Long time scale evolution of radiation-induced defects in Er 2 O 3 Lanchakorn Kittiratanawasin a , Roger Smith b,⇑ a Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK b FZD, Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany article info Article history: Received 26 July 2010 Received in revised form 14 October 2010 Available online 5 November 2010 Keywords: Temperature accelerated dynamics Vineyard method Erbia Point defect diffusion abstract The diffusion of point defects after an irradiation event in Er 2 O 3 is considered. It is shown that saddle point finding methods find one transition that is a rank-2 saddle and a method is proposed to determine the attempt frequency for the transition in this case. Different methods were used to study the point defect transitions which showed good agreement at the lower temperatures but which diverged at high temperature due to anharmonic effects. The diffusion coefficients for the point defects were calculated which indicated that the calculated values could differ by up to a factor of 3 if the exponential prefactor in the Arrhenius expression was not accurately determined. Point defect diffusion in a perfect crystal was shown to be independent of direction. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Many metal oxide materials are radiation tolerant and in a pre- vious paper [1] we investigated the radiation damage induced by collision cascades and calculated the energy barriers for point de- fect diffusion in Er 2 O 3 . The motivation for that study was to inves- tigate how radiation damage in oxide materials having the bixbyite structure differed from other materials such as spinel [2–4] and MgO [5,6]. Materials with the bixbyite structure have been exten- sively investigated by the group of Sickafus et al. [7–10]. The main difference was the higher energy barriers for point defect diffusion in Er 2 O 3 and the fact that the oxygen vacancy was the point defect with the lowest energy barrier whereas in MgO and spinel the iso- lated interstitials had the lowest energy barriers. In that paper the cascades were analysed by molecular dynamics (MD) over picosec- ond time scales after which the dynamics is dominated by rare events. The motion of the point defects which remained at the end of the cascade were then investigated in more detail and the point defect diffusion coefficients were calculated by temperature accelerated dynamics (TAD) [11]. Using this method the diffusion pathways are determined and as the system hops from one basin of attraction to the next, the energy barriers are accurately calcu- lated by the nudged elastic band (NEB) or climbing image nudged elastic band method (CINEB) [12,13]. The TAD method is based on the assumption of harmonic tran- sition state theory with the hopping rate m being given by m ¼ m 0 expðE=kT Þ for a given energy barrier E where m 0 is the (often assumed con- stant) prefactor, T is temperature and k is Boltzmann’s constant. The method works well when there are not lots of small energy barriers surrounding a local minimum energy configuration and takes into account that events which occur at a high temperature may not necessarily occur in the same order as at a lower temper- ature. The method can be fairly expensive on computational time and an alternative method to TAD would be to determine the tran- sitions by a method that finds the saddle points surrounding a local minimum energy configuration by more direct means, e.g. the dimer, ART or RAT methods [14–16] and then determine the prefactor m 0 by the Vineyard method [17]. This should in theory give the same values as those previously determined by TAD and be computationally less expensive, provided the methods can find all relevant transitions so that a full rate catalogue can be produced. In the Vineyard method the prefactor m 0 is calculated from the ratio of the product normal frequencies m i associated of the positive eigenvalues k i (where k i ¼ð2pm i Þ 2 ) of the Hessian matrix of the potential energy function at the minimum and the saddle. For a system on N atoms, m 0 ¼ Q 3N i m min i Q 3N1 i m saddle i ; where the negative eigenvalue at the saddle point does not occur in the product. There is thus an immediate assumption that the saddle point must be rank-1, i.e. have only one negative eigenvalue. This point will be explored later. However, it is worth noting that rank-2 saddles are being increasing investigated in the literature, see for example [18]. Since the main point defect diffusion processes and barriers were determined in the previous paper [1] we concentrate here 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.11.001 ⇑ Corresponding author. Present address: Loughborough University, Loughbo- rough LE11 3TU, UK. E-mail address: R.Smith@lboro.ac.uk (R. Smith). Nuclear Instruments and Methods in Physics Research B 269 (2011) 1712–1719 Contents lists available at ScienceDirect Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb