1. Thermodynamics of the hexagonal
close-packed iron-nitrogen system from
first-principles
PhD Dissertation
Morten B. Bakkedal
Supervisor:
Professor Marcel A. J. Somers
July 2015
Technical University of Denmark
Department of Mechanical Engineering
Section of Materials and Surface Engineering
2.
3. Preface and acknowledgements
The present dissertation is submitted in partial fulfillment of the requirements for the PhD
degree at the Technical University of Denmark (DTU). The presented work has been carried
out 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 Professor
Marcel A. J. Somers.
The PhD project was part of the research project ThInSol (Thermodynamics of Interstitial
Solutions in Cubic and Hexagonal Host Lattices), financially supported by the Danish Council
for Independent Research, Technology and Production Sciences (FTP), under grant number
11-106293.
First-principles calculations were performed at the Computing Center at the Technical
University of Denmark.
I would like to thank my supervisor Professor Marcel A. J. Somers for encouragement
and many enlightening discussions. Thank to Dr. ShunLi Shang and the rest of the Phases
Research Laboratory group at Pennsylvania State University for materials science insight and
productive discussions. Thank to my fellow PhD student Bastian Brink for discussion about
the interpretation of experimental data and for providing numerous references. Credit is also
given to Yi Wang from Pennsylvania State University for the Yphon tool.
Kgs. Lyngby, July 2015
Morten B. Bakkedal
i
5. Abstract
First-principles thermodynamic models are developed for the hexagonal close-packed ε-Fe-N
system. The system can be considered as a hexagonal close-packed host lattice of iron atoms
and with the nitrogen atoms residing on a sublattice formed by the octahedral interstices. The
iron 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 the
atomic numbers and crystal structures as the only input parameters. A complete thermody-
namic description should, at least in principle, include vibrational as well as configurational
contributions. As both contributions are computationally very demanding in first-principles
calculations, the present work is divided in two parts, with a detailed accounts of each of
these contributions.
Vibrational degrees of freedom are described in the quasiharmonic phonon model and the
linear 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 to
describe the system. The quasiharmonic phonon model is generalized to hexagonal systems
and 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 be
obtained at any temperature of interest. The thermal expansion predicted by the generalized
quasiharmonic phonon model is in excellent agreement with experimental data. The model
also allows calculation of the volume–pressure relationship at finite temperature, and good
agreement 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 Monte
Carlo 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 is
reached. Configurational energies are described in an Ising-like cluster expansion determined
from a large database of calculated first-principles energies.
The model provides a description of collective effects of orderings of atoms and phase
transitions observed in large systems. Ensemble average long-range order parameters and the
Cowley–Warren short-range order parameters are calculated and provide evidence of specific
orderings. The intermediate ε-Fe24N10 nitride is predicted as a ground-state structure and
ordering consistent with the structure is observed at finite temperature. An ε → ζ phase
transition 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 crystal
and are compared to recorded M¨ossbauer spectra. Finally, predictions of phase diagrams
from first-principles calculations is demonstrated.
iii
7. Resum´e
Fundamentale termodynamiske modeller udvikles for det heksagonale tætpakkede ε-Fe-N
system. Dette system kan betragtes som best˚aende af et heksagonalt tætpakket værtsgitter
af jernatomer med kvælstofatomer siddende p˚a et undergitter dannet af oktaeder interstielle
pladser. 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 s˚avel som konfigurationelle bidrag. Eftersom begge bidrag er meget tunge i
beregningsmæssig forstand i kvantemekaniske beregninger, er dette arbejde inddelt i to dele
med en detaljeret beskrivelse af hvert af disse bidrag.
Vibrationelle frihedsgrader beskrives i en kvasiharmonisk fonon model og den lineære
responsmetode anvendes til at bestemme kraftkonstanter fra kvantemekaniske beregninger.
Det heksagonale gitter giver særlige udfordringer da kræver to uafhængige gitterparametre
at beskrive dette system. Den kvasiharmoniske fonon model bliver generaliseret til heksago-
nale systemer og en numerisk let h˚andterbar udvidet tilstandsligning udvikles for at beskrive
termodynamiske ligevægtsegenskaber ved positive temperature.
Modellen anvendes p˚a ε-Fe3N specifikt. Gennem den fleksible af modellen kan ligevægts
gitterparameter, kompressibilitetsmodul og den termiske udvidelseskoefficient bestemmes
ved enhver temperatur. Den beregnede termiske udvidelse ved den generaliserede kvasi-
harmoniske model er i glimrende overensstemmelse med eksperimentelle data. Modellen
tillader ogs˚a beregning af rumfang-tryk relationer ved positiv temperature, og god overens-
stemmelse med eksperimentelle data opn˚as ogs˚a i dette tilfælde.
I anden del studeres konfigurationelle frihedsgrader af kvælstofbesætning af de intersti-
tielle pladser gennem termodynamisk stikprøvetagning, ogs˚a kendt som Monte Carlo simula-
tioner, hvor kvælstof atoms tillades at migrere tilfældigt i en stor computerkrystal i henhold til
de relative energier af konfigurationerne indtil kemisk ligevægt opn˚as. De konfigurationelle
energier 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 for
specifikke ordninger. Den mellemliggende ε-Fe24N10 nitrid bestemmes som en grundtilstand-
struktur og ordning konsistent med denne struktur observeres ved positive temperature. En
ε → ζ faseovergang forudsiges med faseovergangsgrænser i glimrende overensstemmelse
med eksperimentelle data. Det lokale jernatommiljø kan eksplicit bestemmes i computerkrys-
tallen og sammenlignes med M¨ossbauer spektre. Slutteligt demonstreres hvordan fasedia-
grammer kan bestemmes fra kvantemekaniske beregninger.
v
11. 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
ix
19. 1Introduction
1.1 Project description
Thermodynamics of Fe-N phases is of crucial importance in understanding material behavior
observed in the practice of nitriding of iron and steels, a thermochemical surface engineering
process 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 during
nitriding [1]. Despite a strong commercial interest and widespread application of nitriding—
and nitrocarburizing in general—it is currently not possible to model the thermodynamics
of 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 host
lattices. 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 thermodynamic
models accounting for long-range order of the nitrogen atoms have been developed and fitted
to experimental data [1, 9, 10]. The ground-state structural stability at 0 K has also been
investigated in first-principles calculations [11, 12].
The advances in computational science and information technology over the last few
decades 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 host
lattice 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 layers
is ABAB . . . , then the interstitial sites are located on C layers between each iron
atom layer, so that the stacking sequence is ACBCACBC . . . .
1
20. 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 numbers
and crystal structures as the only input parameters, thereby allowing predictions of structural
stabilities. The density function theory (DFT) approach [14] have proved to be particularly
successful, and currently allows calculation of energies of systems of ten thousands of atoms
in the most efficient implementations [15].
The project originally aimed at development of thermodynamic models of Fe-Cr-Ni-C-N
systems based on the semi-empirical CALPHAD approach [16], where experimental data are
introduced as model parameters and configurational energies are calculated in a mean-field
approximation [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 calculations
based on the density functional theory approach, where energies of configurational states
expressed as discrete occupation of individual atoms in the crystalline solid are calculated
explicitly.
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 grows
exponentially with the number of components in the system, and systems containing more
than 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 of
freedom for alloy systems are presenting major difficulties in first-principles calculations [24].
Therefore two complementary direction of thermodynamic model development were chosen
for 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. This
particular 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 degrees
of 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 obtained
through 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’s
knowledge 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 response
method 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
21. 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 thermodynamic
properties at non-zero pressure, and good agreement between predicted finite temperature
properties 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], also
known 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 potential
determining 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 expansions
provide a practical pathway to establish the link between quantum mechanics and statisti-
cal physics in alloy systems [31]. Previously, cluster expansions have mostly been employed
to 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-disorder
phase transitions. Recently, however, cluster expansion coefficients have been determined for
real 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 are
identified, including the intermediate ε-Fe24N10 nitride hypothesized earlier [2]. Ordering
consistent 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 thermodynamic
statistical sampling. The phase transition and the predicted phase boundaries are in excellent
agreement with experimental data [2].
Cowley–Warren short-range order parameters [35] are calculated as ensemble averages
of the sampled computer crystals and serve as signatures of specific orderings, providing
evidence for both the ζ phase ordering and for ordering consistent with the ε-Fe24N10 at finite
temperature.
The local environment of the iron atoms can be explicitly calculated in the computer crys-
tal and are compared to experimental data from recorded M¨ossbauer 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 unified
model, and the configurational degrees of freedom is therefore more naive in this approach.
The predicted relationship between chemical potential and nitrogen occupation obtained
from the Gibbs free energy is compared to experimental data from nitriding potentials [1].
3
22. 1 Introduction
1.2 Dissertation outline
First-principles calculations and the density functional theory approach are briefly reviewed
in the remaining sections of this introductory chapter.
The quantum mechanical foundation for finite temperature first-principles calculations
is introduced in Chapter 2. The quantum mechanical electronic problem and the Born–
Oppenheimer approximation are introduced. The procedure for obtaining finite temperature
equilibrium properties in terms of the partition function is described in general, and contribu-
tions from vibrational and electronic excitations are described specifically. The quasiharmonic
phonon 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 is
developed for the hexagonal system, serving as a prerequisite for the generalization of the
quasiharmonic phonon model to hexagonal systems.
The finite temperature quantum mechanical theory and the extended equation of state are
applied to develop the vibration thermodynamic model for the ε-Fe6N2 structure in Chap-
ter 4, allowing prediction of finite temperature equilibrium lattice parameters and thermal
expansion.
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 Schr¨odinger 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 the
reduced Planck constant. The wave function ψ and energy eigenvalue E are to be solved. The
differential operator on the left-hand side is the energy operator, the Hamiltonian, and often
written in the compact form
ˆH =
−¯h2
2m
ˆ 2
+ V, (1.2)
representing kinetic and potential energy, respectively.
As the Schr¨odinger 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 the
Schr¨odinger equation directly, or indirectly as discussed in the following section, are called
first-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
23. 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
ˆH |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 ˆN, whereas real
physical quantities, such as eigenvalues Ni or ensemble averages N = ˆN = ∑i piNi are
marked without “hats”. For the energy operator, the Hamiltonian, the convention is to use
ˆH for the operator, Ei for eigenvalues, and U = ˆH for the internal energy. In mathematical
terms, quantum mechanical operators are typically differential equation operators whereas
physical quantities are scalars, i.e., simple real numbers.
1.4 Density functional theory
Electrons bound to nuclei to form atoms satisfy the multi-body Schr¨odinger equation, which
is a generalization of (1.1),
−¯h2
2m ∑
i
ˆ 2
i + ∑
i
V(ri) + ∑
i<j
U(ri, rj) Ψ(r1, . . . , rN) = EΨ(r1, . . . , rN), (1.4)
where ri = (xi, yi, zi),
ˆ 2
i =
∂2
∂x2
i
+
∂2
∂y2
i
+
∂2
∂z2
i
. (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 is
currently only possible for very simple systems of a few electrons [38, 39, 40]; and that is only
the case when the atomic nuclei are treated as non-quantum particles. The eigenstates—the
many-dimensional wave functions—encode vast amounts of information, far more than the
ground-state energy required to determine structural stability of crystalline systems. Some of
this 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 electron
cloud 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
24. 1 Introduction
Kohn and Sham [14] proved that the multi-body problem (1.4) can be transformed into a
problem of non-interacting electrons described in terms of the much simpler one-dimensional
equation (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 functional
derivative of the related exchange-correlation energy Exc. If ψi and Ei are wave function and
eigenenergy solutions of the non-interacting problem, respectively, with Veff as the potential
in (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)
and
N = 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, exact
ground-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 expressions
for 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 for
plane-wave basis approaches [42] or linearly for the wavelets basis approach [15], much better
than the exponential scaling of the Schr¨odinger equation itself.
The mathematical and implementational details of the density functional approaches are
not given in the present work as these are readily available in textbooks [41, 43, 44]. See also
Baroni 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 Ab
initio Simulation Package (VASP) [42, 45], an efficient DFT solver that works in parallel cluster
computing environments and is implemented using plane-wave basis expansions, especially
suitable for bulk materials. Details of the first-principles calculation parameters are given in
the following chapters.
The thermodynamic statistical sampling is performed using code developed by the author,
running on the same parallel cluster computing environment.
6
25. 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 macroscopic
variables, 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 approach
to 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 a
crystalline 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 the
eigenvalues of the energy operator, the Hamiltonian, which is a partial differential operator
in the electron coordinates. The actual mathematical details of this generally very difficult
problem are not described in the present work. Instead the powerful second quantization
of quantum mechanics is chosen, where these technicalities of the differential equations are
masked 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 ensemble
occupation probabilities are given by the energies of the excited states with the probabilities
decreasing 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 generally
only 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
26. 2 Theoretical foundation
2.2 The electronic problem
The most fundamental quantity in the mathematical formulation of quantum mechanics is the
energy differential equation operator, also known as the Hamiltonian, given in the simplest
form in (1.2). In multi-body electronic problems of crystalline solids it is often decomposed as
ˆH = ˆH0
+ ˆHel
+ ˆHph
+ ˆHel-ph
(2.1)
with static nuclei repulsion ˆH0, an electronic part ˆHel, and a vibrational part ˆHph. The elec-
tron-phonon interaction ˆHel-ph is completely neglected in the present work.1 ˆH0 includes
static Coulomb repulsion between the positively charged nuclei. The electronic and vibra-
tional terms are described in more details in the following sections. The exact mathematical
details are beyond the scope of this work and are generally omitted. Details of the electronic
problem are available in the literature [28, 36].
In solving the eigenstates and eigenenergies of (2.1), it is noted that the nuclei are much
heavier in mass compared to the electrons, and their motions can therefore be ignored in
solving the electronic Schr¨odinger equation. This is the Born–Oppenheimer approximation
[47]: the motion of the nuclei and the electrons can be separated and the electronic and nuclear
problems can be solved with independent wave functions. When accounting for vibrational
contributions of the nuclei in the lattice, the electron cloud is thus assumed to instantaneously
rearrange itself around the moving nuclei in this approximation [28].
It follows that the arrangements of the nuclei can be considered as a parameter of the
quantum 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 the
lattice parameters of the unit cell.
As electronic and vibrational degrees of freedom are assumed decoupled, the eigenstates
of (2.1) can be described with respect to an abstract basis |εω; r ,2 representing vibrational ω
and electronic ε eigenstates, respectively, and the elastic relaxation parameter r.
Finite temperature mixed mode states of a thermodynamic ensemble are described by
the quantum mechanical density operator [48, 43]. In thermodynamic equilibrium with an
external heat reservoir this is given by
ˆρ = e−β ˆH
= ∑
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., ˆH |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 equilibrium
only. 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 ˆρ = ∑
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
27. 2.3 Electronic contribution
The trace, and therefore also Z, is independent of the chosen basis, i.e., the actual choice
of |εω; r is not important. The usual macroscopic thermodynamic potentials and derived
properties 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,
ˆHel
= ∑
kσ
εkσ ˆc†
kσ ˆckσ, (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), ˆc†
kσ is the electron creation operator, ˆckσ
its adjoint annihilation operator, and an energy dispersion relation εkσ.
The electronic contribution is described in a grand-canonical ensemble, controlling the
number of electrons ˆN = ∑kσ ˆc†
kσ ˆckσ by the electronic chemical potential µ. The partition
function is
Zel
= tr e−β( ˆHel−µ ˆN)
= ∏
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
ˆNel
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
ˆNel
= ∑
σ
dε dσ(ε)nF(ε) = ˆNel
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 = ˆc†
kσ |0 cannot be assumed to
diagonalize ˆHel, but by construction this is the case in the density functional approach with
the effective exchange-correlation potential (1.7). In that case the density of states (2.7) and
the corresponding free energy are defined with respect to this transformed system instead of
9
28. 2 Theoretical foundation
0
2
4
6
8
10
12
−15 −10 −5 0 5 10 15 20
Density,eV−1
Energy, ε, eV
εF
d↑
d↓
(a) ε-Fe6N2
0
2
4
6
8
10
12
−15 −10 −5 0 5 10 15 20
Density,eV−1
Energy, ε, eV
εF
d↑
d↓
(b) ε-Fe6N3
Figure 2.1: Calculated spin-polarized electronic density of states dσ for hexagonal
close-packed structures at equilibrium volume. The integrated density below the
Fermi energy εF is the number of electrons ˆNel
T=0
per six-site unit cell. The
electronic contribution to the free energy is evaluated from the density of states
dσ(ε) and Fermi energy εF, both calculated from first-principles as described in
Section 2.3. To obtain a smooth volume-dependent contribution and to maintain
consistency between εF and ˆNel careful interpretation of the VASP output data is
important: 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 of
states dσ(ε) itself is then a piece-wise constant step function. See Appendix D for
more details. It should also be noted that the density of states is calculated by
VASP using an effective DFT potential that diagonalizes ˆHel in the plane-wave
basis and not the true Coulomb potential. It is nonetheless used here as a best
estimate and in anticipation that the electronic contribution to the free energy will
be small far below the Fermi temperature εF/kB 8 × 104 K. This is the usual—
and often unstated—assumption in first-principles thermodynamic calculations
using DFT [49].
the real physical system. This fact is often ignored in actual calculations as the assumption
provides the best available estimate, and since the contribution from electronic excitations is
usually 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 the
output reported by the first-principles calculation.
2.4 Vibrational contribution
Phonons are quantized collective oscillations of the nuclei in the crystal. In the harmonic and
quasiharmonic model the contribution is described by
ˆHph
= ∑
kλ
¯hωkλ
ˆb†
kλ
ˆbkλ +
1
2
, (2.11)
3Obviously, the total positive charge of the nuclei does not change with temperature, so the solid to remain
charge neutral, the total number of electrons must also be kept constant.
10
29. 2.4 Vibrational contribution
where k is the wave vector, λ is the branch index, ˆb†
kλ is the phonon creation operator, and
ˆbkλ 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 defining
the 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 allowed
and the partition function is4
Z = tr e−β ˆHph
= ∏
kλ
e−β¯hωkλ/2
1 − e−β¯hωkλ
, (2.13)
so the contribution to the free energy is
Fph
=
1
2 ∑
kλ
¯hωkλ + β−1
∑
kλ
log 1 − e−β¯hωkλ = β−1
∑
kλ
log 2 sinh
β¯hωkλ
2
. (2.14)
Introducing density of states
d(ω) = ∑
kλ
δ(ω − ωkλ) (2.15)
as for electrons, the free energy can be expressed as
Fph
= β−1
dω log 2 sinh
β¯hω
2
d(ω). (2.16)
The non-zero ground-state energy corresponding to the limit β → ∞ is
Fph
=
¯h
2
dω ωd(ω). (2.17)
Thus the vibrational contribution is non-zero even for T = 0. This is a quantum mechanical
property of the harmonic oscillator originating from the last term in (2.11).5 In the present
work, 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 Hessian
of 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 in
the reciprocal space. If the finite crystal contains N unit cells in total, each with p nuclei, the sum above has 3pN
terms, the number of phonon normal modes. In general this number is very large (in the order of 1023), and the
sum approaches a continuous integral. The assumption is that the physics of a finite crystal resembles that of an
ideal infinite crystal since this number is so exceedingly large, except for a tiny fraction of atoms sufficiently close
to 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 ambient
pressure [51].
11
30. 2 Theoretical foundation
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-consistent
conditions 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 and
polarizations ekλ are the solutions to the eigenvalue equation
D(k)ekλ = ω2
kλekλ (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 given
lattice parameters.
Examples of calculated force constants are given in Figure 2.2 for the ε-Fe-N system. The
resulting densities of states are given in Figure 2.3. More computational details are given in
Chapter 4 and Chapter 5.
12
31. 2.4 Vibrational contribution
−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
Stretchingforceconstant,eV/˚A2
Bendingforceconstant,eV/˚A2
Bond length, ˚A
Fe-N
Fe-Fe
N-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
Stretchingforceconstant,eV/˚A2
Bendingforceconstant,eV/˚A2
Bond length, ˚A
Fe-N
Fe-Fe
N-N
(b) ε-Fe6N3
Figure 2.2: Calculated force constants for hexagonal close-packed structures as a
function of bond length. Stretching force constants (black) and the weaker bending
force constants (blue, secondary axis) for the Fe-N, Fe-Fe, and N-N bonds. The
force constants are calculated for equilibrium lattice parameters.
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18
Density,d,THz−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
Density,d,THz−1
Frequency, ω, THz
(b) ε-Fe6N3
Figure 2.3: The phonon density of states obtained for the force constants given in
Figure 2.2.
13
32. 2 Theoretical foundation
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 the
optical branches are projected onto the three acoustic branches λ. Vibrations are allowed up to
a maximum frequency ωD, often expressed in terms of the characteristic Debye temperature
ΘD, ¯hωD = kBΘD, determined by the vibrational degrees of freedom of the system. From
these assumptions
d(ω) =
V
2π2
ω2
¯hv3
(2.21)
for ω ≤ ωD, and the free energy
Fph
= 3β−1
D3(β¯hωD) (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 the
partition function (2.3). For a given elastic displacement parameter r, the Helmholtz free
energy 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 and
the remaining relative nuclei positions parameter r ,
Fac = min
r
Fr=(a,c,r ), (2.24)
with nuclei at equilibrium positions, i.e., the value of r that minimizes (2.24). If Vac denotes
the volume of the hexagonal unit cell, the Helmholtz free energy expressed in the usual vari-
ables is
F(T, V) = min
Vac=V
Fac, (2.25)
where V is the volume. This corresponds to the pair of lattice parameters that minimizes the
energy for a given fixed unit cell volume.
The expressions above define the bridge between the quantum mechanical description of
the system (subscript notation) and a description in terms of macroscopic variables (function
notation).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 in
Chapter 3, allows deriving thermodynamic properties and other thermodynamic potentials
through 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 is
given by
p(T, V) = −
∂F
∂V
(T, V), (2.26)
6The volume normalized to unit cells is accepted as a macroscopic variable here.
14
33. 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 through
the 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) → (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 energy
is 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.
2.6 Configurational degrees of freedom
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 configurational
state 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 include
configurational degrees of freedom |σεω; r , though it is important to realize that σ is formally
a parameter in the quantum mechanical setting of the Born–Oppenheimer approximation. A
specific example of the parametrization of σ is given in Section 5.3.
Two ensembles are considered in the present work corresponding to whether conditions
on the total number of particles or of constant chemical potential are imposed. Distinguishing
between these two cases is particularly important in the thermodynamic statistical sampling
in Chapter 6.
15
34. 2 Theoretical foundation
2.6.1 Canonical ensemble
In the canonical ensemble heat is allowed to be exchanged with an external reservoir at some
fixed temperature, but the total number of nuclei is assumed fixed. The partition function is
modified to include configurational degrees of freedom,
Z = tr ˆρ = ∑
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 with
an external reservoir. For brevity only one distinct type of particle is considered. The corre-
sponding grand-canonical ensemble Hamiltonian is obtained by the transformation
ˆHG = ˆH − µ ˆN, (2.33)
where ˆN is the particle number operator and µ is the chemical potential. In equilibrium,
imposing constant chemical potential, the density operator is given by ˆρG = e−β ˆHG , and the
corresponding grand-canonical ensemble partition function is given by
ZG = tr ˆρG = ∑
σ
∑
ω
∑
ε
e−β(Eσωε;r−µNσ)
. (2.34)
The mixed mode thermodynamic state of the system is given by the density operator ˆρG,
similar to (2.2), and ensemble averages of a physical quantity ˆA is given by
ˆA G =
tr ˆρG
ˆA
ZG
. (2.35)
defined in terms of the density operator ˆρG 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 ˆN G, closely resem-
bling Lagrange multipliers known from mathematical constrained optimization.
For brevity ˆH, Z, and · are often used to denote grand-canonical ensemble quantities
instead of ˆHG, ZG, and · G, respectively [36]. This practice is followed in the present work
from 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, electronic
and 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],
Z
el+ph
σ = ∑
ω
∑
ε
e−βEσωε;r
(2.36)
16
35. 2.6 Configurational degrees of freedom
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,
F
el+ph
σ;r = −β−1
log Z
el+ph
σ . (2.37)
Similarly, in Born–Oppenheimer approximation, electronic equilibrium can be assumed
to be established at any time as nuclei vibrate, so for a given configuration σ and a vibrational
excitation ω,
Zel
σω = ∑
ε
e−βEσωε;r
(2.38)
and
Fel
σω;r = −β−1
log Zel
σω. (2.39)
Collecting these terms, the grand-canonical partition function (2.34) can be decomposed
in terms of one of these two hybrid potentials,
Z = ∑
σ
e−β(F
el+ph
σ;r −µNσ)
= ∑
σ
∑
ω
e−β(Fel
σω;r−µNσ)
, (2.40)
with the free energies Fel
σω;r and F
el+ph
σ;r appearing in the partition function as the true eigenen-
ergies in (2.34).
Therefore, if F
el+ph
σ;r can be determined, configurational free energy can be calculated in-
dependently of the electronic and vibrational degrees of freedom. This also proves that the
thermodynamic statistical sampling method presented in Chapter 6 is valid even if electronic
and 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 required
for the thermodynamic statistical sampling in Chapter 6 and is therefore given explicitly in
this 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 configuration
σ in the ensemble is proportional to the Boltzmann factor
pσ =
e−β(Eσ−µNσ)
Z
. (2.42)
Hence the ensemble average energy
U = ˆH = ∑
σ
pσEσ, (2.43)
entropy
S = −kB log
ˆρ
Z
= −kB ∑
σ
pσ log pσ, (2.44)
and particle number
N = ˆN = ∑
σ
pσNσ (2.45)
17
36. 2 Theoretical foundation
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 Z
Z ∑
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
37. 3Energy–volume equation of state
The thermodynamic potentials introduced in Section 2.5 establish the link between discrete
quantized 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 of
parameter values, e.g., for smoothly defined lattice parameters, some parametrization of the
thermodynamic potentials are required, known as an equation of state (EOS) [26]. In this
chapter an equation of state is developed for the hexagonal system, where two independent
lattice parameters pose an additional challenge. The hexagonal equation of state serves as one
of the numerical prerequisites to properly study thermal expansion of the ε-Fe6N2 structure
from first-principles.
Before the hexagonal case is treated, the simpler one-dimensional energy–volume case is
shortly reviewed. The hexagonal equation of state is then introduced as an extension to the
one-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 the
many 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 inexact
smooth parameterization of the calculated energies. The smoothness is important for cal-
culation of derived thermodynamic properties. In the present work, the Birch–Murnaghan
equation (BM) [27],
E(V) = E0 +
m−1
∑
i=2
Ei
V
V0
−n/3
− 1
i
, (3.2)
is chosen, which by the binomial theorem is just a polynomial expansion in V−n/3, with
E0, 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
38. 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 as
the 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 the
present 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 in
temperature, a smoothly defined function
(T, V) → 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 consistently
across 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 corresponding
equilibrium values are listed in Table 3.1, where also the bulk modulus and its derivative
with respect to pressure have been calculated for the smoothly defined equation of state as
described 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 with
respect to pressure B0 are parameters instead of derived quantities [26], but this is not chosen in the present work
for simplicity.
20
39. −48.2
−48.1
−48.0
−47.9
−47.8
−47.7
−47.6
66 68 70 72 74 76 78 80 82
8.0
10.0
12.0
14.0
16.0
18.0
Energy,E,eV
Magneticmoment,µ/µB
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 84
11.0
12.0
13.0
14.0
15.0
Energy,E,eV
Magneticmoment,µ/µB
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 90
11.0
11.5
12.0
12.5
13.0
13.5
14.0
Energy,E,eV
Magneticmoment,µ/µB
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 96
6.0
7.0
8.0
9.0
10.0
11.0
12.0
Energy,E,eV
Magneticmoment,µ/µB
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 96
2.0
4.0
6.0
8.0
10.0
12.0
14.0
Energy,E,eV
Magneticmoment,µ/µB
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 96
0.0
2.0
4.0
6.0
8.0
10.0
Energy,E,eV
Magneticmoment,µ/µB
Volume, V, ˚A3
(f) Fe6N5
−94.0
−93.0
−92.0
−91.0
−90.0
84 88 92 96 100 104
4.0
6.0
8.0
10.0
12.0
14.0
16.0
Energy,E,eV
Magneticmoment,µ/µB
Volume, V, ˚A3
(g) Fe6N6
Figure 3.1: First-principles ground-state energies (square) for a selection of unit
cell volumes V for each of the symmetrically distinct configurations of nitrogen
occupations 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 (gray
square) 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.
40. 3 Energy–volume equation of state
σ d E0, eV V0, ˚A3 B0, GPa B0
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, and
derivative of bulk modulus with respect to pressure B0. Occupied (black) and un-
occupied (white) sites are visualized for each configuration. The table lists ground
state values without contributions for electronic and vibrational excitations. The
degeneracy is the number of symmetrically equivalent nitrogen occupations of
the six-site conventional hexagonal unit cell. The sum of the degeneracies satisfies
∑σ dσ = 26, corresponding to the total number of possible nitrogen occupation
configurations of the six sites.
22
41. 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
Energy,E,eV
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 curves
more or less move in parallel as the volume is changed from the equilibrium vol-
ume.
23
42. 3 Energy–volume equation of state
3.2 Extension to hexagonal systems
Of particular interest for the present work is the hexagonal close-packed lattice defined by a
pair 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. The
lattice 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 described
by 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 should
contain the original volume-dependent equation in its formulation as a special case, making
V and the ratio of lattice parameters r = c/a natural independent variables, and the energy
is 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) → (Vac, c/a), where
Vac =
√
3
2
a2
c (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
V
V0
−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 +
1
2
γ(V) r − r0(V)
2
. (3.7)
The quadratic expansion parameter is itself expanded as
γ(V) =
mc−1
∑
i=0
γi
V
V0
−n/3
− 1
i
, (3.8)
with parameters γ0, . . . , γmc−1.
24
43. 3.3 Necessity of the extension and magnetic phase transitions
2.68
2.72
4.2
4.3
4.4
4.5
−66.7
−66.6
−66.5
−66.4
−66.3
−66.2
Energy,eV
V → E(V, r0(V))
r → E(V, r)
a, ˚A
c, ˚A
Energy,eV
(a) ε-Fe6N2
2.68
2.72
4.2
4.3
4.4
4.5
−74.8
−74.6
−74.4
−74.2
−74.0
Energy,eV
a, ˚A
c, ˚A
Energy,eV
(b) ε-Fe6N3
Figure 3.4: First-principles ground-state energies for hexagonal close-packed
structures for various lattice parameters (a, c) (circle) and the fitted extended equa-
tions of state (a, c) → E(Vac, c/a) visualized as a function of lattice parameters
(dotted mesh). Projection V → E(V, r0(V)) through equilibrium lattice parame-
ters as a function of imposed volume (thick blue line) and projection r → 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 equivalently
the equilibrium values of the corresponding lattice parameters (a, c), can be calculated as
these 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 → E(V, r0(V)) = E0(V), (3.9)
which by construction is the minimum energy for any given volume V. This allows derived
thermodynamic 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 thermal
expansion coefficient can also be calculated by introducing smoothly defined temperature
dependence 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-canonical
ensembles 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-state
energies 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
3.3 Necessity of the extension and magnetic phase transitions
The first-principles calculator VASP (cf. Section 1.5) allows automatic relaxation of unit cell
lattice parameters (a, c) as well as position of nuclei relative to the unit cell, i.e., numerically
obtaining 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
44. 3 Energy–volume equation of state
2.60
2.64
2.68
2.72
2.76
4.7
4.8
4.9
5.0
5.1
−87.8
−87.6
−87.4
−87.2
−87.0
−86.8
−86.6
Energy,eV
a, ˚A
c, ˚A
Energy,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 cell
volume constant for a selection of volumes. The pair of lattice parameters that
minimizes the energy subject to the unit cell volume constraint are determined by
VASP’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 equation
of state only in terms of the volume can be problematic. It is unclear whether the
discontinuity in optimal lattice parameters as a function of volume is a result of
numerical issues with the relaxation algorithm or if a volume-induced magnetic
phase transition exists. This can only be resolved by calculating first-principles
energies for more points on the surface of lattice parameters. (a) Resulting energy
as a function of lattice parameters. (b) Projection showing the lattice parameters
only. The discontinuity is clearly visible. Constant-volume curves are shown (thin
dotted 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. See
also 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 large
equation systems is highly non-trivial and numerical issues may arise.
Another potential problem in obtaining a reliable expression for the equation of state is
volume-induced magnetic phase transitions. Magnetic properties are not studied in detail in
the present work, and none of the models introduced in the following chapters take magnetic
properties into account. To monitor and avoid possible magnetic phase transitions, a sudden
change 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 the
fitting 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 the
present work and can therefore safely be discarded. This problem is not observed for the
ε-Fe6N2 structure.
26
45. 3.3 Necessity of the extension and magnetic phase transitions
2.6
2.7
2.8
2.9 4.1
4.2
4.3
4.4
4.5
4.6
6
7
8
9
10
11
12
13
Magneticmoment,µ/µB
a, ˚A
c, ˚A
Magneticmoment,µ/µB
(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-Fe6N3
structure at V 90.4 ˚A. As the model does not account for magnetic properties
only 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 of
lattice parameters (circle); points corresponding to V > 90.4 ˚A are discarded (red
circle). (b) Projection showing the lattice parameters only. The constant-volume
curve is shown (thin dotted line). See also Figure 3.1d.
27
47. 4Thermal expansion of the ε-Fe6N2
structure
The first investigation presented in this dissertation is a thermodynamic vibrational model of
the hexagonal close-packed ε-Fe6N2 structure, denoted in these first-principles calculations by
the size of its primitive unit cell. Specifically, the quasiharmonic phonon model is used aiming
at predicting thermal expansion of the lattice. This structure was chosen for validation of the
model due to the availability of experimental data [2, 3, 4, 5, 7, 8]. The unit cell of the structure
is visualized in Figure 4.1 and Wyckoff positions are listed in Table 4.1. More generally, the
structure 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 presented
in Chapter 5 and Chapter 6.
In the harmonic phonon model, a quadratic energy response is assumed when nuclei are
perturbed from equilibrium positions, corresponding to specifying the second-order deriva-
tives of the energy in nuclei positions; the first-order derivatives vanishes at equilibrium. The
Hamiltonian of this model is given in Section 2.4. The quasiharmonic phonon model is a
generalization of the harmonic model, where the dispersion relation (2.12) depends on unit
cell 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 host
lattice atoms (gray), two interstitial sites occupied by nitrogen atoms (black), and
four vacant interstitial sites (white). The lattice parameter a defines the distance
between atoms in the horizontal close-packed planes (cf. Figure 3.3).
29
48. 4 Thermal expansion of the ε-Fe6N2 structure
6g (x, 0, 0), (0, x, 0), (−x, −x, 0), (−x, 0, 1
2 ), (0, −x, 1
2 ), (x, x, 1
2 ); x = 0.3262a
2c (1
3 , 2
3 , 1
4 ), (2
3 , 1
3 , 3
4 )
a Calculation from [11].
Table 4.1: Wyckoff positions of atoms in the iron sublattice (6g) and occupied
interstitial sites (2c). The space group of the structure is P6322 with space group
number 182.
the present work, this is further generalized by allowing the dispersion relation to depend
on both of the lattice parameters a and c of the hexagonal unit cell. Numerical prerequisites
for the hexagonal generalization are already given in Chapter 3, where a smoothly defined
equation of state parametrization allows numerical minimization of energy to be performed
as a function of lattice parameters and temperature.
Since only vibrational and electronic freedom are considered here, the partition function
of Section 2.2 applies. Hence the thermodynamic potentials introduced in Section 2.5 can be
employed in the mathematical treatment.
The force constants for obtaining the dynamical matrix and the phonon dispersion relation
are calculated in the density functional perturbation theory (DFPT) approach [28], defining a
set 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 approximation
employed 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 full
potential frozen-core PAW method [55, 56], and the exchange-correlation is treated within
the 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 performed
with 11 × 11 × 11 mesh k-points centered on the Γ point to avoid breaking the hexagonal
symmetry. Methfessel–Paxton of first order is used with smearing width 0.2 eV [60]. The
plane wave basis set is truncated at 520 eV and the energy convergence criterion for electronic
self-consistency is 10−5 eV per atom.
First-principles energies are obtained in two steps. Firstly, nuclei positions relative to
the 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 interpolation
of the correlation part of the exchange correlation functional [61]. Ground-state energies are
calculated for 130 distinct pairs lattice parameters spanning a fine mesh around minimum
energy values (cf. Figure 4.7). The large number of calculations is possible as no relaxation of
lattice parameters are performed.
Phonon calculations are carried out by the supercell method [24] in a unit cell of 48 iron
host atoms obtained by repeating the original unit cell 2 × 2 × 2 times. Force constants are
calculated with exact second-order energy derivatives using density functional perturbation
theory (DFPT) [28], also as implemented in VASP. A Γ centered mesh of 4 × 4 × 4 k-points
is 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
49. 4.2 Force constants and vibrational contributions
wave basis set truncation at 500 eV is used for the phonon calculation to make the memory
requirements of larger supercell problem manageable. Accurate relaxation of nuclei posi-
tions relative to the supercell are performed before force constants are obtained by DFPT. As
calculation 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 equilibrium
and 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 the
author’s FuncLib library is used for non-linear optimization.
The inversion of functions required to perform the Legendre from the Helmholtz free
energy to the Gibbs free energy are solved to full machine precision by bisection [54].
4.2 Force constants and vibrational contributions
Second-order changes in energy corresponding to perturbation of pairs of nuclei positions
defined as the force constants in Section 2.4.1 are estimated in the supercell approach using
the linear response method.
For each pair of nuclei, a basis with one basis vector in the direction of the separation of
the nuclei and two basis vectors perpendicular to the first basis vector can be chosen. This
procedure defines a change-of-basis matrix P, and when this is applied to the 3 × 3 force
constants matrix Φ(jj , nn ), stretching and bending force constants can be estimated by the
diagonal entries of the force constants matrix P−1Φ(jj , nn )P in the changed basis. This is
performed for the purpose of illustration and the result is given in Figure 4.2 for equilibrium
lattice 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
Stretchingforceconstant,eV/˚A2
Bendingforceconstant,eV/˚A2
Bond length, ˚A
Fe-N
Fe-Fe
N-N
Figure 4.2: Calculated force constants as a function of bond length. Stretching
force constants (black) and the weaker bending force constants (blue, secondary
axis) for the Fe-N, Fe-Fe, and N-N bonds. The force constants are calculated for
equilibrium lattice parameters.
31
50. 4 Thermal expansion of the ε-Fe6N2 structure
0
1
2
3
4
5
6
7
0 2 4 6 8 10 12 14 16 18 20
Density,d,THz−1
Frequency, ω, THz
a = 2.66 ˚A, c = 4.27 ˚A
a = 2.68 ˚A, c = 4.31 ˚A
a = 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
Frequency,THz
Reduced wave vector
Figure 4.4: Phonon dispersion relation along important symmetry direction of the
hexagonal close-packed unit cell calculated for equilibrium lattice constants. See
symmetry points in Figure 4.5. Three acoustic phonon branches (converging to 0
towards the Γ point) and a number of optical branches.
A
A
Γ
H
K
M
Figure 4.5: The first Brillouin zone of the hexagonal close-packed unit cell [63].
32
51. 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 a
decomposition is not required for defining the dynamic matrix (2.19), which depends on the
total energy and its derivatives only.
The phonon density of states is obtained from the force constants through (2.20). This
allows 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, showing
that the density of states changes continuously with unit cell volume. The phonon dispersion
relation along important symmetry of the hexagonal unit cell directions is given in Figure 4.4
for the equilibrium lattice parameters. Similar dispersion relation curves are obtained for
other lattice parameters.
4.3 Fitting the equation of state
The extended hexagonal equation of state developed in Section 3.2 is employed to predict
finite temperature equilibrium properties from the quasiharmonic phonon model.
A five-parameter Birch–Murnaghan equation of state (3.2) is chosen, augmented with
three additional parameters (mr = 2, mc = 1) to fit the entire surface of lattice parameters
though (3.6) and (3.8); Birch–Murnaghan order parameter n = 2 is chosen. Since energies
have been calculated for a large number of pairs of lattice parameters (cf. Section 4.1), reliable
estimates of eight parameters in the equation of states can be obtained. The parameters are
obtained 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
√
3
2
a2
i ci,
ci
ai
. (4.2)
The fitting is repeated for any temperature of interest, resulting in a smooth temperature
dependence as discussed in Chapter 3. As an illustration of the finite temperature fitting
procedure, the calculated Helmholtz free energies are given in Figure 4.6 for three pairs of
lattice parameters, with similar curves obtained for other lattice parameters. For any given
temperature 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 temperature
and lattice parameters.1
Table 4.2 lists predicted equilibrium lattice parameters, bulk modulus, pressure derivative
of bulk modulus, and thermal expansion evaluated for selected temperatures. Comparison to
experimental data available in the literature is provided [1, 3, 7]. The complete set of equation
of 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 itself
smoothly defined.
33
52. 4 Thermal expansion of the ε-Fe6N2 structure
−69.00
−68.50
−68.00
−67.50
−67.00
−66.50
−66.00
E+Fel+Fph,eV
a = 2.66 ˚A, c = 4.27 ˚A
a = 2.68 ˚A, c = 4.31 ˚A
a = 2.71 ˚A, c = 4.35 ˚A
−0.08
−0.06
−0.04
−0.02
0.00
Fel,eV
−2.50
−2.00
−1.50
−1.00
−0.50
0.00
0.50
0 200 400 600 800 1000
Fph,eV
Temperature, T, K
Figure 4.6: Helmholtz free energy F = E + Fel + Fph for three pairs of lattice
parameters (a, c) as a function of temperature T, obtained by (2.6) and (2.16) from
the calculated phonon densities of states (cf. Figure 4.3; same color coding). Notice
that the energy only changes slightly between the three pairs of lattice parameters.
(Upper) Total Helmholtz free energy, including static elastic energy E. The pair of
lattice 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
53. 4.3 Fitting the equation of state
T, K a0, ˚A c0, ˚A B0, GPa B0 α, 10−5 K−1
295 2.6957 4.3379 197.2 5.5 2.85
2.7100b 4.3748b 172.4d 5.7d 2.92c
300 2.6958 4.3383 197.0 5.5 2.86
2.7108a 4.3783a
468 2.6990 4.3508 189.2 6.1 3.31
2.7132b 4.3892b 3.47c
508 2.6997 4.3543 187.1 6.3 3.40
2.7137b 4.3937b 3.60c
546 2.7004 4.3577 185.0 6.5 3.48
2.7143b 4.3985b 3.72c
588 2.7011 4.3618 182.6 6.6 3.57
2.7148b 4.4035b 3.85c
618 2.7017 4.3649 180.9 6.8 3.64
2.7162b 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 B0, and volumetric thermal expansion coeffi-
cient α as a function of temperature T. Equation of state parameters are given in
Table 4.3.
T, K V0, ˚A3 E0, eV E2/E0 E3/E0 E4/E0 ρ0 ρ1 γ0 η, meV
0† 80.963 −66.653 −1.7630 −1.1262 17.355 1.6066 0.11439 −0.33016 1.13
0 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 0.672
300 81.913 −66.520 −1.7032 −1.2605 27.794 1.6093 0.0041618 −0.30203 0.674
468 82.341 −66.939 −1.6340 −1.7322 27.516 1.6120 −0.068319 −0.28293 0.868
508 82.452 −67.059 −1.6153 −1.8487 27.313 1.6129 −0.087758 −0.27811 0.944
546 82.560 −67.179 −1.5966 −1.9598 27.085 1.6137 −0.10699 −0.27344 1.02
588 82.682 −67.317 −1.5752 −2.0827 26.796 1.6148 −0.12914 −0.26820 1.12
600 82.718 −67.358 −1.5688 −2.1178 26.708 1.6151 −0.13565 −0.26668 1.14
618 82.772 −67.420 −1.5592 −2.1704 26.571 1.6156 −0.14555 −0.26440 1.19
723 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 function
of volume vanishes by construction. The ratio of lattice parameters at equilibrium
volume satisfies r0 = c0/a0 = ρ0. The root-mean-squared fitting error η of the
equation of state is also given.
35
