Electron-vibration energy exchange models in nitrogen-containing plasma flows
aa r X i v : . [ phy s i c s . c h e m - ph ] A p r Electron-vibration energy exchange models in nitrogen-containing plasma flows
V. Laporta a) ,
1, 2 and D. Bruno b) , Department of Physics and Astronomy, University College London,London WC1E 6BT, UK Istituto di Metodologie Inorganiche e dei Plasmi, CNR, 70125 Bari,Italy
The physics of vibrational kinetics in nitrogen-containing plasma produced by colli-sions with electrons is studied on the basis of recently derived cross sections and ratecoefficients for the resonant vibrational-excitation by electron-impact. The tempo-ral relaxation of the vibrational energy and of the vibrational distribution functionis analyzed in a state-to-state approach. The electron and vibrational temperatureare varied in the range of [0–50000] K. Conclusions are drawn with respect to thederivation of reduced models and to the accuracy of a relaxation time formalism. Aanalytical fit of the vibrational relaxation time is given. a) [email protected] b) [email protected] . INTRODUCTION Modeling of high speed flows of reactive gas mixtures is required for the description ofspace vehicles entries in planetary atmospheres, as well as for the interpretation of groundbased experiments in high enthalpy wind tunnels and shock tubes.It has long been recognized that vibrational energy relaxation, beside its obvious role inthe global energy balance, plays a major role in the chemical kinetics of non-equilibriumzones . Among the many processes that influence the molecular vibrational energy, in thispaper we focus on inelastic collisions with free electrons, the so-called electron-vibration(e-V) processes. Even in conditions of weak ionization, as they occur in Earth entries atorbital speed or in Mars entries, excitation by electron-impact is very efficient a mechanismfor populating vibrationally excited molecules. In addition, the aerospace community showsa growing interest for flows with sensibly higher ionization degrees. This is due to theinterest in superorbital speed Earth entries as they happen in the return phase from Lunarand Martian exploration missions , and to the investigation of plasma flow control concepts(see Ref. 4 for a review).For nitrogen-containing plasma flows, the direct process of N vibrational-excitation byelectron scattering is a optically forbidden transition, due to the lack of dipole momentumfor the N molecule. For incident electron energies lower than 4 eV, however, the vibrationaltransition occurs via a resonant process involving the Π g state of the N − ion: e − + N (X Σ + g ; v ) −→ N − ( Π g ) −→ e − + N (X Σ + g ; w ) , (1)where v and w are the initial and final vibrational level of N . By means of the resonantprocess in (1) the vibrational-excitation cross section is enhanced by several orders of mag-nitude. This process is very efficient and makes the vibrational temperature equilibratevery rapidly with the free electron temperature to the extent that many simple modelsassume the N vibrational temperature instantaneously equilibrates with the free electrontemperature. A first serious theoretical study for e-V relaxation by electron-impact has beenproposed by Lee . More refined models have been developed by Mertens and by Bourdonand Vervisch .The experimental data on resonant vibrational-excitation cross sections and rate coeffi-cients are difficult to obtain and, in the literature, they are limited to the first few levels .2n order to build a high temperature model information on the upper vibrational levelsis mandatory. Recently, theoretical calculations have been performed for electron-nitrogenresonant scattering considering all vibrational transitions up to the dissociation limit in theenergy range 0 −
10 eV that show very good agreement with available experimental data.The new theoretical results are used in this paper to set up a kinetic study of electron-impactvibrational energy relaxation in nitrogen-containing plasma with the purpose of assessingthe accuracy of existing models.The paper is organized as follows: in section II the basic equations of the kinetic modelare presented; in section III the full set of rate constants for all vibrational transitions areintroduced and used to assess modeling assumptions commonly used in the literature. Thesolution of the full kinetic model is analyzed in section IV together with a reduced multi-quantum vibrational transition approach. The relaxation time formalism is introduced insection V and the results of its application to the vibrational energy relaxation are comparedto those obtained within the full kinetic model. Section VI summarizes the main conclusionsand perspectives. II. MODELING OF ELECTRON-VIBRATION RELAXATION
In order to model the e-V kinetics a nitrogen gas in contact with a free electron bathis considered. The electrons are considered at equilibrium at the constant temperature T e and the N molecules are supposed to be in the electronic ground state X Σ + g , with a initialvibrational distribution at temperature T v . Due to the very efficient resonant mechanism,the electron-N collisions drive the vibrational distribution towards the equilibrium at theelectron temperature T e . Since the electron affinity of the N atom is negative, the resonantdissociative electron attachment for electron-N is suppressed . Therefore, the effects ofmolecular dissociation on vibrational relaxation are not taken into account.The time evolution of the molecular vibrational distribution function (VDF) is describedby the following equations: ∂n v ∂t = n e X w [ K w,v n w − K v,w n v ] , (2)where n e is the electron density and K v,w ( T e ) is the rate constant for the v → w resonant3ibrational-excitation (RVE) process: K v,w ( T e ) = 2 √ π ( k B T e ) − / Z ǫ σ v,w ( ǫ ) e − ǫ/k B T e dǫ , (3)where σ v,w is the corresponding cross section as a function of the electron energy ǫ . InEq. (2) the sum extends over all 67 vibrational levels of N . The detailed balance principleimposes the following condition among the direct and inverse processes: K v,w n ∗ v = K w,v n ∗ w , (4)where n ∗ v is the equilibrium population of the vibrational level v at temperature T e : n ∗ v ∼ e − ǫ v /k B T e , (5) ǫ v being the energy of the N vibrational level v and k B the Boltzmann constant. The totalN vibrational energy is readily obtained from the VDF: E vib = X v n v ǫ v . (6) III. RATE COEFFICIENTS
In this section the rate coefficients K v,w ( T e ) that enter the kinetic equations (2) and (6)for the resonant process in Eq. (1) are presented. They are taken from Ref. 10 (paper Iin the following) where full details on the calculations and the validation of the model bycomparison with other approaches can be found. The method used in I to calculate theelectron-N resonant scattering is the well-known ‘boomerang model’, an approximationof the nonlocal-complex-potential model . Such a method has been used to study manydifferent systems and a complete description together with the application limits can befound in the papers and references therein. Molecular rotations have been taken intoaccount by adding the centrifugal barrier to the potential, parameterized by the rotationalquantum number J and transitions with ∆ J = 0 have been considered. Results show thatthe rate constants for vibrational-excitation do not depend strongly on rotational excitationso, in the following, only J = 0 is considered.In I, a full set of RVE cross sections among all vibrational levels of N has been calculated.The corresponding rate coefficients have been obtained by averaging each cross section over4 = 0
10 3020 405 500 10000 20000 30000 40000 5000002468
Electron temperature H K L R a t ec o e ff i c i e n t H - c m (cid:144) s ec L v ® v + n = 1 n = 5 n = 10 n = 200 10000 20000 30000 40000 5000010 - - - Electron temperature H K L R a t ec o e ff i c i e n t H - c m (cid:144) s ec L ® + n FIG. 1. Theoretical rate coefficients for electron-N resonant vibrational excitations as a function ofelectron temperature as obtained in Ref. 10. Left: Mono-quantum transitions; Right: n -quantumtransitions starting from the ground vibrational level. a equilibrium electron energy distribution as defined in Eq. (3) in the range [0–50000] K ofelectron temperature. Figure 1 shows typical rate constants for mono-quantum transitions( v → v + 1) and for multi-quantum transitions starting from the ground vibrational level(0 → n ).Since a large number of transitions are involved in the full data set, it is natural to lookfor regularities that allow using scaling relations. On the other hand, in the past boththeoretical and experimental studies on RVE by electron-impact were, as a rule, limited to asmall subset of all possible processes: that required the introduction of scaling-laws amongthe rate coefficients in order to obtain the missing data. Mertens proposed to assumea linear relation between log( K ,v ) and the final vibrational level v . Bourdon et al. haveshown, using the Allan data , that this rule is approximately valid up to v = 7 at T e = 1 eV.In Fig. 2 we report the calculated rate coefficients K ,v as a function of the final vibrationallevel v for different electron temperatures T e , together with a linear extrapolation obtainedusing the first 5 points. A linear relation holds with good accuracy for levels v
5. At higherelectron temperatures this limit can be raised to v ≈
10 but transitions with larger jumpsare overestimated by many orders of magnitude. As we shall see in Sec. IV, transitions withquantum jumps up to ∆ v max = 20 have to be considered for a accurate estimation of thevibrational energy relaxation.For transitions starting from vibrational levels other than the ground state, Gordiets e = T e = T e = - - - Vibrational level v K ® v H T e LH - c m (cid:144) s ec L FIG. 2. Rate coefficients K ,v as a function of the final vibrational level v calculated at differentelectron temperatures. Dashed lines are linear extrapolations obtained using the first 5 points. T e = Vibrational level v K ® n (cid:144) K v ® v + n n = T e = Vibrational level v K ® n (cid:144) K v ® v + n n = FIG. 3. K ,n /K v,v + n as a function of the vibrational level v , for different values of the electrontemperature T e , for mono-quantum (left) and bi-quantum (right) transitions. proposed the following scaling law: K v,v + n = K ,n a v , (7) a being an adjustable parameter. In Fig. 3 the ratio K ,n /K v,v + n as a function of the initialvibrational level v , for different values of the electron temperature and for different values ofthe vibrational jump n , is reported. The results show that the linear relation, Eq. (7), holdsapproximately only in limited regions of the parameters’ space and that the a parameter istemperature-dependent.The scaling-laws proposed by Gordiets and Mertens are heuristic guesses coming fromgas discharge physics where typical conditions are electron temperature around 1 eV andthe gas temperature below 1000 K. Figures 2 and 3 show that indeed linear scaling-lawsare reasonable assumptions when the electron temperature is in the range where the ratecoefficients have a broad maximum (10000 . T e . v = 10. For high speed flows applications, however, models should be valid ina much broader range of the parameters’ space. The linearity deteriorates for low electrontemperatures and high vibrational levels and extrapolation of these relations can lead tosevere overestimation of the rate constants. In the following, only the rate coefficients frompaper I are used. IV. VIBRATIONAL ENERGY RELAXATION
Using the rate coefficients presented in section III, the time evolution of the VDF of N molecules and of the vibrational energy E vib is studied by solving the coupled rate equationsin Eqs. (2). In particular the role of the multi-quantum transitions is investigated. Theinitial condition for the VDF is supposed to be a Boltzmann distribution at temperature T v . At given electron temperature, the rate constants for transitions involving the exchangeof many vibrational quanta are much smaller than the others due to the large energy gap:see for example Fig. 1. It is therefore natural to ask wether it is possible to set a upperlimit to the number of quanta exchanged in order to reduce the amount of data required forthe modeling. Figures 4 and 5 show the ‘full’ solution, i.e. obtained including the full setof multi-quantum transitions, and the effect of considering a restricted range of vibrationaltransitions in the vibrational kinetics on the VDF and on the corresponding E vib for twotypical conditions of heating ( T v = 5000 K and T e = 30000 K) and cooling ( T v = 20000 Kand T e = 2000 K). VDF is normalized to the population of the vibrational ground state andvibrational energy E vib is normalized to its equilibrium value E ∗ vib . Times are normalizedby: τ ( T e ) = 1 n e K , ( T e ) , (8)so that the plots are independent of electron and molecular number densities. Figure 4shows the relaxation of the VDF. The latter, although a Boltzmann distribution at thebeginning (by construction) and at the end (by the equilibrium condition) of the relaxation,shows non-equilibrium character both in heating and cooling conditions. In both cases, highvibrational levels relax more slowly than low ones and the VDF exhibits strong depletion(overpopulation) of high levels with respect to a Boltzmann distribution at the same totalenergy. Large multi-quantum jumps have a strong influence on this relaxation and restrictingthe maximum number of exchanged vibrational quanta overestimates the degree of non-7 e = T v = start equilibriumtime = time = - - - - Ε v H eV L n v (cid:144) n v = T e = T v = equilibriumstarttime = time = - - - - Ε v H eV L n v (cid:144) n v = FIG. 4. Temporal evolution of normalized vibrational distributions function as obtained by differentassumptions on the maximum number of vibrational quanta exchanged in collisions. Full line:all transitions allowed; dashed line: ∆ v max = 10; dot-dashed line: ∆ v max = 20. Left panel: T e = 30000 K; T v = 5000 K. Right panel: T e = 2000 K; T v = 20000 K. T e = T v = Normalized time E v i b (cid:144) E v i b * T e = T v = - Normalized time E v i b (cid:144) E v i b * FIG. 5. Temporal evolution of the normalized vibrational energy obtained by different assumptionson the maximum number of vibrational quanta exchanged in collisions. Full line: all transitionsincluded; dashed line: ∆ v max = 10 levels; dot-dashed line: ∆ v max = 20 levels. Left panel: T e = 30000 K; T v = 5000 K. Right panel: T e = 2000 K; T v = 20000 K. equilibrium by many orders of magnitude. The plot on the right in Fig. 4 shows that thiserror can involve vibrational levels as low as v ∼
10. It is interesting to note that a fairlygood agreement is found in Fig. 5 for vibrational energy, even if significant discrepancies areobserved on the VDFs, as vibrational energy is mostly due to low vibrational levels.Figure 6 reports the maximum error, calculated over all the temporal evolution of E vib ,between the results obtained by considering the full set of rate coefficients and those obtainedincluding only transitions with ∆ v max = 10 and ∆ v max = 20 in the kinetic model. Resultsshow ∆ v max = 10 is a reasonable approximation with a maximum error smaller than 10% in8 IG. 6. Maximum error during the temporal evolution of E vib obtained by considering multi-quantatransitions with ∆ v max = 10 (left) and ∆ v max = 20 (right). a wide region of the parameters’ space. Large errors appear for cooling conditions ( T v > T e )when the starting vibrational temperature is large enough. Although so large vibrationaltemperatures are difficult to realize in practice, this is an indication that very large multi-quantum jumps are involved in the relaxation of strongly populated high vibrational levels.With ∆ v max = 20 the maximum error stays below 1% for most conditions of interest.The vibrational energy relaxation is therefore described with reasonable accuracy bytaking into account only transitions where less than 10 vibrational quanta are exchanged; if aaccurate description of the vibrational distribution is required, however, at least ∆ v max = 20must be considered.To conclude this section, Fig. 7 summarizes the temporal evolution of the vibrationalenergy, as obtained by the full kinetic model, for two electron temperatures 3000 K and12000 K and for initial vibrational temperature in the range T v = 0 to 50000 K to modeldifferent heating and cooling conditions. It is clear that the rate of relaxation is mainlydetermined by the electron temperature. But the initial vibrational distribution also playsa role since different initial conditions relax along different paths.9 v = 0 K10000 K20000 K50000 K5000 K - Normalized time E v (cid:144) E v * T e = T v = 0 K - - Normalized time E v (cid:144) E v * T e = FIG. 7. Time evolution of the normalized vibrational energy, E vib /E ∗ vib , for two different values ofthe electron temperature T e , parameterized on the N initial vibrational temperature T v . V. RELAXATION TIME FORMALISM
The standard way of modeling the vibrational energy relaxation is to describe it by aLandau-Teller (LT) rate equation: dE vib dt = E ∗ vib − E vib τ e , (9)where E ∗ vib is the equilibrium vibrational energy at the electron temperature T e , and τ e isthe e-V relaxation time. Early theoretical work of Lee studied the relaxation from a T v = 0condition assuming a harmonic oscillator model for the N molecule and using the crosssections from Ref. 15. Results can be summarized with the following analytical expressionfor τ e : p e τ Lee e = k B T e ( (cid:18) θ v T e (cid:19) X v K ,v ( T e ) v ) − , (10)where p e = n e k B T e is the electron partial pressure and θ v is the characteristic vibrationaltemperature of the N molecule. The relaxation time defined in Eq. (10) depends only on T e and it does not take into account vibrational excitation. Lee’s formulation has been testedand improved in the works of Mertens and Bourdon et al. . In these works, cross sectionshave been taken from the theoretical work by Huo et al. or from the experimental dataof Allan . The Huo et al. results are ab-initio calculations for 0 → v J = 0 , , . − . J = 50 and initial vibrational quantum number up to v = 12 with changes in10 v = Electron temperature H K L p e Τ e H - J m - s L FIG. 8. Vibrational relaxation time as a function of electron temperature for different values ofthe initial vibrational temperature T v . The values are obtained by solving the full set of kineticsequations. vibrational quantum number of − J = 50have been used in the modeling calculations . The Allan’s results are experimental valuesfor 0 → v transitions up to v = 13 obtained at room temperature. In both cases, crosssections for all other transitions must be obtained by assuming a scaling-law . As a result,analytical expressions have been derived for the relaxation time in Eq. (9) that depends bothon T e and T v .Analyzing the full set of RVE rate coefficients, we concluded in Sec. III that such scaling-laws have strong limitations. It is therefore interesting to investigate the limits of a LTdescription of the vibrational energy relaxation and to provide, where possible, updatedvalues for the relaxation time. To this end, time histories of vibrational energy for differ-ent conditions of electron temperature T e and initial vibrational temperature T v have beencalculated with the full kinetic model. As it will shown in the following, in general, therelaxation is not a linear process. Therefore some arbitrariness is implicit in the definitionof the relaxation time. In this paper the value of τ e was obtained in each case by linearizingthe relaxation rate around its starting point, in order to minimize the discrepancies betweenthe LT and the full model values for the vibrational energy.The final results are reported in Fig. 8 and in particular in Fig. 9 they are compared withthose of Lee and Bourdon et al. for T v = 0. Present results lie between the other two,somewhat closer to the more recent results of Bourdon et al. Due to the resonant character of the cross sections, the relaxation time shows a mini-mum. In Table I the point of the minimum ( T min e , p e τ min e ) of the relaxation time, for some11
000 10000 15000 2000002468101214
Electron temperature H K L p e Τ e H - J m - s L FIG. 9. Vibrational relaxation time as a function of the electron temperature with initial vibrationaltemperature T v = 0 K. Full line: this work; dashed line: results from Ref. 6; dotted line: resultsfrom Ref. 9. Vibrational temperature H K L T e m i n H K L Vibrational temperature H K L p e Τ e m i n H - J m - s L FIG. 10. Best fit of T min e and p e τ min e as a function of the initial vibrational temperature, as givenin Table I. vibrational temperatures, is shown. These points can be fitted by the following laws: T min e ( T v ) = bT v + a , (11) p e τ min e ( T v ) = c + d T v , (12)where the coefficients take the following values: a = 22159 .
90 K ,b = 1 . × K ,c = 1 . × − Jm − s ,d = 5 . × − Jm − sK − . (13)Figure 10 shows the best fit superposed to the calculated points.12 v (K) T min e (K) p e τ min e (10 − Jm − s)0 7055.64 1.291000 7010.24 1.292000 6776.13 1.295000 5968.45 1.3510000 4932.81 1.6520000 3800.00 2.6250000 2189.37 4.11TABLE I. Minimum of relaxation time ( T min e , p e τ min e ), for some initial vibrational temperatures T v . For electron temperature T e > T min e the relaxation time curves in Fig. 8 can be fitted,with 1% accuracy, by the following polynomial expression: p e τ e ( T e , T v ) = p e τ min e ( T v ) + X i =1 c i ( T v ) (cid:0) T e − T min e ( T v ) (cid:1) i , T e > T min e (14)where the coefficients c i ( T v ) are given by: c ( T v ) = 2 . × − − . × − T v + 1 . × − T v ,c ( T v ) = 1 . × − + 2 . × − T v − . × − T v ,c ( T v ) = 1 . × − . (15)The LT equation, Eq. (9), can be integrated analytically to give the vibrational energyrelaxation. For each set of initial conditions ( T e , T v ), a value for the relaxation time iscalculated and kept constant during all relaxation; the resulting vibrational energy and thecorresponding value obtained by solving the full set of kinetic equations were comparedand the maximum deviation was estimated. Figure 11 show the results of this analysisas a contour map. A single relaxation time, as in LT approach, is able to describe thee-V process both in relaxing and exciting conditions when the non-equilibrium is not verystrong ( T e ≈ T v ). The range of validity increases for larger T e . Under strong non-equilibriumconditions, however, large discrepancies (larger than 10%) are seen.In order to explain the results of Fig. 11, the relaxation process was analyzed in greaterdetail for three cases. Table II summarizes the conditions for the cases studied, the value for13 IG. 11. Maximum deviation on the vibrational energy as obtained with a Landau-Teller approachcompared with that obtained by solving the full vibrational kinetic model. the relaxation time and the computed error with respect to the exact solution. Figure 12shows the time evolution of the normalized vibrational energy for two heating conditionswith T v = 0 K and T e = 5000 and 15000 K and a cooling condition with T v = 8000 K and T e = 2000 K. We note that the relaxation predicted by the LT equation is faster than theexact solution both in exciting and relaxing conditions. Figure 13 shows the correspondingnormalized VDF at different times during the relaxation. For the LT model, Boltzmanndistributions at all times during the relaxation are assumed with vibrational temperaturecalculated from vibrational energy. In exciting conditions, at the beginning, high vibrationallevels ( v >
1) relax faster than low levels and get strongly overpopulated. In this conditions,the LT predictions agree well with the ‘full’ result since the low-lying levels make the largestcontribution to E vib . At later stages, the reverse is true: again the LT results agree with the‘full’ results for the first two levels, but now all the rest of the VDF is overestimated. Thisis what makes the LT value for E vib larger. Note that also for case 1, where the relaxationpredicted by the LT equation agrees well (within 2%) with the ‘full’ result, the VDF arestrongly non-equilibrium during most of the relaxation. This can have a strong influencewhen a vibrationally favored chemical reaction ( e.g. dissociation) is considered. In relaxingconditions, instead, the high-energy levels relax slower than low levels. As a result, theLT relaxation is faster than the ‘full’ value. Again, strongly non-equilibrium VDF appearduring most of the relaxation. 14 ase T e (K) T v (K) τ e (s) ǫ max .
023 0 .
012 15000 0 0 .
013 0 .
263 2000 8000 0 .
14 0 . τ e is calculated by linearizing the energyrelaxation rate obtained from the full model and ǫ max is the error on E vib . T e = T v = - Normalized time E v i b (cid:144) E v i b * T e = T v = - Normalized time E v i b (cid:144) E v i b * T e = T v = - Normalized time E v i b (cid:144) E v i b * FIG. 12. Time relaxation of normalized vibrational energy for the cases described in Table II. Fullline: exact solution as obtained by solving the full set of kinetic equations; dashed line: constantrelaxation time in a Landau-Teller approach.
This is the origin of the failure of a relaxation time description. Obviously, the modelingdone in Refs. 6 has been improved in Refs. 7,9 by deriving a relaxation time that dependsboth on T e and T v .Based on the results presented in this work for the relaxation of the VDF, however, wedo not expect the inclusions of the dependence of the relaxation time on T v to significantlyimprove the predicting capabilities of the LT model. Indeed, in Fig. 7 we show the E vib relaxation for two values of T e and for several values of T v . It is apparent that these quantities15 e = T v = starttime = - time = - equilibrium - - - - Ε v H eV L n v (cid:144) n v = T e = T v = equilibriumstart time = - time = - - - - - Ε v H eV L n v (cid:144) n v = T e = T v = equilibriumstarttime = - time = - - - - Ε v H eV L n v (cid:144) n v = FIG. 13. Relaxation of the N vibrational distribution function, normalized to the ground vibra-tional state, for the cases described in Table II. Full line: exact solution as obtained by solving thefull set of kinetic equations; dashed line: constant relaxation time in a Landau-Teller approach. alone are not able to describe the system relaxation towards equilibrium. VI. CONCLUDING REMARKS
Recent calculations of RVE cross sections in nitrogen covering all vibrational transitions up to the dissociation limit have been used to study the vibrational energy relaxation asproduced by these resonant processes. Analysis of the full set of rate constants shows thatapproximate scaling-laws adopted in the past have a limited range of validity.Also, the possibility to model the relaxation by a linear rate equation with a singlerelaxation time has been investigated. The results from a relaxation time formalism andfrom the full kinetic model are compared to conclude that the former may be used withreasonable accuracy ( ∼ e.g. dissociation) play a role.The next step will be the inclusion, in the relaxation kinetic of molecular rotation and ofthe dissociation reaction.Finally, it is worth mentioning that the original cross section data allow the modeling ofRVE processes also in conditions where the electron energy distribution function deviatessignificantly from equilibrium. For the present study only rate coefficients have been used.These are obtained from the cross sections by averaging with a Maxwell distribution for theelectron velocities. ACKNOWLEDGMENTS
The authors wish to thank Prof. M. Capitelli (Universit`a di Bari, Italy) for careful readingof the manuscript and helpful discussions. The research leading to these results has receivedfunding from the European Community’s Seventh Framework Programme (FP7/2007-2013)under grant agreement n ◦ REFERENCES M. Capitelli, et al. Plasma Chemistry and Plasma Processing , 32(3):427–450, 2012. S. T. Surzhikov.
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