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INTRODUCTION
Protactinium is the first element in the actinide series with a 5f electron.1 In some systems (PaO2, UO2, NpO2, PuO2), the presence of 5f electrons leads to an improper description of the electronic and the structural properties by DFT.2–7 Several approaches have been developed to overcome these shortcomings of DFT such as the self-interaction correction (SIC), DFT hybrid approach, DFT + U, GW etc. The DFT + U (LDA + U and GGA + U) approach has been shown to effectively correct many of the deficiencies observed by this class of materials with regards to the band gap.8–10 This approach introduces an on-site Hubbard U and the Hund’s J. Dudarev et al.11 combined these two terms in the rotationally invariant formalism to a single term Ue f f=U−J, referred to as the simplified LSDA + U. The introduction of this adhoc parameter leads to a significant improvement in the description of some 5f systems where DFT fails but the choice of the value of this parameter is debatable. The Ue f f parameter can be obtained: (i) from constrained LDA approach,8 (ii) from constrained RPA,12 (iii) from a self-consistent approach,13 and (iv) from fitting the parameter to experimental observables.
Pa and its oxide are used in scintillators for detecting X-rays, for radioactive dating (determination of ancient artifact), in cathode ray tubes with bright green fluorescence, as high temperature dielectrics for ceramic capacitors, in nuclear weapons (used as support in nuclear chain reactions), etc.15 These materials have downsides which has limited its study, namely (i) the high toxicity and radioactive nature and (ii) the limited availability. These materials are mainly found as a byproduct
in nuclear reactions. Furthermore, these materials are (iii) strongly correlated, giving rise to failures by conventional DFT methods.
The understanding of the electronic and structural properties is paramount to harness and control these materials optimally. The elements in the 5f series with localized electrons possess greater atomic volumes and symmetrical structures compared to those with itinerant electrons.16,17 Hence, Pa exists in the bct, fcc and orthorhombic (-U) phase at standard temperature and pressure (STP), high temperature and high pressure respectively. Pa metal possesses itinerant 5f electrons with characteristic properties such as superconducting and high vaporization.15,18,19 There has been significant interest in the understanding of Pa metal,20 due to its unique behavior as the first element with a 5f electron. The effect of pressure induced phase transitions has been explored theoretically and experimentally. Haire et al.21 experimentally showed that Pa metal undergoes a pressure induced structural phase transition at 77 GPa from the bct structure (spacegroup I4/mmm) to an orthorhombic, -U structure (space group Cmcm) with 30% atomic volume reduction and no subsequent phase transition. This was predicted by theoretical studies,16,22 which attribute this to the increase in 5f electron participation in the bonding.
The studies of protactinium oxides (PaO and PaO2) have been limited to the mode of synthesis and purification for the production of Pa metal.15 Prodan et al.23 used the density functional hybrid approach to study actinide dioxides and obtained a band gap of 1.4 eV for PaO2. To the best of our
knowledge, no further studies have been performed to shed more light on the electronic structure of PaO2. Hence, a proper description of the electronic and structural properties of these oxides within the DFT and DFT + U methods is appropriate to understand the physics of these materials.
A systematic first principles study of the dependence of the effective Hubbard-U parameter on LDA + U and GGA + U functionals in Pa and its oxides is lacking. In this work, the DFT methodology is applied to calculate the electronic and structural properties of Pa metal. Using a semi-empirical fit, we then optimize the Hubbard U parameter to give a proper description of PaO and PaO2. The structural and electronic properties of these oxide materials are determined to investigate the role played by the 5f electrons.
In Section II, a brief description of the theoretical and computational methodologies is presented. In Section III, the results and discussions for Pa and its oxides are presented, followed by the conclusions in Section IV.
THEORY AND COMPUTATIONAL DETAILS
All calculations were performed using density functional theory24 as implemented in the VASP code.25 The electron wave functions were described using the projector augmented wave (PAW) method of Bl¨ochl in the implementation of Kresse and Joubert.26 The LDA27 and PBE28 form of the GGA exchange-correlation potentials were used together with their LDA + U and GGA + U variants. An adequately converged kinetic energy cutoff of 500 eV, 500 eV, 550 eV was chosen to ensure fully converged total energies for Pa, PaO and PaO2 respectively. A spacing of k-points of 0.2/Å fine mesh for the Monkhorst-Pack29 grid is used to sample the Brillouin zone, and Methfessel-Paxton smearing30 with a width of 0.2 eV was used to integrate the band at the Fermi level. The total energy, electronic band structure and density of states (DOS) were calculated using the tetrahedron integration method with Bl¨ochl corrections.parameter was computed by optimizing the lattice parameter with respect to U. This resulted in differences for a range of results involving the physical structure, the electronic structure and the energetics within the GGA and LDA schemes for the oxides. Results presented in this work uses LSDA, GGA, LSDA + U and GGA + U for the oxides but without the U parameter for Pa atom.
1 Introduction
1.1 Actinide systems
1.2 Aims and objectives
1.2.1 Thorium based alloys
1.2.2 Eect of Hubbard U parameter
1.3 Thesis structure .
2 Theoretical Framework
2.1 Many-Body Problem
2.2 Density Functional Theor
2.3 Kohn-Sham equations
2.3.1 Local Spin Density Approximation
2.4 Algorithms used in the implementation of the Kohn-Sham equation
2.4.1 Plane wave formalism
2.4.2 Pseudopotential
2.4.3 Projector augmented wave method .
2.4.4 Brillouin zone integration .
2.4.4.1 Linear tetrahedron method .
2.4.4.2 Special k-points .
2.4.4.3 Fermi level smearing .
2.4.5 Atomic relaxations
2.4.5.1 Hellmann-Feynman theorem
2.5 Elastic Properties .
2.6 Lattice Dynamics
2.7 DFT+U
2.7.1 Rotationally-invariant formulation
2.7.2 A simplifier formulation
2.8 Software code
3 Ab initio studies of Th3N4, Th2N3 and Th2N2(NH)
4 A theoretical study of thorium titanium-based alloys
5 First principles LDA + U and GGA + U study of protactinium and protactinium oxides: dependence on the eective U parameter
6 GGA + U studies of the early actinide mononitrides and dinitrides
7 General conclusions
References
