In materials science, the threshold displacement energy is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. It is also known as "displacement threshold energy" or just "displacement energy". In a crystal, a separate threshold displacement energy exists for each crystallographic direction. Then one should distinguish between the minimum and average over all lattice directions' threshold displacement energies. In amorphous solids, it may be possible to define an effective displacement energy to describe some other average quantity of interest. Threshold displacement energies in typical solids are of the order of 10-50 eV.[1] [2] [3] [4] [5]
The threshold displacement energy is a materials property relevant during high-energy particle radiation of materials.The maximum energy
Tmax
Tmax={2ME(E+2mc2)\over(m+M)2c2+2ME}
where E is the kinetic energy and m the mass of the incoming irradiating particle and M the mass of the material atom. c is the velocity of light.If the kinetic energy E is much smaller than the mass
mc2
Tmax=E{4Mm\over(m+M)2}
In order for a permanent defect to be produced from initially perfect crystal lattice, the kinetic energy that it receives
Tmax
Each crystal direction has in principle its own threshold displacement energy, so for a full description one should know the full threshold displacement surface
Td(\theta,\phi)=Td([hkl])
Td,min=min(Td(\theta,\phi))
Td,ave={\rmave}(Td(\theta,\phi))
An additional complication is that the threshold displacement energy for a given direction is not necessarily a step function, but there can be an intermediate energy region where a defect may or may not be formed depending on the random atom displacements.The one can define a lower threshold where a defect may be formed
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It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Hence theoretical study of the threshold displacement energy is conventionally carried out using either classical [8] [9] [10] [11] or quantum mechanical [12] [13] [14] [15] molecular dynamics computer simulations. Although an analytical description of thedisplacement is not possible, the "sudden approximation" gives fairly good approximationsof the threshold displacement energies at least in covalent materials and low-index crystaldirections
An example molecular dynamics simulation of a threshold displacement event is available in 100_20eV.avi. The animation shows how a defect (Frenkel pair, i.e. an interstitial and vacancy) is formed in silicon when a lattice atom is given a recoil energy of 20 eV in the 100 direction. The data for the animation was obtained from density functional theory molecular dynamics computer simulations.
Such simulations have given significant qualitative insights into the threshold displacement energy, but the quantitative results should be viewed with caution.The classical interatomic potentials are usually fit only to equilibrium properties, and hence their predictive capability may be limited. Even in the most studied materials such as Si and Fe, there are variations of more than a factor of two in the predicted threshold displacement energies. The quantum mechanical simulations based on density functional theory (DFT) are likely to be much more accurate, but very few comparative studies of different DFT methods on this issue have yet been carried out to assess their quantitative reliability.
The threshold displacement energies have been studiedextensively with electron irradiation experiments. Electrons with kinetic energies of the order of hundreds of keVs or a few MeVs can to a very good approximation be considered to collide with a single lattice atom at a time.Since the initial energy for electrons coming from a particle accelerator is accurately known, one can thusat least in principle determine the lower minimum threshold displacement
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There are several complications in interpreting the experimental results, however. To name a few, in thick samples the electron beam will spread, and hence the measurement on single crystalsdoes not probe only a single well-defined crystal direction. Impurities may cause the thresholdto appear lower than they would be in pure materials.
Particular care has to be taken when interpreting threshold displacement energiesat temperatures where defects are mobile and can recombine. At such temperatures,one should considertwo distinct processes: the creation of the defect by the high-energyion (stage A), and subsequent thermal recombination effects (stage B).
The initial stage A. of defect creation, until all excess kineticenergy has dissipated in the lattice and it is back to itsinitial temperature T0, takes < 5 ps. This is the fundamental("primary damage") threshold displacement energy, and also the oneusually simulated by molecular dynamics computer simulations.After this(stage B), however, close Frenkel pairs may be recombinedby thermal processes. Since low-energy recoils just above thethreshold only produce close Frenkel pairs, recombinationis quite likely.
Hence on experimental time scales and temperatures above the first(stage I) recombination temperature, what one sees is the combinedeffect of stage A and B. Hence the net effect often is that thethreshold energy appears to increase with increasing temperature,since the Frenkel pairs produced by the lowest-energy recoilsabove threshold all recombine, and only defects produced by higher-energyrecoils remain. Since thermal recombination is time-dependent, any stage B kind of recombination also implies that theresults may have a dependence on the ion irradiation flux.
In a wide range of materials, defect recombination occurs already belowroom temperature. E.g. in metals the initial ("stage I") close Frenkelpair recombination and interstitial migration starts to happen alreadyaround 10-20 K.[18] Similarly, in Si major recombination of damage happens alreadyaround 100 K during ion irradiation and 4 K during electron irradiation[19]
Even the stage A threshold displacement energy can be expectedto have a temperature dependence, due to effects such as thermalexpansion, temperature dependence of the elastic constants and increasedprobability of recombination before the lattice has cooled down back to theambient temperature T0.These effects, are, however, likely to be much weaker than the stage B thermal recombination effects.
The threshold displacement energy is often used to estimate the total amount of defects produced by higher energy irradiation using the Kinchin-Pease or NRTequations[20] [21] which says that the number of Frenkel pairs produced
NFP
FDn
NFP=0.8{FDn\over2Td,ave
for any nuclear deposited energy above
2Td,ave/0.8
However, this equation should be used with great caution for severalreasons. For instance, it does not account for any thermally activated recombination of damage, nor the well known fact that in metalsthe damage production is for high energies only something like20% of the Kinchin-Pease prediction.
The threshold displacement energy is also often used in binary collision approximation computer codes such as SRIM[22] to estimatedamage. However, the same caveats as for the Kinchin-Pease equationalso apply for these codes (unless they are extended with a damagerecombination model).
Moreover, neither the Kinchin-Pease equation nor SRIM take in any wayaccount of ion channeling, which may in crystalline orpolycrystalline materials reduce the nuclear depositedenergy and thus the damage production dramatically for someion-target combinations. For instance, keV ion implantationinto the Si 110 crystal direction leads to massive channelingand thus reductions in stopping power.[23] Similarly, light ion like He irradiation of a BCC metal like Feleads to massive channeling even in a randomly selectedcrystal direction.[24]