During the slowing down process of energetic atoms in a solid, the kinetic energy of a moving ion is partially transferred to host atoms by elastic collisions. The recoiling atoms, in turn, transfer part of their energy to other atoms, etc. Hence a cascade evolves resulting in the formation of a highly disrupted, very hot, region inside the solid. This phenomenon, which is well known in the case of ion-implantation, is called a "thermal spike". It can be viewed as the short term local melting of the implantation affected region. This melting is followed by a rapid quenching of the liquid phase to form a damaged, amorphous, solid structure. This is the basis of the most widely used computational method to create amorphous materials, including carbon, which consists of rapidly quenching a liquid phase of carbon by ab initio [43,44] or tight-binding  molecular dynamics. By investigating the mean square displacement of the atoms during the cooling of the melt and the time required for it to relax, Marks et. al. [44,46] evaluated the life-time of the thermal spike in diamond to be less than 1 picosecond, for energetic impacts below 400 eV. The calculations of Refs. [44,46] have shown that a rapid quench of liquid carbon can produce a tetrahedrally bonded amorphous carbon, similar to the material produced experimentally [39,42] by the deposition of energetic (of the order of a few tens to hundreds of eV) carbon atoms (Ref. ).
The aim of the present study was not to produce an optimized tetrahedral amorphous carbon (t-aC) material, in contrast to the work of Marks , but rather to generate a damaged region embedded inside a diamond matrix and to investigate its pre- and post-annealing properties. The kinetic energy imparted herein to the diamond carbon atoms (416 eV) is however similar to the ion energy commonly used experimentally for tetrahedral amorphous carbon film growth. It is also similar to that used by Marks in his calculations, and therefore our thermal spike life-time should be of the same order of magnitude as that evaluated in Refs. [44,46]. Marks et. al.  found from mean square atomic displacement data that rearrangement of the atoms occurred while the temperature of the sample fell within the range 5000-3000 K. The time spent in this temperature range was therefore defined to be the cooling time. However, measurements suggest that interstitial diffusion can still occur in diamond at 300 K and vacancy diffusion at 800 K, thus another criterion may have to be employed to define the time for atomic rearrangement. Since atomic rearrangement involves transitions between threefold and fourfold coordinated atoms, the variation of the coordination number with time during and shortly after the bombarding event should define the life-time of the thermal spike.
In order to define the coordination number, we have calculated the radial distribution function g(r) of the structures obtained after successive atomic bombardment. We will return to a detailed discussion of g(r) in the next section. g(r) exhibits a clear gap, centered at about 1.9 Å, separating the first and the second peak, independently of the number of atoms energetically displaced. All atoms within the sphere of radius 1.9 Å are thus assumed to comprise the first nearest neighborhood of a given atom. Hence, the number of neighbors of each atom within a distance of 1.9 Å determines the coordination number. We have calculated the number of threefold coordinated atoms in the whole sample as a function of time, for the case of a single bombarding atom with an initial kinetic energy of 416 eV. Energy dissipative boundary conditions were applied to the sample, with fixed atoms in the outermost layer. The results are shown in figure (9.5) (solid line). The life-time of the thermal spike can thus be estimated to be 0.2 ps, very close to 0.21 ps calculated by Marks , who solved the heat diffusion equation for an ion with an energy of 400 eV impinging on a diamond surface.
The same calculations were performed with free boundary conditions, to study the influence of the boundary on the damaged region. The results are shown in figure (9.5) (dashed line). As can be seen, the results of these calculations are nearly indistinguishable from those obtained for the fixed boundary conditions, showing that similar structural rearrangement occurs within subpicoseconds. Thus, we concluded that our sample is large enough to insure that the cooling time is determined not by the boundary conditions but by the diamond matrix itself that rapidly dissipates the high local temperature.
The next point that needs to be clarified is the temperature range of the melted and subsequently cooled region during the life-time of the thermal spike. In the papers of Marks (Refs. , and ) this temperature range is used to determine the cooling rate. For the present study, we need to define the area that encloses the disrupted region in order to evaluate this temperature range. For this purpose, we defined a radius of gyration where is the center of mass of the threefold coordinated atoms, points to the ith atom and n is the number of threefold coordinated atoms. Rn thus reflects the spatial distribution of the threefold coordinated atoms.
For one bombarding atom, Rn is calculated to be 4.5 Å, and the disrupted region (that contains threefold coordinated atoms) is found to extend roughly up to 2 Rn from . The area that encloses the defects may thus be defined by a sphere of radius 2 Rncentered on . This area contains 540 atoms. The average temperature within this area is shown in figure (9.6) as a function of time for fixed (solid line) and free (dashed line) boundary conditions (with energy dissipative conditions in both of the cases), for a single ion bombardment. As can be seen from figure (9.6), the initial temperature in this region is 5100 K, leading to the local melting of the structure. Within a time of 0.2 ps (the life-time of the thermal spike as calculated above), the temperature decreased to 900 K. Since this cooling rate was found to be similar for the two different boundary conditions, we can conclude that the dissipation of the initial high temperature during the thermal spike is predominantly due to the diamond matrix that encloses the disrupted region, and the remaining energy is dissipated at the boundaries. By applying this procedure, with consecutive bombardment of twelve atoms, we were able to create a heavily damaged region embedded in a diamond matrix.