For description of the vibrational relaxation of CO₂ molecules embedded in Ar matrix the model based on multiphonon transitions is applied. Rates for the VT and VV processes are determined from a fitting of simulated and experimental data. The calculations confirmed that radiative processes influence significantly the vibrational energy relaxation of CO₂ embedded in solid Ar, e.g., the determined rate for energy transfer between n₃ and n₂. mode is significantly lower than that predicted under assumption of the nonradiative relaxation

Cenian, A.

Grigorian, G.

Śliwiński, G.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10, p. 1135–1139The vibrational relaxation of CO2 isolated in solid argonA. Cenian1, G. Grigorian2, and G. Œliwiñski11Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdansk 80-952, PolandE-mail: cenian@imp.gda.pl2Institute of Physics, St. Petersburg State University, St. Petersburg 198504, RussiaFor description of the vibrational relaxation of CO2 molecules embedded in Ar matrix themodel based on multiphonon transitions is applied. Rates for the VT and VV processes are deter-mined from a fitting of simulated and experimental data. The calculations confirmed that radiativeprocesses influence significantly the vibrational energy relaxation of CO2 embedded in solid Ar,e.g., the determined rate for energy transfer between 3 and 2 mode is significantly lower thanthat predicted under assumption of the nonradiative relaxation.PACS: 67.40.FdIntroductionA large number of physical and chemical processesin the solid-state involve the vibrational energy relax-ation (VER) of molecules. A study of this process formolecules isolated in noble-gas matrices has attractedconsiderable attention over past years. Among thestudied systems a spectacular class constitute the CO,NO, O3, CO2 or XeF* in solid Ar [1–4]. The vibra-tional energy transfer in 13C16O2 isolated in solid Arhas been investigated by laser induced fluorescencemethod [2]. The intense radiation observed inthe 16-m region after strong excitation of the state(00°1) was ascribed to the vibrational stimula-ted emission. Also, for processes of the type(1110(2)1,22) and (0001 1 + 2, 32) the relax-ation times VT and V3V were estimated from decaytimes of stimulated emission basing on the assumptionthat nonradiative relaxation prevails.In this work we aim at clarifying the influence ofdifferent relaxation channels on populations of vibra-tional levels using the theory of multiphonon relax-ation proposed by Nitzan et al. [5]. Also results of esti-mation of the VER rate constants for the CO2/Arsystem by comparing the experimental and calculatedtemporal pulse-shape of radiation are discussed.In particular, we refer to the experimental resultsfor the 13CO2 vibrational-energy relaxation in solidargon under conditions of matrix-to-reagent molecularratio M/R = 2000 [2]. The laser excitation of the3 (v = 1) level of carbon dioxide molecules was foundto induce strong emissions (1 + 2, 32)  (1, 22)and (1, 22)  2 in the spectral region around16 m. Their sharp threshold as a function of laser ex-citation density was interpreted as a signature of vi-brational stimulated emission. It is known that theCO2 molecules are trapped in solid argon in two dis-tinct sites: single substitutional, which is stable, and© A. Cenian, G. Grigorian, and G. Œliwiñski, 200300011110(1)033 01110(2)VT1000(1)02201000(2)0110000 0= 0 1 2 3V3V15.8 16.716.2  mm mVVTFig. 1. Diagram of the CO2 vibrational levels with energylower than 3000 K; solid arrows denote the transitions ofvibrationally stimulated emission observed in Ar matrices,dashed arrows represent the exemplary nonradiative transi-tions of the type VT, V3V and VVT.double substitutional, which appears to be an unstablesite [2]. The first one due to the limited free-space ischaracterized by a stronger vibration-phonon couplingand shorter relaxation times (one order of magni-tude). The lasers used in experiments [2], allowed fora good time resolution and a site selective excitation.Both sites were found to be luminescent. The reportedemission spectrum due to stable site consisted of threelines (see Fig. 1). The relaxation times VT and V3Vfor processes of the type (1110(2)  1,22) and(0001 1 + 2, 32) were estimated in [2] from decaytimes of the stimulated emission on the assumptionthat the radiative relaxation may be neglected. In thecase of stable site the respective decay times were 700 and 100 ns.During final stage of paper preparation, we becameaware of the interesting work of Chabbi et al. [6], re-porting the rate constants for the VER processes ofisolated CO2, determined by radiation-pulse fitting-procedure using 6 independent constants, i.e., not re-lated by any scaling law.Model descriptionThe vibrational energy relaxation following strongpulsed excitation is studied by solving the kineticequations describing the time evolution of populationsof different vibrational levels of CO2 moleculesdn dt R R R R Rvv v v v v/     VT VVT V V sp ind3, (1)where nv denotes the population of vibrational level vand the Rv terms describe population changes due tothe VT, VVT and V3V intramolecular processes ofnonradiative relaxation as well as to spontaneous andinduced radiative transitions (the notations are thesame as those of Ref. 2). The intermolecular VV pro-cesses are neglected because of low concentration ofCO2 in Ar matrix. The Boltzmann-type initial vibra-tional-distribution is assumed. Excitation is takeninto account by a sudden increase of initial vibra-tional population (N001) of the 0001 state up to thearbitrary level (fitting parameter).The equations describing evolution of the photonnumber has the form:dNi/dt = R R Ri i isp ind loss  , (2)where Ni is the number of photons of the ith transi-tion (see Fig. 1), Ri corresponds to photon gain (boththrough spontaneous and induced processes) and lossprocesses. The main loss channel is related to the pho-ton leaving the excited region as described in Ref. 2.For spontaneous emission the Einstein coefficientsare applied [7]:Av,v–1 =643413 	v vhc, vv–1>2nn2223,where v,v–1 is the frequency of the v v–1 vibra-tional transition, <vv–1> is the respective ma-trix element of the dipole moment and n is the refrac-tive index. The last term describes generally theeffects of solid environment.The stimulated emission cross-section are derivedfrom expression [8]v,v–1 = c2Av,v–1/82v,v–1n2 ,where  is the FWHM of the spectral line associatedwith the v  v–1 transition of assumed Gaussian pro-file and estimated to be equal to 0.15 cm–1 [2].For discussion of the nonradiative vibrational-re-laxation of a guest molecule in a dense medium severalapproaches have been proposed [5,9–11]. Here, the re-laxation rates are described by the multiphonon relax-ation model proposed by Nitzan et al. [5]. This modelhas proven to provide a satisfactory description of therelaxation for the case of matrix isolated diatomic[12–14]. Accordingly, the rate constant of the vv–1transition is given by [5]Kv,v–1(T) = Kv,v–1(0)F(T), (3)where F(T) = (1 + ~n)Ne2S~n is the temperature coeffi-cient, S is the average vibration-phonon couplingstrength (assumed here to be 1), N = Ev,v–1/hph isthe number of matrix phonons involved in dissipationof the vibrational energy gap Ev,v–1 in the nonradi-ative relaxation process, ph is the average phononfrequency (hph = 64 K was assumed for Ar matrix),~n = [exp (hph/kT) –1]–1 is the phonon occupationnumber. For low temperatures (T  5 K) ~n  0,F(T)  1, and Kv,v–1(T)  Kv,v–1(0). The expressionK AvSN Ev vS Nv v,,( )!1101e(4)corresponds to the relaxation rate at T = 0 K, where Ais a constant related to the value of average phononfrequency and the variation of the interaction poten-tial between CO2 molecules and the surrounding ma-trix atoms.Three types of processes are taken into account, andthe notation corresponds to that of Ref. 2:VT: CO2(1 + 2, 32) + Ar  CO2(1, 22) + Ar,(5)VVT: CO2(1, 22) + Ar  CO2(1, 22 ) + Ar ,(6)V3V: CO2(0001) + Ar  CO2(1 + 2, 32) + Ar ,(7)1136 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10A. Cenian, G. Grigorian, and G. Œliwiñskiwhere CO2(1, 22) and CO2(1, 22 ) denote differ-ent vibrational states of the (1, 22) multiplet atFermi resonance – see, e.g., [15].Due to the strong dependence of the rates (3) uponN, only VT processes of = 1 contribute signifi-cantly to energy relaxation. The other processes (V3Vand VVT) also proceed with the minimum vibrationalenergy gap. The VT and V3V reverse processes respon-sible for energy transfer from translation to vibrationare neglected. However, in the case of VVT processessome rates obtained from the principle of detailedequilibrium are not negligibly small.Results and discussionFor the stable site emission the time evolution ofphoton numbers related to radiative profiles is studiedin order to get best agreement with experimental datareported in [2]. Accordingly, three fitting parametersAVT, AVVT and AV3V, related to the constant A in ex-pression (4) for the rates of nonradiative processes,which correspond to transitions of the VT, VVT andV3V type (Eqs. (5)–(7)) are determined. As a result,all rates for vibrational energy exchange can be calcu-lated. So, this procedure replaces the determination ofthe seven independent rate constants as proposed inRef. 6. In both cases, the initial excitation N001 is anadditional fitting parameter.The sets of equations (1) and (2) are solved usingthe GEAR code for numerical integration. Equationsdescribe populations of all the nine vibrational states,which become populated during relaxation process attemperature T = 5 K, and photon numbers of the threeactive radiative-transitions (1 + 2, 32  1, 22)and (1, 22  2) – see Fig. 1. The terms related toradiation gain and losses were determined from Ein-stein coefficients, the stimulated emission cross-sec-tions and populations of respective levels. Thenonradiative terms were determined by rates (4)scaled by the constants AVT, AVVT and AV3V.It should be stressed again that in [2] the followingrates for the processesVT: CO2(1110(2)) + Ar  CO2(1000(1)) + Ar ,KVT_exp = 1.4·106 s–1 (VT = 700 ns), ( 5 )V3V: CO2(0001) + Ar  CO2(1100(1)) + Ar ,KV3V_exp= 107 s–1 (V3V= 100 ns) ( 6 )were proposed, basing on assumption of nonradiativerelaxation. The saturation condition (N001 = 0.5) forinitial excitation was assumed, also.Results of calculations are presented in Fig. 2–7.Figure 2 demonstrates the evolution of radiation in-tensity determined by rates scaled (by changing AVTand AV3V) to the values as proposed in [2] (see dottedline). The assumed value of excitation is evidently toohigh – as was also confirmed in later considerations onenergy conservation [6]. It was found here that the bestfit is obtained for N001_b = 0.07, KVT_b = KVT_exp andKV3V_b = 0.3KV V3 _exp (see solid line in Fig. 2). A si-milar value 4·106 s–1 for the rate KV3V was proposedin [6]. The signal profile is not very sensitive to theThe vibrational relaxation of CO2 isolated in solid argonFizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 1137N       = 0.5N       = 0.07N        = 0.070010010011.00.80.60.40.20 1.0 1.50.5 2.0Intensity, rel. unitsKKK 0.3 KKKK KV  V_expV  V_expV  V V  V_expVT_expVT_expVT VT_exp333 3==Time,    sFig. 2. Time resolved pulse of vibrational stimulated emis-sion from CO2 located in a stable site of Ar matrix; sym-bols ♦ correspond to the measurements, lines represent si-mulations: solid line – for the best fit parameters, dottedline – for the rates determined in [2] and saturation condi-tion (N001 = 0.5), dashed line – for the rates determinedin [2] and N001 = 0.07.N       = 0.090.050.070011.00.80.60.40.20 0.5 1.0 1.5 2.0Time,   sIntensity, rel. unitsFig. 3. Time evolution of the stimulated emission pulsefrom CO2 located in a stable site of Ar matrix; symbols ♦correspond to the measurements, lines represent simula-tions: solid line – for the best fit parameters, dotted line –for optimal rates and N001 = 0.09, dashed line — for opti-mal rates and N001 = 0.05.rates of quasiresonant processes of the type (7) as longas they exceed the value of KV3V. The optimal valueof the rate for the process VVTCO2(1000(1)) + Ar  CO2(0220) + Ar , ( 7 )KVVT_b  109 s–1 was found and it was assumed forall curves in Fig. 2. Figure 3 presents sensitivity ofthe profile on the parameter of initial excitation.The signal profile depends strongly on the rateKV3V (see Fig. 4). Both the position of maximum andthe slope change with the rate variation. The valueKV3V_b is smaller by a factor of about 1/3 if com-pared to KV3V_exp — this should be related to radia-tive effects neglected in [2]. It is evident that the radi-ative processes influence the kinetic seriously, e.g.,they determine the temporal scale of the pulse profile.This agrees well with the data reported in [6], and itstays in sharp contrast to the previous assumptions[2]. If we decrease further the rate for the V3V pro-cess, an additional maximum on the temporal pulse-shape appears; clear evidence of the independent radi-ation from all 3 transitions.The changes of KVT rate influence the investigatedprofile much less significantly (see Fig. 5). The rateincrease shifts the maximum to smaller value and1138 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10A. Cenian, G. Grigorian, and G. Œliwiñski1.00.80.60.40.20 1.0 1.50.5 2.0Intensity, rel. units1.2 KK0.8 KV  V_bV  V_bV  V_b333Time,    sFig. 4. Same as Fig. 3; solid line — for the best fit pa-rameters, dotted line — for the best fit parameters exceptKV3V = 1.2 KV3V_b , dashed line – for the best fit parame-ters except KV3V = 0.8 KV3V_b .1.00.80.60.40.20 1.0 1.50.5 2.0Time,    sIntensity, rel. units1.2 K0.8 KKVT_bVT_bVT_bFig. 5. Same as Fig. 3; solid line — for the best fit pa-rameters, dotted line — for the best fit parameters exceptKVT = 1.2 KVT_b , dashed line — for the best fit parame-ters except KVT = 0.8 KVT_b .1.00.80.60.40.20 1.0 1.50.5 2.0Time,    sIntensity, rel. units10 K0.1 KKVVT_bVVT_bVVT_bFig. 6. Same as Fig. 3; solid line — for the best fit pa-rameters, dotted line — for the best fit parameters exceptKVVT = 10 KVVT_b , dashed line — for the best fit param-eters except KVVT = 0.1 KVVT_b .1s0.2 s2 s 00011110(2)100 0(2)01101000(1)1110(1)0220033010101010100 1000 2000 3000– 1– 3– 5– 7– 9VED, rel. unitsEnergy, KFig. 7. The Treanor—Likal’ter type of distribution for thebest fit parameters and its time evolution; dotted line —0.2 s, dashed line — 1 s, solid line — 2 s after excita-tion.smoothes all structures. In contrast, the KVT decreaseleads to more pronounce structures. The profile is evenless sensitive to the changes of rates for near resonanttransitions (7) — in Fig. 6 we compare the results forKVVT differing two orders of magnitude. Again, thedecrease of the KVVT rate leads to structure enforce-ment, besides it widens also the profile maximum —compare broken and solid lines.Figure 7 presents the time evolution of vibrationaldistribution function. The characteristic saw-toothdistribution of Treanor—Likal’ter type, as describedpreviously in detail [15], can be observed. The grad-ual saturation of the all laser-active transitions(1 + 2, 32  1, 22) and (1, 22  2) is evident.ConclusionIt was shown that the vibrational relaxation ofCO2 molecules embedded in solid Ar matrix could bewell described by the theory of multiphonon transi-tions. The performed simulations confirmed that radi-ative processes influence significantly the vibrationalenergy relaxation of CO2 embedded in solid Ar, e.g.,the determined KV3V rate is significantly lower thanthat predicted under assumption of the nonradiativerelaxation.1. D. Jasmin, et al., J. Chem. Phys. 108, 2303 (1998).2. H. Chabbi, P.R. Dahoo, et al., Chem. Phys. Lett.285, 252 (1998).3. G. Zerza, G. Œliwiñski, and N. Schwentner, Appl.Phys. A56, 156 (1993).4. C. Crepin, M. Broquler, et al., Laser Chem. 13, 65(1999).5. A. Nittzan, S. Mukamel, and J. Jontner, J. Chem.Phys. 62, 200 (1975).6. H. Chabbi, B. Gautier-Roy, et al., J. Chem. Phys.117, 4436 (2002).7. E.U. Conder and G.E. Shortley, Theory of AtomicSpectra, Cambridge Univ. Press, Cambridge (1967).8. W. Koecher, Solid State Laser Engineering, Springer,Berlin (1996).9. D.J. Diestler, J. Chem. Phys. 60, 2692 (1974).10. J. Jortner, Mol. Phys. 32, 379 (1976).11. S.H. Lin, P.H. Lin, and D. Knittel, J. Chem. Phys.64, 441 (1976).12. I.H. Bachir, R. Charneau, and H. Dubost, Chem.Phys. 163, 451 (1992).13. I.H. Bachir, R. Charneau, and H. Dubost, Chem.Phys. 177, 675 (1993).14. A. Salloum and H. Dubost, Chem.Phys. 189, 179(1994).15. A. Cenian, Chem. Phys. 132, 41 (1989).The vibrational relaxation of CO2 isolated in solid argonFizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 1139

The vibrational relaxation of CO₂ isolated in solid argon

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The vibrational relaxation of CO₂ isolated in solid argon

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)