Energy Minimsation

As with any numerical method, convergence criteria are important parameters. During a DFT simulations, the start is an atomic conguration that is suspected to be near a global minimum, then the code will perform successive iterations of solving the DFT equations, in order to converge

towards the lowest energy solution. The simulation will only stop when successive iterations produce energy results with a change smaller than the convergence criteria, as selected by the user. This means that results obtained by DFT codes can be treated as being no more accurate than their convergence criteria (taking into account systemic errors inherent in DFT and in the code itself).

During a DFT run, there are multiple dierent levels of iterative convergence. The one discussed so far has generally been the electronic convergence (also known as Self Consistent runs), eectively determining if the system's electronic structure is in its ground state. In the simulations run in this work, electronic convergence is considered achieved if the dierence in

energy between successive iterations is less than 10−6 _eV.

A further form of convergence is that of geometry optimisation. As discussed, DFT provides information on the forces and energies present in a given atomic conguration. However, the goal often is to determine what the actual atomic structure of a crystal is. To do this, the initial conguration is input as the atomic positions of an experimentally determined crystal structure. After the electronic convergence has been achieved, the system's ions will be moved via a method specied in the DFT package. A number of dierent methods are available, including the conjugate gradient method, the quasi-Newtonian method and the damped-dynamics method. All of these methods represent a method for nding the geometrical conguration with the lowest energy. The work here uses the quasi-Newtonian scheme (the movement of the ions between steps is proportional to the force they would experience). The actual formalisation implemented in this DFT code is known as the Broyden-Fletcher-Goldfarb-Shanno algorithm [118]. Over successive iterations, ions are gradually moved to occupy the lowest energy state available, as dened by the geometry optimisation convergence criteria. This is a key tool in simulating materials, as it allows DFT codes to predict lattice parameters, symmetry groups and to examine how atoms move when inuenced by nearby defects. Like electronic convergence, a system is considered converged when successive changes produce only a small variation in energy. For this study,

a structural convergence criteria of 10−3 _{eV was considered acceptable, although convergence}

criteria were also based on the force between atoms, the displacement of atoms between iterations and the overall stress tensor. More specic details on what these parameters are set to is discussed with the relevant simulations.

SCF Ionic

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0

Electronic convergence criteria

Ionic convergence criteria

Sy stem Energ y (eV ) Iterations Finish Start

Figure 3.9: An illustration of how the energy of the system is expected to converge as further iterations are performed.

As a result of these criteria, it is important to note that all relaxed criteria are at the most thermodynamically favourable ground state. This is something that must be taken into account, particularly if predicting phenomena that would be expected at reactor operating temperatures, as the distribution of states (as dened by the Boltzmann distribution) would not necessarily be the ground state.

The term convergence has been used throughout this section. For clarication and summa- tion, the main forms of convergence as applicable to the work here are summarised below.

• Electronic Convergence: The phenomenon of how subsequent iterations of DFT calculations

approach the lowest energy (ground) state for the system's electronic conguration.

• Ionic Convergence: The phenomenon of how Electronically Converged systems can have

their geometry iteratively improved, such that their ionic conguration approaches it's lowest energy state. This is sometimes referred to as a relaxed structure.

• Convergence with respect to plane wave cut-o energy: The phenomenon of how a system

deciencies of DFT in general). The system is considered converged to within a given tolerance when increasing the cuto energy does not alter a specied value by more than that tolerance.

• Convergence with respect to K-point number: The phenomenon of how a system with more

K-points will be more accurate in representing a system (excusing the deciencies of DFT in general). The system is considered converged to within a given tolerance when increasing the number of K-points does not alter a specied value by more than that tolerance. This section should have provided an explanation of how DFT can be used to model solid materials and some of the considerations that must be taken on board when doing so. Thus, this thesis can now move onto the more important matters of what results have been found from an application of DFT to Zr alloys and the Zr-H system.

Chapter 4

Alloying Additions in Zirconium

Alloys

4.1 Overview

As discussed in chapter 2, a variety of dierent alloying elements are added to Zr in order to improve its resistance to creep, corrosion and its yield strength (see Table 2.1). This has produced a number of dierent commercially used alloys. All the Zr alloys used exhibit the low temperature, hexagonal-close-packed α-phase. β-phase Zr, with a body-centred-cubic structure

is stable above 863◦_{C, although metastable β-phase Zr does exist below this temperature [33].}

The β-phase, although not the primary component of in-reactor alloys, is still important as an intermediary phase during alloy processing and in Zr-Nb alloys as SPPs.

The alloying additions used will either form SPPs or remain in solid solution. The abundance, size and properties of all of these precipitates have a signicant impact on alloy corrosion resist- ance, mechanical properties and the tendency of the alloy to absorb hydrogen [119, 57, 71, 72, 70]. In order to understand the formation and dissolution of precipitates, a grasp of the various thermodynamic driving forces at work is essential. A knowledge of the relative stabilities of the dierent phases may allow prediction of aspects of the life-cycle of these precipitates and by extension, the properties that they aect in the host alloy. In this study, the alloying elements Cr,

Fe, Nb, Ni, Sn, V and Y are examined in Zr using DFT simulations. Atomistic scale techniques in general and DFT simulations specically, have demonstrated a degree of success when modelling alloys [120]. The thermodynamic driving forces for the formation of various intermetallic phases containing these elements are examined, as well as the solubilities of these elements and their intermetallic compounds in both α and β-Zr. This allows examination of the relative stabilities of various dierent phases in Zr alloys.

The work described in this chapter was published in June 2013 by the peer reviewed Journal of Nuclear Materials [121].