Relaxation of the symmetric vibration mode of the CO2 molecule

The applicability of the Landau-Teller energy-relaxation equation of a harmonic oscillator is verified for the CO₂ molecule in which there is Fermi resonance. It is shown that the distribution over the split levels of the symmetric mode formed by vibrational exchange processes and transitions within multiplets is analogous to the Treanore distribution for a singlemode anharmonic oscillator.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 8–17, May–June, 1975.     

Kinetic equations for vibrational relaxation in a mixture of polyatomic gases

Kinetic equations are derived for the relaxation of the vibrational energy in a mixture of polyatomic gases, which are ones with molecules simulated by harmonic oscillators. The most general case is envisaged, where the energy relaxation occurs not only via vibrational-translational transitions but also via multiquantum vibrational exchange involving an arbitrary number of vibrational modes. The analysis also incorporates the possible degeneracy of each mode when the molecules colliding are the same. An expression is derived that extends previous results [1–6] and that relates the vibrational temperatures in the case of quasiequilibrium. Equations are derived for the vibrational relaxation for the CO₂-N₂ case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 29–37, November–December, 1972.We are indebted to L. A. Shelepin for valuable discussions on the results.     

Kinetics of the deactivation of the vibrations of highly excited oscillators in an inert gas medium,taking account of spontaneous radiation

In the diffusion approximation, the article discusses the kinetics of the process of deactivation of the vibrations of radiating anharmonic and harmonic oscillators in an inert gas medium. Limiting solutions are given for the purely radiational deactivation of a classical Morse oscillator and of a harmonic oscillator. It is shown that, with an increase in the effect of spontaneous radiation, the role of the anharmonic character of the vibrations in the process of deactivation increases; the initial (or arbitrary) distribution relaxes more slowly the higher its energy level, i.e., the greater the effect of the anharmonic character of the vibrations. The results are of importance for systems with a considerable population of the upper vibrational levels of the molecules, which may arise as a result of a chemical reaction or by the optical pumping of a gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 62–67, July–August, 1972.The author thanks N. N. Magretova for carrying out the numerical calculation.     

Nonequilibrium vibration-dissociation phenomena behind a propagating shock wave: Vibrational population calculation

An analysis of nonequilibrium phenomena behind a plane shock is presented concerning the vibrational relaxation and the dissociation of a pure diatomic gas. In the first part, the temperature range is 600 K–2500 K and the dissociation processes are neglected. The population of each vibrational level is computed by solving relaxation and conservation equations. The relaxation process is described by the master equations of each vibrational level. The vibrational transition probabilities appearing in the relaxation equations are calculated analytically and take into account the anharmonicity of molecular vibration and the potential angular dependence. The populations obtained are compared to those calculated using a Treanor model and to those calculated with a nonequilibrium Boltzmann distribution. For moderately high levels significant differences may be observed. The importance of the V-V process is found to be weak for the transitions involving the lowest levels. In the second part, the temperature range is 2500 K–5500K and the dissociation process is taken into account as well as the gas dynamic behavior which did not appear in several recent works. The kinetic equations are transformed to obtain a first order differential system and the resolution of such a system coupled with the conservation equations leads to the population of each vibrational level. The vibrational transition probabilities associated with the atom-molecule interaction are deduced from the cross section calculation used in the first part. The bound-free transition probabilities are obtained, following Marrone and Treanor, assuming that dissociation must occur preferentially from the higher vibrational states: the Marrone and Treanor probability model is extended and employed with an anharmonic oscillator. In the present investigation, behind the shock wave, the evolution of the population distribution expressed as a function of the distance is not monotonous: a lag time appears as shown experimentally in previous works for the macroscopic parameters. For moderately high levels the influence of the anharmonicity and those of the V-V processes appear significant and strongly related. In a general way, in both temperature ranges investigated, the V-V processes reduce the effects of the T-V transfer. Finally the influence of thecharacteristic probability temperature U of Marrone and Treanor is analyzed and a method of determination of local varying U is proposed.     

Saturation of absorption of the powerful radiation of a system of anharmonic oscillators

Recently, the theory of nonequilibrium systems simulated by a set of anharmonic oscillators has received significant development. The investigation of such kinds of systems is especially important in the study of problems associated with the stimulation of chemical reactions and the development of effective molecular lasers. The systematic analysis of the kinetics of anharmonic oscillators assumes the simultaneous solution of a large number of nonlinear equations describing the population balance of the vibrational levels. Realization of this approach is associated with cumbersome numerical calculations and does not permit obtaining a qualitative picture of the behavior of the system as a function of the different parameters (pressure, temperature, etc.). An approximate analytical theory has been formulated in [1, 2] which permits finding the distribution function over the vibrational states with the effects of anharmonicity taken into account. We will employ the approach developed in these papers to describe a system of anharmonic oscillators under conditions of powerful optical pumping. This problem was discussed in [3], where it was found that such a system changes into a saturation mode in the case of high pumping levels. The existence of this mode is explained by the fact that the maximum rate of energy input into a vibrational degree of freedom is determined by the rate of distribution of this energy over all the vibrational levels, i.e., by the constant of V—V-exchange. For sufficiently large pumpings the approximation of the Boltzmann distribution function adopted in [3] in connection with the calculation of the saturation parameters is too crude. The goal of this paper is to derive in explicit form expressions for the vibrational energy supply, the absorbed power, and so on, under saturation conditions without the use of the approximation indicated above [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 10–15, September–October, 1978.     

Inverted population of Co2 molecules in expanding gas streams

The system of equations of hydrodynamics, which describes the process of escape of the mixtures CO₂ + N₂ + He, H₂O from a nozzle, is solved numerically in conjunction with the equations of the kinetics of the excitation of the vibrational degrees of freedom of the molecules. It is found that an inverted population of the CO₂ molecules with respect to the transition [00 °1] – [10 °0], is produced under certain conditions at the exit from the nozzle. The magnitude of the inversion depends both on the nozzle configuration and on the initial values of the gas temperature and pressure. It is shown that for a specified nozzle configuration there exist optimal values of these parameters, at which the inverted population of the CO₂ molecules reaches approximately 10¹⁵ cm^–3.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 24–34, September–October, 1971.     

Analysis of Nonequilibrium Dissociation Kinetics in a Boundary Layer Using Various Reaction Models

Values of the nonequilibrium macroscopic reaction rate for a nonisothermal boundary layer of a monatomic diluent gas are calculated using a number of models for thermal dissociation of diatomic molecules — anharmonic Morse oscillators. Analysis is performed for conditions where the diffusive transfer of excited molecules has a significant effect on the population of their upper vibrational levels, which does not only amount to change in vibrational temperature. Under the joint influence of diffusive transfer of molecules, vibrational exchanges, and reactions involving vibrationally excited particles, the local vibrational distribution functions are substantially nonequilibrium. The kinetic models considered take into account the possible contribution of the energy of molecular translational and rotational degrees of freedom to the energy required to overcome the reaction threshold. The effect of multiquantum vibrational—translational exchanges on the distribution of dissociating molecules in their upper vibrational levels is taken into account approximately.     

Study of vibrational deactivation of molecules of carbon dioxide gas during cooling of stream in a supersonic nozzle

The vibrational temperature of the antisymmetrical type of vibrations (v ₃) of the CO₂ molecule at the exit of a supersonic nozzle is measured in the present work using the method of recording the infrared emission. Freezing in of thev ₃-type vibrations was observed during the flow of undiluted carbon dioxide in a nozzle. In this case the vibrational temperature T₃ considerably exceeded the translational temperature. On the basis of a comparison of the experimental results with calculation it can be concluded that vibrational deactivation of CO₂ molecules occurs three to five times faster than the excitation of the vibrations during heating in a shock wave. All the experiments were conducted under the following conditions: maximum expansion of gas in nozzle A/A* = 115, temperature range 1900–2400 °K, pressure range 1–17.5 atm.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 32–40, November–December, 1973.The authors are grateful to U. G. Pirumov and É. A. Ashratov for the calculation of the nozzle profile and the distribution of streamlines as well as for a discussion of the results.     

Vibrational relaxation of diatomic molecules in a non-Boltzmann thermostat

The action of resonance IR laser radiation on a molecular gas leads, at high-power absorption intensity, to a breakdown in the equilibrium (Boltzmann) energy distribution in the internal degrees of freedom [1]. Under realistic conditions, molecular gases usually are (due to small amounts of impurities or isotopic components) multicomponent systems. In this case resonance IR laser radiation (or other methods of selective action), disturbing the distribution function of the primary gas, does not interact directly with impurities. The problem thus arises of determining the distribution function of the impurity gas interacting with the nonequilibrium (non-Boltzmann) thermostat. The present paper, devoted to the solution of this problem, treats the distribution function of harmonic oscillators A, consisting of a small amount of impurities in a system of harmonic oscillators B with given nonequilibrium distribution functions of vibrational energy. The behavior of a system in a nonequilibrium thermostat was first considered in [2, 3] where, as well as in [4, 5], it was shown that in a non-Maxwellian thermostat with a small amount of harmonic oscillator impurities, a Boltzmann distribution in harmonic oscillator vibrational energies is established under stationary conditions, with a temperature differing from the gas-kinetic temperature of the thermostat, defined in terms of the mean-square velocity. The behavior of a small amount of impurities (heavy monoatomic particles and harmonic oscillators) in a non-Maxwellian thermostat of a light gas was further investigated in [6–8]. Unlike the papers mentioned, the present one considers the behavior of a small amount of harmonic oscillator impurities in a thermostat with a Maxwellian velocity distribution and with a nonequilibrium (non-Boltzmann) distribution in vibrational energies.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1978.     

Effect of laser radiation on the vibrational distribution of CO2

In connection with progress in the field of CO₂ lasers, questions of the vibrational kinetics of molecules of CO₂ have been discussed in many communications. In a majority of cases of practical importance, the distribution of CO₂ is due to processes of vibrational exchange (V-V) on which is based the well-known thermodynamic model [1]. In other cases, the V-V exchange does not determine the vibrational distribution, since the perturbation is small; therefore, it is found sufficient to consider a small number of levels of CO₂ (usually three), whose populations satisfy the linear equations of the balance [2]. There is the possibility of conditions where the vibrations are strongly excited and, at the same time, V-V processes are insignificant (a very small CO₂ impurity in the inert gas, with a high degree of ionization). Then the number of equations becomes large. The present article discusses one such case: the excitation of a steady-state vibrational distribution in a glow discharge by laser radiation, whose solution is rather graphic.     

Calculation of the characteristics of a gas-dynamic laser

The dependence of the radiated power on the characteristics of optical cavities in the case of flow systems has been investigated in a number of papers [1–3], in which it is assumed that population inversion of the laser levels is obtained until entry into the cavity. The operation of a cavity is analyzed in [1] in the geometric-optical approximation with allowance for vibrational relaxation in the gas flow. A simplified system of relaxation equations is solved under steady-state lasing conditions and an expression derived for the laser output power on the assumption of constant temperature, density, and flow speed. The vibrational relaxation processes in the cavity itself are ignored in [2, 3]. It is shown in those studies that the solution has a singularity at the cavity input within the context of the model used. In the present article the performance characteristics of a CO₂-N₂-He gas-dynamic laser with a plane cavity are calculated. A set of equations describing the processes in the cavity is analyzed and solved numerically. Population inversion of the CO₂ laser levels is created by pre-expansion of the given mixture through a flat hyperbolic nozzle. The dependence of the output power on the reflectivities of the mirrors, the cavity length, the pressure, and the composition of the active gas medium is determined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi FiziM, No. 5, pp. 33–40, September–October, 1972.     

Vibrational-translational relaxation of anharmonic oscillators at low temperatures

An approximate analytic solution is found to the problem of the vibrational-translational relaxation of anharmonic oscillators at translational temperatures which are small compared with the energy difference between adjacent levels of the oscillator. The deviation of the obtained distribution from the Boltzmann distribution in the relaxation process is analyzed. A study is made of the behavior of the vibrational energy near equilibrium at temperatures such that dissociation has only a small effect on the rate of vibrational relaxation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 3–8, March–April, 1976.The author wishes to thank M. B. Zheleznyak and A. Kh. Mnatsakanyan for a useful discussion of the work.     

Investigation of the inverted medium of a quasi-stationary CO2 laser with pulsed excitation

The results of an investigation of the inverted medium of a quasi-stationary CO₂ laser is presented. The medium is distinguished by the fact that the time of flight of individual molecules through the discharge gaps is less than the relaxation time of the 00 °1 CO₂ laser level. The emitted power, the gain, the saturation intensity, and the gas temperature are measured. Using the experimental data, the distribution of the molecules in the vibrational and rotational states of the inverted medium is calculated. The maximum power density attained in this experimental model is 25 W/cm³. For comparison, the characteristics of a model in which cold CO₂ is added to the flow of excited nitrogen are investigated. It is shown that in this case the output power level is determined by the efficiency with which the jets are mixed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 23–29, January–February, 1973.The authors thank V. M. Fedorov for useful discussions and A. A. Borynyak for his help with the experiments.     

Thermal dissociation of diatomic molecules in a nonisothermal two-temperature boundary layer

Nonequilibrium thermal dissociation in a nonisothermal boundary layer in a mixture of Morse anharmonic oscillators — molecules of a diatomic gas and its atoms — is considered within the framework of the ladder mechanism. The local nonlinear nonequilibrium corrections to the two-temperature macroscopic dissociation rate, which depend, in particular, on the translational and vibrational temperature gradients and the degree of dissociation, are determined.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 191–201, March–April, 1996.     

Relaxation of anharmonic molecules

The inclusion of anharmonic effects is important in vibrational population inversions in CO-lasers [1, 2], in relaxation processes in jets [3], in thermal dissociation [1], in the kinetics of chemical reactions with high thresholds [4], etc. Usually these effects are studied by including anharmonic corrections to the kinetic constants in the discrete model of single-quantum transitions or in the diffusion approximation [1, 2]. In [5] a method was given of solving the relaxation equations fro arbitrary forms of the rate constants and the spectrum of the molecule. The method is valid when the ratio of the population densities of neighboring levels varies smoothly with quantum number. It was shown in [5] that this approximation can be used to construct analytical solutions for a wide class of problems. In the present paper the method of [5] is extended to the case of equations with variable coefficients. The properties of the solutions for VT-relaxation of anharmonic molecules are analyzed, and the inclusion of sources is considered. A simple method of taking into account multiple-quantum transitions is given, as well as an extension of the method to an arbitrary mixture of gases. The population densities are calculated and the possibility of using our solutions in relaxation gas dynamics is discussed.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, No. 3, pp. 22–31, May–June, 1986.     

Population inversion behind a detonation wave propagating in a variable-density medium

The creation of an active medium by means of detonation has been investigated on a number of occasions. It was suggested that one could use the expansion of the detonation products of an acetylene-air mixture in vacuum [1] or the cooling of the detonation products of a mixture of hydrocarbons and air through a nozzle [2, 3]. In [4], the detonation of a solid high explosive was used to produce population inversion in the gas mixture CO₂-N₂-He(H₂O). Stimulated emission from HF molecules was observed in [5] behind the front of an overdriven detonation wave propagating in an F₂-H₂-Ar mixture in a shock tube. Population inversion behind a detonation wave was studied in H₂-F₂-He mixtures in [6–8] and in H₂-Cl₂-He mixtures in [9] with energy release on a plane and on a straight line in a medium with constant density. Similar problems were solved for shock waves propagating in both a homogeneous gaseous medium [7, 10] and in the supersonic part of an expanding nozzle. In the present paper, we study theoretically population inversion behind an overdriven detonation wave propagating in a mixture (fine carbon particles + acetylene + air) which flows through a hypersonic nozzle. The propagation of detonation in media with variable density and initial velocity was considered, for example, in [11, 12]. Analysis of the gas parameters behind a detonation wave propagating in a medium with constant density (for a given fuel) showed that the temperature difference across the detonation front is insufficient to produce population inversion of the vibrational levels of the CO₂ molecule.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 65–71, January–February, 1980.I am grateful to V. P. Korobeinikov for a helpful discussion of the results.     

Theory of two-component gasdynamic laser

In the present paper a numerical calculation is made of the vibrational relaxation of a binary mixture of molecular nitrogen and carbon dioxide gas. The calculation is performed for the entire range of variation of the concentrations of the components and over a wide range of mixture temperatures and pressures for various geometries of the supersonic part of the nozzle (throat dimensions, degree of expansion). It is shown that population inversion of the CO₂ molecules exists within a certain range of variation of the parameters of the mixture and the nozzle. The population inversion of the vibrational levels and the gain of the gaseous mixture are calculated as functions of these parameters and of distance measured from the critical cross section of the nozzle. The energy characteristics of the two-component gasdynamic laser are optimized.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 23–30, May–June, 1974.     

Influence of the initial spatially inhomogeneous temperature perturbations on vibrational relaxation dynamics

The vibrational relaxation of a nonequilibrium molecular gas (T_V>T) plays an important role in the physics of gas lasers, laser chemistry [1], and plasma chemistry [2]. This paper is devoted to an analysis of the dynamics of V-T relaxation with spatially inhomogeneous perturbations of the translational temperature taken into account.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 6, pp. 77–80, November–December, 1984.     

Propagation of small disturbances in a relaxing and radiating diatomic gas in vibrational nonequilibrium

In radiation gasdynamical problems, where the primary object of investigation is a moving gas, the influence of radiation on the parameters of the gas flow is usually neglected to avoid overcomplication of the problem. The growth and behavior of initial disturbances in a scattering, radiating, absorbing, viscous, heat-conducting gas characterized by local thermodynamic equilibrium has been investigated previously [1]. However, for low pressures (p10^–4 to 10^–3 technical atm) and fairly high temperatures of the active molecular degrees of freedom (T10³ to 3·10³K) the distribution of the molecules among the vibrational levels can differ markedly from the equilibrium distribution due to the or der-of-magnitude closeness of the vibrational relaxation time _c associated with collisions and the radiative deactivation time * of excited molecules [2, 3]. We now analyze normal modes in a vibrationally nonequilibrium medium with allowance for radiation scattering in the vibrational-rotational band. We formulate a dispersion relation and discuss some limiting cases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 168–171, September–October, 1976.The author is grateful to V. I. Kruglov, Yu. V. Khodyko, and M. A. El'yashevich for their interest and discussions.