The vibrational relaxation of H2. II. Measured density changes; master equation study of the mechanism of relaxation

Chemical Physics 40 (1979) 87-107 0 North-Holland Publishing Company

THE VIBRATIONAL RELAXATION OF H,. II. MEASURED DENSITY CHANGES; MASTER EQUATION STUDY OF THE MECHANISM OF RELAXATION John E. DOVE z and Heshel TElTELBAUM $z lhpart~~w~~t of Chenlistry, lJtriversif_vof Toronto, Toronto, Ontario. Corzaclahl5S IAl Received 1 Scptcmbcr

1977

The vibrntiod rclasation of H2 in mixtures with inert gnscs 1x1s been studied in shock WVCS, using law schlicrcn dcnsitometry. The measured total density changes corresponding to the observed rclasation are too large to be accounted for by changes in vibrational cncrgy alone. It is suggcstcd that the rotational degrees of freedom of 112 participate in the relasation process. This is confirmed by master equation studies of the relaxation mechanism, which indicate that nn important role is played by transitions involving simultaneous changes in rotational and vibrational energy.

1. Introduction The vibrational relaxation of H2 is an important test case for theories of molecular energy transfer because accurate intermolecular potentials for several systems involving H, can now be calculated from quantum mechanics. In a previous paper [ I] (henceforth referred to as I) we described experimental measurements of the rate of vibrational relaxation of H, by He, Ne, Ar, Kr and H, at temperatures of 13503000 K. The purpose of the present paper is to discuss the mechanism of the observed relaxation process. In a shock wave experiment, the translational energy of the gas molecules is initially raised very suddenly, and the internal degrees of freedom then relax to equilibrium. Generally it is assumed that rotational relaxation is extremely rapid and does not influence the slower vibrational energy transfer, so that the process measured behind the shock front is vibrational relaxation alone. Then, if the transition probabilities

$ Alcsander von Humboldt Special Rcscnrch Fellow 197677. on leave of absence at lnstitut fiir Physikalischc Chcmic der Universitit, Tammannstrassc 6,‘D-3400 Ciittingen, West Germany; permanent address: University of Toronto. %* Present address: Institut fiir Physikalischc Chcmic dcr UniversitLt, Tammannstnsse 6, D-3400 Cijttingen, West Cermany.

among vibrational levels follow the Landau-Teller law. the measured energy relaxation time T is related to p, ,o. the transition probability per collision for deexcitation from u = 1 to u = 0: by l/r =ZP,,o[l

- exp(-IzY/kT)]

,

(1)

where i! is the collision frequency per molecule. In this way a molecular transition rate between quantized levels can be found from an experimentally measured macroscopic quantity. However, there is now substantial evidence, which we discuss below, that the rotational degrees of freedom participate in the vibrational relaxation of Hz (e.g. refs. [Z-7]). The observed relaxation may thercfore consist of a relatively complex sequence of transitions among vibration-rotation levels. The questions then arise, what is the actual mcchar.ism of the high temperature relaxation process, and is there in fact a direct relationship between the measured relaxation time and a molecular transition rate? Clearly the answers to these questions have important implications for experimental tests of energy transfer theories. In the following sections of this paper, we present and discuss three types of relevant evidence: (i) the density changes measured in our shock wave experiments, (ii) a master equation study of the mechanism of relaxation, and (iii) consideration of the measured relaxation rates themselves.

88

J.E Dove. H. Teitelbaunz / The vibrational relaxation of H> II

2. Measured density changes

When a molecular gas relaxes behind a shock front, the interconversion of internal and translational energy creates density gradients which can deflect a beam of light from, e.g., a cw gas laser. In the experimentsreported in I, this energy interconversion process was studied by measuring the time dependence of the laser beam deflection. Note that an absolute calibration of the deflection sensitivity was not needed to determine relaxation times alone. However, we did in fact make an absolute calibration, so that actual density changes could also be measured. Density gradients in the relaxing gas could not be measured accurately until the disturbance due to the shock front itself had effectively ceased to interact with the laser beam. In our experiments, measurements could begin 1 :Oto 1.5 vibrational relaxation times after shock arrival. In order to determine the total density, the measured density gradients therefore had to be extrapolated to the shock front itself. It will be seen that a very short-lived transient gradient due to an independent extremely rapid rotational relaxation directly behind the shock front itself would not be observed in these experiments and should not influence the extrapolated density changes due to vibrational relaxation. As explained in I, given the composition and initial state of the test gas, and the measured speed of the shock wave, one can calculate the state variables at three hypothetical locations in the shock-heated gas: immediately behind the shock front with only the translational degrees of freedom excited; translation and rotation excited, vibration unexcited; and thermal equilibrium_ Thus from thermodynamic and hydrodynamic considerations alone, we know Ap,,, the density change due to vibrational relaxation alone, and AP,, , the density change expected for vibrational and rotational relaxation combined_ 2.2. Resrdts From each relaxation experiment, we obtained a value of Vi, the intercept of a plot of the logarithm of detector output voltage against time. It may be shown from eq. (B7) of I that the total density change

Appohsassociated with the observed relaxation process is

(Note that the last equation on page 442 of I should be numbered B.5). The quantities on the right hand side of eq. (2) above are all either experimental parameters or may be obtained from literature data, e.g. 0, the Gladstone-Dale constant of the gas; they are defined in I. The derivation of eq. (2) implies that only vibrational relaxation is occurring, so that Apohs should be equal to Ap,. However, fig. 1 shows that in all but a few cases. the extrapolated density changes are larger than expected for vibrational relaxation alone. At the upper end of our temperature range, ApUbs is about four times too large and even approaches Apv+,. Although it is not immediately obvious from fig. 1, the observed density changes actually fall along individual curves, one for each mixture, with a scat- ter of about 15% The mixtures which show the largest

1.0

01.

1

_-____---_

’

1500

I

---

---__-__--____

I

I

I

2000

2500

-___

--&I

I

T (“10 Fig. 1. Observed density changes comp?red with those espccted if only vibrational relasation occurs -, and if vibrational + rotational energy relax togther - --. Mixtures denoted by: o 30% H2,70% Kr; l 50% Hz, 50% Kr; o 72% H2, 28% Rr; + 33% Hz, 33% Rr, 34% Hc; a 30% H2, 70% AI; A 50% Hz, 50% Ar; @34% Hz, 33% Kr, 33% NC; n 25% Hz, 25% Kr. 50% NC.

LE. Dove. Ii. Teitelbaum / The vibrational relaxations of H,. II

differences between 4pPobsand 4pv are generally those with the fastest relaxation times. In calculating values of Apobs, three non-idealities were taken into account. Firstly values obtained from eq. (2) were corrected for the finite width of the laser beam (cf. I). Secondly, a correction was applied for non-unidimcnsional

flow caused by the viscous bound-

ary layer in the flow behind the shock front. Application of this correction

required an estimate of the axial extent

of the shock front, which was obtained from a detailed analysis. [8] of the shock curvature [9,10]. The result of the non-unidimcnsional flow is to create a small shift relative to the time origin. Thirdly, there is the question of the location of the time origin itself. A detailed analysis [3,8] of the interaction of the laser rays with the curved shock front shows that. for most practical purposes, the time origin can bc placed at the schlieren spike minimum on the oscilloscope trace (cf. fig. 1 of I). However, the full analysis shows that the time origin is shifted slightly towards the spike maximum, creating a small additional correction. All of these considerations have no influence on the mcasured value of 7, and the cumulative correction to Aoobs, as compared with an uncorrected value using the schlieren minimum as time origin, is at most 25% and often much less. 23_ Discussion In assessing the significance of these surprising results, measurement errors should be discussed. The thermodynamic variables, as well as the instrument constants of eq. (2), can be obtained to better than 5%. The schlieren minimum is very well marked on the oscilloscope traces, and errors in measuring its position are unlikely to affect &J~,,~ by more than 10%. Of course, because of the exponential form of the signal, 4pobs is very sensitive to the choice of time origin; an error of one relaxation time in location of the origin will alter 4p,,bs by a factor of 2.7. Nevertheless a very substantial change would be needed in order to force Apubs to agree with Apv; in some cases, it would be necessary to locate the time origin significantly beyond even the schlieren maximum, almost in the directly observable part of the exponential decay. Some other workers using the laser schlieren technique have also reported measurements of Apobs_ Kiefer and Lutz [l l] found density changes that (when

89

corrected [12] by an instrumental factor of two) are generally larger than expected, though the authors ascribe no special significance to these deviations. Breshears and Bird [13] have reported measurements of density changes in HCI and DCI. In their experiments, they chose the maximum of the schlieren spike as their time origin [ 141. However, we have analyzed the form of signal to be expected from their detector configuration (which was different from ours), and believe that the time origin should be placed earlier, in which cast Apobs would be greater than Apv for their cases also.

From our results, we conclude that the observed relaxation of Ha is not a simple exponential variation of the vibrational energy alone, starting immediately at the shock frout. Two possible explanations are: (a) some rotational relaxation is taking place at the same time, or (b) the observed relaxation is of vibrational energy alone, but it proceeds at its full rate only after an induction period. It is not possible to distinguish between (a) and (b) from our density measurements alone. Explanation (a) implies that rotational and vibrational relaxation occur at fairly similar rates. At room temperature, vibrational relaxation is certainly much the slower process. However vibrational relaxation speeds up above room temperature, whereas rotational relaxation almost certainly slows down [ 1S-17] _The slowing down evidently results from the influence of increasing energy level spacings as the equilibration of higher rotational levels becomes more important. Though it is not known for certain how slow rotational relaxation is at our shock tube temperatures (1350-3000 K), nevertheless explanation (a) appears a distinct possibility. Explanation (b) implies that the onset of vibrational ralaxation is delayed until som upper rotational levels become populated, indicating that rotationally excited Hs is an important intermediate in the vibrational excitation mechanism. It will be seen that in either case our measured density changes suggest strongly that rotation is playing a significant role in the observed relaxation process. 3. Master equation study From the observations described above, we see that the relaxation process may well involve a complex

90

J.E. Dove, H. Teitelbauttz/ The nibrational telaxatiotz of Hz. II

sequence of transitions among vibration-rotation levels of Hz. Moreover, there may be significant rotational disequilibrium in important regions of the relaxation zone. To investigate the mech&sm further, we

have therefore made a nmerical study in which the relaxation was treated as a chemical kinetic problem, each vibration-rotation level being considered as a distinct species_ The time dependence of the level populations was expressed by a set of differential equations (the “master equation”), one for each species. Our essential objective in this study was to make as realistic a model as reasonably possible, of the high temperature relaxation and to solve the resulting equh-

tions, in order to discover what are the important steps in the relaxation mechanism. We did not set out to achieve precise agreement between experimental and calculated relaxation rates. Nevertheless we took care with the selection of the potential and with the computational aspects of the study, in order not to introduce artifacts which might lead to qualitatively wrong conclusions. As input to the master equation, we needed intcrlevel transition probabilities, which we calculated by quantum mechanics. We selected the H2-He system because good ab initio potentials

are available. More-

over we chose to confine our attention to states of even J on the assumption, which seems likely to be very good at high temperatures, that para-HZ is also a good model for normal H,. 3.1. The pctfetttial etlergy surface The potential energy can be expressed as a sum of intramolecular U(r) and intermolecular Y(r, R, 0) contributions. Here r is the H-H internucIear distance, R is the distance between He and the centre of mass of Hz, and 0 is the angle between the r and R vectors. The H-H interaction was represented by a Morse potential, with the rotationa! motion of Hz included by adding a centrifugal repulsion term so that (following Herzberg [ 181) each rotational state has an effective potential curve

Table 1

Enegies (in cm-l) of some cigcnstatcs of the non-rigid rotating ~MorseosciUator model of H2 J

Energy

18 16 14 12 10 8 6 4 2 0

v=o

v=1

Ll=2

16793 13876 11102 8522 6189 4156 2476 1198 363 0

20936 18018 15245 12665 10332 8299 6619 5341 4506 4143

24826

eigenstates of H,. However, the differences between its energies and the spectroscopic values are less than 5% for the states which we find to be the main contributors to the vibrational relaxation mechanism. For the H2-Hc interaction we used the surface of Wilson et al. 1191. Compared to other ab initio surfaces, this has the advantage that it covers a larger range of internuclear distances and hence can represent even quite highly excited states of the system. We note that Wilson et al. used a somewhat larger basis set than Gordon and Secrest [20], but a smaller one than Tsapline and Kutzelnigg [21] and Raczkowski and Lester [22]. The Wilson and Gordon-Secrest surfaces probably fall off too slowly at large distances, where the larger basis set is most important, but this is not expected to be important for modelling collisions at high temperatures. The Wilson surface agrees well with Raczkowski and Lester at short helium-hydrogen distances. The surface of Wilson et al. [19] was presented as a three body interaction term expressing the deviation of the potential from pairwise additivity. Thus they wrote the totai potential energy for HZ-He as V(r,Rz,Ra)

U..(f) = D,{l - exp [-p(r - r,,)] }* + s

,

(3)

where De,0 and t-eare the usual spectroscopic quantities. This non-rigid rotating Morse oscillator model (cf. table 1) does not precisely reproduce the energy

21909 19136 16556 14223 12190 10510 9232 8397 8034

+

YN@,&.R~)

=

VH-I-&)+ ,

VI++_H(RZ)+ VH+H(RJ) (4)

where R2 and R3 are the two H-He internuclear distances and VN is the deviation from additivity expressed by eq. (3) of ref. [ 191 (with r =- RI)_ To ob-

LE. Dow, H. Teitdbmm /The vibmtiomd relaxatio~tof Hz_ II

tain the complete potential, expressions for VH_” and V,,_, have to be added to V,. In this work, we used the Morse oscillator VH_-H(see above), and the H-He potential determined by Gengenbach et al. [23] from molecular beam experiments. For our purposes, it was convenient to cxprcss the interaction potential between H2 and He, V(r, RS, R3)

- VI.-&).

in the analytic form

V(r, R, 0) = V,-,(r)exp[-a(r)

R]

8 x c a,(r) t1=0

P,l(cos 0) -

(3

This was done by a least-squares fit to a large number of points generated on the original potential surface. The coefficients a,, of the eighth-order Legcndre polynomial P,, express the asphericity of the Hz-He

pofen-

tial, and 01is the steepness parameter of the intermolecular repulsion. Both a,, and (Yare functions of r. For r = 0.767 a (the H-H distance in u = 0), or = 3.0 A-’ and a2 = 0.308, which agree well with other potentials calculated for r = r,,. In the range r = 0.6 - 2.6 A, R = 0.6 - 2.6 a, and 6 = 0 - 90”, the fit to eq. (5) satisfied the original points to within 1% accuracy. At the collision energies studied here, the effect of a van der Waals well is expected to be very slight and to be mainly due to an acceleration of the collision. Thus, as is usual in this type of problem, the transition probabilities were multiplied by the factor exp(&r) to

account for a well of depth E. For Hz-He, e/k = 19.7 K [24] _ 3.2. Cahdation oftransitiott probabilities The essential problem is to calculate the quantum mechanical collisional transition probability between the two levels (u, J) and (v’, J’). This is done by calculating the transition probability as a function of E, the initial energy of approach, and integrating over a maxwellian distribution of E:

srR&(E) is the total collision cross section for that energy. At the temperatures considered in this study, a very large number of channels are open in the collisions, and at present it is not feasible to calculate u by methods which explicitly include coupling between these channels. Accordingly, the method which we use to calculate o as a function of E draws on the distorted wave approximation [25] in the form used by Roberts et al. [26]. Note that rotation was explicitly included in our calculations, as well as vibration, and that rotation and vibration were allowed to couple. Except for the intermolecular potential, which is unique to this study, the method which we used to calculate u is described elsewhere [27], so that we summarize it only very briefly here. The inelastic cross section used in eq. (6), is given by

o(u’, J’ Iu,J; E) = ;

mEu(~r,J’[~,J;E)e-EIliT

Xi 0

nR?ot(E)

?(I[

+ 1) f’!(k) -

(7)

k = 27rpr1//2is the initial wave vector, u and /.J being the initial veIocity and reduced mass of He with respect to Hz. P!(k) is the probability of transition between the two states for the partial wave of orbital angular quantum number I. The summation over partial waves I essentially accounts for collisions of different impact parameters. s is a steric factor which equals 1 for Au = 0 and l/3 otherwise. The use of the modified wave number approximation [28] and several decoupling approximations allowed us to obtain analytical expressions for o in terms of the cross section for the s wave (I= 0). These expressions could then be factored into vibrational and rotational matrix elements, and a translational factor, describing the overlap between initial and final states. Note that

the Mies anharmonic factor [29], which is often omitted from distorted wave treatments, was included in our calculations; it has a considerable effect on the transition probabilities [30]. Values of R,,,(E) for eq. (6) were calculated as the distance of closest approach for a direct collision at energy E. Using these methods, we calculated

P(?J’,J’lU,J; T) = (kzy

91

transition

probabilities

for the follow-

ing types of transition among all levels: dE >

(6)

where u(u’, J’lu, J; E) is the cross section for collisional transitions from (u, J) to (u’, J’) at energy E, and

VT:

Au=--I,AJ=O;

RT:

Au=O,U=-2;

TRV:

Av==+l,AJ=~2.

Also about 90 multiquantum

transitions

were consid-

J.E. Dove, Ii Teitelbaum /The vibrational relaxation of Hz. i1

92

ered for the (v, J) levels considered likely to be important to the relaxation mechanism. The types of transition were:

lations and were calculated using detailed balancing. As an example of the results of these calculations, the J dependence of several types of transition probability involving the levels (u =

Av=O,AJ=-4;Av=Q,aT=-6;Av=Q,Ai=-8;

0,J)at3000 K is

shown in fig. 2.

~=--1,hl=+4;Av=-1,~=+6;Au=-1,~=+8. For transitions with [AJ( > 2, we used the more generai expression [31], for the rotational matrix elements

(8) where

II is an even integer

in the range

1.I - J’[
S

min(J +J.‘, 8), and the 3-j function

x (L - 2I)!(L -

[

_I

dn(u,J)/df -

for this system is

= c {k(v,J]v’,J’)rz(v’,J’) U’,J’

k(lJ’,J’lv,J)n(v,J)}

k(v’,J’lv,J)

(L/2)! x (L/2 -J)!(L/2

The master equation

,

(10)

where n(u,J) is the concentration of molecules in the level (u, J) and k(v’,J’]u,J) is the pseudo-first order rate constant for collisional transitions from (u,J) to (v’,J’) at temperature T:

2J’)!(L- 2n)!W

(L + I)!

3.3. Solution of the master equation

(9)

- J’)!(LI:! - n)! *

and I, =J + J,’ + n,. In.all cases, transition probabilities for the reverse processes (TV, TR, and VRT, including multiquantum jumps) were included in our calcu-

=z”JP(v’,J’lv,J;

T) _

Z,,J = rrR~(8kT/rr~)1~2[Hel and is the collision frequency of an Hz molecule in the level (u,J) with He molecules at concentration [He]. The collision diameter R, is calculated for an initial relative energy of kT. In considering the solutions of the master equation, it is convenient to define the quantity rSHO by l/rSHo = k(O, 0 il,o) [ 1 - exp(-hu/kT)]

V-T

(111

,

02)

where Izu is the energy of the (1,O) level. TSHOcan be considered as the relaxation time of a simple harmonic oscillator which relaxes by TV/VT processes only and which has the same value of P(0,0 11, 0; T) as our anharmonic molecule. The master equation was then integrated on an IBM 370/l 6.5 computer, using Gear’s method [32], to give the time dependence of the (0, J) level populations in a simulated shock wave experiment at 3000 K. The program also computed periodically the net

flux of molecules between all pairs of levels among transitions can occur. This information enabled us to trace out the main relaxation pathways and to

which 16_ 2u 2~ 28 32 J pii. 2. Transition probabilities for p-H~(u = 0, J) t- He p-H,(av,J + AJ) + He at 3000 K. The probabilities are shown for three types of transition: (a) T ++R (L\J = i2, AU = 0); (b) TctV(Au=5l,M=O);(c)T,R+V(hl=--2,Au=+l); (d)T-R(aT=+4j;(e)T,R-V(Au=l,hl=-4). 0

L

12

interpret

the relaxation

mechanism_

From the level populations, the vibrational energy, rotational energy, and total internal energy Etot=Eeb+ Erot,were calculated as a function of time. Then a

phenomenological (not necessarily constant) vibratio-

JX Dove, H. Teirelbam

/The vibrational relaxation of H,. II

nal relaxation time rvib may be defined by analogy with the Bethe-Teller law: Tvib --

_

E;b

7SHO

-

&ib

(13)

-

d&ib/d[f/Two]

Trot andTtot may be defined similarly.

3.4. Results atzd discussion The calculated time dependence of 5;ib, 7rot and Ttot is shown in fig. 3. Note that the time scale is expressed

in units of T_, the final relaxation

time of

the internal energy; this is the relaxation time that would be measured in a shock tube vibrational relaxation experiment. Fig. 4 shows the computed molecular inter-level net fluxes at three instants during the relaxation process. Initially, rotation and vibration . . relax qmte Independently. 7rot is very small (fig. 3),

I

1

’

.-

a

I

a

I

I

’

-.-.-.-.-.-.-.---_-.

-_ 10-2

;.. ;

--\; c

I 10-G

,

I

,

I

;

1

I

I

10-L

t &Jo-2 ’ lo2 Fig. 3. The transient behaviour of vibrationaland rotational

relasation times in the p-Hz/He system as a function of time (10~ . -. scale) at 3000 K. - - - - is the simulc harmonic oscillator relaxation time redo usingP(O,011, 0)as PI0 in eq. (1); -is the vibrational r&ration time, - - - the rotational reiasation time, and .*-the relaxation time of the total internal energy, calculated by solution of the master equation. The time scale is normalized by the final relaxation time TV, and the relaxation time by TSHO.

93

since rotational relaxation is taking place by very fast TR processes among the lowest J states of u = 0 (fig. 4a). At the same time, vibrational relaxation is taking place mainly by relatively very slow TV processes

among the lowest (tl, J) levels, with a time constant very close to TSHO.hi fact, initially drib is slightly greater than ?-SHO;the reason for this has already been given [33] _ As the upper J levels of u = 0 begin to be populated, the relatively fast TRV processes begin to contribute substantially to the vibrational relaxation (fig. 4b), and T,,+bfalls (fig. 3). At the Same time, transitions among the more widely spaced upper J levels of u = 0 start to have a substantial influence on the rotational relaxation, which slows down, i.e. 7rot increases. Eventually, after about t/T, = 1, vibration and rotation relax together in a concerted manner, and 7,ib, Trotand rtot become indistinguishable. It is this latter region, Wllere 7vib = Trot = rt,t, that evidently corresponds to the region which we studied experimentally_ At equilibrium at 3000 K, the largest population of any vibrationally excited (u, J) level is in the level (1,4). Moreover, the three IeveIs (1, 2), (1,4) and (I,@ together account for slightly more than one half of the total equilibrium vibrational energy at this temperature. Therefore we can get an insight into the vibrational relaxation mechanism by studying the routes by which the lower J levels of u = 1 become populated, starting from the lower J levels of u = 0. Fig. 4 shows our computed inter-level net fluxes at three instants during the relaxation. Because of the small transition probabilities for AU= +l processes among the low J levels (cf. fig. 2), initially vibrational relaxation is very slow (fig. 4a). As the upperJ levels of u = 0 become populated, rapid vibrational relaxntion by TRV processes begins (fig. 4b). We see from fig. 4c that the favoured route for the fully developed vibrational relaxation is predicted to be rotational excitation up the Jlevels of u = 0 up to about J= 12 and 14, followed by a TRV transition (simultaneous changes in u and J) to u = 1, and then rotational decxcitation down the J ladder (fig. 4~). Similarly, excitation to ZJ= 2 occurs mainly by TRV transitions from the upper/ levels of u = I. Thus a net downwards flux is predicted in the J ladders of vibrationally excited states. Consequently, whereas the population ratios of successive J levels of u = 0 correspond to a temperature less than the translational temperature (3000 K),

94

LE. Dove, H. Teitelbaum / T%e vibrational relation

18

l

.

.

16

l

.

.

of

Hz_ /I

. . .

. v =1

v:2

v=o

(a)

v=l

v=2

(c)

Fig. 4. Computed major inter-level fluscs during relasation of

J .

18

. . . . 8

.

6

. .

8, by He at 3000 K. The points represent v, J energy lcvcls for v = 0 to 2 end J = 0 to 18. with J increasing upwards and u increasing to the right. The numbers corresuond to net flus, in the direction of the arrows, in arbitrary u&; the units arc, however, consistent from one diagram to another. The notation 1.7 (+7) means 1.7 X 107. The fluxes are shown at the following times: (a) t/r_ = 0.0001;(b) t/r- = 0.001;(cl r/To.= 0.3. Vertical lines represent TR/RT processes (nv = 0, al= *2), horizontal lines TV/VT processes (au = 21, AJ = 0) and the diagonally sloping lines TRV/VRT processes (au = ? 1, A/ = ~2). At very early times, the main event is a very fast relaxation of the lower J!evcls of u = 0; vibrational relaxation is then very slow, with TV and TRV processes contributing about equally. As the upper J levels of u = 0 begin to be populated (b), much more rapid TRV processes from these levels begin to be important, and the rate of vibrational rekation increases considerably. The TRV relaxation mechanism is fully developed in (c). It will be seen that in u = 1 and u = 2, the net flux is gcnerillly downwards [except at quite high val-

ues of& note the upward flus from u = l.J= 10 and 12. in (b)]. Fluxes of multiquantum processes are omitted but do not significantly change the overall relaxation mechanism.

a

.

v=2 (b)

in u = 1 and IJ= 2 many of these ratios correspond to temperatures above 3000 K. In fact, some very high transient rotational “temperatures” are found. For example, at t/r- = 0.006, the ratio of (1,12) to (1,

J.E Dodge.H. Teirelbaum / The vibrational relaxotior~of Hz_ II

95

14,161 used master equation methods to study both rotational and vibrational relaxation of HZ--He and HZ-H, at low temperatures, calculating his transition probabilities by an effective potential coupled-channel quantum mechanical method, and obtained good agreement with experimentally measured relaxation rates. He too concluded that rotation plays a significant role in vibrational relaxation.

IO0-

_________.___--w

We conclude that the following sequence of processes can account qualitatively for our experimental observations on vibrational relaxation of H2 in shock

u 10-L-

waves:

10-6-

10-aI

,

1

I

I

I

I

I

10-x 10“ 10-s trr, Eig. 5. Computed rates of changeof internal energy (105scale) for H, in He at 3000 R as a function of time (log SC&): d.&b/d(f/r.x) - - - ; dE~Otld(r1T3 -; dEtOt/d(f/rm) _*a. 10-7

10) corresponds to over 6000 K. However by the time the relaxation has settled to a concerted process, these disequilibria have become very much less pronounced. At t/r_ = 2.0, the ratio of (1, 12) to (1,lO) corresponds to only 3004 K, and that of (0, 12) to (0, 10) corresponds to 2999 K. The calculated time dependence of the rates of change of rotational, vibrational and total energy is shown in fig. 5. Whereas clE,,,/dt falls continuously from f = 0, dEvib/dr is predicted IO accelerate markedly after an induction period in which the molecules partly relax rotationally. Several other workers have made master equation studies of the relaxation of HZ. However, only Pritchard [34,35] has considered the same temperature regime as in the present work. He did not calculate his transition probabilities a priori but instead analyzed the behaviour of the master equation solutions using various reasonable patterns of transition probabilities. He concluded, as we do, that rotation-vibrarion transitions are important in the relaxation process. Rabitz

(1) First. there occurs a rapid rotational relaxation of the low J levels only; during this initial phase, vibrational energy transfer is very slow and takes place mainly by TV processes requiring large energy jumps. This is followed by (2) a somewhat slower rotational relaxation process which populates the higher J levels of IJ= 0. Vibrational relaxation by simultaneous u and J changes from these levels begins to be important. This leads into (3) a phase in which vibration and rotation are relaxing together in a concerted process. In this process, vibrational relaxation occurs mainly by relatively fast TRV transitions from upperJ levels. These transitions are much more nearly resonant than the TV processes (cf. table l)_ Phase (3) corresponds to the region which was directly observable in our experiments. The predicted acceleration of relaxation by the TRV processes is very substantial, by a factor of 47 at 3000 K. Apparently, therefore, the contribution of TV processes to the observed relaxation is practically negligible. On a quantitative basis, however, the results are not completely satisfactory. The calculated vibrational relaxation time for HZ-He at 3000 K is too long by a factor of 20. Moreover, while our model does predict an induction

period for vibrational

relaxation

and an

excess extrapolated density change, the predicted values are much too small. Nevertheless the results are considerably better than from a standard distorted wave calculation for this system, made without taking rotation explicitly into account, which leads to a predicted relaxation rate which is too low by a factor of 1000. Also, of course, the latter calculation predicts

96

J.E. Dow. K Teitelbnum/ The vibrationalrelaxationof !S,. II

no excess extrapolated density change. This suggests, therefore, that the inclusion of rotation in our calculations was a considerable move in the right direction, but that our method of calculation still undercstimates the effects of rotation on the vibrational relaxation process. We stated above that our objective was to clarify the mechanism rather than to obtain quantitative agreement with experiment. Nevertheless, clearly the probable reason for these quantitative discrepancies should be examined, to see whether an improved calculation would be likely to lead to a qualitatively different result. We will consider the magnitudes of the transition probabilities for each type of process in turn. TV transitions The single quantum transition probabilities are small, so that a careful calculation by the distorted wave (DW) method should give good results, at leasi for lowJ_ However, in the present model (cf. fig. 2), rotation of the molecule does not affect the TV transition probabilities. In reality, dynamical effects are expected to increase the transition probabilities at high J, as is found in classical trajectory calculations on Ha-He [36] _ Inclusion of Au = 52 transitions had no significant effect on our calculated relaxation rates. However these transition probabilities are probably underestimated, and in practice they would somewhat increase the relaxation rate, though without changing the basic mechanism. RT transitions At low J, the Ai = C? transition probabilities approach unity, and are expected to be overestimated by the DW method. However for 1AI1 > 4, the transition probabilities may well be too small. An improved calculation would probably give a somewhat slower rotational relaxation and a larger incubation time for vibrational relaxation. TR V traiuitions The approximations made, in order to obtain manageable expressions for the cross sections, partly decouple rotation and vibration and give transition probabilities which are undoubtedly too low. Removal of these approximations would increase the coupling of rotation and vibration in the relaxation process, and speed up the final relaxation rate. Support for these statements is provided by close-coupling calculations [4,6,37] on Ha-Hc. Moreover, the rates of TRV transitions with 1AJl > 2 are almost certainly underestimated by the DW method; thus the shortening of the final relaxation time, by

these transitions, should be substantially greater than the 10% which we found. In summary, therefore, we conclude that further improvements in the computation of the transition probabilities would give better quantitative agreement with experiment and would confirm the qualitative picture of the mechanism provided by our master equation calculations. We also examined the applicability of eq. (I), the Landau-Teller expression fcir r, to our results. WC find that, after the initial transient period, this exprcssion is very well obeyed. However P,,, is not the transition probability from (1,O) to (O,O), but is averaged over initial rotational states and summed over fina ones. For our conditions, a thermal equilibrium avcrage was quite adequate. Thus our present answer to one of the questions posed in the introduction is affirmative, in that the observed relaxation rate can be related to a transition probability for u = 1 + 0. However, the relevant transition probability is now a highly averaged quantity, and cannot readily be deconvoluted to give transition probabilities between individual (u, J) levels. Moreover this answer might have to be modified if a revised calculation were to lead to much greater disequilibria among rotational states. In such a case, it is much less certain that there will be a direct relationship [38-401 between the observed relaxation rate and a transition probability, even if the latter is

averaged over the actual non-equilibrium

distribution.

4. General discussion

We discuss additional evidence relevant to the relaxation mechanism. This evidence comes from (i) other theoretical studies of the relaxation rate, and (ii) experimenta! studies of isotope effects. 4.1. Theoretical calcltlatiom of vibratiotralrelaxation times of H2

Vibrational energy transfer in H, has been the subject of a great deal of attention from a theoretical point of view. In this section, we briefly review calculations of vibrational relaxation time, rviu, concentrating on Ha-Ha and Ha-He because of the availability of ab initio potential energy surfaces for these systems. The results of energy transfer calculations can be

97

J.E. Dove, II. Teitelbaum f The vibrational relasation of’H2. II

sensitive to quite subtle details of the potential and even to different analytical fits of the same surface [6,41-43]. Nevertheless we find a fairly cIear general trend in the theoretical results for rvib, namely that when the effects of rotation are carefully taken into account, the calculations often agree fairly well with experiment. However if only the vibrational degree of freedom is included, the calculated rates are typically several orders of magnitude too small. 4.1.1. Hz-H2 Our experimental results for H2-H2 vibrational relaxation are compared in fig. 6 with the results of three T + V calculations. The three-dimensional distorted wave calculations of Salkoff and Bauer [44] and Calvert [45] apparently predict the absolute values (but not the temperature dependence) of pTH2_F12 rather well. Parker’s semiclassical calculations 1461 underestimate ,W somewhat. However the inclusion of the Mies anharmonic factor alters this picture substantially; for OL= 4.2 A-*, this will reduce [29] PI0 by a factor of 1 X 10e2 to 7 X 10b2, depending on the collision energy. For the CY zz 3.5 a-’ used, e.g., by Salkoff and Bauer,P~ wou!d increase by at least a factor of 100. Parker’s calculation additionally does not

use symmetrized velocities; it is known that correct velocity symmetrization (see, e.g., the review of Rapp and Kassal [47]) brings semiclassical treatments into agreement with quantum calculations. We conclude that T + V theories, without inclusion of rotational effects, predict that pure H2 should relax much more slowly thanactually observed_ When rotation is explicitly included. Billing Sdrcnsen [48] finds good agreement with experiment at low temp&atures, though Ducuing and co-workers 1491 judge the agreement to bc fortuitous since the D2/Hz isotope effect is not correctly predicted at extrelnely low temperatures. 4.1.2. Hz-He Our experimental HZ-He relaxation times are compared with a number of theoretical treatments in fig. 7. The main common feature is that T + V treatments show a steeper temperature dependence than the experiments.

(T OK)+_

(T 0K)-‘3

Fig. 6. Vibrational relaxation times of pure Hz: comparison between experiment and theory: - - - Calvert [45]; Salkoff and Bauer [44] --.-;..- Rrkcr 1461.

I+. 7. Vibmtional relaxation times for Hz dilute in He: present espcrimcnts; -.- SSH theory; -SSH theory with anharmonicity correction: ... “proper” distorted wave C&I&tion;--- Shin theory [52]; -o- Billing S&cnscn theory [48]. l’hc four bracketed lines are Alcsandcr’s [43] effective potcntial calculation using different potential imer_eysurfaces or analytical fits to the same surface.

98

LE. Dove, H. Teitelbotmr / The vibrational relaxation of Hz_ II

From the Hz-He potential of Wilson et al. [ 191, we can extract an exponential repulsion parameter of

3.017 A-’ at 5 kcal mole-’ for u = 1. This value is confirmed by molecular beam experiments of Amdur [SO], which gave an orientation-averaged potential at intermolecular distances similar to those used in our energy transfer calculations; a refit of their power law expression gives a! = 3.02 ? 0.06 8-r in the range 1.4-l .8 A. Using this value of o in a standard SSH calculation [ 511 gives a value of pr which at first sight agrees moderately well with experiment_ However when the Mics anharmonic factor [29] is included, as it should be, the calculated relaxation rates become a factor of 1000 too low. This entirely matches our conclusion (section 3.5) that the distorted wave (DW) method gave much too low relaxation rates if only VT and TV transitions were considered. (The SSH treatment is essentially the DW method, approximated to give a closed form expression.) Shin’s semiclassical twodimensional calculation [52] of P,, using the early Krauss and Mies potential [53], also gave a relaxation time that was very much too slow (fig. 7). Recently, he has improved his calculation [54] and reports excelIent agreement with our experiments. However, his treatment apparently still uses the uncorrected classical trajectory, and has been analyzed [Sj] and found to be inaccurate, so that its status must be in some doubt. In general, then the calculated values ofP,o (T + V; J= 0) are too low to account for the observed HzHe relaxation rates. When effects of rotational excitation are included, the situation changes dramatically, as will be outlined in the following paragraphs. Generally, TRV transitions (u, J) + (u’, 3’) have the major effect, but initial rotational energy also enhances the rate of (u,J) b (u’, J) transitions. One exception is a calculation [48,56] which finds that steps like (0, J) + (1 ,.I - 2) have only a 10% effect on the relaxation rate, but which nevertheless agrees with our measurements of 7. However, we anticipate that use of an anharmonicity correction [30] in these calculations would substantially reduce the importance ofTV transitions ]27] relative to TRV. The most accurate theoretical treatments of rotational and vibrational energy transfer in HZ-He come from quantum mechanical close coupling calculations [57]. Because of the many channels open at high temperatures, such calculations are currently only feasible

at relatively low temperatures. However, McCuire [58-641 and Rabitz [65--691 have developed the coupled states approximation and the effective potential method, which are p.owerful techniques for reducing the required computational effort. Both methods have proved to be reasonably accurate approximations to the full close coupling calculation [6,60-641 if a large enough basis set of states is chosen. This subject has recently been reviewed by Toennies [70] _ Even with these improved methods, the computation of vibrational relaxation rates at shock tube temperatures is still a formidable task, and most studies have concentrated on low temperature relaxation. Calculations, all including TRV processes, have been made by McCuire [6], Toennies [37], Alexander [6, 43,711 and Rabitz [4,5]. While there are significant differences between the different authors, they generally agree on (a) the importance of vibration-rotation energy transfer, e.g. (1,4) + (0,6), even at low temperatures; i.e. the contribution of transitions with Ai# 0 far outweighs that of Al= 0; (b) I Nl> 2 is not unfdvoured; (c) usually, breathing sphere calculations do not adequately predict energy transfer rates (but see refs. [5,72]). Alexander [43] has made calculations over the temperature range of our experiments, and agrees reasonably well (fig. 7) with our value of 7. However this was not a master equation study, but used instead an equilibrium assumption for the rotational states. Consequently it does not give any direct information about transient or nonequilibrium behaviour. Rabitz and Zarur [4] have made a master equation study at lower temperatures than our experiments, and found relaxation to occur by TRV routes, as discussed in section 3.4 above. 4.i.3. Hz-Ne Our experiments [l] showed that vibrational relaxation by Ne is even faster than by He; thus Ne contrasts with Ar and Kr which relax H2 more slowly than does He. Moreover, the temperature dependence of relaxation by Ne is less than for other collision partners [l] _These observations may be connected with the finding 1731 that the H,--Ne potential is more anisotropic than for HZ--He, which will enhance TR and TRV processes_ In support of this, we note that rotational relaxation rates of Ha by inert gases [74, 751, measured near to room temperature, also follow the sequence Ar < He < Ne.

99

.I.E. Dove, H. Teitelbmtrn / The vibrational relaxation of 112. II

4.2. Experitnentally measured isotope effects Interaction potentials involving different isotopic species are identical, to within the Born-Oppenheimer approximation. Consequently it is found that the ratios of TV relaxation rates, for the hydrogen isotopes, can be readily predicted_ Deviations from these predictions may indicate a different relaxation mechanism. A large variety of classical, semiclassical, and quantum TV calculations predict that PI0 depends on the mass and potential parameters as follows: P IOOc0~~1WvWr2/p~U6aV~~ Xexp(-54n4n1v2/&2X.T)113 exp(W2kT)

,

(14)

where v and p are the frequency and reduced mass of the oscillator, y takes into account any asymmetry in the mass distribution of the molecule, and LYis the intermolecular repulsion parameter. At shock tube temperatures, there is substantial agreement between Bird and Breshears [76], Moreno [77], and Kiefer and Lutz [ 111 for pi(D2-D2). and excellent agreement between our work and that of Kiefer and Lutz [I I] for &HZ-HZ). At lower temperaturcs, the relaxation of H, and D2 has been carefully studied by Ducuing’s group [49,78-831. Relaxation of HD has been studied at shock tube temperatures by Simpson et al. [84], and at lower temperatures by Hopkins and Chen [85]. Ratios of relaxation rates for these three isotopic species are therefore available over a large temperature range. For the predicted ratio of TV relaxation rates of H2 and DZ, the dependence onol cancels, but it does not completely do so for HD. We have therefore taken a value of o1= 3.49 II-‘, from molecular beam measurements [86]. Our conclusions are not sensitive to the exact value of 0. Then it may be shown that: p7(HD-ND) P@-Hz) = csch(2993/T) 0.913 csch(2592/T)e~p[(518/T)‘/~]

faster than either H, or D,, although it is intermediate in mass between the two of them. This anomaly was first noticed by Hopkins and Chen [85] who further pointed out that the effect could be explained as due to R * V energy transfer. Very rapid vibrational relaiation, and apparently anomalous isotope effects, have been observed in other hydrides [ 13,87-931 and have been explained as due to R *V transfer tiiSing out of the special form of the mass distribution_ Essentially, in a hydride HX, the centre of mass is close to the X nucleus. Thus, for a given rotational eriergy there is an unusua!ly high peripheral velocity of rotation of the H atom. This high velocity dynamically favours interconversion of rotational and vibrational energy in a collision. The asymmetric mass distribution of HD is thus seen as the factor which speeds up

the relaxation. However, table 2 shows that the periphenl rotation velocities of H, and D, are not very much less than that of HD. Thus rotation may well enhance the vibrational relaxation rates of H2 and D2, though evidently

less than that of HD. We note, incidentally, that the relaxation rates increase in the order of the peripheral roration rates, namely D2 < H2 < HD. Table 3 shows that the ratios of the H2 hnd D2 relaxation times also deviate froni iredictions, especially at low temperatures. In part, t&&e deviations arise from low temperature quahtdmeffects which are not taken into account by eq. (14). A correction at 55 K leads to a predicted value [3] of FT(H2-H2)/ p7(D2-D2) = 0.0067, still not in agreement with experiment. Effects of an attractive portion of the intermolecular potential do not change the conclusion significantly [83], and it is shown that, taking experimental error limits into account 1491, in the studied

Tilblc 2

Coomparisonof the translational and peripheral rotational velocities of 112, HD,and D2 at 300 K -

P@-W = csch(2116/T) At room temperature, the predicted ratio of relaxation times is p7(H2-Hz) : ~T(HD-HD) : p7(D,-D2) = 1 .OO : 15.80 : 18.59, and at 2000 K 1.00 : 1.25 I 1.67. The measured values are 1 .OO: 0.76 : 4.16 and 1.OO: 0.56 : 1.41, respectively. HD relaxes much

El2 HD D2

vtrans ‘) (lo5 cm/s)

90t b, (105 Up/s)

2.520 2.298 l.782

1.831 2.039 1.300

%:_I,,, = (8RT/n4~2. trot = wr averaged over the rotalional

distribution.

.’

. ..

100

LE. Dove. H. Teitelbaum / The vibrational relaxation of Hz. II

Table 3 Isotope effect in the vibrational self-rclasation of hydrogen

Table 4

T

r W)

;dH,-H,)hdD,-D,)

WI

3000 2000 1350 296

55

expt.

refs

0.653 0.597 0.505

0.93 0.68 0.46

[Ill, 111 [111* 111 t111* [II

0.0_517

0.13

1.19 x IO--’

0.015

[781,1791 1781s[791

temperature range neither the absolute value of the isotope effect nor its temperature dependence can be explained simply with a T + V mechanism. According to three-dimensional T G V theories, for heavy collision partners M ~ sinh(2116/T)

pr(Da-M)

sinh(2993/T)

and for collisions

pr&-He) pr( D2-He)

’

cspt.

rcfs.

300

0.932

1.03

300 1350

0.932 2.36

0.39 1.23

[78l, 1791 (7619III

2000

2.17

1.23

3000

1.90

1.23

L761,111 L76l. 111

WI, 1941

ortho- and para-Hz by Ducuing [49,83] _It was clearly shown that Hz in J = 1 relaxes vibrationally more rapidly than does H2 in J = 0, regardless of whether it is self-relaxation or relaxation by He. The authors suggest that processes such as

in AE transferred to translation more than compensates for the large changes in angular momentum. However, it should be pointed out that, while the experiments

127/T)“3]

x sinh(2116jT) sinh(2993jT)

pr(H2-Hc)/pr(D~-Hc)

are taking place, and consider that the reduction

with the

= 0.619 exp[(ll

effect in the vibrational relasation of hydrogen by He

talc.

Cilk.

_m(H,-M)

Isotope

’

do show that rotation influences (17)

usingB = 3.02 8-r. The experimental

results are shown in tables 4 and 5. On the whole, T + V theories seem to predict rather well the observations that &Hz-M) < p7(D2-M) for M # He, and pT(H,-He) > pT(D,-He) except below 310 K, and that the temperature dependence is greater for D2 than for H2 relaxa-

the dynamics

of vibra-

tional energy transfer, they do not directly prove that changes in Jare important during the collisions. In summary therefore, studies using hydrogen isotopes and spin isomers show that T * V theory does not fully explain vibrational energy transfer_ In the case of HD, rotational effects are enhanced by mass asymmetry, and this is already widely accepted. In the case of H2, while the discrepancy noted in any single

tion. Nevertheless table 4 shows that for He collisions, the quantitative agreement with the ratio of rates predicted by T +V theory is not good. Relative to D,, H, relaxes faster than expected, so that the difference between the Hz-He and D2-He rates is less than predicted. The precision of the measurements seems too good for the discrepancy to be due to experimental error. Moreover, as shown by table 5, for heavy collision partners M, H2 again relaxes faster than expected in comparison with D,, so that the difference between Hz-M and Dz-M rates is more than predicted. The most direct experimental evidence on the effect of rotation is not strictly an isotope study at all, but comes from some very elegant experiments on

Table 5 Isotope effect in the vibrational rclasntion of hydrogen by heavy

M

Ar

Kr

inert gses, M T (K)

pr(H2-h0/~7fD2-hf) talc.

espt.

refs

1350 2000 3000

0.505 0.600 0.653

0.32 0.35

LlllsIII

0.41

[111.[11

1350

0.505

0.17

2000 3000

0.600 0.653

0.29

0.51

IllI, 111 1761.(11 [761,[II 1761,ill

J.E. Dove, H. Teitelbaunt / The vibrational relax-ation of Hz. II

comparison with Da might be attributed to experimental error, cumulatively the evidence is that the relaxa-

tion of H, is faster, relative to Da, than expected from T + V transfer. One interpretation of this tinding is that rotational effects arc especially strong in the vibrational relaxation of Ha, even when compared to Da. Alternatively, Ha and D2 may relax in a generally similar manner, but a different theoretical treatment should be applied, which would predict an isotope ratio different from that of the T + V theories.

J.E.D. thanks Professor J. Troe and his colleagues in the Institute for Physical Chemistry of the University of Ciittingen for their kind hospitality during the

period in which this paper was written. He also gratefully acknowledges the award of a Special Research Fellowship by the Alexander von Humboldt Foundation. This research was supported by grants from the National Research Council of Canada and the Defence Research Board. H.T. also thanks the National Research Council for the award of a Scholarship_ We are grateT.E. Hull and W. Enright

111 ] J.H. Kicfer and R.W. Lutz. J. Chem. Phys. 44 (1966)

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3532. 1171C.D. Billing, Chem. Phys. 20 (1977) 35. [ 1 S] G. Herzbcrp, Molecular spectra and molecular structure, Vol. 1 (Van Nostrand, Princeton, 1964) pp. 425-426. [I91 C.W. Wilson, R. Kapral and G. Burns, Chem. Phys. Lcttcrs 24 (1974) 488. 1201 M.D. Gordon and D. Secrest, J. Chem. Phys. 52 (1970)

Acknowledgement

ful to Professors

101

for advice

about methods of numerical integration, and to Dr. David G. Jones for developing the computer program and for many stimulating discussions.

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