Interpretation of Rate Experiments With Resolved Quantum Levels BY TUCKER CARRINGTON National Bureau of Standards, Washington 25, D.C. Received 24th January, 1962 It is now possible to study the steady-state fluorescence and chemiluminescence of small molec- ules in resolved quantum levels, or to observe the time history of the vibrational or rotational relaxation of molecules suddenly produced in a non-equilibrium distribution. These experiments require interpretation in terms of rates or probabilities of vibrational or rotational energy exchange involving specified quantum states. If it is assumed that the state of a molecule after a collision depends only on its state immediately before the collision, and not on its previous history, or on the time at which the collision occurs, the sequence of collisions suffered by one molecule becomes a Markov chain, and the appropriate mathematical formalism may be directly applied. This estab- lishes a relationship between the observable transition rates and the more abstract concept of probability of transition per collision.The mathematical results are illustrated in discussing certain experiments . The rise of non-equilibrium molecular and chemical kinetics in theoretical and more recently experimental studies has given insight into the processes of energy transfer between molecules in resolved quantum levels. Where we once measured rate constants as a function of temperature, thus assuming equilibrium, and later measured relaxation times, often in complex systems where their physical sig- nificance is small, we are now entering a period when several experimental techniques make possible the observation and eventually the calculation from experimental data of transition rates between resolved rotational and vibrational quantum levels.This focuses attention on the statistical concepts which relate the rates or proba- bilities of transitions involving specific quantum states to the observed distribution of molecules among these states. N 0 N- EQUILI B R I U M EXPERIMENT S Before discussing statistical models and current experiments, we may review briefly earlier work on non-equilibrium molecular and chemical kinetics. The study of unimolecular reactions was perhaps the first clear entrance of non-equilib- rium effects, particularly in the fall-off of the rate at low pressures, and in the negative activation energy for the reverse reaction at these pressures.Another important area of study has been the so-called relaxation times for rotational or vibrational energy transfer, measured by the techniques of sound dispersion and absorption, shock waves, and impact tube. In a system of many available energy levels, a single relaxation time has significance only when the system is close to equilibrium, and in general bears no simple relationship to transition rates linking particula levels.1 In a series of experiments, McKinley, Garvin and Boudart,2 Garvin,3.4 and Cashion and Polanyi 5 3 6 have studied spectroscopically infra-red chemiluminescence from reactions of the type A+ BC-+AB + C in which AB may have vibrational energy amounting almost to the entire exothermicity of the reaction.In these experiments one resolves vibrational and rotational structure, and so one can study the population of these levels as a function of pressure and diluents. From these data, crude 44T. CARRINGTON 45 calculations of “ initial distributions ” or rates of population of the various levels by the reaction itself have been made, thus “ subtracting ” from the observed distribu- tions the effects of collisions following the initial reaction. The experiments have been designed primarily to get information about this initial distribution, character- istic of the chemical reaction itself, rather than about the rates of dissipative processes tending to establish equilibrium distributions of rotational and vibrational energy in the molecules after they are formed.In most spectroscopic studies of non- equilibrium systems, the separation of these two processes is extremely difficult. Experiments carried out by Norrish’s group have dealt with the same problem, and have recently been reviewed.7 Flash-lamp techniques are used, and the time resolution thus obtained eliminates some of the averaging over the distribution of free times between collisions which is involved in steady-state experiments. The use of optical absorption spectroscopy allows the determination of population in the ground state as well as in excited states of the molecules observed. In most respects, however, the interpretation of the steady-state experiments and flash-lamp experiments is very similar.8 MOLECULAR COLLISIONS AS A MARKOV CHAIN We now consider the statistical treatment of molecules which can exchange rotational and vibrational energy with a heat bath.In particular, we consider a system of identical, non-interacting molecules, capable of existing in a large number of accessible energy levels. These are dilutely dispersed in an inert gas, a heat bath of fixed temperature, with which they can exchange energy. A given molecule, in the course of its collisions with inert molecules, undergoes random transitions among these energy levels, a sort of random walk, which, however, has the property that transitions toward the levels most populated at equilibrium are somewhat more frequent than those in the opposite direction. This random sequence of transitions can be described as a Markov process because the state of a molecule after its next collision depends only on its state just before that collision, and not on its past history.DISCRETE AND CONTINUOUS TIME DEPENDENCE In connection with observation of transitions from level j to i in a given experi- (a) the rate of transitions, wij, transitions/sec per molecule ; (b) an arbitrarily assumed collision rate, z, collisions/sec per molecule, corres- ponding to an arbitrarily assumed collision cross-section ; (c) the probability of transition per collision, pij ; (d) the probability of transition in time interval 0 to t, Xij(t). mental system, four concepts describing the transition are of interest : We consider first only two levels, and then extend these ideas to the matrices in- volved when many levels are treated.If we consider only one kind of transition, we measure a mean rate w in the ex- perimental system. It is generally desirable to express this rate in a dinensionless form, by comparing it with another rate. For this purpose we assume an arbitrary collision cross-section and from this and the density of heat bath molecules we calculate the mean collision rate z. The quantity w/z has some meaning as a probability of transition per collision when it is less than unity, but it must be kept in mind that some collisions in which the distance of closest approach is equal to, or less than, that corresponding to the arbitrarily chosen cross-section will not lead46 RATE EXPERIMENTS WITH RESOLVED QUANTUM LEVELS to transition, and some more distant collisions will induce transitions.The probability of transition in a given collision is never precisely unity. If we ask for the probability of a transition in a given time interval t, the obvious answer is X(t) = pzt, the probability of transition per collision times the number of collisions in the time interval t. This will have meaning as a probability only when pzt< 1. The simplest form for X(t) for arbitrary t is obtained by assuming an ex- ponential distribution of free times between effective collisions. The treatment then resembles that of an optical absorption coefficient. The fraction of molecules which undergo a transition (are absorbed) in time t is X(t) = 1 -exp (- wt). For small t this approaches X(t) = wt = pzt, as we should expect.The simple ideas just discussed are easily extended to the general case of many available energy levels. We define the matrix A having as elements the rates wij, and express A as zQ, where z is a constant scale factor representing a maximum collision rate, and the off-diagonal elements of Q are qij = W i j / z . Since transitions j to j are unobservable in the present model, the diagonal elements of Q are deter- mined by conservation of number of molecules. This gives q j j = -2qij so that Q has column sums zero. The problem now is to establish relationships between the matrices A (hence Q), P and X(t). We again set pij = WQ-/Z, but now z must be chosen large enough so that &j< 1 for all j. The diagonal elements are then defined by pij = 1 - 2 pij = 1 +ajj/z.This definition makes qgj = pij for i#j, and in fact Q = P-1. We now consider the relation between the matrices X(t) and P. The elements Xij give the fraction of molecules in state i at time t, given that all molecules were in state j at time zero. It represents the observed time behaviour of the system, and can also be considered as the probability of finding a molecule in state i, given that it was initially in state j . This matrix X(t) has been frequently discussed,*. 9 and is given by X(t) = exp (At). Substituting P for A and expanding in the limit t+O gives X(t)-,l+z(P-1)t. If this limiting expression may be used for t as large as t = l/z, we have X(l/z)wP. This says that the probability of any transition during a period equal to the mean free time is approximately equal to the probability of that transition per collision.This approximation gets better and better as we increase z, since the Wij are fixed and the qij thus gets smaller. Increasing z, however, means including relatively distant encounters as collisions, so that the concept of collision covers more different types of encounters and is less well defined. Consider again a number of molecules all of which are in the same quantum level at time zero. They undergo collisions and collision induced transitions randomly in time. We may investigate the applicability of a model in which each molecule makes collisions separated by the time interval l/z, i.e., at a regular rate. Knowing the matrix P, we can calculate the distribution of molecules among the available levels after any number of collisions.If x(n) is a vector in which each element xi(n) gives the fraction of molecules in level i after n collisions, where x(0) is the initial distribution. It is of interest to compare this result with the correct treatment based on collisions occurring at random intervals. In this case The difference between the two models lies in the fact that all the molecules will i i # j i#5 x(n) = P"x(O), (1) x(t) = exp (At)x(O), (2)T. CARRINGTON 47 not make the same number of collisions in any given time interval. We may co,m- pare (1) and (2) by setting t = n/z and expanding : (3) x(n) = P”x(0) = (l+Ajz)”x(O) = (l+nA/z+n(n-1)A2/2z2+ . . .)x(O), x(t = nlz) = exp (An/.) = (1 + nAlz + n 2A2/2z2 + .. .)x(O). (4) The two expressions then tend to agree in the limit t = 0, particularly when z is large and the elements of A are small. For the real physical systems we are dis- cussing, the matrix P has the property lim Pn = POo, so that the distribution ap- proaches a limiting equilibrium value as time, or number of collisions, increases. The two forms (3) and (4) therefore agree also in the limit t+m. The assumption that each molecule makes exactly the average numberlof collisions in any period of time may lead to significant errors in some cases, particularly in steady-state experiments. 10 A detailed comparison of random and regular collision models treating fluorescence in comets has been made elsewhere.11 n--tc;o FORMAL DESCRIPTION OF EXPERIMENTS INVOLVING RESOLVED QUANTUM LEVELS The concepts just presented can be used to give a concise formal description of models which are idealizations of experiments of the types described in the pre- ceding references.2-7 Discussing again a heat bath in which molecules of interest are dilutely dispersed, suppose that we have in some volume under observation an initial distribution xo, a vector in which each element xp gives the fraction of molec- ules initially in level i. Suppose that these molecules are removed by processes unspecified, except that the rate of removal of molecules in level i is zi per second per molecule.We provide further that molecules are constantly introduced into the system at a total rate c per unit volume, with a fraction xi(0) going into level i.This influx of molecules begins suddenly at time zero and is steady thereafter. We wish to investigate the time dependence and steady state of this system. Denoting the total number of molecules in unit volume by n, we have dx . CXi(0) 1 = CA..x.-z.x.+ -, n U J 1 1 dt dxldt = (A - Z)x + (c/n)x(O), or, in matrix notation, in which 2 is a diagonal matrix having elements zi.12 In the steady state we must have (c/n)zxi(0) = cln = ‘&xi. i i (7) Putting this value of c into (5) gives an equation for calculating the steady-state distribution, in which the superscript s labels the steady-state distribution and the matrix z is given by Both A and z have the property that all column sums are zero, hence the determinant of (A+z) is zero and a solution to (8) exists, determined by the requirement ZXP = 1.(A + ;)x(’) = 0 zi j = z jxi(o) - Zi6i j . (8) (9) * 148 RATE EXPERIMENTS WITH RESOLVED QUANTUM LEVELS If all the zi are equal, eqn. (8) reduces to where z is l/zi. If in eqn. (8) one assumes that every collisional transition probab- ility is proportional to the corresponding radiative transition probability, with always the same proportionality constant k, each term is multiplied by a factor (1 + kn), where n is the density of molecules in the heat bath. This cancels everywhere, and there is no effect of inert gas pressure.5 If x(~) is known from observation and a transition probability model is assumed, eqn. (8) can be written in the form (1 - zA)dS) = x(O), (10) ~ ( 0 ) = (Z - A)x'"/? (8') convenient for calculating x(0).Here Z is the average, Z = Zi&). i In terms of the steady-state solution, the differential eqn. (6) can be written, dx/dt = (A - Z)(X - x(')), (x - x@)) = exp (A - z)t(x" - x(~)), and its solution is as is readily verified by differentiation. The eigenvalues of A are not greater than zero 9 and those of Z are just the zt, so that the matrix exponential 13 goes to zero as t becomes large, and of course approaches unity as t goes to zero. This ensures that the solution (12) has the necessary properties in these limits. If there is no steady input, and no removal of molecules from the system, we have the simple relaxation equation The form of (1 2) indicates that the " quenching " processes zi which remove molecules from the system will speed up the attainment of the steady state.When Z is large, there will be little time for the relaxation processes described by A so, for example, there may be relatively little rotational or vibrational energy transfer during the observed lifetime of the molecules of interest. The model just discussed is applicable to the experiments of Polanyi596 and Garvin 3 , 4 if the energy levels involved are excited vibrational levels only. The " quenching " processes zi are then all those which produce transitions to the ground vibrational level. Since these molecules are unobservable and we ignore their col- lisional re-excitation to higher levels, they are essentially removed from the system. The model also describes in an idealized way, the flash-lamp experiments of Norrish's group.7 Here in the simplest case, the steady input is zero, 2 = 0, and all levels are included in the distributions of interest. The initial distribution xo relaxes to thermal equilibrium according to eqn.(1 3). The matrix equations just presented are, of course, entirely equivalent to the cor- responding systems of scalar equations describing the same model, and the matrix formulation introduces no new physical content. It is, however, useful in several ways. It divides the complexity of the problem into two parts, each of which may be more easily understandable and tractable than the sea of algebraic equations with which one is otherwise flooded. The first part is the formulation of the problem in matrix notation. In many cases it is possible to do this directly, without resort to the much more cumbersome scalar notation.The second part is the solution of the equations by the techniques of matrix algebra. Here the " book- keeping " required to keep track of the many energy levels is done automatically, and the many mathematical results concerning the properties of matrices can be applied. The matrix formulation is also convenient for use with high-speed electronic x = (exp At)x". (13)T. CARRINGTON 49 computers, for example, in predicting experimental results which would follow from some assumed set of transition probabilities. With the experimental pre- cision now available, it is in general not possible to calculate transition rates from experimental data, especially when transitions are not restricted to nearest neighbour levels? In fact, the observed distributions are quite insensitive to changes in the transition rate values.This is because there are in general a great many ways of going from statej to some other state k in, say, 5 transitions. Many of these possible sequences of transitions will not involve some particular transition pq, even when p or q may lie between j and k. Any particular transition probability ppq will not play a strong role in determining the results. What is observed is an average of arbitrarily many Markov chains of variable length, but usually short, since the ob- served distributions are generally far from equilibrium. TREATMENT OF EXPERIMENTAL RESULTS The treatment just outlined can be applied to the interpretation of several experi- ments involving observation of collision effects on molecules in resolved quantum levels.We now discuss two such examples. VIBRATIONAL DISTRIBUTION I N CHEMILUMINESCENCE In the steady-state experiments of Garvin and of Polanyi, the main interest is in deriving the relative rates of population of the several vibrational and rotational levels produced by the reaction under study. To do this, one has to allow for, or in some sense subtract, the effects of subsequent collisions tending to bring the ex- cited molecules into equilibrium with the heat bath. To do this, one has to assume a set of rate coefficients or transition probabilities for the transfer of vibrational or rotational energy among all the levels involved. The treatment developed in the preceding section can be used conveniently to compare results obtained with several different transition-probability models.As an illustration of this, we consider the work of Garvin on the reaction H + 0 3 = 0 2 + OH, wher6 the piime indicates vibiational excitation. We consider three sets of transition probabilities. In model a, we use radiative transition prob- abilities, as calculated by Garvin, and assume that the collisional transition prob- abilities are proportional to these. This model involves some relaxation of the harmonic oscillator selection rule Av = 1, but nearest neighbour transitions are still greatly predominant. In model b, nearest neighbour transitions are still favoured, but not so much as in a. The absolute values are chosen as to give, for each upper level, the same total probability of leaving that level in b as in a.In model c, transi- tions from a given upper level to all lower levels are equally probable, and the total transition probability from each upper level is normalized as before. Applying these three models to Garvin’s observed vibrational distribution 4 gives, using eqn. (S‘), three “ estimates ” of the input distribution, x(O), shown in fig. 1. The very drastic change in the transition-probability model from a to c has only the effect of smoothing out the x(0) distribution somewhat. This is strong indication that if accurate transition probabilities were known, the input distribution derived from them would not be very different from those shown in the figure. It should be emphasized that the model used here ignores collisions between two vibrationally excited OH molecules, such collisions being rendered negligible by the heat bath.Broida 14 has proposed that collisions of this type, OH’ + OH’-+OH* + OH, pro- ducing an electronically excited OH molecule, are important in these experimental systems.50 RATE EXPERIMENTS WITH RESOLVED QUANTUM LEVELS I I I MONOCHROMATICALLY EXCITED FLUORESCENCE We mention now a group of experiments in which the excitation mechanism is clearly known and fixed, and interest centres on collision processes by which rotational and vibrational energy is exchanged with a heat bath. These are steady-state fluor- escence experiments in which a single quantum level of rotation and vibration is excited by illumination of a gas with a well-chosen sharp emission line from an arc or discharge tube, which overlaps an absorption line of the gas.One then observes fluorescence from this initial level and from neighbouring levels populated from it by collisions. Many experiments of this type have been done,ls-ls but they have not allowed detailed interpretation. An experiment of this type with OH has recently been reported,lo and one is now in progress with NO. I 0 1 2 3 4 5 6 7 8 9 vibrational quantum number FIG. 1.-Relative rates of production of vibrationally excited OH molecules in the reaction HfO3 -+ OH’+02, calculated from experimental results of Garvin et al., ref. (4). Three different vibrational transition probability models are used. a consists of the radiative transition probabilities, and strongly favours nearest neighbour transitions.In c jumps of any Av are equally likely, and b is intermediate. The curve with no maximum is the observed distribution, ref. (4). The monochromatically excited fluorescence of NO was studied by Kleinberg and Terenin,lg using the cadmium line at 2144A. They obtained electronic and vibrational energy transfer collision efficiencies, but did not resolve the rotational structure. Experiments of this type are now being repeated with adequate rota- tional resolution.20 In the upper electroiiic state 2C the level v’ = 1, K’ = 13 is excited, with about two-thirds of the molecules having electron spin in the opposite direction to that of the rotational angular momentum. For NO+NO collisions, rough preliminary results give collision efficiencies of 1.3 for electronic quenching, 0-5 for vibrational energy transfer u’ = 1 to v’ = 0, and 1.5 for rotational energy transfer out of K’ = 13.The two spin states in rotational levels other than 13 seem to be roughly equally populated. One also observes fluorescence from the levelT . CARRINGTON 51 v’ = 0, populated by vibrational energy transfer collisions. A study of the rotational structure of this level should give information about the simultaneous changes in rotational and vibrational quantum number. Our observed efficiency of quenching from the first vibrational level is about 3 times of that found by Basco, Callear and Norrish.21 Addition of argon markedly increases the amount of rotational energy transfer in the v’ = 1 level.This is presumably because argon is much less effective as a collision partner than NO in removing electronic and vibrational energy, so molecules live longer in v’ = 1 and undergo more rotational transfer collisions. Addition of chemically inert gases thus allows one to change the effective time-scale for rotational relaxation. Concerning the rotational-energy distributions, one can test any transition- probability model by using the model and the known initial level to compute the distribution to be expected. The inverse problem, computing a transition-probability model from the experimental results, is in general much more difficult.8 It is a pleasure to acknowledge helpful conversations with David Garvin. 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