Joint ICTP-IAEA Workshop on Fusion Plasma Modelling using Atomic and Molecular Data January 2012
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1 Joint ICTP-IAEA Workshop on Fusion Plasma Modelling using Atomic and Molecular Data January 2012 Qunatum Methods for Plasma-Facing Materials Alain ALLOUCHE Univ.de Provence, Lab.de la Phys. des Inter.Ioniques et Moleculaires, Marseille, cedex 20 FRANCE
2 Quantum methods for plasma-facing materials A.Allouche Physique des Interactions Ioniques et Moléculaires Marseille, FRANCE A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling
3 When ab initio calculations? Physical sputtering, high energy particles impinging! no use Molecular dynamics! the systems DFT (ab initio MD) can investigate are several orders of magnitude smaller than classical MD Quantum theory!!! when chemistry is concerned because chemistry is essentially a problem of electrons (exchange of electrons, sharing electrons..) Fundamental mechanisms of hydrogen atoms retention and release Barrier of activation and Transition states (diffusion, trapping,..), QT gives the barrier to overcome, the problem of HOW is dynamics Structure of materials and alloys (C/Be/W) Electronic structure: DOS, spectroscopy... A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling
4 Quantum method The time-independent Schrödinger equation H! = E! For a many-body system where H is an operator n H =! 1 n # 2 " + V & ) i N $ % i ' ( + ) ) 1 i=1 i=1 j<i r ij The SE can be solved exactly for very few systems, for atoms and molecules the variational method allows to develop the approximate wave function! in an orbital basis set " (most often gaussian-type orbitals). A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 3
5 Hartree-Fock The Hartree-Fock method makes four major simplifications: * The Born-Oppenheimer approximation is inherently assumed. The full molecular wavefunction is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons. * Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic. * The variational solution is assumed to be a linear combination of a finite number of basis functions. The finite basis set is assumed to be approximately complete. * Each energy eigenfunction is assumed to be describable by a single Slater determinant, an antisymmetrized product of one-electron wavefunctions (i.e. orbitals). A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 4
6 Molecular orbital The Fock operator: J: Coulomb (classical) K: Exchange (quantum) Correlation energy!e correl = E exact " lim(e HF ) basis A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 5
7 Calculating the correlation energy: Interaction of Configurations Configuration interaction (CI) is a post Hartree-Fock method for solving the non-relativistic Schrödinger equation within the Born-Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination of Slater determinants used for the wave function. In terms of a specification of orbital occupation interaction means the mixing (interaction) of different electronic configurations He ground state (1s 2 ) 2s He excited states: 1s The solution can be perturbation (Møller-Plesset, MP2) or variational (Full CI, CCSD, CCSD(T),..) Problem: cutting the series? A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 6
8 Density functional theory In DFT the key variable is the particle density, which for a normalized! is given by for a given ground-state electron density it is possible, in principle, to calculate the corresponding ground-state wave-function!! 0 is a unique functional of n(0). Every quantity O is a functional of n(0) In the Schrödinger equation T(n) and U(n) are called universal functionals, V(n) is called a non-universal functional, as it depends on the system under study A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 7
9 First, one considers an energy functional that doesn't explicitly have an electron-electron interaction energy term Thus, one can solve the so-called Kohn-Sham equations of this auxiliary non-interacting system which yields the orbitals that reproduce the density of the original many-body system. The effective single-particle potential can be written in more detail as: where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term is called the exchange-correlation potential The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas, therefore approximations must be found A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 8
10 Solid state DFT applications A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 9
11 Computational details PBE GGA functional + ultrasoft pseudopotentials and plane wave Bloch functions Systems ~ 100 atoms Energy profiles (PES): each configuration is fully optimized (BFGS gradient optimization) excepted the selected reaction coordinates. The barriers of activation are calculated using the Nudged Elastic Band theory BZ sampling 6x6x1 (surface) 6x6x6 (bulk) 10
12 Graphite A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 11
13 Multiple adsorption Two patterns : 2 weakly bonded doublets 1 Strongly bonded quartet No single adsorption A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 12
14 Phonons calculation vs. HREELS Experimental Spectra Phonon at # point A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 13
15 Beryllium - tungsten mixed material A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 14
16 The goal: giving a quantum description of the first steps of Be/W alloying starting from surface reactions W on Be! alloy formation With Ch.Linsmeier s permission Be on W! no stable alloy A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 15
17 Energy profiles Be on/in W(100) Adsorption energy -2.3 ev Trapping energy: endothermic Adsorption energy -4.2 ev Trapping energy -4.8 ev W on/in Be(0001) A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 16
18 Be(0001) surface + W atom DOS 1/ The Fermi energy falls in the middle of the partly filled W(5d) and Be(2s2p) bands 2/ Negligible Be(0001) surface reconstruction 3/ The surface and sub-surface DOSs are not very different Exothermic W adsorption & trapping Stable alloy formation A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 17
19 W(100) surface + Be atom DOS Non-interacting system - Atomic Be (2s) 2 (2p) 0 the Be(2p) band is empty - W(5d) band is half filled Binding Be - W(100) needs: electron transfer Be(2s)! Be(2p) then sp 2 hybrid orbitals whose projected DOS will extend on both sides of E F combination Be(sp 2 ) - W(5d) A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 18
20 W(5d) DOS Because of surface reconstruction, the surface DOS is very different from those of inner layers The shortest is the distance between the maximum of W(d) band to the Fermi energy, the most efficient the interaction is A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 19
21 A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 20
22 With coverage: Vertical interaction Lateral interaction Vertical interaction is predominant Be clusterization A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 21
23 Be(0001) oxidation Molecular oxygen ground state is a spin-triplet state (~ bi-radical). Its dissociation could be activated 1. The barrier is due to T~S conical intersection A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 22
24 Multiple dissociative adsorption The minimum is reached for 3 molecules A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 23
25 Hexagonal superstructure The oxidized layer tends to dissociate from the crystal A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 24
26 The DOS interpretation Be - O Be - Be In average, each oxygen atom gains 0.2 electrons from the surrounding beryllium atoms included in the layer. Be - O On the total each O(2p) shell now bears nearly 5 electrons A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 25
27 Zoom on the surface layer 2pz Oxygen nonbonding energy levels localized above and below the surface Pauli repulsion A.Allouche: Quantum methods Be - O bonds Be - Be interlayer bonds Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 26
28 A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 27
29 Diffusion processes in beryllium: contribution to H/D desorption mechanisms M. Reinelt & Ch. Linsmeier 1 kev D + ions implantation with Ch.Linsmeier s permission A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 28
30 Mono-vacancy in Be Formation energy: 0.96 ev Activation barrier to vacancy diffusion: TS = 0.72 ev isotropic A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 29
31 Be diffusion in beryllium crystal Self interstitial formation energy 4.18 ev Low barriers to diffusion ev ev ev ev Anisotropy of the Be crystal A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 30
32 H interstitial Formation energy 1.82 ev Diffusion // to the hexagonal plane: TS 0.41 ev Normal to the plane: TS 0.74 ev A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 31
33 H in vacancy Activation energies //! 1.70 ev " 0.85 ev (Interstitial diffusion energy from 0.41 ev to 0.74 ev)!! 1.60 ev " 0.63 ev A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 32
34 Hydrogen trapping in beryllium atomic vacancy n H Detrapping TS! (ev) Trapping TS " (ev) H-H Å Vacancy radius (Å) 0H H H H H H H H 2 isolated in a vacancy is not stable. Solution energy in Vc < in bulk A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 33
35 Relations H - vacancy Formation energy of a vacancy: - in the bulk: 0.96 ev - close to an interstitial H ev It is also well known that vacancies can be created at the surface during H desorption A.Allouche: Quantum methods Joint ICTP-IAEA Workshop on Fusion Plasma Modelling 34
36 Model for H/D desorption H $E in Vc < $E in bulk trapped first in Vc The regular interstitial Hs desorb first M. Reinelt & Ch. Linsmeier 1 kev D + ions implantation Desorption creates Vc at surface T Vc migration A ALLOUCHE, M OBERKOFLER, M REINELT AND CH LINSMEIER J. Phys.Chem. C 114, 3588 (2010) M OBERKOFLER, M REINELT, A ALLOUCHE, S LINDIG AND CH LINSMEIER Phys.Scr. T 138, (2009) A.Allouche: Quantum methods Some Hs interact with new Vc. Some others desorb from Vc
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