specific configuration (�90%) includes only

one N atom in the composition range of the c 0 phase,

indicating that long range order (LRO) is present in

the c 0 phase.

3.5. Distribution of N atoms in the c-Fe[N] phase

The possible presence of short range order (SRO) ofN atoms in c-Fe[N] phase can be evaluated by compar-

ing the site occupancies of tetrahedron clusters, as calcu-

lated from the tetrahedron cluster probability qabcdijkl

obtained from CVM for a fixed composition and tem-

perature, with those for a random distribution [3]. The

fractions of tetrahedron clusters occupied by 0–4 N

atoms as calculated by CVM and random model for

an N content of 10 at.% and at T = 900 K are shownin Fig. 7. The CVM results indicate that the main con-

tributions are the tetrahedron clusters with 0 and 1 N

atom; as compared to a random distribution, significant

differences are observed.

The degree of SRO of N atoms in the c phase can be

deduced from the Mossbauer spectroscopy data. The

fractions of Fe atoms surrounded by 0, 1,. . .,4 N atoms

obtained from experimental Mossbauer data [30–32], de-noted by A0, A1,. . .,A4, respectively, are given in Fig. 8

together with those calculated (by CVM) pertaining to a

Fig. 7. Fraction of tetrahedron clusters (containing 0–4 N atoms)

occurring in the c-Fe[N] phase for an N content of 10 at.% and at

900 K as obtained from the CVM and for a random distribution.

Fig. 8. The Fe atom surroundings by N, An (n = 0, 1, 2), as obtained

by the CVM (lines) at 900 K and the corresponding Mossbauer data

(symbols).

S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264 263

temperature of 900 K. For the CVM calculations, the

temperature has little influence on the clusters distribu-

tions. The calculations of A0, A1,. . .,A4 are given in de-

tail in [3]. Fig. 8 shows that the CVM calculated resultsagree very well with the experimental results as obtained

from Mossbauer data. The calculations and the experi-

ments show that the Fe surrounded by 0 and 1 N atom

(i.e., A0 and A1) are dominant. The Fe atom surround-

ing A2 is negligible small (fraction less than 0.003).

4. Conclusions

By using ab initio calculations (i.e., the L/APW + lo

method, including the magnetic contributions), the

Debye–Gruneisen model (to describe the vibrational

contributions) together with the tetrahedron approxima-

tions of the CVM, the FCC based c-Fe[N] and

c 0-Fe4N1�x phases have been described successfully.

The equilibrium properties at 0 K and high tempera-ture (e.g., 900 K) including the relative internal energy,

the volume, the bulk modulus and the pressure deriva-

tive of bulk modulus for a series of FCC based Fe–N

compounds have been obtained by using ab inito calcu-

lation and Debye–Gruneisen model, which agree well

with the available experimental data. Among the com-

pounds Fe4N (L12, D022, D023), Fe4N (L12) is the stable

one with the largest volume, which is affected by the lar-ger average magnetic moment of Fe atoms. The vibra-

tional influences increase the equilibrium volume and

decrease the bulk modulus for the FCC based Fe–N

compounds.

The c-Fe[N]/c 0-Fe4N1�x phase boundaries calculated

with vibrational affects agree well with the experimental

data. The calculated c/(c + c 0) phase boundary is sensi-

tive to the equilibrium volume of Fe (A1, high spin)

and Fe4N (L12, D022, D023). The calculated lattice

parameters of the c 0 phase agree well with the experi-mental ones, whereas for the c phase the calculated val-

ues are about 2% greater than the experimental ones.

The calculated thermal expansion coefficients are in

rather good agreement with experimental data, although

the calculated values for the c phase are underestima-

tions because the anti-Invar effect is not taken into ac-

count. The present cluster distribution variables of the

c phase correspond very well with the available data ob-tained by Mossbauer spectroscopy. The main Fe atom

surroundings for the c phase are the clusters with 0

and 1 N atom: in the c phase short range order occurs,

while in the c 0 phase long range order is present.

Acknowledgements

The authors gratefully acknowledge the Foundation

for Fundamental Research of Matter (FOM) and the

Netherlands Institute of Metals Research (NIMR) for

their financial support. This work was also sponsored

by the National Computing Facilities Foundation

(NCF) for the use of supercomputer facilities.

Appendix A

If the equation of state (EOS) of internal energy is of

the form of Eq. (4), then the equilibrium volume (Ve),

bulk modulus (Be) and the pressure derivative of the

bulk modulus ðB0eÞ can be estimated by the following

formulae,

V e ¼ �4c3 � 9bcd þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc2 � 3bdÞð4c2 � 3bdÞ2

qb3

; ðA:1Þ

Be ¼2ð9d þ 5cV 1=3

e þ 2bV 2=3e Þ

9V 2e

; ðA:2Þ

B0e ¼

54d þ 25cV 1=3e þ 8bV 2=3

e

27d þ 15cV 1=3e þ 6bV 2=3

e

ðA:3Þ

and, conversely from the known Ee; V e; Be and B0e, the

coefficients in Eq. (4) can be obtained from following

formulae,

a ¼ Ee �9

2Beð�4þ B0

eÞV e; ðA:4Þ

b ¼ 9

2Beð�11þ 3B0

eÞV 4=3e ; ðA:5Þ

264 S. Shang, A.J. Bottger / Acta Materialia 53 (2005) 255–264

c ¼ 9

2Beð10� 3B0

eÞV 5=3e ; ðA:6Þ

d ¼ 9

2Beð�3þ B0

eÞV 2e : ðA:7Þ

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