Departments of Mathematics, Physics and DACSO MASTER THESIS Modelling for Science and Engineering Computational study of defects in tungsten-crystals by means of density functional theory Alba Gordo´ Vilaseca September 2018 Computational study of defects in tungsten-crystals by means of density functional theory Alba Gordó Vilaseca Master in Modelling for Science and Engineering Universitat Autònoma de Barcelona Under the direction of Dr. Stephan Mohr and Prof. Dr. Mervi J. Mantsinen Barcelona Supercomputing Centre September 2018 Acknowledgements En primer lloc, he d’agrair al Dr. Stephan Mohr haver-me brindat l’oportunitat d’endinsar-me en la seva recerca i treballar al seu costat, així com la seva pacient i intensa dedicació en la direcció i supervisió d’aquest treball. M’agradaria donar les gràcies també al Marc Eixarch Fernández, antic alumne de l’Stephan, el treball del qual, [1], ha estat la base d’aquesta tesi. Agraeixo a la Prof. Dr. Mervi J. Mantsinen la possibilitat que em va donar de pertànyer al Grup de Fusió del Centre de Supercomputació de Barcelona, així com la seva constant atenció i interès pel meu treball. Vull donar les gràcies al Dr. Xavier Sáez i al Dr. Albert Gutiérrez per la seva infinita paciència i la seva ajuda constant, així com al Dani Gallart, pels seus consells i comprensió. També vull tenir un record especial per a la resta de membres del Grup de Fusió, amb els qui he compartit llargues estones de dinar i de qui he après una pila de coses. Vull agrair a EUROfusion el fet de dotar-nos de les hores de CPU necessàries per realitzar les simulacions dutes a terme al superordinador Marconi, a Itàlia. També, vull donar les gràcies al Chris, de qui em vull tan a prop, a les dues àvies i a les meves preuadíssimes amigues. Per últim, res d’això no hagués estat possible sense el suport incondicional dels meus pares ni sense la curiositat que m’han estimulat des que era petita; tampoc sense els meus germans, a qui em sento profundament lligada i a qui admiro tant. Abstract The characterisation of neutron resistant materials for the plasma facing component armour in a fusion reactor has been established as one of the eight assignments needed to pursue in order to make fusion electricity feasible [2]. So far, tungsten is being regarded as the ‘baseline’ material for this purpose [2]. Understanding the effect of irradiation on materials requires developing atomic-scale models for radiation defects, which describe how the defects evolve and interact [3]. This Master’s thesis tackles the computational study of vacancies in tungsten crystals by means of Density Functional Theory in an implementation of the BigDFT code [4]. As a continuity of the work developed by Marc Eixarch [1], particular emphasis is given to the method used to calculate the energy of systems of different size. This energy is first calculated from scratch for those systems and afterwards using previously generated support functions. The aim of this new approach is to overcome a possible dependence of the "convergence rate" of the support functions with the system size and try to generate a set of consistent results, and it is also faster. A detailed analysis of the correlation between size of the cells, method used and calculated energies has been carried out. Table of contents List of figures ix List of tables xi 1 Introduction1 1.1 Scientific context . .1 1.2 Essentials on fusion research . .2 1.3 Tungsten as a material . .4 2 Density functional theory and its implementation in BigDFT5 2.1 The density functional theory (DFT) . .5 2.1.1 The Hohenberg-Kohn theorems . .7 2.1.2 The Kohn-Sham method . .9 2.2 Solid-state physics in a nutshell . 11 2.3 Numerical methods and further approximations . 13 2.3.1 Pseudopotentials . 14 2.3.2 Linear scaling approach . 15 2.3.3 Particularities of metallic systems . 17 2.4 Wavelets . 18 2.4.1 The Haar wavelet family, an iconic approach to wavelets . 18 2.4.2 Three dimensional wavelets . 21 2.5 BigDFT; the code . 21 2.5.1 Implementation of wavelets and resolution levels . 22 2.5.2 Implementation of the linear scaling approach . 23 3 Study of point defects by means of BigDFT 27 3.1 Brief introduction to point defects . 27 3.2 Ab initio W-crystal simulations by means of BigDFT . 29 viii Table of contents 3.3 Full system approach . 30 3.4 Fragment approach . 32 3.4.1 Atomic fragments . 35 3.4.2 Onion-alike cluster fragments . 36 3.4.3 Two cluster fragments . 39 3.5 Brief study of the CPU time consumption . 41 4 Conclusions and future work 43 4.1 Conclusions . 43 References 45 List of figures 1.1 Magnetic confinement devices: tokamak and stellarator . .3 1.2 Representations of different atoms distributions; minimal cell and lattice . .4 2.1 Example of a Wigner-Seitz cell . 12 2.2 Representation of the pseudopotential approximation . 15 2.3 Density of states in a metal . 17 2.4 The Haar scaling function f and the wavelet y ............... 19 2.5 Piecewise function by means of the scaling-function of the Haar wavelet . 19 2.6 Scaling function as linear combination of a scaling function and a wavelet . 19 2.7 Scaling function f and wavelet y functions contribution . 20 2.8 Least asymmetric Daubechies wavelet family of order 2m = 16 . 22 2.9 Visualisation of the fine and coarse regions of an example molecule . 23 2.10 Flowchart of the minimal basis approach . 26 3.1 Scheme of two different point defects in a 250-atom tungsten crystal . 28 3.2 Formation energies of various perfect and defect configurations . 31 3.3 Flowchart of the fragment approach . 33 3.4 Visualisation of the fragments . 34 3.5 Energy per atom for different perfect crystals through aromic fragmentats . 35 3.6 VFE for different atomic fragmentation for the 250-atoms crystal . 36 3.7 VFE for different cluster fragmentations for the 250-atoms crystal . 37 3.8 VFE for different cluster fragmentation and system sizes . 38 3.9 VFE for different cluster fragmentation and system sizes (bis) . 38 3.10 Visualisation of some two cluster shells systems . 39 3.11 VFE for different two cluster-shell fragmentations for the 250-atoms crystal 40 3.12 VFE for different 2 cluster fragmentations for the 250-atoms crystal (bis) . 40 3.13 CPU time since initialisation of the template and fragment calculations . 41 3.14 CPU time since initialisation of the template and fragment calculations . 42 List of tables 2.1 Three-dimensional scaling function and seven three-dimensional wavelets . 21 3.1 Basic parameters used in the full system approach . 30 3.2 VFE for different cells in the full system approach . 31 Chapter 1 Introduction 1.1 Scientific context The next decades are crucially important to guiding the world on a path towards lower greenhouse gas emissions. However, the world’s population will continue to grow and the proportion of population living in cities is expected to keep increasing [5], which means that even more sustainable energy will be needed later in the century. Nuclear fusion is one of the few zero greenhouse gas emissions energy sources with the potential of replacing fossil fuels. In addition to that, fusion energy is inherently safe, with zero possibility of a meltdown scenario and no long lived waste, and depends on a virtually infinite fuel: water. But so far, this is just a promise and commercial stations are unlikely to materialise before 2050 [6]. Nuclear fusion is the process by which two or more atomic nuclei come close enough to form one or more different atomic nuclei and subatomic particles (neutrons or protons). It is the process that powers stars and trough which the different elements are created. Among all fusion processes, the main reaction fuelling the stars is the combination of light hydrogen atoms (H) to produce the heavier element helium (He) and this is as well the process being regarded to as an energy source on the Earth. In particular, due to its high cross section, the most efficient fusion reaction in the laboratory is the reaction between two hydrogen isotopes; 2 3 deuterium 1H and tritium 1H. 2 3 4 1 1H + 1H ! 2He + 0n (1.1) 5 The intermediate product of the reaction is an unstable 2He nucleus, which immediately 4 ejects a neutron with 14:1MeV. The recoil energy of the remaining 2He nucleus is 3:5MeV, so the total energy liberated is 17:6MeV [7]. 2 Introduction At very high speeds, when the ionised hydrogen atoms collide, the natural electrostatic repulsion that exists between their positive charges is overcome and the atoms fuse. The mass of the resulting He atom, however, is not the exact sum of the initial H atoms; as Einstein’s famous formula E = mc2 describes, some mass m is lost and a great amount of energy E is released [8]. In the Sun core, this process takes place at temperatures of around 1;5 × 107 K. However, without the benefit of gravitational forces at work in the Universe and the resulting pressure in the Sun, achieving fusion on Earth has required a different approach and, as explained in Section 1.2, even higher temperatures [6]. Apparently, no known material can cope with that heat, and hence the process needs to be confined in vacuum. Nevertheless, due to the direct contact with its hot edges,the identification and study of suitable materials to construct the container of the nuclear fusion vacuum is of central interest.