Permissions for Reuse

We study the translational, vibrational, and rotational energy spectra of atoms and molecules reflected or sputtered from deuterated amorphous carbon surfaces by impact of low-energy (1–30 eV) deuterium atoms. Both the rovibrational and translational energies of sputtered deuterium molecules are found to be close to 1 eV over the whole impact energy range, with approximate equipartition between rotational and vibrational modes, particularly at the higher impact energies. Sputtered carbon-containing molecules are vibrationally energetic, with rovibrational energies in the range of 1.5–2.5 eV; translational and rotational motions are less energetic, close to 0.5 eV, but hotter, with more energy per degree of freedom. The energy distributions of ejected molecules confirm the partial thermalization of the impact cascade. We also study the angular spectrum of the velocity of the outgoing particles as well as their angular momentum. While the velocity vectors are described well by a cosine distribution, a preferred direction of rotation is found at the lowest energies, with the angular momenta preferentially oriented parallel to the surface. ©2008 American Institute of Physics

Contents

• INTRODUCTION
• MOLECULAR DYNAMICS SIMULATION DETAILS
• AVERAGE ROVIBRATIONAL AND TRANSLATIONAL ENERGIES OF EJECTED PARTICLES
• ENERGY DISTRIBUTION OF EJECTED PARTICLES
• ANGULAR DISTRIBUTIONS OF EJECTED PARTICLES
• CONCLUSIONS
• ACKNOWLEDGMENTS
• REFERENCES
• FIGURES
• FOOTNOTES

Inelastic transitions resulting from atomic collisions in the plasma edge (scrape-off layer and divertor) regions of the tokamak plasma can have a major impact on the energy transfer and exhaust from the plasma core.¹ Various inelastic interactions of hydrogen ions and atoms with vibrationally and rotationally excited hydrogen molecules and molecular ions, especially charge transfer, are critical due to their significantly enhanced transition probabilities over the corresponding processes with ground-state molecules (Ref. 2, and references therein). Although vibrationally excited states are produced by recombination, i.e., e⁻+H, it is believed that the main source of rovibrationally excited H₂ is hydrogen molecules sputtered from the hydrogenated carbon surface of the divertor tiles.³

Rovibrationally excited molecules are also important in possible mechanisms for enhancing the volumetric recombination of divertor plasma in formation of the plasma detachment regime,^4,5,6 which is necessary for the reduction in heat loads on divertor plates, one of the most important tasks in today's fusion energy research. Heat load reduction schemes include the use of collision processes involving rovibrationally excited hydrogen molecules and hydrocarbon molecular plasma impurities.⁷ The latter represents an alternative to the standard recombination schemes based on vibrationally excited molecular hydrogen. Assuming that there is a significant fraction of vibrationally excited hydrogen molecules in the divertor region, plasma recombination mechanisms such as molecule-assisted recombination (MAR) and molecule-assisted dissociation (MAD) were proposed.⁸ For example, MAR starts with a charge exchange reaction H₂(j,v)+H⁺H(j,)+H and is followed by dissociative recombination with a plasma electron. The MAR based on vibrationally excited H₂ molecules is a sink for molecules, which are only efficiently regenerated at the wall surface.

The MAD chain of reactions can also start from a vibrationally excited hydrocarbon, C_xH_y, which might be even more effective in the creation of the detached plasma layer than H₂ due to the “recycling” features of the C_xH_y MAR: the products of the MAD with more complex hydrocarbons are often themselves hydrocarbon molecules, ions, or radicals which can again undergo MAR/MAD processes.⁷

A significant weakness in assessments of the importance of MAR, both hydrogen molecule and hydrocarbon based, is in the theoretical models describing the kinetics of vibrational and rotational excitations of hydrogen molecules coupled to transport of H and C_xH_y, and, in particular, a lack of knowledge of the rovibrational distribution of these molecules, both when created at the carbon surface and deeper in the plasma. Here we study only one aspect of these complex atomic processes: the distribution of rovibrational and translational energies of the particles as they are ejected from carbon surfaces upon bombardment by deuterium atoms.

Interest in the MAR processes extends well beyond the fusion community. For example, they are very important for building our basic understanding of the plasma constituents in many technical applications. Moreover, since H, H₂, C_xH_y, and their isotopes are among the dominant species in the universe they involve some of the most important atomic reactions in astrophysics.⁹ For example, the H₂+D⁺HD+H⁺ reaction is a major source of HD found in diffuse stellar clouds and may have played a dominant role in HD formation in the early universe. In addition, hydrocarbon chemistry plays a central role in the gas phase chemistry crucial to our understanding of the structure and evolution of star-forming regions.¹⁰

In most presently operating medium- and large-size fusion tokamak devices the plasma-facing components in the divertor contain carbon-based materials. The interaction of divertor hydrogenic plasma with these materials leads, primarily through chemical erosion, to significant amounts of hydrocarbon molecules which enter the plasma. The composition of these molecules depends on the energy of the impact particles, the temperature of the surface, its microstructure, and the level of hydrogenation. At higher impact energies more complex hydrocarbons are emitted.¹¹ In this paper, using a molecular dynamics (MD) approach,^{12,13,14,15,16} we focus on the states of the molecules sputtered from carbon by deuterium impact, which is important information for realistic modeling of the edge plasma chemistry, but has not been intensively studied

Details of our molecular dynamics approach are presented in Sec. II. The dependence of the average translational and rovibrational energies of sputtered particles on impact energy is studied in Sec. III, while the detailed distributions of the various energy modes, as well as their angular properties, are presented in Secs. IV s5, respectively. Finally, Sec. VI contains our conclusions.

In this section we provide a brief description of the MD simulations with attention to a few improvements over our previous work.¹¹ We study the classical many-body dynamics of an amorphous hydrogenated carbon surface, using classical potentials designed to treat covalent bonding and reactivity. We use the reactive empirical bond-order (REBO) potential,¹⁷ which is a member of the Tersoff–Brenner classical bond-order family of potentials¹⁸ and provides a good, empirical description of the covalent bonds for nonpolar systems. We note that the 2002 parametrization¹⁷ of this potential used here, which was fit to vibrational frequencies of molecular hydrocarbons and the elastic constants of bulk carbon-based materials, is well suited to describe both the bulk a-C:D substrate and the smaller molecular species that are ejected from the sample.

The target surface in our MD simulations is deuterated amorphous carbon (a-C:D). This reflects the fact that the graphitic divertor plate becomes amorphous and heavily deuterated at long plasma exposure times, particularly at the high fluxes (up to 10²⁵  m⁻² s⁻¹) representative of ITER operating conditions. The a-C:D MD simulation surface is also representative of the carbon films deposited elsewhere in the reactor. Although the bulk material is known to have a D:C ratio of 0.4%,¹⁹ simulations have showed that at low impact energies this ratio can be substantially elevated at the interface.^11,20,21,22 In previous work, comparing sputtering yields from simulations and beam experiments, we have found it crucial to reproduce the experimental conditions as accurately as possible. Consequently, we pay particular attention to the preparation of the target surfaces, as in previous studies.^{11,20,21,22,23}

Like in our previous studies¹¹ the surface preparation was performed in several phases. The initial sample consisted of a hydrogenated amorphous carbon sample with a density of 2.0  g cm⁻³ and hydrogen/carbon ratio of 0.4 (700 deuterium atoms and 1750 carbon atoms). This sample was prepared by heating a pure carbon (diamond) bulk sample to 10⁴  K, allowing it to remain molten for several tens of picoseconds, and quenching it back down to 300 K at a rate of 500  K ps⁻¹. Deuterium was then introduced into the simulation cell by substitution, at which point the system was again heated to a temperature of 10⁴  K for 10 ps and requenched to 300 K over 62 ps. After this the periodic boundary conditions were removed from the simulation cell in the z direction (perpendicular to the interface), generating a two-dimensional periodic slab. This surface was then relaxed at 300 K for 100 ps. In this way we generated an initial, “virgin” a-C:D surface that was the starting point for preparation of individual surfaces by D bombardment at different impact energies, for use in the sputtering studies. This resulting sample was a cube of approximately 26.5  Å in each direction

The surfaces were prepared by MD simulation of the cumulative bombardment of D at normal incidence, starting from a virgin surface, homogeneously deuterated at a D:C ratio of 0.4 and equilibrated. As reported earlier,^{11,20,21,22,23} the yields of ejected particles reach a quasisteady state after some initial transient period. One key improvement in the current study over previous work is in the range of fluences examined. Our earlier calculations used surfaces prepared by fluences of 1.1×10²⁰–2.0×10²⁰  m⁻² (800–400 D impacts on the ~7  nm² surface). Here we extend deeper into the steady-state region, using surfaces prepared by D fluences of 1.4×10²⁰–2.8×10²⁰  m⁻² (1000–2000 D impacts). The upper boundary of this region is illustrated by vertical lines in Fig. 1. Over the chosen range of fluences the surface is significantly damaged relative to the initial, equilibrium structure, especially at higher impact energies. The interface contains a large number of D-rich filamentous structures. Figure 1 shows that the position of the C interface shifts as the surface is eroded. Within fluences of 1.4×10²⁰  m⁻² (1000 impacts) the interface becomes substantially enriched in deuterium. This D-rich layer remains nearly constant in width, although the width varies with impact energy, and evolves downward as the surface is eroded. Significantly, some unmodified bulk material remains until fluences of 2.8×10²⁰  m⁻² (2000 impacts), even at the highest impact energy of 30 eV. Note also that even as the surface is eroded, low-density surface swelling and long chains of atoms can extend above the original position of the interface.

Figure 1.

The second improvement introduced in the current study was to reduce the nominal flux at which the surface was prepared. In the simulations of cumulative impacts, the surface evolves freely for a period of time after each impact, and is then equilibrated at 300 K using a Langevin thermostat to allow dissipation of the deposited thermal energy. In previous simulations,¹¹ we performed 1 ps of free evolution. In the present calculations we doubled the free evolution time, using 2 ps free evolution, followed by 1 ps of the thermostat application. This is equivalent to a nominal flux of 4.7×10²⁸  m⁻² s⁻¹. These two modifications to the surface preparation procedure, both intended to better reproduce experimental conditions, did not result in statistically significant differences in the yields of ejected methane or acetylene, cases where a comparison was available.¹¹

The present focus is on prompt emission for impact energies of 30 eV and below, where we find that the ejection of the majority of hydrocarbons occurs within 5 ps after an impact (see Fig. 2). We detect ejections when a species crosses (completely) a removal plane that is set to 3  Å above the surface structures (the range of the short-ranged REBO potential is 2  Å). Because the structure of the filamentous surface deformations varies with impact energy, the position of this removal plane also varied with the energy (up to ~30  Å at 30 eV). At the highest impact energy of 30 eV all of the ejected D atoms (either reflected or sputtered) and 65% of the D₂ were detected within 5 ps, as shown in Fig. 2. The ejection of hydrocarbons was significantly slower, with only about 50% of the sputtered species detected in the first 5 ps. Consequently, sputter yields were calculated for 30 eV impacts using up to 35 ps collection times. At lower energies, the ejections occur substantially faster due to smaller penetration of the projectiles, and 5 ps collection times are used. The reported sputter yields are thus a lower bound, although the error is expected to be small, and decreases with decreasing impact energy. Most of the sputtering chemistry happens close to end of the penetration cascade. The longer escape times from greater end-of-range depths for these products to reach the interface explains the longer collection times needed for higher energy impacts. In addition, the presence of heavier (slower moving) hydrocarbons increases with impact energy.¹¹ Figure 3(b) shows a clear trend toward later ejections for heavier sputtered hydrocarbons.

Figure 2.

Figure 3.

Finally, while our previous studies of the sputtering yields covered a range of 7.5–30 eV/D of impact energy,¹¹ here we consider impact energies of as low as 1 eV, providing predictions not available to experiments (where the lowest measured values are 10 eV with D⁺ impact). As before, for each energy and each surface considered, we generate 4096 random impacts for energies of 10 eV or less, and 2048 impact trajectories for higher impact energies. All impacts were at normal incidence. The results were averaged over impacts and over six different steady-state surfaces prepared with fluences in the range of 1000–2000 impacts, thus providing statistics of between 12 000 and 24 000 impacts for each energy.

The fate of the impacting D atoms is illustrated in Fig. 4, which indicates the fraction of D atoms that are retained in the deuterated surface or ejected—whether as “reflected” D atoms, or sputtered as either D₂ or within a hydrocarbon. All curves are the calculated yields and involve no normalization. The reflected yield includes both the actual reflected projectiles and the D atoms sputtered from the surface. Therefore, the ejected particles are not only impacting atoms but also part of erosion of the previously accumulated D atoms. When we reach the state in which the total number of D atoms in the sample is decreasing by erosion, the yield of emitted D should be bigger than one. This occurs at sufficiently high fluences, as illustrated in Fig. 5. However, having in mind the finite size of the simulation cell, we have chosen to work with surfaces for which the bottom of the simulation cell is still intact (see Fig. 1). For some of these surfaces the rate of change in the number of D atoms between 1000 and 2000 impacts is positive, for some negative, as can be observed in Fig. 5. This leads to a rate of implantation that is on average slightly higher (~5%) than the rate of erosion, i.e., with a total D yield slightly smaller than one. The fact that our total yield is very close to one reflects that we are in the vicinity of the saturation regime. An additional reason why the yield is slightly smaller than at high impact energies is that particle emission can take longer to collect. We used a 30 ps collection time for 30 eV impacts, but a 5 ps simulation for all other energies. The smaller simulation time at 20 eV may lead to a small (~10%–15%) underestimation of the total D emission yield.

Figure 4.

Figure 5.

There is a distinct difference between the behavior at low impact energies of 3 eV and below, and the behavior at 10 eV and higher. This transition is determined by the 4–5 eV necessary for covalent bond dissociation in a-C:D. At the lowest energy examined (1 eV), almost all D atoms are reflected. The reflection coefficient decreases steeply to 0.5 at impact energies of only 3 eV; most of the remaining D atoms are either retained in the surface or reemitted as D₂ molecules. Above 5 eV, the reflection coefficient continues to decrease, although much more slowly. A larger fraction of the D atoms is retained in the substrate at higher energies, not surprisingly. The D₂ yield is largest at 5–10 eV, and decreases gradually as the impact energy increases above 10 eV. The continued decrease in D reflection is due both to the increased D retention (implantation) as well as growth in the yield of hydrocarbons that carry off multiple D atoms.

We applied a Langevin thermostat to the bottom 2  Å of the simulations cell in x and y dimensions (with a time constant of 100 fs), while constraining v_z=0 for these atoms, to prevent overall motion of the cell after the impact. The influence of the bottom thermostat, is shown in Fig. 6, which displays the average temperature change in the simulation cell (plus the impact particle) as function of time after an impact of D for two energies (10 and 30 eV). We display results for three cases: (1) no thermostat applied, (2) a thermostat applied in the x and y directions at the bottom “2 Angstroms” and v_z=0, and (3) a thermostat applied in all three Cartesian dimensions. The target cell was in thermal equilibrium at 300 K prior to impact. The initial elevated temperature is due to the kinetic energy (KE) of the impinging particle. The impact particle thermalizes with the topmost layers of the cell within ~100  fs, and then it takes about 5 ps (somewhat less for 30 eV, somewhat more for lower energies) for the impact-induced heat to reach the thermostated bottom part of the cell, initiating the cell temperature decay. Having in mind the cell dimensions, this corresponds to a few angstroms per picosecond for the velocity of the thermally excited wavefront. Once the thermostat starts removing energy form the cell, the temperature decays as T(t)~t^−0.035, almost independently of the impact energy, and takes about 30 ps to “cool” the cell to the thermostat temperature. We note that power law provided a better fit than the expected exponential decay.

Figure 6.

In conclusion, from the observed speed of thermal conduction no substantial cooling of the cell (heated by a single particle impact) takes place within 5 ps. Since most of the sputtered particles were emitted within 5 ps, the kinetic and rovibrational temperatures of the emitted particles are not affected unphysically by the thermostat, and are simulated in conditions that approximate those of the experimental system. Note that the extended simulation time for 30 eV is due to the time that very slow detached particles need to reach the collection plane at 30  Å. The distant collection, in turn, is required by the growth of extended filamentous structures at higher impact energies.

While molecular simulations have access to atomistic detail of each collision, some level of insight into the sputtering process can be obtained by examining properties that are accessible experimentally, such as the energetics of the sputtered particles. As discussed in Sec. I, both translational and internal energies of sputtered particles are of importance for modeling the transition dynamics of collisional atomic processes in the divertor plasma, and for MAR, in particular.

Figure 7(a) shows how the energy of the sputtered particles at the time of detection is partitioned between internal (rovibrational) and center-of-mass (c.m.) translational modes, for molecular deuterium, comparing it with the KE of D atoms in Fig. 7(b) as a function of impact energy. The c.m. translational KE increases with increasing impact energy for D, which is not surprising as long as there is non-negligible contribution from reflections. However the translational KE saturates at about 1 eV for D₂. The rovibrational KE of sputtered D₂ molecules decreases above 7.5 eV, at a level substantially below the D₂ dissociation energy. The enhanced ejection energy at impact energies of 5–10 eV is related to the threshold energy of 4–5 eV for C–C bond breaking.

Figure 7.

The partitioning of the KE of ejected hydrocarbons is somewhat different, as shown in Fig. 8. The rovibrational KE is much larger than the translational energy, due to the greater number of vibrational degrees of freedom in the polyatomic hydrocarbon ejecta. Once again, there is evidence of some structure at impact energies near the 5 eV C–C bond energy, although poor statistics due to the small number of ejections at low energies do not allow this to be conclusively distinguished from random noise. At higher energies, the translational and rovibrational energies of the ejecta vary only weakly with impact energy. The translational KE is roughly 0.5 eV, as observed previously.¹¹

Figure 8.

The KEs can also be described as temperatures. Although temperatures are strictly only meaningful under conditions of at least local equilibrium, complete thermalization can be assumed, the calculation can nonetheless be revealing in illustrating the nonequilibrium behavior. In a molecule with N atoms, the “temperatures” associated with the various motions are defined by j(kT/2)=E, where j is the number of degrees of freedom for a particular mode with energy E, and k is the Boltzmann constant. Specifically, j=3 for the c.m. translational motion, j=3 for rotational motion of nonlinear molecules (j=2 for linear D₂, CH, and C₂), and j=3N−6 for vibrational motion (or 3N−5 for linear molecules). Figure 9 shows the temperatures associated with translational and rovibrational motions for various ejecta. This confirms that the ejections are nonequilibrium processes, with incomplete equilibration between the various motions. In every case the translational motion is consistently hotter than the rovibrational motion (although this difference is small for CD₃ ejecta), indicating that the ejection process is more efficient at transferring energy into c.m. translation than into rotational or vibrational degrees of freedom. The ejection temperatures are relatively constant at higher energies (except for CD₃), consistent with ejection occurring from a stage where the collision cascade has generated local rovibrational excitations averaging around 2500 K (for hydrocarbons) to 4000 K (D₂) and translational temperatures of 4000–6000 K. These temperatures are considerably higher than the 300 K substrate temperature and also quite a bit lower than the impact KEs (corresponding to 200 000  K at 10 eV, for example). It makes sense that the lighter species are ejected from a pool with higher local rovibrational temperature, as these species are more likely to be ejected in primary collisions near the surface where the impact energy has been dissipated less. The translational temperatures differ from the rovibrational temperatures by no more than a factor of 2 (and by much less than these temperatures differ from either the 300 K substrate temperature or the impact kinetic energies corresponding to ~10⁵  K), indicating that while ejection takes place before complete local equilibration, some partial thermalization has occurred.

Figure 9.

The averages displayed in Fig. 9(a) mask a great degree of variation, which is shown in Fig. 10. The rovibrational KE of the ejecta increases with molecular size, due to the larger number of vibrational modes. However the rovibrational temperature is relatively independent of molecular mass, further supporting the idea that ejecta are emitted from the collision cascade at a fairly characteristic state of energy dissipation.

Figure 10.

Because the rotational KE is not a conserved quantity (it is coupled to both vibrational and conformational energies), it is useful to characterize the rotational motion through the angular momentum, shown in Fig. 11. The angular momentum L can be calculated from the rotational energy, E_rot=(L²/2I), where I is the instantaneous value of the moment of inertia. This quantity is conserved for an ejected molecule that no longer interacts with the substrate. The heavier hydrocarbons have higher angular momenta than lighter CD₃ and D₂ molecules, the latter of which have L values of about 0.5  amu (Å)²/fs (~150  a.u.) independent of impact energy. These high values of the angular momentum (in units of ) justify the classical description of the rotational motion of the ejected molecules. The angular momentum of larger hydrocarbons changes rapidly at low energies near the ~5  eV threshold for breaking C–C bonds, but is relatively constant at higher energies. Despite the fact that hydrocarbons have higher angular momenta than light D₂ molecules, the distribution of hydrocarbon angular momenta does not appear to change dramatically with increasing molecular weight.

Figure 11.

A complete description of the state of sputtered and reflected particles includes the differential sputtering yields as a function of the various constants of motion of the ejected particles. For example, a large amount of information is available in the literature concerning the translational energy distributions of sputtered particles due to particle bombardment (see, e.g., Refs. 24,25,26,27, and references therein). However, most of these works refer to physical sputtering and are not obviously applicable to the chemical sputtering processes discussed here. Nevertheless, we find that many of the same models used in physical sputtering can describe our data.

One of the oldest models for physical sputtering is that of collision cascades leading to a Thompson probability density^24,25,26,27 of ejected particles given by

where E_T is the surface binding energy of the ejected particle, n is around 2–3, and E_s is the energy of the ejected (sputtered) particle. Another common model is that of thermal spike processes in which the impinging ion creates a local “hot” area in the solid along the incoming cascade^24,25,26,27 from which particles are thermally sputtered following a Maxwell–Boltzmann distribution

where E_M=kT corresponds to the local temperature T. Finally, in contrast to translational energies, the internal vibrational and rotational spectra of molecules in thermal equilibrium also follow a Boltzmann distribution^28,29

where the different density of states for rotation and vibration accounts for the different prefactor when compared to the Maxwell–Boltzmann distribution, Eq. (2), for translational motion.

In the following we pragmatically adopt these functional forms and use them as fitting functions of (E). We apply the least-squares fitting, nonlinear parameter estimation method, and treat E_T, n, and E_M as free parameters. It turns out that even in cases where functional form (1) provides a good fit, no satisfactory physical meaning can be ascribed to the parameters E_T and n, which are taking arbitrary fitting values. However, the Boltzmann energies E_M are more physically reasonable, supporting the idea of at least a partial thermalization.

Figure 12 shows the distribution of ejected translational kinetic energies for deuterium atoms at three representative energies. At impact energies well below the threshold for sputtering (breaking of the C–D bond) the spectrum of ejected D atoms contains only inelastically reflected particles, which can be fitted to a single Maxwell distribution. Thus, for an impact energy of E_i=1  eV, a Maxwell distribution with E_M~0.17  eV (=0.17E_i) fits the data well. For an impact energy of E_i=3  eV, a single Maxwell distribution gives E_M~0.43  eV (=0.14E_i), but the fit is less satisfactory because of the presence of some sputtered D atoms (see Fig. 1). At an impact energy of 15 eV, well above the C–D dissociation energy, the distribution can be described successfully only by a superposition of two Maxwell distributions, one with E_M=0.114  eV and another one with E_M=2.04  eV (=0.136E_i). The more energetic piece of the distribution is likely due to reflections, as indicated by the impact energies returning a fraction of the impact energy that is consistent with that observed at nonreactive impact energies. The less energetic distribution, on the other hand, likely corresponds to D atoms sputtered from the surface upon impact.

Figure 12.

At all energies the value of E_M for reflected particles represents 13%–17% of the impact energy. In other words, the reflected projectile loses on average more than 80% of its initial energy. This energy loss exceeds the maximum that can occur in a binary D–C collision, clearly demonstrating the many-body nature of this collision process, which involves multiple target excitations.

The behavior is more complicated for molecular ejecta, as should be expected. The distributions of c.m. translational energy of sputtered D₂ molecules are shown in Fig. 13 at three representative energies. A Thompson-like distribution (E_T=1.64  eV and n=5.9, atypical for a Thompson distribution) provides a reasonable fit for 3 eV D impacts, except in the high-energy tail of the distribution. Poor fit in the high-energy tail is typical for Thompson-like distributions in all cases studied here. For higher impact energies of 10 and 20 eV, the D₂ translational energy is described well with a double Boltzmann distribution [Eq. (3), with E_M=0.63 and 3.8 eV for E_i=10  eV and E_M=0.50 and 4.95 eV for E_i=20  eV]. In each case, the majority of the ejections occur with a Boltzmann distribution with E_M of about 0.5 eV, while a second distribution at characteristic energies of 4–5 eV is necessary to describe the tail of the distribution.

Figure 13.

Finally, we show the c.m. energy distributions for sputtered hydrocarbons in Fig. 14. A variety of different fitting functions can model these distributions. Very few hydrocarbons are observed at low impact energies. At impact energy of 10 eV, a Thompson-like distribution provides a good fit. At 30 eV, a Thompson-like distribution fails to fit the high-energy tail of the distribution, as is observed in many cases. Thus we are required to use a Boltzmann distribution in addition to the Thompson-like form in order to provide a good description of the tail.

Figure 14.

For the rovibrational energy distributions, typically a single Boltzmann or Maxwell function is adequate, as seen in Fig. 15 for hydrocarbons (DC's) and in Fig. 16 for D₂, with ejection of D₂ at higher energies being an exception that requires a double distribution. In general, a Maxwell–Boltzmann form tends to provide a better fit at low energies.

Figure 15.

Figure 16.

The individual vibrational and rotational energies are not conserved quantities for a single molecule. However, the probability density of each of these energies reaches a stationary distribution after several vibration/rotation periods even for a single molecule, and converges quickly to a well-defined average when summed over the full ensemble of ejected particles. In Fig. 17 we present several illustrative cases for the distributions of vibrational (D₂) and rotational (hydrocarbons) energies. A Boltzmann distribution describes the vibrational energies of ejected D₂ at low impact energy, while a Thompson-like distribution must be supplemented with a Boltzmann distribution in order to provide a good description of the tail distributions at impact energy of 10 eV. On the other hand, a single Thompson distribution succeeds in describing the distribution of rotational energies of sputtered hydrocarbons, even for 20 eV impact energy.

Figure 17.

Overall, it is clear that the Thompson model, while sometimes providing an accurate fit to the data, is not a good description for the underlying collision processes, due to the unphysical values of E_T and n obtained. The Maxwell–Boltzmann model is more frequently applicable, and with reasonable parameters, reinforcing the view that the ejections occur from a local region that is perhaps partially thermalized, but certainly not fully equilibrated.

Another set of properties of the ejected particles that both provides insight into the sputtering process, and could potentially be investigated experimentally, are the angular distributions of both the linear and angular momenta of the ejected particles. Because the target is amorphous and we only consider D impacts normal to the surface (which defines the z axis), the angular distributions are independent of the azimuthal angle in the (x,y) plane. Therefore, only the dependence on the polar angles _P (of the ejected particle's c.m. momentum P) and _L (of the particle's angular momentum L), with respect to the surface normal, are described. These distributions are defined as

where N(,) is the number of particles detected in the interval (−/2,+/2) and C is a normalization constant such that the maximum value of dN/d lies on the unit circle.

The angular distributions for the c.m. momentum vector for various impact particles and impact energies of D are presented in Fig. 18. For physical sputtering and normal incidence on an amorphous solid the angular distributions of the ejected momenta are expected to follow dN/d_p~cos _P.^19,20,21 The present distributions for D ejection approximately mimic this behavior, particularly at high energies. However, the behavior for D₂ and hydrocarbons is somewhat more sharply peaked around ejection normal to the surface than cos _P.

Figure 18.

The angular distributions of angular momentum are not expected to follow a cos _L behavior. For random emission processes such as activated thermal desorption, one would expect to find a uniform (isotropic) distribution, dN/d_L~constant. Figure 19 shows that the angular distributions for D₂ are indeed close to uniform at high energies, but are closer to a sin _L dependence at the lowest impact energies. This indicates that the ejected particles have a preferential direction of rotation, the angular momentum being parallel to the surface (perpendicular to the surface normal). That is, a molecule leaving the surface along the surface normal will tend to tumble end over end more often than it will spin as a propeller. This makes sense for low impact energy ejections resulting from a primary collision, as secondary collisions would be required to redirect the normally directed linear momentum of the impacting atom into directions parallel to the surface. The same is qualitatively true for the ejected hydrocarbons but clear trends are difficult to visualize due to poorer statistics.

Figure 19.

The average c.m. translational energy of ejected hydrocarbon particles is found to be about 0.5 eV, but the sputtered molecules typically have higher rovibrational energies reaching ~2  eV, which are dominated by the multiple vibrational modes. If equipartition is assumed in order to convert these energies into temperatures, it is found that the temperatures of the different degrees of freedom are much more alike in magnitude, although the translational motions are consistently hotter than the rovibrational modes. The same is true for D₂, with approximate equipartition between the rotational and vibrational modes of energy (around 0.5 eV and 4000 K each), but with translational motions that are considerably hotter (around 1 eV and 6500 K at high impact energies. The various components of the energy are only weakly dependent on the D impact energy, and the degree of thermalization also appears to be fairly uniform across ejected hydrocarbons of different masses.

The distributions of the translational energy of the ejected particles are described well by a double Boltzmann/Maxwell distributions, indicating different energetics for reflection and sputtering. The rovibrational energy distributions could be described well by generalized Thompson, Boltzmann, and/or Maxwell distributions, where the Thompson-like model typically overestimates the tails of the distributions. The empirical nature of these fits is a consequence of the complex dynamics of chemical sputtering, indicating that only partial thermalization of the ejected particles is reached in most cases, and that these models are too simplistic to provide a full understanding of the sputtering process at these energies.

The angular distributions (per unit solid angle) of the ejected particle momenta show that particles are preferentially emitted along the surface normal. A different behavior is found for the angular momentum distributions of the sputtered molecules which preferentially rotate about axes parallel to the surface at the lowest impact energies, becoming isotropic for the higher energies

We acknowledge support by the Office of Fusion Energy Sciences (P.S.K.) and the Office of Basic Energy Sciences (C.O.R.) of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725 with UT-Battelle, LLC, and partial support through SciDAC. S.J.S. acknowledges financial support for this work by the Department of Energy (Contract No. DEFG0201ER45889), the National Science Foundation (Contract No. CHE0239448) and the DOD (47539-CH-MUR). This research was performed in great part using 512-1024 processors of the Cray X1E computer located at ORNL, through DoE INCITE under Project No. SC18392.

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Fig. 1. (Color online) Density of carbon and deuterium as function of D fluence, for various impact energies. A depth of zero represents the midpoint of the initial surface. The color coding reflects the fraction of the maximum density (see scale on the right). First citation in article

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Fig. 2. (Color online) Time evolution of the cumulative fraction of ejected particles collected, for D impact energies of 5 eV (upper solid line), 15 eV (dashed red line), and 30 eV (lower solid line), for ejection of D (a), D₂ (b) and hydrocarbon (c). First citation in article

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Fig. 3. Collection times for sputtered hydrocarbons of different masses. First citation in article

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Fig. 4. (Color online) Yield of ejected deuterium per impact of D. Note that (D  retained)+(D  ejected)+(2  D₂  sputtered)+(D  sputtered  with  DC's)=1 and (total  D)=1−(D  retained). First citation in article

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Fig. 5. (Color online) Evolution of number of D atoms in the simulation cell with increasing fluence. First citation in article

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Fig. 6. (Color online) Decay of the average temperature change of the simulation cell for (a) 10 eV and (b) 30 eV impacts of D. First citation in article

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Fig. 7. (Color online) Average kinetic energies of D₂ (a) and D (b) ejecta as a function of D impact energy. The D₂ KE is partitioned into translational and rovibrational components. First citation in article

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Fig. 8. (Color online) Average energies of ejecta as a function of impact energy of D for (a) all hydrocarbons and (b) ejected CD₃. The error margins are standard errors obtained from six independent target surfaces. First citation in article

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Fig. 9. (Color online) Average temperatures of sputtered molecules associated with the rovibrational and c.m. translational motions. Vibrational data in (a) were not calculated since the number of vibrational degrees of freedom depends nontrivially on the geometrical chemical structure of ejected hydrocarbon. First citation in article

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Fig. 10. Sputtered hydrocarbon rovibrational energies as function of the molecular mass, for D impact energies of (a) 5 eV, (b) 10 eV, and (c) 20 eV. The rovibrational temperatures as function of mass for 20 eV impact energy are shown in (d). First citation in article

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Fig. 11. (Color online) (a) Average angular momentum of sputtered molecules as function of the D impact energy, and (b) distribution of angular momentum with mass of the molecule. The error bars represent standard errors. First citation in article

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Fig. 12. (Color online) Distributions of kinetic energies of reflected/sputtered D atoms for selected impact energies. First citation in article

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Fig. 13. Distributions of the translational energy of sputtered D₂ molecules for selected impact energies. First citation in article

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Fig. 14. Translational energy distributions of ejected hydrocarbons at impact energies of (a) 10 eV and (b) 30 eV. First citation in article

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Fig. 15. (Color online) Distributions of rovibrational energy of sputtered hydrocarbons at impact energies of (a) 10 eV and (b) 20 eV. First citation in article

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Fig. 16. Distributions of rovibrational energy of sputtered D₂ molecules at impact energies of (a) 2 eV and (b) 20 eV. First citation in article

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Fig. 17. (Color online) Distributions of vibrational and rotational energies of sputtered molecules at selected impact energies. First citation in article

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Fig. 18. (Color online) Distributions of solid angles of ejected particle momentum, dN/d_P. The thick dashed lines represent an angular distribution dN/d_P~cos _P. The same symbol-color coding is used at all pictures, even in cases when some energies are not present. First citation in article

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Fig. 19. (Color online) Distributions of ejected molecules as a function of the solid angle of the angular momentum (dN/d_L). First citation in article

^*Electronic mail: krsticp@ornl.gov.

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