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Thermodynamics of the hexagonalclose-packed iron-nitrogen system fromfi rst-principles

Morten B. BakkedalJuly 2015

Thermodynamics of the hexagonalclose-packed iron-nitrogen system from

first-principlesPhD Dissertation

Morten B. Bakkedal

Supervisor:Professor Marcel A. J. Somers

July 2015

Technical University of DenmarkDepartment of Mechanical Engineering

Section of Materials and Surface Engineering

Preface and acknowledgements

The present dissertation is submitted in partial fulfillment of the requirements for the PhDdegree at the Technical University of Denmark (DTU). The presented work has been carriedout at the Section of Materials and Surface Engineering, Department of Mechanical Engineer-ing (MEK), during the period from May 2012 to July 2015 under the supervision of ProfessorMarcel A. J. Somers.

The PhD project was part of the research project ThInSol (Thermodynamics of InterstitialSolutions in Cubic and Hexagonal Host Lattices), financially supported by the Danish Councilfor Independent Research, Technology and Production Sciences (FTP), under grant number11-106293.

First-principles calculations were performed at the Computing Center at the TechnicalUniversity of Denmark.

I would like to thank my supervisor Professor Marcel A. J. Somers for encouragementand many enlightening discussions. Thank to Dr. ShunLi Shang and the rest of the PhasesResearch Laboratory group at Pennsylvania State University for materials science insight andproductive discussions. Thank to my fellow PhD student Bastian Brink for discussion aboutthe interpretation of experimental data and for providing numerous references. Credit is alsogiven to Yi Wang from Pennsylvania State University for the Yphon tool.

Kgs. Lyngby, July 2015

Morten B. Bakkedal

i

Abstract

First-principles thermodynamic models are developed for the hexagonal close-packed ε-Fe-Nsystem. The system can be considered as a hexagonal close-packed host lattice of iron atomsand with the nitrogen atoms residing on a sublattice formed by the octahedral interstices. Theiron host lattice is assumed fixed.

The models are developed entirely from first-principles calculations based on fundamen-tal quantum mechanical calculation through the density functional theory approach with theatomic numbers and crystal structures as the only input parameters. A complete thermody-namic description should, at least in principle, include vibrational as well as configurationalcontributions. As both contributions are computationally very demanding in first-principlescalculations, the present work is divided in two parts, with a detailed accounts of each ofthese contributions.

Vibrational degrees of freedom are described in the quasiharmonic phonon model and thelinear response method is applied to determine force constants from first-principles calcula-tions. The hexagonal lattice poses a special challenge as two lattice parameters are required todescribe the system. The quasiharmonic phonon model is generalized to hexagonal systemsand a numerically tractable extended equation of state is developed to describe thermody-namic equilibrium properties at finite temperature.

The model is applied to ε-Fe3N specifically. Through the versatility of the model, equi-librium lattice parameters, the bulk modulus, and the thermal expansion coefficient can beobtained at any temperature of interest. The thermal expansion predicted by the generalizedquasiharmonic phonon model is in excellent agreement with experimental data. The modelalso allows calculation of the volume–pressure relationship at finite temperature, and goodagreement with experimental data is obtained also in this case.

In the second part, configurational degrees of freedom of the nitrogen occupation of the in-terstitial sites are investigated by thermodynamic statistical sampling, also known as MonteCarlo simulations, where nitrogen atoms are allowed to migrate randomly in a large com-puter crystal according to relative energies of the configurations until chemical equilibrium isreached. Configurational energies are described in an Ising-like cluster expansion determinedfrom a large database of calculated first-principles energies.

The model provides a description of collective effects of orderings of atoms and phasetransitions observed in large systems. Ensemble average long-range order parameters and theCowley–Warren short-range order parameters are calculated and provide evidence of specificorderings. The intermediate ε-Fe24N10 nitride is predicted as a ground-state structure andordering consistent with the structure is observed at finite temperature. An ε → ζ phasetransition is predicted with phase boundaries is excellent agreement with experimental data.The local environment of the iron atoms can be explicitly calculated in the computer crystaland are compared to recorded Mossbauer spectra. Finally, predictions of phase diagramsfrom first-principles calculations is demonstrated.

iii

Resume

Fundamentale termodynamiske modeller udvikles for det heksagonale tætpakkede ε-Fe-Nsystem. Dette system kan betragtes som bestaende af et heksagonalt tætpakket værtsgitteraf jernatomer med kvælstofatomer siddende pa et undergitter dannet af oktaeder interstiellepladser. Værtsgitteret af jernatomer antages fast.

Modellerne er udviklet udelukkende fra fundamentale kvantemekaniske ligninger gen-nem tæthedsfunktionalteoritilgangen with atomnummer og krystalstruktur som eneste para-metre. En fuldstændig termodynamisk beskrivelse skal, i det mindste i princippet, inklud-erer vibrationelle savel som konfigurationelle bidrag. Eftersom begge bidrag er meget tunge iberegningsmæssig forstand i kvantemekaniske beregninger, er dette arbejde inddelt i to delemed en detaljeret beskrivelse af hvert af disse bidrag.

Vibrationelle frihedsgrader beskrives i en kvasiharmonisk fonon model og den lineæreresponsmetode anvendes til at bestemme kraftkonstanter fra kvantemekaniske beregninger.Det heksagonale gitter giver særlige udfordringer da kræver to uafhængige gitterparametreat beskrive dette system. Den kvasiharmoniske fonon model bliver generaliseret til heksago-nale systemer og en numerisk let handterbar udvidet tilstandsligning udvikles for at beskrivetermodynamiske ligevægtsegenskaber ved positive temperature.

Modellen anvendes pa ε-Fe3N specifikt. Gennem den fleksible af modellen kan ligevægtsgitterparameter, kompressibilitetsmodul og den termiske udvidelseskoefficient bestemmesved enhver temperatur. Den beregnede termiske udvidelse ved den generaliserede kvasi-harmoniske model er i glimrende overensstemmelse med eksperimentelle data. Modellentillader ogsa beregning af rumfang-tryk relationer ved positiv temperature, og god overens-stemmelse med eksperimentelle data opnas ogsa i dette tilfælde.

I anden del studeres konfigurationelle frihedsgrader af kvælstofbesætning af de intersti-tielle pladser gennem termodynamisk stikprøvetagning, ogsa kendt som Monte Carlo simula-tioner, hvor kvælstof atoms tillades at migrere tilfældigt i en stor computerkrystal i henhold tilde relative energier af konfigurationerne indtil kemisk ligevægt opnas. De konfigurationelleenergier beskrives i en Ising-lignende klyngeekspansion bestemt fra en stor database af kvan-temekanisk beregnede energier.

Modellen given en beskrivelse af kollektive fænomener som ordninger af atomer og fase-overgange observeret i store systemer. Ensemblegennemsnit langtrækkende ordensparame-tre og Cowley-Warren kortrækkende ordensparametre beregnes og giver indikationer forspecifikke ordninger. Den mellemliggende ε-Fe24N10 nitrid bestemmes som en grundtilstand-struktur og ordning konsistent med denne struktur observeres ved positive temperature. Enε → ζ faseovergang forudsiges med faseovergangsgrænser i glimrende overensstemmelsemed eksperimentelle data. Det lokale jernatommiljø kan eksplicit bestemmes i computerkrys-tallen og sammenlignes med Mossbauer spektre. Slutteligt demonstreres hvordan fasedia-grammer kan bestemmes fra kvantemekaniske beregninger.

v

Contents

1 Introduction 11.1 Project description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 First-principles calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Computational platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theoretical foundation 72.1 First-principles thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The electronic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Electronic contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Vibrational contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Debye model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Thermodynamic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Configurational degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6.1 Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6.2 Grand-canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6.3 Separating degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . 162.6.4 Occupation probabilities, internal energy, and entropy . . . . . . . . . . 17

3 Energy–volume equation of state 193.1 One-dimensional energy–volume case . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Extension to hexagonal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Necessity of the extension and magnetic phase transitions . . . . . . . . . . . . 25

4 Thermal expansion of the ε-Fe6N2 structure 294.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Force constants and vibrational contributions . . . . . . . . . . . . . . . . . . . . 314.3 Fitting the equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Lattice parameters and thermal expansion . . . . . . . . . . . . . . . . . . . . . . 384.5 Volume–pressure relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.6 Negative pressure correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Partition function approach 455.1 The ε-Fe-N system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Thermodynamic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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5.3 Configurational state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.5 Structure screening and selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.6 Force constants and vibrational contributions . . . . . . . . . . . . . . . . . . . . 515.7 Fitting the equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.8 Helmholtz free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.9 Gibbs free energy and chemical potential . . . . . . . . . . . . . . . . . . . . . . 565.10 Lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.11 Possible two-phase region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.13 Feasibility of extension to the ε-Fe-C-N system . . . . . . . . . . . . . . . . . . . 61

5.13.1 Combinatorial explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.13.2 Non-separation of energies . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Thermodynamic statistical sampling 656.1 Markov chain algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2 Cluster expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2.1 Mean-field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Generic hexagonal close-packed system . . . . . . . . . . . . . . . . . . . . . . . 706.4 The ε-Fe-N system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.5 Review of earlier work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.6 Database of structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.7 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.8 Ground-state structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.8.1 ε-Fe6N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.8.2 ε-Fe6N3 and ζ-Fe8N4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.8.3 Intermediate ε-Fe24N10 nitride . . . . . . . . . . . . . . . . . . . . . . . . 88

6.9 Chemical potential and the sampling procedure . . . . . . . . . . . . . . . . . . 896.10 Site occupations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.10.1 Intermediate ε-Fe24N10 nitride . . . . . . . . . . . . . . . . . . . . . . . . 936.11 Long-range order parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.12 Cowley–Warren short-range order parameters . . . . . . . . . . . . . . . . . . . 996.13 ε→ ζ phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.13.1 Proposed ζ-Fe16N6 structure . . . . . . . . . . . . . . . . . . . . . . . . . 1066.14 Local environment of iron atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.14.1 Distinct sextet hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.14.2 Sextet-sextet interaction within the close-packed plane hypothesis . . . 1126.14.3 Sextet-sextet neighbor count hypothesis . . . . . . . . . . . . . . . . . . . 115

6.15 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.16 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.17 Equilibrium ζ phase lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . 1186.18 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7 Conclusions 121

A Chemical potential from experimental data 123

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B Thermodynamics of crystalline solids in chemical equilibrium with a reservoir 125

C Alternative database cluster expansion 127

D Integration of the electronic density of states 133

E Additional properties of the Gorsky–Bragg–Williams approximation 137

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Contents

x

List of Figures

1.1 ε-Fe-N unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electron density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Electronic density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Force constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Phonon density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Equations of state for Fe6N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Hexagonal lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Equation of state energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Relaxation of lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 Magnetic phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 ε-Fe6N2 unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Force constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Phonon density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Phonon dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 First Brillouin zone of the hexagonal close-packed unit cell . . . . . . . . . . . . 324.6 Helmholtz free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.7 Equation of state energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.8 Equation of state projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.9 Volume as a function of temperature . . . . . . . . . . . . . . . . . . . . . . . . . 384.10 Volumetric thermal expansion coefficient . . . . . . . . . . . . . . . . . . . . . . 394.11 Lattice parameters as a function of temperature . . . . . . . . . . . . . . . . . . . 404.12 Lattice parameters as a function of pressure . . . . . . . . . . . . . . . . . . . . . 414.13 Volume as a function of pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.14 Ratio of lattice parameters as a function of pressure . . . . . . . . . . . . . . . . 42

5.1 ε-Fe-N unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2 Partition function interstitial configurations . . . . . . . . . . . . . . . . . . . . . 495.3 Helmholtz free energy without vibrational contributions . . . . . . . . . . . . . 515.4 Force constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.5 Equation of state energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.6 Equation of state energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.7 Equation of state energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.8 Equation of state projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.9 Helmholtz free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

xi

List of Figures

5.10 Gibbs free energy and chemical potential . . . . . . . . . . . . . . . . . . . . . . 575.11 Equilibrium lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.12 Symmetrically distinct carbon unit cells . . . . . . . . . . . . . . . . . . . . . . . 635.13 Carbon equations of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1 Configurational state vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Mean-field configurational state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Generic hexagonal close-packed clusters . . . . . . . . . . . . . . . . . . . . . . . 706.4 Phase diagram of generic hexagonal close-packed system . . . . . . . . . . . . . 726.5 Statistical sampling chemical potential of generic hexagonal close-packed system 726.6 Hexagonal close-packed units cell . . . . . . . . . . . . . . . . . . . . . . . . . . 736.7 Fe-N phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.8 Configuration A and Configuration B . . . . . . . . . . . . . . . . . . . . . . . . 756.9 Mean-field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.10 Gorsky–Bragg–Williams Gibbs free energy . . . . . . . . . . . . . . . . . . . . . 776.11 Gorsky–Bragg–Williams chemical potential . . . . . . . . . . . . . . . . . . . . . 776.12 Gorsky–Bragg–Williams site occupations . . . . . . . . . . . . . . . . . . . . . . 786.13 Database unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.14 Clusters in the ε-Fe6N2 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.15 Cluster expansion clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.16 Energies of database structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.17 Magnetic moments of database structures . . . . . . . . . . . . . . . . . . . . . . 846.18 Unit cell lattice vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.19 ε-Fe6N2 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.20 ε-Fe6N3 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.21 ζ-Fe8N4 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.22 ε-Fe24N10 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.23 Statistical sampling chemical potential . . . . . . . . . . . . . . . . . . . . . . . . 906.24 Site occupations in the computer crystal . . . . . . . . . . . . . . . . . . . . . . . 926.25 Enlarged unit cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.26 Site occupations of 24-site unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . 956.27 Site occupations of alternative 12-site unit cell . . . . . . . . . . . . . . . . . . . 956.28 Computer crystal planes with yN = 5

12 . . . . . . . . . . . . . . . . . . . . . . . . 966.29 Order parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.30 Clusters in the ε-Fe6N3 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.31 Cowley–Warren short-range order parameters . . . . . . . . . . . . . . . . . . . 1016.32 Additional Cowley–Warren short-range order parameters . . . . . . . . . . . . 1026.33 ε→ ζ phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.34 ε→ ζ phase transition lattice parameters . . . . . . . . . . . . . . . . . . . . . . 1046.35 Computer crystal at phase transition at T = 573 K . . . . . . . . . . . . . . . . . 1046.36 Computer crystal at phase transition at T = 373 K . . . . . . . . . . . . . . . . . 1056.37 Computer crystal at T = 723 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.38 Proposed ζ-Fe16N6 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.39 Mossbauer spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.40 Iron atom neighborhoods of interstitial sites . . . . . . . . . . . . . . . . . . . . . 1086.41 Nitrogen location disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.42 Sextet occurrence as a function of temperature . . . . . . . . . . . . . . . . . . . 111

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6.43 Sextet occurrence as a function of yN . . . . . . . . . . . . . . . . . . . . . . . . . 1116.44 Sextets of perfectly ordered structures . . . . . . . . . . . . . . . . . . . . . . . . 1136.45 Statistical sampling chemical potential sensitivity analysis . . . . . . . . . . . . 1166.46 Statistical sampling phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.47 Expansion of the orthorhombic unit cell . . . . . . . . . . . . . . . . . . . . . . . 1186.48 Expansion of ζ-Fe8N4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.1 Nitriding potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C.1 Alternative database unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127C.2 Energies of alternative database structures . . . . . . . . . . . . . . . . . . . . . . 128C.3 Chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130C.4 Computer crystal at phase transition at T = 573 K . . . . . . . . . . . . . . . . . 131C.5 Cowley–Warren short-range order parameters . . . . . . . . . . . . . . . . . . . 132

D.1 Interpolation of electronic density of states . . . . . . . . . . . . . . . . . . . . . 135D.2 Number of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

E.1 Gibbs free energy of Configuration A and Configuration B . . . . . . . . . . . . 137E.2 Site occupations Configuration A and Configuration B . . . . . . . . . . . . . . 139

xiii

List of Figures

xiv

List of Tables

3.1 Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Wyckoff positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Lattice parameters, bulk modulus, and expansion coefficient . . . . . . . . . . . 354.3 Equation of state parameters for Fe6N2 . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Energy and lattice parameters as a function of pressure . . . . . . . . . . . . . . 41

5.1 Structure screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2 Equation of state parameters for Fe6N3 . . . . . . . . . . . . . . . . . . . . . . . . 545.3 Equation of state parameters for Fe12N4 . . . . . . . . . . . . . . . . . . . . . . . 545.4 Equation of state parameters for Fe12N5 . . . . . . . . . . . . . . . . . . . . . . . 545.5 Lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.6 Number of symmetrically distinct configurations . . . . . . . . . . . . . . . . . . 61

6.1 Generic hexagonal close-packed cluster expansion . . . . . . . . . . . . . . . . . 706.2 Majority and minority sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3 Cluster expansion coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.4 Cluster expansion structure energies . . . . . . . . . . . . . . . . . . . . . . . . . 846.5 Site occupations from neutron scattering . . . . . . . . . . . . . . . . . . . . . . . 936.6 Cowley–Warren short-range order parameters for perfectly ordered structures 1006.7 Mossbauer sextet relative occurrences . . . . . . . . . . . . . . . . . . . . . . . . 1086.8 Sextets of nitrogen atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.9 Sextets of perfectly ordered structures . . . . . . . . . . . . . . . . . . . . . . . . 1136.10 Sextet-sextet interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.11 Sextet-sextet out-of-plane interaction . . . . . . . . . . . . . . . . . . . . . . . . . 115

C.1 Cluster expansion coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xv

List of Tables

xvi

1Introduction

1.1 Project description

Thermodynamics of Fe-N phases is of crucial importance in understanding material behaviorobserved in the practice of nitriding of iron and steels, a thermochemical surface engineeringprocess applied to enhance wear and corrosion performance of steels. One phase of major im-portance is the iron-based ε-Fe2N1−z nitride, 0 < z ≤ 1

3 , developing on steel surfaces duringnitriding [1]. Despite a strong commercial interest and widespread application of nitriding—and nitrocarburizing in general—it is currently not possible to model the thermodynamicsof this phase from first-principles calculations. Crystallographically the phase can be con-sidered as an interstitial solid solutions of nitrogen in a hexagonal close-packed metal hostlattices. The system is visualized in Figure 1.1.

The ε-Fe-N system has been studied experimentally [2, 3, 4, 5, 6, 7, 8] and thermodynamicmodels accounting for long-range order of the nitrogen atoms have been developed and fittedto experimental data [1, 9, 10]. The ground-state structural stability at 0 K has also beeninvestigated in first-principles calculations [11, 12].

The advances in computational science and information technology over the last fewdecades have shifted materials research and development from empirical approaches to a

ab

c

Fe

Figure 1.1: The conventional hexagonal close-packed unit cell of the ε-Fe-N sys-tem. Nitrogen atoms are allowed to occupy the interstitial sites (black) of the hostlattice of iron atoms (gray). Six iron atoms and six interstitial sites exist per con-ventional unit cell. If the close-packed stacking sequence of the iron atom layersis ABAB . . . , then the interstitial sites are located on C layers between each ironatom layer, so that the stacking sequence is ACBCACBC . . . .

1

1 Introduction

new paradigm based on integrated computational-prediction and experimental-validation[13].

First-principles methods based on fundamental quantum mechanical equations have en-abled accurate calculations of electronic structures and total energies with the atomic numbersand crystal structures as the only input parameters, thereby allowing predictions of structuralstabilities. The density function theory (DFT) approach [14] have proved to be particularlysuccessful, and currently allows calculation of energies of systems of ten thousands of atomsin the most efficient implementations [15].

The project originally aimed at development of thermodynamic models of Fe-Cr-Ni-C-Nsystems based on the semi-empirical CALPHAD approach [16], where experimental data areintroduced as model parameters and configurational energies are calculated in a mean-fieldapproximation [17]. The aim was changed during the progress of the project to the more am-bitious goal of developing a thermodynamic model entirely from first-principles calculationsbased on the density functional theory approach, where energies of configurational statesexpressed as discrete occupation of individual atoms in the crystalline solid are calculatedexplicitly.

Thermodynamic models of configurational degrees of freedom based entirely on first-principles calculations have only recently emerged as viable options [18, 19, 20, 21, 22, 23],due to computational complexity of such approaches. The computational complexity growsexponentially with the number of components in the system, and systems containing morethan a few components are currently not feasible to describe in first-principles models. There-fore the investigation was restricted to a detailed description of the binary ε-Fe-N system.

Thermodynamic models incorporating both vibrational and configurational degrees offreedom for alloy systems are presenting major difficulties in first-principles calculations [24].Therefore two complementary direction of thermodynamic model development were chosenfor the present work.

Vibrational thermodynamic model

In the first part the vibrational thermodynamics of the ε-Fe6N2 structure is investigated, de-noted here by the size of its primitive unit cell for consistency with other calculations. Thisparticular structure is chosen due to extensive availability of experimental data in the litera-ture [2, 3, 4, 5, 7, 8]. Perfect configurational ordering is assumed, and configurational degreesof freedom are ignored in this part.

An accurate vibrational thermodynamic model is developed for the hexagonal system.First-principles vibrational calculations are performed in the quasiharmonic phonon model[25]. A smoothly defined parametrization of calculated first-principles energies is obtainedthrough an equation of state [26, 27], thereby allowing mathematically well-defined thermo-dynamic potentials to be obtained from the finite number of first-principle calculations.

The two independent lattice parameters of the hexagonal system pose a special challenge.The quasiharmonic phonon model is generalized to describe this situation, to the author’sknowledge the first time such a generalization has been developed. As a numerical pre-requisite an extended hexagonal equation of state is proposed for the two-dimensional case.Phonon dispersion relations of the quasiharmonic model is determined by the linear responsemethod in a large supercell [24] as closed-form expression for the second order energy deriva-tive through density functional perturbation theory [28].

The model allows predicting equilibrium lattice parameters at any temperature interest.

2

1.1 Project description

This allows prediction of the thermal expansion coefficient and excellent agreement with ex-perimental data [3] is obtained. The versatile model also allows predicting thermodynamicproperties at non-zero pressure, and good agreement between predicted finite temperatureproperties and high-pressures experimental data [7] is also obtained in this case.

Thermodynamic statistical sampling

In the other complimentary direction of model development, an accurate account of com-plex configurational ordering is investigated by thermodynamic statistical sampling [29], alsoknown as Monte Carlo simulations, of nitrogen occupations of interstitial sites in a large com-puter crystal. The configurational space is randomly sampled with the chemical potentialdetermining the number of nitrogen atoms in the grand-canonical ensemble.

The energies of the statistical sampling are approximated in a Ising-like cluster expan-sion model with coefficients calculated from first-principles energies [30]. Cluster expansionsprovide a practical pathway to establish the link between quantum mechanics and statisti-cal physics in alloy systems [31]. Previously, cluster expansions have mostly been employedto study generic model systems with the expansion coefficients chosen explicitly to demon-strate specific properties of generic model systems [32, 33], predicting complex physical be-havior such as short-range ordering (SRO), long-range ordering (LRO), and order-disorderphase transitions. Recently, however, cluster expansion coefficients have been determined forreal systems from extensive databases of first-principles calculations [18, 19, 20, 21, 22, 23],corresponding to the approach chosen in the present work.

First-principles energies are calculated for an extensive database of large structures al-lowing cluster expansion coefficients to be obtained [34]. Several ground-state structures areidentified, including the intermediate ε-Fe24N10 nitride hypothesized earlier [2]. Orderingconsistent with this structure is also identified in finite temperature statistically sampled com-puter crystals.

A ζ-Fe8N4 structure of orthorhombic ordering is predicted as a ground-state structure.The orthorhombic unit cell can be described by the hexagonal lattice if special symmetry con-ditions are imposed, allowing prediction of an ε→ ζ phase transition by the thermodynamicstatistical sampling. The phase transition and the predicted phase boundaries are in excellentagreement with experimental data [2].

Cowley–Warren short-range order parameters [35] are calculated as ensemble averagesof the sampled computer crystals and serve as signatures of specific orderings, providingevidence for both the ζ phase ordering and for ordering consistent with the ε-Fe24N10 at finitetemperature.

The local environment of the iron atoms can be explicitly calculated in the computer crys-tal and are compared to experimental data from recorded Mossbauer spectra [1, 10].

Hybrid thermodynamic model

In addition to these complementary approaches, a hybrid model is investigated by approx-imating the grand-canonical partition function directly. The model incorporates configura-tional contributions as well as electronic and vibrational contributions in the same unifiedmodel, and the configurational degrees of freedom is therefore more naive in this approach.The predicted relationship between chemical potential and nitrogen occupation obtainedfrom the Gibbs free energy is compared to experimental data from nitriding potentials [1].

3

1 Introduction

1.2 Dissertation outline

First-principles calculations and the density functional theory approach are briefly reviewedin the remaining sections of this introductory chapter.

The quantum mechanical foundation for finite temperature first-principles calculationsis introduced in Chapter 2. The quantum mechanical electronic problem and the Born–Oppenheimer approximation are introduced. The procedure for obtaining finite temperatureequilibrium properties in terms of the partition function is described in general, and contribu-tions from vibrational and electronic excitations are described specifically. The quasiharmonicphonon model is introduced. A description of configurational degrees of freedom is given.

The equation of state is introduced in Chapter 3, and the extended equation of state isdeveloped for the hexagonal system, serving as a prerequisite for the generalization of thequasiharmonic phonon model to hexagonal systems.

The finite temperature quantum mechanical theory and the extended equation of state areapplied to develop the vibration thermodynamic model for the ε-Fe6N2 structure in Chap-ter 4, allowing prediction of finite temperature equilibrium lattice parameters and thermalexpansion.

The hybrid model extension of the thermodynamic model is also investigated in grand-canonical ensemble using the direct partition function approach in Chapter 5.

The remaining part of the dissertation is devoted to the thermodynamic statistical sam-pling and cluster expansion approach in Chapter 6.

1.3 First-principles calculations

The time-independent Schrodinger equation in the simplest one-particle case

[−h2

2m

(∂2

∂x2 +∂2

∂y2 +∂2

∂z2

)+ V(x, y, z)

]ψ(x, y, z) = Eψ(x, y, z), (1.1)

where x, y, z are coordinates of the particle, m its mass, V is the external potential, and h is thereduced Planck constant. The wave function ψ and energy eigenvalue E are to be solved. Thedifferential operator on the left-hand side is the energy operator, the Hamiltonian, and oftenwritten in the compact form

H =−h2

2m∇2 + V, (1.2)

representing kinetic and potential energy, respectively.As the Schrodinger equation is fundamental to the quantum description of the physical

world—the relativistic version of this equation known as the Dirac equation correctly pre-dicts the magnetic moments of the electron to 15 significant digits1—methods that use theSchrodinger equation directly, or indirectly as discussed in the following section, are calledfirst-principles methods; no fitting to experimental data is required.

Throughout the present work operators and wave function are formulated mathemati-cally in the compact second quantization of quantum mechanics [37], a mathematical abstrac-

1This is a often quoted result in textbooks to demonstrate the accuracy of the quantum description, e.g., in [36].Relativistic means incorporating the special theory of relativity.

4

1.4 Density functional theory

tion that hides the tedious details of the underlying differential equations. Thus eigensolu-tions to (1.1) are written in the abstract notation

H |i〉 = Ei |i〉 , (1.3)

where Ei is the energy eigenvalue of the eigenstate |i〉.In general, quantum mechanical operators are marked with “hats” as in N, whereas real

physical quantities, such as eigenvalues Ni or ensemble averages N = 〈N〉 = ∑i piNi aremarked without “hats”. For the energy operator, the Hamiltonian, the convention is to useH for the operator, Ei for eigenvalues, and U = 〈H〉 for the internal energy. In mathematicalterms, quantum mechanical operators are typically differential equation operators whereasphysical quantities are scalars, i.e., simple real numbers.

1.4 Density functional theory

Electrons bound to nuclei to form atoms satisfy the multi-body Schrodinger equation, whichis a generalization of (1.1),

[−h2

2m ∑i∇2

i + ∑i

V(ri) + ∑i<j

U(ri, rj)

]Ψ(r1, . . . , rN) = EΨ(r1, . . . , rN), (1.4)

where ri = (xi, yi, zi),

∇2i =

∂2

∂x2i+

∂2

∂y2i+

∂2

∂z2i

. (1.5)

is the second-order partial differential operator, V is the external potential, and U is the po-tential of electron-electron repulsion.

Solving this directly as an eigenvalue problem is computationally very demanding and iscurrently only possible for very simple systems of a few electrons [38, 39, 40]; and that is onlythe case when the atomic nuclei are treated as non-quantum particles. The eigenstates—themany-dimensional wave functions—encode vast amounts of information, far more than theground-state energy required to determine structural stability of crystalline systems. Some ofthis can be discarded by folding the wave function,

ρ(r) = N∫

dr2 . . .∫

drN |Ψ(r, r2, . . . , rN)|2, (1.6)

so that the spatially defined electron density is obtained, measuring the density of the electroncloud around the nuclei as a function of position in space r.

Figure 1.2: Electron density in the three-dimensional space. Illustration from [41].

5

1 Introduction

Kohn and Sham [14] proved that the multi-body problem (1.4) can be transformed into aproblem of non-interacting electrons described in terms of the much simpler one-dimensionalequation (1.1). This is achieved by introducing the effective external potential

Veff[ρ](r) = V(r) + e2∫

dr′ρ(r′)|r− r′| + Vxc[ρ](r), (1.7)

where e is the electron charge and the correction Vxc is introduced obtained as the functionalderivative of the related exchange-correlation energy Exc. If ψi and Ei are wave function andeigenenergy solutions of the non-interacting problem, respectively, with Veff as the potentialin (1.1), the energy of the multi-body problem is

E[ρ] = ∑i

Ei −e2

2

∫dr∫

dr′ρ(r)ρ(r′)|r− r′| + Exc[ρ]−

∫dr ρ(r)Vxc(r), (1.8)

where electron density of the non-interacting satisfies

ρ(r) = ∑i|ψi(r)|2 (1.9)

andN =

∫dr ρ(r) (1.10)

to keep to total number of electrons fixed.Thus the energy is expressed in terms of the electron density ρ through a coupled set of

equations. The ground-state energy is solved self-consistently such that (1.8) is minimized.The electron density is a function itself, so the energy E[ρ] is a functional, i.e., a function

of functions, and for this reason such approaches are called density functional theory (DFT).Provided the correct expression for the exchange-correlation functional Exc is known, exactground-state energy of the multi-body problem can in principle be obtained. Unfortunately,the exchange-correlation functional is generally not known exactly and various approxima-tions have to be used. These are often based on limiting cases where more precise expressionsfor the exchange-correlation functional can be obtained.

One important advantage of Kohn–Sham equations is that the numerical difficulty in solv-ing the problem—the computational time—scales quadratically with the number of atoms forplane-wave basis approaches [42] or linearly for the wavelets basis approach [15], much betterthan the exponential scaling of the Schrodinger equation itself.

The mathematical and implementational details of the density functional approaches arenot given in the present work as these are readily available in textbooks [41, 43, 44]. See alsoBaroni et al. [28] on which this short review is based for more details.

1.5 Computational platform

First-principles calculations throughout the present work are performed using the Vienna Abinitio Simulation Package (VASP) [42, 45], an efficient DFT solver that works in parallel clustercomputing environments and is implemented using plane-wave basis expansions, especiallysuitable for bulk materials. Details of the first-principles calculation parameters are given inthe following chapters.

The thermodynamic statistical sampling is performed using code developed by the author,running on the same parallel cluster computing environment.

6

2Theoretical foundation

In this chapter a rigorous mathematical framework for describing thermodynamics—partic-ularly configurational degrees of freedom—in a setting of first-principles calculations is pre-sented, as such a formal treatment seem to be lacking in the literature. The required mathe-matical formalism for describing smoothly defined thermodynamic potentials in macroscopicvariables, such as volume and temperature, in terms of the discrete set of eigenstates and en-ergy eigenvalues of the fundamental quantum mechanical equations is established.

The grand-canonical ensemble and the grand-canonical ensemble Hamiltonian are intro-duced. The application of the grand-canonical Hamiltonian itself is a standard textbook exer-cise [36], but is almost always applied to electronic systems only. The same general approachto the configurational contribution to the thermodynamic potentials as electronic and vibra-tional contributions is employed.

2.1 First-principles thermodynamics

The quantum mechanical problem of determining energies and wave functions of atoms in acrystalline solid is solved by treating the atomic cores, the nuclei, as classical charged particles,and the only electrons as quantum mechanical particles. The energy is determined from theeigenvalues of the energy operator, the Hamiltonian, which is a partial differential operatorin the electron coordinates. The actual mathematical details of this generally very difficultproblem are not described in the present work. Instead the powerful second quantizationof quantum mechanics is chosen, where these technicalities of the differential equations aremasked by the convenient creation and annihilation operators [37, 46].

The effect of finite temperature, i.e., T > 0, is that also non-ground states become occu-pied. In the thermodynamic equilibrium setting assumed in the present work, the ensembleoccupation probabilities are given by the energies of the excited states with the probabilitiesdecreasing exponentially as energies are increased from the ground state.

The Fermi temperature, defining the extent to which electronic excited states are occupied,is often an order of magnitude larger than the melting point of the crystal [36], so generallyonly the electronic ground-state energy is of interest. Approximations exist to quantify elec-tronic free energies from the density functional approach for finite temperatures.

7

2 Theoretical foundation

2.2 The electronic problem

The most fundamental quantity in the mathematical formulation of quantum mechanics is theenergy differential equation operator, also known as the Hamiltonian, given in the simplestform in (1.2). In multi-body electronic problems of crystalline solids it is often decomposed as

H = H0 + Hel + Hph + Hel-ph (2.1)

with static nuclei repulsion H0, an electronic part Hel, and a vibrational part Hph. The elec-tron-phonon interaction Hel-ph is completely neglected in the present work.1 H0 includesstatic Coulomb repulsion between the positively charged nuclei. The electronic and vibra-tional terms are described in more details in the following sections. The exact mathematicaldetails are beyond the scope of this work and are generally omitted. Details of the electronicproblem are available in the literature [28, 36].

In solving the eigenstates and eigenenergies of (2.1), it is noted that the nuclei are muchheavier in mass compared to the electrons, and their motions can therefore be ignored insolving the electronic Schrodinger equation. This is the Born–Oppenheimer approximation[47]: the motion of the nuclei and the electrons can be separated and the electronic and nuclearproblems can be solved with independent wave functions. When accounting for vibrationalcontributions of the nuclei in the lattice, the electron cloud is thus assumed to instantaneouslyrearrange itself around the moving nuclei in this approximation [28].

It follows that the arrangements of the nuclei can be considered as a parameter of thequantum mechanical problem with the positions of the nuclei described by an elastic dis-placement parameter r, so that the wave function eigensolutions only depend on the remain-ing positions of the electrons. In a crystalline solid the parameter r implicitly contains thelattice parameters of the unit cell.

As electronic and vibrational degrees of freedom are assumed decoupled, the eigenstatesof (2.1) can be described with respect to an abstract basis |εω; r〉,2 representing vibrational ωand electronic ε eigenstates, respectively, and the elastic relaxation parameter r.

Finite temperature mixed mode states of a thermodynamic ensemble are described bythe quantum mechanical density operator [48, 43]. In thermodynamic equilibrium with anexternal heat reservoir this is given by

ρ = e−βH = ∑ω

∑ε

|εω; r〉 e−βEωε;r 〈εω; r| , (2.2)

where Eωε;r is the energy eigenvalue of the state |εω; r〉, i.e., H |εω; r〉 = Eωε;r |εω; r〉, β =(kBT)−1 is reciprocal temperature, kB is the Boltzmann constant, and T is absolute tempera-ture. It must be emphasized that this mixed mode state describes thermodynamic equilibriumonly. The exact mechanism and timescale of equilibration are not of interest.

The partition function Z is defined as the trace of the density operator,

Z = tr ρ = ∑ω

∑ε

e−βEωε;r . (2.3)

1While electron-phonon interaction is crucially important in describing superconductivity, electronic and vi-brational degrees of freedom are often assumed to be decoupled. See discussion below.

2“Abstract” refers to “unspecified” here; the exceedingly complicated mathematical details of the eigensolu-tions of the differential operator (2.1) are omitted.

8

2.3 Electronic contribution

The trace, and therefore also Z, is independent of the chosen basis, i.e., the actual choiceof |εω; r〉 is not important. The usual macroscopic thermodynamic potentials and derivedproperties can be obtained from the partition function. Details are given in Section 2.5.

2.3 Electronic contribution

The electronic part of the Hamiltonian is expressed in a plane-wave basis,

Hel = ∑kσ

εkσ c†kσ ckσ, (2.4)

where k is the wave vector, σ = ↑, ↓ is the electronic spin (σ is the standard symbol for elec-tronic spin; the notation is used in this section only), c†

kσ is the electron creation operator, ckσits adjoint annihilation operator, and an energy dispersion relation εkσ.

The electronic contribution is described in a grand-canonical ensemble, controlling thenumber of electrons N = ∑kσ c†

kσ ckσ by the electronic chemical potential µ. The partitionfunction is

Zel = tr e−β(Hel−µN) = ∏kσ

(1 + e−β(εkσ−µ)

), (2.5)

and the electronic free energy is then

Fel = −β−1 ∑σ

∫dε dσ(ε) log

(1 + e−β(ε−µ)

), (2.6)

where the spin-polarized electronic density of states

dσ(ε) = ∑k

δ(ε− εkσ) (2.7)

has been introduced.The ground-state number of (valence) electrons in the crystalline solid satisfies

〈Nel〉∣∣T=0 = ∑

σ

∫ εF

−∞dε dσ(ε) = ∑

kσ

∫ εF

−∞dε δ(ε− εkσ) = ∑

kσ

1εkσ<εF , (2.8)

where εF is the Fermi energy. The chemical potential µ is determined by the condition

〈Nel〉 = ∑σ

∫dε dσ(ε)nF(ε) = 〈Nel〉

∣∣T=0, (2.9)

to maintain constant number of electrons in the crystal,3 where

nF(ε) =1

eβ(ε−µ) + 1(2.10)

is the Fermi–Dirac distribution. This condition must be satisfied in the calculation.In general, plane-wave basis spanned by the states |kσ〉 = c†

kσ |0〉 cannot be assumed todiagonalize Hel, but by construction this is the case in the density functional approach withthe effective exchange-correlation potential (1.7). In that case the density of states (2.7) andthe corresponding free energy are defined with respect to this transformed system instead of

9


0

2

4

6

8

10

12

−15 −10 −5 0 5 10 15 20

Den

sity

,eV−

1

Energy, ε, eVεF

d↑d↓

(a) ε-Fe6N2

0

2

4

6

8

10

12

−15 −10 −5 0 5 10 15 20

Den

sity

,eV−

1

Energy, ε, eVεF

d↑d↓

(b) ε-Fe6N3

Figure 2.1: Calculated spin-polarized electronic density of states dσ for hexagonalclose-packed structures at equilibrium volume. The integrated density below theFermi energy εF is the number of electrons 〈Nel〉

∣∣T=0 per six-site unit cell. The

electronic contribution to the free energy is evaluated from the density of statesdσ(ε) and Fermi energy εF, both calculated from first-principles as described inSection 2.3. To obtain a smooth volume-dependent contribution and to maintainconsistency between εF and 〈Nel〉 careful interpretation of the VASP output data isimportant: Values are given as integrated density of states Dσ(ε) =

∫ ε−∞ dε′dσ(ε′)

and intermediate values are assumed to be linearly interpolated; the density ofstates dσ(ε) itself is then a piece-wise constant step function. See Appendix D formore details. It should also be noted that the density of states is calculated byVASP using an effective DFT potential that diagonalizes Hel in the plane-wavebasis and not the true Coulomb potential. It is nonetheless used here as a bestestimate and in anticipation that the electronic contribution to the free energy willbe small far below the Fermi temperature εF/kB ' 8× 104 K. This is the usual—and often unstated—assumption in first-principles thermodynamic calculationsusing DFT [49].

the real physical system. This fact is often ignored in actual calculations as the assumptionprovides the best available estimate, and since the contribution from electronic excitations isusually dominated by other contributions far below the Fermi temperature.

The spin-polarized density of states and the Fermi energy are calculated in the first-prin-ciples code. Examples of such calculations are given in Figure 2.1 for the hexagonal close-packed ε-Fe-N system. See Appendix D for discussion of the correct interpretation of theoutput reported by the first-principles calculation.

2.4 Vibrational contribution

Phonons are quantized collective oscillations of the nuclei in the crystal. In the harmonic andquasiharmonic model the contribution is described by

Hph = ∑kλ

hωkλ

(b†

kλbkλ +12

), (2.11)

3Obviously, the total positive charge of the nuclei does not change with temperature, so the solid to remaincharge neutral, the total number of electrons must also be kept constant.

10


where k is the wave vector, λ is the branch index, b†kλ is the phonon creation operator, and

bkλ is its adjoint annihilation operator.The relation establishing the energy εkλ for a given wave vector k is known as the disper-

sion relation,εkλ = hωkλ, (2.12)

depending implicitly on the elastic displacement parameter r to allow quasiharmonic oscilla-tions. Specifically, ωkλ is determined independently for each pair of lattice parameters (a, c)in the decomposition r = (a, c, r′), with the remaining nuclei position parameter r′ definingthe Hessian as described in Section 2.4.1. Introducing this dependence is crucial in the gener-alization of the quasiharmonic phonon model.

Phonons, being bosonic quasiparticles, are not conserved, and hence cannot have a non-zero chemical potential. In finite crystals only a finite number of vibrational states is allowedand the partition function is4

Z = tr e−βHph= ∏

kλ

e−βhωkλ/2

1− e−βhωkλ, (2.13)

so the contribution to the free energy is

Fph =12 ∑

kλ

hωkλ + β−1 ∑kλ

log[1− e−βhωkλ

]= β−1 ∑

kλ

log[

2 sinhβhωkλ

2

]. (2.14)

Introducing density of statesd(ω) = ∑

kλ

δ(ω−ωkλ) (2.15)

as for electrons, the free energy can be expressed as

Fph = β−1∫

dω log[

2 sinhβhω

2

]d(ω). (2.16)

The non-zero ground-state energy corresponding to the limit β→ ∞ is

Fph =h2

∫dω ωd(ω). (2.17)

Thus the vibrational contribution is non-zero even for T = 0. This is a quantum mechanicalproperty of the harmonic oscillator originating from the last term in (2.11).5 In the presentwork, this changes ground-state equilibrium lattice parameters slightly compared to calcula-tions performed without accounting for vibrational contributions.

2.4.1 Linear response

The force constant matrix is determined for each pair of lattice parameters (a, c) as the Hessianof the elastic energy E(r′) = E0;r=(a,c,r′) in the remaining elastic displacement variables r′, i.e.,

4The set of wave vectors k satisfying boundary conditions in a macroscopic crystal spans a very fine mesh inthe reciprocal space. If the finite crystal contains N unit cells in total, each with p nuclei, the sum above has 3pNterms, the number of phonon normal modes. In general this number is very large (in the order of 1023), and thesum approaches a continuous integral. The assumption is that the physics of a finite crystal resembles that of anideal infinite crystal since this number is so exceedingly large, except for a tiny fraction of atoms sufficiently closeto the boundary. See full derivation of partition function in [50].

5Direct experimental evidence of this zero-point energy is observed in properties of liquid helium at ambientpressure [51].

11


the positions of the nuclei,

Φ(jj′, nn′) =∂2E

∂ujn∂uj′n′′ , (2.18)

defined for pairs of nuclei jn and j′n′, where ujn is the position of the nucleus j in unit cell n.The Hessian itself is calculated in the first-principles code by generalizing the self-consistentconditions of density functional theory approach to yield analytical expressions for second-order energy derivatives in the density functional perturbation theory (DFPT) approach [28].

The dynamic matrix D(k) follows with 3× 3 blocks defined as

D(jj′, k) = (mjmj′)−1/2 ∑

n′Φ(jj′, 0n′)eik·(rj′n′−rj0 ) (2.19)

for phonon wave vector k, where mj is the mass of nucleus j. Phonon frequencies ωkλ andpolarizations ekλ are the solutions to the eigenvalue equation

D(k)ekλ = ω2kλekλ (2.20)

from which the density of states is determined by integrating (2.15). A finite unit cell smooth-ing scheme must be used for actual calculations [52].

The force constants (2.18) are calculated with the nuclei positions r′ corresponding to min-imum energy values for each pair of lattice parameters (a, c) in the decomposition r = (a, c, r′)of the elastic displacement parameter, i.e., with nuclei positions r′ fully relaxed for any givenlattice parameters.

Examples of calculated force constants are given in Figure 2.2 for the ε-Fe-N system. Theresulting densities of states are given in Figure 2.3. More computational details are given inChapter 4 and Chapter 5.

12


−6

−4

−2

0

2

4

6

1 2 3 4 5 6 7−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Stre

tchi

ngfo

rce

cons

tant

,eV

/A

2

Ben

din

gfo

rce

cons

tant

,eV

/A

2

Bond length, A

Fe-NFe-FeN-N

(a) ε-Fe6N2

−6

−4

−2

0

2

4

6

1 2 3 4 5 6 7−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Stre

tchi

ngfo

rce

cons

tant

,eV

/A

2

Ben

din

gfo

rce

cons

tant

,eV

/A

2

Bond length, A

Fe-NFe-FeN-N

(b) ε-Fe6N3

Figure 2.2: Calculated force constants for hexagonal close-packed structures as afunction of bond length. Stretching force constants (black) and the weaker bendingforce constants (blue, secondary axis) for the Fe-N, Fe-Fe, and N-N bonds. Theforce constants are calculated for equilibrium lattice parameters.

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18

Den

sity

,d,T

Hz−

1

Frequency, ω, THz

(a) ε-Fe6N2

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20

Den

sity

,d,T

Hz−

1

Frequency, ω, THz

(b) ε-Fe6N3

Figure 2.3: The phonon density of states obtained for the force constants given inFigure 2.2.

13


2.4.2 Debye model

In the simpler Debye model a perfectly linear dispersion relation is assumed, ωkλ = vk,with sound velocity v and length k of the wave vector k, in a reduced zone scheme where theoptical branches are projected onto the three acoustic branches λ. Vibrations are allowed up toa maximum frequency ωD, often expressed in terms of the characteristic Debye temperatureΘD, hωD = kBΘD, determined by the vibrational degrees of freedom of the system. Fromthese assumptions

d(ω) =V

2π2ω2

hv3 (2.21)

for ω ≤ ωD, and the free energy

Fph = 3β−1D3(βhωD) (2.22)

follows from (2.16), where D3(x) =∫ x

0 dt t3/(et − 1) is the Debye function.

2.5 Thermodynamic potentials

Thermodynamic potentials defined in terms of macroscopic variables are obtained from thepartition function (2.3). For a given elastic displacement parameter r, the Helmholtz freeenergy is

Fr = −β−1 log Z, (2.23)

or if decomposing r = (a, c, r′), with lattice parameters (a, c) of the hexagonal unit cell andthe remaining relative nuclei positions parameter r′,

Fac = minr′

Fr=(a,c,r′), (2.24)

with nuclei at equilibrium positions, i.e., the value of r′ that minimizes (2.24). If Vac denotesthe volume of the hexagonal unit cell, the Helmholtz free energy expressed in the usual vari-ables is

F(T, V) = minVac=V

Fac, (2.25)

where V is the volume. This corresponds to the pair of lattice parameters that minimizes theenergy for a given fixed unit cell volume.

The expressions above define the bridge between the quantum mechanical description ofthe system (subscript notation) and a description in terms of macroscopic variables (functionnotation).6 The Helmholtz free energy is minimized at equilibrium when constant tempera-ture and volume are imposed.

The established smoothly defined energy–volume relationship (2.25), whether defined di-rectly from the partition function or described by the equation of state expansion defined inChapter 3, allows deriving thermodynamic properties and other thermodynamic potentialsthrough a series of Legendre transforms [53]. The most important derivations are given ex-plicitly below.

The external hydrostatic pressure needed to deform the structure to a given volume V isgiven by

p(T, V) = − ∂F∂V

(T, V), (2.26)

6The volume normalized to unit cells is accepted as a macroscopic variable here.

14

2.6 Configurational degrees of freedom

and the isothermal bulk modulus is

B(T, V) = V∂2F∂V2 (T, V) = −V

∂p∂V

(T, V). (2.27)

The equilibrium volume V0 for a given pressure p0 and temperature T is defined throughthe relation p(T, V0) = p0. The pair of thermodynamic conjugate variables V and p is estab-lished by the change-of-variables mapping

(T, V) 7→ (T, p(T, V)). (2.28)

The volumetric thermal expansion coefficient is determined using the inverse of the Jaco-bian of change-of-variables mapping,

α(T, p0) =1

V(T, p0)

∂V∂T

(T, p0) = −∂p∂T

(T, V0)(

V0∂p∂V

(T, V0))−1

=∂p∂T

(T, V0)1

B(T, V0),

(2.29)where V0 = V(T, p0) is the temperature and pressure dependent equilibrium volume and

∂p∂T

(T, V) = − ∂2F∂V∂T

(T, V) (2.30)

is the mixed derivative known from the Maxwell relations.Finally, the Gibbs free energy is obtained through the Legendre transform

G(T, p) = F(T, V(T, p)) + pV(T, p), (2.31)

from (2.25) and the inverse of the change-of-variables mapping (2.28). The Gibbs free energyis minimized at equilibrium when constant temperature and pressure are imposed.

The thermodynamic potentials are given in general form including configurational de-grees of freedom in Section 5.2.


Configurational degrees of freedom are introduced by allowing nuclei to move between lat-tice sites of the crystal. The arrangements of the nuclei are described by the configurationalstate σ, parameterizing nuclei positions in the crystal such that small deviations form ideal-ized lattice positions are ignored.

The basis introduced in Section 2.2 to describe states of the system is extended to includeconfigurational degrees of freedom |σεω; r〉, though it is important to realize that σ is formallya parameter in the quantum mechanical setting of the Born–Oppenheimer approximation. Aspecific example of the parametrization of σ is given in Section 5.3.

Two ensembles are considered in the present work corresponding to whether conditionson the total number of particles or of constant chemical potential are imposed. Distinguishingbetween these two cases is particularly important in the thermodynamic statistical samplingin Chapter 6.

15


2.6.1 Canonical ensemble

In the canonical ensemble heat is allowed to be exchanged with an external reservoir at somefixed temperature, but the total number of nuclei is assumed fixed. The partition function ismodified to include configurational degrees of freedom,

Z = tr ρ = ∑σ

∑ω

∑ε

e−βEσωε;r , (2.32)

where Eσωε;r is the eigenenergy of |σεω; r〉.

2.6.2 Grand-canonical ensemble

In grand-canonical ensemble particles as well as heat are also allowed to be exchanged withan external reservoir. For brevity only one distinct type of particle is considered. The corre-sponding grand-canonical ensemble Hamiltonian is obtained by the transformation

HG = H − µN, (2.33)

where N is the particle number operator and µ is the chemical potential. In equilibrium,imposing constant chemical potential, the density operator is given by ρG = e−βHG , and thecorresponding grand-canonical ensemble partition function is given by

ZG = tr ρG = ∑σ

∑ω

∑ε

e−β(Eσωε;r−µNσ). (2.34)

The mixed mode thermodynamic state of the system is given by the density operator ρG,similar to (2.2), and ensemble averages of a physical quantity A is given by

〈A〉G =tr ρG A

ZG. (2.35)

defined in terms of the density operator ρG and the derived partition function ZG.Compared to the canonical ensemble, the chemical potential µ is the exogenous variable

that determines grand-canonical ensemble average number of particles 〈N〉G, closely resem-bling Lagrange multipliers known from mathematical constrained optimization.

For brevity H, Z, and 〈 · 〉 are often used to denote grand-canonical ensemble quantitiesinstead of HG, ZG, and 〈 · 〉G, respectively [36]. This practice is followed in the present workfrom this point on when no ambiguity exists.

2.6.3 Separating degrees of freedom

It is often useful to express the grand-canonical partition function (2.34) in terms of configura-tional degrees of freedom only. During the lifetime of a particular configuration σ, electronicand vibrational equilibrium will be attained very rapidly and can be assumed to be estab-lished at any time with a corresponding well-defined partition function [29],

Zel+phσ = ∑

ω∑

ε

e−βEσωε;r (2.36)

16


identical to (2.3) of the non-configurational electronic problem. A hybrid potential corre-sponding to the Helmholtz free energy is introduced though this partition function,

Fel+phσ;r = −β−1 log Zel+ph

σ . (2.37)

Similarly, in Born–Oppenheimer approximation, electronic equilibrium can be assumedto be established at any time as nuclei vibrate, so for a given configuration σ and a vibrationalexcitation ω,

Zelσω = ∑

ε


andFel

σω;r = −β−1 log Zelσω. (2.39)

Collecting these terms, the grand-canonical partition function (2.34) can be decomposedin terms of one of these two hybrid potentials,

Z = ∑σ

e−β(Fel+phσ;r −µNσ) = ∑

σ∑ω

e−β(Felσω;r−µNσ), (2.40)

with the free energies Felσω;r and Fel+ph

σ;r appearing in the partition function as the true eigenen-ergies in (2.34).

Therefore, if Fel+phσ;r can be determined, configurational free energy can be calculated in-

dependently of the electronic and vibrational degrees of freedom. This also proves that thethermodynamic statistical sampling method presented in Chapter 6 is valid even if electronicand vibrational degrees of freedom are included.

2.6.4 Occupation probabilities, internal energy, and entropy

The configurational contribution to the free energy is of particular interest as that is requiredfor the thermodynamic statistical sampling in Chapter 6 and is therefore given explicitly inthis section. Assuming the partition function decomposed as described in Section 2.6.3,

Z = ∑σ

e−β(Eσ−µNσ), (2.41)

where Eσ is the hybrid potential, the probability pσ of encountering a particular configurationσ in the ensemble is proportional to the Boltzmann factor

pσ =e−β(Eσ−µNσ)

Z. (2.42)

Hence the ensemble average energy

U = 〈H〉 = ∑σ

pσEσ, (2.43)

entropy

S = −kB

⟨log

ρ

Z

⟩= −kB ∑

σ

pσ log pσ, (2.44)

and particle numberN = 〈N〉 = ∑

σ

pσNσ (2.45)

17


are obtained.The grand potential defined through the partition function can then also be expressed in

terms of internal energy, entropy, and particle number,

Ω = −β−1 log Z

= −β−1 log ZZ ∑

σ

e−β(Eσ−µNσ)

= ∑σ

e−β(Eσ−µNσ)

Z

(Eσ + β−1 log

e−β(Eσ−µNσ)

Z− µNσ

)

= ∑σ

pσ

(Eσ + β−1 log pσ − µNσ

)

= U − TS− µN.

Thus the more familiar expression of the grand potential Ω is obtained.

18

3Energy–volume equation of state

The thermodynamic potentials introduced in Section 2.5 establish the link between discretequantized states of the electronic quantum mechanical problem and smoothly defined macro-scopic variables. As it is not feasible to calculate first-principles energies for a continuum ofparameter values, e.g., for smoothly defined lattice parameters, some parametrization of thethermodynamic potentials are required, known as an equation of state (EOS) [26]. In thischapter an equation of state is developed for the hexagonal system, where two independentlattice parameters pose an additional challenge. The hexagonal equation of state serves as oneof the numerical prerequisites to properly study thermal expansion of the ε-Fe6N2 structurefrom first-principles.

Before the hexagonal case is treated, the simpler one-dimensional energy–volume case isshortly reviewed. The hexagonal equation of state is then introduced as an extension to theone-dimensional case.

3.1 One-dimensional energy–volume case

The Helmholtz free energy at some fixed temperature T as defined from (2.25),

E(V) = F(T, V), (3.1)

can be determined from first-principles for any given unit cell volume V, thereby, in princi-ple, defining a smooth function. It is infeasible to calculate this for more than a small num-ber of points; therefore, a smooth energy–volume relationship is established by one of themany equations of state presented in the literature [26], and may be regarded as an interpo-lation between explicitly calculated first-principles energies, or, more precisely, as an inexactsmooth parameterization of the calculated energies. The smoothness is important for cal-culation of derived thermodynamic properties. In the present work, the Birch–Murnaghanequation (BM) [27],

E(V) = E0 +m−1

∑i=2

Ei

[(VV0

)−n/3

− 1]i

, (3.2)

is chosen, which by the binomial theorem is just a polynomial expansion in V−n/3, withE0, V0, E2, . . . , Em−1 as parameters. The order parameter n is some fixed integer. As the first-order expansion in V around V0 vanishes, E0 and V0 are identified as the equilibrium en-

19

3 Energy–volume equation of state

ergy and equilibrium volume, respectively, and the other parameters are higher-order non-equilibrium corrections.1 This formulation is preferred to the direct polynomial expansion asthe latter tends to be numerically unstable with very large coefficients. In the original Birch–Murnaghan equation of state the order parameter n = 2 is used; this is also the choice in thepresent work.

Since the expansion (3.2) can be performed for any temperature T, and since the elec-tronic contribution (2.6) and the vibrational contribution (2.16) both are smoothly defined intemperature, a smoothly defined function

(T, V) 7→ E(V)|T (3.3)

can be assumed in both temperature and volume. The smoothness in temperature is pre-served for the parametrized equation of state if the fitting procedure is applied consistentlyacross temperatures.

Examples of fitted one-dimensional energy–volume equations of state are given in Fig-ure 3.1 and Figure 3.2 for ground-state energies of the ε-Fe-N system. The correspondingequilibrium values are listed in Table 3.1, where also the bulk modulus and its derivativewith respect to pressure have been calculated for the smoothly defined equation of state asdescribed in Section 2.5. Details of the fitting procedure are given in Section 4.3.

1It is possible to parametrize the equation of state so that also the bulk modulus B0 and its derivative withrespect to pressure B′0 are parameters instead of derived quantities [26], but this is not chosen in the present workfor simplicity.

20

−48.2

−48.1

−48.0

−47.9

−47.8

−47.7

−47.6

66 68 70 72 74 76 78 80 828.0

10.0

12.0

14.0

16.0

18.0E

nerg

y,E

,eV

Mag

neti

cm

omen

t,µ

/µ

B

Volume, V, A3

(a) Fe6

−57.6

−57.5

−57.4

−57.3

−57.2

−57.1

68 70 72 74 76 78 80 82 8411.0

12.0

13.0

14.0

15.0

Ene

rgy,

E,e

V

Mag

neti

cm

omen

t,µ

/µ

B

Volume, V, A3

(b) Fe6N

−66.7

−66.6

−66.5

−66.4

−66.3

−66.2

−66.1

74 76 78 80 82 84 86 88 9011.0

11.5

12.0

12.5

13.0

13.5

14.0

Ene

rgy,

E,e

V

Mag

neti

cm

omen

t,µ

/µ

B

Volume, V, A3

(c) Fe6N2

−74.8

−74.6

−74.4

−74.2

−74.0

−73.8

78 80 82 84 86 88 90 92 94 966.0

7.0

8.0

9.0

10.0

11.0

12.0

Ene

rgy,

E,e

V

Mag

neti

cm

omen

t,µ

/µ

B

Volume, V, A3

(d) Fe6N3

−81.3

−81.2

−81.1

−81.0

−80.9

−80.8

80 82 84 86 88 90 92 94 962.0

4.0

6.0

8.0

10.0

12.0

14.0

Ene

rgy,

E,e

V

Mag

neti

cm

omen

t,µ

/µ

B

Volume, V, A3

(e) Fe6N4

−87.8

−87.6

−87.4

−87.2

−87.0

−86.8

−86.6

82 84 86 88 90 92 94 960.0

2.0

4.0

6.0

8.0

10.0

Ene

rgy,

E,e

V

Mag

neti

cm

omen

t,µ

/µ

B

Volume, V, A3

(f) Fe6N5

−94.0

−93.0

−92.0

−91.0

−90.0

84 88 92 96 100 1044.0

6.0

8.0

10.0

12.0

14.0

16.0

Ene

rgy,

E,e

V

Mag

neti

cm

omen

t,µ

/µ

B

Volume, V, A3

(g) Fe6N6

Figure 3.1: First-principles ground-state energies (square) for a selection of unitcell volumes V for each of the symmetrically distinct configurations of nitrogenoccupations of interstitial sites of the conventional hexagonal unit cell (lowest en-ergy configurations only; cf. Table 3.1). A four-parameter Birch–Murnaghan equa-tion of state (3.2) is fitted to the calculated energies (solid line). Some points (graysquare) are discarded as discontinuities in the magnetic moment (blue circle, sec-ondary axis) indicate possible volume-induced magnetic phase transitions. Ener-gies, volumes, and magnetic moments are per six-site conventional unit cell. Lat-tice parameters are relaxed using VASP’s optimization algorithm for each volume;this is refined for the hexagonal lattice in Section 3.2.


σ d E0, eV V0, A3 B0, GPa B′0

Fe6 1 −48.2 72.2 161 4.40

Fe6N 6 −57.5 75.2 163 4.73

Fe6N2 6 −66.7 80.9 198 5.70

6 −66.0 80.3 186 5.81

3 −64.6 82.3 164 3.46

Fe6N3 6 −74.7 84.3 239 2.91

2 −73.4 86.6 146 3.16

12 −72.8 85.6 162 6.45

Fe6N4 6 −81.3 87.1 272 5.11

6 −80.6 88.3 212 5.15

3 −80.0 88.2 190 7.07

Fe6N5 6 −87.7 91.7 288 4.27

Fe6N6 1 −93.2 99.2 284 4.11

Table 3.1: Nitrogen occupation configuration σ, degeneracy d, equilibrium en-ergy per unit cell E0, equilibrium volume per unit cell V0, bulk modulus B0, andderivative of bulk modulus with respect to pressure B′0. Occupied (black) and un-occupied (white) sites are visualized for each configuration. The table lists groundstate values without contributions for electronic and vibrational excitations. Thedegeneracy is the number of symmetrically equivalent nitrogen occupations ofthe six-site conventional hexagonal unit cell. The sum of the degeneracies satisfies∑σ dσ = 26, corresponding to the total number of possible nitrogen occupationconfigurations of the six sites.

22

3.1 One-dimensional energy–volume case

−67.0

−66.5

−66.0

−65.5

−65.0

−64.5

−64.0

74 76 78 80 82 84 86 88 90

Ene

rgy,

E,e

V

Volume, V, A3

Figure 3.2: First-principles ground-state energies (square) for the three symmetri-cally distinct Fe6N2 structures (cf. Table 3.1) and the fitted four-parameter Birch–Murnaghan equations of state (solid line). Notice that the equation of state curvesmore or less move in parallel as the volume is changed from the equilibrium vol-ume.

23


3.2 Extension to hexagonal systems

Of particular interest for the present work is the hexagonal close-packed lattice defined by apair of two independent lattice parameters (a, c) as visualized in Figure 3.3.

c

a

Figure 3.3: Lattice parameters a and c of the hexagonal close-packed lattice. Thelattice parameters are the dimensions of the primitive hexagonal unit cell. The vol-ume of the unit cell is given by (3.5), and the unit cell can equivalently be describedby the volume and the ratio r = c/a.

A full description of thermodynamic equilibrium properties of the system requires a hex-agonal extension of the equation of state in two variables. Ideally, such an extension shouldcontain the original volume-dependent equation in its formulation as a special case, makingV and the ratio of lattice parameters r = c/a natural independent variables, and the energyis therefore defined as the function

E(V, r) = Fac, (3.4)

determined uniquely by the Helmholtz free energy (2.24) from the inverse of the change-of-variables mapping (a, c) 7→ (Vac, c/a), where

Vac =

√3

2a2c (3.5)

is the volume of the primitive hexagonal unit cell.The energy is expanded in the following steps. Firstly, the equilibrium value of the ratio

of lattice parameters r0(V) = arg minr E(V, r) for a given volume V is expanded as

r0(V) =mr−1

∑i=0

ρi

[(VV0

)−n/3

− 1]i

, (3.6)

with parameters ρ0, . . . , ρmr−1. Secondly, a correction to E0(V) from the original volume-dependent equation of state (3.2) is introduced to second order in r, noting that ∂E

∂r (V, r0(V))vanishes,

E(V, r) = E0(V)

[1 +

12

γ(V)(r− r0(V)

)2]

. (3.7)

The quadratic expansion parameter is itself expanded as

γ(V) =mc−1

∑i=0

γi

[(VV0

)−n/3

− 1]i

, (3.8)

with parameters γ0, . . . , γmc−1.

24

3.3 Necessity of the extension and magnetic phase transitions

2.68

2.724.2

4.3

4.4

4.5

−66.7

−66.6

−66.5

−66.4

−66.3

−66.2

Ene

rgy,

eV

V 7→ E(V, r0(V))

r 7→ E(V, r)

a, A

c, A

Ene

rgy,

eV

(a) ε-Fe6N2

2.68

2.724.2

4.3

4.4

4.5

−74.8

−74.6

−74.4

−74.2

−74.0

Ene

rgy,

eV

a, A

c, A

Ene

rgy,

eV

(b) ε-Fe6N3

Figure 3.4: First-principles ground-state energies for hexagonal close-packedstructures for various lattice parameters (a, c) (circle) and the fitted extended equa-tions of state (a, c) 7→ E(Vac, c/a) visualized as a function of lattice parameters(dotted mesh). Projection V 7→ E(V, r0(V)) through equilibrium lattice parame-ters as a function of imposed volume (thick blue line) and projection r 7→ E(V, r)for some fixed unit cell volume V = 82.6 A3 (thin red line).

Having established this parametrization, the equilibrium values of (V, r), or equivalentlythe equilibrium values of the corresponding lattice parameters (a, c), can be calculated asthese change with vibrational excitations as a function of temperature.

The original volume-dependent equation of state is recovered from the quadratic expan-sion (3.7) by the projection

V 7→ E(V, r0(V)) = E0(V), (3.9)

which by construction is the minimum energy for any given volume V. This allows derivedthermodynamic properties as pressure p, bulk modulus B, and its derivative B′ to be cal-culated directly from the simpler expansion (3.2) as described in Section 2.5. The thermalexpansion coefficient can also be calculated by introducing smoothly defined temperaturedependence as explained above.

The tractability of the proposed parameterization also allows obtaining equilibrium ener-gies with various constraints imposed as required for applications with the grand-canonicalensembles partition function in Chapter 5. However, the preference for tractability in pre-dictions in the chosen parameterization makes the fitting procedures somewhat complicated,and non-linear optimization is required.

Examples of fitted hexagonal equations of state are given in Figure 3.4 for ground-stateenergies of the ε-Fe-N system. More details of the ε-Fe-N system, finite temperature applica-tions, and the fitting procedure are given in Chapter 4 and Chapter 5


The first-principles calculator VASP (cf. Section 1.5) allows automatic relaxation of unit celllattice parameters (a, c) as well as position of nuclei relative to the unit cell, i.e., numericallyobtaining shape and positions such that the energy is minimized. In obtaining an energy–volume equation of state, a selection of volumes is selected, and the other parameters are re-laxed using VASP’s conjugate-gradient algorithm [54]. Contrary to the case for face-centered

25


2.602.64

2.682.72

2.76 4.74.8

4.95.0

5.1

−87.8

−87.6

−87.4

−87.2

−87.0

−86.8

−86.6

Ene

rgy,

eV

a, A

c, A

Ene

rgy,

eV

(a)

4.70

4.80

4.90

5.00

5.10

5.20

2.56 2.60 2.64 2.68 2.72 2.76 2.80

c,A

a, A

(b)

Figure 3.5: Relaxation of lattice parameters of ε-Fe6N5 while keeping the unit cellvolume constant for a selection of volumes. The pair of lattice parameters thatminimizes the energy subject to the unit cell volume constraint are determined byVASP’s conjugate-gradient relaxation algorithm. The structure ε-Fe6N5 is not im-portant for the present work, but included here to illustrate that fitting an equationof state only in terms of the volume can be problematic. It is unclear whether thediscontinuity in optimal lattice parameters as a function of volume is a result ofnumerical issues with the relaxation algorithm or if a volume-induced magneticphase transition exists. This can only be resolved by calculating first-principlesenergies for more points on the surface of lattice parameters. (a) Resulting energyas a function of lattice parameters. (b) Projection showing the lattice parametersonly. The discontinuity is clearly visible. Constant-volume curves are shown (thindotted line); the points are therefore the minimum energy pairs of lattice param-eters along each of the lines as determined by VASP’s relaxation algorithm. Seealso Figure 3.1f.

cubic and the body-centered cubic lattices, where the unit cell shapes are completely deter-mined by the unit cell volumes, the hexagonal system has an additional degree of freedom,making the relaxation and fitting of equations of state more complicated.

Also, implementing a conjugate-gradient energy minimization algorithm for very largeequation systems is highly non-trivial and numerical issues may arise.

Another potential problem in obtaining a reliable expression for the equation of state isvolume-induced magnetic phase transitions. Magnetic properties are not studied in detail inthe present work, and none of the models introduced in the following chapters take magneticproperties into account. To monitor and avoid possible magnetic phase transitions, a suddenchange in magnetic moment is taken as a proxy for a volume-induced magnetic phase transi-tion away from the expected ferromagnetic spin structure of the ε phase at ambient pressure[3], and a single magnetic phase can be studied by explicitly excluding these points in thefitting of the equation of state.

This is demonstrated for the ε-Fe6N3 structure in Figure 3.6, where only unit cells of vol-umes V ≤ 90.4 A are included; unit cells volumes larger than this are not relevant for thepresent work and can therefore safely be discarded. This problem is not observed for theε-Fe6N2 structure.

26


2.62.7

2.82.9 4.1

4.24.3

4.44.5

4.6

6

7

8

9

10

11

12

13

Mag

neti

cm

omen

t,µ

/µ

B

a, A

c, A

Mag

neti

cm

omen

t,µ

/µ

B

(a)

4.1

4.2

4.3

4.4

4.5

4.6

2.6 2.7 2.8 2.9

c,A

a, A

V = 90.4 A

(b)

Figure 3.6: Possible volume-induced magnetic phase transition of the ε-Fe6N3structure at V ' 90.4 A. As the model does not account for magnetic propertiesonly points satisfying V ≤ 90.4 A are included in the fitting of the equation of state.This problem is not observed for ε-Fe6N2. (a) Magnetic moment as a function oflattice parameters (circle); points corresponding to V > 90.4 A are discarded (redcircle). (b) Projection showing the lattice parameters only. The constant-volumecurve is shown (thin dotted line). See also Figure 3.1d.

27


28

4Thermal expansion of the ε-Fe6N2

structure

The first investigation presented in this dissertation is a thermodynamic vibrational model ofthe hexagonal close-packed ε-Fe6N2 structure, denoted in these first-principles calculations bythe size of its primitive unit cell. Specifically, the quasiharmonic phonon model is used aimingat predicting thermal expansion of the lattice. This structure was chosen for validation of themodel due to the availability of experimental data [2, 3, 4, 5, 7, 8]. The unit cell of the structureis visualized in Figure 4.1 and Wyckoff positions are listed in Table 4.1. More generally, thestructure is also one of the ground states of the ε-Fe-N system as confirmed experimentally[3, 5] and by earlier first-principles calculations [11], and a thermodynamic model for the ε-Fe6N2 serves as the starting point of the first-principles investigation of that phase presentedin Chapter 5 and Chapter 6.

In the harmonic phonon model, a quadratic energy response is assumed when nuclei areperturbed from equilibrium positions, corresponding to specifying the second-order deriva-tives of the energy in nuclei positions; the first-order derivatives vanishes at equilibrium. TheHamiltonian of this model is given in Section 2.4. The quasiharmonic phonon model is ageneralization of the harmonic model, where the dispersion relation (2.12) depends on unitcell volume. This allows for thermal expansion of equilibrium unit cell volumes to be ac-counted for since the phonon free energy (2.14) depends on volume in this generalization. In

ab

c

Fe

N

Vacancy

Fe

Figure 4.1: The conventional hexagonal close-packed unit cell with six iron hostlattice atoms (gray), two interstitial sites occupied by nitrogen atoms (black), andfour vacant interstitial sites (white). The lattice parameter a defines the distancebetween atoms in the horizontal close-packed planes (cf. Figure 3.3).

29

4 Thermal expansion of the ε-Fe6N2 structure

6g (x, 0, 0), (0, x, 0), (−x,−x, 0), (−x, 0, 12 ), (0,−x, 1

2 ), (x, x, 12 ); x = 0.3262a

2c ( 13 , 2

3 , 14 ), (

23 , 1

3 , 34 )

a Calculation from [11].

Table 4.1: Wyckoff positions of atoms in the iron sublattice (6g) and occupiedinterstitial sites (2c). The space group of the structure is P6322 with space groupnumber 182.

the present work, this is further generalized by allowing the dispersion relation to dependon both of the lattice parameters a and c of the hexagonal unit cell. Numerical prerequisitesfor the hexagonal generalization are already given in Chapter 3, where a smoothly definedequation of state parametrization allows numerical minimization of energy to be performedas a function of lattice parameters and temperature.

Since only vibrational and electronic freedom are considered here, the partition functionof Section 2.2 applies. Hence the thermodynamic potentials introduced in Section 2.5 can beemployed in the mathematical treatment.

The force constants for obtaining the dynamical matrix and the phonon dispersion relationare calculated in the density functional perturbation theory (DFPT) approach [28], defining aset of self-consistent equations to obtain second-order partial derivatives of ground-state en-ergies. This allows formally exact calculations of force constants, assuming the approximationemployed in the exchange-correlation functional are valid.

4.1 Computational details

First-principles density functional theory calculations are performed with the Vienna Ab ini-tio Simulation Package (VASP) [42, 45]. The electron-ion interactions are described by the fullpotential frozen-core PAW method [55, 56], and the exchange-correlation is treated withinthe generalized gradient approximation (GGA) [57] of Perdew–Burke–Ernzerhof (PBE) [58].Monkhorst–Pack sampling [59] of basis set wave vectors in the Brillouin zone is performedwith 11 × 11 × 11 mesh k-points centered on the Γ point to avoid breaking the hexagonalsymmetry. Methfessel–Paxton of first order is used with smearing width 0.2 eV [60]. Theplane wave basis set is truncated at 520 eV and the energy convergence criterion for electronicself-consistency is 10−5 eV per atom.

First-principles energies are obtained in two steps. Firstly, nuclei positions relative tothe fixed unit cell are relaxed in 45 relaxation steps using the conjugate-gradient algorithm[54]. Secondly, accurate energies are calculated with the Vosko–Wilk–Nusair interpolationof the correlation part of the exchange correlation functional [61]. Ground-state energies arecalculated for 130 distinct pairs lattice parameters spanning a fine mesh around minimumenergy values (cf. Figure 4.7). The large number of calculations is possible as no relaxation oflattice parameters are performed.

Phonon calculations are carried out by the supercell method [24] in a unit cell of 48 ironhost atoms obtained by repeating the original unit cell 2× 2× 2 times. Force constants arecalculated with exact second-order energy derivatives using density functional perturbationtheory (DFPT) [28], also as implemented in VASP. A Γ centered mesh of 4× 4× 4 k-pointsis used for this calculation. To obtain accurate force constants, the energy convergence cri-terion for electronic self-consistency is lowered to 10−8 eV per atom. A slightly lower plane

30

4.2 Force constants and vibrational contributions

wave basis set truncation at 500 eV is used for the phonon calculation to make the memoryrequirements of larger supercell problem manageable. Accurate relaxation of nuclei posi-tions relative to the supercell are performed before force constants are obtained by DFPT. Ascalculation of force constants is significantly more expensive computationally, this is only per-formed for 62 distinct pairs of lattice parameters, excluding points far away from equilibriumand chosen in anticipation of preference for larger lattice parameters for higher temperatures.

The phonon density of states integration is performed with Yphon [52]. Ipopt [62] and theauthor’s FuncLib library is used for non-linear optimization.

The inversion of functions required to perform the Legendre from the Helmholtz freeenergy to the Gibbs free energy are solved to full machine precision by bisection [54].


Second-order changes in energy corresponding to perturbation of pairs of nuclei positionsdefined as the force constants in Section 2.4.1 are estimated in the supercell approach usingthe linear response method.

For each pair of nuclei, a basis with one basis vector in the direction of the separation ofthe nuclei and two basis vectors perpendicular to the first basis vector can be chosen. Thisprocedure defines a change-of-basis matrix P, and when this is applied to the 3 × 3 forceconstants matrix Φ(jj′, nn′), stretching and bending force constants can be estimated by thediagonal entries of the force constants matrix P−1Φ(jj′, nn′)P in the changed basis. This isperformed for the purpose of illustration and the result is given in Figure 4.2 for equilibriumlattice parameters. Similar results are obtained for the remaining lattice parameters.

−6

−4

−2

0

2

4

6

1 2 3 4 5 6 7−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Stre

tchi

ngfo

rce

cons

tant

,eV

/A

2

Ben

din

gfo

rce

cons

tant

,eV

/A

2

Bond length, A

Fe-NFe-FeN-N

Figure 4.2: Calculated force constants as a function of bond length. Stretchingforce constants (black) and the weaker bending force constants (blue, secondaryaxis) for the Fe-N, Fe-Fe, and N-N bonds. The force constants are calculated forequilibrium lattice parameters.

31


0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20

Den

sity

,d,T

Hz−

1

Frequency, ω, THz

a = 2.66 A, c = 4.27 Aa = 2.68 A, c = 4.31 Aa = 2.71 A, c = 4.35 A

Figure 4.3: Phonon density of states for a equilibrium lattice parameters (black),for slightly smaller lattice parameters (red), and for slightly larger lattice parame-ters (green), respectively, showing continuous change in the density of states.

0

5

10

15

20

Γ K M Γ A H K

Freq

uenc

y,T

Hz

Reduced wave vector

Figure 4.4: Phonon dispersion relation along important symmetry direction of thehexagonal close-packed unit cell calculated for equilibrium lattice constants. Seesymmetry points in Figure 4.5. Three acoustic phonon branches (converging to 0towards the Γ point) and a number of optical branches.

A

A

Γ

H

KM

Figure 4.5: The first Brillouin zone of the hexagonal close-packed unit cell [63].

32

4.3 Fitting the equation of state

Only the total energy as a function of all nuclei positions can be calculated from first-principles, and an exact decomposition in individual contributions can therefore not be ob-tained; e.g., bending one bond may implicitly stretch another bond, and visa versa. Such adecomposition is not required for defining the dynamic matrix (2.19), which depends on thetotal energy and its derivatives only.

The phonon density of states is obtained from the force constants through (2.20). Thisallows the vibrational contribution to the Helmholtz free energy to be calculated by (2.16).Phonon densities of states are given in Figure 4.3 for three pairs of lattice parameters, showingthat the density of states changes continuously with unit cell volume. The phonon dispersionrelation along important symmetry of the hexagonal unit cell directions is given in Figure 4.4for the equilibrium lattice parameters. Similar dispersion relation curves are obtained forother lattice parameters.


The extended hexagonal equation of state developed in Section 3.2 is employed to predictfinite temperature equilibrium properties from the quasiharmonic phonon model.

A five-parameter Birch–Murnaghan equation of state (3.2) is chosen, augmented withthree additional parameters (mr = 2, mc = 1) to fit the entire surface of lattice parametersthough (3.6) and (3.8); Birch–Murnaghan order parameter n = 2 is chosen. Since energieshave been calculated for a large number of pairs of lattice parameters (cf. Section 4.1), reliableestimates of eight parameters in the equation of states can be obtained. The parameters areobtained by minimizing the squared deviations

η =

√√√√ k

∑i=1

(Efit(Vi, ri)− E(Vi, ri)

)2

k(4.1)

of Helmholtz free energies at

(Vi, ri) =

(3√

32

a2i ci,

ci

ai

). (4.2)

The fitting is repeated for any temperature of interest, resulting in a smooth temperaturedependence as discussed in Chapter 3. As an illustration of the finite temperature fittingprocedure, the calculated Helmholtz free energies are given in Figure 4.6 for three pairs oflattice parameters, with similar curves obtained for other lattice parameters. For any giventemperature of interest, the corresponding free energies are used in the minimization in (4.1)and (4.2), resulting in a smoothly defined hexagonal equation of state in both temperatureand lattice parameters.1

Table 4.2 lists predicted equilibrium lattice parameters, bulk modulus, pressure derivativeof bulk modulus, and thermal expansion evaluated for selected temperatures. Comparison toexperimental data available in the literature is provided [1, 3, 7]. The complete set of equationof state parameters and fitting errors is given in Table 4.3 for reference.

1Mathematically, the function is composed entirely of smoothly defined function and is therefore itselfsmoothly defined.

33


−69.00

−68.50

−68.00

−67.50

−67.00

−66.50

−66.00

E+

Fel+

Fph,e

V

a = 2.66 A, c = 4.27 Aa = 2.68 A, c = 4.31 Aa = 2.71 A, c = 4.35 A

−0.08

−0.06

−0.04

−0.02

0.00

Fel,e

V

−2.50

−2.00

−1.50

−1.00

−0.50

0.00

0.50

0 200 400 600 800 1000

Fph,e

V

Temperature, T, K

Figure 4.6: Helmholtz free energy F = E + Fel + Fph for three pairs of latticeparameters (a, c) as a function of temperature T, obtained by (2.6) and (2.16) fromthe calculated phonon densities of states (cf. Figure 4.3; same color coding). Noticethat the energy only changes slightly between the three pairs of lattice parameters.(Upper) Total Helmholtz free energy, including static elastic energy E. The pair oflattice parameters with lowest energy is exchanged at T = 250 K. (Middle) Elec-tronic contribution Fel. (Lower) Vibrational contribution Fph. The vibrational con-tribution is large compared to the electronic contribution. The vibrational contri-bution is positive in numerical terms for T < 375 K and negative for T > 375 K.

34


T, K a0, A c0, A B0, GPa B′0 α, 10−5 K−1

295 2.6957 4.3379 197.2 5.5 2.852.7100b 4.3748b 172.4d 5.7d 2.92c

300 2.6958 4.3383 197.0 5.5 2.862.7108a 4.3783a

468 2.6990 4.3508 189.2 6.1 3.312.7132b 4.3892b 3.47c

508 2.6997 4.3543 187.1 6.3 3.402.7137b 4.3937b 3.60c

546 2.7004 4.3577 185.0 6.5 3.482.7143b 4.3985b 3.72c

588 2.7011 4.3618 182.6 6.6 3.572.7148b 4.4035b 3.85c

618 2.7017 4.3649 180.9 6.8 3.642.7162b 4.4065b 3.95c

a Experimental data by Somers et al. [1].b Experimental data by Leineweber et al. [3].c As determined from fit to experimental data by Leineweber et al. [3].d Experimental data by Niewa et al. [7] (ambient temperature assumed).

Table 4.2: Equilibrium lattice parameters (a0, c0), equilibrium bulk modulus B0,bulk modulus pressure derivative B′0, and volumetric thermal expansion coeffi-cient α as a function of temperature T. Equation of state parameters are given inTable 4.3.

T, K V0, A3 E0, eV E2/E0 E3/E0 E4/E0 ρ0 ρ1 γ0 η, meV0† 80.963 −66.653 −1.7630 −1.1262 17.355 1.6066 0.11439 −0.33016 1.130 81.490 −66.237 −1.7716 −0.73486 26.018 1.6070 0.076263 −0.32357 0.733

295 81.901 −66.510 −1.7050 −1.2473 27.784 1.6092 0.0060788 −0.30256 0.672300 81.913 −66.520 −1.7032 −1.2605 27.794 1.6093 0.0041618 −0.30203 0.674468 82.341 −66.939 −1.6340 −1.7322 27.516 1.6120 −0.068319 −0.28293 0.868508 82.452 −67.059 −1.6153 −1.8487 27.313 1.6129 −0.087758 −0.27811 0.944546 82.560 −67.179 −1.5966 −1.9598 27.085 1.6137 −0.10699 −0.27344 1.02588 82.682 −67.317 −1.5752 −2.0827 26.796 1.6148 −0.12914 −0.26820 1.12600 82.718 −67.358 −1.5688 −2.1178 26.708 1.6151 −0.13565 −0.26668 1.14618 82.772 −67.420 −1.5592 −2.1704 26.571 1.6156 −0.14555 −0.26440 1.19723 83.099 −67.805 −1.4996 −2.4748 25.668 1.6190 −0.20708 −0.25081 1.45† Without zero-point vibrational energy (cf. Section 2.4).

Table 4.3: Equation of state parameters as described by (3.2), (3.6), and (3.8) corre-sponding to the temperatures listed in Table 4.2; Birch–Murnaghan order parame-ter n = 2 is chosen. Notice that the first-order energy expansion term as a functionof volume vanishes by construction. The ratio of lattice parameters at equilibriumvolume satisfies r0 = c0/a0 = ρ0. The root-mean-squared fitting error η of theequation of state is also given.

35


The calculated first-principles energies and surface of energies as a function of lattice pa-rameters obtained from the fitted hexagonal equation of state are given in Figure 4.7. Similarsurfaces are obtained for finite temperatures. The projection of V 7→ E(V, r0(V)) defining theequilibrium energy as a function of volume is given in Figure 4.8 at T = 0 and T = 600 K,clearly showing the expected thermal expansion predicted by the quasiharmonic phononmodel, i.e., increasing volume at which the lowest energy is reached. A perpendicular projec-tion r 7→ E(V, r) for fixed volume V = 82.6 A3 and varying values of the ratio r = c/a is alsogiven, showing good fit to first-principles energies also along this direction.

The fitting errors in Table 4.3 show that it is possible to fit the entire energy surface withthree additional parameters while obtaining a small fitting error η ' 1 meV (per unit cell; or0.1 meV per atom). Also, the fittings error obtained here are almost the same as that of a purefive-parameter Birch–Murnaghan equation through points restricted to equilibrium latticeparameters defined from r0(V), with very similar equilibrium curves V 7→ E(V, r0(V)).

36

2.68

2.724.2

4.3

4.4

4.5

−66.7

−66.6

−66.5

−66.4

−66.3

−66.2

Ene

rgy,

eV

V 7→ E(V, r0(V))

r 7→ E(V, r)

a, A

c, A

Ene

rgy,

eV

(a)

4.15

4.20

4.25

4.30

4.35

4.40

4.45

4.50

2.62 2.64 2.66 2.68 2.70 2.72 2.74 2.76 2.78

c,A

a, A

(b)

Figure 4.7: (a) First-principles ground-state energies for hexagonal close-packedstructures for various lattice parameters (a, c) (circle) and the fitted extended equa-tions of state (a, c) 7→ E(Vac, c/a) visualized as a function of lattice parameters(dotted mesh). Projection V 7→ E(V, r0(V)) through equilibrium lattice parame-ters as a function of imposed volume (thick blue line) and projection r 7→ E(V, r)for some fixed unit cell volume V = 82.6 A3 (thin red line). Energies and volumeare normalized to the conventional unit cell (six iron atoms). The quasiharmonicphonon model allows phonon free energies to be calculated for each of these pointsand the extended equation of state to be refitted for any T > 0. (b) Projection ofcalculated pairs of lattice parameters.

−66.68

−66.64

−66.60

−66.56

−66.52

−66.48

Ene

rgy,

eV

T = 0

−67.36

−67.32

−67.28

−67.24

−67.20

−67.16

78 79 80 81 82 83 84 85 86

Free

ener

gy,e

V

Volume, V, A3

T = 0

T = 600 K

−66.635

−66.630

−66.625

−66.620

−66.615

−66.610

−66.605

−66.600

−66.595

1.54 1.56 1.58 1.60 1.62 1.64 1.66

Ene

rgy,

eV

r = c/a

Figure 4.8: (Upper) First-principles energies grouped by unit cell volume V, i.e.,energies for multiple pairs of lattice parameters (a, c) with the same unit cell vol-umes Vac have been calculated. The view corresponds to the surface plot of Fig-ure 4.7 seen from the side along the equilibrium curve V 7→ E(V, r0(V)) (blueline). The equilibrium curve of the fitted extended equation of state accurately fitsthrough the lowest energy points for each V. (Lower) Same calculation repeatedfor the Helmholtz free energy Fσ;ac at T = 600 K. A smaller selection of pointshas been calculated than for T = 0. (Right) First-principles energies for volumeV = 82.6 A3. The view corresponds to the surface plot of Figure 4.7 seen alongthe constant-volume curve r 7→ E(V, r) (brown line). Similar fits are obtained forother values of V and for T > 0. Equilibrium ratio r = c/a is correctly reproducedby the fitted equations of state. Excellent fit also in this direction.


4.4 Lattice parameters and thermal expansion

The equation of state extended to hexagonal systems, as described in the previous section,makes it possible to calculate equilibrium values of (V, r) or, equivalently, the correspondinglattice parameters (a, c), as a function of temperature, corresponding to the minimum on thesurface of energies in Figure 4.7a. The equilibrium volume

V(T) = arg minV′

F(T, V ′) (4.3)

as a function of temperature is given in Figure 4.9 and compared with experimental datapublished in the literature [1, 3].

The volumetric thermal expansion coefficient is given by

α(T) =1

V(T)dVdT

(T), (4.4)

corresponding to (2.29) at zero pressure. The predicted expansion coefficient is given in Fig-ure 4.10 as a function of temperature. A functional form

V(T) = V(T0)e∫ T

T0(α0+α1T′)dT′ (4.5)

has been suggested by Fei [64] to obtain an estimate of the expansion coefficient from exper-imental data of lattice parameters, which by the definition (4.4) is a first-order polynomialexpansion of α(T) in temperature T. Figure 4.9 and Figure 4.10 give the resulting fit of (4.5)to experimental data and the corresponding expansion coefficient (4.4), respectively.

81.0

81.5

82.0

82.5

83.0

83.5

84.0

84.5

85.0

0 100 200 300 400 500 600 700

Vol

ume,

V,A

3

Temperature, T, K

Figure 4.9: Predicted equilibrium volume V (solid line) as a function of temper-ature T. Experimental data obtained by Leineweber et al. [3], heating (triangleup) and cooling (triangle down) phases, respectively, with (4.5) fitted to the values(dotted blue line), and by Somers et al. [1] (diamond). Negative pressure correction(dashed line); details are given in Section 4.6.

38

4.4 Lattice parameters and thermal expansion

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 100 200 300 400 500 600 700

Exp

ansi

onco

effic

ient

,α,1

0−5 K

−1

Temperature, T, K

Figure 4.10: Predicted volumetric thermal expansion coefficient α (solid line) as afunction of temperature T compared to fit to experimental data (dotted blue line);see Figure 4.9 for description. The difference at high temperatures may be due toanharmonicity.

The pair of lattice parameters (a, c) that minimizes the Helmholtz free energy (3.4) is givenin Figure 4.11. The relative deviation in the present work compared to experimental lattice pa-rameters at ambient temperature is (−0.53%,−0.84%). This compares to (−0.81%,−1.30%)obtained by Shang et al. [11] and (+0.50%,−0.89%) obtained by Shi et al. [12] with the optimalchoice of the empirical parameter in the GGA+U exchange-correlation functional.

A better agreement of the present model with experimental data as compared to earlierwork is obtained since expansion of the unit cell due to vibrational excitations is accountedfor. The difference between predicted equilibrium values at T = 0 in the present work andthose in [11] is due to the quantum effect of zero-point vibrations originating from the lastterm in (2.11).

The predicted unit cell volume given in Figure 4.9, the thermal expansion coefficient givenin Figure 4.10, and the lattice parameters given in Figure 4.11, obtained from the generalizedquasiharmonic phonon approximation show good agreement with predicted thermal expan-sion compared to experimental data in the literature [1, 3].

39


2.690

2.695

2.700

2.705

2.710

2.715

2.720

2.725

a,A

4.320

4.335

4.350

4.365

4.380

4.395

4.410

4.425

0 100 200 300 400 500 600 700

c,A

Temperature, T, K

Figure 4.11: Predicted lattice parameters (solid line). Experimental data obtainedby Leineweber et al. [3], heating (triangle up) and cooling (triangle down) phases,respectively, and by Somers et al. [1] (diamond). Negative pressure correction(dashed line).

4.5 Volume–pressure relationship

The tractability of the model also allows prediction of equilibrium properties for imposedpositive pressure p > 0 at any given temperature. The equilibrium volume V(T, p) is ob-tained from the inverse of the change-of-variables mapping (2.28), implemented numericallyas discussed in Section 4.1. The procedure is illustrated in Figure 4.12, where equilibriumlattice parameters are obtained for selected pressures.

In Table 4.4 unit cell volume and lattice parameter ratio c/a are listed and compared tohigh-pressure experimental data obtained by Niewa et al. [7]. Notice that the sample wasprepared at high-temperature and quenched to ambient temperature. Other experimentaldata in the literature suggest configurational disordering of interstitial site occupations at ele-vated temperatures [5], which would be locked in the quenching process. The present modeldoes not account for such configurational disordering. Also notice that the volume–pressureexperimental data are obtained from a sample with a slightly higher nitrogen concentrationwith the analytical composition Fe3N1.05O0.017.

Figure 4.13 shows the predicted volume as a function of pressure in good agreement withthe experimental data. Figure 4.14 shows the predicted ratio of lattice parameters c/a as

40

4.5 Volume–pressure relationship

4.20

4.22

4.24

4.26

4.28

4.30

4.32

4.34

4.36

4.38

2.62 2.64 2.66 2.68 2.70 2.72

c,A

a, A

p = −3.5 GPa

p = 0 GPa

p = 4.7 GPa

p = 10.4 GPa

p = 15.4 GPa

p = 18.5 GPa

Figure 4.12: Parametric plot of lattice parameters (a, c) as a function of pressurep at T = 295 K. The curve shown here corresponds to the equilibrium curve inFigure 4.7a (thick blue line), except that the lattice parameters are calculated forT = 295 K here. Equilibrium lattice parameters are given for a selection of pres-sures and corresponding volumes. Comparison to experimental data is providedin Table 4.4.

T, K p, GPa F, eV V, A3 a, A c, A c/a295 −3.5 −66.49 83.5 2.7129 4.3653 1.6091295 0.0 −66.51 81.9 2.6957 4.3379 1.6092

83.5b 2.7100b 4.3748b 1.6143b

83.7a 1.6125a

295 4.7 −66.48 80.1 2.6753 4.3053 1.609381.5a 1.6141a

295 10.4 −66.39 78.1 2.6531 4.2699 1.609479.4a 1.6154a

295 15.6 −66.25 76.3 2.6332 4.2381 1.609578.3a 1.6164a

295 18.5 −66.15 75.4 2.6220 4.2202 1.609577.4a 1.6184a

a Experimental data by Niewa et al. [7] (ambient temperature assumed).b Experimental data by Leineweber et al. [3].

Table 4.4: Predicted energy F, volume V, lattice parameters a and c, and the ratioof lattice parameters c/a as a function of temperature T and pressure p. Negativepressure included to illustrate the negative pressure experimental correction; thenegative pressure is chosen to match experimental unit cell volume obtained byLeineweber et al. [3], i.e., V = 83.5 A3 (cf. Section 4.6).

41


74

76

78

80

82

84

0 5 10 15 20

Vol

ume,

V,A

3

Pressure, p, GPa

Figure 4.13: Predicted unit cell volume (solid line) at T = 295 K as a function ofpressure. Experimental data obtained by Niewa et al. [7] (square) and by Leinewe-ber et al. [3] (triangle). Negative pressure correction (dashed line); this is obtainedby shifting the volume–pressure curve by −3.5 GPa horizontally. The change involume as pressure is increased is in excellent agreement with the experimentaldata.

1.608

1.610

1.612

1.614

1.616

1.618

1.620

0 5 10 15 20

c/a

Pressure, p, GPa

Figure 4.14: Predicted ratio c/a lattice parameters (solid line) at T = 295 K as afunction of pressure. Experimental data obtained by Niewa et al. [7] (square) andby Leineweber et al. [3] (triangle). Negative pressure correction (dashed line). Themodel does not predict significant change in the ratio c/a of lattice parameters afunction of imposed pressure; for the same reason the effect of the negative pres-sure correction is diminishing.

42

4.6 Negative pressure correction

a function of pressure. No significant variation in c/a is predicted by the model, while aslightly increasing value of c/a as the pressure is increased is indicated by the experimentaldata. Moreover, the predicted bulk modulus and its derivative with respect to pressure listedin Table 4.2 are in good agreement with experimental data and provide significant improve-ments compared to previous first-principles calculations [7].

4.6 Negative pressure correction

The exchange-correlation functionals in the density functional theory approach often lead tooverbinding, predicting lattice parameters that are too small, and applying negative pressurehas been proposed in the literature as a pragmatic correction to improve predictions by intro-ducing only one experimental parameter [65], though it must be noted that the overbinding ismore pronounced with the simpler linear density approximation exchange-correlation func-tional than the generalized gradient approximation used in the present work [44]. Anotherapproach is the semi-empirical parameters introduced in GGA+U exchange-correlation func-tionals as has been demonstrated in the prediction of ground-state properties of the ε-Fe6N2structure by Shi et al. [12]. In that case, however, vibrational contributions were ignored,making the agreement with experimental data of the empirically corrected lattice parameterscomparable the to uncorrected lattice parameters in the present work.

The negative pressure approach is briefly tested in this section due to its simplicity andbecause the numerically tractable formulation of the proposed model enables easy calculationof thermodynamic properties at any pressure. It is also included to allowing comparison withthe alternative semi-empirical GGA+U exchange-correlation approach.

The calculations are performed similarly to the pressure-dependence discussed in Sec-tion 4.5, except that a negative pressure p < 0 is applied instead. In the present work, thesame negative pressure correction is used for all temperatures, with p = −3.5 GPa chosen tocorrespond to the unit cell volume at T = 295 K from experimental data obtained by Leinewe-ber et al. [3] as listed in Table 4.4. Unit cell volumes at other temperatures, the ratio of latticeparameters r = c/a, and the thermal expansion coefficient are not otherwise corrected.

The predicted volume as a function of temperature is given in Figure 4.9 in the negativepressure correction. Excellent agreement to experimental data is obtained for a range of tem-peratures, despite that the correction is defined at T = 295 K only. This is due to the accurateprediction of thermal expansion. Both of the lattice parameters are also predicted in the neg-ative pressure correction and are given in Figure 4.11. Good agreements are also achievedin this case. However, it is not possible to obtain perfect agreement at T = 295 K with onlyone correction parameter such as the imposed negative pressure. Notice that the thermalexpansion coefficient given in Figure 4.10 agrees slightly better in the uncorrected model.

From Table 4.4, the experimental value of r = c/a at T = 295 K is 1.6143 [3]. The predicteduncorrected p = 0 value is 1.6092, almost identical to the value of 1.6091 where the negativepressure correction has been introduced. This is compared to 1.5920 is obtained using theoptimal value of the empirical parameter in the GGA+U exchange-correlation functional [12].

43


4.7 Conclusions

A vibrational thermodynamic model was developed for the hexagonal close-packed ε-Fe6N2structure, based on density functional theory first-principles calculations through the general-ized quasiharmonic phonon model and the extended equation of state previously developedfor the hexagonal systems.

Finite temperature equilibrium lattice parameters and thermal expansion coefficient werecalculated and good agreement with experimental data was obtained. Predicted finite tem-perature volume–pressure relationship is also in good agreement with high-pressure experi-mental data.

44

5Partition function approach

The partition function described in fairly general terms in Chapter 2 is very fundamental infinite temperature quantum physics and is used extensively to describe thermodynamic prop-erties of subatomic particles (such as electrons and photons) or quasi-particles (phonons) [36].In this chapter, the partition function is applied to investigate the atomic configurational de-grees of freedom in crystalline solids in canonical or grand-canonical ensembles, i.e., specifi-cally when the thermodynamic mixed mode states of the system include atomic arrangementsas specified by the configurational state σ.

The unified description of the thermodynamics of any degree of freedom through thepartition function allows explicit incorporation of electronic and vibrational degrees of free-dom, while configurational occupancies are determined by an externally controlled chemicalpotential. The advantage of this approach is that it allows closed-form expressions for ther-modynamic potentials, such as the Gibbs free energy, in a first-principles model as long as theconfigurational space is restricted to a feasible size.

5.1 The ε-Fe-N system

The ε-Fe-N system is similar the Fe6N2 structure investigated in details in Chapter 4, exceptthat the nitrogen occupation of the interstitial sites is no longer assumed fixed, but allowedto vary depending on the chemical potential and temperature. The iron host lattice is still as-sumed fixed and fully occupied by iron atoms, and no chemical potential is therefore requiredto control the amount of iron occupation. The unit cell is given in Figure 5.1. Only the groupof interstitial sites is relevant for configurational description of the system.

First-principles ground-state energies (i.e., at T = 0) of structures corresponding to spe-cific individual occupations of interstitial sites have been calculated by Shang et al. [11]. Pre-viously developed thermodynamic models of the ε-Fe-N system include the Gorsky–Bragg–Williams approach to configurational entropy by Kooi et al. [9], where model parameters areobtained as fits to thermochemical data. This model have been applied to study nitrogen ab-sorption and ordering of occupation of interstitial sites experimentally by Somers et al. [1] andby Pekelharing et al. [10].

The thermodynamic model developed for the Fe6N2 structure in Chapter 4 is defined interms of the partition function (2.3), where only electronic and vibrational degrees of freedom

45

5 Partition function approach

ab

c

Fe

(a) (b)

Figure 5.1: (a) The hexagonal close-packed unit cell showing iron host lattice andsix interstitial sites highlighted. (b) The arrangement of the interstitial sites defin-ing the configurational state σ. The twelve-site unit cell is obtained by stacking theunit cell twice in the [001] direction.

are included,Z = ∑

ω∑

ε

e−βEωε;r , (5.1)

and where Eωε is the energy and β is the reciprocal temperature. The model for ε-Fe-N systemdeveloped in this chapter corresponds to the grand-canonical ensemble partition function(2.34), where configurational degrees of freedom are also included,

Z = ∑σ

∑ω

∑ε

e−β(Eσωε;r−µNσ) = ∑σ

e−β(Fσ;r−µNσ), (5.2)

where µ is the chemical potential and Nσ is number of nitrogen atoms in configuration σ; thelatter expression is given in terms of the hybrid potential

Fσ;r = −β−1 log ∑ω

∑ε


including electronic and vibrational contributions introduced in Section 2.6.3. Thus essen-tially the same fundamental thermodynamic equations are employed in both cases.

It is prohibitively expensive computationally to consider the fully general configurationalstate σ from first-principles and the size of the configuration space must be reduced signifi-cantly. The approximations are discussed in Section 5.3 and Section 5.5.

Of particular interest is prediction of the relationship between the externally controlledchemical potential µ and the resulting fraction of occupied interstitial sites yN. The chemicalpotential is not directly available from the experimental data. Instead it is obtained indirectlyfrom the nitriding potential

rN =pNH3

p3/2H2

(5.4)

defined in terms of partial pressures pNH3and pH2

of a NH3/H2 mixture. Details of the pro-cedure for obtaining the chemical potential from the nitriding potential data are given inAppendix A.

The model is somewhat simplistic in the approach to configurational degrees of freedom.A much more sophisticated model is investigated in Chapter 6. Despite of the greatly reducedconfiguration space, the approach provides a reasonable good description of the mapping ofthe chemical potential µ and yN—possibly since the included configurations maps the local

46


environment of iron atoms as demonstrated in the thermodynamic statistical sampling—andthe model allows explicit inclusion of vibrational as well as configurational degrees of free-dom.


The thermodynamic potentials are obtained similarly to the potentials introduced in Sec-tion 2.5, but are generalized to incorporate configurational degrees of freedom. In this casethe starting points is grand potential obtained from the grand-canonical ensemble partitionfunction (5.2). For a given elastic displacement parameter r, the grand potential is

Ωr = −β−1 log Z, (5.5)

or if decomposing r = (a, c, r′), with lattice parameters (a, c) and relative nuclei positionsparameter r′, the grand potential in the usual thermodynamic variables is obtained

Ωac = minr′

Ωr=(a,c,r′), (5.6)

with nuclei at equilibrium positions, i.e., the value of r′ that minimizes (5.6).The ensemble average number of particles N as a function of temperature T, lattice pa-

rameters (a, c), and chemical potential µ is obtained as

N = 〈N〉 = tr ρNZ

=1Z ∑

σ

Nσe−β(Fσ;r=(a,c,r′)−µNσ), (5.7)

where N is the number operator and Nσ the configurational eigenvalues, i.e., the number ofparticles in one particular configurational state σ.

The pair of thermodynamic conjugate variables µ and N is established by the change-of-variables mapping

(T, a, c, µ) 7→ (T, a, c, N). (5.8)

The Helmholtz free energy is obtained from the inverse of this mapping by the Legendretransform

Fac = Ωac + µN. (5.9)

This can be expressed in the usual variables by letting Vac denotes the volume of the hexagonalunit cell,

F(T, V, µ) = minVac=V

Fac. (5.10)

The Gibbs free energy is obtained similarly to (2.31) by introducing the pressure

p(T, V, N) = − ∂F∂V

(T, V, N), (5.11)

through the inverse of the the change-of-variables mapping

(T, V, N) 7→ (T, p, N). (5.12)

by the Legendre transform

G(T, p, N) = F(T, V, N) + pV. (5.13)

47


The chemical potential at constant pressure is obtained as

µ(T, p, N) =∂G∂N

(T, p, N). (5.14)

One advantage of the direct partition function approach is that all of these transforms canbe calculated explicitly.

5.3 Configurational state

The atomic arrangements have so far been described by an abstract configurational state vari-able σ. The iron host lattice is assumed fixed and fully occupied, and can be ignored inthe configurational description; thus the configuration can be described as a spin-line vec-tor σ = (σ1, . . . , σn), where

σi =

+1, N atom at lattice position ri,−1, unoccupied,

(5.15)

enumerating all n interstitial sites in the crystal, with the coordinates above being idealizedpositions, that is, ignoring the elastic relaxation r.

While the thermodynamics described so far derived from the partition function (2.34) isformally exact, given that the Hamiltonian H and its eigenenergies Eσωε;r are accurate, theconfiguration space is prohibitively large, containing 2n different configurations. Thereforean approximation must to introduced to proceed.

At least two options exist for reducing the computational burden. The first is choosing asufficiently large unit cell, performing thermodynamics of the full configuration space of thatunit cell, and, finally, approximating the partition function of the system by

Z = Zk0, (5.16)

with k = n/m, and where Z0 is the partition function defined on the unit cell with m sites,corresponding to quasi-independent subsystems, i.e., energies of the subsystems computedwith periodic boundary conditions imposed. The approach is pursued in this chapter. Theother option is performing random sampling of the full state σ, also known as Monte Carlosimulation, a method often employed in conjunction with cluster expansion methods [29].This is studied in Chapter 6.4.

The advantage of the first option is its computational simplicity and the tractability ofthe thermodynamic potentials and the Legendre transforms as described in Section 5.13.1.Also it is expected to be the only feasible option of a detailed account of vibrational contribu-tions, as the quasiharmonic phonon approximation would be numerically very expensive inthe sampling approach. Thus the investigation in this chapters includes contributions fromvibrational and electronic degrees of freedom.

Compared to the Gorsky–Bragg–Williams approach to configurational entropy [1], whereatoms are assumed to interact only through an average background field of other atoms, theapproximation of quasi-independent subsystems as given in (5.16) does account explicitly forinteraction between pairs of atoms within the chosen unit cell. See Section 6.5 for more details.

48



The same first-principles calculation parameters as used for the Fe6N2 structure are appliedto the remaining structures in the partition function approach. See Section 4.1 for details.

The inversion of functions required to perform the Legendre transformations are solvedto full machine precision by bisection [54]. This is the case when transforming between thechemical potential and particle number pair of conjugate variables and the volume and pres-sure pair of conjugate variables.1

5.5 Structure screening and selection

(a) σα (b) σβ (c) σγ (d) σδ

Figure 5.2: Four interstitial configurations σα, σβ, σγ, σδ, also denoted by Fe6N2,Fe6N3, Fe12N4, and Fe12N5, respectively, corresponding to the sizes of the primi-tive unit cells; σα is identical to the ordered ε-Fe6N2 structure studied in Chapter 4.Occupied sites (black) and vacant sites (white). Notice that σδ can be obtained asthe stacking σα + σβ. Similarly, σγ is the stacking 2× σα with one displaced nitro-gen atom. See also Table 5.1.

As discussed in Section 5.3 a sufficiently large unit cell is required to reliably determine theconfigurational thermodynamics of the system. A twelve-site unit cell is chosen obtained bystacking the conventional hexagonal unit cell given in Figure 5.1 twice in the [001] direction,and therefore contains four layers of interstitial site planes with twelve iron atoms and twelveinterstitial sites in total.

The configurational state (5.15) of this chosen unit cell has twelve entries when accountingfor translational symmetry, corresponding to the arrangement of twelve sites in Figure 5.1b;158 symmetrically distinct configurations exist in total. This unit cell is chosen as the maximalfeasible unit cell size to avoid the combinatorial explosion for larger sizes.

As calculating the Helmholtz free energy Fσ;r=(a,c,r′) for any lattice parameters (a, c) andfor each of these symmetrically distinct configuration σ is not feasible, the problem is re-duced as follows. Firstly, only configurations with occupations in the range 1

3 ≤ yN ≤ 12

are considered. Secondly, a screening of structures are performed by calculating energies at(a, c) = (2.69 A, 4.31 A), corresponding to the average of the equilibrium ground-state latticeparameters of the Fe6N2 and Fe6N3 structures as determined by Shang et al. [11]. These con-figurations are expected to dominate in the equilibrium. Thirdly, vibrational contributions areignored in this initial screening. The structures and the calculated ground-state energies arelisted in Table 5.1. The Helmholtz free energy Fac obtained from the grand-canonical ensemblepartition function without vibrational contributions is given in Figure 5.3.

1The approach of the bisection method is accepted as an essentially closed-form solution, as it is fast, can becalculated to arbitrarily high precision, and can be fully automated.

49


σ d E, eV Symbol

Fe12N4 6 −133.31 2× σα

12 −133.17 σγ

48 −132.46

24 −132.31

12 −132.28

12 −132.02

48 −131.78

48 −131.78

σ d E, eV Symbol

Fe12N5 24 −141.32 σδ

12 −140.82

48 −140.52

Fe12N6 6 −149.21 2× σβ

24 −148.64

Table 5.1: Two of the twelve-site unit. configurational degeneracy d, ground-stateenergy E, and designated symbols for the four included structures. See also Ta-ble 3.1 for the six-site unit cell.

50


−8.34

−8.32

−8.30

−8.28

−8.26

−8.24

−8.22

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50−8.6

−8.4

−8.2

−8.0

−7.8

−7.6

−7.4

−7.213

38

512

1124

12

Hel

mho

ltz

free

ener

gy,F

,eV

Che

mic

alpo

tent

ial,

µ,e

V

yN

Figure 5.3: First-principles energies Fσ;r=(a,c,r′) per atom (square and triangle) andthe Helmholtz free energy Fac calculated from the grand-canonical ensemble par-tition function (5.2) through (5.9) for (a, c) = (2.69 A, 4.31 A) at T = 723 K. Vibra-tional contributions are ignored in this preliminary search for structures to includein more detailed calculations. Only the four lowest energy structures (square) hadnon-negligible contributions to the Helmholtz free energy as calculated by the par-tition function, corresponding to the configurations σα, σβ, σγ, σδ in Figure 5.2. No-tice that yN is given both fractional and decimal representation.

This preliminary investigation indicates that only four structures participate with non-negligible contributions. These four structures σα, σβ, σγ, σδ are shown in Figure 5.2, and aredenoted by Fe6N2, Fe6N3, Fe12N4, and Fe12N5, respectively, corresponding to the sizes of theprimitive unit cells. Thus only contributions from these explicitly given configurations areconsidered in this model when considering other lattice parameters and including vibrationalcontributions in the general quasiharmonic phonon model.

Larger unit cells, e.g., large enough to be consistent with the Fe24N10 nitride hypothesizedby Jack [2], cannot be employed in the present model, due to the combinatorial explosionwhen the number of sites is increased further. The thermodynamic statistical sampling inChapter 6 allows studying more complex properties of configurational degrees of freedom inthe cluster expansion approximation. In particular, strong ordering of nitrogen occupationsaround iron atoms consistent with the configurations σα and σβ, respectively, is predicted.Thus even in a model that provides a more detailed account of configurational degrees offreedom, preferences for the configurations σα and σβ are predicted. See Section 6.14 fordetails.


Force constants and the vibrational contributions are calculated as discussed in Section 4.2,where the Fe6N2 structure is already treated. Force constants for the remaining Fe6N3, Fe12N4,and Fe12N5 structures are given in Figure 5.4 for equilibrium lattice parameters for each of the

51


−6

−4

−2

0

2

4

6

1 2 3 4 5 6 7−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Stre

tchi

ngfo

rce

cons

tant

,eV

/A

2

Ben

din

gfo

rce

cons

tant

,eV

/A

2

Bond length, A

Fe-NFe-FeN-N

(a) Fe6N3

−6

−4

−2

0

2

4

6

1 2 3 4 5 6 7−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Stre

tchi

ngfo

rce

cons

tant

,eV

/A

2

Ben

din

gfo

rce

cons

tant

,eV

/A

2

Bond length, A

Fe-NFe-FeN-N

(b) Fe12N4

−8

−6

−4

−2

0

2

4

6

8

1 2 3 4 5 6 7−1.0

−0.5

0.0

0.5

1.0

Stre

tchi

ngfo

rce

cons

tant

,eV

/A

2

Ben

din

gfo

rce

cons

tant

,eV

/A

2

Bond length, A

Fe-NFe-FeN-N

(c) Fe12N5

Figure 5.4: Calculated force constants as a function of bond length. Stretchingforce constants (black) and the weaker bending force constants (blue, secondaryaxis) for the Fe-N, Fe-Fe, and N-N bonds. The force constants given above arefor the equilibrium lattice parameters; similar results are obtained for other lat-tice parameters. Notice that stretching one particular bond can implicitly bendanother bond; hence some stretching force constants are calculated negative andsome bending force constants are calculated positive.

structures, respectively. Similar force constants are obtained for any of the considered latticeparameters.

5.7 Fitting the equations of state

The extended equation of state is fitted as described in Section 4.3 for the Fe6N2 structure. Thefitted parameters for a selection of temperatures are given in Table 5.2, Table 5.3, and Table 5.4for the Fe6N3, Fe12N4, and Fe12N5 structures, respectively. The fitted equations of state arevisualized in Figure 5.5, Figure 5.6, and Figure 5.7.

52

5.7 Fitting the equations of state

2.68

2.724.2

4.3

4.4

4.5

−74.8

−74.6

−74.4

−74.2

−74.0

Ene

rgy,

eV

a, A

c, A

Ene

rgy,

eV

(a)

4.10

4.15

4.20

4.25

4.30

4.35

4.40

4.45

4.50

2.60 2.65 2.70 2.75 2.80 2.85

c,A

a, A

(b)

Figure 5.5: Equation of state for Fe6N3. Notice that energies are per six-site unitcell. Compare with Figure 4.7.

2.68

2.724.2

4.3

4.4

4.5

−133.2

−133.0

−132.8

−132.6

−132.4

−132.2

Ene

rgy,

eV

a, A

c, A

Ene

rgy,

eV

(a)

4.10

4.15

4.20

4.25

4.30

4.35

4.40

4.45

4.50

4.55

2.55 2.60 2.65 2.70 2.75 2.80

c,A

a, A

(b)

Figure 5.6: Equation of state for Fe12N4. Notice that energies are per twelve-siteunit cell.

2.68

2.724.2

4.3

4.4

4.5

−141.4

−141.2

−141.0

−140.8

−140.6

−140.4

Ene

rgy,

eV

a, A

c, A

Ene

rgy,

eV

(a)

4.10

4.15

4.20

4.25

4.30

4.35

4.40

4.45

4.50

4.55

2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90

c,A

a, A

(b)

Figure 5.7: Equation of state for Fe12N5. Notice that energies are per twelve-siteunit cell.

53

T, K V0, A3 E0, eV E2/E0 E3/E0 E4/E0 ρ0 ρ1 ρ2 γ0 η, meV0† 84.295 −74.716 −1.9272 0.26196 11.064 1.5747 0.24310 8.1343 −0.22629 1.490 84.969 −74.228 −1.8860 0.95287 −7.6680 1.5730 0.27202 6.8741 −0.21664 0.740

295 85.369 −74.514 −1.7822 2.5433 −38.671 1.5701 0.38275 5.0269 −0.19520 0.925300 85.382 −74.525 −1.7798 2.6000 −39.542 1.5700 0.38648 4.9752 −0.19464 0.931468 85.881 −74.980 −1.7164 5.1299 −70.539 1.5658 0.55194 3.0044 −0.17469 1.18508 86.014 −75.110 −1.7087 5.9098 −78.114 1.5645 0.60444 2.4653 −0.16977 1.24546 86.144 −75.242 −1.7049 6.7115 −85.284 1.5630 0.65978 1.9264 −0.16506 1.31588 86.292 −75.394 −1.7050 7.6635 −93.139 1.5612 0.72768 1.2990 −0.15983 1.39600 86.334 −75.439 −1.7058 7.9478 −95.365 1.5606 0.74846 1.1134 −0.15833 1.41618 86.399 −75.507 −1.7079 8.3840 −98.685 1.5597 0.78084 0.82940 −0.15609 1.45723 86.787 −75.932 −1.7384 11.145 −117.50 1.5534 1.0017 −0.96944 −0.14296 1.65† Without zero-point vibrational energy (cf. Section 2.4).

Table 5.2: Equation of state parameters for the Fe6N3 structure. Notice that en-ergies and volumes, including fitting errors, are per six-site unit cell. See alsoFigure 4.3.

T, K V0, A3 E0, eV E2/E0 E3/E0 E4/E0 ρ0 ρ1 γ0 η, meV0† 160.88 −133.18 −1.6470 −1.9919 −3.4549 1.6014 0.052996 −0.33253 3.930 161.95 −132.36 −1.6447 −4.4174 −43.175 1.6066 0.077815 −0.30601 1.20

295 163.01 −132.91 −1.5623 −3.7249 −25.495 1.6093 −0.0091108 −0.28813 0.958300 163.04 −132.93 −1.5602 −3.7136 −25.136 1.6094 −0.011525 −0.28767 0.953468 164.12 −133.77 −1.4833 −3.5694 −15.097 1.6130 −0.10237 −0.27079 0.813508 164.40 −134.02 −1.4624 −3.5996 −13.416 1.6142 −0.12671 −0.26637 0.795546 164.67 −134.26 −1.4413 −3.6463 −12.088 1.6153 −0.15082 −0.26205 0.784588 164.98 −134.54 −1.4164 −3.7143 −10.918 1.6167 −0.17864 −0.25715 0.777600 165.07 −134.62 −1.4089 −3.7362 −10.640 1.6172 −0.18682 −0.25573 0.776618 165.20 −134.75 −1.3974 −3.7709 −10.266 1.6179 −0.19929 −0.25357 0.776723 166.04 −135.53 −1.3226 −3.9954 −9.0741 1.6224 −0.27704 −0.24060 0.786† Without zero-point vibrational energy.

Table 5.3: Equation of state parameters for the Fe12N4 structure. Notice that ener-gies and volumes, including fitting errors, are per twelve-site unit cell.

T, K V0, A3 E0, eV E2/E0 E3/E0 E4/E0 ρ0 ρ1 γ0 η, meV0† 164.41 −141.35 −1.6939 −1.9241 −1.6862 1.5935 0.23322 −0.26573 3.030 165.70 −140.44 −1.6832 −4.6839 −43.594 1.5916 0.13908 −0.25392 1.33

295 166.87 −141.02 −1.6018 −3.2485 −27.581 1.5905 0.19192 −0.25856 0.989300 166.90 −141.04 −1.6001 −3.2143 −27.103 1.5905 0.19318 −0.25865 0.981468 168.10 −141.93 −1.5518 −2.3490 −9.6313 1.5895 0.23429 −0.26132 0.798508 168.40 −142.18 −1.5421 −2.2508 −5.2135 1.5893 0.24323 −0.26183 0.787546 168.69 −142.44 −1.5330 −2.2022 −0.97397 1.5890 0.25130 −0.26227 0.789588 169.01 −142.73 −1.5228 −2.2004 3.7369 1.5888 0.25972 −0.26271 0.804600 169.10 −142.82 −1.5198 −2.2100 5.0845 1.5887 0.26203 −0.26283 0.811618 169.24 −142.95 −1.5152 −2.2329 7.1051 1.5886 0.26540 −0.26300 0.824723 170.06 −143.77 −1.4845 −2.5663 18.795 1.5879 0.28322 −0.26382 0.930† Without zero-point vibrational energy.

Table 5.4: Equation of state parameters for the Fe12N5 structure. Notice that ener-gies and volumes, including fitting errors, are per twelve-site unit cell.

5.8 Helmholtz free energy

−74.72

−74.68

−74.64

−74.60

−74.56

−74.52

Ene

rgy,

eV

T = 0

−75.44

−75.40

−75.36

−75.32

−75.28

−75.24

80 81 82 83 84 85 86 87 88

Free

ener

gy,e

V

Volume, V, A3

T = 0

T = 600 K

−74.695

−74.690

−74.685

−74.680

−74.675

1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63

Ene

rgy,

eV

r = c/a

Figure 5.8: Equation of state for Fe6N3. (Upper) First-principles energies groupedby unit cell volume V, i.e., energies for multiple pairs of lattice parameters (a, c)with the same unit cell volumes Vac have been calculated. The view corresponds tothe surface plot of Figure 5.5 seen from the side along the equilibrium curve V 7→E(V, r0(V)) (blue line). The equilibrium curve of the fitted extended equation ofstate accurately fits through the lowest energy points for each V. (Lower) Samecalculation repeated for the Helmholtz free energy Fσ;ac at T = 600 K. (Right) First-principles energies for volume V = 82.6 A3. The view corresponds to the surfaceplot of Figure 5.5 seen along the constant-volume curve r 7→ E(V, r) (red line).A good fit is slightly harder to obtain for Fe6N3 compared to the other structures(cf. Figure 4.8).

A good fit of the Fe6N3 structure is harder to obtain than for the remaining structures, andone additional parameter in the equation of state (3.6) is required by taking mr = 3. Despiteof this, the fitting error η is relatively high for this structure as listed in Table 5.2. Projectionsof the fitted equation of state for the Fe6N3 structure is given in Figure 5.8.

5.8 Helmholtz free energy

Predictions of chemical potentials at fixed pressure and equilibrium lattice parameters bothrequire calculations of the Helmholtz free energy Fac as an intermediate step, before the Legen-dre transform (5.13) can be performed. With the fitted equations of state the grand-canonicalensemble Helmholtz free energy can be calculated for any pair of lattice parameters (a, c).

As an illustration, Helmholtz free energies have been calculated for three pairs of latticeparameters, corresponding to equilibrium values at T = 723 K for the Fe6N2, Fe6N3, andFe12N5 structures. The result is given Figure 5.9.

55


−8.49

−8.48

−8.47

−8.46

−8.45

−8.44

−8.43

−8.42

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Hel

mho

ltz

free

ener

gy,F

,eV

yN

a = 2.70 A, c = 4.38 Aa = 2.74 A, c = 4.35 Aa = 2.78 A, c = 4.32 A

Figure 5.9: First-principles energies Fσ;r=(a,c,r′) per atom (square) and the corre-sponding Helmholtz free energy Fac calculated from the grand-canonical ensemblepartition function (5.2) through (5.9) three pairs of lattice parameters at T = 723 K(line). Vibrational and electronic contributions are included. Notice that yN isgiven both fractional and decimal representation.

5.9 Gibbs free energy and chemical potential

Having obtained a smoothly defined closed-form expression for the Helmholtz free energyFac for any pair of lattice parameters, the Gibbs free energy can be obtained from the Legen-dre transform (5.13). The corresponding chemical potential µ required to maintain a specificnitrogen occupation yN is given by (5.14). The predicted Gibbs free energy and chemical po-tential at pressure p = 0 and temperature T = 723 K is given in Figure 5.10.

The predicted chemical potential is compared with experimental data obtained by Somerset al. [1], where the interstitial occupation yN is obtained for a given nitriding potential (5.4).The chemical potential can be obtained under ideality assumptions from the nitriding po-tential as described in Appendix A. Notice that in this process a freedom in choosing anunobservable reference energy exists, resulting in an unknown additive constant in the chem-ical potential. This constant is chosen to obtain best possible agreement with the predictedchemical potential. The predicted chemical potential is generally in good agreement with theexperimental data when comparing the slope, i.e., the remaining degree of freedom. Similarresults are obtained for other temperatures.

The model predicts rapidly increasing chemical potential as yN → 12 , the high-symmetry

point where half of the interstitial sites are occupied. Limited experimental evidence supportsthis behavior such as the asymptotic behavior of yN as a function of the applied nitridingpotential rN as obtained by Somers et al. [1]. Moreover, the experimental method did notallow values yN > 0.4937 to be reached, which might be an indication of an increasinglycostly energy requirement to reach such concentration, as predicted by the present model. SeeAppendix A for more details on the asymptotic behavior. The similarly predicted behavior ofthe chemical potential as yN → 1

3 has not been tested experimentally.

56

5.10 Lattice parameters

−8.49

−8.48

−8.47

−8.46

−8.45

−8.44

−8.43

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50−8.6

−8.4

−8.2

−8.0

−7.8

−7.613

38

512

1124

12

Gib

bsfr

eeen

ergy

,G,e

V

Che

mic

alpo

tent

ial,

µ,e

V

yN

Figure 5.10: First-principles free energies Fσ;r=(a,c,r′) per atom at equilibrium latticeparameters (square) and the Gibbs free energy G calculated from grand-canonicalensemble partition function (black line) at T = 723 K and p = 0. The calculatedchemical potential is shown on the secondary axis (blue line, secondary axis). Ex-perimental data from Somers et al. [1] (blue circle, secondary axis) obtained fromthe nitriding potential as described in Appendix A. Notice that in this process afreedom in choosing an unobservable reference energy exists, resulting in an un-known additive constant in the chemical potential.

To investigate this further from a first-principles perspective much larger unit cells arerequired. The thermodynamic statistical sampling employed in conjunction with a clusterexpansion in Chapter 6 predicts a similar behavior around specific high-symmetry points.It is hypothesized that small kink around yN = 5

12 an artifact of the size of the chosen unitcell, and that this would be smoothed if the computation could be performed on much largerunit cells. (Notice, however, that a similar kink, although much weaker, is predicted by thethermodynamic statistical sampling in Section 6.9 at lower temperatures.)

5.10 Lattice parameters

The explicit expression (5.9) for the Helmholtz free energy Fac allows equilibrium values of thelattice parameters to be obtained as a function of yN at any temperature, i.e, the value of (a, c)that minimizes Fac. The predicted equilibrium lattice parameters are given in Figure 5.11 atT = 300 K and is compared to experimental data obtained by Somers et al. [1]. The values arealso listed in Table 5.5. Similarly to the Fe6N2 structure discussed in Section 4.4, the predictedvalues are generally smaller than the values obtained from experimental data.

The predicted equilibrium lattice parameter a is generally in good agreement with ex-perimental data. However, the model does not predict expansion in the lattice parameters cas a function of yN. The present model provides an improvement compared to earlier first-principles calculations by Shang et al. [11], as this model includes contributions from vibra-tional excitations in the quasiharmonic phonon approximation.

57


2.68

2.70

2.72

2.74

2.76

2.78

a,A

4.32

4.34

4.36

4.38

4.40

4.42

4.44

13

38

512

1124

12

c,A

yN

Figure 5.11: Predicted equilibrium lattice parameters at T = 300 K calculated us-ing the partition function (solid line) and equilibrium lattice parameters of theselected structures (square). Experimental data by Somers et al. [1] (circle). Noticethat the lowest energy structure at yN = 1

3 has the largest equilibrium lattice pa-rameters and thus dominates the equilibrium lattice parameters as predicted bythe partition function. Reasonable agreement with experimental data for latticeparameter a; for the lattice parameter c the agreement is best for yN ' 1

3 .

58

5.11 Possible two-phase region

yN a0, A c0, A0.3335 2.6958 4.3381

2.7103a 4.3730a

0.3354 2.6964 4.33822.7134a 4.3843a

0.3500 2.7009 4.33812.7138a 4.3822a

0.3745 2.7088 4.33682.7229a 4.3937a

0.3892 2.7137 4.33532.7328a 4.3961a

0.4157 2.7230 4.33232.7407a 4.4054a

0.4334 2.7297 4.33332.7491a 4.4125a

0.4569 2.7390 4.33352.7576a 4.4133a

0.4928 2.7531 4.32862.7641a 4.4200a

0.4949 2.7539 4.32812.7684a 4.4236a

a Experimental data by Somers et al. [1].

Table 5.5: Equilibrium lattice parameters as a function of yN at T = 300 K.

5.11 Possible two-phase region

Pekelharing et al. [10] has suggested that the range 0.39 ≤ yN ≤ 0.482 is a two-phase regionwith phases of orderings consistent with the Fe6N2 and Fe6N3 structures, respectively, inequilibrium.

This is not supported by the model developed in this chapter. The calculations predictone continuous phase as the calculated Gibbs free energy is a convex function. The Gibbs freeenergy in two-phase regions is obtained as the convex hull extension of the energies of theindividual phases—also known as the common tangent method [66]—with the chemical po-tential obtained as its derivative consequently being constant throughout two-phase regions.This is neither the case for the chemical potential calculated from the partition function asevident from Figure 5.10, nor is it the case for the experimental data of chemical potentials,assuming ideality of the ammonia gas as explained in Appendix A.

Notice that a narrow two-phase region is identified by thermodynamic statistical sam-pling in Chapter 6. In that case the other phase is a ζ phase defined on an orthorhombiclattice.

59


5.12 Conclusions

The direct partition function method and the extended equation of state allow explicit cal-culations of equilibrium lattice parameters and chemical potential as a function of both thenitrogen occupation of interstitial sites yN and temperature, where vibrational excitations areaccounted for in the quasiharmonic phonon approximation. The model provides good agree-ment with experimental data and provides improvements to previous first-principles calcu-lations.

Detailed accounts of vibrational excitations have been provided though the equation ofstate extended to the hexagonal system. In order for the calculations to be practical, themodel is somewhat naive in the approach to configurational degrees of freedom, and theconfigurational space is reduced significantly. A much more realistic configurational model isdeveloped in Chapter 6 using thermodynamic statistical sampling at the cost of ignoring thevibrational and the electronic degrees of freedom. The model developed in this chapter andthe thermodynamic statistical sampling model therefore provide two complementary direc-tions of thermodynamic model development, incorporating vibrational and configurationaldegrees of freedom, respectively.

60

5.13 Feasibility of extension to the ε-Fe-C-N system


It has been investigated whether a possible extension to the ε-Fe-C-N is feasible, where threepossibilities exist for each interstitial site: occupied by nitrogen, occupied by carbon, or va-cant, with yC and yN denoting fractions of interstitial sites occupied by carbon and nitrogenatoms, respectively. This requires that the configurational space can be reduced to a practicalsize as discussed for the ε-Fe-N system in Section 5.5.

Two problems exist in achieving these requirements and is described in the followingsections. Due to these problems the investigations of the ε-Fe-C-N system were not continuedfurther. The thermodynamic statistical sampling presented in Chapter 6 is expected to bemore suited for such an investigation, but would require the cluster expansion methodologyto be extended to ternary systems and a substantially larger database of structures.

5.13.1 Combinatorial explosion

The first problem is the combinatorial explosion in number of symmetrically distinct con-figurations to consider as the number of alloying elements and unit cell size are increased.Compared to the binary case where interstitial sites may be vacant or occupied by a nitrogenatom, having 13 symmetrically distinct configuration of the six-site unit cell. In the ternarysystem, where carbon atoms are introduced as well, the number of symmetrically distinct con-figurations increases to 92 for the six-site unit cell. The effect is even more pronounces for thetwelve-site unit cell considered for the Fe-N system. In this case the number of symmetricallydistinct configurations increases from 158 to 12769. See Table 5.6.

The symmetry reduction calculation of the twelve-site Fe-C-N system takes several daysto complete compared to a few seconds for the six-site system, and therefore defines the up-per limit of feasibility with current computational resources, i.e., symmetry reduction of theconfigurational space is prohibitively expensive for larger systems of more than twelve inter-stitial sites.

System Configurations Configurations(six sites) (twelve sites)

ε-Fe-N 13 158ε-Fe-C-N 92 12769

Table 5.6: Number of symmetrically distinct configurations for ε-Fe-N and ε-Fe-C-N systems, respectively. The number grows rapidly as the unit cell size is in-creased. The simplest case of the ε-Fe-N system with six sites is given in full detailsin Table 3.1.

5.13.2 Non-separation of energies

In addition to the combinatorial explosion and contrary to the case for the Fe-N system wheremany configurations could be excluded as providing insignificant contributions to the grand-canonical ensemble partition function, a similar screening for significant configurations forthe Fe-C-N system could not exclude configurations to nearly the same extend.

This is demonstrated in Figure 5.12 and Figure 5.13, where possible configurations cor-responding to yC = yN = 1

6 are identified. The lowest-energy configurations and the corre-

61


sponding equilibrium energies are given in Figure 5.12. The energy as a function of volume isgiven in Figure 5.13 for the calculated configurations. The energies of all of these configura-tions are almost identical and reduction of the configurational space is therefore not possibleto the same extend as for the Fe-N system.

From the energies of the structures in Figure 5.12 it is concluded that nitrogen and car-bon atoms have a strong preference for not being nearest-neighbor, and that carbon-carbonnearest-neighbors is preferred to carbon-nitrogen nearest-neighbors; both of these are pre-ferred to nitrogen-nitrogen nearest-neighbors, which is not seen in any of the lowest-energystructures. The lowest energy structure given in Figure 5.12a corresponds to σα

N/C occupiedby both nitrogen and carbon atoms. The second-lowest energy structure given in Figure 5.12ccorresponds to the stacking σα

N + σαC, i.e., where one of the σα is occupied by nitrogen atoms

and the other is occupied by carbon atoms.In conclusion, whereas most of the 158 symmetrically distinct twelve-site Fe-N configura-

tions listed in Table 5.6 could be excluded from the calculation, this is not to be the case forthe 12769 symmetrically distinct twelve-site Fe-C-N configurations and the computational re-sources required to perform the required first-principles calculations are therefore expectedto increase significantly. The direct partition function approach is therefore prohibitively ex-pensive and this method is not pursued further.

Notice that energies tend to move in parallel as unit cell volume is changed away fromequilibrium volume. This is important in the statistical sampling approach discussed in Chap-ter 6.

62


(a)2E0 = −134.18 eV

(b)2E0 = −132.90 eV

(c)E0 = −134.12 eV

(d)E0 = −134.10 eV

(e)E0 = −134.08 eV

(f)E0 = −134.03 eV

(g)E0 = −133.50 eV

(h)E0 = −133.48 eV

Figure 5.12: When carbon is introduced as an interstitial element, the numberof symmetrically distinct configurations increases. (a)–(b) Lowest energy Fe6CNstructures. Interstitial sites occupied by nitrogen (black), carbon (gray), or vacant(white). (c)–(h) When the unit cell size is increased, significantly larger number ofpossible configurations exists. Identified lowest energy Fe12C2N2 structures.

63


−134.2

−134.0

−133.8

−133.6

−133.4

−133.2

−133.0

−132.8

−132.6

154 156 158 160 162 164 166 168

Ene

rgy,

E,e

V

Volume, V, A3

Figure 5.13: Energy as a function unit cell volume for lowest-energy Fe-C-N con-figurations for yC = yN = 1

6 . Many energetically very close-lying equations ofstate of Fe6CN and Fe12C2N2 structures. Calculated for six-site (black) and twelve-site unit cells (orange). Volumes and energies are given per twelve-site unit cell.The equations of state correspond to configurations from Figure 5.12. Notice thatthe equation of state curves more or less move in parallel as the volume is changedfrom the equilibrium volume.

64

6Thermodynamic statistical sampling

In order to account for collective effects of ordering of atoms in large systems, such as config-urational phase transitions, thermodynamic statistical sampling—often referred to as MonteCarlo simulation after the famous casinos of Monaco—is often employed. For brevity onlythe binary case—such as a single type of atom and vacancies—is considered, so that one vari-able, the particle number N or the chemical potential µ in the conjugate pair, is sufficient fordescribing the thermodynamic equilibrium state.

The starting point is the grand-canonical ensemble partition function (2.34). In order to re-duce the complexity of the problem only configurational degrees of freedom are considered,and the electronic and vibrational degrees of freedom are either completely ignored or incor-porated through hybrid potentials as explained in Section 2.6.3. In both cases the partitionfunction is reduced to

Z = ∑σ

e−β(Eσ;r−µNσ), (6.1)

where Eσ;r and Nσ are the energy and number of particles of configuration σ, respectively, µ isthe chemical potential, β = (kBT)−1 is reciprocal temperature, kB is the Boltzmann constant,and T is absolute temperature. The elastic displacement parameter r = (a, c, r′) is decom-posed in lattice parameters (a, c) and the remaining relative atomic positions r′. The relativepositions are ignored, and atoms are assumed to be at equilibrium positions.

The terminology used in this chapter is that a structure is the full description of the atomicarrangements of the system, including the geometry and atomic positions specified the elasticdisplacement parameter. The corresponding configuration σ, on the other hand, is the ideal-ized state vector of the atomic occupancies of the sites in a structure. The statistical sampling

7→

+1 −1+1

+1

−1+1

−1+1

−1

+1+1

+1+1

−1+1

+1

−1+1

−1+1

−1

+1−1

+1 −1

7→ σ = (+1,−1,+1,+1,+1, . . . )

Figure 6.1: The procedure for obtaining the idealized configurational state vectoris illustrated by the mapping above of real atomic positions, to idealized latticepositions, and finally to the configurational state vector σ.

65

6 Thermodynamic statistical sampling

is concerned the configurational state only. The description in terms of the idealized statevector is illustrated in Figure 6.1.

It must be noted that the partition function and the grand potential Ω = −β−1 log Z aredefined only for a given fixed pair of lattice parameters (a, c). Thus temperature–volumephase diagrams are the natural objects to study in first-principles predictions; temperature–pressure phase diagrams as obtained from experimental data measured at constant pres-sure, are much harder to calculate, as these requires an additional Legendre transform ofthe Helmholtz free energy.

The probability of encountering a particular configuration σ in a grand-canonical ensem-ble is proportional to the Boltzmann factor,

pσ =1Z

e−β(Eσ;r−µNσ), (6.2)

Thus the absolute probability of encountering one specific configuration requires knowledgeof the energies of all the other possible configurations through the normalization by Z.

The ensemble average of some physical quantity A, represented by its quantum mechan-ical operator, is given by

〈A〉 = ∑σ

pσ Aσ, (6.3)

where Aσ is the value of A associated with configuration σ.One physical quantity of particular interest is the number of atoms N, so that (6.2) and

(6.3) define the relationship between chemical potential µ and ensemble average number ofatoms N = 〈N〉. This can be exploited to determine the Helmholtz free energy F(T, V, N) bythermodynamic integration, if its value is known for a single thermodynamic reference state[67], e.g., if the energy is known at T = 0, the Helmholtz free energy can be determined forany T > 0.

6.1 Markov chain algorithm

Unfortunately, as the total number of configurations is exceedingly large—in the order of21023

for a macroscopic crystal—calculating (6.3) directly is impossible. In thermodynamicstatistical sampling averages are instead approximated by sampling a representative subsetof configurations, such as the Metropolis–Hastings algorithm for alloy systems [68, 69, 70].The procedure is a follows: A sequence of sample configurations

σ0, σ1, . . . , σt, σt+1, . . . (6.4)

is to be generated by a Markov chain process such that

〈A〉 = limn,m→∞

1m

n+m

∑t=n+1

Aσt , (6.5)

i.e., so that ensemble averages can be evaluated in terms of simple unweighted averages withrespect to the trajectory Aσt of this sequence. It can be proved that sequences generated us-ing the following algorithm provide convergence of (6.5) to correct ensemble averages [69]:

66

6.1 Markov chain algorithm

Firstly, some initial configuration σ0 is chosen. Then, inductively, transition from σt to σt+1 isgiven by the Markov chain transition probability

P(σt → σt+1) =

e−β(

Eσt+1−Eσt−µ(N

σt+1−Nσt ))

Eσt+1 − µNσt+1 ≥ Eσt − µNσt ,1 otherwise,

(6.6)

where σt+1 is restricted to be obtained from σt by replacing a single atom. In binary systemsthe configurational state σ can be conceived of as a spin-like vector with entries +1 and −1for each lattice site of crystal. A spin-flip is when one of the entries of state vector is changedfrom −1 to +1 or from +1 to −1. Thus the distance between σt and σt+1 is assumed to beexactly one spin-flip. Changing to a lower energy configuration is always accepted by (6.6),whereas the probability of changing to a higher energy configuration decreases exponentiallywith the energy difference.1

The sampled sequence is truncated at some point and the practically of the method restson whether the error

1m

n+m

∑t=n+1

Aσt − 〈A〉 (6.7)

can be made small enough without excessive calculations. The Markov chain transition prob-abilities are not uniquely determined by (6.3) and the requirement (6.5), and other choicesexist [69]. The present algorithm is expected to be efficient as energetically favorable configu-rations are visited exponentially more often.2

Physically, (6.5) can be thought of as some equilibration phase of n time-steps followed byan estimation phase of m time-steps when equilibrium is reached. The algorithm depends onenergies Eσ and particle numbers Nσ only, not on the physical quantity in question. Thus thesame sequence can be used to evaluate the ensemble average of several physical quantitiessimultaneously.

The initial configuration σ0 is—at least mathematically—not important for the correctnessof (6.5). However, in the present work two limiting cases have been tested: initializing froma perfectly long-range ordered state and from a randomly chosen state. Long-range orderedstates tend to avoid inconvenient anti-phase boundaries for temperatures below critical tem-perature of long-range ordering [33], and this initialization is therefore generally used.

The algorithm can be generalized to include non-equilibrium nuclei positions [71], andthus, in principle, describing vibrations and diffusion of atoms.

As a final remark it must be noted that as long as only averages such as (6.3) are of inter-est, then the parameters of the problem as well as the quantities to be determined are bothperfectly deterministic, i.e., not random variables in probability theory terms. One couldtherefore hope for some mathematical transform to eliminate the intermediate stochastics of

1The similar transition probability in the canonical ensemble is

P(σt → σt+1) =

e−β(E

σt+1−Eσt ) Eσt+1 ≥ Eσt ,1 otherwise,

and σt+1 is restricted to be obtained from σt by a double spin-flip to make the total number of +1 and −1 spinsunchanged. The canonical ensemble is only employed briefly in Section 6.14.

2Efficient approximations of (6.3) have been studied since the beginning of the computer age; one of the veryfirst electronic computers, the 1952 MANIAC computer, was constructed specifically with this purpose under thedirection of Metropolis himself.

67


the problem formulation, and the need for statistical sampling, such as the Feynman–Kacformula that expresses statistical averages of continuously defined stochastic diffusion pro-cesses through ordinary non-stochastic partial differential equations [72, 73]. However, suchtransforms depend intimately on the continuous properties of the stochastic processes, andthe discrete nature of the configuration space makes the existence of any such transformsunlikely.

6.2 Cluster expansion

The approximation scheme given in Section 6.1 is not very useful by itself, as calculatingtransition probabilities (6.6) from first-principles is far too expensive computationally. Insteadthe energies can be approximated by a cluster expansion as described in this section.3

Cluster expansions are generalized Ising models [74], where the energy of a particularconfiguration σ is expanded as an infinite series of increasingly large clusters,

E(σ) = ∑α

JαΦα(σ), (6.8)

where α is the cluster, Jα is the expansion coefficient, also known as the effective cluster inter-action (ECI), and Φα is the cluster function in a formally exact infinite expansion [30].

A cluster α = (p1, . . . , pn) is an unordered collection of n = |α| lattice sites throughoutthe crystal. The simplest non-trivial clusters are pair clusters of two sites. Examples of suchclusters are given for the hexagonal close-packed system in Figure 6.3.

In the binary system the cluster function is given by

Φα(σ) = θ(σp1) · · · θ(σpn), (6.9)

where

θ(σp) =

+1, A atom at site p in σ,−1, B atom at site p in σ.

(6.10)

The expansion is easily generalized to systems of any number of components by generalizingthe cluster function (6.9) and (6.10) [29, 30].

The expansion includes a special degenerate empty cluster (n = 0) denoted α0 with acluster correlation function satisfying Φα0(σ) = 1 for any configuration σ. Point clusters(n = 1) contribute with energies only depending on the type of atom (A or B). Thus we mayrewrite (6.8) as

E(σ) = E0 + E1(NA − NB) + ∑|α|≥2

JαΦα(σ), (6.11)

where E0 = Jα0 and E1 = ∑|α|=1 Jα are zeroth-order and first-order energy contributions asfunctions of composition, respectively, and NA and NB are number number of A and B atoms,respectively.

Higher-order clusters serve as corrections to the lower-order energy expansion. Thereforepair clusters (n = 2), as the simplest non-trivial clusters, merely express preference for seg-regation4 (Jα < 0) or ordering (Jα > 0), since the pair cluster correlation function is positive

3In cluster expansions, the function notation E(σ) is often used for the energy instead of the quantum mechan-ical notation Eσ to emphasize dependence as an expansion in the configuration σ.

4Preference for atoms being segregated (or clustered) in domains of like atoms.

68

6.2 Cluster expansion

for any like combination of atoms (AA and BB) and negative for any unlike combination (ABand BA).

In practical applications the expansion (6.8) is truncated after a finite number of terms.Thermodynamic properties are determined by applying the Markov chain algorithm to sys-tems where the energies are expressed in terms of this expansion, and the configurationalstates are described in terms of occupations of lattice sites in very large computer crystals.

Complex physical behavior such as short-range ordering (SRO), phase transition betweenphases of different long-range orderings (LRO), and order-disorder phase transitions can bepredicted from even the simplest pair cluster expansions as demonstrated for generic face-centered cubic systems by Ackermann et al. [32] and for hexagonal close-packed systems byCrusius and Inden [33]. The method has also been used in conjunction with first-principlescalculations to study ordering in real non-model systems, including the Fe and Fe-Cr systems[18, 20, 21], the Fe-Ni, Fe-Cr, and Cr-Ni systems and the ternary Fe-Cr-Ni system [23], andother non-iron based system [19, 22].

6.2.1 Mean-field approximation

The cluster expansion model can be solved in a mean-field approximation where atoms areassumed to interact with an average background field of other atoms instead of interactionwith individual atoms. The large discretely defined configurational state σ is then reduced toa small continuously defined state of average occupations of sites in a small unit cell, typicallychosen to being able to represent an expected ordering and containing fewer than eight sites.The procedure for reduction of the discretely defined state is illustrated in Figure 6.2.

A formally exact infinite recursive formula exists for the configurational entropy of a sys-tem with configurational energies expressed as a cluster expansion [75]. The Gorsky–Bragg–Williams approximation of configurational entropy shortly reviewed in Section 6.5 can beregarded as a truncation of this recursion at the first non-trivial term [31], and, more gen-

(a) (b)

Figure 6.2: Representations of the configurational state for a two-dimensional sys-tem. Obtained as a single layer from an actual statistical sampling of the ε-Fe-N system (cf. Figure 6.36a). (a) Large unit cell and discrete configurational stateσ = (−1,+1,−1,+1,+1, . . . ). (b) The corresponding mean-field representationwith the reduced average configurational state σ = (0.00, 0.53, 0.84) and the unitcell (solid line). Fractional site occupations are illustrated by different intensities(dark to light gray).

69


erally, the CALPHAD approach [16] can be considered as mean-field approximated clusterexpansions with some approximation of the configurational entropy. Such an approxima-tion is required in the mean-field approach as individual configurational states are no longervisited by random sampling.

Methods using this approximation are also known as cluster variation methods, as mini-mum energy occupations are often solved variationally [29]. The method has been explicitlydemonstrated for the hexagonal close-packed system [76]. One important advantage of theapproach is that energy minimization can be solved by function minimization, such as usingthe non-linear solver Ipopt [62], typically ensuring correct and rapid convergence. However,the complexity of the configurational degrees of freedom is greatly reduced and unexpectedorderings can be inhibited by the choice of unit cell the configurational state.

Except for this short review, the approximation has not attempted in the present work asthe statistical sampling approach is expected to be more accurate [29].

6.3 Generic hexagonal close-packed system

In this section the generic hexagonal close-packed system of Crusius and Inden [33] is brieflystudied. The binary system corresponds to A and B atoms located on the iron atom sites of theε-Fe-N system and with the interstitial sites completely unoccupied. The cluster expansion isgiven in Table 6.1 and the two clusters are visualized in Figure 6.3.

α1 α2

Figure 6.3: Nearest-neighbor pair clusters between sites in the hexagonal close-packed lattice (red). This primitive unit cell corresponds to that of the host latticeof iron atoms (gray) alone. The stacking sequence of the two alternating iron atomclose-packed planes is ABAB . . . , when the interstitial sites are excluded. The clus-ter expansion energy (6.8) is calculated with contributions from all symmetricallyequivalent clusters throughout the crystal, i.e., from clusters that can be obtainedby any space group operation from these clusters.

α Coordinates mα Jα

α1 (0, 0, 0); (1, 0, 0) 6 14 W11

α2 (0, 0, 0); ( 23 , 1

3 , 12 ) 2 1

4 W12

Table 6.1: Pair clusters α1 and α2 and the corresponding cluster expansion co-efficients defined for this system. The coordinates of one of the mα symmetri-cally equivalent clusters per primitive unit cell are given. The clusters accountfor nearest-neighbor interaction within the close-packed plane and between twoplanes, respectively.

70

6.3 Generic hexagonal close-packed system

This brief diversion serves multiple purposes. Firstly, being a very simple system, it isincluded to demonstrate how a phase diagram can be obtained from statistical sampling andthat complex ordering can be predicted from even the simplest non-trivial cluster expansion.Secondly, it serves as validation of the implementation of the statistical sampling code on asystem that has been studied in great details. Finally, it is an important demonstration of thedifferent kinds of functional relationships between the chemical potential and fraction of Batoms xB to be expected.

A sequence of configurations (6.4) is generated by the Markov chain algorithm and theensemble average fraction of B atoms

xB =〈NB〉

〈NA〉+ 〈NB〉(6.12)

is calculated. This is repeated for a range of chemical potentials µ and the resulting relation-ship between µ and xB is given in Figure 6.5. The resulting phases corresponds closely to thephase diagram predicted by Crusius and Inden [33] in Figure 6.4.

Multiple phases of distinct orderings are obtained. The functional relationship between µand xB is particularly interesting. Since the chemical potential is constant throughout a two-phase region, values of xB in the interior of such a region cannot be reached by choosing thechemical potential alone. Instead a discontinuity in xB exists, corresponding to the values atthe edges of the two-phase regions.

Near phase boundaries within phases linear relationships are predicted, whereas nearhigh-symmetry points within phases, xB = 1

4 in particular, the chemical potential increasessteeply, only to flatten out again after that point. This corresponds to the similar observationmade for the ε-Fe-N system using the direct partition function approach in Section 5.8. A morecomplete investigation of this would require implementation of thermodynamic integration[67, 77]. In the short-range ordered (SRO) region for 0.32 ≤ xB ≤ 0.42, the relationship ismostly linear as adding one additional atom only effects the system locally.

71


Figure 6.4: Phase diagram of the generic hexagonal close-packed A-B systemexpressed by the nearest-neighbor cluster expansion given in Table 6.1. Calcu-lated by Crusius and Inden [33] for W11/W12 = 0.8. Normalized temperatureτ = kBT/(W11/4) with τ = 1.1 highlighted (red rectangle). Short-range order-ing (SRO) is predicted above the A2B phase (shaded). (Left) Expressed in termsof fraction of B atoms xB. (Right) Expressed in terms of the chemical potential µ.This does not allow formation of two-phase regions (cf. Figure 6.5).

0

2

4

6

8

10

12

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

18

316

14

516

38

716

12

Che

mic

alpo

tent

ail,

µ/(W

11/

4)

xB

D019 SRO B19

Figure 6.5: Correspondence between the chemical potential µ and the resultingaverage interstitial occupation yN in the statically sampled grand-canonical en-semble calculated trough (6.5). The predicted phase boundaries (dashed line) ac-curately agree with the calculations of Crusius and Inden [33] (cf. Figure 6.4). No-tice the distinct profiles of the chemical potential µ as a function of xB for variousphases and around the high-symmetry value xB = 1

4 within the D019 phase. Thesame behavior is predicted for the ε-Fe-N system in Figure 5.3 and Figure 6.23a.The chemical potential is mostly linear in the short-range ordering (SRO) rangeas introducing additional atoms only affects the system locally. The A2B phase(two-dimensional ordering in the close-packed planes) is not stable at τ = 1.1.

72



The configurational thermodynamics of the ε-Fe-N system is investigated using the clusterexpansion and thermodynamic statistical sampling approach. The iron sites in the lattice areassumed fully occupied, and the iron atoms can therefore be considered as inactive specta-tor atoms. The system is then essentially a binary system and the methodology developedin the preceding sections can be applied directly, with nitrogen atoms and vacancies beingrepresented by A and B atoms, respectively, and with spins +1 and −1 assigned. The con-figurational state σ is fully described by the occupation of the interstitial sites, and the ironatoms contribute only to the energies of the states as a static background field.

The cluster expansion is obtained from first-principles energies calculated for a large data-base of structures. The structure selection procedure and the cluster expansion fitting is de-scribed in details below.

The direct partition function approach introduced in Chapter 5 allows explicit incorpora-tion of electronic and vibrational degrees of freedom. This is not feasible with the thermo-dynamic statistical sampling approach, and electronic and vibrational degrees of freedom aretherefore ignored in this investigation. However, the statistic sampling approach is expectedto provide a much more detailed account of the configurational degrees of freedom.

The average fraction of interstitial sites occupied by nitrogen atoms can be formally de-fined in terms of ensemble averages (6.3) of the particle number operators NN and NFe,

yN =〈NN〉〈NFe〉

, (6.13)

where 〈NFe〉 is assumed fixed and included for normalization only, so the nitrogen numberoperator can simply be denoted by N = NN with the corresponding chemical potential µ. Therange of compositions satisfying 1

3 ≤ yN ≤ 12 is of particular interest as the hexagonal close-

packed arrangement is expected to be stable in this range according to the phase diagramobtained by Jack [2] given in Figure 6.7. The unit cell of the system is given in Figure 6.6.

ab

cFe

(a)a′b′

c′

B1

B2

A1

A2

C1

C2

Fe

(b)

Figure 6.6: (a) The primitive hexagonal close-packed unit cell with two iron atoms(gray) and two interstitial sites (black). If the close-packed stacking sequence ofthe iron atom layers is ABAB . . . , then the interstitial sites are located on C layersbetween each iron atom layer, so that the stacking sequence is ACBCACBC . . .(cf. Figure 6.3). Lattice vectors of the primitive unit cell a, b, c. (b) The conventionalhexagonal unit cell with six interstitial sites denoted by A1, A2, B1, B2, C1, C2. Thelattice vectors of the conventional unit cell satisfy a′ = a− b, b′ = a + 2b, c′ = c(two different projections).

73


Figure 6.7: Phase diagram of the binary Fe-N system compiled by Jack [2] with therange of interest highlighted (red rectangle). The hexagonal close-packed ε phaseis stable throughout most of the composition range 1

3 ≤ yN ≤ 12 considered in

the present work. In a narrow band around yN ' 12 an orthorhombic ζ phase is

observed with a corresponding narrow two-phase ε + ζ region. The γ′ phase isinhibited in the statistical sampling by the choice of unit cell.

The computer crystal used for the statistical sampling is represented by 16× 16× 8 con-ventional unit cells containing 12288 interstitial sites in total. Periodic boundary conditionsare imposed to avoid surface effects. The computer crystal is chosen large enough that theinteractions between one atom and its resulting mirrored atoms in the neighboring cells arediminishing.

In principle, to obtain a temperature–pressure phase diagram, the thermodynamic sta-tistical sampling should be performed for a fine mesh of lattice parameters (a, c), similarlythe Legendre transform performed in the partition function approach in Section 5.9 to ob-tain the Gibbs free energy from the Helmholtz free energy. As the statistical thermody-namic sampling for one pair of lattice parameters is already very expensive numerically,this is obviously infeasible. Instead a temperature–volume phase diagram is calculated and(a, c) = (2.71 A, 4.31 A) is chosen, corresponding to the equilibrium lattice parameters foryN = 5

12 at T = 0 as obtained in Section 5.10. The investigation presented in this chapteris more concerned with possible stable orderings of the crystal lattice than determining anaccurate relationship between yN and the chemical potential µ. If the more or less parallelmovements of the equations of state curves for different configurations in Figure 3.2 and Fig-ure 5.13 can be assumed in general, the effects of using lattice parameters slightly differentfrom equilibrium values are expected to be similar for different configurations, and the stableorderings would therefore not change significantly.

74

6.5 Review of earlier work

Another approach is to ignore the dependence on lattice parameters in the partition func-tion (6.1) and thereby eliminating the need for a proper Legendre transformation to obtaintemperature–pressure phase diagrams. Physically, this corresponds to a crystalline solidwhere stresses from non-matching lattice parameters are completely ignored. This approx-imation is employed by Wrobel et al. [23] in the first-principles thermodynamic statisticalsampling analysis of the Fe-Cr-Ni system.


The Gorsky–Bragg–Williams approximation of configurational entropy has been employedin earlier work to predict ordering of the ε-Fe-N system [1, 9, 10]. The model corresponds tothe simplest non-trivial instance of the mean-field approximation introduced in Section 6.2.1,and serves as a natural starting point for the thermodynamic statistical sampling to provideinsight into possible orderings. The model is briefly reviewed in this section; fitting parame-ters from earlier work are used here to explicitly calculate site occupations, Gibbs free energy,and chemical potential.

In this model, the average occupation, varying from 0 if fully unoccupied to 1 if fullyoccupied, of the six interstitial sites A1, A2, B1, B2, C1, C2 given in Figure 6.8 describe the con-

B1

B2

A1

A2

C1

C2

Configuration A

B1

B2

A1

A2

C1

C2

Configuration B

Figure 6.8: Configuration A and Configuration B are ordered configurations withmajority occupation at the indicated nitrogen sites and minority occupation at thevacant sites, respectively. These two configurations correspond to the lowest en-ergy configurations listed in Table 3.1. Compare with Figure 5.2.

Fe

B1

B2

A1

A2

C1

C2

Figure 6.9: Configuration B in the mean-field approximation. Fractional site occu-pations are illustrated by different intensities (dark to light gray). The sites A1 andB2 are almost fully occupied and C1 and C2 are almost unoccupied. The sites A2and B1 are completely unoccupied. The actual calculation is given in Figure 6.12b.This is contrasted to thermodynamic statistical sampling approach, where an largecomputer crystal is chosen, and sites are discretely occupied, i.e., either occupiedor unoccupied. Compare with Figure 6.2.

75


figurational state y,y =

(A1 y, A2 y, B1 y, B2 y, C1 y, C2 y). (6.14)

The total nitrogen occupation fraction is then given by

yN =16 ∑

k=A1,A2,B1,B2,C1,C2

ky. (6.15)

The model assumes nearest-neighbor interaction in a mean-field approximation, and thefollowing expression is obtained for the Gibbs free energy,

G(T, yN) = G0Fe + yNG0

N + WC

(yN −

26(A1 yA2 y + B1 yB2 y + C1 yC2 y

))

+ WP

(yN −

16(A1 yB1 y + A1 yC1 y + B1 yC1 y + A2 yB2 y + A2 yC2 y + B2 yC2 y

))

+16

RT ∑k=A1,A2,B1,B2,C1,C2

(ky log ky +

(1− ky

)log(1− ky

)), (6.16)

where G0Fe and G0

N are (insignificant) reference energies and WP and WC are exchange energieswithin the basal plane of the hexagonal unit cell and in the direction perpendicular to theclose-packed plane, and R is the gas constant. The exchange energies WP/(RT0) = −4.0 andWC/(RT0) = −3.5 are obtained by Pekelharing et al. [10] by fitting the model to experimentaldata at reference temperature T0 = 723 K.

Configuration A Majority sites: A1 y and B2 y = C2 yMinority sites: A2 y and B1 y = C1 y

Configuration B Majority sites: A1 y = B2 yMinority sites: A2 y = B1 y and C1 y = C2 y

Table 6.2: The two distinct configurations considered. The conditions imposed onthe majority and minority sites follow from the symmetries of the unit cell (seeFigure 6.8). With these assumptions the chemical potentials calculated in [10] isreproduced exactly.

Two distinct configurations, Configuration A and Configuration B, that minimize theGibbs free energy (6.16) are identified defined by the conditions listed in Table 6.2, and theglobal minimum of the Gibbs free energy is obtained by the convex hull extension. The result-ing Gibbs free energy is given in Figure 6.10 and its derivative, the chemical potential, is givenin Figure 6.11. At T = 723 K the model predicts a narrow two-phase region 0.349 ≤ yN ≤ 0.36.Configuration A minimizes the Gibbs free energy for yN ≥ 0.36. Thus the experimental datacorrespond to Configuration A for the given model parameters.

The configurational states, i.e., the occupation vector y, of Configuration A and Configu-ration B are explicitly solved by minimizing the Gibbs free energy subject to the conditionslisted in Table 6.2 using the non-linear constrained optimizer Ipopt [62] as implemented inthe author’s FuncLib library and given in Figure 6.12. The unit cell in this mean-field approx-imation corresponding to Configuration B is visualized in Figure 6.9.

Additional properties of the Gorsky–Bragg–Williams approximation of configurationalentropy are discussed in Appendix E.

76


−8

−6

−4

−2

0

2

4

6

0.30 0.35 0.40 0.45 0.50

Gib

bsfr

eeen

ergy

,G,m

eV

yN

B AA + B

Figure 6.10: Calculated Gibbs free energy of Configuration A and Configuration Bat T = 723 K defined relative to Fe6N2 and Fe6N3 reference states. The narrowtwo-phase region is determined by the convex hull extension, or equivalently thecommon tangent method [66], and corresponds to the global minimum of theGibbs free energy (6.16).

−8.30

−8.25

−8.20

−8.15

−8.10

−8.05

−8.00

−7.95

−7.90

0.30 0.35 0.40 0.45 0.50

Che

mic

alpo

tent

ial,

µ,e

V

yN

B AA + B

Figure 6.11: Chemical potential of Configuration A and Configuration B. The two-phase region has constant chemical potential. The insignificant reference valueG0

N is chosen to correspond to the value obtained from the first-principles calcu-lation. Experimental data from [10] (circle); model parameters are chosen to fitexperimental data.

77


0.0

0.2

0.4

0.6

0.8

1.0

0.30 0.35 0.40 0.45 0.50

Site

occu

pati

on

yN

A1 y

B2 y = C2y

B1 y = C1y

(a) Configuration A

0.0

0.2

0.4

0.6

0.8

1.0

0.30 0.35 0.40 0.45 0.50

Site

occu

pati

on

yN

A1 y = B2y

C1 y = C2y

A2 y = B1y

(b) Configuration B

Figure 6.12: Site occupations as a function of yN at T = 723 K. Majority sites haveoccupations above yN and minority sites below, respectively; see Table 6.2. A2 y isalmost zero in Configuration A and is omitted.

78

6.6 Database of structures


The cluster expansion coefficients used for the statistical sampling are determined from adatabase of structures, those energies are calculated from first-principles. The procedure forobtaining database structures and expansion coefficients is described in this section.

Firstly, a unit cell for the database structures is chosen. Periodic boundary conditions areimposed to avoid surface effects. Ideally, the size of the unit cell has to be large enough tobe able to determine isolated effects of exchanges of atoms, so that the interactions with mir-rored images of the exchanged atoms in the neighboring cells are diminishing, an unwantedartifact produced by the periodic boundary conditions. On the other hand, the computationalresources limit the size of the unit cell. A unit cell with 96 interstitial sites in four interstitiallayer planes is chosen as a compromise between these two considerations. The chosen unitcell is depicted in Figure 6.13.

This unit cell is consistent with the expected orderings of the ε-Fe6N2 and ε-Fe6N3 nitrides,i.e., Configuration B and Configuration A given in Figure 6.8, respectively. Moreover, Jack [2]has predicted the existence of a distinct ζ phase for yN ' 1

2 described by an orthorhombic unitcell and has hypothesized an intermediate ε-Fe24N10 nitride, both of which can be described

ab

Figure 6.13: Interstitial sites in (001) layer planes of the hexagonal close-packedlattice (black dots; iron atoms are ignored) and various unit cells. Primitive hexag-onal unit cell (black) with lattice vectors a and b (thick lines; lattice vector c isorthogonal to the plane), conventional hexagonal unit cell (blue rhombus), andorthorhombic unit cell (red rectangle). The database unit cell (large green rhom-bus) is chosen to be consistent with both hexagonal and orthorhombic orderings.Hence perfect hexagonal and orthorhombic orderings can be represented by thedatabase unit cell; the orthorhombic unit cell is repeated exactly three times inthe vertical [110] direction (i.e., a + b) and twice in the horizontal [110] direction(i.e., a− b). In general, the orthorhombic unit cell breaks the hexagonal symmetryand may change its lattice parameter in the [110] direction, but this is inhibitedin the computer crystal. To account for possible ordering in the [001] direction,the database unit cell includes four distinct layers of (001) planes and contains 96interstitial sites in total. The computer crystal used for sampling is much larger.

79


Fe

(a) Φα(σ) = +1

Fe

(b) Φα(σ) = −1

Figure 6.14: Two of the clusters (red) included in the cluster expansion with re-spect to the ε-Fe6N2 structure. The clusters are symmetrically equivalent as de-termined by the symmetries of the parent lattice, with the common equivalenceclass of clusters denoted by α2,2 and assigned the coefficient Jα2,2 in the clusterexpansion (6.8). In this case the clusters represent nearest-neighbor interactionswithin the close-packed plane. The contribution to the expanded energy is thesum of cluster functions (6.9) defined for the given structure. (a) Two like atomsresulting in the cluster function Φα(σ) = +1. (b) Two unlike atoms resulting inΦα(σ) = −1.

by the chosen unit cell. To account for possible ordering in the [001] direction, the unit cellincludes four layers of interstitial planes.

Secondly, low-energy structures are selected from an initial statistical sampling of a sim-plified nearest-neighbor cluster expansion obtained from the energies listed in Table 3.1.5 Theselection process is repeated as more accurate expansion coefficients are obtained to includestructures omitted in the first iteration. 128 symmetrically distinct structures are selected intotal.6

Thirdly and finally, the cluster expansion is fitted to calculated first-principles energiesof the structures in the database. The number of cluster expansion coefficients that can bemeaningfully distinguished is limited by the number of structures in the database. The fol-lowing symmetrically distinct clusters are included in the expansion: the empty cluster α0,the point cluster α1, pair clusters α2,1, . . . , α2,13, triple clusters α3,1, . . . , α3,6, four-point clustersα4,1, . . . , α4,5, five-point clusters α5,1, . . . , α5,2, and six-point clusters α6,1, . . . , α6,2. The clustersare illustrated in Figure 6.15. The contribution to cluster expansion from the α2,2 cluster withrespect to the ε-Fe6N3 structure is demonstrated in Figure 6.14. The fitting procedure of Walleand Ceder [34] is applied so that clusters determined to provide only negligible reduction inthe fitting error are explicitly excluded by taking coefficients to be identically zero.

The clusters and the fitted cluster expansion coefficients are listed in Table 6.3. First-principles energies of the database structures and the corresponding cluster expanded en-ergies are given in Figure 6.16. Predictions of ground-state structures are kept unchanged formost values of yN, i.e., the first-principles calculation as well as the cluster expansion pre-

5More precisely, zeroth- and first-order reference energies E0 and E1 as well as interaction between nearest-neighbor interstitial atoms within the close-packed plane and between two planes, corresponding to the clustersα0, α1, α2,1, and α2,2 in Table 6.3.

6The database unit cell was enlarged after the discovery of a distinct ζ phase, and therefore approximatelyhalf of the structures are sampled from a four-layer 48-site unit cell and half of the structures from a two-layer48-unit cell. First-principles calculations of all structures are performed on the 96-site unit cell to avoid numericaltruncation errors.

80


α0 α1 α2,1 α2,2 α2,3 α2,4

α2,5 α2,6 α2,7 α2,8 α2,9 α2,10

α2,11 α2,12 α2,13 α3,1 α3,2 α3,3

α3,4 α3,5 α3,6 α4,1 α4,2 α4,3

α4,4 α4,5 α5,1 α5,2 α6,1 α6,2

Figure 6.15: The clusters included in the cluster expansion. A cluster in a un-ordered collection on lattice points (red), i.e., interstitial sites for this system,defined throughout the crystal. The complete set of clusters includes any sym-metrically equivalent cluster obtained by space group operations from one ofα0, α1, α2,1, . . . , α6,2, e.g., six symmetrically equivalent clusters exists of α2,2 perprimitive unit cell, resulting in 36864 symmetrically equivalent clusters in the com-puter crystal in total. Coordinates of the clusters are listed Table 6.3.

81


−8.335

−8.330

−8.325

−8.320

−8.315

−8.310

−8.305

−8.300

−8.295

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

1748

38

1948

512

2148

1124

2348

12

Ene

rgy,

E,e

V

yN

(a) First-principles energies per atom of the 96-site structures in the database usedto fit the cluster expansion coefficients. For yN = 1

3 the distance between thelowest energy and second-lowest energy is larger than for other values of yN anda particularly strong preference for ordering is expected here.

−0.010

−0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

13

1748

38

1948

512

2148

1124

2348

12

Ene

rgy,

E−

E ref

,eV

yN

ε-Fe6N2 ζ-Fe8N4

ε-Fe6N3

ε-Fe24N10

(b) Cluster expansion fitted energies per atom (circle) with fitting errors (line seg-ment) of structures in the database. Data points in each box correspond to thesame value of yN. Energies of four important structures are indicated (cf. Sec-tion 6.8). A root-mean-squared fitting error of 0.96 meV per atom is obtained. Thecluster expansion is given in Table 6.3. Energies are relative to reference ener-gies obtained for ground-state structures at yN = 1

3 and yN = 12 , i.e., defined as

Eref = 6[( 12 − yN)Eε-Fe6N2

+ (yN − 13 )Eζ-Fe8N4

]. Improvements of cluster expan-sion fits are not possible without including significantly more clusters, making thestatistical sampling more demanding computationally.

Figure 6.16: Database of structures used to obtain the cluster expansion.

82


α Coordinates mα dα/a Jα, eVα0 1 0.00 −24.30α1 (0, 0, 0) 2 0.00 −3.428α2,1 (0, 0, 0); (0, 0, 1

2 ) 2 1.00 0.3167α2,2 (0, 0, 0); (1, 0, 0) 6 1.22 0.09280α2,3 (0, 0, 0); (1, 0, 1

2 ) 12 1.58 0.02315α2,4 (0, 0, 0); (0, 0, 1) 2 2.00 0.06930α2,5 (0, 1, 0); (1, 0, 0) 6 2.12 0.009874α2,6 (0, 1, 0); (1, 0, 1

2 ) 12 2.35 0.004963α2,7 (0, 0, 0); (1, 0, 1) 12 2.35 0.005391α2,8 (0, 0, 0); (2, 0, 0) 6 2.45 −0.002814α2,9 (0, 0, 0); (2, 0, 1

2 ) 12 2.65 0.0005945α2,10 (0, 1, 1); (1, 0, 0) 6 2.92 0.002774α2,11 (0, 1, 0); (1, 0, 1) 6 2.92 0α2,12 (0, 0, 0); (0, 0, 3

2 ) 2 3.00 0α2,13 (0, 0, 0); (2, 0, 1) 12 3.16 0.003646α3,1 (0, 0, 0); (1, 0, 0); (1, 1, 0) 4 1.22 0α3,2 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0) 24 1.58 0α3,3 (0, 0, 0); (0, 1, 0); (1, 1, 1

2 ) 12 1.58 0α3,4 (0, 0, 0); (1, 0, 0); (1, 1, 1

2 ) 12 1.58 0α3,5 (0, 0, 0); (0, 0, 1

2 ); (0, 0, 1) 2 2.00 0.06663α3,6 (0, 0, 0); (0, 0, 1); (1, 0, 1

2 ) 12 2.00 0.003853α4,1 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0); (1, 0, 12 ) 6 1.58 0

α4,2 (0, 0, 0); (0, 1, 12 ); (1, 1, 0); (1, 1, 1

2 ) 12 1.58 0α4,3 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0); (1, 1, 12 ) 12 1.58 0

α4,4 (0, 0, 0); (0, 1, 0); (1, 1, 0); (1, 1, 12 ) 12 1.58 0

α4,5 (0, 0, 0); (0, 0, 12 ); (1, 0, 0); (1, 1, 0) 12 1.58 0

α5,1 (0, 0, 0); (0, 0, 12 ); (0, 1, 0); (1, 1, 0); (1, 1, 1

2 ) 12 1.58 0α5,2 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0); (1, 1, 0); (1, 1, 12 ) 12 1.58 0

α6,1 (0, 0, 0); (0, 0, 12 ); (0, 1, 0); (0, 1, 1

2 ); (1, 1, 0); (1, 1, 12 ) 2 1.58 0

α6,2 (0, 0, 0); (0, 0, 12 ); (1, 0, 0); (1, 0, 1

2 ); (1, 1, 0); (1, 1, 12 ) 2 1.58 0

Table 6.3: Cluster expansion fitted to calculated first-principles energies of thestructures in the database. The symmetrically distinct clusters α0, α1, α2,1, . . . , α6,2are included in the cluster expansion. For the cluster α, mα symmetrically equiv-alent clusters exist per primitive unit cell, each with contribution Jα in the expan-sion (6.8) of the energy. A root-mean-squared fitting error of 0.96 meV per atom isobtained. The coordinates and the maximal separation distance dα of one of thesymmetrically equivalent clusters are given. See visualization in Figure 6.15. Theexpansion coefficient Jα is identically zero for some of the clusters as the contribu-tion is determined to be insignificant, i.e., the difference in fitted database structureenergies are indistinguishable from the case where non-zero values are allowed.The system favors ordering of atoms as the most significant pair-cluster coefficientare positive, i.e., Jα2,1 > 0 and Jα2,2 > 0, corresponding to nearest-neighbor inter-action within the close-packed plane and between two planes, respectively. Thismerely states a preference for separation of nitrogen atoms. The contribution fromthe static background of iron atoms is included in Jα0 .

83


yN Structure E, eV Efit, eV Efit − E, eV13 ε-Fe6N2 −8.3311 −8.3293 0.0018512 ε-Fe24N10 −8.3221 −8.3210 0.001112 ζ-Fe8N4 −8.3016 −8.3033 −0.0016

ε-Fe6N3 −8.2988 −8.2997 −0.0009

Table 6.4: First-principles energies and fitted cluster expansion energies of selectedimportant structures. Energies are given per atom. A root-mean-squared fittingerror of 0.96 meV per atom is obtained. Structures are denoted according to theprimitive unit cells. The structures are discussed and visualized in Section 6.8.

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

1748

38

1948

512

2148

1124

2348

12

Mag

neti

cm

omen

t,µ

/µ

B

yN

Figure 6.17: Magnetic moment per atom of structures in the database predicted bythe first-principles calculations. The magnetic moment stays inside a narrow band,indicating that the same magnetic phase is predicted throughout the databasestructures (cf. Figure 3.1), and that magnetic properties can safely be ignored in thestatistical sampling. The predicted magnetic moments per atom decline slowly asthe number of nitrogen atoms is increased.

dicts the same lowest energy structures, with one notable exception for yN = 1124 . Energies of

selected important structures are listen in Table 6.4.The cluster expansion model does not account for magnetic properties and it has be veri-

fied that no abrupt changes of the magnetic properties exist for the structures in the databaseto ensure that this can be safely ignored, i.e., no magnetic phase transitions. Therefore, mag-netic moments have also been calculated for each of the structures in the database. Figure 6.17shows the required gradual change in magnetic moments. More complicated magnetic clus-ter expansion models for the Fe and Fe-Cr system have previously been developed [21], butis not attempted in the present work due to the significant additional complexity introducedin such models.

84



First-principles energies are generally calculated as described in Section 4.1. The databasestructures are significantly larger than the previous calculations, contains up to 144 atoms intotal per structure. Therefore the following changes of first-principles calculation parametersare performed: A self-consistency energy convergence condition of 10−4 eV per atom is cho-sen. This is the VASP default value and is chosen in anticipation of a cluster expansion fittingerror an order of magnitude larger than this. A plane-wave basis truncation energy of 400 eVis chosen. 4× 2× 3 Γ centered k-points are used for the large 96-site irregularly shaped unitcell.

Unit cell lattice parameters are kept fixed in the calculations. Atomic positions relative tothe unit cell are relaxed before first-principles energies are calculated.

The author’s own implementation of the thermodynamic statistical sampling is used, im-plemented in the C# language, and running on Linux computer clusters with code generatedby the Mono compiler. The code is validated by comparison to previous calculations as de-scribed in Section 6.3. Pseudo-random numbers for the Markov chain algorithm (6.6) aregenerated using a fast linear congruential generator with period 232 [54]. This is assumedsufficiently accurate for the statistical sampling.

The cluster expansion coefficients are determined from first-principles energies of struc-tures in the database using the Alloy Theoretic Automated Toolkit (ATAT) [34].

6.8 Ground-state structures

Before proceeding to the thermodynamic statistical sampling, several observations must benoted for the identified ground state structures in the database.

Formally proving that a particular structure is a ground-state structure from fundamentalquantum physical equations, i.e., that no structure with the same composition and lowerenergy exists, is a highly non-trivial problem to solve mathematically. If energies are insteadassumed expressed in terms of a cluster expansion, the problem reduces substantially and hasbeen solved exactly in certain favorable cases [78, 29]. Such formal proofs are far outside thescope of the present work, and observations from the structures in the database obtained by

ab

(a)

a′b′

(b)

b′′

a′′

(c)

Figure 6.18: Relations between unit cells and the corresponding lattice vectors inthe (001) planes; each of the unit cells contain two such planes. Compare withFigure 6.13. (a) Two-site primitive hexagonal unit cell. (b) Six-site conventionalhexagonal unit cell. (c) Eight-site orthorhombic unit cell. The imposed hexagonalsymmetry b′′ =

√3a for the lengths of the lattice vectors is generally not satisfied

for the orthorhombic unit cell.

85


the iterative procedure outlined in Section 6.6 are accepted as representative, i.e., it is assumedthat no important ground state structure is missed by the iterative procedure.

Further improvements of the cluster expansion fits of ε-Fe6N2, ε-Fe24N10, ζ-Fe8N4 energiesare not possible without negatively affecting the fits of the remaining structure energies to asignificant extend or without introducing significantly more clusters in the expansion, i.e.,more clusters than given in Figure 6.15.

Almost perfect ordering in the [001] direction, i.e., along the c lattice vector, is generallypredicted for any of the studied temperatures T and nitrogen occupation yN, so that the com-puter crystal is composed of an alternating sequence of (001) layer planes, and that any twoeven layer planes and any two odd layer planes are generally identical. Therefore the sizeof the computer crystal is chosen somewhat smaller in this direction than would otherwisehave been necessary and contains 16 of these planes in total. Moreover, it generally sufficesto visualize one even layer plane and one odd layer plane to describe the computer crystal.

6.8.1 ε-Fe6N2

The ε-Fe6N2 is confirmed as a ground-state structure. The structure is visualized in Fig-ure 6.19. Other structures obtained by a single double-spin flip from ε-Fe6N2 differ in generalsignificantly in energy from this ground-state structure, indicating a particular strong order-ing for yN = 1

3 . The large energy differences are readily evident from Figure 6.16.The lattice vectors of the conventional hexagonal unit cell a′, b′, c′ are given in terms of

the primitive hexagonal unit cell lattice vectors by

a′ = a− b,b′ = a + 2b,c′ = c.

(6.17)

See illustration in Figure 6.18b.

Fe

(a) (b)

N, even

N, odd

Vacant

Figure 6.19: The ε-Fe6N2 ground-state structure. (a) Three-dimensional view.Occupied interstitial sites (black), vacant interstitial site (white), and iron atoms(gray). (b) Occupations of interstitial sites in alternating even and odd layer (001)planes with nitrogen in even layers (blue), nitrogen in odd layers (green), or bothlayers unoccupied (white), and the unit cell (solid line). Iron atoms are insignifi-cant due to the two perfectly alternating layers.

86

6.8 Ground-state structures

The structure is the same as the structure studied in details in Chapter 4. The space groupof the structure is P6322 with space group number 182.

6.8.2 ε-Fe6N3 and ζ-Fe8N4

Contrary to the initial expectation, the ε-Fe6N3 structure proved not to be a ground-state struc-ture. The structure is visualized in Figure 6.20.

Instead several structures with distinct ordering in the [110] direction were predicted,specifically, invariant under translation by a − b. The is consistent with the ζ phase con-

Fe

(a) (b)

N, even

N, odd

Figure 6.20: The ε-Fe6N3 structure. (a) Three-dimensional view. Occupied inter-stitial sites (black), vacant interstitial site (white), and iron atoms (gray). (b) Oc-cupations of interstitial sites in alternating even and odd layer (001) planes withnitrogen in even layers (blue) or nitrogen in odd layers (green) and the unit cell(solid line). Iron atoms are insignificant due to the two perfectly alternating layers.

a′′

b′′

c′′

(a) (b)

N, even

N, odd

Figure 6.21: The ζ-Fe8N4 ground-state structure. (a) Three-dimensional view.Occupied interstitial sites (black), vacant interstitial site (white), and iron atoms(gray). (b) Occupations of interstitial sites in alternating even and odd layer (001)planes with nitrogen in even layers (blue) or nitrogen in odd layers (green) andthe unit cell (solid line). Iron atoms are insignificant due to the two perfectly alter-nating layers. Distinguishing pattern is highlighted (dotted line).

87


jectured by Jack [2]. The ζ phase unit cell lattice vectors a′′, b′′, c′′ are given in terms of theprimitive hexagonal close-packed unit cell lattice vectors as

a′′ = 2a + 2b,b′′ = −a + b,c′′ = c.

(6.18)

See illustration in Figure 6.18c.By enlarging the database unit cell as described in Figure 6.13, ordered structures of both

the ε phase and the ζ phase can be described. The identified ground-state ζ-Fe8N4 structureis visualized in Figure 6.21. The lengths of the lattice vectors enforced by the symmetry of thehexagonal close-packed lattice satisfies b′′ =

√3a. This symmetry is broken in the orthorhom-

bic unit cell in general. A more details investigation of this is given in Section 6.17.

6.8.3 Intermediate ε-Fe24N10 nitride

The ε-Fe24N10 structure is visualized in Figure 6.22. The primitive unit cell of this structurecontains 24 interstitial sites in two (001) plane layers and is therefore larger than the otherordered structures considered here. From the visualization it is clear that the unit cell of thestructure can be considered as composed of two ε-Fe6N3 and two ε-Fe6N2 structures alignedin this four times larger unit cell. No other structure with the same composition and lowerenergy is encountered and the structure is therefore concluded to be a ground-state structureas hypothesized already by Jack [2].

The energy of this structure is given in Table 6.4. Notice that the first-principles energy islower that the fitted cluster expansion energy, making this structure slightly less favorable inthe statistical sampling than it would have been with the first-principles energy. Better fittingof this structure is impossible without either including significantly more clusters than listed

(a) (b)

N, even

N, odd

Vacant

Figure 6.22: The intermediate ε-Fe24N10 nitride. (a) Three-dimensional view.Occupied interstitial sites (black), vacant interstitial site (white), and iron atoms(gray). (b) Occupations of interstitial sites in alternating even and odd layer (001)planes with nitrogen in even layers (blue), nitrogen in odd layers (green), or bothlayers unoccupied (white), and the unit cell (solid line). Iron atoms are insignif-icant due to the two perfectly alternating layers. Distinguishing pattern is high-lighted (dotted line).

88

6.9 Chemical potential and the sampling procedure

in Table 6.3 or by accepting significantly worse fits of most other structures. In particular, it isexpected that more six-point clusters could be included to account for more complicated long-range ordering of this structure compared to the other ground state structures. However, thecascading increment in number of clusters from every subcluster of such a six-point cluster(as required by the fitting algorithm [34]) would make the statistical sampling algorithm sig-nificantly slower and would require more database structures to obtain reliable estimates ofexpansion coefficients.

Ordering consistent with the ε-Fe24N10 structure is encountered to some extend in thefinite temperature statistical sampling. See Section 6.10.1 and Section 6.12 for more details.

The ε-Fe24N10 intermediate nitride has also been investigated by neutron diffraction byLeineweber et al. [4], testing for both P6322 symmetry (corresponding to the symmetry of theideally ordered ε-Fe6N2 structure) and the lower P312 symmetry. The distribution of nitrogenatoms could not be fully resolved and the investigation was inconclusive.


The chemical potential µ is used as the exogenous variable in the thermodynamic statisticalsampling. Other variables, such as the ensemble average occupation

yN =〈NN〉〈NFe〉

, (6.19)

depend on µ as endogenous variables through (6.2) and (6.3). The sampling is performed insteps by decreasing µ from −7.4 eV to −9.2 eV.

Equilibration and proper separation of phases proved to be very hard for this system. Toseparate the ζ and ε phases as discussed in Section 6.8.2, approximately 104 successful spin-flips per sites are required, i.e., number of trial spin-flips being accepted by (6.6) and normal-ized by number of sites in the computer crystal; the total number of trial spin-flips is muchlarger, depending on temperature and preference for ordering, and is in the order of 1010

for the computer crystal in total. This very high number is almost two orders of magnitudesmore than recommended by Crusius and Inden [33]. It is likely that the ζ phase would emergemore clearly if the hexagonal close-packed symmetry imposed on the computer crystal couldbe relaxed and the orthorhombic lattice could expand in the [110] direction (cf. Figure 6.18c).The equilibrium orthorhombic unit cell lattice parameters are determined in Section 6.17.

After equilibration, the ensemble averages (6.5) are estimated by performing additionally2× 104 successful spin-flips per site. The procedure is repeated for temperatures T = 723 K,T = 573 K and T = 373 K. The predicted relationship between µ and the resulting value ofyN is given in Figure 6.23. Notice the smoothly calculated relationship. This is possible toobtain despite the randomly sampled nature of the calculations due to a very large numberof spin-flips (law of large numbers).

The model predicts several distinct phases separated by corresponding two-phase regions.The interior of two-phase regions cannot be reached by controlling the chemical potentialalone as the chemical potential is constant throughout such regions. Hence no data pointswithin two-phase regions are obtained.

Of particular interest for the present work is the emergence of a distinct ζ phase withorderings consistent with an orthorhombic unit cell. This is in excellent agreement with thephase diagram obtained by Jack [2] in Figure 6.7, where the phase boundaries are observed

89

−9.2

−9.0

−8.8

−8.6

−8.4

−8.2

−8.0

−7.8

−7.6

−7.4

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Che

mic

alpo

tent

ial,

µ,e

V

yN

(a) T = 723 K

−8.6

−8.4

−8.2

−8.0

−7.8

−7.6

−7.4

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Che

mic

alpo

tent

ial,

µ,e

V

yN

ε ζε + ζ

(b) T = 573 K

−8.6

−8.4

−8.2

−8.0

−7.8

−7.6

−7.4

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Che

mic

alpo

tent

ial,

µ,e

V

yN

ε ζε + ζ

(c) T = 373 K

Figure 6.23: Correspondence between the chemical potential µ and the resultingaverage interstitial occupation yN in the statically sampled grand-canonical en-semble. (a) No two-phase region predicted at this temperature. Predictions foryN ≤ 1

3 are without the scope of the present work and is an extrapolation from thedatabase of structures; included to demonstrate the functional shape of the chemi-cal potential around the high-symmetry value yN = 1

3 , similarly to the predictionsin Section 6.3. (b) The model predicts an ε → ζ phase transition at µ = −7.80 eV,with a corresponding two-phase region 0.4865 ≤ yN ≤ 0.4948. (c) The modelpredicts an ε → ζ phase transition at µ = −7.89 eV, with a corresponding two-phase region 0.4750 ≤ yN ≤ 0.4937. The two-phase region is significantly widerfor this temperature. Excellent agreement occurs with the phase diagram obtainedfrom experimental data in Figure 6.7 (notice that the γ′ phase is not considered).Additional minor phase boundaries are predicted. See Section 6.13.1 for details.


as an abrupt change in lattice parameters and breaking of the hexagonal lattice parametersymmetry. See more details in Section 6.13. Data points are sampled at increased densityat yN ' 0.48 to investigate possible phase transitions. At T = 373 K an additional phaseconsistent with a ζ-Fe16N6 structure is predicted. More details are given in Section 6.13.1.

The chemical potential is evaluated assuming constant volume and is therefore not di-rectly comparable to experimental data obtained at constant pressure. However, the chemicalpotential obtained in this section closely resembles that obtained from the Helmholtz poten-tial calculated by the direct partition function approach in Section 5.5 (cf. Figure 5.3); the lattercan be Legendre transformed to obtain the Gibbs potential due to the closed-form formulationof the partition function, and is in good agreement with experimental data from the literature[1, 10].

The thermodynamic statistical samplings at T = 723 K are continued until yN = 0.30is reached. This is an extrapolation of the database structures included to demonstrate thefunctional shape of the chemical potential considered as a function of yN around the high-symmetry value yN = 1

3 , corresponding to the Fe6N2 ground-state structure. The result issimilar to the observations for the generic hexagonal close-packed system in Section 6.3, andcan be considered as a smearing of the T → 0 limiting case where only equilibriums of trueground-state structures are possible (and and the chemical potential therefore is piece-wiseconstant step function).7

7Or alternatively, when extending phase diagrams to T → 0, phases are points of zero widths around theground states [34]. The remaining areas are spanned by two-phase regions with constant chemical potential.

91


6.10 Site occupations

In this section the statically sampled computer crystal is tested for orderings consistent withthe majority and minority sites of Configuration A or Configuration B defined in Table 6.2. Inthe Gorsky–Bragg–Williams approximation reviewed in Section 6.5, nitrogen occupations aredefined as exogenous variables for the sites A1, A2, B1, B2, C1, C2. In the statistical sampling,on the other hand, these are calculated as averages of occupations of interstitial sites of eachof the possible translations of the conventional unit cell in Figure 6.6b in the sampled com-puter crystal. The somewhat naive approach chosen in this section allows direct comparisonwith the predictions obtained using the Gorsky–Bragg–Williams approximation. Other morerefined methods are introduced in the following sections.

It turns out that site occupations cannot be defined meaningfully for ensemble averages.The reason for this is the following: Majority sites—say A1—may change spontaneously be-tween each of the symmetrically equivalent orientations of the crystal (and this is expectedto happen eventually if the visited subset of the configuration space is large enough). Thereis no way of enforcing correct orientation of the computer crystal in the sampling procedure.Therefore, when calculating ensemble averages by (6.5), configurations where A1 is indeeda majority site are mixed with configurations where the equivalent site A2 happened to be amajority site by the random sampling procedure.

0.0

0.2

0.4

0.6

0.8

1.0

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Site

occu

pati

on

yN

A1 y

A2 y

B1 y

B2 y

C1 y

C2 y

Figure 6.24: Average nitrogen occupations of the sites A1, A2, B1, B2, C1, C2 of thecomputer crystal at T = 723 K. Fractions of occupied sites in a single instance σt ofthe sampled sequence of configurations (6.4) in a computer crystal obtained afterequilibration of 100 successful spin-flips per site. Compare with site occupations ofConfiguration A and Configuration B obtained from the Gorsky–Bragg–Williamsapproximation in Figure 6.12.

92


yNA1 y A2 y B1 y B2 y C1 y C2 y

0.3403 0.9986 0.0000 0.0000 0.9991 0.0241 0.02160.9865a 0.0042a 0.0042a 0.9865a 0.0273a 0.0273a

0.9065b 0.0052b 0.0052b 0.9065b 0.1093b 0.1093b

a Experimental data obtained by Liapina et al. [5] from neutron scatteringafter being slowly cooled from T = 573 K.

b Same as above. Quenched from T = 573 K.

Table 6.5: Site occupations compared to experimental data obtained by Liapina etal. [5] from neutron scattering of powder of analytical composition Fe3N1.021, cor-responding to yN = 0.3403. Very strong ordering consistent with Configuration Bis predicted. The experimental data confirm ordering consistent with Configura-tion B and indicate weaker ordering than predicted by the statistical sampling.

To counter this, one might attempt, for each σt in the sequence of sampled configurations(6.4), to reorient the computer crystal in one of the symmetrically equivalent orientations tomaintain majority and minority sites according to Table 6.2, e.g., by reorienting σt such thatA1 y is largest and C1 y is smallest. A reorientation defined like this is always strictly defined,but such a scheme is problematic: Ensemble averages of A1 y and B2 y should be exactly iden-tical in Configuration B, but the reorientation procedure outlined here produces A1 y > B2 y assmall variations always assign the largest occupation number to A1 y.

Nonetheless, insight can still be gained by analyzing a single computer crystal instanceσt in the sequence of sampled configurations, most importantly because such an approachallows direct comparison with the results obtained from Gorsky–Bragg–Williams approxima-tion. In this section, average site occupations are obtained after equilibration by 100 success-ful spin-flips per site. This is sufficient at T = 723 K considered here for comparison withthe Gorsky–Bragg–Williams approximation, as equilibration is fast at this high temperature.After each such iteration, the chemical potential is lowered slightly and the equilibration pro-cedure is repeated. The sampling is initialized from the ordered structure ε-Fe6N3, consistentwith Configuration A. The resulting site occupations are given in Figure 6.24 as a function ofyN.

Very strong ordering consistent with Configuration B is predicted for yN ' 13 with site

occupations A1 y = B2 y = 1. Experimental data by Liapina et al. [5] for powder of analyt-ical composition Fe3N1.021 obtained by neutron scattering after quenching from T = 573 Kconfirm the Configuration B ordering and site occupations A1 y = B2 y = 0.9065 are obtained,significantly weaker than the ordering predicted by statistical sampling. The site occupationsA1 y = B2 y = 0.9865 are obtained when the sample is slowly cooled to ambient temperature.See full list of site occupations in Table 6.5.

6.10.1 Intermediate ε-Fe24N10 nitride

Interesting is also the noisy looking region centered at yN ' 512 in Figure 6.24. An intermediate

ε-Fe24N10 nitride was hypothesized by Jack [2] and identified as a ground state structure inSection 6.8.3. In order to investigate ordering consistent with this structure, the original unitcell is enlarged four times, and the 24 sites

A1, A′1, A′′1 , A′′′1 , B1, B′1, B′′1 , B′′′1 , . . . , C2, C′2, C′′2 , C′′′2 (6.20)

93


B1

B2

A1

A2

C1

C2B′1

B′2

A′1

A′2

C′1

C′2

B′′1

B′′2

A′′1

A′′2

C′′1

C′′2

B′′′1

B′′′2

A′′′1

A′′′2

C′′′1

C′′′2

(a)

B1

B2

A1

A2

C1

C2

B∗1

B∗2

A∗1

A∗2

C∗1

C∗2

Fe

(b)

Figure 6.25: Sites in the enlarged unit cells. Compare with the original sites inFigure 6.6b. (a) Twice as large in the [110] and [120] directions. The original sixsites are mirrored and denoted A1, A′1, A′′1 , A′′′1 , B1, B′1, B′′1 , B′′′1 , . . . , C2, C′2, C′′2 , C′′′2 .(b) Twice as large in the [001] direction. Mirrored sites A1, A∗1 , B1, B∗1 , . . . , C2, C∗2 .

are obtained by mirroring the original six sites as defined Figure 6.25a. Similarly, the twelvesites

A1, A∗1 , B1, B∗1 , . . . , C2, C∗2 (6.21)

are obtained by enlarging the unit cell twice as defined in Figure 6.25b.The resulting average occupations of the 24 sites are given in Figure 6.26. The occupations

are almost exactly grouped four-by-four, e.g., so that the average occupation of A1 is almostidentical to the average occupations of the mirrored sites A′1, A′′1 , and A′′′1 . Similarly, theaverage occupations of the twelve sites are given in Figure 6.27, and again the occupationsare grouped two-by-two by the six sites of the original unit cell. Thus the ordering is stilldescribed in terms of the six site unit cell; no ordering consistent with these larger unit cells areobserved at T = 723 K, such as ordering consistent with the ε-Fe24N10 structure in particular.

The origin of the apparent noisiness is instead that the symmetries in occupations of ma-jority sites A1 and B2 and in minority sites C1 and C2 is broken, so that the coupling

A1 y = B2 y, C1 y = C2 y, A2 y = B1 y, (6.22)

corresponding to Configuration B, is not as strongly defined for yN ' 512 in the sampled

computer crystal. Thus when evaluating the fractional site occupations other instances ofthe sampled computer crystal σt, the occupations tend to fluctuate around average values.Contrarily, no indications of any break of symmetries between any of the group of sites exist,e.g., between A1 and its mirrored sites A′1, A′′1 , A′′′1 .

Representative planes of the computer crystal can also be useful in the investigation oforderings at a selection of temperatures. Such planes are given in Figure 6.28 for yN = 5

12 .Clearly no ε-Fe24N10 ordering is observed at T = 723 K. As the temperature is lowered toT = 575 K and T = 373 K the distinctive patterns of the ε-Fe24N10 structure are clearly visible,while significant disordering still exists.

94


0.0

0.2

0.4

0.6

0.8

1.0

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Site

occu

pati

on

yN

A1 y

A2 y

B1 y

B2 y

C1 y

C2 y

Figure 6.26: ε-Fe24N10 ordering tested. Tested by enlarging the six-site unit cellfour times as defined in Figure 6.25a. Average site occupations are calculated forthe computer crystal for each of the original sites and their respective mirroredsites (same colors). No indication of this ordering at T = 723 K as the average siteoccupations are grouped almost exactly four-by-four, so that the additional free-dom in site occupations is not employed. The grouping is strongly defined evenin the noisy looking part around yN ' 5

12 . Some non-random ordering appear foryN ≤ 1

4 , but this is outside the scope of the present work.

0.0

0.2

0.4

0.6

0.8

1.0

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Site

occu

pati

on

yN

A1 y

A2 y

B1 y

B2 y

C1 y

C2 y

Figure 6.27: An alternative ε-Fe12N5 ordering tested. Tested by enlarging the six-site unit cell twice as defined in Figure 6.25b. This corresponds to the structureincluded in the partition function approach in Section 5.5. Average site occupa-tions are calculated for the computer crystal for each of the original sites and theirrespective mirrored sites (same colors). No indication of this ordering at T = 723 Kas the average site occupations are grouped almost exactly two-by-two, so that theadditional freedom in site occupations is not employed.

95


(a) T = 723 K and yN = 512

N, even

N, odd

Vacant

Fe

(b) T = 573 K and yN = 512 (c) T = 373 K and yN = 5

12

Figure 6.28: Representative neighboring (001) planes of sampled computer crys-tals for yN = 5

12 obtained at a selection of temperatures. Iron atoms (gray dot)are indicated as the computer crystal is not composed of two planes in a perfectalternating sequence. (a)–(b) Significant disordering. (c) The sampled computercrystal clearly resembles the perfectly ordered ε-Fe24N10 structure as temperatureis lowered to T = 373 K (cf. Figure 6.22). Some disordering still exists.

96

6.11 Long-range order parameters

6.11 Long-range order parameters

The degree of ordering of a specific configuration σ compared to a given perfectly orderedreference structure can be defined in terms of the long-range parameter introduced by Nixand Shockley [79]. The order parameter is defined for a binary alloy of A and B atoms as

S = γα + γβ − 1 =γα − xA

yβ=

γβ − xB

yα, (6.23)

where α and β are sites of A atoms and B atoms in the perfectly ordered reference structure, yα

and yβ relative number of α and β sites, γα and γβ are the fractions of α and β sites occupiedby the right atom in σ, and xA and xB are the atomic fractions of A and B, respectively [35].From this definition −1 ≤ S ≤ 1, with S = 1 if and only if σ is perfectly ordered withγα = γβ = 1, and S = 0 for completely disordered structures. Generally, S < 1 is obtained fornon-stoichiometric compositions and for finite temperature T > 0.

In the sampled computer crystal, the order parameter is calculated for each symmetricallyequivalent choice of α and β sites of the perfectly ordered reference structure and the maximalvalue is used for calculation of the order parameter. This modification is necessary as specificchoices of majority and minority sites cannot be imposed on the computer crystal in the sta-tistical sampling. The order parameter is meaningfully defined as an ensemble average withthis modification.

This approach is different from that of Crusius and Inden [33], where the symmetriesare incorporated explicitly in a single order parameter formula; the latter approach provedunfruitful for the ε-Fe-N system and is expected to result in very complex expressions forlarge reference structures such as ε-Fe24N10. In both cases, a globally defined order parametermay fail to capture ordering for some types of lattice defects, such as stacking faults, in anotherwise perfectly ordered crystal.

The long-range order parameter is applied to the ε-Fe-N system and orderings consistentwith ε-Fe6N2, ε-Fe6N3, ε-Fe24N10, and ζ-Fe8N4 are tested. The predicted ensemble averageorder parameters are given as a function of yN in Figure 6.29.

The computer crystal at yN = 13 is determined to be perfectly ordered consistent with the

ε-Fe6N2 structure, i.e., Sε-Fe6N2= 1 for any of the considered temperatures, as already dis-

cussed in Section 6.10. Simultaneously, significant ε-Fe6N3 ordering is also predicted withSε-Fe6N3

= 23 . This is due to overlapping of majority sites of the two structures. Hence distin-

guishing orderings with respect to these reference structures with the globally defined orderparameters can be problematic.

The order parameter fails to capture ζ phase ordering at the temperatures of interest, prob-ably because too much disordering exists and since ordering is possible in multiple directionswith the imposed hexagonal symmetry. This is discussed in more details in Section 6.13. Theorder parameter Sζ-Fe8N4

is significantly larger for T = 573 K compared to T = 723 K.Ordering consistent with ε-Fe24N10 is predicted to increase at yN ' 5

12 in agreement withthe hypothesized intermediate ε-Fe24N10 nitride. The ordering is clearly visible in the repre-sentative planes given in Figure 6.28.

The explanation of the deviation of order parameters at T = 373 K and yN ' 38 in Fig-

ure 6.29c is that a distinct phase emerges at this temperature. The ordering in this phase isconsistent with an ε-Fe16N6 structure. See Section 6.13.1 for more details.

97

0.0

0.2

0.4

0.6

0.8

1.0

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Ord

erpa

ram

eter

yN

ε-Fe24N10

ε-Fe6N3ε-Fe6N2

ζ-Fe8N4

(a) T = 723 K

0.0

0.2

0.4

0.6

0.8

1.0

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Ord

erpa

ram

eter

yN

ε-Fe24N10

ε-Fe6N3ε-Fe6N2

ζ-Fe8N4

(b) T = 573 K

0.0

0.2

0.4

0.6

0.8

1.0

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Ord

erpa

ram

eter

yN

ζ-Fe16N6

ε-Fe24N10 ε-Fe6N3ε-Fe6N2

ζ-Fe8N4

(c) T = 373 K

Figure 6.29: Predicted ensemble average order parameters defined relative to theperfectly ordered ε-Fe6N2, ε-Fe6N3, ε-Fe24N10, and ζ-Fe8N4 reference structures,respectively. (a) Despite no ε → ζ phase transition is predicted for T = 723 K, theordering changes significantly for yN ' 0.495, possibly due to a narrow almostinvisible phase transition. (b) Different ordering at yN ' 5

12 . The ζ phase is toodisordered to allow any ordering of significance to be captured. (c) The deviationat yN ' 3

8 is likely a distinct phase of ordering consistent with a ζ-Fe16N6 structure(cf. Section 6.13.1).

6.12 Cowley–Warren short-range order parameters

6.12 Cowley–Warren short-range order parameters

Another class of order parameters is the locally defined Cowley–Warren short-range orderparameters [35]. These can conveniently be defined in terms of the clusters introduced inSection 6.2 through the cluster function (6.9). Specifically, for a pair-cluster α = (α1, α2) and agiven configurational state σ, the cluster correlation function is the average

ρα(σ) = 〈Φα′(σ)〉α′∼α, (6.24)

including any cluster α′ that is symmetrically equivalent to α, i.e., that can be obtained fromα through space group operations defined on the lattice.8 It must be emphasized that 〈 · 〉α′∼α

denotes the average as obtained through these operations for a single fixed instance of thecrystal as described by the configurational state σ and not the usual ensemble average. Fora binary alloy of A and B atoms Φα(σ) is 1 if the sites contain atoms AA or BB and −1 is thesites contain atoms AB or BA.

Fe

(a) Φα(σ) = −1

Fe

(b) Φα(σ) = +1

Figure 6.30: Two symmetrically equivalent α2,2 clusters (red) with respect to theε-Fe6N3 structure. Nearest-neighbor occupations within the close-packed planeare in the ratio two-to-one of unlike and like atoms, respective, resulting in theaverage ξα2,2 = − 1

3 for this structure. Compare with Figure 6.14. (a) Two unlikeatoms resulting in the cluster function Φα(σ) = +1. (b) Two like atoms resultingin Φα(σ) = −1.

The Cowley–Warren short-range order parameter with respect the pair of neighboringsites as specified by the cluster α is defined by the ensemble average of the cluster correlationfunction through the normalization

ξα =〈ρα〉 − (xA − xB)

2

4xAxB, (6.25)

resulting in a number between −1 and 1, i.e., 〈 · 〉 is the usual ensemble average. The nor-malization is introduced so that 1 is obtained for perfect correlation, −1 for perfect anti-correlation, and 0 for complete randomness, even if xA = xB = 1

2 is not satisfied. The orderparameter calculation is illustrated for the ε-Fe6N3 structure in Figure 6.30.

The locally defined Cowley–Warren order parameters are more robust to limited disorder-ing than the globally defined order parameters introduced in Section 6.11, especially latticedefects such as as stacking faults of an otherwise perfectly ordered crystal. Moreover, con-trary to the site occupations discussed in Section 6.10, the order parameters are well-defined

8The space group is the combination of the translational symmetry of a unit cell and the point group symmetryoperations of reflection, rotation, and the screw axis and glide plane symmetry operations [80].

99


Structure α2,1 α2,2 α2,3 α2,4 α2,5 α2,6 α2,7 α2,8 α2,9 α2,10 α2,11 α2,12 α2,13

ε-Fe6N2 − 12 − 1

214 1 1 − 1

2 − 12 − 1

214 1 1 − 1

2 − 12

ε-Fe6N3 −1 − 13

13 1 1 −1 − 1

3 − 13

13 1 1 −1 − 1

3

ε-Fe24N10 − 57 − 13

351135 1 23

35 − 57 − 13

35 − 15

1135

2335

2335 − 5

7 − 15

ε-Fe12N5 − 57 − 13

351135

2335 1 − 5

7 − 1335 − 13

351135

2335

2335 − 5

7 − 1335

ζ-Fe8N4 −1 − 13

13 1 1

3 − 13 − 1

3 − 13

13

13

13 −1 − 1

3

ζ-Fe16N6 − 35 − 19

451345 1 29

45 − 35 − 19

45 − 1145

1345

2945

2945 − 3

5 − 1145

Table 6.6: Cowley–Warren short-range order parameters for perfectly orderedstructures. See illustrations of clusters in Figure 6.15. Notice that except for ε-Fe12N5, all of the structures have perfect order between two even layer planes andtwo odd layer planes and the order parameters corresponding to the cluster α2,4are +1. Likewise, even and odd layer planes in ε-Fe6N3 and ζ-Fe8N4 have exactlyopposite occupations and the order parameters corresponding to the cluster α2,1both equal −1. Similarly, the other order parameters are calculated by inspectingthe geometric arrangements of the ordered structures. Compare with ensembleaverage parameters of the computer crystal in Figure 6.31 and Figure 6.32.

in thermodynamic ensembles, and provide a more systematic approach to determining spe-cific orderings than the representative planes of the computer crystal.

The Cowley–Warren order parameters can be considered as signatures of specific order-ings. The order parameters are explicitly calculated for the perfectly ordered ε-Fe6N2, ε-Fe6N3,ε-Fe24N10, and ζ-Fe8N4 structures corresponding to the pair-clusters a2,2, . . . , a2,13 and arelisted in Table 6.6. Included are also the ε-Fe12N5 structure from Chapter 5 and the proposedζ-Fe16N6 structure from Section 6.13.1. The predicted ensemble average order parameters atT = 573 K are given in Figure 6.31 and a subset of the predicted order parameters at T = 723 Kand T = 373 K are given in Figure 6.32.

The predicted ensemble average order parameters provide clear indication of ζ phase or-dering for yN ' 1

2 since the ensemble average parameters are in good agreement with that ofthe perfectly ordered ζ-Fe8N4 structure. The ordering is distinguished by the clusters α2,5, α2,6,α2,10, and α2,11 in particular, where the order parameters of the perfectly ordered structuresare different.

The order parameters are also generally consistent with the ε-Fe24N10 intermediate nitride,especially at T = 373 K.

The cluster α2,4 defines correlation between two neighboring even layer sites or two neigh-boring odd layer planes in the [001] direction, and therefore ξα2,4 = +1 for structures com-posed of a perfectly alternating sequence of two distinct (001) layer planes. A slight deviationfrom this is predicted in the sampled computer crystal, particularly for yN ' 5

10 . Interestingly,the ordering in the [001] direction is best described by the ε-Fe12N5 structure discussed inChapter 5.

Ordering perfectly consistent with the proposed ζ-Fe16N6 in Section 6.13.1 is predictedyN ' 3

8 at T = 373 K. This is not the case for temperatures higher than this.Long-range ordering is generally predicted to be preserved as the order parameters re-

main non-zero as the cluster lengths increase.

100

−1

0ξ α

2,1

0

1

ξ α2,

3

0

1

ξ α2,

5

ε-Fe6N2

ε-Fe24N10

ε-Fe6N3

ζ-Fe8N4

−1

0

ξ α2,

7

0

1

ξ α2,

9

0

1

ξ α2,

11

−1

0

13

38

512

1124

12

ξ α2,

13

yN

−1

0

ξ α2,

2

0

1

ξ α2,

4

ζ-Fe12N5

−1

0

ξ α2,

6

−1

0ξ α

2,8

0

1

ξ α2,

10

−1

0

13

38

512

1124

12

ξ α2,

12

yN

Figure 6.31: Ensemble average Cowley–Warren short-range order parameter pre-dicted for the computer crystal at T = 573 K (small dots) compared with orderparameters for perfectly ordered structures ε-Fe6N2, ε-Fe6N3, and ε-Fe24N10 (redsquares), ε-Fe12N5 (orange diamond), and ζ-Fe8N4 (blue circle). For yN ' 1

2 the ζphase provides significantly better agreement with the sampled ensemble averageorder parameters than the ε phase, i.e., ζ-Fe8N4 is generally in much closer agree-ment with sampled ensemble average values than ε-Fe6N3. This is the case forα2,5, α2,6, α2,10, and α2,11 in particular. The order parameters are generally consis-tent with the ε-Fe24N10 intermediate nitride. Notice from α2,4 that the simulationpredicts ordering in the [001] direction slightly different from each of the perfectlyordered structures (all of which contain two alternating (001) layer planes only).


0

1

ξ α2,

5

0

1

13

38

512

1124

12

ξ α2,

10

yN

−1

0

ξ α2,

6

0

1

13

38

512

1124

12

ξ α2,

11yN

(a) T = 723 K

0

1

ξ α2,

5

ε-Fe6N2

ε-Fe24N10

ε-Fe6N3

ζ-Fe8N4ζ-Fe16N6

0

1

13

38

512

1124

12

ξ α2,

10

yN

−1

0

ξ α2,

6

0

1

13

38

512

1124

12

ξ α2,

11

yN

(b) T = 373 K

Figure 6.32: Ensemble average Cowley–Warren short-range order parameter pre-dicted for the computer crystal (small dots) compared with order parametersfor perfectly ordered structures ε-Fe6N2, ε-Fe6N3, and ε-Fe24N10 (red squares),ε-Fe12N5 (orange diamond), ζ-Fe8N4 (blue circle), and ζ-Fe16N6 (green triangle).For yN ' 1

2 the ζ phase provides significantly better agreement with the sampledensemble average order parameters than the ε phase, i.e., ζ-Fe8N4 is generallyin much closer agreement with sampled ensemble average values than ε-Fe6N3.(a) Predicted order parameters are closed to those of ζ-Fe8N4 even at this temper-ature. (c) Perfect ζ phase ordering is not predicted even at this lower temperature.The deviation at yN ' 3

8 is likely a distinct phase of ordering consistent with anε-Fe16N6 structure (cf. Section 6.13.1).

102

6.13 ε→ ζ phase transition


The hexagonal close-packed lattice chosen for the computer crystal and the thermodynamicstatistical sampling allow the orthorhombic ζ ordering to be represented as discussed in Sec-tion 6.6. Indications of an ε→ ζ phase transition have already been obtained by the Cowley–Warren short-range order parameters in particular. The phase transition with the movementsof nitrogen atoms between even and odd layer planes is illustrated in Figure 6.33.

ε ε→ ζ ζ

N, even

N, odd

Figure 6.33: The ε→ ζ phase transition. Idealized orderings for each of the phasescorresponding the ε-Fe6N3 and ζ-Fe8N4 structures (i.e., for yN = 1

2 ). The move-ments of nitrogen atoms between even and odd layer planes during the phasetransition are illustrated [2].

The predicted phase transition boundaries are given in Figure 6.23 and are in excellentagreement with phase transition boundaries obtained from experimental data as an abruptchange in lattice parameters observed by Jack [2]. These lattice parameters are given in Fig-ure 6.34 for reference.

The ζ phase is inhabited in the computer simulation as lattice parameters consistent withthe hexagonal close-packed lattice are enforced. Thus if the lattice was allowed to expandin the [110] direction (corresponding to along the b′′ lattice vector of the orthorhombic unitcell) as suggested from the experimental data, it is conjectured that energies of configurationsconsistent with ζ phase ordering would be comparably more favorable. See Section 6.17 formore details on the ζ phase equilibrium lattice parameters.

Representative planes of the sampled computer crystal are given in Figure 6.35 and Fig-ure 6.36 for T = 573 K and T = 373 K, respectively. The phase transition is not clearly visiblefor T = 723 K. For comparison, representative planes of the sampled computer crystal at thistemperatures are also given in Figure 6.37, corresponding to approximately the same valueof yN. Notice that the close-packed hexagonal lattice allows ζ phase ordering in multiple di-rections. All of the sampled computer crystals of the ζ phase contain lattice defects where thedistinguishing ζ phase patterns are shifted or rotated. This could be the explanation for thefailure of the globally defined order parameter to capture ζ phase ordering in Section 6.11. Itis hypothesized the more well-defined ordering ζ phase would be observed if the hexagonalsymmetry was broken and ordering was enforced in one particular direction. See discussionin Section 6.17.

The original database unit cell was not consistent with the ordering of the ζ phase. None-theless, the ε → ζ phase transition was predicted, indicating robustness in the predicting ofthe ζ phase ordering. See Appendix C for details.

103


2.64

2.68

2.72

2.76

2.80

a,A a = b/

√3

b/√

3

a

ε ζε + ζ

4.34

4.36

4.38

4.40

4.42

4.44

0.25 0.30 0.35 0.40 0.45 0.50

c,A

yN

ε ζε + ζ

Figure 6.34: The ε→ ζ phase transition as identified by an abrupt change in latticeparameters. Experimental data obtained by Jack [2] (circle) with fitted polynomialexpansions (line). The symmetry b =

√3a of the lattice parameters of the hexago-

nal close-packed lattice is broken in the ζ phase (cf. Figure 6.18); a slight change inthe lattice parameter c is also observed. The samples were prepared at T ' 700 Knear the ε→ ζ phase transition.

(a) T = 573 K and µ = −7.805 eV (b) T = 573 K and µ = −7.800 eV

N, even

N, odd

Vacant

Fe

Figure 6.35: Phase transition at T = 573 K. Representative neighboring (001)planes of the sampled computer crystals. Distinguishing patterns are highlighted(dotted line). (a) Just on the ε side of the ε → ζ phase transition (cf. Figure 6.23b).(b) Just on the ζ side of the ε → ζ phase transition. The ζ ordering is possible inmultiple directions for orthorhombic lattice parameters consistent with the hexag-onal unit cell (cf. Figure 6.13). Notice the very small difference in chemical poten-tial µ. In both cases significant disorder exist as temperature is elevated and asperfect ordering is not possible for yN ' 0.48. Only a few double-unoccupied andno double-occupied sites exist.

104



N, even

N, odd

Vacant

Fe

Figure 6.36: Phase transition at T = 373 K. Representative neighboring (001)planes of the sampled computer crystals. Distinguishing patterns are highlighted(dotted line). Substantial change in the ordering of computer crystal for the smallchange in chemical potential for this temperature. (a) Just on the ε side of theε → ζ phase transition (cf. Figure 6.23c). (b) Compared to 573 the ζ phase patternare more ordered, with the patterns aligned in the same direction (horizontally).

T = 723 K and µ = −7.725 eV

N, even

N, odd

Vacant

Fe

Figure 6.37: The computer crystal at T = 723 K where no ε → ζ phase transitionis predicted. Representative neighboring (001) planes of the computer crystal.Chemical potential µ = −7.725 eV, corresponding to yN = 0.4906. Significantdisordering exists. Distinguishing patterns of both the ε and ζ phases are visi-ble. Compare with the computer crystal at phase transition boundaries at lowertemperatures in Figure 6.35 and Figure 6.36.

105


6.13.1 Proposed ζ-Fe16N6 structure

From the Figure 6.23c it is evident that the model predicts at least one additional distinct phasewith two corresponding clearly visible two-phase regions at T = 373 K. This is supported bythe discontinuities in the order parameters in Figure 6.29c and in the Cowley–Warren short-range order parameters in Figure 6.32b.

Representative planes of the computer crystal in Figure 6.38 obtained for yN = 38 . The or-

dering of the phase could be consistent with a ζ-Fe16N6 structure with a unit cell as indicated,corresponding to the high-symmetry point yN = 3

8 .The unit cell of this structure corresponds to the orthorhombic unit cell repeated twice in

the vertical [110] direction (cf. Figure 6.13) and the structure is therefore not consistent withthe database unit cell. Hence the energy of the structure is not calculated explicitly and is notincluded in the fitting procedure of the cluster expansion. The much larger computer crystalis large enough to allow the ordering of the phase.

Multiple similar minor phase transitions are likely to be hidden in the noise of the sam-pling process.

T = 373 K and yN = 38

N, even

N, odd

Vacant

Fe

Figure 6.38: Representative neighboring (001) planes of sampled computer crys-tals for yN = 3

8 obtained at T = 373 K. Iron atoms (gray dot) are indicated as thecomputer crystal is not composed of two planes in a perfect alternating sequence.Ordering could be consistent with a ζ-Fe16N6 structure. The unit cell of the pro-posed structure (dashed line) is not consistent with the chosen database unit cell,and the structure is therefore not calculated explicitly in the database. Very lim-ited disordering and visibly less pronounced than for the ε-Fe24N10 structure inFigure 6.28c.

106

6.14 Local environment of iron atoms


The strong magnetic properties of iron allow signatures of the local environment of the ironatoms—such as how nitrogen atoms are distributed around iron atoms—to be obtained ex-perimentally from Mossbauer spectroscopy [1, 10, 81], and therefore offer an opportunity tocompare with prediction from the statistically sampled computer crystal, where the exact dis-tribution of atoms can be evaluated directly at any instant.

Two of the Mossbauer spectra recorded by Somers et al. [1] are given in Figure 6.39 forreference. The total nitrogen occupation of interstitial sites yN is determined independentlyby other methods; in this case from thermogravimetry.

Possible explanations of the origin of the distinct Mossbauer components are investigatedusing the sampled computer crystals in the following sections from a thermodynamic statis-tical sampling perspective.

6.14.1 Distinct sextet hypothesis

Each iron atom in the lattice is surrounded by six nearest-neighbor interstitial sites, whichconveniently, in the setting of cluster expansions, can be identified as a six-point cluster or asextet, specifically corresponding to the α6,1 cluster. Two symmetrically equivalent sextets ex-ist per primitive hexagonal unit cell, one for each iron atom, and are visualized in Figure 6.40.The full list of possible sextet occupation configurations is given in Table 6.8 grouped in 13distinct groups of symmetrically equivalent configurations according to symmetries of thehexagonal unit cell.

Somers et al. [1] and Pekelharing et al. [10] have conjectured that the components of thespectra correspond to distinct sextet occupation configurations, i.e., to distinct nitrogen occu-pations of the six nearest neighbor sites of the iron atoms. Thus the four components in thespectrum in Figure 6.39b correspond to four distinct sextets and, moreover, for yN = 0.4151 tobe maintained the most satisfactory fit in obtained if two of the sextets are occupied by exactlythree nitrogen atoms [1]; the resulting sextet occurrences are given in Table 6.7 for reference.

(a) yN = 0.3303 (b) yN = 0.4151

Figure 6.39: Mossbauer spectra with fitted curves recorded by Somers et al. [1]for varying total nitrogen interstitial occupations yN as determined from thermo-gravimetry. (a) ε phase reference powder with yN = 0.3303. A single componentis visible in the spectrum. (b) ε phase foil with yN = 0.4151. Four distinct compo-nents visible; three of them clearly visible.

107


Figure 6.40: Iron atoms (gray) and the six nearest-neighbor interstitial sites (red)constituting a sextet (red line). The sextets can be identified with the cluster α6,1(cf. Figure 6.15). Two symmetrically equivalent sextets per primitive unit cell. Thecomputer crystal contains 12288 sextets in total. Notice that the sites of the sextetsare shared with sites of neighboring sextets. See full list of possible sextet occupa-tion configurations in Table 6.8.

yN I II III (a) III (b)0.3303 0 1 0 00.3351 0.12 0.68 0.201 00.3795 0.072 0.578 0.221 0.1290.4151 0.01 0.454 0.327 0.2040.4568 0 0.262 0.308 0.430

Table 6.7: Relative occurrences of the four distinct sextets identified from Moss-bauer spectra by Somers et al. [1] for varying total number of occupied interstitialsites yN. The sextets are occupied by 1, 2, 3, and 3 nitrogen atoms, respectively.Notice the significant occurrences from two distinct sextets occupied by three ni-trogen atoms.

In order to get an acceptable fit to the recorded spectrum in Figure 6.39b, at least fourcomponents are required, and to maintain the correct value of the independently determinedtotal nitrogen occupation yN, the nitrogen occupation of each of the component can be in-ferred. The details of this fitting procedure is given in [1] and is beyond the scope of thepresent work. The result is given in Table 6.7 for reference.

Mossbauer spectra are recorded at temperature T = 4.2 K after fast cooling of the samplessynthesized at T = 723 K [1].9 However, it is highly unlikely that configurational equilibriumcorresponding to this very low temperature is reached within the timescale of the experi-ment, and the samples are expected to have atomic configurations locked somewhere in thetemperature range from 300 K to 600 K, depending on the energy barriers for the atoms toreach equilibrium arrangements; the latter may itself depend on yN. Computer calculation isperformed at fixed temperature T = 373 K. The exact temperature is not important as highdegree of local ordering is obtained even for elevated temperatures.

In order to analyze sextet occurrences of the computer crystal, symbols are chosen for thesymmetrically distinct sextet configurations,

N0, N1, N2, N′2, N′′2 , N3, N′3, N′′3 , N4, N′4, N′′4 , N5, N6, (6.26)

as defined in Table 6.8. The symbols are chosen according to the calculated first-principlesenergies for the conventional hexagonal unit cell listed in Table 3.1, such that the unprimed

9The Curie temperature of ε-Fe-N varies from about 560 K for yN = 13 to less that 4.2 K for yN ' 1

2 [82].

108


Symbol Degeneracy Sextet configurations

N0 1

N1 6

N2 6

N′2 6

N′′2 3

N3 6

N′3 2

N′′3 12

N4 6

N′4 6

N′′4 3

N5 6

N6 1

Table 6.8: Possible nitrogen occupation configurations of sextets surrounding ironatoms, visualized with occupied (black) and unoccupied (white) sites. The 64 pos-sible sextet occupation configurations are grouped according to symmetries of theconventional hexagonal unit cell. The degeneracy is the number of symmetricallyequivalent sextets in each group. Symbols are assigned according to the calculatedfirst-principles energies listed in Table 3.1. Thus only the lowest energy configu-rations N0, N1, N2, N3, N4, N5, and N6 are expected to occur in the T → 0 limit.Contrarily, in the perfectly disordered T → ∞ limit (ignoring decomposition ofthe crystal lattice), the energies are unimportant, and the probability of encounter-ing a particular sextet is proportional to its degeneracy, e.g., it is twice as likely toencounter N′′3 as N3 despite its higher calculated energy, simply because it is morelikely to encounter the former arrangement by chance.

109


symbol indicates the lowest energy configuration (or ground state), one prime indicates firstexcited state, and two primes indicate second excited state.

It must be emphasized that the terminology introduced above only applies to the nitrogenoccupations in the computer crystal as a group of six neighboring interstitial sites surroundingthe iron atoms. Thus no specific magnetic model that couples occupations with Mossbauerspectra is implied.

Specifically, Shang et al. [83] have conjectured that the two distinct Mossbauer spectrumcomponent associated with three neighboring nitrogen atoms are are explained by a disorderin the location of the nitrogen atoms, and that some N3 sextets have been replaced by N′3,corresponding to the double spin-flip in the Markov chain terminology. This is illustrated inFigure 6.41.

N3

→

N′3

Figure 6.41: The disorder suggested by Shang et al. [83]. Some of the N3 sextetsare replaced by N′3.

Figure 6.42 gives the occurrences the 13 distinct sextets as a function of temperature forfixed nitrogen occupation yN = 5

12 as obtained by canonical ensemble statistical sampling.Very similar results are obtained for other values of yN, though obviously with different rel-ative occurrences of the N2 and N3 sextets. For low temperatures corresponding to the ex-pected range discussed above only the sextets N2 and N3 are predicted to occur. In particular,neither of N′3 nor N′′3 is predicted in non-negligible quantities. Figure 6.43 gives the occur-rences of the sextets a as function of yN at fixed temperature T = 373 K. Only the N2 and N3sextets are encountered.

From these observations it is concluded that the hypothesis of distinct sextet and the disor-der in Figure 6.41 in particular are strongly rejected according the thermodynamic statisticalsampling. The hypothesis of two distinct sextets with three nitrogen atoms is not supportedby the present model as virtually no N′3 sextets are predicted from the statistical sampling.

110


0.0

0.1

0.2

0.3

0.4

0.5

0.6

500 1000 1500 2000 2500 3000

Sext

etoc

curr

ence

Temperature, T, K

N3

N2

N′2

N1

Figure 6.42: The local nearest-neighbor nitrogen neighborhood of iron atoms. Sta-tistically sampled canonical ensemble average sextet occurrences of the 13 sym-metrically distinct sextets of interstitial sites of Table 6.8 as a function of temper-ature for fixed composition yN = 5

12 . The result is obtained by keeping track thearrangements of nitrogen atoms at the six interstitial sites (the sextets) surround-ing each iron atom and grouped by symmetrically equivalent arrangements. Gen-erally, only the sextets N1, N2, N′2, and N3 are encountered in the simulated com-puter crystal, and for T ≤ 500 K only the sextets N2 and N3 are encountered. Inthat case the ratio of occurrences of these sextets is such that yN = 5

12 is main-tained. The sextet N′3 is not observed due to a combination of its high energy andlow degeneracy (cf. Table 3.1). Similar results are obtained for other values of yN.See sextet occurrences for perfectly ordered structures in Table 6.9.

0.0

0.2

0.4

0.6

0.8

1.0

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Sext

etoc

curr

ence

yN

N3

N2

Figure 6.43: Perfect sextet ordering in the computer crystal at T = 373 K as afunction of yN. Only the sextets N2 and N3 are encountered. No data points in theε + ζ two-phase region. Notice that the points can be connected by two perfectlystraight lines, indicating very strong ordering of sextets.

111


6.14.2 Sextet-sextet interaction within the close-packed plane hypothesis

The perfectly ordered ε-Fe24N10 structure is of particular interest. The sextet occurrences ofthis structure are equally divided between the N2 and N3 sextets. However, the N3 sextets canbe further subdivided in exactly two groups according to the arrangements of the six nearest-neighbor sextets within the close-packed (001) planes. The arrangements are illustrated inFigure 6.44. Similar calculations are performed for each of the considered perfectly orderedstructure in Table 6.9.

A mapping between the sextet-sextet arrangements and the distinct components of theMossbauer spectra given in Table 6.7 is therefore hypothesized with reasonable agreementfor the ε-Fe6N2 and ε-Fe24N10 structures, assuming that finite temperature disordering andthe possible emergence of N1 sextets could account for the relatively small deviations.

The hypothesis is tested on the computer crystal by calculating grand-canonical ensembleaverage occurrences of the symmetrically distinct sextet-sextet configurations. The result isgiven in Figure 6.10 for T = 373 K and yN = 5

12 . Very complex ordering is predicted for theinteractions between neighboring sextets with significant occurrences of multiple symmet-rically distinct sextet-sextet configurations. This is the case for the nearest-neighbors of theN3 sextet as well as the N2 sextets, contrary the situation for the perfectly ordered ε-Fe24N10structure (cf. Table 6.9c); in both cases with approximately equal occurrences of the first fewsextet-sextet configurations.

Thus if the sextet-sextet orderings are as listed in Table 6.10 for yN = 512 , the Mossbauer

spectra must be decompose into significantly more than four distinct components and thespecific orderings predicted by the statistical sampling could be difficult to meaningfully dis-tinguish in experimental data.

It must be noted that the ε-Fe24N10 is less energetically favorable in the fitted cluster ex-pansion than for the first-principles energies due to a fitting error obtained in the truncatedcluster expansion (cf. Table 6.4). Hence if improved cluster expansion fits could be obtained,it is likely that the occurrences of the sampled computer crystal is slightly closed to that of theperfectly ordered ε-Fe24N10 structure and that stronger sextet-sextet ordering is predicted.

112


(a) (b) (c)

N, even

N, odd

Vacant

Fe

Figure 6.44: Sextets of the perfectly ordered ε-Fe24N10 structure. Iron atoms (graydot) and the corresponding sextets (red triangle). (a) N2 sextet (solid line) in thecenter surrounded by N2-N2-N3-N2-N3-N3 sextets (dotted line). (b) N3 sextet inthe center surrounded by N2-N2-N3-N3-N3-N3 sextets. (c) N3 sextet in the centersurrounded by N2-N2-N2-N2-N3-N3 sextets. The close-packed plane of iron atomsis shifted one plane compared to the other sextets.

Occurrence Sextet Neighbor sextets Summary1 N2 N2-N2-N2-N2-N2-N2 6×N2

(a) ε-Fe6N2


(b) ε-Fe6N3

Occurrence Sextet Neighbor sextets Summary12 N2 N2-N2-N3-N2-N3-N3 3×N2 + 3×N314 N3 N2-N2-N3-N3-N3-N3 2×N2 + 4×N314 N3 N2-N2-N2-N2-N3-N3 4×N2 + 2×N3

(c) ε-Fe24N10


(d) ζ-Fe8N4

Table 6.9: Sextets of perfectly ordered structures grouped by the six neighboringsextets within the close-packed plane. Neighboring sextets are given in correctsequential order so that the six neighbors form a hexagon in the given order. No-tice that none of the higher-energy sextet configurations N′2, N′′2 , N′3, N′′3 , N′4, N′′4are encountered for these perfectly ordered structures. In particular, also the or-thorhombic ζ-Fe8N4 contains N3 sextets only. See illustration of ε-Fe24N10 sextetsin Figure 6.44.

113


Occurrence Sextet Neighbor sextets Summary0.1195 N2 N2-N2-N3-N2-N3-N3 3×N2 + 3×N30.0965 · N2-N2-N2-N3-N3-N3 3×N2 + 3×N30.0952 · N2-N2-N3-N3-N3-N3 2×N2 + 4×N30.0742 · N2-N2-N3-N2-N2-N3 4×N2 + 2×N30.0443 · N2-N2-N2-N3-N2-N3 4×N2 + 2×N30.0429 · N2-N2-N2-N2-N2-N3 5×N2 + N30.0253 · N2-N2-N2-N2-N3-N3 4×N2 + 2×N30.0053 · N2-N2-N2-N2-N2-N2 6×N20.0001 · N′2-N2-N3-N3-N3-N3 N2 + N′2 + 4×N30.0001 · N′2-N2-N3-N2-N2-N3 3×N2 + N′2 + 2×N30.0001 · N′2-N2-N3-N2-N3-N3 2×N2 + N′2 + 3×N30.0001 · N2-N′2-N3-N2-N3-N3 2×N2 + N′2 + 3×N30.0001 N2 N2-N2-N3-N′2-N3-N3 2×N2 + N′2 + 3×N30.0001 N′2 N2-N2-N3-N2-N3-N3 3×N2 + 3×N30.0770 N3 N2-N2-N2-N2-N3-N3 4×N2 + 2×N30.0711 · N2-N3-N3-N3-N3-N3 N2 + 5×N30.0700 · N2-N2-N3-N3-N3-N3 2×N2 + 4×N30.0698 · N2-N2-N3-N2-N3-N3 3×N2 + 3×N30.0536 · N2-N2-N2-N3-N3-N3 3×N2 + 3×N30.0430 · N2-N2-N2-N2-N2-N3 5×N2 + N30.0320 · N2-N3-N3-N2-N3-N3 2×N2 + 4×N30.0287 · N2-N3-N2-N3-N3-N3 2×N2 + 4×N30.0231 · N2-N2-N2-N3-N2-N3 4×N2 + 2×N30.0202 · N3-N3-N3-N3-N3-N3 6×N30.0066 · N2-N2-N2-N2-N2-N2 6×N20.0006 · N2-N3-N2-N3-N2-N3 3×N2 + 3×N30.0001 · N′2-N2-N2-N2-N3-N3 3×N2 + N′2 + 2×N30.0001 · N′2-N2-N3-N3-N3-N3 N2 + N′2 + 4×N30.0001 N3 N2-N′2-N2-N2-N3-N3 3×N2 + N′2 + 2×N3

Table 6.10: Ensemble average occurrences of sextets subdivided by symmetricallydistinct arrangement of sextets neighbors in the close-packed plane in correct se-quential order. Calculated for T = 373 K and yN = 5

12 . The higher-energy sextetN′2 is predicted to occur with very small probability. The predicted arrangementsof sextets in the computer crystal are far more complex than for the perfectly or-dered structures listed in Table 6.9.

114


6.14.3 Sextet-sextet neighbor count hypothesis

In Section 6.14.2 sextet-sextet ordering within the close-packed (001) plane was considered.If out-of-plane atoms are also included, each iron atom is surrounded by twelve neighbor-ing iron atoms; six within the close-packed plane and six out-of-plane neighbors. Anothersimilar explanation of the distinct components of the Mossbauer spectra is offered by Chenet al. [81] by conjecturing that the components map to the total number of nitrogen atoms ofthese twelve neighboring iron atoms, not considering specific symmetrically equivalent con-figurations.10 They explicitly assume that more than two components corresponding to theN3 sextet exist and ideally should be included in fits of the Mossbauer spectra, noting thatthis large number of components cannot reliably be resolved from the experimental data.

This can be compared to the computer crystal by performing a simple count of the twelveneighbor sextet configurations. Also in this case the ordering obtained from the statisticalsampling is complex. Table 6.11 lists the occurrences of the distinct sextets grouped by thetwelve nearest-neighbor sextets. Multiple distinct neighborhoods exist for both the N2 andN3 sextets. The higher-energy sextet N′2 was ignored in this count.

Occurrence Sextet Neighbor sextets0.1522 N2 7×N2 + 5×N30.1007 · 6×N2 + 6×N30.0924 · 8×N2 + 4×N30.0599 · 5×N2 + 7×N30.0449 · 9×N2 + 3×N30.0230 · 10×N2 + 2×N30.0194 · 4×N2 + 8×N30.0055 · 3×N2 + 9×N30.0052 · 11×N2 + 1×N30.0009 · 2×N2 + 10×N30.0004 · 12×N20.0001 N2 1×N1 + 6×N2 + 5×N30.0903 N3 4×N2 + 8×N30.0900 · 5×N2 + 7×N30.0840 · 6×N2 + 6×N30.0662 · 3×N2 + 9×N30.0596 · 7×N2 + 5×N30.0409 · 8×N2 + 4×N30.0291 · 2×N2 + 10×N30.0196 · 9×N2 + 3×N30.0092 · 1×N2 + 11×N30.0046 · 10×N2 + 2×N30.0012 · 12×N30.0005 N3 11×N2 + 1×N3

Table 6.11: Ensemble average occurrences of sextets subdivided by the twelvesextets neighbors. Calculated for T = 373 K and yN = 5

12 . The higher-energy N′2sextet is ignored.

10Thank to Bastian Brink for suggesting this reference.

115


6.15 Sensitivity analysis

Unsurprisingly, the exact location of the phase boundaries are very sensitive to even smalldifferences in predicted computer crystal energies. An alternative database of structures, ob-tained from a smaller unit cell and with lower-precision first-principles calculation param-eters, was initially used for the thermodynamic statical sampling. The smaller unit cell ofthe alternative database was used before the orthorhombic ζ phase ordering was discovered(cf. Figure 6.23b). As extensive calculations were performed based on this alternative databasebefore it was replaced by the current database of structures, an opportunity is offered to studythe effect of slight changes in cluster expansion parameters. The full alternative database isgiven in Appendix C.

The phase boundaries of the two cluster expansions are given in Figure 6.45. In both casesthe ζ phase is predicted for yN ' 1

2 , but the two-phase region ε + ζ is shifted towards largervalues of yN for the alternative cluster expansion. Similarly, two-phase region is predicted for

−8.0

−7.9

−7.8

−7.7

−7.6

Che

mic

alpo

tent

ial,

µ,e

V

ε ζε + ζ

−8.0

−7.9

−7.8

−7.7

−7.6

0.45 0.46 0.47 0.48 0.49 0.50

Che

mic

alpo

tent

ial,

µ,e

V

yN

ε ζε + ζ

Figure 6.45: Chemical potential µ as a function of yN predicted by statistical sam-pling at T = 573 K. (Upper) Same as in Figure 6.23b. The ε → ζ phase transitionis predicted at µ = −7.80 eV with a two-phase region 0.4865 ≤ yN ≤ 0.4948.(Lower) Obtained using the alternative database. The phase boundaries changesubstantially. The ε → ζ phase transition is predicted at µ = −7.86 eV with atwo-phase region 0.4779 ≤ yN ≤ 0.4883. See also Appendix C. Notice that lower-precision first-principles calculation parameters have been used for the alternativedatabase, which could explain some of the difference.

116

6.16 Phase diagram

T = 723 K, whereas the alternative database cluster expansion predicts a narrow two-phaseregion at this temperature. See Appendix C for more details.

Other predicted properties, such as the short-range order-parameters in Section 6.12 andthe sextet-sextet ordering in Section 6.14, are not sensitive to the change of database and thesame predictions are obtained in both cases, proving robustness in these predictions.

It must be emphasized that the final database of structures as described in Section 6.6contains more structures and that first-principles energies are calculated to higher precision.

6.16 Phase diagram

Phase diagrams are notoriously hard to accurately predict from first-principles as the phaseboundaries depend on small differences in energies as demonstrated in Section 6.15. Nonethe-less, the predicted phase diagram is given in Figure 6.46 for completeness and to demonstratethe capabilities of the model. Phase boundaries are interpolated between calculated tempera-tures. The phase diagram provides excellent agreement with the phase diagram compiled byJack [2] (cf. Figure 6.7). Notice, however, that the γ′ phase is completely absent as that phaseis prohibited by the chosen computer crystal unit cell.

Vibrational contributions have been completely ignored in the thermodynamic statisticalsampling in the present work. Including these contributions through the hybrid potentialpartition function (2.40) may also change the location of phase boundaries. However, calcu-

300

400

500

600

700

800

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Tem

pera

ture

,T,K

yN

ε ζε + ζ

Figure 6.46: Phase diagram predicted by the thermodynamic statistical sampling.For each chosen chemical potential, the average interstitial occupation yN corre-sponds to one and only one phase. The two-phase region ε + ζ contains no sam-pled points, as the interior of such a region cannot be reached by choosing thechemical potential alone. Compare with the phase diagram of Jack in Figure 6.7and with the phase diagram of the generic hexagonal system in Figure 6.4. Noticethat the γ′ phase is completely absent as that phase is prohibited by the chosencomputer crystal unit cell. Phase boundaries are interpolated between calculatedtemperatures.

117


lation of vibrational contributions of the structures in the database is not attempted as it isexpected to be very expensive computationally.

The calculation of a temperature–pressure phase diagram is not attempted in the presentwork for two principal reasons.

Firstly, such a calculation from first-principles is expected to be significantly more com-plex than the temperature–volume phase diagram as the thermodynamic statistical samplinghas to be repeated for a fine mesh of lattice parameters to determine reliable estimates of theGibbs free energy through the Legendre transform. As the computational resources requiredto reach sufficient equilibration of the computer crystal are already extensive for a single pairof lattice parameters as measured in number of spin-flip steps per site and the computerutilization—the calculations require several months of running time—such a generalizationwould be at least an order of magnitude more expensive and is therefore prohibitively expen-sive.

Secondly, as is evident from the sensitivity analysis in Section 6.15, phase boundaries arevery sensitive to small differences in predicted energies and attempting to resolve dependenceon pressure therefore easily results in nonsensical predictions. The phase diagram predictedfor the chosen lattice parameters is accepted as an approximation for other lattice parameters.

6.17 Equilibrium ζ phase lattice parameters

In general, the orthorhombic unit cell breaks the hexagonal symmetry and the close-packingcondition on the lengths of the lattice vectors b′′ =

√3a, so that the distances between all sites

in the (001) are no longer equal. This is illustrated in Figure 6.47.Such a break of symmetry is not possible to incorporate in the computer crystal and the

statistical sampling, and the calculated ζ phase energies are expected to be higher—and there-fore less favorable—than they would otherwise have been. The relation (6.18) defining thelattice vectors of the orthorhombic unit cell in terms of the lattice vectors of the primitivehexagonal unit cell is modified by introduction an additional degree of freedom,

a′′ = 2a + 2b,b′′ = x(−a + b),c′′ = c,

(6.27)

Figure 6.47: Illustration of the expansion in the b′′ direction of the orthorhombicunit cell, breaking the hexagonal symmetry b′′ =

√3a and the close-packing prop-

erty. The expansion is exaggerated. Compare with the unit cells in Figure 6.13 andFigure 6.18.

118

6.18 Conclusions

−8.306

−8.304

−8.302

−8.300

−8.298

−8.296

−8.294

−8.292

0.98 0.99 1.00 1.01 1.02 1.03 1.04

Ene

rgy,

E,e

V

b′′/(√

3a)

Figure 6.48: First-principles calculated energies of the ζ-Fe8N4 structure as a func-tion of b′′ (square) with a fitted equation of state (solid line). The equilibriumenergy corresponds to a predicted expansion of b′′ of 2% relative to the value

√3a

with the hexagonal symmetry is imposed. Compare with the expansion from ex-perimental data in Figure 6.34.

where x = b′′/(√

3a) is the expansion in the horizontal direction of the orthorhombic unit cellrelative to the close-packed case.

To quantify the effect of the break of symmetry, first-principles energies of the ζ-Fe8N4ground-state structure are calculated, keeping a, b, and c fixed and unchanged, but allowing xto vary. The result is given in Figure 6.48. The equilibrium energy corresponds to a predictedexpansion of 2%. The expansion is in good agreement with the expansion obtained fromexperimental data discussed in Section 6.13.

The energy per atom for the equilibrium lattice parameters is 0.0041 eV lower than thewith hexagonal symmetry imposed, i.e., corresponding to x = 1. This is significantly largerthan the root-mean-squared fitting error of the cluster expansion (cf. Table 6.4).

6.18 Conclusions

The statistical sampling and the cluster expansion approach provides an accurate accountfor configurational degrees of freedom in crystalline solids. Compared with previously em-ployed simpler approximations, very detailed descriptions of occupations of individual sitescan be obtained in this model. A detailed account of the ε-Fe-N system has been providedin this chapter with predictions in good agreement with experimental data. Particularly in-teresting is the prediction of the ε → ζ phase transition and the excellent agreement with theexperimental data of phase boundaries, though it noted that phase boundaries are very sen-sitive to small changes in energies. The intermediate ε-Fe24N10 nitride has been predicted forthe first time in a thermodynamic model.

The cluster expansion approach is easily generalized to ternary systems such as the ε-Fe-C-N system. Substantially larger databases of structures and more terms in the clusterexpansion are required to reliably describe this additional degree of freedom, and ternary

119


systems are therefore computationally more expensive partly due to the larger number offirst-principles calculations required and partly due to increased number of clusters in thesampling. Moreover, two chemical potentials instead of just one can be chosen independentlyin ternary systems.

Very complex ordering of nitrogen occupations of the local environment of iron atoms waspredicted. The six nearest-neighbor nitrogen sites were predicted to be strongly ordered withonly two possible configurations, one with two nitrogen atoms and one with three nitrogenatoms, strongly rejecting the simplest hypothesis of the origin of the observed Mossbauerspectra. The longer-ranged ordering was predicted to be far more complex and a simplemodel was not offered. Detailed follow-up Mossbauer investigations could be combined withthe explicitly predicted sextet-sextet orderings to test of improved fits of recorded spectracould be obtained by incorporating this information.

120

7Conclusions

The present work was focused on the hexagonal ε-Fe-N system. Detailed first-principlesthermodynamic models were developed for the system using multiple complementary ap-proaches.

The first approach focused on a detailed description of the vibrational degrees of free-dom. Vibrations were described in the quasiharmonic phonon model and the linear re-sponse method was applied to determine force constants from first-principles calculations.The hexagonal lattice posed a special challenge as two lattice parameters are required to de-scribe the system. In order to generalize the quasiharmonic approach to hexagonal systems,a numerically tractable extended equation of state was proposed, providing accurate descrip-tion of entire energy surface as a function of the two independent lattice parameters. Thelinear response method was applied to an extensive set of pairs of lattice parameters to ob-tained a reliable equation of state for any temperature of interest.

The versatile model is applied to the ε-Fe6N2 structure in particular. Improved agreementto equilibrium lattice parameters compared to earlier first-principles calculation is provided,and first-principles calculations were performed for the first time at finite temperature forthe hexagonal lattice. Excellent agreement with experimental data of thermal expansion co-efficient was obtained. The model also allows calculation of volume–pressure relationship atfinite temperature, and also in this case good agreement with experimental data is obtained.In this respect the present work is a major improvement.

In the second and complementary part of the present work, thermodynamic statisticalsampling was employed for studying the configurational degrees of freedom for the ε-Fe-Nsystem, where the iron host lattice was assumed fixed and nitrogen atoms were allowed tooccupy interstitial sites depending on an externally controlled chemical potential. Finite tem-perature ensemble average properties were approximated from truncated sampling process,assuming chemical equilibrium with an external reservoir.

An account of the very complex configurational degrees of freedom was provided, allow-ing description of collective effects of ordering of atoms and phase transitions observed inlarge systems such as crystalline solids, and in this regard provides an improvement com-pared to the semi-empirical CALPHAD approach, where the configurational entropy can becalculated explicitly in a crude mean-field approximation. A prediction of an ε → ζ phasetransition is obtained in good agreement with experimental data. The sampling approachproved to provide more insight than the hybrid model of the direct partition function ap-proach despite its inexact nature.

121

7 Conclusions

The method has emerged in recent years as a viable alternative to the CALPHAD ap-proach and allows explicit incorporation of first-principles calculations. The method can beextended and is applicable to more complicated systems such a the ternary ε-Fe-C-N systemor for studying short-range ordering in the body-centered cubic Fe-Cr system. The approachhas the potential of achieving wider applicability as computational power increases.

The hybrid approach was also applied. The advantage of the approach is that configura-tional as well as vibrational degrees of freedom can be included explicitly and that Legendretransforms therefore can be calculated explicitly to obtained Helmholtz and Gibbs free en-ergies. The allows comparison with experimental data obtained at constant pressure. Goodagreement with nitriding potential data is obtained. The approach is somewhat naive in thedescription of configurational degrees of freedom compared to the sampling approach.

122

Appendix AChemical potential from experimental

data

An equilibrium state of ε-Fe2N1−z for 0 < z ≤ 13 cannot be attained with pure nitrogen gas

at atmospheric pressure, because equilibrium partial pressures of nitrogen amount to severalgigapascals for this iron nitride for temperatures in the range 500–1000 K [1]. The chemicalpotential of nitrogen in an NH3/H2 mixture, µN,g, can be defined on the basis of the hypo-thetical equilibrium

NH3 −− 12 N2 +

32 H2, (A.1)

whereµN,g =

12

µN2= µNH3

− 32

µH2. (A.2)

Assuming ideality of the NH3/H2 mixture with partial pressures pNH3and pH2

,

µNH3= G0

NH3+ RT log

pNH3

p0 ,

µH2= G0

H2+ RT log

pH2

p0 ,

where G0NH3

and G0H2

are reference energies at some reference pressure p0, so it follows

µN,g = G0NH3− 3

2G0

H2+

12

RT log p0 + RT log rN, (A.3)

with the nitriding potential is defined as

rN =pNH3

p3/2H2

. (A.4)

At equilibrium of the solid with the surrounding gas the chemical potential of the solidsatisfies µN,s = µN,g = µN. If the chemical potential µN is expressed in energy per atom(instead of energy per mole), we take R = kB, where kB = 8.62× 10−5 eV/K in atomic units.

If standard atmospheric pressure is taken as reference pressure, (A.3) results in an physi-cally unobservable and generally temperature dependent reference level

G0 = G0NH3− 3

2G0

H2. (A.5)

123

A Chemical potential from experimental data

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

0.0 0.2 0.4 0.6 0.8 1.0

y N

Nitriding potential, rN , Pa−1/2

Figure A.1: Experimental data of yN as a function of the nitriding potential rN atT = 723 K obtain by Somers et al. [1]. Notice the asymptotic behavior as yN → 1

2 ,requiring an increasing high nitriding potential to reach yN ' 1

2 .

In Section 5.1 the value G0 = −8.285 eV is used at T = 723 K to obtain best possible agreementwith first-principles calculations of energies where such reference energies can be obtained.

For reference, correspondence between the nitriding potential rN and yN at T = 723 Kobtain by Somers et al. [1] is given in Figure A.1.

124

Appendix BThermodynamics of crystalline solids in

chemical equilibrium with a reservoir

Thermodynamics is usually introduced by writing fundamental relations of potentials in dif-ferential form or, alternatively, by introducing potentials as proper functions in an axiomaticframework [84]. Here, the latter approach is preferred, as it is easier to do this in a mathemat-ically sound way.

The Gibbs free energy of the total system of a reservoir (r) and a crystal (s) where intersti-tial sites are available for occupation by atom A is decomposed as

G(NA(r), NA(s), NV) = Gr(NA(r)) + Gs(NA(s), NV) (B.1)

for fixed temperature and pressure, assuming only thermal interaction, i.e. exchange of heatand particles. The number of occupied sites is NA(s) and NV is the number of vacancies.

At equilibrium the energy is minimized subject to the constraints

NA(r) + NA(s) = NA (B.2)

andNA(s) + NV = Ns, (B.3)

corresponding to the total number of atoms of the system (particle conservation) and totalnumber of sites, respectively. Introducing Lagrange multipliers µA and µV , minimum is at-tained when the gradient of the unconstrained Lagrange function vanishes,

Λ(NA(r), NA(s), NV , µA, µV) = G(NA(r), NA(s), NV)

+ µA(NA − NA(r) − NA(s)) + µV(Ns − NA(s) − NV),

which is satisfied whenµA(s) − µV = µA(r) = µA, (B.4)

where µA(r) =∂Gr

∂NA(r), µA(s) =

∂Gs∂NA(s)

, and µV = ∂Gs∂NV

are the chemical potentials of atoms in thereservoir and solid, respectively.

Physically, due to the structural constraint, an atom A can be added to solid only by si-multaneously removing a vacancy V. This may be indicated by the chemical reaction

A(r) + V −− A(s), (B.5)

125

B Thermodynamics of crystalline solids in chemical equilibrium with a reservoir

with the equilibrium condition (B.4). Here, µA(s) and µV are so-called virtual chemical po-tential corresponding to structural elements [85]; only the difference µA(s) − µV is physicallyobservable, e.g. through the chemical potential of A in the reservoir µA(r) at equilibrium.

We may also consider the energy of the solid to be a function of number of occupied sitesNA(s) only,

Gs(NA(s)) = Gs(NA(s), Ns − NA(s)), (B.6)

where the structural constraint is contained implicitly so that no virtual chemical potential isrequired, and the chemical potential

µA(s) =∂Gs

∂NA(s)(B.7)

is related to the virtual chemical potentials by

µA(s) = µA(s) − µV . (B.8)

Thus the same result is obtained for chemical equilibrium as before,

µA(s) = µA(r) = µA. (B.9)

As the physically unobservable virtual chemical potentials are not defined in first-princi-ples calculations, this approach is is used in the present work.

126

Appendix CAlternative database cluster expansion

An alternative database of structures was initially used for fitting of cluster expansion coef-ficients. The 48-site unit cell of the database structure is given in Figure C.1 and is half thesize in the [110] direction of the unit cell used for the final database of structures as describedin Section 6.6. This smaller database unit cell is not consistent with the orthorhombic unitcell and the ζ phase ordering. Despite of no ζ phase structure energies being included in theinitially fitted cluster expansion, ζ phase ordering was predicted in the much larger computercrystal in the statistical sampling process. As this unexpected distinct ordering was discov-ered, the database unit cell was changed, and the entire fitting and sampling procedure wererepeated. Moreover, the first-principles energies of the structures in this initial database werecalculated with lower precision of first-principles calculation parameters, with plane-wavebasis truncation energy of 350 eV per atom and 3× 3× 3 k-points.

ab

Figure C.1: Alternative database unit cell (large green rhombus). The unit cell isnot consistent with the orthorhombic unit cell (red rectangle) as the orthorhombicunit cell is contained a non-integral number of times in the [110] direction in thedatabase unit cell. This database unit cell was used before the orthorhombic ζphase ordering was discovered. Compare with Figure 6.13.

127

C Alternative database cluster expansion

As extensive calculations were performed using this alternative database and as the re-sults are useful in determining sensitivity of predictions—phase boundaries in particular—inthe estimated cluster expansion parameters, the resulting cluster expansion is presented here.The location of phase boundaries are compared in Section 6.15.

94 symmetrically distinct structures are included in the alternative database. The calcu-lated first-principles energies and the fitted cluster expansion energies are given in Figure C.2.A lower root-mean-squared fitting error of 0.71 meV per atom is obtained for the alternativedatabase compared to the database of structure in Section 6.6. This is likely due to lowernumber of structures in the database to fit with the same number of cluster expansion coeffi-cients and as most structures in the database have orderings consistent with the conventionalhexagonal unit cell.

−0.010

−0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

13

1748

38

1948

512

2148

1124

2348

12

Ene

rgy,

E−

E ref

,eV

yN

ε-Fe6N2

ε-Fe6N3

Figure C.2: Cluster expansion fitted energies per atom (circle) with fitting errors(line segment) of structures in the alternative database. Data points in each boxcorrespond to the same value of yN. Energies of two important structures areindicated. A root-mean-squared fitting error of 0.71 meV per atom is obtained.The cluster expansion is given in Table C.1. Energies are relative to the referencesenergy Eref = 6[( 1

2 − yN)Eε-Fe6N2+ (yN − 1

3 )Eζ-Fe8N4]. Notice that ε-Fe6N3 proved

not to be a ground-state structure.

128

α Coordinates mα dα/a Jα, eVα0 1 0.00 −24.22α1 (0, 0, 0) 2 0.00 −3.317α2,1 (0, 0, 0); (0, 0, 1

2 ) 2 1.00 0.3227α2,2 (0, 0, 0); (1, 0, 0) 6 1.22 0.09710α2,3 (0, 0, 0); (1, 0, 1

2 ) 12 1.58 0.03146α2,4 (0, 0, 0); (0, 0, 1) 2 2.00 0.01741α2,5 (0, 1, 0); (1, 0, 0) 6 2.12 0.01035α2,6 (0, 1, 0); (1, 0, 1

2 ) 12 2.35 0.004507α2,7 (0, 0, 0); (1, 0, 1) 12 2.35 0.005072α2,8 (0, 0, 0); (2, 0, 0) 6 2.45 −0.003292α2,9 (0, 0, 0); (2, 0, 1

2 ) 12 2.65 0.00002302α2,10 (0, 1, 1); (1, 0, 0) 6 2.92 0.002457α2,11 (0, 1, 0); (1, 0, 1) 6 2.92 0α2,12 (0, 0, 0); (0, 0, 3

2 ) 2 3.00 0α2,13 (0, 0, 0); (2, 0, 1) 12 3.16 0.003173α3,1 (0, 0, 0); (1, 0, 0); (1, 1, 0) 4 1.22 0α3,2 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0) 24 1.58 0α3,3 (0, 0, 0); (0, 1, 0); (1, 1, 1

2 ) 12 1.58 0α3,4 (0, 0, 0); (1, 0, 0); (1, 1, 1

2 ) 12 1.58 0α3,5 (0, 0, 0); (0, 0, 1

2 ); (0, 0, 1) 2 2.00 0.01734α3,6 (0, 0, 0); (0, 0, 1); (1, 0, 1

2 ) 12 2.00 −0.004392α4,1 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0); (1, 0, 12 ) 6 1.58 0

α4,2 (0, 0, 0); (0, 1, 12 ); (1, 1, 0); (1, 1, 1

2 ) 12 1.58 0α4,3 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0); (1, 1, 12 ) 12 1.58 0

α4,4 (0, 0, 0); (0, 1, 0); (1, 1, 0); (1, 1, 12 ) 12 1.58 0

α4,5 (0, 0, 0); (0, 0, 12 ); (1, 0, 0); (1, 1, 0) 12 1.58 0

α5,1 (0, 0, 0); (0, 0, 12 ); (0, 1, 0); (1, 1, 0); (1, 1, 1

2 ) 12 1.58 0α5,2 (0, 0, 0); (0, 0, 1

2 ); (1, 0, 0); (1, 1, 0); (1, 1, 12 ) 12 1.58 0

α6,1 (0, 0, 0); (0, 0, 12 ); (0, 1, 0); (0, 1, 1

2 ); (1, 1, 0); (1, 1, 12 ) 2 1.58 0

α6,2 (0, 0, 0); (0, 0, 12 ); (1, 0, 0); (1, 0, 1

2 ); (1, 1, 0); (1, 1, 12 ) 2 1.58 0

Table C.1: Cluster expansion fitted to calculated first-principles energies of thestructures in the alternative database. For the cluster α, mα symmetrically equiva-lent clusters exist per primitive unit cell, each with contribution Jα in the expansionof the energy. A root-mean-squared fitting error of 0.71 meV per atom is obtained.The coordinates and the maximal separation distance dα of one of the symmetri-cally equivalent clusters are given. Compare with Table 6.3.

129


−8.8

−8.6

−8.4

−8.2

−8.0

−7.8

−7.6

−7.4

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Che

mic

alpo

tent

ial,

µ,e

V

yN

ε ζε + ζ

(a) T = 723 K

−8.8

−8.6

−8.4

−8.2

−8.0

−7.8

−7.6

−7.4

0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50

13

38

512

1124

12

Che

mic

alpo

tent

ial,

µ,e

V

yN

ε ζε + ζ

(b) T = 573 K

Figure C.3: Correspondence between the chemical potential µ and the resultingaverage interstitial occupation yN in the statically sampled grand-canonical en-semble. Distinct phases marked (dashed lines). (a) The alternative database clus-ter expansion predicts a ε → ζ phase transition at T = 723 K and µ = −7.8 eV,with a corresponding narrow two-phase region 0.4814 ≤ yN ≤ 0.4851. (b) Themodel predicts a ε → ζ phase transition at µ = −7.86 eV, with a correspondingtwo-phase region 0.4779 ≤ yN ≤ 0.4883.

130

Equilibration and sampling is performed as described in Section 6.9. The correspondencebetween the externally controlled chemical potential µ and the average nitrogen occupationis given in Figure C.3. An ε→ ζ phase transition is also predicted for the alternative databasecluster expansion. Representative planes of the computer crystal the phase boundaries aregiven in Figure C.4.

The predicted Cowley–Warren short-range order parameters given in Figure C.5 are al-most identical to the order parameters predicted in Section 6.12.


N, even

N, odd

Vacant

Fe

Figure C.4: Phase transition at T = 573 K. Representative neighboring (001)planes of the sampled computer crystal at T = 573 K. Distinguishing patternsare highlighted (dotted line). (a) Just on the ε side of the ε → ζ phase transition(cf. Figure C.3). (b) Just on the ζ side of the ε→ ζ phase transition. The ζ orderingis possible in multiple directions for orthorhombic lattice parameters consistentwith the hexagonal unit cell. Compare with Figure 6.35.

131


−1

0

ξ α2,

1

0

1

ξ α2,

3

0

1

ξ α2,

5

ε-Fe6N2

ε-Fe24N10

ε-Fe6N3

ζ-Fe8N4

−1

0

ξ α2,

7

0

1

ξ α2,

9

0

1

ξ α2,

11

−1

0

13

38

512

1124

12

ξ α2,

13

yN

−1

0

ξ α2,

2

0

1

ξ α2,

4

ζ-Fe12N5

−1

0

ξ α2,

6

−1

0

ξ α2,

8

0

1

ξ α2,

10

−1

0

13

38

512

1124

12

ξ α2,

12

yN

Figure C.5: Ensemble average Cowley–Warren short-range order parameter pre-dicted for the computer crystal at T = 573 K (small dots) compared with orderparameters for perfectly ordered structures ε-Fe6N2, ε-Fe6N3, and ε-Fe24N10 (redsquares), ε-Fe12N5 (orange diamond), and ζ-Fe8N4 (blue circle). Calculated for thealternative database cluster expansion coefficients. Compare with Figure 6.31.

132

Appendix DIntegration of the electronic density of

states

To obtain a smooth volume-dependent electronic contribution to the free energy and to main-tain consistency between the the Fermi energy εF and 〈Nel〉 careful interpretation of the VASPoutput data is important. The documentation states—somewhat vaguely—that the integrateddensity of states is to be linearly interpolated, not the density of states itself. More precisely,the ensemble average number of electrons is given by

〈Nel〉 =∫

dεd(ε)nF(ε) (D.1)

whered(ε) = d↑(ε) + d↓(ε) (D.2)

is the total electronic density of states,

nF(ε) =1

eβ(ε−µ) + 1(D.3)

is the Fermi–Dirac distribution, and µ is the electronic chemical potential. The integrateddensity of state is defined as

D(ε) =∫ ε

−∞dε′d(ε′). (D.4)

The Fermi energy εF is defines so that

〈Nel〉∣∣T=0 =

∫ εF

−∞dε d(ε) (D.5)

and consistency between the number of electrons and the reported values of εF and d musttherefore be maintained. Only a finite number of density of states points are reported by VASPand interpolation between calculated values is assumed. Since both the integrated densityof states and the density of states itself are reported at least two interpolations schemes arepossible. In the present work, linear interpolation between reported values of the integrateddensity of states D is assumed, resulting in the density of states d being a piece-wise constantstep function.

133

D Integration of the electronic density of states

The two possible interpolation schemes are illustrated in Figure D.1 for the ε-Fe6N2 struc-ture. In Figure D.2 the calculated number of electron are given for both interpolation schemesand the reported Fermi energy. The interpolation schemes used in the present work results inthe most stable number of electrons as a function of unit cell volume.

Notice that for a finite crystal, the mathematically correct density of states is composed ofDirac δ functions on the form

d(ε) = ∑kσ

δ(ε− εkσ), (D.6)

and is therefore not a smoothly defined function. This is due to the quantized energy levelsof the system.

134

0

2

4

6

8

10

12

14

−15 −10 −5 0 5 10 15 20

Den

sity

,d,e

V−

1

Energy, ε, eVεF

6.0 6.5 7.0 7.5 8.0 8.5 9.08

9

10

11

12

Figure D.1: Two interpolation schemes of the reported total densities of statesd = d↑ + d↓ for the ε-Fe6N2 structure. Interpolation of the density of states itself(brown) and of the integrated density of states (black).

61.8

62.0

62.2

62.4

62.6

62.8

63.0

63.2

63.4

63.6

63.8

76 78 80 82 84 86 88 90 92 94

Ele

ctro

nco

unt,〈N

el〉

Volume, V, A3

Figure D.2: Number of electrons calculated from the two different interpolationschemes. The expected number of electrons is 63 for the ε-Fe6N2 structure. Inter-polation of the density of states itself (brown circle) and of the integrated densityof states (black square); the latter resulting in a much more stable number of elec-trons.

135

D Integration of the electronic density of states

136

Appendix EAdditional properties of the

Gorsky–Bragg–Williams approximation

The Gorsky–Bragg–Williams approximation of configurational entropy and the estimated ex-change energies of Section 6.5 are extrapolated to lower temperature to illustrate a situationof a wider two-phase region, i.e., assuming same values of WP and WC in (6.16). The resultingGibbs free energies of the two configurations are given in Figure E.1 at T = 300 K. In theground-state limit T → 0, the two-phase region is expected to span the entire interval as onlythe pure Fe6N2 and Fe6N3 structures are stable.

The model also predicts an order-disorder phase transformation at the temperature TC.Above this temperature, sites are no longer divided between preferentially occupied andpreferentially unoccupied sites. It must be emphasized that the model is not calibrated topredict this critical temperature with any accuracy, and this extrapolation in included only to

−4

−2

0

2

4

6

8

10

0.30 0.35 0.40 0.45 0.50

Gib

bsfr

eeen

ergy

,G,m

eV

yN

B AA + B

Figure E.1: Calculated Gibbs free energy of Configuration A and Configuration Bat T = 300 K. The two-phase region is determined by the convex hull extension.Energies are defined relative to references states Fe6N2 and Fe6N3 with atomicunits chosen for comparison to first-principles calculation.

137

E Additional properties of the Gorsky–Bragg–Williams approximation

demonstrate the capabilities of the model. The site occupations as a function of temperaturefor Configuration A and Configurations B, respectively, is given in Figure E.2.

138

0.0

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000 1200 1400 1600 1800 2000

Site

occu

pati

on

Temperature, T, K

A1 y

B2 y = C2y

B1 y = C1y

A2 y

512

TC

(a) Configuration A

0.0

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000 1200 1400 1600 1800 2000

Site

occu

pati

on

Temperature, T, K

A1 y = B2y

A2 y = B1y

C1 y = C2y 512

TC

(b) Configuration B

Figure E.2: Site occupations as a function of temperature T for yN = 512 . Ma-

jority sites have occupations above yN and minority sites below, respectively; seeTable 6.2. Above the order-disorder transformation critical temperature TC all ofthe occupations ky are identical to yN. Notice that the model parameters have notbeen calibrated with respect to accuracy in the predicted order-disorder tempera-ture; this purely serves as a demonstration of one of the important properties of themodel. The predicted critical temperature is TC = 1930 K, but the phases decom-pose long before reaching this temperature according to ε-FexN −−−− 1

2 N2 + xFe.

139

E Additional properties of the Gorsky–Bragg–Williams approximation

140

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