studies of material properties using ab initio and...

ACTA

UNIVERSITATIS

UPSALIENSIS

UPPSALA

2008

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 422

Studies of Material Properties usingAb Initio and Classical MolecularDynamics

LOVE KO I

ISSN 1651-6214ISBN 978-91-554-7154-5urn:nbn:se:uu:diva-8626

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Front page illustration

Billiards is a game that requires skill, stamina, determination and patience.

Thus, the game is actually close to physics research, where these qualities are

necessary. In the hard-sphere approach, the balls can be seen as atoms

interacting, bouncing, clustering or spreading, and the cue is the technique

used to influence the atoms/balls, whether it is experiment or theory. Here,

the balls in the upper triangle can be seen as crystalline, whereas the lower

triangle show atoms/balls in a molten state. Actually, this represents a

coexistence phase setup, which is a technique in molecular dynamics

calculations when melting should be determined.

List of Papers

This thesis is based on the following papers, which are referred to in the text

by their Roman numerals.

I The impact of system restriction in molecular dynamicsapplied to the melting of Ne at high pressureL. Koci, A. B. Belonoshko and R. Ahuja

Submitted to Computational Materials Science

II Mechanical stability of TiO2 polymorphs under pressure: abinitio calculationsL. Koci, D. Y. Kim, J. S. de Almeida, M. Mattesini, E. Isaev and

R. Ahuja

Submitted to Journal of Applied Physics

III Dynamical stability of the hardest known oxide and the cubicsolar material: TiO2D. Y. Kim, J. S. de Almeida, L. Koci and R. Ahuja

Applied Physics Letters 90, 171903 (2007)

IV Anomalous elastic behavior of superconducting metals athigh pressureL. Koci, Y. Ma, A. Oganov, P. Souvatzis and R. Ahuja

Submitted to Physical Review B

V Study of the high pressure helium phase diagram usingMolecular DynamicsL. Koci, R. Ahuja, A. B. Belonoshko and B. Johansson

Journal of Physics: Condensed Matter 19, 016206 (2007)

VI Towards accurate melting temperatures from Monte Carlosimulations for neon and argon: from clusters to the bulkE. Pahl, L. Koci, F. Calvo, R. Ahuja and P. Schwerdtfeger

In manuscript

VII Ab initio and classical molecular dynamics of neon melting athigh pressure

v

L. Koci, R. Ahuja and A. B. Belonoshko

Physical Review B 75, 214108 (2007)

VIII Melting of Na at high pressure from ab initio calculationsL. Koci, R. Ahuja, L. Vitos and U. Pinsook

Accepted for publication in Physical Review B

IX Simulation of shock-induced melting of Ni using moleculardynamics coupled to a two-temperature modelL. Koci, E. M. Bringa, D. S. Ivanov, J. Hawreliak, J. McNaney,

A. Higginbotham, L. V. Zhigilei, A. B. Belonoshko, B. A.

Remington and R. Ahuja


Virtual Journal of Ultrafast Science 5 Issue 8 (2006)

X Ab initio calculations of the elastic properties of ferroperi-clase Mg1−xFexO (x≤0.25)L. Koci, L. Vitos and R. Ahuja

Physics of the Earth and Planetary Interiors 164, 177 (2007)

XI Molecular dynamics calculation of liquid iron properties andadiabatic temperature gradient in the Earth’s outer coreL. Koci, A. B. Belonoshko and R. Ahuja

Geophysical Journal International 168, 890 (2007)

XII Molecular dynamics study of liquid iron under high pressureand high temperatureL. Koci, A. B. Belonoshko and R. Ahuja


XIII Ab initio calculations of hcp and bcc Fe at extreme conditionsL. Koci, A. B. Belonoshko and R. Ahuja

In manuscript

Reprints were made with permission from the publishers.

Publications not included in this thesis

Ab initio and classical molecular dynamics calculations of thehigh-pressure melting of NeL. Koci, A. B. Belonoshko and R. Ahuja

Accepted for publication in Journal of Physics

vi

A Graph Theoretical Approach to the Shunting ProblemG. Di Stefano and L. Koci

Electronic Notes in Theoretical Computer Science 92, 16 (2004)

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . 32.1.1 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . 52.1.2 The Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Exchange correlation energy . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Self-consistent Kohn-Sham equations . . . . . . . . . . . . . . . 8

2.2 Ab initio Computational methods . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 The PAW method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.4 The EMTO method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Potential design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.1 Two-body potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.2 Many-body potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Integration algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 The Verlet algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Nosé-Hoover algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 One-phase and two-phase simulations . . . . . . . . . . . . . . . 243.4.2 Radial distribution function . . . . . . . . . . . . . . . . . . . . . . . 263.4.3 Lindemann criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4.4 Mean square displacement . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Ab initio or classical molecular dynamics? . . . . . . . . . . . . . . . . . . . 31

4.1 The ab initio and classical combination . . . . . . . . . . . . . . . . . . 325 Equation of state and compression . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.1 Murnaghan EOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2.2 Birch-Murnaghan EOS . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Elasticity and hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.1 TiO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.1.1 EOS, elastic constants and bulk moduli . . . . . . . . . . . . . . 436.1.2 Phase stabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Group-V and V I metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Melting at high pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.2 Interatomic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.3 Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.3.1 He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.3.2 Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3.3 Na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.1 Rankine-Hugoniot relations and equation of state . . . . . . . . . . 598.2 The two-temperature model . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.2.1 The combined TTM-MD model . . . . . . . . . . . . . . . . . . . . 618.3 Shock-induced melting of Ni with MD . . . . . . . . . . . . . . . . . . 62

9 Inner Earth studies: (Mg,Fe)O and Fe . . . . . . . . . . . . . . . . . . . . . . . 65

9.1 MgO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669.2 (Mg,Fe)O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679.3 Fe at outer core conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

10 Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

x

1

Chapter 1

Introduction

he connection between a flint stone from the stone age, a compass

from the dawn of our era, a transistor from last century to a su-

percomputer of today is that they all serve as practical tools. The

knowledge of materials evolves with the improvements of exper-

imental techniques, refined theoretical methods and enhanced computational

performance. Experiments, theories and calculations/simulations are three ar-

eas for the understanding of matter, where each area is correlated to one an-

other in terms of agreement, evaluation, performance and so on. A positive

circle can be obtained: research gives better materials which lead to better

measurement equipment and computers which help research!

The work performed in this thesis are based on calculations. Although the re-

sults obtained from computations only can be validated by experiments, there

are certain benefits. For example, experimental studies of matter at extreme

conditions (i.e. high pressure and high temperature) are difficult and some-

times expensive. As an extreme example, a Russian attempt of drilling into

the Earth began in 1970 and ended in 1994. The final depth reached was 12

300 meters, not even 0.2 % of the Earth’s radius. Considering more usual

experimental environments to reach extreme conditions such as the diamond-

anvil cell technique or the use of gas guns, theoretical calculations can serve

to understand and compliment the experiments, as well as being tests for new

theories.

The two main paths of conducting computational simulations on condensed

matter are first- principles methods (ab initio) and classical methods. Ab initiomethods, evolved from the basis of density functional theory (DFT), are based

on quantum mechanics. Although considering the benefit of supercomputers,

the computational costs for these calculations are still challenging. Molec-

ular dynamics (MD) based on classical models provide the ability of simu-

lating hundred of thousands of atoms. The fast computations arise from the

fact that the interatomic potential is empirically set from experiments or first-

principles data, or combinations thereof. In this thesis, ab initio and classical

MD calculations have been performed mainly in extreme condition environ-

2 CHAPTER 1. INTRODUCTION

ments. However, the theoretical framework to mimic the nature of atoms and

molecules can be applied to, practically, any system. Therefore, the same the-

ories and simulation programs as used in this work have also been used for e.g

the simulation of transmembrane proteins, hydrogen storage calculations, the

dynamics in HIV-1 protease, the study of ice/water transitions, etc. The fact

that MD is a multidisciplinary field is fantastic, and yields great collaborating

opportunities for researchers from neighboring fields of interest.

The scope of this thesis is to examine material properties such as hardness,

elasticity, diffusion, melting and phase transitions. After this introduction,

chapter 2 describes the basic machinery of density functional theory (DFT).

This theory, developed in 1964-65 by Walter Kohn, lead to the 1988 Nobel

Prize. Chapter 3 describes the basics of molecular dynamics. It is explained

how the atom potentials yield interatomic forces and how the atomic posi-

tions are updated through Newton’s equation. Furthermore, it is shown how

the obtained data from MD can be analyzed. Chapter 4 discusses the inter-

esting relation between classical and ab initio molecular dynamics. As the

theoretical results from this work often are compared to experiments, chap-

ter 5 gives a short description of experimental techniques to generate extreme

conditions in the sense of pressure and temperature. Thermodynamical equa-

tions relating equation of state to pressure are also presented. The elasticity

of matter is presented in chapter 6: the application to TiO2 is fascinating, as

it has been predicted to be one of the hardest existing oxides. Furthermore,

with the group-V metals V, Nb, Ta as potential superconductors, their erratic

elastic behavior under pressure is studied. In chapter 7, the results from the

melting of the rare gases He, Ne and the alkali metal Na are shown. Chap-

ter 8 also shows melting results, but from MD simulations that mimic shock

wave experiments. In the last chapter, the properties of abundant elements and

compounds in the Earth’s interior are shown. This is highly interesting, as

an enhanced knowledge of these elements can lead to a deeper (in a double

sense!) understanding of our, and other planets’, dynamic behavior.

3

Chapter 2

Density functional theory

o find the electronic structure of a material, one need to solve the

time independent Schrödinger equation

Hψ(Ri,rn) = Eψ(Ri,rn) , (2.1)

where H is the Hamiltonian, E is the energy and ψ(Ri,rn) is the wave func-

tion, representing the probability amplitude for finding a particle at a given

point in space at a given time. In principle, knowing the atomic structure of

electrons and ions, the equation yields the energy E and the wave function

ψ . The solution of the Schrödinger equation is known as first principles or

ab initio. The nomenclature refers to the fact that only the laws of physics are

obeyed, and no empirical inputs are used. However, considering that N ∼ 1023

electrons and ions for a small piece of matter, this partial differential equation

is not computationally tractable. To overcome this problem, Density Func-tional Theory (DFT) was developed during the last century. Being a magnumopus, it is the primary tool for electronic structure computations in theoretical

physics and chemistry. The main idea of DFT is that the many electron system

can be treated as a one electron system, where the energy is calculated from

the electron density.

2.1 The Born-Oppenheimer approximation

The Hamiltonian of Eq. 2.1 may be written

H =12 ∑

i

P2i

Mi+

12 ∑

n

p2n

m+

12 ∑

i�= j

ZiZ je2

|Ri −R j| + (2.2)

+12 ∑

n�=n′

e2

|rn − rn′ |−∑

in

Zie2

|Ri − rn| ,

where index i and j run on nuclei, n and n′ on electrons, Ri and Pi are positions

and momenta of the nuclei, rn and pn of the electrons, Zi the atomic number

4 CHAPTER 2. DENSITY FUNCTIONAL THEORY

of nucleus i, Mi its mass and m the electron mass. The first two terms in the

equation are the kinetic energies of the nuclei and the electrons and the third,

fourth and fifth term represent the coulombic potential between nuclei-nuclei,

electron-electron and nuclei-electron. However, Eq. 2.2 is not possible to solve

practically, as the computation time would be too long. Thus, due to the fact

that Born and Oppenheimer [26] noted that the movements of the electrons

are about 100 times faster than those of the nuclei, one can factorize the wave

function into

ψ(Ri,rn) = Ξ(Ri) Φ(rn,Ri) , (2.3)

where Ξ(Ri) describes the nuclei and Φ(rn,Ri) describes the electrons. Thus,

we get two separate equations as

HelΦ(rn,Ri) = V (Ri)Φ(rn,Ri) , (2.4)

where

Hel =12 ∑

n

p2n

m+

12 ∑

i�= j

ZiZ je2

|Ri −R j| +12 ∑

n�=n′

e2

|rn − rn′ |−∑

in

Zie2

|Ri − rn| (2.5)

and [∑

i

P2i

2Mi+V (Ri)

]Ξ(Ri) = E Ξ(Ri) . (2.6)

Thus, the difference between the true Hamiltonian in Eq. (2.2) and the latter

expressions is that the nuclei in Eq. (2.4) are regarded as fixed, resulting in

an energy V (Ri) called the interatomic potential which will be put into Eq.

(2.6), to give the motion of the nuclei (note that all electronic effects are in-

corporated in V (Ri)). Although being simplified by the Born-Oppenheimer

method, sine qua non the calculations would be severe, the evaluation of Eq.

(2.4) to get V (Ri) is still challenging, and will be discussed in section 3.1. Fur-

thermore, in some cases, one must be cautious with the fixed nuclei position

due to quantum effects: a consequence of the uncertainty principle that pre-

vents particles from being in a state of absolute rest. One such a case could be

molecular dynamics performed on a light mass system at low temperatures.

Specifically for this thesis, Ne is highly interesting as it is an intermediate

quantum system providing valuable insight when compared to other quantum

and classical systems. A validity test for the classical approximation in Eq.

(2.6) can be done based on the de Broglie thermal wavelength [53]

Λ =

√2π h2

MkBT, (2.7)

where M is the atom mass, T the temperature and h the Planck constant. The

approximation is justified if Λ� am, where am is the nearest neighbor distance

for the nuclei. Shown in Fig. 2.1 is the ratio Λ/am as a function of T for Ne.

2.1. THE BORN-OPPENHEIMER APPROXIMATION 5

For the low PT domain, the Λ/am ratios were found to be approximately 0.1.

However, at higher pressure and temperature, the ratios decrease significantly.

Although am becomes smaller at higher pressures, the increase in T ensures

that Λ decreases.

Nearest neighbor distance (Angstrom)

Tem

pera

ture

(K

)

2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5500

1000

1500

2000

2500

3000

3500

4000

0.03

0.04

0.05

0.06

0.07

0.08

Figure 2.1: The ratio of the de Broglie wavelength and the nearest neighbor distance

am for Ne as a function of temperature T .

2.1.1 The Hohenberg-Kohn theorems

The Hohenberg-Kohn approach is applicable to any interacting particle system

in an external potential Vext(r) and fixed nuclei due to the Born-Oppenheimer

approach with the Hamiltonian (omitting nuclei-nuclei interaction)

H =−h2

2me∑

i∇2

i +∑i

Vext(ri)+12 ∑

i�= j

e2

|ri − r j| . (2.8)

DFT is based upon the two following theorems of Hohenberg and Kohn:

Theorem 1 For any system of particles in an external potential Vext (r), thepotential is uniquely determined by the ground state density n0(r).

Given the ground state density n0, the many-body wave functions are deter-

mined as the Hamiltonian is known. Therefore, all properties about the system

are known.

Theorem 2 For any external potential Vext (r), the energy E in terms of thedensity n0(r) can be found. The density minimizing the energy E is the groundstate density n0(r).


Figure 2.2: The schematic form of the theorems by Hohenberg and Kohn. Given an

external potential Vext(r), the solution of the Schrödinger equation gives the solu-

tions Ψi(r), including Ψ0(r) which gives the ground state density n0(r). The loop is

closed from the density to the potential by the Hohenberg-Kohn theorems as stated.

Thus, having the ground state electron density n0(r), the minimum energy can

be found. However, the problem remains nontrivial regarding the calculation

of the density. A schematic version of the theorems are shown in Fig. 2.2.

2.1.2 The Kohn-Sham equations

Having in mind that the ground state density n0(r) determines all properties

of the many-body wave functions, the theorems do not state how to find the

functionals. Actually, only the one-electron system, i.e. the hydrogen atom,

is entirely known. The theory given by Kohn and Sham includes the treat-

ment of the many-body problem as an independent particle problem, using an

exchange-correlation functional. The ansatz of Kohn and Sham is based on

the following: Considering an auxiliary system of non-interacting particles,

the ground state density n0(r) is shared by both systems, i.e. a density that

completes the schematic loop shown in Fig. 2.2 as well as the auxiliary loop

for the independent particles. The ground state energy, written as a function

of the electron density with the Kohn-Sham approach, is

EKS = Ts[n]+12

∫d3rd3r′

n(r)n(r′)|r− r′| + (2.9)

+

∫drVext (r)n(r)+EII +Exc[n] ,

2.1. THE BORN-OPPENHEIMER APPROXIMATION 7

where Ts[n] is a term containing the kinetic energies of non-interacting elec-

trons, the second term is the Hartree energy of the self-interactive electron

density n(r), the third term is the energy due to the potential from the nuclei

with the electron density and EII is the nuclei-nuclei interaction. Exc[n] is the

exchange correlation energy where the many-body effects are accumulated,

and it is the one and only uncontrollable approximation in DFT. Having Exc,

the effective potential for the auxiliary system is

Ve f f = Vext +

∫ n(r′)|r− r′|dr′+

dExc(n(r))dn(r)

. (2.10)

2.1.3 Exchange correlation energy

The effect of exchange and correlation in the Kohn-Sham approach is ex-

pressed as a functional of the electron density, Exc[n]. Although this unknown

functional could be very complex, two rather simple approximations as de-

scribed in the following sections have been applied successfully.

The local density approximationThe assumption that Exc in Eq. 2.9 is equal to the exchange-correlation en-

ergy in an electron gas with the same density n(r) leads to the local density

approximation (LDA):

Exc =

∫n(r)εxc(n(r))dr , (2.11)

where εxc is the energy per electron at r, only depending on the density n(r)Since LDA is derived from the homogeneous electron gas, one might think

that the validity would be strictly limited. Per contra, ab initio calculations

with the LDA method are one of the most frequently used tools in materials

science and has been successfully applied in describing material properties.

However, the LDA show some systematic errors as e.g. the underestimation

of equilibrium volumes by a few percent.

Generalized gradient approximationThe generalized gradient approximation (GGA) also includes the gradient of

the electron density, ∇n(r), in the third term of Eq. (2.10), i.e.

Exc =

∫n(r)εxc(n(r), |∇n(r)|)dr . (2.12)

Perdew and Wang [91] have shown a great improvement to LDA, especially

for the transition metal series. However, the improvement is not consistent as

some calculations often lead to worse results. Furthermore, one well-known

deficiency for both LDA and GGA regards bandgap calculations, where the

approximations usually give underestimated energies.


n(r)

Self-consistency criterion true?

Δ n(r) < σ

Veffective

(r)=Vext

(r)+VH

[n(r)]+Vxc

[n(r)]

Hψi=εψ

i

n(r)=Σ ci|ψ

i(r)|

2

Kohn-Sham equation

Self-consistency criterion

Effective potential

Electron density

Initial density guess

Etot

Self-consistency criterion false?

↓

↓

↓

↓

→

→←

Figure 2.3: The schematic form of the self-consistency loop based on the theorems by

Hohenberg and Kohn. Given an initial guess of the electron density, the loop continues

in the way V j → n j → V j+1 → n j+1... and so on until convergence of the potential

and the density has been reached.

2.1.4 Self-consistent Kohn-Sham equations

Given an effective potential Ve f f and a density n(r), the independent particle

Schrödinger equations must be solved such that the potential and the density

are consistent. The easiest way of realizing the solution to the Kohn-Sham

equations in section 2.1.2 is a schematic diagram, where an initial density

n(r) is guessed. The density Ve f f can be computed (Eq. (2.10)) and inserted

into the Kohn-Sham equation as

[− 1

2∇2 +Ve f f (r)

]Ψi(r) = εiΨi(r) . (2.13)

The yielded wave functions Ψi(r) are then used to calculate the electron den-

sity

n(r) = ∑i

ci|Ψi(r)|2 , (2.14)

where the calculated density n(r) is put back into the loop if no convergence

has been reached with its prior value in this iterative process.

2.2. AB INITIO COMPUTATIONAL METHODS 9

2.2 Ab initio Computational methods

2.2.1 Bloch’s theorem

Considering the Kohn-Sham Eq. 2.13, the many-body effects have been in-

cluded in the effective potential Ve f f . However, to tackle the problem of in-

finitely many electrons, the periodicity of the ions in a perfect crystal makes

the potential invariant under lattice translations. Then, it is only necessary to

consider the number of electrons within the unit cell. From Bloch’s theorem

[67], the one-particle wavefunction can be written

Ψk(r) = eik·ruk(r) , (2.15)

with k as the wave vector and uk(r) as a function with the lattice periodicity.

To clarify this, one can define G ·R = 2πm, where G are the reciprocal lattice

vectors, R is a real space lattice vector and m is an integer. Then, the periodic

function uk(r) can be expanded in plane waves as

uk(r) = ∑G

ckeiG·r, (2.16)

which, inserted in Eq. (2.15), becomes

Ψk(r) = ∑G

ck,Gei(k+G)·r , (2.17)

where ck,G are the sought coefficients. If the limit is set to G ≤ Gmax, the

corresponding cutoff energy is found from

Ecuto f f =hG2

max

2me, (2.18)

Only a finite number of energy levels will be occupied at each k-point, and

one only needs to consider a finite number of electrons. Although there is an

infinite number of k-points, relatively few waves are sufficient to determine

the groundstate density of the solid: primo, only the k-points in the first Bril-

louin zone (BZ) have to be considered, due to the periodicity. Secondo, the

k-point subspace inside the BZ can be reduced due to rotation and inversion

symmetries. Terzo, wavefunctions from closely located k-points are almost

identical, and can be replaced by just one point. Thus, the the total energy can

be calculated from a discrete number of k-points. However, one must keep in

mind that more exact calculations can be achieved by using a denser k-point

grid and/or a higher Ecuto f f .

2.2.2 Pseudopotentials

The use of plane waves is a convenient way to construct the wave function

in the periodic solid. However, the core electrons, which are low in energy,


remain relatively unchanged compare to the rapid changes of the valence

electrons, which are involved in the chemical bonding. Therefore, the frozencore approximation can be applied, meaning that the core electrons are fixed

whereas the the Kohn-Sham equations are solved for the valence states in a

plane wave basis set. A pseudopotential can replace the true potential close to

the core, and in this way, the many-plane waves that would have been neces-

sary to describe the atomic-orbital-like cores can be avoided. The result is an

effective speedup of the calculation.

The pseudopotential should be soft, meaning that the expansion of the valence

wave functions should be allowed by using few plane waves. Furthermore, a

good transferability of the potential is important for valid calculations in other

environments, where the potentials for crystals and atoms are different. Shown

in Fig. 2.4 is the pseudo-wavefunction Ψpseudo, the exact function Ψx and their

respective potentials Vpseudo and Vx. Due to its softness, the Ψpseudo does not

reproduce the all electron wavefunction inside the core radius rc, but well out-

side. To accurately describe the true valence density, the charge density of the

pseudo-wavefunctions should, of course, be as close as possible. To strive for

this accuracy, the norm within the rc radius can be forced to be the same:∫ rc

0|Ψpseudo(r)|2dr =

∫ rc

0|Ψx(r)|2dr . (2.19)

If the norm conserving criterion is removed, the pseudopotentials are said to

be ultrasoft. The advantage with this approach is that a large value of rc can

be used. As a consequence, the plane-wave cut-off in the calculations can be

reduced, leading to faster calculations. However, caution must be taken, as

an augmentation term has to be added in the core region to compensate for

the non-norm conservation. Furthermore, the transferability suffers from the

removal of the norm conservation.

2.2.3 The PAW method

The projector augmented wave (PAW) method was developed in the early

1990’s by Blöchl [108]. The uniqueness of this method is that it combines the

simple, plane-wave pseudopotential approach with the versatility of the linear

augmented plane wave (LAPW) method [5]. Just as for the pseudopotential

method, projectors and auxiliary functions are introduced. However, the PAW

approach keeps the full all-electron wavefunction close to the nucleus, eval-

uating a combination of integrals of smooth functions throughout space plus

contributions from radial integration over muffin-tin spheres. For MD pur-

poses, the PAW is generally regarded as very accurate [31]. The basic ideas

with PAW can be shown by relating the true wave function Ψ with the smooth

part of a valence function Ψ inside a region (corresponding to the part r ≤ rcfor the pseudopotentials in Fig. 2.4) as

Ψ = T Ψ . (2.20)


Figure 2.4: The schematic form of the pseudo-wavefunction Ψpseudo, the exact func-

tion Ψx and their respective potentials Vpseudo and Vx. The white part of the inserted

billiard ball can be thought of the ion core for which r ≤ rc, where the pseudo-

wavefunctions are applied. At the white-green boundary r = rc, the functions co-

incide.


By expansion of the smooth functions,

|Ψ〉 = ∑m

cm|Ψm〉 , (2.21)

and with the aid of Eq. 2.20, the all-electron function is written

|Ψ〉 = T |Ψ〉 = ∑m

cm|Ψm〉 . (2.22)

With the expansion, Eq. 2.22 can be written

|Ψ〉 = |Ψ〉+∑m

cm(|Ψm〉− |Ψm〉) . (2.23)

As T is linear, the coefficients cm are given by a projection in each sphere,

cm = 〈pm|Ψ〉 . (2.24)

By inserting Eq. 2.24 in Eq. 2.23, one gets

|Ψ〉 = |Ψ〉+∑m

(|Ψm〉− |Ψm〉)〈pm|Ψ〉 , (2.25)

and the extraction of Ψ yields

T = 1+∑m

(|Ψm〉− |Ψm〉)〈pm| , (2.26)

There are many possible choices for the projections pm, and they can be close

to the ones used in the pseudopotentials. However, the difference from the

pseudopotential approach is that Eq. 2.26 still involves the all-electron wave-

function Ψ. Any operator A in the original all electron space can be trans-

formed into A as

A = T †AT , (2.27)

and the Kohn-Sham equation can be written

(H − ε S|Ψ〉 = 0 , (2.28)

with H = T †HT as the pseudopotential hamiltonian and S = T †T as the over-

lap operator.

2.2.4 The EMTO method

The recently developed exact muffin-tin orbitals (EMTO) theory [7, 6, 9], is an

improved Korringa-Kohn-Rostoker (KKR) [8, 105] method. It allows an exact

calculation of the one-electron Kohn-Sham states, and consequently the one-

electron total energy, for optimized overlapping muffin-tin (MT) potentials by

using a Green’s function formalism. The difference between EMTO and the

usual muffin-tin based KKR methods lies in the overlapping spheres, which


can be used for an accurate one-electron potential. The effective potential in

Eq. (2.13) is written

V (r) = V0 +∑R

[VR(rR)−V0] , (2.29)

where R runs over the lattice sites and VR(rR) are spherical potentials which

become equal to V0 on the muffin-tin (MT) radius boundary. In the case of

non-overlapping MT:s, VR(rR) is equal to the averaged integrated potential

within the sphere with radius SMT ,

VR(rR) =1

4π

∫V (r)dr, |rR| ≤ SMT , (2.30)

where rR ≡ r−R. Outside the MT radius, in the interstitial zone ω , the con-

stant potential is the spacial average of the potential,

V0 =1ω

∫V (r)dr . (2.31)

For fixed SMT radii, VR(rR) and V0 are optimized by minimizing their mean

square deviation.

For a more accurate representation of the full potential, the spheres should

overlap. [6] To solve Eq. (2.13) for the MT potential in Eq. (2.30) the wave

function is expanded in a complete basis set

Ψ j(r) = ∑RL

ΨRL(ε j,rR)vRL, j , (2.32)

where ΨRL are the MT orbitals for every site R and L denotes the orbital

and magnetic quantum number l and m, respectively. The coefficients vRL, jare constructed from the condition that Eq. (2.32) solves Eq. (2.13) in the

entire space. At R, inside the potential sphere, ΨRL(ε ,rR) are composed of

normalization functions, partial waves and spherical harmonics functions as

ΨRL(ε ,rR) = NRl(ε)φRL(ε ,rR)YL(rR) . (2.33)

ψRL(ε ,rR) are found from the solutions of the radial Schrödinger equation

with the potential VR(rR),

δ 2[rRφRl(ε ,rR)]

δ r2R

=[ l(l +1)

r2R

+VR(rR)− ε]rRφRL(ε ,rR) . (2.34)

In the interstitial zone,[∇2 +κ2

]ΨRL(κ ,rR) = 0, κ2 ≡ ε −V0 , (2.35)

as the potential is approximated by V0. The boundary condition for Eq. (2.35)

is that the spherical waves behave like real harmonics functions YL(rR) on the


non-overlapping spheres with radius aR centered at each R. The expansion of

ΨRL(κ ,rR) with the functions can be written

ΨRL(κ ,rR) = fRl(κ ,rR)YL(rR)δRR′δLL′ +∑L′

gR′l′(κ ,rR′)×YL′(rR′)SR′L′RL(κ) ,

(2.36)where the "head" and "tail" functions fRl(κ ,rR) and gRl(κ ,rR) are linear com-

binations of spherical Bessel and Neumann functions [112, 1] and SR′L′RL(κ)are the expansion coefficients found from the Dyson equation [10]. To match

the conditions between φRL(ε ,rR)YL(rR) and ΨRL(κ ,rR), an additional free-

electron solution is introduced as

ϕRL(ε ,r) = fRl(κ ,rR)+gRl(κ ,rR)DRl(ε) , (2.37)

where DRl(ε) is the logarithmic derivative of ϕRL(ε ,r) at rR = aR. Eq. 2.37

makes the ψRL(ε ,rR) waves continuous and differentially smooth at SMT and

ΨRL(κ ,rR) continuous at aR. The exact muffin-tin orbitals are constructed as

the superposition of the screened spherical waves, the partial waves and the

free-electron solution

ΨRL(ε ,r) = ΨRL(κ ,rR)+[NRl(ε)φRL(ε ,rR)−ϕRL(ε ,r)

]YL(rR) , (2.38)

where the last two terms are truncated outside SMT . If the screened spherical

wave component from a trial wave function around site R cancels the free-

electron solution component ϕRL(ε ,r)YL(rR), the trial function will be the so-

lution of the Kohn-Sham equation with the potential V (r). This happens if the

kink cancellation equation

∑RL

aR′[SR′L′RL(κ2

j )−δR′RδL′LDRl(ε j)]vRL, j = 0 , (2.39)

is satisfied for all R′ and l′ ≤ lmax. By solving Eq. 2.39, the single-electron

energies and wave functions can be found.


Figure 2.5: The schematic presentation of the coherent potential approximation, CPA.

CPAThe EMTO-CPA (coherent potential approximation) method is a mean-field

approach to treat substitutional random systems. It neglects the short-range

order and local lattice relaxation effects, but, at the same time, it gives highly

accurate total energies for completely disordered solid-solution [109, 77, 78,

106]. By keeping the accuracy for both ordered and disordered systems at

the same footing, this method represents an ideal tool for tracing the effect of

dopands on the thermodynamical and mechanical properties of solid-solutions

[90, 110, 68, 111], including minerals [39, 38, 113, 40]. The basics of CPA is

shown in Fig. 2.5 for a binary alloy of components 1 and 2 and concentrations

c1 and c2, respectively. The effective (coherent) potential will give the same

scattering properties as the averaged one from the alloy components.

In this thesis, the EMTO-CPA has been applied to examine the equation of

state and elastic properties of Mg1−xFexO.

17

Chapter 3

Molecular Dynamics

he aim of Molecular Dynamics (MD) is to compute the motions of

particles in solids, liquids and gases, where the motion describes

how the position and velocities change with time. In MD, the laws

of classical mechanics are followed, with the well known New-

ton’s law

Fi = miai , (3.1)

for each atom i in a system of N atoms, where mi is the mass of the atom, ai its

acceleration and Fi is the force acting upon it. For each MD step, the atomic

accelerations are integrated into new positions which yield new forces, and so

on. Hence, the system is deterministic, in contrast to e.g. Monte Carlo meth-

ods. Given initial positions and velocities, one could think that the dynamics

are completely determined. This statement is however not true due to integra-

tion time steps and rounding errors, resulting in trajectories deviating from

the ’true’ ones. Moreover, as the forces Fi usually are obtained as the gradient

of a potential function, a relevant question arises how to find this potential,

depending on the positions of the particles. The realism of the simulation is

therefore dependent on the potential considering the conditions under which

the simulation is being run.

This section describes the basics of classical MD. Instead of using the inte-

gration of the electronic and ionic systems as shown in chapter 2, interactions

where electrons are not considered explicitly will be examined.

18 CHAPTER 3. MOLECULAR DYNAMICS

3.1 Potential design

The forces in MD are found from the negative gradient of the potential func-

tion V depending on the atomic coordinates as

Fi = −∇V (r1, ...,rN) . (3.2)

Searching for a potential function V , it must agree with the complex behav-

ior of the material considered, where the nuclei and the electrons determine

the properties. Finding the potential V (Ri) by means of Eq. (2.4) and Eq.

(2.5) is called first-principles or ab initio molecular dynamics. However, this

technique requires massive computer resources, as the computations can be

extremely time consuming. In this section, the focus is instead set on the se-

lection of appropriate potentials that will mimic the behavior of the "true"

systems in a realistic way. The first step is to select an analytical form of the

potential. The form was earlier an approximation to the molecular cluster as

a two-body term potential. Today, many-body potentials are being set, with

the aim to capture as much as possible of the authentic physics at the expense

of computer time. A typical form is constructed by many functions, e.g. geo-

metrical properties as angles between atoms. Furthermore, the parameters for

the potential chosen are to be set, which is very important for the discussed

resemblance to the true system.

The fitting of the potential parameters can be based on first-principles theory

or to experiments, giving priority to realism rather than to connections with

first-principles. The transferability indicates that the potential, generated for

a given atomic configuration, should generate others accurately. Having said

this, two environments differing vastly as e.g. a bulk metal and a diatomic

molecule can of course not be assumed to follow one common specific poten-

tial.

3.1.1 Two-body potentials

Dominating the bonding characters of closed shell systems such as rare gases,

the weak van der Waals forces could be explained by a simple model: consider

two atoms separated by the distance r. The dipole moment of the first atom, p1,

will yield an electric field proportional to p1/r3. The induced dipole moment

for the second atom will be p2, expressed as

p2 = αE ∼ αp1

r3 , (3.3)

where α is the polarizability. The multiplication of the two dipole moments

divided by r3 to yield the interaction energy gives

p1p2

r3 ∼ αp21

r6 . (3.4)

At short distances, however, the repulsion of the ion cores comes into play. A

strong repulsive term at small interatomic distances refer to the Pauli exclusion

3.1. POTENTIAL DESIGN 19

Figure 3.1: In [a], a Lennard-Jones potential as a function of interatomic radius. In

[b], there is a sample of 15 atoms, randomly distributed in a simulation box in 2 D. The

arrows indicate the direction and largeness of the forces at the instant configuration.

The forces are found from Eq. 3.2 with the L-J potential in Eq. 3.5. The interatomic

distances are found from the minimum image criterion, meaning that distances to

atoms from periodic multiplications also are taken into consideration. The value of σwas chosen such that there is both interatomic attraction and repulsion.

principle that two electrons cannot have the same quantum numbers, as the

electron clouds start to overlap at small r. As the power generally chosen is

(1/r)12, the most commonly used pairwise interaction is the Lennard-Jones

potential, given by the expression

ΦLJ(r) = 4ε [(σr

)12 − (σr)6] . (3.5)

Depending only on two parameters, ε for the depth of the potential well and σto determine the radius of minimum interaction, the function is one of the sim-

plest in MD. The potential shows a small attraction at long distances, where

the negative term (1/r)6 is dominant, and a strong repulsion at small distances

due to the term (1/r)12.


As shown in Eq. (3.5), the Lennard-Jones potential has an infinite range. Thus,

it is necessary to introduce a cut-off distance Rc to save computer time, where

Rc is the distance between two atoms. However, there will be a small "jump"

for the potential, which may interrupt with the simulation. Therefore, a small

shift is introduced to make the potential curve smooth at the cut-off distance,

i.e.

V (r) =

{ΦLJ(r)−ΦLJ(Rc) if r ≤ Rc

0 if r > Rc .(3.6)

Obviously, ΦLJ(Rc) should be close to zero, such that the overall potential is

shifted as little as possible.

3.1.2 Many-body potentials

To suit the behavior of metals, an approach using many-body theory is more

convenient than the two-body potential. The physical aspect of this is that

the bonds become weaker as the atomic density increases. For example,

Ec/kBTm, defined as the ratio between the cohesive energy and the Boltzmann

constant times the melting temperature, is about 10 for two-body systems but

about 30 for metals. Thus, metals have an ’extra cohesion’ with respect to

pairwise systems, which is less effective in comparison to two-body forces in

keeping the system crystalline. Furthermore, the ratio between the vacancy

formation energy and the cohesive energy, Ev/Ec is between 1/4 and 1/3 for

metals. For pairwise systems, if relaxations are neglected, the ratio is 1.

In the manuscripts XI and XII of this thesis, the Sutton-Chen method

[46],[102] was used to model Fe, and in manuscript IX, the model described

by Zhou [117] for Ni was used. For both potentials, the repulsive part is

treated as in the two-body approach, but the attractive part is a superposition

of contributions from the neighboring atoms as

V (r) =12

N

∑i�= j

Φ(ri j)+F(ρi) , (3.7)

where Φ(r) is the two-body part and F(ρ) is the superposition function of a

"generalized coordination", ρ . The coordination function is written as

ρi =N

∑i�= j

ρ(ri j) , (3.8)

where ρ(ri j) is a short-ranged, decreasing function of distance. The particular

form of the functions Φ, ρi and F(ρ) for the Sutton-Chen model are

Φ(ri j) = ε(ari j

)n ,

ρ(ri j) = (ari j

)m ,

3.1. POTENTIAL DESIGN 21

F(ρi) = −εCN

∑i=1

√ρi . (3.9)

For the Zhou model, the Φ form is

Φ(ri j) =Ae−α(ri j/re−1)

1+(ri j/re −κ)20 −Be−β(ri j/re−1)

1+(ri j/re −λ )20 , (3.10)

with

ρ(ri j) =fee−β(ri j/re−1)

1+(ri j/re −λ )20 , (3.11)

where A,B,α,β ,κ ,λ ,re and fe are adjustable parameters. F(ρ) is built as

sums at three different electron density ranges, ρn = 0.85ρe, ρ0 = 1.15ρe and

ρ0 ≤ ρ as

F(ρi) =3

∑i=0

Fni(ρρn

−1)i, ρ < ρn, ρn = 0.85ρe , (3.12)

F(ρi) =3

∑i=0

Fi(ρρe

−1)i, ρn ≤ ρ < ρ0, ρ0 = 1.15ρe ,

F(ρi) = Fe[1− ln(ρρe

)η ] · ( ρρe

)η , ρ0 ≤ ρ .

In the above equations, Fni,Fi and Fe are all adjustable parameters. The choices

of ρn = 0.85ρe and ρ0 = 1.15ρe where ρe is the equilibrium electron density

is motivated by the possibility of fitting all equilibrium properties in the range

ρn ≤ ρ < ρ0.


3.2 Integration algorithms

3.2.1 The Verlet algorithm

In MD, the Verlet algorithm, or variations of it, is the most widely used scheme

for integrating the newtonian equations of motion [52]. The positions r(t) are

evaluated by means of a Taylor expansion to the third order both backwards

and forwards, i.e.,

r(t +Δt) = r(t)+v(t)Δt +12

a(t)Δt2 +16

b(t)Δt3 +O(Δt4)

r(t −Δt) = r(t)−v(t)Δt +12

a(t)Δt2 − 16

b(t)Δt3 +O(Δt4) , (3.13)

where v(t), a(t) and b(t) are the first, second and third derivative of the posi-

tion with respect to the time t. If the two expressions are added, we get

r(t +Δt) = 2r(t)− r(t −Δt)+a(t)Δt2 +O(Δt4) , (3.14)

resulting in a third order algorithm, even though no third order derivatives are

present. The Verlet scheme offers simplicity and good stability for moderately

large time steps. However, variations might be convenient to also include ve-

locities. As previously described, the ionic acceleration in Eq. (3.14) is found

from

ai =Fi

mi=

−∇V(ri(t))mi

, (3.15)

where V(r(t)) is the potential. Although the velocities v(t) are not explicitly

needed in Eq. (3.14), their evaluation by e.g.

v(t) =r(t +Δt)− r(t −Δt)

2Δt, (3.16)

implies an error proportional to O(Δt2), which can be problematic when the

kinetic energy is computed. Variations of the Verlet algorithm, such as the

leap-frog algorithm (Eq. (3.21)) overcomes this velocity scaling problem.

3.3 Nosé-Hoover algorithm

To keep a constant pressure and temperature in the simulations, the Nosé-

Hoover thermostat and barostat can be used. The modified equations of motion

are [98]

Δr(t)Δt

= v(t)+η(r(t)−R0)

Δv(t)Δt

=F(t)m

− [χ(t)+η(t)]v(t)

Δχ(t)Δt

=1τ2

T(

TText

−1)

3.4. MELTING 23

Δη(t)Δt

=1

NkBTextτ2P

V (t)(P−Pext)

ΔV (t)Δt

= 3η(t)V (t) , (3.17)

where η and χ are the barostat and temperature friction coefficients, R0 is the

center of mass of the system, τP and τT time constants, T the temperature, Textthe desired temperature, P the instantaneous pressure, Pext the desired pres-

sure and V the system volume. The Gibbs free energy G of the system is the

conserved quantity

GNPT = GNV T +PextV (t)+3NkBText

2η(t)2τ2

P . (3.18)

The temperature T is updated from the kinetic energy of the atoms,

T =2Ekin

3NkB=

13NkB

N

∑i

p2i

m, (3.19)

whereas the pressure P is updated from the virial equation

P =2N3V

〈Ekin〉+ 13V

⟨ N

∑i

N

∑j>i

Fi j · ri j

⟩. (3.20)

From Eq. (3.17), the integration algorithm for χ ,η ,v and r with the leap-frogscheme is

χ(t +12

Δt) = χ(t − 12

Δt)+Δtτ2

T(

TText

−1)

χ(t) =12

[χ(t − 1

2Δt)+ χ(t +

12

Δt)]

η(t +12

Δt) = η(t − 12

Δt)+1

NkBTextτ2P

V (t)(P−Pext)Δt

η(t) =12

[η(t − 1

2Δt)+η(t +

12

Δt)]

v(t +12

Δt) = v(t − 12

Δt)+[F(t)

m− (χ(t)+η(t))v(t)

]Δt

v(t) =12

[v(t − 1

2Δt)+v(t +

12

Δt)]

r(t +Δt) = r(t)+Δt(

v(t +12

Δt)+η(t +12

Δt)[r(t +12

Δt)−R0])

r(t +12

Δt) =12

[r(t)+ r(t +Δt)

]. (3.21)

3.4 Melting

Although studied theoretically for almost a century, melting is still enigmatic.

Understanding how and why a crystalline solid melts is as important as what


determines the temperature at which this happens. There are numerous crite-

ria for determining the melting. However, the Lindemann criterion [73] from

1910 and the Born criterion [25] from 1939 are two of the most cited through-

out literature, presented later in this chapter.

The melting temperature Tm for a pressure P is defined by the condition where

the Gibbs free energy of the solid and liquid are equal, i.e.

Gsolid(P,Tm) = Gliquid(P,Tm) . (3.22)

The fact that the solid structure is more ordered than the liquid is captured in

the entropy. For all known materials (except He), the solid phase at the melting

point has lower entropy than the liquid phase,

ΔSm = R ln(

Wl

Ws

), (3.23)

where ΔSm is the entropy difference between the two phases, R is the gas

constant and Wl and Ws are all the possible configurations referring to the total

energy at any instant. Thus, Wl ≥ Ws, yielding a positive ΔSm which can be

interpreted as the liquid phase being more randomized.

In the manuscripts, the radial distribution function, the Lindemann criterion

and the mean square displacement have been used as configuration indicators

and are presented in the three following subsections.

3.4.1 One-phase and two-phase simulations

Melting can be divided into two subcategories: homogeneous, where the melt-

ing initiates in the bulk region, resulting in atomic displacements from the lat-

tice positions, and heterogeneous, where the atoms diffuse at the surfaces [32].

If periodic boundary conditions (PBC) are used, the surface effects are absent.

Therefore, a temperature high enough to force atomic displacements implies

homogeneous melting. However, the overheating to find the melting point can

be substantial compared to the ’true’ melting temperature, Tm. This is due to

hysteresis effects associated with the need for a initial liquid seed to appear.

A liquid configuration can start to grow only when this seed contains a few

atoms. Belonoshko et al. [17] have recently stated that the relation between

Tm and the superheating limit (Ts) is Ts/Tm = 1 + ln(2/3), i.e. a difference of

approximately 23 %. By placing a solid structure on top of a molten config-

uration, one avoids this hysteresis effects. An example of a two-phase setup

is shown in Fig. 6.5, where 864 atoms in a perfect fcc lattice were heated

to liquify. Then, they were cooled down and placed on top of another per-

fect 864-atom lattice. If the simulation is performed at a temperature which is

higher than Tm, the liquid structure starts to conquer the solid one. Per con-tra, if T < Tm the entire atomic configuration will start to solidify during the

simulation.

3.4. MELTING 25

Figure 3.2: A fcc crystal of 864 atoms (one-phase) is shown in 1a). A MD temper-

ature T below the limit of super heating Ts will not be sufficient to melt the crystal,

even though T > Tm, as shown in 1b). In 2a), a 864-atom crystal is put together with a

molten structure (two-phase), containing the same number of atoms. This system will

solidify if T < Tm (2b) whereas the phase will become liquid if T > Tm (2c). By nar-

rowing the temperature interval iteratively, the melting temperature can be estimated

at the specific pressure in the simulation.


3.4.2 Radial distribution function

There is a concern as to whether simulations that are performed near the melt-

ing curve imply solid or liquid structures. One of the most well known tools

for examining atomic configurations is the radial distribution function (RDF).

The function g(r) can be defined in the following way [52]:

g(r) =∑n

k=1 Nk(r,dr)n(1

2 N)ρV (r,dr), (3.24)

where n is the total number of time steps of the equations of motion, N(r,dr)the number of atoms found in a spherical shell at distance r and with thickness

dr, ρ the system density and V (r,dr) is the spherical shell volume.

For high symmetric crystal structures such as e.g. fcc, bcc and hcp, the g(r)forms a sequence of delta symbols, where a lattice can be distinguished from

its distribution function. For example, the fcc lattice has fewer atomic pairs

separated by a factor of√

2 but more separated by√

3 compared to the sim-

ple cubic lattice. To visualize the bcc/fcc discrepancy by means of the g(r),a perfect bcc lattice consisting of 1024 He atoms was created by multiply-

ing the bcc unit cell containing 2 atoms 8 times (2× 8× 8× 8) in each of

the three orthogonal directions. To resemble the density of the matter at the

simulation conditions, the density was set to ρ=6.44 Å3/atom. Three simula-

tions with constant pressure and constant temperature (NPT) were performed

at 16 GPa and 200 K, 300 K and at 425 K, respectively. The radial distribu-

tion function g(r) was evaluated for the three simulations after 20 000 time

steps, as shown in Fig. 3.3. The number of neighbors for the first shells are

12,6,24,12,24,8,48,6 for fcc and 8,6,12,24,8,6,24,24 for bcc. The first peak

for the fcc structure is clearly higher than that of the bcc structure, indicating

the number of atoms at the nearest neighbor distance. Due to few atoms in

the fifth and sixth shells for bcc, the g(r) shows a minimum at 4.6 Å, whereas

the fcc lattice has peaks at the third and fourth shells, manifesting in peaks at

3.6 and 4.25 Å. The molten configuration at 425 K is identified by a damped,

oscillating behavior. Although the g(r) can give a fingerprint of the structure,

at some conditions, it does not give the same clear distinction between the

phases as shown in Fig. 3.3.

3.4.3 Lindemann criterion

The first theory explaining bulk melting was presented by Lindemann [73],

using the vibrations of the atoms as a criterion. As the thermal vibrations in-

crease when the bulk temperature increases, the atoms start to invade the space

of their nearest neighbors. According to Lindemann, this disturbance yields

melting when the root mean square vibration amplitude√

< u2 > exceeds at

least a tenth of the nearest neighbor distance. The melting temperature can

then be written as

Tm = 4π2m < u2 > /kB , (3.25)

3.4. MELTING 27

Figure 3.3: The radial distribution function for three NPT simulations for He at 16

GPa, 200K; 16 GPa, 300 K and 16 GPa, 425 K. The arrows from below indicate the

peak positions in a perfect bcc lattice, whereas the arrows from above indicate perfect

fcc peaks. The correspondence between the arrows and the peaks from the simulations

clearly separates the structures: the low temperature simulation (200 K) results in a

fcc configuration, whereas a higher temperature (300 K) implies a solid-solid phase

transformation to the bcc structure before melting (425 K).


where m is the atomic mass. The vibrational based model only refers to simple

structures, i.e. closed packed configurations. More complex molecules exhibit

a vibrational complexity which rules out any simple rule of lattice stability,

determined by the vibrational amplitudes of the molecular centers of mass.

Furthermore, the model starts from the solid alone, although the melting tran-

sitions involve liquid and solid phases [18] as described in section 3.4.1.

3.4.4 Mean square displacement

The mean square displacement (MSD) is yet another melting criterion. For a

species of N particles, the MSD is calculated as [92]

< |r(t)− r(0)|2 >=1

NNt

N

∑n=1

Nt

∑t0|rn(t + t0)− r(t0)|2 , (3.26)

where r(t) is the atomic position at time t and r(0) is the initial atomic position.

The diffusion parameter D is calculated using the Einstein relation

< |r(t)− r(0)|2 >= 6Dt . (3.27)

For a solid, the MSD is almost constant in time, compared to an almost linear

behavior for a fluid. Thus, melting can be detected by D, showing a kink at

Tm. However, one must be careful when calculating the diffusion from MD

with PBC: as atoms close to the edges can leave the simulation box and enter

on the opposite site, a mathematically enhanced diffusion will be the result,

although this is not the physical case. This becomes obvious when looking at

the Na atoms in Fig. 3.4 [a]. So, to be on the safe side, one should just use

the atoms well inside the box, or shift back those atomic trajectories that show

characteristic PBC ” jumps”.

3.5 Structure factor

The comparison of experimental and theoretical results is highly important.

The g(r), as presented in section 3.4.2, can be converged to the more exper-

imentally related structure factor. In the early 20th century, W. H. and W. L.

Bragg found that crystalline substances gave characteristic patterns from X-

rays [11]. The intensity of a Bragg peak is proportional to |S|2, where the

structure factor S is defined as

S(G) = ∑j

f je−iG·r j . (3.28)

In the above equation, G is the wave vector change k−k′and r j is the vector

to the center of the atom j. The atomic form factor f j is defined as

f j =

∫dV n j(ρ)e−iG·ρ , (3.29)

3.5. STRUCTURE FACTOR 29

Figure 3.4: Atomic displacement as a function of time and space for Na at approx-

imately 14 GPa. Each dot represents the x and y coordinate in the simulation box,

where the choice of a 2-dimensional presentation is due to enhanced visualization. As

a dot is shown every 5th time step, the time difference between two adjacent dots is 5

fs. From left to right, the temperature is 800 K (a), 900 K (b) and 1000 K (c), keeping

the pressure almost constant. As the atoms are relatively fixed in their position in (a),

the atoms start to leave their atomic sites to diffuse as seen in (b) and (c).

where n j(ρ) is the electron concentration. Assuming that r and G make the

angle α , Eq. 3.29 becomes

f j = 4π∫

r2 d(cosα)

2n j(r) · sinGr

Grdr . (3.30)

Inserting Eq. 3.30 into Eq. 3.28, one obtains

S(G) = ∑j[4π

∫r2 d(cosα)

2n j(r) · sinGr

Grdr]e−iG·r j . (3.31)

Now, the structure factor can be re-written in terms of the g(r) from Eq. 3.24

and the particle density ρ as

S(q) = 4πρ∫ ∞

0[g(r)−1]

sin(qr)qr

r2dr . (3.32)

31

Chapter 4

Ab initio or classical moleculardynamics?

he development of first-principles (ab initio) methods have

been reported in numerous articles and books, and the ongoing

progress of electronic structure calculations is highly important

in several scientific fields. The improvement of numerical

algorithms in combination with an increasing performance of computer

systems continue to cut simulation times. However, ab initio simulations are

still limited to operate only on relatively small systems.

Representing the other extreme of the spectra, classical MD is relatively fast.

As the electronic effects are incorporated in the interatomic potentials, and

thus no electrons are treated explicitly in the calculations, the simulations can

sometimes treat millions of atoms.

Table 4.1 shows the CPU times for the first 10 steps in a typical molecular

dynamics calculation. The ab initio molecular dynamics (AIMD) results

are shown as a function of both the number of atoms and processors used.

To compare, a calculation on a single processor with classical dynamics is

shown. Although there are O(N) algorithms, classical MD calculations often

scale O(N2) as every atom interacts pairwise with the atoms found inside the

cutoff volume. On the other hand, the ab initio calculations scale O(N 3).Therefore, when comparing calculation times for similar system sizes, the

AIMD yields significantly longer times than the classical approach. To

shorten the simulation times, parallel calculations, where the scale is O(P)with P as the number of processors, are sometimes essential.

Certainly, when any material property cannot be truly reproduced without the

explicit presence of electrons, the classical approach is not sufficient. As an

example, the Ne metalization is very different compared to the heavier gases

Ar, Kr and Xe as the energy band gap between the valence 2p states and the

unoccupied 3d states is large. For argon, krypton and xenon however, the pstates are very well hybridized with the corresponding d states. Therefore,

32CHAPTER 4. AB INITIO OR CLASSICAL MOLECULAR DYNAMICS?

the comparison of these gases at extreme conditions when studied with abinitio or classical methods is highly interesting.

4.1 The ab initio and classical combination

By fitting the results of a first principles calculation to a model, a classical MD

simulation can be performed to operate on a big system, combining the advan-

tageous aspects of the first-principles and the classical methods. If the model

allows a precise energy calculation to be made, in agreement with first princi-

ples calculations, the two techniques are equivalent. As reliable ab initio data

is available today for many systems, MD models can be developed to authenti-

cally simulate atomic configurations without the need to refer to experimental

results. The parameters for the embedded atom Sutton-Chen model as used

for Fe (section 3.1.2) were found by such a fitting. Furthermore, other studies

regarding e.g. two-phase melting [31] and thermodynamic integration [101]

have also combined the promising aspects of classical and ab initio MD. In

the former study by Cazorla et al. [31], the energies from AIMD calculations

on the solid and the liquid were matched to a classical reference function.

In this way, a large system of coexisting solid and liquid could be calculated

by means of the classical model. Furthermore, differences in the free energy

between the AIMD and the classical results were used to correct the melting

curve.

Sugino and Car [101] used a similar technique when studying the melting of

Table 4.1: CPU times in seconds for the first 10 steps in an arbitrary MD calculation as

a function of method, processors and atoms. The classical MD times agree perfectly

with the O(N) scaling of the code. The ab initio calculations should scale as O(N 3)and inversely to O(P), but show some irregularities.

Atoms

Method Proc. 2 4 16 32 54 108

Ab initio MD 2 3.0 9.3 160 699 2417 3167

4 2.1 5.5 84 346 1233 2106

8 1.8 3.6 64 263 894 1197

12 1.5 3.6 38 159 507 745

16 1.8 3.9 28 126 373 495

Atoms

6912 16384 32000 108000 186624 500000

Classical MD 1 1.1 2.9 5.6 21 37 102

4.1. THE AB INITIO AND CLASSICAL COMBINATION 33

Si by thermal integration. The chemical potential difference between the DFT

and the classical system, Δμ , can be calculated along a thermodynamic path

λ as

Δμ =

∫ 1

0

∂∂λ

∂λ =

∫ 1

0〈H1 −H0〉λ ∂λ , (4.1)

where H0 and H1 are the classical and DFT Hamiltonians and the bracketed ex-

pression is the temporal average from the Hamiltonian H(λ ) = λ (H1 −H0)+H0 along the λ -trajectory. In this way, short MD runs can be used to calculate

Δμ .

35

Chapter 5

Equation of state andcompression

ound in any basic physics book, the pressure is found as the force

F acting on a surface A. On Earth, the pressure is 1 atmosphere

which is approximately 100 kPa, whereas the pressure at the center

of the Earth is astonishingly ∼3500000 times higher, i.e. 350 GPa.

As more than 90% of the matter in the solar system exists at P>100 GPa, it is

easy to understand that pressures determine how stars and planets are formed.

5.1 Experimental techniques

The theoretical results presented in this thesis are often compared to experi-

mental findings. Therefore, it is important to give an insight in the experimen-

tal setups and how the experiments are performed. Static loading (diamond

anvil cells (DAC), multi-anvil cells) and shock loading (gas guns) are the two

main techniques to generate ultra-high pressures. In the DAC, pressure is ob-

tained by applying a load to the back of two opposing diamonds, as shown

at the top in Fig. 5.1. As the areas of the diamonds usually are less than 0.20

mm2, the pressure applied to the back becomes significantly magnified. The

gasket, constructed from a hard material foil with a hole, prevents the sam-

ple from extruding. The volume at high pressure, usually in the order of 10−3

mm3, is compressed uniaxially by the diamonds. A pressure medium (alco-

hol at lower P/T ratios, high density gas1 at higher P/T) surrounds the sample

to ensure hydrostatic pressure. Alongside the sample is another sample with

well known high-pressure properties to serve as a calibration. In the DAC,

pressures in the range of the Earth’s inner core, i.e. P>300 GPa, can be ob-

tained [80, 74].

1Noble gases such as He and Ne are often used for this purpose. The high PT results of these

gases are shown in chapter 7.

36 CHAPTER 5. EQUATION OF STATE AND COMPRESSION

For temperatures below 1000 K, one can heat the DAC by placing a resistive

furnace around the diamonds. The temperature is then measured using ther-

mocouples close to the diamond tips. To reach higher temperatures up to 4000

K, the sample can be heated by lasers. The rays pass through the diamonds

and are absorbed by the sample with the benefit that undue heating of the dia-

monds can be avoided.

The multi-anvil cells, as shown in the middle of Fig. 5.1, have a lower pressure

range (P<25 GPa) compared to the DAC, but can on the other hand contain

bigger sample sizes in the order of cm3. For the heating, an electric current

passes through a furnace within the assembly to generate temperatures up to

2000 K.

In the case of gas-guns, pressures in a TPa range can be reached, although the

duration is on a millisecond scale. The setup can be seen at the bottom of Fig.

5.1. By igniting the charge, the first-stage piston compresses the H2 gas, until

the rupture of the disc. This accelerates the projectile and it hits the sample at

several km/s. The result is a high-pressure shock wave, and chapter 8 presents

more on this topic.

5.2 Equation of state

An equation of state (EOS) is a formula describing the connection between

various properties of a system as e.g. pressure, temperature, volume and en-

ergy. In this thesis, properties such as the energy as a function of volume

E(V ), equilibrium volume V0 the bulk modulus B and its derivative B′

have

been important for analyzing the matters considered.

5.2.1 Murnaghan EOS

The pressure P is defined as the negative derivative of the energy E with re-

spect to the volume V , i.e.

P = −(δE

δV

)T

. (5.1)

Furthermore, the bulk modulus B and its pressure derivative B′ are

B = −V( δP

δV

)T

(5.2)

and

B′=

(δBδP

)T

, (5.3)

respectively. As the pressure impact on B′

is small, the integration of Eq. 5.2

with B′ � B

′0 yields

B = B0 +B′0P . (5.4)

5.2. EQUATION OF STATE 37

Figure 5.1: In the DAC (top), pressure is applied to the back of the diamonds by

tightening the base plate. The high magnification of the pressure is due to the small

area of the diamond tips pressing on the sample. To obtain a hydrostatic pressure,

a medium surrounds the sample and the pressure calibration sample. The MAX80

is taken as an example for the multi-anvil cell apparatus (middle), where six tungsten

carbide anvils can compress samples of 0.5 cm3. At the bottom, the double-stage light

gas-gun is shown, where the compressed H2 gas shoots the projectile along the barrel

and into the sample.

38 CHAPTER 5. EQUATION OF STATE AND COMPRESSION

Using Eqs. 5.2 and 5.4 gives

δVV

= − δPB0 +B′

0P. (5.5)

Integrating Eq. 5.5, the pressure is found as

P(V ) =B0

B′0

((V0

V

)B′0 −1

). (5.6)

The integration of Eq. 5.7 with respect to the volume yields the Murnaghan

EOS for the energy as a function of volume as

E(V ) = E0 +B0VB′

0

( (V0V )B

′0

B′0 −1

+1)− B0V0

B′0 −1

. (5.7)

5.2.2 Birch-Murnaghan EOS

Introduced by Birch [22, 23], the third-order EOS have the following bulk

modulus B, pressure P and energy E:

B =32

B0

[73

x73 − 5

3x

53

][1+ χ(x

23 −1)

]+

32

B0

[x

73 − x

53

][1+

23

χx23

], (5.8)

P =32

B0

[x

73 − x

53

][1+ χ(x

23 −1)

], (5.9)

and

E = E0 +32

B0V0

[32(χ −1)x

23 +

34(1−2χ)x

43 +

12

χ x63 − 2χ −3

4

], (5.10)

where x=V0/V and χ= 34(B

′0 −4) have been added for convenience.

39

Chapter 6

Elasticity and hardness

y applying a force to any solid body, there is a deformation (per-

haps the most intuitive example is that of a spring). One can define

the hardness as the resistance of the material to such a deforma-

tion. In the atomistic scale, the elastic constants correspond to the

rigidity of a crystal. The complexity of the crystal determines the number of

elastic constants associated with that lattice. For example, a high symmetry

configuration like the body-centered cubic lattice (bcc) has three independent

constants, whereas the hexagonal phase has five.

Just as the force of a spring obeys Hooke’s law, the deformation can be ex-

pressed as the energy gained by applying the pressure from the strains. If a

Taylor expansion of the energy E(V,η) is performed, where V is the volume

and η a matrix connected to a distortion matrix,

E(V,η) = E(V0,0)+V0

(∑i j

τi jηi j +12 ∑

i jklci jklηi jηkl

)+O(η3) , (6.1)

with V0 as the equilibrium volume and τi j being elements in the stress tensor,

the elastic constants Ci jkl can be found. The eulerian distortion matrix e is

given as

e =

⎛⎜⎝

αxx αxy αxz

αyx αyy αyz

αzx αzy αzz

⎞⎟⎠ . (6.2)

To connect η and e, the points R1 and R2 can be defined, separated by R in

space as shown in Fig. 6.1. Assume that R1 and R2 are almost at the same

point, just slightly altered. Then, the relation

R2 = R1 +R , (6.3)

will, after small distortions of R1 and R2 yield

dR2 = dR1 +dR (6.4)

40 CHAPTER 6. ELASTICITY AND HARDNESS

R

R

R12

dR

dRdR12

Figure 6.1: Schematic figure of two points R1 and R2 in space with the separation R.

or

dR2 = dR1 +dRdR1

dR1 = dR1(1+∇R) . (6.5)

Assuming ∇R to be constant, one can relabel it into the distortion matrix e in

Eq. 6.2. Furthermore, one can evaluate the scalar product difference between

R2 and R1 as

dR22 −dR2

1 = dR21(2e+ e2) = 2dR2

1η . (6.6)

Now, η is defined as the Lagrangian distortion matrix. Thus, the relation be-

tween the Eulerian matrix in Eq. 6.2 and the Lagrangian matrix in Eq. 6.6

is

η = e+ e12

e . (6.7)

For both matrices, it is trivial to see that in the case of no strain, η = e = 0.

Furthermore, for magnitudes of the strains in e on the percentage scale, η ∼ e.

Due to the symmetry of the lattice, the use of the Voigt notation simplifies

the representation. The elements xx,yy,zz,yz,xz, and xy are replaced by the

integers 1,2, ..,6. The Taylor expansion from Eq. 6.1 can therefore be written

E(V,η) = E(V0,0)+V0

(∑

iτiηi +

12 ∑

i jci jηiη jξiξ j

), (6.8)

where i runs from 1,2, ...,6. As e.g. xz = zx for symmetric reasons, the inserted

factor ξ = 1 for the diagonal elements i = 1,2 or 3 and ξ = 2 otherwise. As the

Voigt notation run from 1,2, ..,6, the 6×6 elastic constant matrix C is written

41

-0.05 -0.025 0 0.025 0.05Distortion (e)

En

erg

y (

arb

. u

nit

s)

0 GPa10 GPa20 GPa30 GPa40 GPa50 GPa60 GPa70 GPa

Figure 6.2: Energies (arb. units) for the high-pressure cotunnite phase of TiO2 as a

function of distortion. The increase of strain and the pressure results in an increase

in energy, where the curves have been shifted for enhanced readability. By fitting the

curvatures to a polynomial as a function of strain e, the shear moduli can be evaluated.

as

C =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

c11 c12 c13 c14 c15 c16

c21 c22 c23 c24 c25 c26

c31 c32 c33 c34 c35 c36

c41 c42 c43 c44 c45 c46

c51 c52 c53 c54 c55 c56

c61 c62 c63 c64 c65 c66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (6.9)

In this thesis, the most frequently occurring moduli are c11, c12, c44 and

c′=(c11-c12)/2. To find these constants, distortions can be applied according to

the rule

a′ = [I+ ε(e)]a , (6.10)

where I is the 3×3 identity matrix, a (a′) are the undistorted (distorted) lattice

vectors, and ε(e) is the strain component matrix. For volume conserving

deformations, det ε(e) = 0, and the first order terms in Eq. (6.8) disappear.

To find the modulus c′, the orthorhombic deformation


Figure 6.3: Shown to the left is an undistorted simple cubic lattice, where the ho-

mogeneous compression (expansion) are shown as red arrows in the underlying xyz-

diagram. In the middle, the tilted x and y axes from Eq. 6.12 yield the deformation

to calculate the c44 constant. Analogously for the rightmost images, the deformation

gives c′

from Eq. 6.11.

εo(e) =

⎛⎜⎝

e 0 0

0 −e 0

0 0 e2

1−e2

⎞⎟⎠ , (6.11)

can be applied. Analogously, a monoclinic deformation

εm(e) =

⎛⎜⎝

0 e/2 0

e/2 0 0

0 0 e2

1−e2

⎞⎟⎠ , (6.12)

can yield c44. The Swedish billiard balls in Fig. 6.3 show the strains for c44(middle) and c

′(right).

6.1 TiO2

The search for hard materials is a highly studied area, and involves researchers

from several fields. One of the aims is to replace expensive diamond tips in

tools used for cutting and drilling of e.g. steel and rock. Zirconium diox-

ide (ZrO2) is one of the most well known diamond substitutes, and has a

reported bulk modulus of 444 GPa. However, recent studies have shown a

similar B=431 GPa for the high-pressure cotunnite phase of titanium diox-

ide (TiO2,titania), and as TiO2 is 35 % lighter than ZrO2, a further research is

6.1. TIO2 43

Figure 6.4: Schematic figure of the dye-sentizised solar cell (DSSC). When the dye

(S) absorbs light (hν), it is promoted into an electronically excited state (S�). The

electron released is adsorbed into the semiconductor TiO2 film (1), and is transported

by diffusion through the film to the anode, and the circuit (2). The dye cation (S+)

reacts with a reduced species of a redox couple, usually I−/I−3 (3). The carrier of the

positive charge, I−3 , releases the charge at the Pt cathode, and reduces back to I− (4).

The process is regenerative, i.e. there is no net change in the composition of the cell.

highly motivated. The possibility to quench this high-pressure phase could im-

ply a substitution of ZrO2 as artificial diamond. Furthermore, TiO2 has several

important applications in industry: dye-sensitized solar cells (Fig. 6.4) benefit

from the oxide’s corrosion resistance and low bandgap and dynamic random

access memories improve with the use of TiO2 compared to SiO2 due to high

dielectric constants.

6.1.1 EOS, elastic constants and bulk moduli

Total energy calculations were performed in the framework of the density

functional theory (DFT) as it is implemented in the QuantumESPRESSO code

[14] in conjunction with the plane-wave (PW) basis set and ultrasoft pseu-

dopotentials. By relaxing the TiO2 rutile structure (Fig. 6.5 a) at ambient pres-

sure, the parameters were found to be a=4.68 Åand c=3.01 Å . For pyrite and

fluorite (Fig. 6.5 b and c) the parameters were found to be 4.94 and 4.88 Å,

respectively, whereas for cotunnite (d), a=5.46, b=3.16 and c=6.30 Å. The

comparison of theory and experiment is shown in Table 6.1. The calculations

for rutile overestimate the lattice parameter a by 2.0 % and c by 1.7 % com-

pared to experimental data. This is consistent with the findings of Muscat etal. [86] who have reported the trend of GGA overestimating both a and c.

Furthermore, this deviation from the equilibrium volume with the exchange-

correlation functional has been observed for two other dioxides, namely SiO2and ZrO2. It is also worth noting that the c/a ratio from the calculations in this


Figure 6.5: The phases of TiO2 studied here: a) rutile, b) pyrite, c) fluorite and d)

cotunnite. The big, red spheres represent the O atoms, and the small, blue spheres the

Ti atoms.

6.1. TIO2 45

Table 6.1: Structural parameters of rutile, fluorite, pyrite and cotunnite TiO2 at 0 GPa(unless specified). Lengths are in Å and volumes in Å3 for two TiO2 formula units. Forrutile, a=b and for fluorite and pyrite, a=b=c.

Rutile a b c VolumePBE (this work) 4.681 3.005 65.855

PW91 [86, 83, 72] 4.624-4.690 2.981-2.992 63.821-65.768Exp. [27, 56] 4.587-4.594 2.954-2.959 62.154-62.435

FluoritePBE (this work) 4.882 58.220

PBE [104] 4.833 56.375PW91 [86] 4.897 58.706

B3LYP [104] 4.824 56.065Exp. [81] 4.870 57.750

PyritePBE (this work) 4.942 60.340

PBE [104] 4.911 59.310PW91 [86] 4.894 58.592

B3LYP [104] 4.893 58.630

CotunnitePBE (this work, 0 GPa) 5.456 3.158 6.303 54.303PBE (this work, 60 GPa) 5.187 3.003 5.994 46.683

Exp. [41] (61 GPa) 5.163 2.989 5.966 46.266

work, 0.642, is in perfect agreement with the experimentally found ratio 0.644

from both Burdett et al. [27] and Isaak et al. [56]

For fluorite, the parameters and volumes calculated in this work agree with

the cited GGA data, whereas a slight overestimation is seen compared to LDA

data and Hartree-Fock theory. For pyrite, the same trend is shown as for the

fluorite calculations. The compression of the cotunnite structure at ambient

conditions to 60 GPa indicates an almost perfect match, as the deviation from

experiment is less than 1%.

The c44 calculations as a function of pressure up to 70 GPa for the pyrite,

fluorite and cotunnite phases are shown in Fig. 6.6. The cotunnite structure

indicates the highly interesting results of a very high rigidity compared to the

cubic types. The calculated bulk modulus B0 and its derivative B′

from Eq.

5.8 for the rutile phase shown in Table 6.2 is in good agreement with both

theory and experiment. Furthermore, the inequality [15] 13 (c12 +2c13) < B0 <

13 (2c11 + c33) for the bulk modulus is also fulfilled for mechanical stability.

For the cubic forms pyrite and fluorite, the calculated B0 data are somewhat

low compared to LCAO-HF calculations. Although the bulk modulus for the

fluorite phase is overestimated compared to experiment [81], it is well be-

low the the remarkably high B0=395 GPa as reported recently by Swamy and


Table 6.2: Bulk properties (in GPa) of the rutile, pyrite, fluorite and cotunnite phasesof TiO2 at 0 GPa.

Method Rutile Pyrite Fluorite CotunnitePBE (this work) B0 200 239 246 272 (427±7)

B′

5.75 4.19 4.41 4.09PBE [104] B0 215±1 220±4 395±4

B′

5.35±0.16 4.86±0.11 1.75±0.05B3LYP [104] B0 224±8 258±2 390±4

B′

5.64±0.90 4.35±0.04 2.06±0.06LCAO-HF[86, 93, 94] B0 239-304 318±10 331±10 380±10

LCAO-LDA[86, 93, 85] B0 209-264PW-LDA[83, 35, 50] B0 240-244 282-287

Exp.[41, 81, 84, 49, 56] B0 211-230 202±5 431±10B′

6.76 1.3±0.1 1.35±0.1

Muddle [104]. For cotunnite, Eq. 5.8 yields B0=272 GPa with B′=4.09. How-

ever, using the experimentally found B′=1.35 ± 0.1 [41], the bulk modulus

would become B0=427± 7 GPa. Dubrovinsky et al. [41] predict cotunnite to

be the most stable phase at pressures above 70 GPa, having a lower Gibbs free

energy than the OI (space group Pbca) and MI (P21/c) phases. Furthermore,

the authors have reported the possibility of preserving the cotunnite type at

ambient pressure by cryogenic quenching.

6.1.2 Phase stabilities

Although Fig. 6.6 fulfills one of the criteria for mechanical stability (c44>0)

for fluorite, pyrite and cotunnite, there are still possible dynamical instabili-

ties. From phonon density of states (PDOS) calculations with an ab initio force

constant method, the pyrite phase shows broad amplitudes in the negative fre-

quency regime (Fig. 6.7 a), indicating instability for all pressures studied. The

∼-5 THz peak, shifting to lower frequencies at higher pressures, corresponds

mostly to Ti vibrations. Similarly, the oxygen peak moves to more negative

frequencies which means that the Ti-O coupling also contributes to the pyrite

instability. For fluorite, the ∼-4 THz peak at ambient pressure (Fig. 6.7 b) de-

creases rapidly, and disappears completely at 55 GPa. This indicates that the

fluorite phase is dynamically stable at high pressure. For cotunnite, (Fig. 6.8)

the phase appears to be stable at P>8 GPa. This suggests that even though

rutile or anatase might have lower static total energies at ambient conditions,

the cotunnite phase could still be quenched down to relatively low pressures.

6.1. TIO2 47

0 10 20 30 40 50 60 70Pressure (GPa)

0

50

100

150

200

250

300

350

400

c 44 (

GP

a)CotunnitePyrite

Fluorite

Figure 6.6: The elastic constant c44 as a function of pressure for the pyrite, fluorite

and cotunnite phases of TiO2.

Figure 6.7: PDOS of the TiO2 pyrite (a) and fluorite (b) phases as a function of fre-

quency and pressure.


Figure 6.8: PDOS of the cotunnite TiO2 phase as a function of frequency and pressure.

6.2 Group-V and VI metals

The superconducting properties of the group-V elements vanadium (V), nio-

bium (Nb) and tantalum (Ta) are highly interesting as the metals have a rel-

atively high transition temperature (Tc). Furthermore, molybdenum (Mo) and

tungsten (W), neighbors of Nb and Ta in the periodic table, are often used

as equation of state (EOS) reference materials at high pressure and/or at high

temperature. The reasons for the use of these metals have been their very high

melting points at ambient pressure in combination with the stability of the bcc

structure up to extremely high pressures. However, newly found experimental

results for V which question the stability of the bcc phase have renewed the

interest in the vanadium and chromium group metals.

As a starting point, the EOS for the metals were examined. In Table 6.3, the

VASP calculations show a slight underestimation of 3 percent for the equilib-

rium volume V0, whereas for the other studied metals Nb, Ta, Mo and W, V0is consequently somewhat overestimated, with, at most, the same percentage.

The bulk modulus and its derivative, B and B′, are relatively well reproduced

for all metals.

6.2. GROUP-V AND VI METALS 49

Table 6.3: Properties of bcc V, Nb,Ta, Mo and W compared to experiment. Volumesare in Å3/atom and bulk moduli in GPa.

Reference V0 B′

B0

V GGA,PAW (this study) 13.49 3.75 182Exp. [36] 13.905 3.5(2) 195(3)

Exp. [107] 3.5(5) 162(5)Nb GGA,PAW (this study) 18.32 3.85 174

Exp.[37] 17.98Exp. [66] 17.98 3.4(3) 168(4)

Ta (this study) 18.11 3.67 211Exp. [34] 18.04 3.52 194

Mo (this study) 16.01 4.22 254Exp. [64, 37] 15.58 4.5 261

W (this study) 16.13 3.89 329Exp. [34] 15.86 4.3 296

From the elastic constants c11, c12 and c44 in Fig. 6.9, the group-VI elements

Mo and W show a steady, increasing behavior as a function of pressure. On

the other hand, the results for the group-V elements V, Nb and Ta, are much

more irregular which is a highly interesting phenomenon. To explain this, the

Fermi surface sheets of the elements were analyzed. The Brillouin zone cross

sections of the Fermi surfaces of V in the central {100} and {110} planes as

a function of compression are shown in Fig. 6.10 for the lattice parameters 1,

0.95, 0.90 and 0.85 a0, respectively. The hole-pocket-shaped second band

around the Γ point shrinks with pressure, indicating that the motion of this

point is towards the Fermi level. The nesting vector q relation to the elastic

constant in the long wavelength limit is

c44 = ω2(q)ρ/q2 , (6.13)

where ω is the phonon frequency and ρ the density. Kohn anomalies, showing

infinities in δω/δq for the transverse acoustic modes, have been found for the

vanadium group metals [87, 46]. More recently, Suzuki and Otani [103] found

this anomalous behavior of phonon frequency softening at higher pressure for

V, and at P >∼ 130 GPa, the frequencies became imaginary.

Shown in Fig. 6.11, the length of the inserted nesting vector q for V, Nb and

Ta decreases with pressure, and disappears at 247, 74 and 275 GPa, respec-

tively. The shrinking nesting vectors are therefore a possible reason for the

c44 irregularities seen for V, Nb, and Ta. Mo and W, on the other hand, show

vector magnitudes close to the ambient one throughout the entire studied pres-

sure range. Furthermore, at the pressures where q=0, the c44:s for the group-Velements start to show a more steady behavior, and increase monotonically

(Δc44/ΔP > 0) as shown in Fig. 6.9. From Fermi surface calculations for V,


Figure 6.9: The elastic constants c11 (upper left panel), c12 (upper right) and c44(lower panel) for V, Nb, Mo, Ta and W as a function of pressure.

Figure 6.10: Fermi surface cross sections for V for the lattice parameters 1, 0.95, 0.90

and 0.85a0 corresponding to the pressures 0, 37, 104 and 217 GPa.

6.2. GROUP-V AND VI METALS 51

Figure 6.11: The decreasing nesting vector q for V, Nb, Mo, Ta and W as a function

of pressure. In the long wavelength limit, c44 = ω2(q)ρ/q2, and the elastic constant

weakens as ω2(q) decreases faster than q2.

the energy EF is close to a critical point Ec, showing a peak in the electronic

density of states. At this energy, the Fermi surface undergoes a topological

transition shaped as a saddle point. In Fig. 6.12 (a), the Fermi surface and its

cut are shown for ε(k) = EF > Ec. As the Fermi energy is shifted past the

critical point Ec = EF − 46 meV, a neck is developed along the Γ−N sym-

metry direction connecting the inner sheet around the Γ-point with the disc

shaped sheet around the N-point. This neck is shown in Fig. 6.12 (b), where

the Fermi level has been shifted 60 meV. The electronic topological transition

(ETT) contribution to the elastic constants c∗i j can, to the leading order in E,

be calculated as

c∗i j =1V

∂ 2Eband

∂εi∂ε j≈

≈ 14π2V |E∗|3/2

[− EF

2E−1/2

]∂ E∂εi

∂ E∂ε j

(6.14)

where εi are strain components and V is the volume. The equation clearly

shows the effect when E = EF − Ec is very small: E−1/2 increases rapidly

which means that the first term in the bracketed expression decreases due to

the minus sign, lowering the c∗i j. Therefore, the underestimation of the c44 for

V and Nb at ambient conditions might be due to the calculations sensitivity to

the fine details of the Fermi surface around the ETT. Furthermore, the Fermi

level EF moves away monotonically from the critical point Ec with pressure.

As the ETT effect could be significant at lower pressures, the contribution

should, on the other hand, be small at high P.


Figure 6.12: In (a), the Fermi surface and its cut for vanadium are shown for ε(k) =EF > Ec, whereas in (b), the Fermi surface and its cut is shown for ε(k) = EF −0.06eV < Ec. Here Ec is the critical energy of an electronic topological transition in

which a neck develops along the Γ−N symmetry direction.

For the vanadium neighbor chromium, with two extra d band electrons com-

pared to V, the c44 constant is stiffer at ambient conditions (100 GPa for Cr

whereas the experimental data for V show ∼ 43 GPa). This is also consistent

with the finding of this work, where the extra d electron from Nb to Mo and

Ta to W manifests in a much higher c44, as shown in Fig. 6.9. These results

show that the elastic constant softening in V, Nb and Ta arises from the Fermi

surface nesting and the electronic topological transition.

53

Chapter 7

Melting at high pressure

n this chapter, the MD melting of the noble gases He and Ne and

the alkali metal Na is presented. The results are taken from the

manuscripts I, V, VI, and VIII.

7.1 Introduction

Although rare in the Earth’s atmosphere, helium and hydrogen constitute 95

% of the matter in the solar system [95] and the planets Jupiter and Saturn

are believed to consist mainly of these two substances [115]. Furthermore,

considering that neon is the fifth most abundant chemical element in the

universe by mass, the properties of He and Ne are highly interesting at high

pressures.

Just as the stability of He and Ne makes experimental studies relatively

straightforward, the alkali metal sodium (Na) has often been regarded as a

’simple’ metal due to its electronic structure. However, recent experimental

findings for Na have reported an extraordinary big melting temperature drop

of 700 K between 31 and 120 GPa. Therefore, the high-pressure properties of

Na seem far more complex than previously expected.

7.2 Interatomic potentials

Two interatomic potentials and parameter settings for 4He were considered:

the Buckingham form from Ross and Young [95] and the Aziz [12] potential.

The Buckingham potential is defined as

V (r) = − Ar6 +Be−Cr . (7.1)

The adjustable parameters A,B and C were fitted to equation of state data of

high pressure liquid and solid helium [115], using full-potential linear muffin-

54 CHAPTER 7. MELTING AT HIGH PRESSURE

tin orbital (FPLMTO) electronic structure calculations. The Aziz potential is

based on a combination of ab initio calculations of the self-consistent-field

Hartree-Fock repulsion between closed shell systems, an empirical estimate of

the correlation energy and semiempirically determined dispersion coefficients

C6, C8 and C10. It has the form

V ∗(x) = Ae−αx − (C6

x6 +C8

x8 +C10

x10 )F(x) , (7.2)

where

F(x) =

{e(−D

x −1)2for x < D

1 for x ≥ D

and x = r/rm. For Ne, the Lennard-Jones potential shown in Eq. (3.5) was

used, and table 7.1 shows the examined potential parameters for He and Ne.

Table 7.1: The Buckingham [115] and Aziz [12] potential parameters for He and theL-J parameters [16] for Ne.

He Buckingham A B C

113.09 37101 4.4148

Aziz A α C6 C8 C10 D ε /k (K) rm (Å)

0.5449e6 13.353 1.373 0.425 0.178 1.241 10.8 2.9673

Ne L− J ε /k (K) σ35.1 2.72

7.3 Melting

7.3.1 He

Both for He and Ne, the PT melting properties were found from MD calcula-

tions based on the two-phase coexistence method (section 3.4.1). Comparing

the Aziz and Buckingham melting curves for He with the data of Ross and

Young, [95] Loubeyre and Hansen [75] and Mao et al., [79] the Aziz poten-

tial shows a better agreement to the referred data, shown in Fig. 7.1 (a). At

the same pressure, the Buckingham potential melts at a lower temperature

than the Aziz due to its softer potential shape. In Fig. 7.1 (b), the melting and

phase diagram with the Buckingham potential is shown. For low pressures,

an increase in temperature implies a transition from the fcc crystal to a liq-

uid structure. However, for higher pressures, the fcc phase transforms into the

bcc structure before reaching the liquid phase at higher temperatures, a result

which was detected from Fig. 3.3. Although starting from a setup with the

fcc and liquid configurations, the bcc structure is more favorable than any of

7.3. MELTING 55

0 5 10 15 20 25Pressure (GPa)

0

100

200

300

400

500

Tem

per

ature

(K

)

Aziz (this work)Buckingham (this work)

Aziz (Loubeyre and Hansen)

Buckingham (Ross and Young)

Exp. (Mao et al.)

(a)

10 15 20 25 30 35 40Pressure (GPa)

200

250

300

350

400

Tem

per

ature

(K

) Liquid Bcc Fcc

(b)

Figure 7.1: In (a), the two-phase He melting with the Buckingham [95] and Aziz [12]

potentials (this work), and the melting curves by Ross and Young [95], Loubeyre and

Hansen [75] and Mao et al. [79] are shown. In (b), the phase boundaries for He as a

result of two-phase simulations with the Buckingham potential [95] are shown. For

lower pressures, the melting implies a transition from the fcc crystal to a liquid struc-

ture. For higher pressures, the fcc transforms into the bcc structure before reaching

the liquid phase at higher temperatures. The bcc structure was sporadically observed

at a pressure as low as 12 GPa. However, as the gap between the fcc and the liquid

in the pressure and temperature range is small, the determination of the structure is

difficult. By increasing the temperature and pressure, observations from 22 GPa and

340 K more consistently revealed the bcc structure.

these at certain temperature and pressure conditions, implying that its Gibbs

free energy is indeed lower. For the Aziz potential, no bcc phase was found

for the pressure and temperature range shown in (b), nor for higher pressures

and temperatures, following the melting curve up to 25 GPa.

7.3.2 Ne

Before addressing the high pressure properties, the ambient pressure

melting point for neon was studied by means of parallel tempered

Monte Carlo (MC) simulations of the ’magic’ number clusters NeN(N = 13,55,147,309,561,923) with the use of an extended Lennard-Jones

(ELJ) potential. From a linear extrapolation of Tm as a function of N−1/3, the

bulk melting (N → ∞) was found to be 26.6 K, as shown in Fig. 7.3. Quantum

effects for the light Ne system could lower the bulk melting somewhat, as

previous studies on smaller cluster sizes have shown a ∼ 10% decrease of

Tm. Thus, the experimental data of 24.6 K could therefore be accurately

reproduced.

Regarding the high-pressure region, X-ray diffraction studies at room tem-

perature by Hemley et al. [54] show no solid-solid phase transitions up to 110

GPa. Therefore, only the fcc phase was considered in the applied pressure (P)


(a) (b) (c)

Figure 7.2: The Mackay magic number clusters with (a) 13, (b) 55 and (c) 147 atoms,

respectively.

10 12 14 16 18 20 22 24 26 28Melting temperature (K)

0

0.1

0.2

0.3

0.4

0.5

N-1

/3

Ne13

55

309561

923

147

Figure 7.3: Size dependence of the Ne melting temperatures after 10 million MC

steps. The cluster sizes are inserted in the figure by their respective melting point, as

well as experimental melting (dotted line) [45].

0 25 50 75 100 125 150 175Pressure (GPa)

0

500

1000

1500

2000

2500

3000

3500

4000

Tem

per

atu

re (

K)

L-J (one-phase)

L-J (two-phase)

VASP (one-phase)

Exp-6 (Vos et al.)

Figure 7.4: The Ne melt line for the 1728-atom system (864+864, two-phase, 40 000

time steps) with the L-J interaction from Bellisent-Funel et al. [16] Furthermore, the

melt line from one-phase 108-atom simulations over 3000 time steps with the same

potential is shown.

7.3. MELTING 57

and temperature (T ) range. The melting overshoot in Fig. 7.4 from the AIMD

compared to the L-J two-phase system in the pressure range 25 to 100 GPa

is approximately 200 K. The phenomenon is due to the difference between

homogeneous and heterogeneous melting. This could also indicate that ab ini-tio two-phase calculations, of which the melting accuracy has been reported

[19, 20, 2, 70], could show melting close to the classical two-phase results.

7.3.3 Na

Pressure applied to a solid sample usually helps negate thermal agitation. This

makes negative melting (dTm/dP < 0) rare, although water/ice is a well known

example. 1 Apart from this fundamentally interesting phenomenon for Na, the

metal could very well show superconducting properties as was discovered for

its periodic table neighbor, Li. In combination with the Na negative melting,

the metal could then possibly show "superfluidity" at high pressure, but, at

modest temperatures.

Recent experimental data from Gregoryanz et al. have shown a big melting

temperature drop (∼700 K) in the extraordinary wide pressure range (31-118

GPa) [51]. However, certain difficulties have prevented studies of alkali met-

als, since these metals are very reactive especially at extreme pressures and

temperatures. Therefore, the comparison of experimental results with theoret-

ical calculations is indeed very important.

For the MD calculations, a 128-atom bcc supercell was constructed multiply-

ing the two-atom unit cell in all three orthogonal directions 4×4×4. Analo-

gously for the fcc structure, a 108-atom cell was created from the four-atom

unit cell in a 3× 3× 3 grid. The calculations were performed in a one-phase

setup to avoid extensive calculations, but with the awareness of possible over-

heating effects.

Atomic analysis showed an increase in coordination number for the liquid

compared to the solid phase with 211 atoms (47 GPa), 296 atoms (67 GPa)

and 291 atoms (99 GPa), equivalent to the mean addition of 1.9, 2.7 and again

2.7 neighbors per atom for the melting at the three pressures. This result em-

phasizes a denser liquid phase. From the Clausius-Clapeyron equation

dTm

dP=

dVm

dSm, (7.3)

where Tm, Vm and Sm is the temperature, volume and entropy at melting, this

implies negative melting.

The melting curve from the AIMD calculations and the experimental data

from Gregoryanz et al. [51] are shown in Fig. 7.5. There is a good agreement

between the AIMD melting and the experimental results at pressures up to

1This is, however, not an explanation why one can skate on ice: the pressure required is far

greater than that actually produced by ice skates. Instead, the possibility to skate is due to

surface effects.


Figure 7.5: The melting data of Na. The AIMD and Lindemann results from this work

are compared to the experimental findings of Gregoryanz et al. [51]

50 GPa, but at higher pressures, the MD gradually overestimates the melting

temperatures. This discrepancy could be due to the crystal superheating. To

understand the physical origin of the negative melting, the c44, c11 and c12 for

bcc and fcc as a function of pressure were calculated. The three cubic elastic

constants were used to compute the polycrystalline shear modulus G. From Gand the bulk modulus B, the pressure dependent Debye temperature ΘD was

evaluated, and inserted into the a ΘD-dependent Lindemann criterion,

TLm = const ×V 2/3Θ2D , (7.4)

where V is the volume. TLm is in agreement with the experimental findings,

although there is a slight overestimation of the melting temperatures through-

out the studied pressure range. It is most surprising that the negative gradient

dTLm/dP ∼ −5.9 K/GPa found from the TLm for P > 50 GPa is close to the

experimental value of dTm/dP ∼−7.8 K/GPa for P > 30 GPa. Thus, the con-

clusion could be drawn that the negative melting in Na could be related to the

softening in the single crystal elastic constants above P∼ 50 GPa.

59

Chapter 8

Shock waves

or decades, shock waves have been used to study the behavior

of materials under extreme conditions. In Fig. 7.4, the schematic

setup of a shock-wave experiment with a gas gun was shown, but

shocks can also be generated with molecular dynamics. One sur-

face of the system, the piston, is driven inward along the sample axis at a

constant velocity UP, leading to a shock wave with velocity US. A simple,

one-dimensional shock front typically induces a step rise in pressure, as the

sample goes from a pre-shocked state to a post-shocked state. A schematic

shock front is shown in Fig. 8.1 (left). If the pressure is found above the

Hugoniot elastic limit (HEL), shown as point B in Fig. 8.1 (right), a single

plastic wave propagates. If the pressure is found under the PHEL at A, a sin-

gle elastic wave propagates. At high pressure conditions, atoms in the lattice

can respond irreversibly, and the kinetics of defects can lead to e.g. solid-solid

phase transformations or melting. If the pressure is at an intermediate point

(C), two shocks propagate as a plastic wave and an elastic wave. The elastic

wave, meaning that the shocked lattice will relax back to its original configu-

ration, is faster than the plastic wave. The plastic wave travels at the speed of

the shock US, whereas the elastic precursor travels at the longitudinal speed of

sound.

8.1 Rankine-Hugoniot relations and equation of state

The Rankine-Hugoniot equations govern the behavior of shock waves. The

idea is to consider a one-dimensional, steady flow of a fluid subject to the

Euler equations and require that mass, momentum, and energy are conserved.

After a unit time, the piston has moved the distance UP and the shock US. As-

suming the cylinder area A, the initial mass AUS is compressed to A(US−UP),leading to the density increase ρ0 to ρ . Thus, the mass conservation equation

is

A(US −U0)ρ0 = A(US −UP)ρ = m . (8.1)

60 CHAPTER 8. SHOCK WAVES

Shock direction

Pre

ssu

re

Volume

Pre

ssu

re

HugoniotRayleigh line

U

V E

P

V E P U

U

P

HH

H

00 00

S C

B

A

HELP

V0

Figure 8.1: To the left, a shock wave is moving with speed US, as a result of the

piston velocity UP. P0, V0 and E0 is the pre-shocked pressure, volume and energy,

whereas PH , VH and EH is the pressure, volume and energy after the shock. As a

single shock propagates, the schematic figure represents either a high-pressure wave

or a low-pressure wave. In the case of an intermediate pressure, a plastic and an elastic

wave could propagate. To the right, the P-V diagram of the shock Hugoniot initially

at V0 with the loading along the Rayleigh line, with PA = PHEL being the maximum

shock pressure where the material acts elastically. PB is the minimum shock pressure

to overdrive the HEL, i.e. a single shock wave. At the intermediate point C, there is a

plastic and an elastic shock wave.

As the piston applies a force A(PH −P) to the material, the momentum per

unit time is found as m(UP−U0). Inserting this equation into Eq. (8.1), we get

PH −P0 = ρ0(Us −U0)(Up −U0) . (8.2)

Using the volume as the inverse density, the volume relation from Eq. (8.1)

becomesVH

V0= 1− Up −U0

Us −U0. (8.3)

The compress work that the piston does on the material per unit time is APUP,

being equal to the sum of the kinetic energy 1/2Aρ0USU2P and the internal

energy Aρ0US(E −E0). The conservation of energy implies that

PUP = ρ0US(12

U2P +E −E0) . (8.4)

By means of Eq. (8.1), (8.2) and (8.4), the energy relation can be written as

EH −E0 =12(PH +P0)(V0 −VH) . (8.5)

The experimentally determined linear relation between the the shock wave

velocity US and the piston velocity UP is found as

US = C +SUP , (8.6)

8.2. THE TWO-TEMPERATURE MODEL 61

where C is the speed of sound in the bulk at zero pressure and S is the linear

coefficient in the relation between Us and Up. A major difference between ho-

mogeneous deformation and shock loading is that the total volumetric strain

behind the shock front, ε =UP/US, is constant. The shock pressure, PH , is also

constant behind the shock front. The coefficient S is used to solve Eq. (8.2).

The pressure-volume line from P0,V0 to PH,VH is the Raleigh line, whereas

the shock-compression curve is called the Hugoniot. Comparing shocks to

isothermal and static compressions at the same volume, the shocks cause

higher thermal and total pressures. This is due to the net irreversible energy,

found as the total specific energy under the Raleigh line subtracted by the

reversible energy from the entropic compression under the isentrope.

8.2 The two-temperature model

The two-temperature model (TTM) was developed in 1956 by Kaganov, Lif-

shits, and Tanatarov to describe the electron-phonon relaxation in metals. Dur-

ing the last decade the TTM has been widely used to explain the energy loss

by electrons excited by femtosecond laser pulses. The rapidly equilibrated

electrons transfer heat to the vibrating atoms (the phonons). The time for this

event, which is on the picosecond scale, is governed by the electron-phonon

coupling. When there is a thermal equilibrium between the ions and the elec-

trons, the heat from the surface to the bulk can be described as the com-

mon thermal diffusion. To treat the different temperatures of the electrons and

the phonons, the TTM considers two coupled nonlinear differential equations

[57],

Cel(Tel)δTel

δ t= ∇[Kel(Tel)∇Tel]−G(Tel −Tph)+S(z, t)

Cph(Tph)δTph

δ t= ∇[Kph(Tph)∇Tph]+G(Tel −Tph) , (8.7)

where Cel and Cph are the heat capacities for electrons and phonons, K is the

heat conductivity, T is the temperature, G is the electron-phonon coupling

constant and S is a source term from the laser pulse. The system of Eq. (8.7)

is solved by applying a finite difference scheme.

8.2.1 The combined TTM-MD model

By combining the two-temperature approach with MD, the heat conduction

equation for the phonons in Eq. (8.7) is replaced with the atomic equations of

motion in the MD method. The diffusion equation for the electron temperature

Tel in Eq. (8.7) is solved simultaneously with the MD,

Cel(Tel)δTel

δ t=

δδ z

[(Kel(Tel)

δδ z

Tel

]−G(Tel −Tph)+S(z, t)


miai = Fi +ξ mivTi , (8.8)

where

ξ =1n

n

∑k=1

GVN(T kel −Tph)

/∑

imi(vT

i )2 . (8.9)

In the above equations, Fi is the interatomic force, mi and vi is the atomic

mass and velocity and VN is the volume per cell. The thermal velocities vTi are

defined as vTi = vi −vc, i.e. the atomic velocities vi subtracted by the velocity

of the collective motion of the atoms in a cell, vc. The additional second term

in Eq. (8.8) to the newtonian equation is the electron-phonon coupling. The

coupling, together with the phonon temperature Tph, is defined for each cell,

and the summation in Eq. (8.9) is performed over all the atoms of a cell. The

phonon temperature Tph in Eq. (8.8) is found from the average kinetic energy

of the atoms, ET ,

Tph =2ET

3kB= ∑

i

mi(vTi )2

3kBn, (8.10)

where n are all atoms in a cell. The time step in the finite difference scheme

for the integration of Eq. (8.8) holds the von Neumann stability criterion

ΔtvN ≤ (Δz)2 Cel(Tel)

2Kel(Tel,Tph), (8.11)

where Δz is the spatial difference. Typically, ΔtvN is smaller than the time

steps required for the MD equations. If the factor is n, i.e., ΔtMD = nΔtvN , the

accumulated electron-phonon energy is written as

ΔEel−ph =n

∑k=1

ΔtvNGVN(T kel −Tph) . (8.12)

The coupling in Eq. (8.9) is therefore changed by the accumulated energy in

Eq. (8.12), leading to a change in the atomic motion in Eq. (8.8).

8.3 Shock-induced melting of Ni with MD

In manuscript IX, MD simulations of shock-induced melting of Ni are per-

formed with an embedded-atom model (EAM). As MD simulations only con-

tain lattice heat conduction, the inclusion of electronic heat conduction and

electron-phonon coupling is of interest. By using non-equilibrium molecular

dynamics (NEMD) simulations with the MD+TTM approach as described, the

electronic contribution to the temperature distribution is studied. The electron-

phonon coupling constant G in Eq. (8.8) is dependent on the density and tem-

perature. Experimental fitting have shown that a constant G is sufficient, and

a comparison has been made with the generally accepted G = G0 as well as a

ten time increase G = 10×G0, leaving every thing else the same. In Fig. 8.2, a

8.3. SHOCK-INDUCED MELTING OF NI WITH MD 63

Figure 8.2: A shock wave moving along the x axis with speed US as a function of the

piston velocity UP. To the left, UP=1.0 km/s, and behind the shock there are disloca-

tion loops. To the right, UP=4.0 km/s, where a liquid configuration is found behind

the shock front. Some atoms found to the left have passed through the "permeable"

piston. The coloring is related to the atomic movement [65].

snapshot is shown for a shock moving from left to right, just as the schematic

shock in Fig. 8.1. The upper plot shows a relatively low piston velocity, which

does not imply melting. However, dislocations are produced, and for shocks

along <100> there are four different, equally active slip planes where dislo-

cation loops are generated. This effect has previously been reported for both

pair and many-body potentials [69]. For a higher piston velocity, there is a

liquid structure behind the shock front easily verified by comparison to the

ordered single crystal ahead of the shock.

Due to to a fast electronic heat conduction and electron-phonon coupling,

there is a possibility of a lattice temperature decrease behind the shock, and

an increase immediately ahead of the shock. This ”pre-heating” of the lattice

could change the melting pressure. However, the effect was shown to be very

small, as the melting pressures Pm was found to be 275±10 GPa, 285±10 GPa,

and 290±10 GPa for MD, MD+TTM and MD+TTM×10, respectively. This

was confirmed by visual inspection of snapshots, in color to show crystalline

regions and defects (Fig. 8.2). The liquid-solid co-existence simulations pre-

dict a lower melting temperature compared to the MD and the MD+TTM at

the same pressure. Although superheating should be limited in shock simula-

tions due to dislocations, recent results indicate a maximum bulk superheating

of 20 %. Therefore, the higher melting temperatures for the MD simulations

compared to the co-existence simulations are reasonable. As shown in Fig.

8.3, the increase in the melting pressure is low when the electronic contri-

butions are included. However, materials possessing a bigger electron-phonon

coupling or materials undergoing longer time scale simulations than presented


0 50 100 150 200 250 300 3500

2

4

6

8MDMD+TTMMD+TTMx10SGmodelLindemannmelt line

T lat(103K)

PH(GPa)

Figure 8.3: P-T diagram for MD, MD+TTM and MD+TTMx10. The solid black line

shows the liquid-solid coexistence simulations with the EAM potential. The shock

temperature from the Steinberg-Guinan (SG) model [99], [82] and the Lindemann

melt line are also presented.

here, might show bigger melting pressure effects. The modeling of an accu-

rate electron-phonon coupling may have consequences on thermodynamical

properties as e.g. resistivity, thermal conductivity and heat capacity [48].

65

Chapter 9

Inner Earth studies: (Mg,Fe)Oand Fe

cut through the Earth, as shown in Fig. 9.1, reveals the crust,

mantle and core regions. Ferropericlase Mg1−xFexO is believed

to be the second most abundant mineral in the Earth’s mantle, and

deeper down, in the core, it is believed that iron (Fe) is the main

component. Therefore, the properties of iron and iron-alloys at high pressure

and high temperature are interesting and important for the understanding of

the Earth’s interior.

Figure 9.1: A schematic view of the Earth’s interior. The crust and mantle are com-

posed of solid silicates and oxide minerals. The mantle is divided into the upper man-

tle (depth<410 km), the transition zone (410-670 km) and the lower mantle (670-

2890 km) [30]. The liquid outer core (2890-5150) consists mainly of iron but with ∼10 wt. % lighter alloying elements. The solid inner core extends to the center of the

Earth at 6371 km.

66 CHAPTER 9. INNER EARTH STUDIES: (MG,FE)O AND FE

9.1 MgO

Abundant minerals in the Earth’s lower mantle are believed to be Mg-Fe

silicate perovskite (Mg,Fe)SiO3, rock-salt structured ferropericlase (Mg,Fe)O

and calcium silicate perovskite CaSiO3. Although several properties of MgO

have been studied extensively both theoretically and experimentally, works

regarding the inclusion of Fe are scarce. Therefore, high-pressure calculations

of MgO were performed for two reasons: primo, to benefit from the many

references available on MgO, secondo, to test the EMTO calculation setup

before inserting any iron into the compound of Mg1−xFexO.

The EOS as shown in Fig. 9.2 is found slightly above other theoretical

[89, 63] and experimental data [43]. The small volume overestimation is

∼2.5% compared to the averaged reference data.

The calculated bulk modulus B and the elastic constants c44, c11 and c12 for

MgO are shown in Table 9.1. The calculated c44 is in excellent agreement

with both theoretical and experimental data. For c11 and c12, the results

agree well with theory but show a somewhat erratic behavior compared to

experiment. However, this discrepancy is within the errors which could arise

for elastic constant calculations. Therefore, the conclusion was drawn that

the EMTO method correctly describes the properties of MgO. Based on

numerous studies, one could also assume that when small concentrations of

Fe are introduced into the compound, the EMTO method still gives feasible

results.

4 4.25 4.5 4.75 5 5.25 5.5 5.75 6

Volume (cm3/mol )

0

20

40

60

80

100

Pre

ssu

re (

GP

a)

MgO (this work)

MgO (Oganov and Dorogokupets)

MgO (Karki et al.)MgO (exp., Duffy et al.)

Figure 9.2: The EOS for MgO. The EMTO calculations are compared to theoretical

data by Oganov and Dorogokupets (2003) and Karki et al. (1997) and experiment by

Duffy et al. (1995) The volumes have been scaled by a factor of 0.5 due to the EMTO

setup.

9.2. (MG,FE)O 67

9.2 (Mg,Fe)O

The electronic spin transition of iron in minerals found at the Earth’s mantle

conditions has been reported to have a significant impact on several geophysi-

cal properties such as the density and thermal conductivity. In Fig. 9.3 (a), the

magnetic moment as a function of pressure and iron composition is shown.

For 6.25 and 12.5% Fe, the magnetic moment is almost vanished at 60 GPa.

The results agree perfectly compared to spin state measurements of ferroper-

iclase Mg0.95Fe0.05O and Mg0.83Fe0.17O, showing the high-spin to low-spin

transition to increase with Fe content and to occur between 46 and 55 GPa

and between 60 and 70 GPa, respectively [13, 71]. For the Fe-richer struc-

tures, 18.8% Fe shows a transition just below 90 GPa, whereas for 25%, the

behavior is smoother, without any clear transition. This implies a possible

co-existence of high-spin (HS) and low-spin (LS) states as reported both the-

oretically and experimentally. Another recent study has indicated the stability

of the HS state for magnesiowüstite throughout the Earth’s mantle [55]. More

specifically, the transition pressure increases with iron concentration and tem-

perature for (Mg,Fe)O as ΔPtr/ΔT ∼ 0.18, 0.20 and 0.31 GPa/K for 17, 25 and

40 % Fe, respectively. In this work, the studied pressure range for the elastic

constants in this work was below 100 GPa with an iron composition ≤ 20%. In

order to avoid the numerical difficulties near the magnetic transition, the self-

consistent elastic constant calculations for the solid solutions were performed

in the HS state using the fixed spin approximation. In Fig. 9.3 (b), the increase

of Fe leads to a volume increase in the EOS, evaluated at the same pressure.

This is expected, as the molar volume of wüstite is 9% higher than periclase at

ambient conditions. Furthermore, the volume expansion as a function of iron

content agrees with the comparison of the experimental data for MgO with that

of Mg0.6Fe0.4. The calculated c44 as a function of Fe content is shown in Fig.

Table 9.1: Bulk modulus and elastic constant calculations for MgO. The EMTOmethod (this work) is compared to VASP calculations with the GGA from Oganovand Dorogokupets (2003) and LDA calculations from Karki et al. (1997) Included isalso experimental DAC data from Sinogeikin and Bass (1999) and Zha et al. (2000)All data presented are in GPa.

c44 c11 c12P B Bc EMTO GGAa LDAb Exp.c Exp.d EMTO GGAa LDAb Exp.c Exp.d EMTO GGAa LDAb Exp.c Exp.d

8 177 194 152 148 144 164 161 316 335 359 371 365 107 103 105 104 10520 226 242 160 157 154 176 171 417 440 464 466 480 130 120 123 124 11036 289 306 169 168 164 189 571 580 602 565 144 143 148 16457 361 390 176 180 173 203 738 744 770 645 172 170 178 23983 454 494 183 191 183 923 945 968 220 203 220a[89]b[63]c[116]d[97]


(a) (b)

Figure 9.3: In (a), the magnetic moment of (Mg,Fe)O as a function of composition

and pressure is shown. For the iron poor compounds (6.25 and 12.5% Fe), the mag-

netic moment vanishes at approximately 60 GPa. The somewhat wider spin transition

range for Mg0.83Fe0.17O [71] is inserted as vertical lines in the figure, separating

the high-spin (HS) from the low-spin (LS) region. For the richer Fe phases, 18.8%Fe shows a transition just below 90 GPa, whereas for 25% Fe, the transition occurs

within a wide (∼100 GPa) pressure window. The EOS as a function of the Fe con-

tent in Mg1−xFexO is shown in (b), where the volumes have been scaled by a factor of

0.5. The effect of increasing the amount of iron into the ferropericlase yields a volume

increase, consistent with the higher molar volume for FeO compared to MgO.

9.4, and the trend from the experimentally observed c44 with increasing iron

percentage is very well reproduced by the EMTO results. In Fig. 9.5 (a), the

Δc44/ΔP for Mg0.9Fe0.1O is increasing with a negative second derivative. On

the other hand, for Mg0.8Fe0.2O, Δc44/ΔP is slightly decreasing after approx-

imately 26 GPa. Combined X-ray diffraction and spectroscopy experiments

have reported the transformation of the cubic Mg0.8Fe0.2O to a rhombohedral

structure at 35±1 GPa [62]. The softening of the c44 could therefore be a sign

for the phase transition observed by experiment.

In Fig. 9.5 (b), (c11-c12) increases with pressure for all compounds.

However, similar to c44, the gradient Δ(c11-c12)/ΔP decreases with the

amount of Fe. Experimental findings in a low pressure region report a quite

stable Δc11/ΔP with iron composition as Δc11/ΔP=9.35(13) for MgO [61] and

Δc11/ΔP=9.3(2) and 9.6(4) for Mg0.76Fe0.24O and Mg0.44Fe0.56O, respectively

[60]. However, Jackson et al. (2006) have shown the lower gradient 8.35

for ferropericlase with 6 % Fe. As Δc11/ΔP Δc12/ΔP, the trend of the

(c11-c12) pressure gradient can be assumed to follow the one of c11. From a

polynomial fit to the experimental data, the elastic constants as a function

of iron composition x can be written (c11-c12)=192 − 184x + 139x2 − 51x3

GPa [61]. For the concentrations studied in this work, x = 0,0.10 and 0.20,

the function yields P=192, 175 and 160 GPa respectively, i.e. a lowering of

pressure with Fe content.

9.2. (MG,FE)O 69

0 0.05 0.1 0.15 0.2 0.25Fe / (Mg+Fe)

100

110

120

130

140

150

160

C44 (

GP

a)

this workexp., Jacobsen et al.exp., Jackson et al. (1978)

exp., Jackson et al. (2006)

Figure 9.4: The shear constant c44 as a function of (Mg,Fe)O composition. A soften-

ing with iron content is shown both for theory (this work) as well as for experiment

[58, 61, 59]

0 20 40 60 80 100Pressure (GPa)

100

120

140

160

180

200

C44 (

GP

a)

MgO

Mg0.9

Fe0.1

O

Mg0.8

Fe0.2

O

(a)

0 20 40 60 80 100Pressure (GPa)

200

300

400

500

600

700

800

(C1

1-C

12)

(G

Pa)

MgO

Mg0.9

Fe0.1

O

Mg0.8

Fe0.2

O

(b)Figure 9.5: In (a), c44 as a function of pressure and (Mg,Fe)O composition is pre-

sented. Consistent with the results in [a], the effect of iron is a softening of the elastic

constant. MgO and Mg0.9Fe0.1O show increasing data as a function of pressure. A

kink is found for the c44 for Mg0.8Fe0.2O at 26 GPa, which could be due to a phase

transition. In (b), the (c11-c12) as a function of pressure and Mg1−xFexO compound

is shown. For MgO, the results from this work in the low pressure range is some-

what underestimated compared to experiment [97], as shown in Table 9.1. The results

from Mg0.9Fe0.1O and Mg0.8Fe0.2O are consistent with earlier findings, implying a

softening with iron content.


0 2 4 6 8 10Interatomic distance ( )

0

0.5

1

1.5

2

2.5

3

RD

FAmbient, 1800 K

27 GPa, 2600 K42 GPa, 2700 K

50 GPa, 2700 K

58 GPa, 2900 K

Å

(a)

0 2 4 6 8 10q (1/ )

-2

0

2

4

6

Str

uct

ure

fac

tor

S(q

)

Å

Ambient

27 GPa

42 GPa

50 GPa

58 GPa

(b)

Figure 9.6: The radial distribution functions (a) and the structure factors (b) for five

different pressure and temperature conditions (the structure factors have been shifted

for enhanced readability.)

9.3 Fe at outer core conditions

As liquid iron is the predominant matter in the outer core of the Earth,

manuscripts XI and XII have focused on liquid Fe properties. Firstly, an

initial body centered cubic (bcc) lattice was simulated at 58 GPa and 6000 K,

yielding a molten configuration. The melt was cooled down to five different

pressures and temperatures for easier comparison to experimental data [96].

For all five structures, the radial distribution functions shown in Fig. 9.6

(a) indicate molten structures, as there are no significant lattice peaks. The

structure factor calculations compared to experimental data [96] indicate that

there is a more accurate match for the first peak as pressure increases, as

seen in Fig. 9.6 (b). For the second peak, the low pressure EAM data show a

better agreement to the experimental data. However, as the first peak is the

most significant peak and that the experimental data show erratic behaviors

for higher q values, the EAM potential is more suitable for high pressures.

Further NPT calculations were performed in a grid in the PT diagram with

P ranging from the core mantle boundary (CMB) pressure 125 to 325 GPa,

being the approximate inner core boundary pressure (ICB). The temperatures

varied from 3500 to 7100 K, as the temperature at the core side of the CMB

is estimated to be between 4000 and 5000 K [10] and the ICB temperature

is estimated to be 7100 K [21]. Evaluating the thermal expansion α and

the specific heat capacity cp from temperatures above melting down to the

melting temperatures, the adiabatic gradient equation was calculated at the

ICB. Earth data for the pressure and gravity as a function of terrestrial depth

was taken from the PREM [44]. The grid of calculated temperature gradients

for the melting PT points were fitted to a z polynomial, and the resulting T (z)plot was evaluated after integration, as shown in Fig. 9.7 (a). For comparison,

a calculation was made with a 500 K lower initial temperature at the ICB.

9.3. FE AT OUTER CORE CONDITIONS 71

2500 3000 3500 4000 4500 5000Depth (km)

3500

4000

4500

5000

5500

6000

6500

7000

Tem

per

ature

(K

)

EAM Adiabat (1)

EAM Adiabat (2)

EAM Adiabat with el. cont.Melting curve (Belonoshko et al)

Melting (Ma et al)

Adiabat (Anderson et al)

(a)

125 150 175 200 225 250 275 300 325 350 375Pressure (GPa)

9.5

10

10.5

11

11.5

12

12.5

13

13.5

Den

sity

(g/c

m³)

Ab initio(1)

Ab initio(2)

Exp

PREM EAM

(b)

Figure 9.7: Shown in (a) is the iron adiabat temperature as a function of terrestrial

depth in the liquid iron outer core from the EAM compared to the adiabat of Anderson

[10], the iron melting curve [21] and experimental melting [76]. Indices (1) and (2)

show the EAM with different initial temperatures, starting from the ICB. The adiabat

from the EAM with the electronic contribution is also presented. Shown in (b) is the

iron density as a function of pressure along the adiabat from the EAM, PREM [44],

the ab initio (1) [29], the ab initio (2) data and the extrapolated experimental data

from 150 GPa [4].

Following the calculated densities along the adiabat in (b), the specific heat

CVe for fcc iron is 1.7 ± 0.3R [24],[114]. From the MD data, the specific

heat is close to the classical limit, Cv = 3R per atom. If the total Cv = 5R is

assumed to be an upper limit for the temperature and pressure at the ICB,

the increase of specific heat leads to a decreasing temperature gradient.

Therefore, the adiabatic curve with the electronic contribution and the initial

temperature of 7100 K at the ICB is found above the EAM adiabat at the

same initial temperature, as shown in Fig. 9.7 (a).

The EAM densities along the adiabatic curve were calculated and compared

with the PREM data, the ab initio data from Vocadlo et al. [29], and Alfè

et. al [4], and the extrapolated experimental data at 150 GPa [4]. The

shift in density compared to the PREM occurs due to the lighter element

impurities in the outer core compared to the pure iron calculations with the

EAM method, shown in Fig. 9.7 (b). A density decrease of 10.3±3.0 wt%in comparison with pure iron is suggested for the outer core [4], where the

lighter elements could be S (or Si) to 10.0±2.5 % and O to 8.0±2.5 %.

From Fig. 9.7 (a), the density discrepancy found at e.g. 200 GPa is 8.2 wt%,

slightly underestimating the contents of lighter elements, but still well within

the error estimations [4]. With the same relationship between S (or Si) and O

as proposed by Alfè, the molar percentages of S (or Si) and O would decrease

to 7.9 % and 6.3 %, respectively.

The diffusion parameter D along the adiabatic curve in Fig. 9.7 was cal-

culated for the EAM, in agreement with the ab initio calculations from the


Table 9.2: EAM calculated diffusion parameters for temperatures and pressures alongthe adiabatic curve.

P (GPa) T (K) D (10−9m2s−1)

125 5000 8.9

150 5450 8.5

175 5750 7.8

200 6000 7.0

225 6200 7.0

250 6400 6.9

275 6600 6.3

300 6800 5.8

325 7100 5.2

ICB where D=5;4-5 [3], [29]. The adiabat for T=6000 K at the ICB [10] has

T=4300 K at the CMB, where the calculations shown here shows a shift of

the curve with approximately 1000 K. Thus, the diffusion parameters in Ta-

ble 9.2 are higher. However, if the referred adiabat is followed, the diffusion

parameter D calculated at the temperatures and pressures of the adiabat is

5.0±1.0×10−9m2s−1. This stable behavior is also found in the study by Alfè

et al. [3]. The diffusion parameters in Table 9.3 were calculated for a wider

range of temperatures and pressures, showing a satisfactory agreement to the

cited results.

Due to impurities, the true adiabatic temperature in the Earth’s core might

be slightly different from the adiabat calculated from pure iron. For cast iron

(< 4% C), the thermal expansion is somewhat lower, and the specific heat is

somewhat higher [88]. This yields the assumption that the temperature gra-

dient might be slightly lower with the lighter element impurities discussed in

earlier sections, i.e. an adiabatic curve slightly above the proposed adiabatic

curves in Fig. 9.7.

Furthermore, one can discuss the possible range of T at the CMB by analyzing

the possible scenarios provided by the data in Fig. 9.7. If, instead of Tm = 7100K [21] and the Tm = 6500 K we should use the Tm = 6000 K [10], then the

adiabatic T at the CMB can be lowered down to 4750 K. The effect of the

impurities is likely to increase this number. Therefore, 5000 K seems to be a

quite reasonable estimate of the T at the core side of the CMB.

The possible stabilization of the bcc phase at extreme conditions was also

studied. Initially, the ab initio calculations were performed for the hexagonal

close-packed (hcp) phase, to benefit from more experimental and theoretical

studies. Molecular dynamics was performed for the hcp-Fe 128-atoms system

at T=1250 K and at volumes corresponding to the highest pressure of approx-

9.3. FE AT OUTER CORE CONDITIONS 73

Table 9.3: EAM calculated diffusion parameters (in 10−9m2s−1) for temperatures andpressures compared to Alfé et al. [3] (in brackets).

T(K)

P (GPa) 4300 5000 6000 7000

132 5.5 (5.2) x x x

140 x 7.0 (7.0) x x

151 x x 10.0 (10.0) x

170 x x 8.5 (9) x

181 x x x 11.1(11)

251 x x 5.7(6) x

264 x x x 8.7(9)

imately 220 GPa. Shown in Fig. 9.8 is the equation of state for hcp-Fe. The

AIMD calculations in this work agree well with experimental data [42], per-

formed at the same temperature T=1250 K. There is an approximate density

shift of 0.25 cm3/mol throughout the studied pressure range between the two

inserted experimental curves. This yields a thermal expansion coefficient of

4.00e−5 K−1 at 22 GPa which is in perfect agreement with α=3.88e−5 K−1 at

the same pressure reported by Funamori et al. [47] Furthermore, EOS calcu-

lations showed a good match between the equilibrium volume, bulk modulus

and its derivative from this work with other theoretical studies [100] and ex-

periment [42]. Thus, the calculations for the hcp phase can reproduce previous

theoretical and experimental findings both at 0 K and at elevated temperatures.

This is important before examining the bcc phase at extreme conditions, where

the comparison is much more difficult due to scarce data.

The bcc 128-atoms systems were simulated in NVE ensembles at different

initial temperatures, and the stress tensors were examined as a function of

temperature and time step as shown in Fig. 9.9 (the data have been shifted

for enhanced visibility). The stresses fluctuate in a narrow pressure range for

the high temperature calculations T=3400 and 2900 K. This implies that the

simulation is hydrostatic, and that the bcc-phase is stable. However, for the

calculations from T=2400 K down to T=1100 K, the tensors are no longer

compact. These pressure splits are most apparent for the lower temperatures,

and suggest the bcc instability. The T=2600 K calculation seem to show a

both hydrostatic and non-hydrostatic behavior, which could be interpreted as

the bcc stability limit. This is in agreement with the results of Vocadlo et al.[28] who find the deviation from the bcc phase at temperatures below 3000 K

and ∼ 260 GPa.


0 50 100 150 200 250 300Pressure (GPa)

4

4.5

5

5.5

6

6.5

7

Volu

me

(cm

3/m

ol)

Theory, this work (T= 1250 K)

Theory, this work (T= 0 K)

Exp., Dewaele et al. (T=300 K)

Exp., Dubrovinsky et al. (T=1250 K)

Figure 9.8: The equation of state showing volume as a function of pressure for hcp-Fe.

The filled circles (this work) are in reasonable agreement with the cited experimental

data [33],[42].

0 250 500 750 1000Time steps

0

50

100

150

200

250

300

350

Str

ess

tenso

rs (

GP

a)

T=3400 K

T=2900 K

T=2600 K

T=2450 K

T=2300 K

T=1700 K

T=1100 K

Figure 9.9: The stress tensors as a function of temperature for bcc Fe. At T > 2600

K, the bcc is stable as the tensors fluctuate in a narrow pressure range. For T <2600

K, the tensors split, which is an indication of the crystal instability.

75

Chapter 10

Sammanfattning

opplingen mellan en flintsten från stenåldern, en kompass från

medeltiden, en transistor från förra seklet och en superdator från

dags dato är att de alla är, eller har varit, praktiska hjälpmedel.

Allt sedan Hedenhös har människan strävat efter att förbättra

åtråvärda materialegenskaper som hårdhet, slitstyrka, färg, elasticitet etc.

Att denna strävan är minst lika stark idag är lätt att inse, speciellt när man

begrundar utvecklingen inom exempelvis elektroniken.

Kunskaper om material utvecklas med hjälp av förbättrade experiment,

teorier och datorbaserade beräkningar, och dessa tre är starkt förknippade

med varandra med avseende på överensstämmelse, utvärdering, prestanda

o.s.v. Intressant nog erhålles en positiv cirkel: forskning ger bättre material

och ger i sin tur bättre mätutrustning och datorer som förbättrar forskningen!

För en djupare insikt inom materialteknik krävs att materialen studeras med

grundläggande fysik på en atomär nivå. Matematik och fysik går hand i hand,

och en av de mest fascinerande aspekterna med teoretisk materialforskning

är att dessa teorier och formler uttryckta i de program som används faktikst

fungerar för att uppskatta en eller flera materialegenskaper.

Resultaten i denna avhandling om metaller, metallegeringar och gaser är

baserade på beräkningar/simulationer. Trots att kvaliteten på dessa endast

kan bli utvärderade med jämförelsen till experiment, finns det många fördelar

som bl.a. en ökad överblick, förenklade system, ekonomiska vinningar etc.

Som ett exempel kan man nämna experimentella studier av material under

extrema förutsättningar (tex. höga tryck/höga temperaturer) som kan vara

svåra och dyra att genomföra. Ett ryskt försök att borra ett rekordlångt hål

ner i jorden började 1970 och slutade 1994. Det slutliga djupet blev ”bara”12 300 meter, bara 0.2 % (!) av jordens radie (i kontrast till detta skrivs

dagligen modellbaserade artiklar om ämnens och legeringars egenskaper

under tryck och temperaturer som motsvarar jordens medelpunkt). Om

man beaktar vanligare experiment för att uppnå extrema förutsättningar

som diamantpressar eller gaskanoner kan beräkningar hjälpa att förstå och

komplettera dessa.

76 CHAPTER 10. SAMMANFATTNING

Beräkningarna i avhandlingen kan i sin tur delas upp i två generella

riktningar: första-princip-beräkningar (lat: ab initio) och klassiska metoder.

Ab initio, som betyder ”från början”, byggs upp av en teori som utvecklades

under 1964-65 av Walter Kohn och ledde till 1988 års Nobelpris. Här följs

Schrödingerekvationen vilken innebär att inga förutbestämda parametrar

används, till skillnad från modeller i klassiska metoder. Fördelen med

ab initio-beräkningar, förutom den mycket tilltalande aspekten att ingen

empirisk kunskap måste tillföras, är att de är mycket exakta. Nackdelen är

att de tar lång tid att genomföra-även fast superdatorer används kan oftast

bara färre än 500 atomer användas i beräkningarna. Klassiska metoder kan

däremot simulera system med hundratusentals atomer, eftersom interaktionen

är modellerad i förväg. Problemet med modellerna är att de kan ha en

begränsad förmåga att reproducera ”verkligheten” och att de kan vara svåra

att matcha mot experimentella resultat. Detta gör att jämförelsen mellan de

två teknikerna är mycket intressant, vilket denna avhandling också belyser.

Även fast ab initio- och de klassiska beräkningarna i denna tes ofta har utförts

i extrema förhållanden, är den teoretiska basen densamma för, praktiskt

taget, vilket system som helst. Samma program har nämligen använts för

att exempelvis simulera transmembranprotein, vätelagring, dynamiken i

HIV-1-proteaser, studerandet av is/vatten-övergångar, o.s.v. Det fantastiska

med detta är att dörrar öppnas för ett ökat samarbete mellan forskare inom

relaterade områden.

77

Acknowledgments

his will be a long section....

First of all I would like to thank my supervisor, prof.

Rajeev Ahuja who gave me the opportunity to start

my PhD studies at Physics IV. He has always helped me out when

I have come with questions regarding my research, and he has a

great feeling for what’s hot and what’s not. I want to acknowledge

him and our ”father of physics”, prof. Börje Johansson. You have

really created a nice atmosphere in the group! Furthermore, I would

also like to thank ”Ångan high” for being such an inspiring and creative place.

I’m grateful to my MD guru/wizard Mr. Whitefoot ”Tolja” Be-lonoshko - Not only for our discussions about MD simulations and

manuscript preparations, but also for some unforgettable Russian stories, told

with a lot of humor... My graduation would have been impossible without

him! It has been nice to profit from the expertize of L. Vitos and his EMTO

code as well as from E. Isaev regarding the ESPRESSO code. Speaking

about that, I’m grateful to the coffee machine at the second floor for giving

me with that special Italian feeling when pressing the cappuccino button (hm.)

Perhaps the highlights of my PhD studies have come from my

visits abroad: E. Bringa (Livermore, USA) taught me shock waves and

how to make an Argentinian barbecue, A. R. Oganov and Y. Ma (ETH

Zürich, Switzerland) showed me Swiss perfection as well as one of the

best ripping table-tennis Chinese-gripped forehands seen in the Alps, and

P. Schwerdtfeger (Massey, New Zealand) convinced me that upside-down

chemistry actually works.

As the PhD studies are so much more than just studies, I espe-

cially want to thank Anders, Cecilia, Mikael, Oscar and Petros for those

late party nights, so often finishing with a well-deserved kebab roll at the

next Guide Rouge candidate-Johannesgrillen.

Theoretical studies can often lead to long chained-to-the-desk

hours. Therefore, I want to thank Norrlands Nations Fotboll for letting me

78 CHAPTER 10. SAMMANFATTNING

think I’m Christian ”Bobo” Vieri1, Stallet/Svettis for postponing the advent

of the beer belly and the ”Rise-and-shine-Friday-morning-play-at-07.00”rinkbandy team where tackling/tripping/interfering a professor is considered

perfectly legitimate, right Olle?

A warm embracement is exclusively directed to my family and to

Karin, the sweetest thing to come out of Småland since FrödingeOstkakaT m.

As it is my firm belief that ”less is more” so easily can be trans-

lated to ”worse is better”, I’d like to excuse my behavior in terms of

’one-liners’ and appalling sense of humor for spoiling the lunches of Anden,Anders, Andreas, Björn, Carlos, Cecilia, Francesco, Fräddy Boy, Johan,Jonas, Lars, Mattias, Mikael, Moyses, Ola, Oscar, Petros, Peter, Poojaand Torbjörn.

Uppsala, a sunny day in March 2008

1b. 12 July 1973 in Bologna, Italy. 1.85 m, 82 kg. 49 Italian caps, 23 goals.

79

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