Molecular dynamics simulations of threshold displacement energies in Fe

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Abstract

We compare systematically the threshold displacement energy surface of 11 interatomic potentials in Fe. We discuss in detail different possible definitions of threshold displacement energies, and how they relate to different kinds of experimental threshold displacement energies. We compare the threshold results to experiments, and find that none of the 11 tested potentials agrees fully with experiments. However, all the potentials predict some qualitative features in the same way, most importantly that the threshold energy surface close to the 1 0 0 crystal direction is flat and that the largest threshold energies occur around very roughly the 1 2 3 crystal direction.

Introduction

One of the basic quantities defining the radiation resistance of a material is the threshold displacement energy, i.e. the energy needed to displace an atom in a material to create a stable Frenkel pair. The concept of threshold displacement energy was probably devised by Wigner in the early 1940s, as reported by Burton [1], and already in 1949 it appeared as a functional parameter in Seitz’s model to treat elastic collisions [2], where it was assessed as equal to the sum of the cohesive energy plus the formation energy of the Frenkel pair (in total about 25 eV). Since then it has played a key role in radiation damage theory. For example, if the amount of radiation-induced defects increases linearly with energy, the damage level can be well predicted by the Kinchin–Pease (or its variation NRT [3]) equation which states that the amount of damage is proportional to the ratio of the nuclear deposited energy and an effective threshold displacement energy Ed [4]. Even in materials where the Kinchin–Pease equation is not valid, typically dense metals, the damage level is often given in terms of a cascade efficiency which is the actual number of defects compared to the Kinchin–Pease prediction [5], [6]. Because of this, it is of importance to know the value of the threshold displacement energy in any material where irradiation effects are of interest.

The threshold displacement energy has been studied both experimentally and by computer simulations in a wide range of materials (see e.g. [7], [8], [9], [10] and references therein). From an application point of view, of particular interest is the threshold displacement energy in Fe. Radiation damage in Fe-based materials is of great interest because the main structural materials in fission and fusion reactors are steels. In addition, it is possible to use ion implantation to harden steels. Thus it is surprising that the threshold displacement energy in Fe has in fact been studied less than in many other materials. The average threshold displacement energy most frequently used for Fe, the so-called NRT or ASTM standard, is 40 eV [11]. Its source in literature review papers [7], [8] is often cited to be [10], but this paper is also a review and bases its value on the MD simulations carried out by Erginsoy et al. in 1964 [12]. There is one experiment by Lucasson which gives an average threshold energy of 24 eV for Fe [13], but in his later review papers even Lucasson himself does not use this value [10], apparently because the result is dependent on the choice of the damage model. The experiments which do exist give only the threshold energy along the low-index lattice directions 1 0 0 [14], 1 1 0 and 1 1 1 [15], [16], not the average over all directions, that would be most appropriate for the effective threshold displacement energy used in the Kinchin–Pease formulation.

There have been significant advances in not only computer capacity but also in the understanding of interatomic interactions since 1964 [17], [18], [19], including additional theoretical works on the threshold displacement energy in Fe. Agranovich and Kirsanov [20] studied the threshold energies close to 1 0 0 and 1 1 1 including thermal displacements (in contrast to the work of Erginsoy et al. that was carried out at 0 K) and obtained threshold energies of 18 eV around 1 0 0 and 26 eV around 1 1 1, in fairly good agreement with the experiments of Lomer and Pepper [15]. Apparently the first systematic simulations of threshold energies in Fe employing many-body potentials were carried out by Bacon et al. [21] who in 1993 simulated threshold energies with the Finnis–Sinclair potential [17], modified in the repulsive part [22]. They obtained thresholds of 18 eV around 1 0 0, 30 eV around 1 1 0 and >70 eV around 1 1 1 at 0 K. Soon after this, Doan and Vascon [23] adjusted another Fe potential [24] with a repulsive potential in a manner which gave good agreement with experiments [23]: 21 eV around 1 0 0, 31 eV around 1 1 0 and 18.5 eV around 1 1 1. Also several other, less detailed, studies of the threshold displacement energies have been carried out in the context of adjusting the repulsive part of the potentials to have a realistic high-energy part (see Section 2.3). However, none of the works on the threshold energy in Fe have affected the NRT standard. Moreover, the works have used slightly different (and sometimes poorly documented) definitions of what the threshold energy is, especially regarding whether it is calculated in the exact crystallographic direction, or in some angular interval around it to account for electron beam spreading. Hence it is of interest to review the threshold energies given by different models using the same threshold energy definitions for all the potentials.

In the current paper we systematically reexamine the issue of the threshold displacement energy in Fe. We simulate the full three-dimensional threshold energy surface using 11 different interatomic potentials, taking care that all non-physical simulation parameters (such as the simulation cell size) are chosen so that they do not affect the end result. We compare the results of all potentials with each other and experiment. We also discuss the original simulations by Erginsoy et al. in view of the present simulations.

Section snippets

Definition of threshold displacement energy

It might seem to be straightforward to define a threshold displacement energy of a material. However, one can in fact define several different threshold displacement energies depending on the viewpoint and the experimental situation one wishes to model. Since distinguishing between these is important for understanding some of the results of this paper, we review here different possible definitions.

The most straightforward distinction comes from consideration of irradiation geometry. First of

Results for different threshold definitions

We used the Ackland potential (“ABC”) to test how the different threshold definitions described above affect the result quantitatively. We obtained values of Ed,avepp=63±1eV, Ed,aveav=44.8±0.4eV, and Ed,avema=32.4±1.2eV. The corresponding median values are Ed,medpp=44eV, Ed,medav=41eV, and Ed,medma=33eV. The differences in the values can be understood as follows. The “ma” values are always the lowest because they are based on the minimum in an interval. The “pp” values are the largest because

Conclusions

We have shown that contrary to a common assumption, the damage production curve as a function of increasing recoil energy for a given atom is not a monotonously rising curve, but can in fact sometimes go to zero after being one already. The probability of this is large enough that it can affect an average threshold significantly.

The methodology used here allows systematic testing of future Fe potentials against threshold displacement experiments.

In our comparison of threshold energies

Acknowledgements

This research was supported by Association Euratom-TEKES, by the Academy of Finland under project No. 205729, and by a travel grant from the Magnus Ehrnrooth foundation. We thank M.Sc. Niklas Juslin for carrying out the threshold energy calculations of the FS-CB potential. This research was made possible by generous computer capacity available on the “mill” and “ametisti” Linux GRID clusters of the Departments of Physical Sciences, Chemistry and the Helsinki Institute of Physics.

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