22 nd International Symposium on Plasma Chemistry July 5-10, 2015; Antwerp, Belgium Analysis of recombination and relaxation of non-equilibrium air plasma generated by short time energetic electron and photon beams M. Maulois, M. Ribière, O. Eichwald and M. Yousfi Université Paul Sabatier de Toulouse, CNRS, Laplace, 118 Route de Narbonne, FR-31062 Toulouse Cedex, France CEA/DAM, FR-46500 Gramat, France Abstract: The present contribution is devoted to the study of the evolution, recombination and relaxation of non-equilibrium air plasma generated by short time energetic photon and electron beams. We described the reaction scheme used and the numerical method modeling the density evolution of the plasma species. The role of the main species and reactions during the different stages of the air plasma evolution are identified and analyzed. Keywords: Energetic electron and photon interactions, Non-equilibrium air plasma, collision cross-sections 1. Introduction The problem of energy deposition of electron and photon beams in gases and the analysis of the associated plasmas is of interest in the areas of for instance the interactions of radiation with matters [1], the physics of upper atmosphere [2], the electron-beam generated lasers [3] or electrical discharges [4], the x-ray radiography [5]. Non equilibrium air plasmas generated by energetic photon and electron beams can also be, as in the framework of the present work, of great interest in the electromagnetic perturbations of electronic devices due to plasma-induced electromagnetic field (see e.g., [6] showing the non-negligible effects of the plasma-inducedelectromagnetic field). We have taken into account high energy photon and electron beams simultaneously released in ambient air during a short time (0.5 ns). Photon energy is chosen equal to 0.35 MeV during this short time and the initial electron energy to 1 MeV. During this short time of particle beam application, strong ionization and dissociation of the air molecules are expected thus leading to air plasma generation with a certain ionization degree. After this first stage of plasma formation lasting 0.5 ns, there is a second stage dominated by recombination processes where secondary electrons having sub-ionization energy interact with the air plasma during its slow relaxation towards its initial conditions. The choice of such short time for particle beams is aimed to validate the considered chemical kinetics reaction scheme and the associated basic data. Section 2, following this introduction, is devoted to the description of the energy profile of the particle beams and the considered reaction scheme for the study of the formation of the evolution of the present air plasma initially composed by synthetic air (i.e., 80% N 2 and 20% O 2 ). It is also devoted to the associated basic data (reaction coefficients) taken either from literature or calculated from collision cross sections. Formalism of chemical kinetics model and the numerical scheme are described in section 3 while the obtained results on the plasma formation and relaxation are discussed and analyzed in section 4. 2. Energy profile of particles, reaction scheme and basic data 2.1 Energy profile of particles When short time pulsed beams of primary energetic electrons and photons interact with air, energetic particles of the beams lose energy mainly through inelastic collision processes. The evolution of photon energy during the beam application can be obtained from Monte Carlo simulation taken into account the dominant inelastic photons collisions corresponding to the present work energy range (photo-ionization, Compton collision and in a less degree pair production). However, in the case of a very short time of the pulsed beam, the mean photon energy can be assumed as quasi-invariant even though the evolution of the mean photon energy E ph ( from an initial energy E ph (t=0) can be obtained from the following relation considering the energy losses due to collisions with the background gas: E ( = E ( t = 0)e ph ph vph, tott (1a) where v ph,tot is the product of the background gas density N, the light velocity c and the total collision cross-section σ ph,tot (N x c xσ ph,tot ). Furthermore, the time evolution of the electron mean energy E e ( from an initial energy E e (t=0) of primary electrons can be approximated by the following relation obtained from an analytic solution of the conservation equation of electron mean energy assuming homogeneous medium, constant total collision frequency v e,tot and without considering the action of external forces [7]: 3 3 Ee ( = kt B gas + Ee ( t 0) kt B gas e 2 = 2 2m v e, tot t M (1b) P-I-2-45 1
m and M are respectively the electron and gas masses. It is noteworthy that the electron energy relaxes in a long time scale towards gas energy 3/2 k B T gas (Tgas and k B being gas temperature and Boltzmann constan. It is also possible to consider a more rigorous time evolution of the electron mean energy from the solution of a more complete energy conservation equation. Such equation can be written by considering individually each process (elastic, attachment, recombination, excitation, ionization, Bremsstrahlung radiative loss and also the possible source term S ph due to photon impacts. This formalism is detailed in reference [8]. These relations (1a) and (1b) of the time evolution of photon E ph ( and electron E e ( energies have been considered in the case of short time pulsed particle beam. 2.2 Reaction scheme of air plasma As soon as the particle beams impact the background gas, this leads to the generation of different plasma species that in turn can disappear or generate new species following various collision processes. The electron creations by photon impacts are due to photo-ionization, Compton collision and pair production while energetic electron impacts lead to air ionization, dissociation and also creation of long living excited states (or metastables) that can in turn be ionized or dissociated thus generating new or further species. Therefore, the species thus formed through interactions with air molecules or atoms by high energetic particles are secondary electrons and photons, single and multi-ionized atoms and molecules, dissociated molecules and metastable states. All these species constitute a kind of non-equilibrium plasma with ionization degree depending on the energy and the time duration of the initial pulsed particle beams. Obviously the different plasma species can interact with background air and also with themselves. These interactions, occurring mainly in a longer time scale than the short time beam duration, can be for instance recombination, charge transfer, electron detachment, ion conversion, stepwise and Penning ionizations. During this longer time scale, there are also collisions occurring between secondary electrons and background gas. The reactions corresponding to all these primary and secondary processes that have been considered in the present work are detailed in [8]. In fact, we have considered 25 species (photons: hν, electrons: e -, neutrals in background states: N 2, N, O 2, O, NO, positive ions: N + 2, N ++ 2, N +, N ++, O + 2, O ++ 2, O +, O ++, N 2 O + 2, N + 4, N + 3, NO +, O + 4, negative ions: NO -, O - 2, O -, and nitrogen metastables: N 2 (A 3 Σ + u) and N 2 (a Π g )) interacting following 175 reactions. The main creation and loss processes between the 25 considered species can be summarized as following: - Reactions involving photons as photo-ionization processes leading to single and double ions and Compton interactions - Reactions involving electrons as ionizations by electron impacts leading to single and double ions, electronic dissociation, nitrogen metastable formation, stepwise ionizations, electron attachments, radiative electron recombination, two- and three-body electron recombination - Electron detachments - Recombination involving positive and negative ions - Two and three body charge transfers - Recombination involving metastable and neutral - Penning ionizations - Two and three body neutral recombination. 2.3 Basic data The determination of the density evolution during the considered time scale, lasting between the plasma generation towards the plasma recombination and its relaxation, for the different plasma species needs the a priori knowledge of the reaction coefficients of each considered interaction. In the case of the photon and electron impacts taken into account in the present reaction scheme, the collision cross sections are generally known and come from the literature. Therefore, using the known data of collision cross sections, the reaction coefficients can be calculated from relation (2a) for a given photon reaction number i having a photon mean energy E ph ( already determined from previous relations (1) or (2): k ( E ( ) = σ ( E ( ) ϕ( E ( ) ph, i ph ph, i ph ph k ph,i is the reaction coefficient of first order ( s -1 ) of the photon reaction number i while σ ph,i (in cm 2 ) is the corresponding collision cross section depending on photon energy. As photon velocity is constant, there is no need to consider a photon distribution as for instance in the case of electrons since we considered mono-energetic photons (i.e., corresponding to a Dirac photon distribution at energy E ph () and the reaction coefficient is therefore directly obtained from the photon flux ϕ (in cm -2 s -1 ). The photon flux is representative of the studied case of energy deposition of the considered ionizing radiation beams; ϕ is obtained from Monte Carlo simulation method using similar procedures and processes as PENELOPE code [9]. In the case of reaction coefficient k e,j (in cm 3 /s) corresponding to the electron impact number j for an electron mean energy E e ( already determined from previous relations (1) or (2), we can write: 2 k E t = f E t d m σ e e e e 1 2 e, j( e( )) e, j( e) e ( e( ), e) e (2a) (2b) σ e,i is the associated collision cross-section depending on electron kinetic energy e e and f(e e (, e e ) is the electron distribution function assumed Maxwellian with a mean energy E e (. The reactions coefficients versus mean energy are displayed in figure 1 in the case of photons and in figure 2 in the case of electrons. The collision cross-sections of photon impacts used to calculate the photon collision frequencies for N 2 and O 2 displayed in figure 1 are taken from reference [10]. The 2 P-I-2-45
references for the other species are detailed in [8]. It is noteworthy that when it was necessary such collision cross sections are extended towards high energy range (up to several MeV) in coherence with appropriate high energy formalisms. The collision cross-sections of electron impacts used for the calculation of electron reaction coefficients displayed in figure 2 are detailed in reference [8]. Here also collision cross-sections have been extended when that was necessary towards high energy range from appropriate formalisms. Furthermore, in the case of reactions involving heavy particles as charge transfer, recombination between positive and negative ions, electron detachment, reaction between excited species and neutral recombination, the reaction coefficients are taken from the literature using Arrhenius formulae for the reaction number h between heavy species at temperature T s. Fig. 1. Calculated single photo-ionization collision frequencies of O 2, NO, N 2, O and N versus photon energy with double photo-ionization collision frequencies of O and N versus photon energy. Fig. 2. Calculated ingle ionization coefficients by electron impacts of NO, N 2, O 2, O and N with double ionization coefficients by electron impacts of N, O, NO, N 2 and O 2 versus electron mean energy. Due to the short time duration of the energetic particle beams, the temperature T s of each considered heavy species is assumed close to the initial air temperature. In fact, a real increase of the temperature of heavy species requires a time scale much more higher, of several microseconds in air at atmospheric pressure in order to accumulate a huge number of interactions between more particularly free electrons and ions. Indeed, it is known [11] that such Coulomb interactions, particularly efficient when ionization degree increases, highly contribute to the ion heating thus leading to a global temperature rise of the neutral species due to their interactions with the heated ions. 3. Chemical kinetics model: formalism and numerical method Assuming that the space gradients can be neglected (mean volume approximation), the evolution of the density of the plasma species generated by the ionizing radiation (energetic photons and electrons) is governed by a strongly coupled ordinary system of first order time dependent differential equations. This system can be written in a general form as: dn( / dt = f ( n(, n ( is the vector of densities n i (, i varying from 1 to the total number of considered particles (26 in our case). f ( n(, is the vector of source terms f i ( n(, which takes into account all the reactions involved in the creation and loss of the corresponding particles i. For a given species i, f i ( n(, can be written as follow in for instance the case of two body kinetics reactions: f n(, = k ( n ( n ( k ( n ( n ( ) i ( gain, j 1 j 2 j loss, m 1m 2m t j m k gain,j ( is the reaction coefficient of the reaction j involved in the creation of the species i, while n 1j ( and n 2j ( are the densities of the two species interacting in the two body reaction j. k loss,m ( is the reaction coefficient of the reaction m involved in the disappearance of the species i, while n 1m ( and n 2m ( are the densities of the two species interacting in the two body reaction j. In this case, the reaction coefficients are expressed in cm 3 /s. In the case of a three body reaction, the reaction coefficient is expressed in cm 6 /s and the corresponding source term is the product of a reaction coefficient with the densities of the 3 interacting species. The numerical solution of such strongly coupled system of equations requires some precautions in order to avoid the numerical instabilities because of the stiffness of those equations. Stiffness is due to the very different magnitudes of the reaction coefficients leading to very different time scales for the evolution of the various considered. Similar numerical method, already used elsewhere [12] and [13] in non-equilibrium plasma chemistry, is detailed in reference [14]. P-I-2-45 3
4. Results and discussions The initial conditions, before the impacts of energetic photon and electron beams, required for the chemical kinetics model concerns: - initial background gas composed by synthetic air at atmospheric pressure (i.e., 80% N 2 and 20%O 2 ) and ambient temperature (300 K) - initial density assumed equal to 10 3 cm -3 for the most considered plasma species. Energy profiles of photon and electron beams are already given in section 2 and the reaction coefficients of each considered process are evoked in section 2. Figure 3 display time evolution of electron density with some neutral densities when ambient air is irradiated by the short time (0.5 ns) photon and electron beams while figure 4 corresponds to the case with only the electron beam (1 MeV for initial energy). As expected, the creation of new electron by photon processes starts early, at about femtosecond for Compton processes while electron ionization occurs 2 decades later at tenth of picosecond (about 0.1 ps). This is simply due to the higher magnitude of photon collision frequency (around 10 15 s -1 ) in comparison to the electron one (7x10 12 s -1 ). Furthermore, these results lead to several remarks. N 2 ionization by electron impacts at the same energy is much higher (around 10-18 cm 2 ). The magnitude of electron density is also correlated to the magnitude of the neutral target species. Indeed, in the case of Fig. 3 with photon and electron beams, molecule dissociation start early due to photon interactions and the densities of neutral species (N 2, O 2, N and O) are necessary lower. This obviously favors a lower electron source term. For instance, at the end of irradiation (0.5 ns), N 2 density in Fig. 3 is around 10 6 cm -3 and O 2 density around 1.9 x10 5 cm -3 while without photon, these densities are higher, around 3 decades for both molecules. Results on ion densities (not shown here) confirm this trend since gas irradiation with both photon and electron beams generally leads to stronger ionization processes and therefore higher ion densities. At the end of the irradiation (0.5ns), due to the strong dissociation processes, the magnitude of the neutral densities n displayed in figs 3 and 4 follow this order: n N > n O > n N2 > n O2 > n NO. Then after the beam irradiation we assumed that the air plasma is governed by secondary electrons having sub-ionization energy without any effect of photon interactions which are neglected. In these conditions, the air plasma enters mainly in a recombination and relaxation stage. For instance, the time evolution of electron density that obviously depends on the magnitude of secondary electron energy (in both tested cases: 1 ev in fig. 3 and 10 ev in fig. 5) relaxes during this transient stage towards to its initial value under mainly the effect of two and three body electron recombination processes. Fig. 3. Electron density with neutral species densities in the case of both photon beam with initial energy of 0.35 MeV and electron beam with initial energy of 1 MeV applied during 0.5ns (secondary electron energy = 1 ev). During the beam irradiation corresponding to the air plasma formation, electron density at the end of the beam (0.5 ns) is of about 4.9 x 10 19 cm -3 when photon and electron beams are both applied (Fig. 3) while it becomes about 8% higher (about 5.3 x 10 19 cm -3 ) in the case of the alone electron beam (Fig. 4). As the source term of electron density depends both on collision frequency and the densities of colliding species (particularly N 2, O 2, N and O), the magnitude of electron density can be first correlated to the magnitude of the associated collision cross sections. For instance, single N 2 photo-ionization is about 5x10-27 cm 2 at 0.35 MeV and N 2 Compton cross-section of about 5x10 24 cm -27 cm 2 at 0.35 MeV while Fig. 4. Electron density with neutral species densities in the case of only electron beam with initial energy of 1 MeV applied during 0.5 ns (Secondary electron energy = 1 ev). 4 P-I-2-45
Fig. 5. Electron density with neutral species densities in the case of both photon beam with initial energy of 0.35 MeV and electron beam with initial energy of 1 MeV applied during 0.5 ns (Secondary electron energy = 10 ev). Last, this coherent relaxation behavior shows to a kind of validation of the chosen kinetics reaction scheme and the associated basic data. 5. References [1] L.V. Spencer and U. Fano. Phys. Rev., 93, 1172 (1954) [2] A.E.S. Green et al. J. Geophys. Res., 82, 5104 (1977) [3] S. Rockwood et al. IEEE J. Quant. Elec., 9, 120 (1973) [4] J. Bretagne et al. J. Phys. Appl. Phys., 14, 1225 (1981) [5] D. R. Welch et al. Phys. Plasmas 1994 11, 751 (2004) [6] D. Benyoucef and M. Yousfi. Plasma Sources Sci. Technol., 23, 044007 (2014) [7] M. Yousfi and A. Chatwiti. J. Phys. D, 20, 1457 (1987) [8] M. Maulois et al. to be published during 2015 [9] J. Sempau, J.M. Fernandez-Varea, E. Acosta and F. Salvat. Nucl. Instr. Meth. Phys. Res. B, 207 107-123 (2003) [10] J.A. Fennelly and D.G. Torr. Atom. Data Nucl. Data Tables, 51, 321 (1992) [11] E. Carpene et al. in Laser Process. Mater. (P. Schaaf; Ed.) (Berlin Heidelberg: Springer), pp. 21 47 (2010) [12] O. Eichwald et al. J. Appl. Phys., 82, 4781 (1997) [13] O. Lamrous el al. J. Appl. Phys., 79, 6775 (1996) [14] W.H. Press et al. Numerical Recipies in FORTRAN. (Cambridge: Cambridge University Press) (1992) P-I-2-45 5