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Vibration-to-vibration energy transfer in COCO collisions
Volume30,
is
CHEMICAL PHYSICS LETTERS
number 2
Jmuwy
L9:5
: : . :
VIBR+%iION-T&‘IBRATION
:
.,
EN@%Y %4NSFE<
._
IN CO-CO COLLISTOI\IS”
Received 10 May 1474 Revised manuscript received 19 Atigust 1974
Vibration-t~v~ration
energy transfer rates for collision between two CO molecules arc calculated. Rcsul~ obtained and theoretical valtys. it is pointed out that none of the existing wIcuIatians gives a complete clccount of the available experimental data.
are compared with recent experiments
1. Introduction
,,
The rate constant for the vibration-to-vibration
energy transfer process
CO(~=~~-l)~CO(~=~)~CO(~=~)+CO(v=O)+[26.9(n-1)-O.09(r~2-l)]~-1 (0 n has been the subject of several experimental studies [l--6]. Three of these experimental studies obtain the room temperature rate constant for the tr = 2 case. The study of Stephenson [3] gives k2 = (1.26 -t 0.3) X IO5 sWLtocr-I, while that of Powell [6] yields k2 = (0.6 2 0.3) X 105 s-l torr -I. Each of these experimental values, which differ host by a factor of 2, is well outside the other’s error limits. A third experimental study of Sackett et aI. [5], bedause of its prel&ninary ,nature, has large error bars, k2 = (8.6 4 2.5) X IO4 s-I torr’l, and is therefore ofno help in deciding which of the two experiments is more reliable. On the theoretica.? side, the present author knows of three studies [7-Y]. Jeffers and Kelley [7] have used earlier work of the present author [IO] to obtain dipoledipole and dipole-quadrupole rates for process (i).‘The disquieting feature.about this calculation is th2.t tie, Cross section for a shorter range dipole-quadrupole interaction decreases more rapidly with energy mismatch than that for longer range dipole-dipole interaction. For example, the ratio (k;/k;)aa for dipole-dipole interaction at 300 K is about 2 whereas the same ratio for dipoIe-quadrupole interaction (k;fk;)&, is abou? 100. ms resdt is in direct conflict with the rather well established fact that the power spectrum of a shorter-range, for@ should fdIl less steeply with energy mismatch than that of a longer range ‘force [11,12]. Tam [S] has used earlier work of the present au&or [lo], using modified procedufes for averaging over impact parameter-and Maxwe&Boltzniann velocity distribution, to compute energy transfer cross sections for process (1). Tam’s result using his method is somewhat lower than that’he obtains usin& &m&hod of ref. [lo]. Tam’s results, in contrast.to those of feffers and Keliey, suggest that dipole-dipole coupling explains the @incock-Smith results for fii-k;t at room temperature. These hvo calculations use the same equation to compute cross sections for process (I). S&e their results dp not agree, at least one has to involve errors L-w~ et al. [9] have used dipole-~poi~,~teraction’as the energy transfk causing,p+s, but have introduced a parameter which allows arbitrary AJ transitions, still retaking * Work’supported b~&&t AFOSR-73;2475B. $:VisitingProfessor of Physiis.. .. _’
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Volume 30, number 2. ..
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- CHEM1CALPHYSIC.i ‘,,
LETTERS, ,_
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15 J&ry
1975
the straight line approximation. Before .&blisiling the causes oflimitations of the e,tisting theory, it is nofclear to’the present author, that introducing adjustable.parameters is the_correct’th@g to do: Secondly, execution of .; the c~alculation is not faultIess. For example, LevOn et aI.,ignore ihe term containirig’{2; the second or,der modi.fied Bessel.,function of the third kind in the dipole-dipole path integrals [lo]. !$-On et al.;:tLSingone adjustable parameter, obt& good agreement with the work of Wittig and $nith who measure reverse of ki’to k10 at 100 K. However, their calculatedvalue of k, is&-naller, by a factor ofnbotit six, than that measured by Stephenson and M&burg [4]. In,view of the.unsatisfactory situation regardins the late constAts fy pro(1) and +Aeir practi. cd importance in CO lasers, it was consrdered desirable to undertake a calculation.of process (1). In this letter we km-y out a straight path Born calculation for dipoleAipole, dipol+quadrupole and dipole+ctupole interactions. In addition, we carry out a limited amount of distorted-wave Born calculation [13] for process (1) using spherically symmetric Lennard-Jones (6-12) potential as the distorting potential.. :
of the identity
2. Coqsqueices
of CO molecules
‘. This calculation treats CO molecules
as distinguishable
particles.
The question
then arises: are the processes
co(V=1)+cO(Y=l)‘+co(v=2)+co(Y=o) and...
(2)
..
,. CO(v = 1) + CO(Y = 1) -* CO(Y = 0) + CO(V = 2)
(3)
two distinct channels which contribute equally to the rate constant k 2? To answer this question we recall 1141 that odd partial wavesin the wavefunction for relative motion change sign upon inter-changing our CO molecules whereas even partial waves remain unchanged odd partial wayes, we can write
under this operation.
Defining
Q, and $,-as the sum over even and
$+ 7 2_112~,[d,(v~ilm,)Q2(u2i2mz) -i 011v21’p23 &Jyljly)I end
(4)
: ‘. IL, = 2-“?-Qd[Q,(vlilnl,)92(Yzi2nz2)-~1(~2j21112192(Yljl~~~)l
Eqs. (4) and (5)‘thus give the properly
symmetrized
wavefunction
.
(5)
unless all three internal
quantum
molecultx 1 and 2 are equal. In the event that u1 = v2, )I = j2 and m, = ~2, the properly symmetrized tions are ‘.
,-
numbers of wavefunc-
..
++ = 2”28,1q3i,ml
1 @#lp91
>
(6)
where an extra factor of 2l/’ is put in to obtain the same incide.nt current density aCin eq. (4) The dipole‘dipole operator ‘. .).!; ‘PJ ‘dd.-
:
T-
.
6Qm9’4
.. (7)
6
does not change sign on interchanging CO molecules. This is because operation of interchange corresponds to interchanging the two dipole moments, and reversing the sign of r. The transitions caused by the dipole operator, .‘there{ore, obey the seIection ruies + t, + and - + -. The matrix element of the dipoleAipole operator when two ‘.molecules in the initial state have the same three quantum numbers can be written as
. ...
i~:lv,,l\o+}=(~~l(~~~(~~;i;m;)~z(v;j~m~) + ~~cY;i;m;)~2(Y;i;~~z~)].lv~~i~~~~~~~~~j~*~~~~~~~?
: .’
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i m,)4, .’ : ~ ‘. :.:. ~2~~~lC~,(~;l;~~~)~~(vf~m’)IV 22 2 .dd I@1(u 11 1 ,2 (V ,11i m 1,il-CJ..‘. ., _. -, , . .’ .:_. ,. ,.. .;’: ‘, ‘32 .. .’ ‘: ;::1, : .‘_ ‘. ; 1.. : ‘:. ‘._ .’ _’(. ” : ,’ ;‘. ; : .-. .I._ .,
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30, number 2
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CHEhlICAL PHYS1CS.LEITER.S
15 Januar); 1975
:
ff the weights of the even ahd odd partial waves are the same, which is the situation commomy encountered in problems involving molecuiar c&sions, the ~~t-h~d side of-eq. (S} is 2?2 times the matrix elements that would have been obtained if we had not symmetrized the.wavefunction and assumed that eq. (2) or eq. (3) describe the exit channels. The cross section would then be twice that obtained using eq. (2) or eq. (3), and ~h~.~s~n~is~able.particle.scheme:~f two molecules with the same vibrational quantum~collide, the probabilim that they have
the same] and m quantum numberiS$/HI For.a collisionbetweentwo’03 molecu!es,this probability is about 1.4/T. Ewn at lOOK, use of eq. .(I) would enhance the crosssection by only about 1.4% In.the,event that the two colliding molecules do not have the same quantum
numbers
in the initial
or the final states, we can write
where the first ten-n is the direct term and the second the exchange term. Because of the existence of selection rules in the dipolar transitions in CO-CO coUisi&s (Af = +I, hnz = 0 ,*I) in the first Born approtiation, the exchange ter-k in eq. (9) is either zero or about,equal to the direct term, In the former case, the energy transfer cross section is given by either process (2) or (3), whereas in the_latter case the cross section is given by the sum of the cross sections for processes (2) and (3). The probability that the latter situation prevails in collision between two CO molecI&s, is roughly 3B/kTor about 8.4/T. Thus in our calculation, which computes the rates for process (2) or (3), there is fess than 10% enhancement in the collision cross section, due to the identity of molecules. This rather small effect, which we will ignore, is due to the rigorous selection rules that our calculation assumes. In calculations.withoui selection rules [IS], the use of unsymmetrized wavefimctions~may introduce a much larger error.
3.
computations
For the straight path calculation we will use the equations derived earlier [lo] by the present author and Dr. used were the hard sphere diameter, d = 3.65 .k, and the dipole moment matrix element [16]
Brau. The molecuIaiparameters
there is no similar study for the dependence of the quadrupole moment on the vibrational coordinate, a similar dependence was assumed [ 17,181
Although
=12x 6.0 x 10-3D2A2 * IOll&-I_@ =n~(0~Q11)12 The spectroscopic parameters used for determining the energy jevels we&taken from Herzberg [19]. For the distorted wave calculation, the equation used was derived earlier by the present author and Dr. Schlossberg [13].The distortion potential was taken to be the Lennard-Jones 6-12 potential [20] V(r) =4e[(~I/r)~~ 7 (&)6], with weLI : depth e = 1.4 X 10-l: erg and d = 3.65 A, the value of the hard.sphere diameter in the straight path approximation.Because of the expense involved in the distorted w&e klculation, this calculation was performed for only X log c;m-‘. This conesponds io the most probable. one value of the relative velocity, corresponding to k =-1.3 relative speed between two CO moIecules at 300 K.‘Tabfe 1 compares the results of the straight path calculation with the distorted wave calculation for dipole-qua&pole coupling. Table 2 m&es similar comparison for dipoledipole coupling. The first thing to notice is ‘that the ratio for the .dipaIe-dipole rate constank (L&k& % 1.6 whereas the same ratio for dipole-quadrupole coupling, i.e., (k;lk;)dq, is 1.0. At higher vaIues of the energy rriismatch, ‘m’contrast to Jeffers-Kelley cakulation;dipo!e~quadrupol~ coupling,makes a greater contributi.on to the reaction cross section. We believe that correct reflection’of the physics requires that shorter range force gives a larger. crc)ss Section’ f&large en&& energy mi&atch. Also evident from these tables is &hatthe straight line calculation, folIowing the recipe given in ref.- [IO], is remarkably a&rate under the &cumstance that, the ;elatiGe. .I -
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Table:1 Comption of the strai&t path and distorted wave kc&tion for the energ.’ transfer probability per collision using .’ dipdle-qua&pole coupling for the process 3) ...CO(V=1)+CO(Y=li-1)‘CO(Y=O)iCO(~~~~) ~: + [26.9(~-1) - O:O?(n’-_l)] cm-’ Disioztedware
15 hluary.1975
CtiEMICAz, PHYSICS LETTERS
‘Volume 30?.nZmber 2 .; 1
'-
,’ ..,. Table 2 .’ . Comparistin of the stitight pith.and distorted wave &lc~lation for the energy transfer probability’ for collision using dipole-di;!ole coupting for the process y” ,CO(v=1)+CO(v=n-l)+CO(v=O)+CO(v=n) .. -’ +[26.9(n,l) - 0.09(r;2-1)] cm-’ Distorted wave
Sttip.Jltpathcalcllkion
_-
cslglation ,. k= 1.3X~I09’crn-’
k= 1.3X log cm-’
7-7 300 K
11
k=1.3x
2 3 4
8.5X IO4 1.2x’10-3
8.5 x lo+. l.iX 10-3
7.5x io-4
1.0x 10-j
-2 3
1.3x 10-Z 1.6 x 10-Z
5
1.3x 10-s 1.1x-10-3
n
6
1.3x 10-3
a
7.8X lo4 $5 x 103
9.
2.2x
7.
1.0
:,i: ,13. 14 15 16
.’
‘.
a) Rotational
:
T=300K
i.5X lo-* 1.9X 10-Z
1.3 x lo-
-2
1.7 x 10-Z 1.7 x 10-Z
‘4
1.7x 10-z
20x
-5
1.1 x 10-z
1.3 x 10-2
1.1 x 10-2
9.6 X 103-
6
4.3 x 10-T
4.7 x 10-X
7.6X lo4
6.8X,10q
7
1.2x 10-3
4.2X
4.0X
8
2.7 X lo4
4.7x 10-3 1.1x 10-3 2.0 x 104
9
6.4 x 1O-5
3.3 x 10-s
6.0x
IO4
1.9.x 104 6:s x 10-s 2.0x 10-s 5.4x 106 1.5 x 106 5.8 x 1O-7 2.8X lo-’ !.3[X 10-7
temperature
.k= 1.3 x 109 cm-’
109cm-l
1.1 x 10-3
;;I;
1.1 x 16-3
IO-$.
..
1.1 x lo+
;I;;
8.6x 1O-5 3.0 x.10-s s-7x 106 3.1 x 106 i.1 x 10m6 3.7x 10-7 1.3 x lo-’
Straight path calculation
calculation
lo4
1.9x 104 7.6 X lo-’ 2.6 x 1O-s
a) Rotational
temperature
10-z.
1.3 x lo29 x 104
1O-5
to bc 300 K.
of the gas was taken
-
of thegas was taken to be 340 k.
energy is of the ordar of the well depth. The skight lint: calculation used here differs from that in ref..[lO] in that the expressions presently used for avera@ng over velocity gave good accuracy for x < 20, whereas previously.these expressions were accurate for x < 10, where X =-UT, w is the.absolute value of the energy mismatch in radian/s, and 7 is the time duration of thk collision. Table 3 gives’the calculated values of the rates k, as a function of temperature, aid compares the cakulated and experimental results. The calculated dipole-dipole .’ probabilities are in fair agreement with those obtained by Jeffers-Ke!Iey, except for n = 2,3 and 4 at 300 K; the. latter valuk for these cues are smaller than the preserkcalculation by abput 50%. For reasons mentioned earlier -we belietie that the Jeffers-Kelley calculation for dipole-quadrupole rates may involve errors; the results, however, are not affected because ‘Theseprobabilities are smzdl. Our results for the d-d calculation begin to be smaller than t@e experimental values for II = 8 at room temperature. At lower temperatures, the disagreement be@ns to appear for smaller values of n. The discrepancy between our and Tam’s results may be due to &fact that Tam averages &v&ra Maxwell-Boltzrnann velocity distribution from zero ta infmity, in spite of the fact that he cornputes the rate constants in the endothermic direction. He thereby overestimates the rates by about exp(AE/kT), where’ AE is the energy mismatch, k the Boltzmarm constant, and T the temperature in K. Tam’s results and the present calculation agree rather well upon inclusion of this correction factor. To see if higher multipole moments play a role in the energy transfer processes considered irl this paper, a straight path calculation foi dipole-octupole coupling was performed. The octupole moment matrix element for the O-1 transition, was obtained from the value of its derivative evalu:lted at equilibrium i&&clear distance [la]. The matrix element between higher vibrational levels was compu:ted using the relatibn .. IGz&-!)/~= rzlUio7]O)12 = II X O.SS X 10m2 D2 A4 .: kuxlational
. The values ofP, at 300.K’for dipole-octupole coupling are 1.1 X 10-p, 9.7 X lo-!, 7:s X 10e5, for n = 7, 8 and ‘9, r&pe_&ively. Since the quadrupolkquadrupole coupling appears tb be of the same strength as @pole-octupole coupling,‘.it appears that oniy dipole-dipole coupling plays an imporkuQ role in,thr,.energyltransfer processes .
y.-,.,
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Volumej0, numb& 2
CHEMICAL PHYSICS LETTERS
IS Jznua~
1975
.. Table 3
co,~pafison of experimental
and calculated valdes of probabili ties p&r co&ion, P, for the energf llksfer - 0.090z2-111 cm-’
proCX%S’)
cocv =c-1) +co(~ =1)+ co(~ =0) +co@ =0) +[26.901-1) T= 100K
T=250K
.‘.
T=300K
,!
calculated
experimentaib)
calculated
experimentill
2’
3.6 x 10-Z
3.9 x 10-2 [4]
1.6 x 1O-2
1.7 x 10-2 [4]
cl
3.4 X 1W2 1231 3
3.5 x 10-z
4.1 x 10-2 -
2.0 x 10-2
4
;.ox
1o-2
1.7 x 10-z 1.5 x 10-Z
2.0 x 10-2
I.?? 10-z
x 1o-2
x 1G-2
calculsted
expkimental
1.4 x 10-Z ).
1.8x lo-“[3]’ 1.2 x 10-Z 151 8.4 x 10-3[6].
1.6 x 10-2
1.2 x k-2
1.8 x 10-z
1.2
x 10-z
i.lX
iv2
1.0x 8.2X
m-2 10-3
5
1.6 x 10-3
7.1 x IO-3 9.5 x 10-3
1.0
6
3.2 x 104
2.8X LO-3 5.7x IO-3
3.8X 1o-3
6.6 x 1O-3
4.8 x 10-3,
6.0 x 10-3 4.3 X 10-3
7
5.6 x 10-S
9.0 x LO-4 3.1 x IO-3
6.8 x lo4
3.2 x 10”
2.G x 1(Y3
3.1 x IO” 2.5 x 10-S
8
8.0 x lo*
1.7 x 10-X
3.9 X IO4
1.6 x 1O-3
6.9 x IO+
1.7 x 10-S 1.4 x 10-z
8.2 x 10L4
1.3 x IO4
9.1
: 9
9.9
x 10-7
1.1
1.2x
x 10”’
2.5
lG-2
x 10-4
d,
9.5 x lo”?; 9.0 x 10-j
3) The cakulated values use s&S&t path approximation. b) For experiments at 100 K for tz = 3-9, the first entry is from Powell [23 J, while the second is from Wttig and Smith (21. c) Ex&aim&ts for n = 4-P were performed at 254 K by Wittig and Smith [2].. d) For expetiments at 300 K tk n = 3-9, the fist entry is frbm POWII [6], whi~e’thc second is from Hancock and Smith [I].
considered here. The failure of the rnu~t~po~arnear-resonant calculation at large energy mismatches is entirely expected. It is likely that for large values of II the cross section, aS pointed out by Jeffers and Ke!ley [7], is determined by short range forces. These authors obtain good agreement with the results of Hancock and Smith [l] at 300 I(. Their results are, however, about an order of magnilude smaller than those’of the Witfig-Srntith [2J experiments at 100 K. It is also possible that cross sections for large values of n are given by second order processes 121,221. A large amount of experimental work at different temperatures is becoming available to put these and other theories
[15] to the test.
Acknowledgement
.:
.:
The author is thankful to Dr. Hart for help with the computations and to Drs.Powell and Stephenson for Ietting me see their work before publication. It is a pleasure to acknowledge the hospitality and kindness of Professor A:‘Javan and other members of the Infrared and Optical Lasers Laboratory at M.I.lT_ ..
..
.’
Note added in proof
-:,
1’ Since the manuscript was wiitteri, Hancock-et ai:[24] _’ .,. ‘, .: ._ .: ,.,
have measuredP2 = (I .52 ? 0.04) X 10q2 at room’teni-
‘265 ..
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30; number.2
‘,
CHEiIIcAL
,;
_: ..
PHYSICj I&TE$S,
‘...
,l5 January 1975
:
“:pe&e m a&emint tiith Stephenson’s’reflllt.I3]: Fti+ki s;d T&z~$[25] have determined P2Lpg, at ‘room ‘, temperature,.,to be 8.7(-3J 1.4(-P), 1.3(-Z); 1.3(-2j, 7.5(-3); 4.0(-3), 2.2(-3), 1.8(-3j, respectively; the
‘1Gm3er iqthe brackets denotes powers df 10. Fushiki‘and Ts~~&iya’sP2 agrees ,with. that of Powell [6] and is almost i factor df2’miaUer than that of Stepherisoh and that of fiancbck et al. P4-Pg Clues bf !?T tie consistent‘: ,. l$hi&er~hm thoie of Hancock id Smith [l].., ‘. .-
:
,,:.
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:il] i;. Hancockand LW.hl._Smith, Cl&n. khys. Letters 8 (1971) 41.,. ,,-. 1. [2] C. Wittig +nd’I.W.M. Smith, Chem. Phys. Letters 16 (1972) 212. ‘. .[3] J.C. Stephenson, AppL Phys. Letters 22 (1973).576. .[4] Y.C. Stephenson and ELR Mosburg Jr., J. C&em. Phys. 60 (1974) 3562. ’ [SJ P.R. Sackett, A. Hordvik and H. Schlossbcrg, Appl. Phys Letters 22 (1973) 367. [61.-F&T. PowelI, J. C&m. Phys. 59 (1973) 4937.. [71 W.Q. Jeffers and I.D. Kelley,‘J. Chcm. Phys. 55 (1971) 4433. [8] W.G. Tani, Can J:Phys 50 (1972) 2691. 19.1 hi. Lev-On, W.E. Palke and RC. hlillikan, Chcm. Phys: Leiters 24 (1974) 59. [IO] RD., Sharma and CA. Btiu, J. Chem. Phys 50 (19691924. [ 111 R.D. Sharma, Phys. Rev. 177’(1969) 102. [12] N.F. Mott a@d H.SW @assey, The theory of atomic’collisions (Chrendon Press, London, 1965)~~. 806-808. [l?] R.1). Sharma and ,e. Schlcssberg, @em. Phys Letters 20 (1973) 5. [14] K. Gottfried, Quaritum mechanics; Vol. 1, Fundamentals iBcnjami7, New, York, 1956) pp. 341-343. [15] T.A. Dillon. and J.C. Stephenson, Phys. Rev. .4 6 (19?2) 1460. [ 161 L.k Young and W.J.. Each&‘!. Chem. phys. 44 (1966) 4195. : 1171 RX. Nesbet, I. Chem. Phys 40 (1964) 3619. [IS] D.B. Newmann and 2. Wassermpn, private communication. 1191 .G. Heizberg, Spectia of diatbmic molecules (Van Nostrand, Prince!on, 1950). [ 201 J-0. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular theory $g.~ses ar?d liquids (Wiley, NewYork, 1959) pp. 11 IO- 1111. .[21].RD. Sharma, Phys. Rev. A 2-(1970) i73. [22] R.D. Sha-a and C.W. Kern, J. Chcm. Phys. 55 (1971) 117i. [23] KT. Powell. unptiblished results. [24] G. Hancock, Star tid S. Green, .I. them. Phys., to be published. [25] Fushiki and S. Tsuchiya, to be published.
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is
CHEMICAL PHYSICS LETTERS
number 2
Jmuwy
L9:5
: : . :
VIBR+%iION-T&‘IBRATION
:
.,
EN@%Y %4NSFE<
._
IN CO-CO COLLISTOI\IS”
Received 10 May 1474 Revised manuscript received 19 Atigust 1974
Vibration-t~v~ration
energy transfer rates for collision between two CO molecules arc calculated. Rcsul~ obtained and theoretical valtys. it is pointed out that none of the existing wIcuIatians gives a complete clccount of the available experimental data.
are compared with recent experiments
1. Introduction
,,
The rate constant for the vibration-to-vibration
energy transfer process
CO(~=~~-l)~CO(~=~)~CO(~=~)+CO(v=O)+[26.9(n-1)-O.09(r~2-l)]~-1 (0 n has been the subject of several experimental studies [l--6]. Three of these experimental studies obtain the room temperature rate constant for the tr = 2 case. The study of Stephenson [3] gives k2 = (1.26 -t 0.3) X IO5 sWLtocr-I, while that of Powell [6] yields k2 = (0.6 2 0.3) X 105 s-l torr -I. Each of these experimental values, which differ host by a factor of 2, is well outside the other’s error limits. A third experimental study of Sackett et aI. [5], bedause of its prel&ninary ,nature, has large error bars, k2 = (8.6 4 2.5) X IO4 s-I torr’l, and is therefore ofno help in deciding which of the two experiments is more reliable. On the theoretica.? side, the present author knows of three studies [7-Y]. Jeffers and Kelley [7] have used earlier work of the present author [IO] to obtain dipoledipole and dipole-quadrupole rates for process (i).‘The disquieting feature.about this calculation is th2.t tie, Cross section for a shorter range dipole-quadrupole interaction decreases more rapidly with energy mismatch than that for longer range dipole-dipole interaction. For example, the ratio (k;/k;)aa for dipole-dipole interaction at 300 K is about 2 whereas the same ratio for dipoIe-quadrupole interaction (k;fk;)&, is abou? 100. ms resdt is in direct conflict with the rather well established fact that the power spectrum of a shorter-range, for@ should fdIl less steeply with energy mismatch than that of a longer range ‘force [11,12]. Tam [S] has used earlier work of the present au&or [lo], using modified procedufes for averaging over impact parameter-and Maxwe&Boltzniann velocity distribution, to compute energy transfer cross sections for process (1). Tam’s result using his method is somewhat lower than that’he obtains usin& &m&hod of ref. [lo]. Tam’s results, in contrast.to those of feffers and Keliey, suggest that dipole-dipole coupling explains the @incock-Smith results for fii-k;t at room temperature. These hvo calculations use the same equation to compute cross sections for process (I). S&e their results dp not agree, at least one has to involve errors L-w~ et al. [9] have used dipole-~poi~,~teraction’as the energy transfk causing,p+s, but have introduced a parameter which allows arbitrary AJ transitions, still retaking * Work’supported b~&&t AFOSR-73;2475B. $:VisitingProfessor of Physiis.. .. _’
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261 ..~-
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Volume 30, number 2. ..
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- CHEM1CALPHYSIC.i ‘,,
LETTERS, ,_
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15 J&ry
1975
the straight line approximation. Before .&blisiling the causes oflimitations of the e,tisting theory, it is nofclear to’the present author, that introducing adjustable.parameters is the_correct’th@g to do: Secondly, execution of .; the c~alculation is not faultIess. For example, LevOn et aI.,ignore ihe term containirig’{2; the second or,der modi.fied Bessel.,function of the third kind in the dipole-dipole path integrals [lo]. !$-On et al.;:tLSingone adjustable parameter, obt& good agreement with the work of Wittig and $nith who measure reverse of ki’to k10 at 100 K. However, their calculatedvalue of k, is&-naller, by a factor ofnbotit six, than that measured by Stephenson and M&burg [4]. In,view of the.unsatisfactory situation regardins the late constAts fy pro(1) and +Aeir practi. cd importance in CO lasers, it was consrdered desirable to undertake a calculation.of process (1). In this letter we km-y out a straight path Born calculation for dipoleAipole, dipol+quadrupole and dipole+ctupole interactions. In addition, we carry out a limited amount of distorted-wave Born calculation [13] for process (1) using spherically symmetric Lennard-Jones (6-12) potential as the distorting potential.. :
of the identity
2. Coqsqueices
of CO molecules
‘. This calculation treats CO molecules
as distinguishable
particles.
The question
then arises: are the processes
co(V=1)+cO(Y=l)‘+co(v=2)+co(Y=o) and...
(2)
..
,. CO(v = 1) + CO(Y = 1) -* CO(Y = 0) + CO(V = 2)
(3)
two distinct channels which contribute equally to the rate constant k 2? To answer this question we recall 1141 that odd partial wavesin the wavefunction for relative motion change sign upon inter-changing our CO molecules whereas even partial waves remain unchanged odd partial wayes, we can write
under this operation.
Defining
Q, and $,-as the sum over even and
$+ 7 2_112~,[d,(v~ilm,)Q2(u2i2mz) -i 011v21’p23 &Jyljly)I end
(4)
: ‘. IL, = 2-“?-Qd[Q,(vlilnl,)92(Yzi2nz2)-~1(~2j21112192(Yljl~~~)l
Eqs. (4) and (5)‘thus give the properly
symmetrized
wavefunction
.
(5)
unless all three internal
quantum
molecultx 1 and 2 are equal. In the event that u1 = v2, )I = j2 and m, = ~2, the properly symmetrized tions are ‘.
,-
numbers of wavefunc-
..
++ = 2”28,1q3i,ml
1 @#lp91
>
(6)
where an extra factor of 2l/’ is put in to obtain the same incide.nt current density aCin eq. (4) The dipole‘dipole operator ‘. .).!; ‘PJ ‘dd.-
:
T-
.
6Qm9’4
.. (7)
6
does not change sign on interchanging CO molecules. This is because operation of interchange corresponds to interchanging the two dipole moments, and reversing the sign of r. The transitions caused by the dipole operator, .‘there{ore, obey the seIection ruies + t, + and - + -. The matrix element of the dipoleAipole operator when two ‘.molecules in the initial state have the same three quantum numbers can be written as
. ...
i~:lv,,l\o+}=(~~l(~~~(~~;i;m;)~z(v;j~m~) + ~~cY;i;m;)~2(Y;i;~~z~)].lv~~i~~~~~~~~~j~*~~~~~~~?
: .’
.’
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i m,)4, .’ : ~ ‘. :.:. ~2~~~lC~,(~;l;~~~)~~(vf~m’)IV 22 2 .dd I@1(u 11 1 ,2 (V ,11i m 1,il-CJ..‘. ., _. -, , . .’ .:_. ,. ,.. .;’: ‘, ‘32 .. .’ ‘: ;::1, : .‘_ ‘. ; 1.. : ‘:. ‘._ .’ _’(. ” : ,’ ;‘. ; : .-. .I._ .,
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30, number 2
:
CHEhlICAL PHYS1CS.LEITER.S
15 Januar); 1975
:
ff the weights of the even ahd odd partial waves are the same, which is the situation commomy encountered in problems involving molecuiar c&sions, the ~~t-h~d side of-eq. (S} is 2?2 times the matrix elements that would have been obtained if we had not symmetrized the.wavefunction and assumed that eq. (2) or eq. (3) describe the exit channels. The cross section would then be twice that obtained using eq. (2) or eq. (3), and ~h~.~s~n~is~able.particle.scheme:~f two molecules with the same vibrational quantum~collide, the probabilim that they have
the same] and m quantum numberiS$/HI For.a collisionbetweentwo’03 molecu!es,this probability is about 1.4/T. Ewn at lOOK, use of eq. .(I) would enhance the crosssection by only about 1.4% In.the,event that the two colliding molecules do not have the same quantum
numbers
in the initial
or the final states, we can write
where the first ten-n is the direct term and the second the exchange term. Because of the existence of selection rules in the dipolar transitions in CO-CO coUisi&s (Af = +I, hnz = 0 ,*I) in the first Born approtiation, the exchange ter-k in eq. (9) is either zero or about,equal to the direct term, In the former case, the energy transfer cross section is given by either process (2) or (3), whereas in the_latter case the cross section is given by the sum of the cross sections for processes (2) and (3). The probability that the latter situation prevails in collision between two CO molecI&s, is roughly 3B/kTor about 8.4/T. Thus in our calculation, which computes the rates for process (2) or (3), there is fess than 10% enhancement in the collision cross section, due to the identity of molecules. This rather small effect, which we will ignore, is due to the rigorous selection rules that our calculation assumes. In calculations.withoui selection rules [IS], the use of unsymmetrized wavefimctions~may introduce a much larger error.
3.
computations
For the straight path calculation we will use the equations derived earlier [lo] by the present author and Dr. used were the hard sphere diameter, d = 3.65 .k, and the dipole moment matrix element [16]
Brau. The molecuIaiparameters
there is no similar study for the dependence of the quadrupole moment on the vibrational coordinate, a similar dependence was assumed [ 17,181
Although
=12x 6.0 x 10-3D2A2 * IOll&-I_@ =n~(0~Q11)12 The spectroscopic parameters used for determining the energy jevels we&taken from Herzberg [19]. For the distorted wave calculation, the equation used was derived earlier by the present author and Dr. Schlossberg [13].The distortion potential was taken to be the Lennard-Jones 6-12 potential [20] V(r) =4e[(~I/r)~~ 7 (&)6], with weLI : depth e = 1.4 X 10-l: erg and d = 3.65 A, the value of the hard.sphere diameter in the straight path approximation.Because of the expense involved in the distorted w&e klculation, this calculation was performed for only X log c;m-‘. This conesponds io the most probable. one value of the relative velocity, corresponding to k =-1.3 relative speed between two CO moIecules at 300 K.‘Tabfe 1 compares the results of the straight path calculation with the distorted wave calculation for dipole-qua&pole coupling. Table 2 m&es similar comparison for dipoledipole coupling. The first thing to notice is ‘that the ratio for the .dipaIe-dipole rate constank (L&k& % 1.6 whereas the same ratio for dipole-quadrupole coupling, i.e., (k;lk;)dq, is 1.0. At higher vaIues of the energy rriismatch, ‘m’contrast to Jeffers-Kelley cakulation;dipo!e~quadrupol~ coupling,makes a greater contributi.on to the reaction cross section. We believe that correct reflection’of the physics requires that shorter range force gives a larger. crc)ss Section’ f&large en&& energy mi&atch. Also evident from these tables is &hatthe straight line calculation, folIowing the recipe given in ref.- [IO], is remarkably a&rate under the &cumstance that, the ;elatiGe. .I -
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263
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Table:1 Comption of the strai&t path and distorted wave kc&tion for the energ.’ transfer probability per collision using .’ dipdle-qua&pole coupling for the process 3) ...CO(V=1)+CO(Y=li-1)‘CO(Y=O)iCO(~~~~) ~: + [26.9(~-1) - O:O?(n’-_l)] cm-’ Disioztedware
15 hluary.1975
CtiEMICAz, PHYSICS LETTERS
‘Volume 30?.nZmber 2 .; 1
'-
,’ ..,. Table 2 .’ . Comparistin of the stitight pith.and distorted wave &lc~lation for the energy transfer probability’ for collision using dipole-di;!ole coupting for the process y” ,CO(v=1)+CO(v=n-l)+CO(v=O)+CO(v=n) .. -’ +[26.9(n,l) - 0.09(r;2-1)] cm-’ Distorted wave
Sttip.Jltpathcalcllkion
_-
cslglation ,. k= 1.3X~I09’crn-’
k= 1.3X log cm-’
7-7 300 K
11
k=1.3x
2 3 4
8.5X IO4 1.2x’10-3
8.5 x lo+. l.iX 10-3
7.5x io-4
1.0x 10-j
-2 3
1.3x 10-Z 1.6 x 10-Z
5
1.3x 10-s 1.1x-10-3
n
6
1.3x 10-3
a
7.8X lo4 $5 x 103
9.
2.2x
7.
1.0
:,i: ,13. 14 15 16
.’
‘.
a) Rotational
:
T=300K
i.5X lo-* 1.9X 10-Z
1.3 x lo-
-2
1.7 x 10-Z 1.7 x 10-Z
‘4
1.7x 10-z
20x
-5
1.1 x 10-z
1.3 x 10-2
1.1 x 10-2
9.6 X 103-
6
4.3 x 10-T
4.7 x 10-X
7.6X lo4
6.8X,10q
7
1.2x 10-3
4.2X
4.0X
8
2.7 X lo4
4.7x 10-3 1.1x 10-3 2.0 x 104
9
6.4 x 1O-5
3.3 x 10-s
6.0x
IO4
1.9.x 104 6:s x 10-s 2.0x 10-s 5.4x 106 1.5 x 106 5.8 x 1O-7 2.8X lo-’ !.3[X 10-7
temperature
.k= 1.3 x 109 cm-’
109cm-l
1.1 x 10-3
;;I;
1.1 x 16-3
IO-$.
..
1.1 x lo+
;I;;
8.6x 1O-5 3.0 x.10-s s-7x 106 3.1 x 106 i.1 x 10m6 3.7x 10-7 1.3 x lo-’
Straight path calculation
calculation
lo4
1.9x 104 7.6 X lo-’ 2.6 x 1O-s
a) Rotational
temperature
10-z.
1.3 x lo29 x 104
1O-5
to bc 300 K.
of the gas was taken
-
of thegas was taken to be 340 k.
energy is of the ordar of the well depth. The skight lint: calculation used here differs from that in ref..[lO] in that the expressions presently used for avera@ng over velocity gave good accuracy for x < 20, whereas previously.these expressions were accurate for x < 10, where X =-UT, w is the.absolute value of the energy mismatch in radian/s, and 7 is the time duration of thk collision. Table 3 gives’the calculated values of the rates k, as a function of temperature, aid compares the cakulated and experimental results. The calculated dipole-dipole .’ probabilities are in fair agreement with those obtained by Jeffers-Ke!Iey, except for n = 2,3 and 4 at 300 K; the. latter valuk for these cues are smaller than the preserkcalculation by abput 50%. For reasons mentioned earlier -we belietie that the Jeffers-Kelley calculation for dipole-quadrupole rates may involve errors; the results, however, are not affected because ‘Theseprobabilities are smzdl. Our results for the d-d calculation begin to be smaller than t@e experimental values for II = 8 at room temperature. At lower temperatures, the disagreement be@ns to appear for smaller values of n. The discrepancy between our and Tam’s results may be due to &fact that Tam averages &v&ra Maxwell-Boltzrnann velocity distribution from zero ta infmity, in spite of the fact that he cornputes the rate constants in the endothermic direction. He thereby overestimates the rates by about exp(AE/kT), where’ AE is the energy mismatch, k the Boltzmarm constant, and T the temperature in K. Tam’s results and the present calculation agree rather well upon inclusion of this correction factor. To see if higher multipole moments play a role in the energy transfer processes considered irl this paper, a straight path calculation foi dipole-octupole coupling was performed. The octupole moment matrix element for the O-1 transition, was obtained from the value of its derivative evalu:lted at equilibrium i&&clear distance [la]. The matrix element between higher vibrational levels was compu:ted using the relatibn .. IGz&-!)/~= rzlUio7]O)12 = II X O.SS X 10m2 D2 A4 .: kuxlational
. The values ofP, at 300.K’for dipole-octupole coupling are 1.1 X 10-p, 9.7 X lo-!, 7:s X 10e5, for n = 7, 8 and ‘9, r&pe_&ively. Since the quadrupolkquadrupole coupling appears tb be of the same strength as @pole-octupole coupling,‘.it appears that oniy dipole-dipole coupling plays an imporkuQ role in,thr,.energyltransfer processes .
y.-,.,
264’ .-. $ :
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Volumej0, numb& 2
CHEMICAL PHYSICS LETTERS
IS Jznua~
1975
.. Table 3
co,~pafison of experimental
and calculated valdes of probabili ties p&r co&ion, P, for the energf llksfer - 0.090z2-111 cm-’
proCX%S’)
cocv =c-1) +co(~ =1)+ co(~ =0) +co@ =0) +[26.901-1) T= 100K
T=250K
.‘.
T=300K
,!
calculated
experimentaib)
calculated
experimentill
2’
3.6 x 10-Z
3.9 x 10-2 [4]
1.6 x 1O-2
1.7 x 10-2 [4]
cl
3.4 X 1W2 1231 3
3.5 x 10-z
4.1 x 10-2 -
2.0 x 10-2
4
;.ox
1o-2
1.7 x 10-z 1.5 x 10-Z
2.0 x 10-2
I.?? 10-z
x 1o-2
x 1G-2
calculsted
expkimental
1.4 x 10-Z ).
1.8x lo-“[3]’ 1.2 x 10-Z 151 8.4 x 10-3[6].
1.6 x 10-2
1.2 x k-2
1.8 x 10-z
1.2
x 10-z
i.lX
iv2
1.0x 8.2X
m-2 10-3
5
1.6 x 10-3
7.1 x IO-3 9.5 x 10-3
1.0
6
3.2 x 104
2.8X LO-3 5.7x IO-3
3.8X 1o-3
6.6 x 1O-3
4.8 x 10-3,
6.0 x 10-3 4.3 X 10-3
7
5.6 x 10-S
9.0 x LO-4 3.1 x IO-3
6.8 x lo4
3.2 x 10”
2.G x 1(Y3
3.1 x IO” 2.5 x 10-S
8
8.0 x lo*
1.7 x 10-X
3.9 X IO4
1.6 x 1O-3
6.9 x IO+
1.7 x 10-S 1.4 x 10-z
8.2 x 10L4
1.3 x IO4
9.1
: 9
9.9
x 10-7
1.1
1.2x
x 10”’
2.5
lG-2
x 10-4
d,
9.5 x lo”?; 9.0 x 10-j
3) The cakulated values use s&S&t path approximation. b) For experiments at 100 K for tz = 3-9, the first entry is from Powell [23 J, while the second is from Wttig and Smith (21. c) Ex&aim&ts for n = 4-P were performed at 254 K by Wittig and Smith [2].. d) For expetiments at 300 K tk n = 3-9, the fist entry is frbm POWII [6], whi~e’thc second is from Hancock and Smith [I].
considered here. The failure of the rnu~t~po~arnear-resonant calculation at large energy mismatches is entirely expected. It is likely that for large values of II the cross section, aS pointed out by Jeffers and Ke!ley [7], is determined by short range forces. These authors obtain good agreement with the results of Hancock and Smith [l] at 300 I(. Their results are, however, about an order of magnilude smaller than those’of the Witfig-Srntith [2J experiments at 100 K. It is also possible that cross sections for large values of n are given by second order processes 121,221. A large amount of experimental work at different temperatures is becoming available to put these and other theories
[15] to the test.
Acknowledgement
.:
.:
The author is thankful to Dr. Hart for help with the computations and to Drs.Powell and Stephenson for Ietting me see their work before publication. It is a pleasure to acknowledge the hospitality and kindness of Professor A:‘Javan and other members of the Infrared and Optical Lasers Laboratory at M.I.lT_ ..
..
.’
Note added in proof
-:,
1’ Since the manuscript was wiitteri, Hancock-et ai:[24] _’ .,. ‘, .: ._ .: ,.,
have measuredP2 = (I .52 ? 0.04) X 10q2 at room’teni-
‘265 ..
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30; number.2
‘,
CHEiIIcAL
,;
_: ..
PHYSICj I&TE$S,
‘...
,l5 January 1975
:
“:pe&e m a&emint tiith Stephenson’s’reflllt.I3]: Fti+ki s;d T&z~$[25] have determined P2Lpg, at ‘room ‘, temperature,.,to be 8.7(-3J 1.4(-P), 1.3(-Z); 1.3(-2j, 7.5(-3); 4.0(-3), 2.2(-3), 1.8(-3j, respectively; the
‘1Gm3er iqthe brackets denotes powers df 10. Fushiki‘and Ts~~&iya’sP2 agrees ,with. that of Powell [6] and is almost i factor df2’miaUer than that of Stepherisoh and that of fiancbck et al. P4-Pg Clues bf !?T tie consistent‘: ,. l$hi&er~hm thoie of Hancock id Smith [l].., ‘. .-
:
,,:.
‘. -,
.
,:
:-
CRefq&&s’ .’
-:
”
,:
:.
:
:il] i;. Hancockand LW.hl._Smith, Cl&n. khys. Letters 8 (1971) 41.,. ,,-. 1. [2] C. Wittig +nd’I.W.M. Smith, Chem. Phys. Letters 16 (1972) 212. ‘. .[3] J.C. Stephenson, AppL Phys. Letters 22 (1973).576. .[4] Y.C. Stephenson and ELR Mosburg Jr., J. C&em. Phys. 60 (1974) 3562. ’ [SJ P.R. Sackett, A. Hordvik and H. Schlossbcrg, Appl. Phys Letters 22 (1973) 367. [61.-F&T. PowelI, J. C&m. Phys. 59 (1973) 4937.. [71 W.Q. Jeffers and I.D. Kelley,‘J. Chcm. Phys. 55 (1971) 4433. [8] W.G. Tani, Can J:Phys 50 (1972) 2691. 19.1 hi. Lev-On, W.E. Palke and RC. hlillikan, Chcm. Phys: Leiters 24 (1974) 59. [IO] RD., Sharma and CA. Btiu, J. Chem. Phys 50 (19691924. [ 111 R.D. Sharma, Phys. Rev. 177’(1969) 102. [12] N.F. Mott a@d H.SW @assey, The theory of atomic’collisions (Chrendon Press, London, 1965)~~. 806-808. [l?] R.1). Sharma and ,e. Schlcssberg, @em. Phys Letters 20 (1973) 5. [14] K. Gottfried, Quaritum mechanics; Vol. 1, Fundamentals iBcnjami7, New, York, 1956) pp. 341-343. [15] T.A. Dillon. and J.C. Stephenson, Phys. Rev. .4 6 (19?2) 1460. [ 161 L.k Young and W.J.. Each&‘!. Chem. phys. 44 (1966) 4195. : 1171 RX. Nesbet, I. Chem. Phys 40 (1964) 3619. [IS] D.B. Newmann and 2. Wassermpn, private communication. 1191 .G. Heizberg, Spectia of diatbmic molecules (Van Nostrand, Prince!on, 1950). [ 201 J-0. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular theory $g.~ses ar?d liquids (Wiley, NewYork, 1959) pp. 11 IO- 1111. .[21].RD. Sharma, Phys. Rev. A 2-(1970) i73. [22] R.D. Sha-a and C.W. Kern, J. Chcm. Phys. 55 (1971) 117i. [23] KT. Powell. unptiblished results. [24] G. Hancock, Star tid S. Green, .I. them. Phys., to be published. [25] Fushiki and S. Tsuchiya, to be published.
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