Collision-induced molecular excitation: Comparison of the collinear model with three-dimensional calculations

CHEhlICAL

PHYSICS

1 hlarcll 1975

LETTIiRS

6QLLISIO~-LVDUtiED MOL&ULAR EXCITATION: COMPARISON OF THE COLLINEAR MODEL WITH’TH&-DIMENSIONAL

CALCULATIONS

H.J. KGRSCH and ti. PHILLPP Inrtitut

fir

i7eorerische

Physik 1, Universitiir Mhster.

Aliinrter,

Ge~&ony

Received5 December 19%

Energy loss spectn linear

YZbrational

for threedimensional

extitation

probabilities

backward scattering in.a hard-sphere The

collision

+&km

is treated

h-, the

collidon model are compared qu-hm

mechulicd

hp~ls

with cob nppro~~a_

tie= The COG~dl&n model turned out to be u~3ble to describe the threedimensional bachlard scattering, Disagreement was found in the structure of the energy loss spectra and even in the averageenern transfer.

Cqllinear models have been widely used in study $8 inelastic atom-diatom collisions [ l]. On one hand they serve as test models for the development of numerical methods’and approximations, and or? the other tie restiIts of one-dimensional calculations 2re used in discussions of the mechanism of energy transfer or to interprete experimental results, which are - of course - three-dimensional [I]. The collinear model is ma@ly used to explain three-dimensional backward,scattering [2-71, mostly to get information about the vibrational excitation of the tiolecule. Because. of the wide spreading of coIlinear models it is of great interest to compare onedimensional descriptions of inelastic scattering proCeLseS with

three-dimensional

ones.

The’foUawing arguments ape often used to justify the collinear model: the collinear collision configuration is the most effective one for vibrational excita.tion; backward scattering in the three-dimensional c2se is mainly due .to the collinear configuration; rota: tional excitation is uqprobable in the case of backward scatfering. Theoretical i&estigations on the validity of the collinear-model are rare. Recently classical traject.ory cal-cu!ations have been done for the two- or three-dimensional c2se [g71CJ]. Near!y all of these calculations are ‘.

‘.

-.

restricted to central collisions of the‘atom and the molecule [8, lo], or to two-dimensional collisions, in which the atom collides centrally with one of the molecular atoms [9]. The extremely non-collinear collision configurations are eliminated by these restrictions. Nevertheless it turned out, that some of the assumptions, on which the collinear model is based, are incorrect: for example a C, con6guration of the atoms proved to be more effective for the energy transfer than the collinear one [lo]. Collisions which are not central have been rarely investigated, but they are of great‘importance for coupled vibrational and rotational excitation. T?II: exact quantum mechanical solutions of the one-dimensional [ 1 l] 2nd the three-dimensional [ 121 scattetig problem which are available do not yet allow a computin. This is due to tie fact that the

tiee4mensional calculationsare restrictedto very small-:olIision energies (at most the first vibrationaLly excited state can be included). For this reason it is of interest to compare the onedirnebsional model with the.three-dimensional backward :;cattering in a model, which CZIR be solved quantum nlechanicdy (ai Ieast in a good approximation). The question of interest is the following: Is it possible to get any information about the three-dimensional case f&m the c&near. model?

Volume ,31, hmber

L bIJXl1 1975

CHEMICAL PHYSICS LETTERS

2

2. Theory

ground state (i = 0) initially factor F;fi-!,,j is given by:

We describe the atom-diatom collision in a very simplified way as a collision between three hard billard-balls, two of which (the “niolecule”) are bound by an interaction potential (harmonic oscillator). Although not realistic - at least at small collision energies - this model is otten used because of its simplicity (it should be stressed that an exact quantummechanical solution.of this three-body problem is by’ no means trivial). For this hard-sphere excitation the quantum mechanical impulse approximation (IA) is a good approach both one-dimensionally [ 13- 151 and three-dimensionally [ 161. In the IA one assumes the momentum exchange between the colliding atom and

the moleculeto occur momentarilywith only one of the molecularatoms, the other one behaves as a spectator during this collision. The application of the IA to moIecuIar scattering processes has been recently described in detail [16]. In this paper we only refer to the most important results. All formulas and results are given for the center-of-mass system and for simplicity we assume the case of a homonuclear molecule.

F,?i,Jap)

the sD
= (2j’ + 1)

+Q) denotes the radial wavefunction of the molecule specified by the quantum numbers K andj (approximated.by a hzmopic oscillator wavefunction). ijt is the spherical Bessel function of the fist kind.

The target form factor depends only on f-he magnitude of the momentum transfer. In the collinear case the problem can be solved exactly [I 1,171, but for a hard-sptiereintemction the IA has proved

to be an excellent

approximation

COIII-

pared with the expensive exact calculations [13]. Therefore we use the results of the IA for the collinear transition probabilities from vibrational state rz to state n’ [13]f?: P n’n = (1 +m)2(lp’-p12/4p’p) +m

The differential cross section for a transition from one molecular state (marked by the vibrational quantum number n and the rotaticmal quantum numberj) &;,b) denotes the wavefunction of +&ernolecuIar state (harmonic oscillator). The integral in eq. (3) can be solved analytically [!3]. In the IA multiple sczttering is neglected. It is supposed, that the IA is a good

to a final state (n’,i’) is -intheIA-givenby[16]: do(ny’t-nj$)ldiZ

= 4~~(2nfh)~@‘/p)

X lPl($p’ - mp/2( 1 +m);$ X F;71nj(p’ -p)

- mp’/2( 1 +m)) I2

.

(1)

B is the scattering angle, p the initial and p’ the foal momentum of the colliding atom. The well-known-

mass-parameter m

approach for high collision energies 2nd smzll mass ratio m (for details see refs. [ 13,161).

3. Results

is the ratio betweenthe massof

this atom

and the total mass of the system (in tie molecule), p is the reduced mass of the scattering system. The half-on-shell matrix elements? of the two-body T-matrix can be evaluated case of a homonuclear

easily for the hard-sphere interaction potential (for details see ref. [16]). If the molecule is in the rotational ? It fiould be mentioned

that there are two ver$ons of the IA: The post- and priopfon, Here we folloy the propoti of ref. [ 161and use the prior-form in the case of excitation and the post-foml in the ccse of de-excitation.

It is not the aim of this paper to andyse the au. ence of specific parameters of the cakion system. For this reason all following caicuhtions refer to the system Lie-N,. From test calculations for other systems it turned out that the information we got from

Li*-N2 can be carried over to other systems (one has to t+e care that the conditions of validity of the 1A are fulfilled, i.e., especially the sma.Kmass ratio m). .‘i’he Ii’-EJ, scattering potential has been calcut? The factor 4 is misskg in ref. [131.

297

Volye

&-l,“k

CHEMICAL

2

;

l&ed iti an SCF appro&atiqn [18]; in dur model ‘. @&l$ibns tkk hard-sphere radius is fitted to this potefitial (foi details see ref$6]). The reIatWe cross .. :

.

turned otit to be ‘not very sensitive to a varia-.

se&ions tion

1 .hlarch 1975

PHYSICS LETTERS

of the hard-sphere

radius. In all calculations

we

.supposed the molecule td be initially in the rotational

ground state.

‘:

loss spectra

3-I. G0fl.

The distribution of the energy transferred in the collision to the possible vibrational and rotational. states of the mqlecule at fsed scattering’angle and fmkd co.llision energy, i.e., the differential cross section ai a function of the. energy transfer is called an ., ener,gy Ioss spectrum. Such energy Ioss spectra are of : interest because they can b’emeasured directly 6 a be& sdatte& experiment [4,‘18,19]. Detailed ,theo- r&icalJy calculated energy loss spectra in which all rotitional and vibrational foal states are taken into account, are given in ref. [ 161. Only the vibrational states of th& molecule are accessible in onedimensional cakulations. Fig. 1 shotis the collinear transition probabibties

-2

energy

do@‘?+

(rii”)

< (n’ +$)

.

increase with the initial quantum

(5)

*w-interval centeied at AE = (H’- n)tiw. Fig. 1 shows of the energy transfer in the case of backward scattertig (0 =. 180”). A comparison with ‘the col.Lnear.transition probabiJities Pn;l leads to the

From these results it is clear that detailed

the ene+gy scale, the broadening of the sph-um &,cj .&thedecrease of the mean energy.transfer with

_’ ‘:--. . .

.. :.

.

;

(for de-

struc-

&lothk comparisbtibetweenthe one-dimegsiorql

increasing ini+tl -vibrational quantum number )I. ‘. :‘I.:

number

t&es of the collinear P,vn cannot be carried over to the three-dimensional case; especially the information i:bout the influence of initial molecular excita‘tion on the &ekSy.t&nsfe,r requires a full threedimellsibnal treatment:

‘.

results:

.(a) Common features of the backward scattering and .: the .doUinear case are the approximate position on

2%

6

tails see the discussion in ref. [13]). These oscillations do not appear in the case of three-d;Jnen.sional backward scattering. (c) In the case of collinear scattering the mean energy transfer decreases more rapidly than the threedimensiopal one with increasing initial excitation.

Pi,: as a function

‘.:

5

atsly different, particultirly ‘in the case of higher initial vibrational excitation. The collinear energy loss spectra show characteristic oscillattins, which

(4)

’

Pi,: describes the probability of a transition intp an

following

4

(b) The shape of the energy loss spectra is consider-

W/da

WdW,,t

< A$ii~

3

transfer AC for different initial vibrational quantum numbers n = 0, 1. 2. The dots at integer values of AC denote the colLinear xansition probabilities. The results are given for the system,LiC-NT at collision energy Ek = 2 eV.

In this formula (07’) stands for a sum over all states ‘. which belong to energy transfers (II’-+)

2

Fis 1. Energy loss spectra for backward scattering (averaged OVEI emr,oy transfer intervalsfiw) as a function of the energ

of Ek = 2 eV as a functi&

of the energy transfer AE. In addition the threedimensional cross sections show the rotational states of the molecule. To cotipare them with the onediniensional results the three-dimensional cross settion is averaged over eliergy transfer intervals fiw: n’n

1

AC/fiW

[eq. (3)] for various vibrational

sia&s and a Enetic

p3JJ = C

-1.0

:

‘,

,,

,’

j,:,

,.I

.:,

... . I_:

; ‘_,,

:

‘..

_;

Volume 3 ;. numScr 2

CHEMICAL PHYSICS LETTERS Table 1 .. Quantum mechanigl

and the three-dimensional results is Tossible, if the backward-scattering cross sections are summed over

all final rotational states wilh iied quantum

in -

final vibrational

sions

nO)/d!i-l

(ddW,,,

1

=I,

(6)

u

Fig. 2 shbws these probabilities for vibrational excitation in the case of backward scattering as a function of the final vibrational

quantum

number

most of the transferred

energy is rotational

zz

and G3

tionsdefined

for three-

forcollincarcnl&

quvltum

numbei

n = 0, I,2

areclsssic;rl OF semiclsssicalapprotis-

in eqs

I1

Ek

!7)-(9)

SD

SD

x&

icz

Es

1eV

0 1 .2

0.513 0.433 0.329

0.485 0.339 0.178

0.555 0.555 0.555

0.515 0.434 0.352

0.482 0.336 0.190

2eV

0 1 2

1.069 0.985 0.916

1.036 0.878 0.697

1.111 X111 1.111

1.070 0.989 0.905

1.038 0.892 0.746

II’, compared

with the one-dimensional P,,;,. There is no similarity in the results. From fig. 2 one can see that the most probable transitions are vibrationally elastic, i.e.,

h;'D

for the syslemLI -Nz at collision energiesEk = I eV

and 2 eY, and initial vibrational

da(n’+

.I

average ener,oy transfer zsD

dimenrion~bbackward,+~tte~~d

number:

3;3;_ C

L Slarch 1975

energy.

3.2. Average erler& trarrsfer We have shown that the detailed structures of the collinear transition probabilities do not agree wiih the results of three-dimensional backward scattering. It is interesting whether they agree at least on an average. The mean energy transfers Z3D (in the case of threedimensional backward scattering) and zlD (in the collinear case) are shown in table 1 for two collision energieS Ek = 1 eV and 2 eV and for initial vibrational quantum

numbers

n = 0, 1,2. One can see quantita-

E,

= [4m/(l+

the initial account.

&?]

excitation In a classical

Ek

(7)

of the molecule calculation

is not taken

into

eq. (7) can be de-

rived easily both in the collinear case and in the case of three-dimensional backward scattering. In all OK calculations G1 proved to be an upper bound for the average energy transfer. The second formula

,

a5

tive verification of the impression of fig. I: The collinear mean values decrease more rapidly with increasing quantum number IL The other coIumns of table 1 show theoretical v&es for the energy transfer which turn out from classical or semiclassical considerations. In the well-known formula:

0.4

z,

0.3

is derived from.a classical consideration in the coIlinear case [15] and the final formula:

0.2

G3 = [4m/(l +m)2] [Ek-fiw(n+;)((I--fm)l

= [4m/( 1+ m)2] [Ek - fiw(n ++)]

(8)

(9)

was given recently in a paper on the semicIassi& 0.1

0

0

1

2-3

4

5

6

7

n’ Fig. 2 Comparison of the collinear tramition probabilities I’,.,#,, for vibrational excitation and the excitation probabilities P$, [defined in q. (6)] for three-dimensional backward scat~tering 25a function of the fmaltiibrational quantum num-

ber II’. The Llitial vibrational qutitum number is n= 0. The reSU~CSY~@VYM forrhesystemLiC-N2ntcDUiZienergyEk= : 2 eV.

treatment of the collinear hard-sphere collision [20]. A comparison shows that the semiciassical E3 is in very good ggreement &fi the collinertr quantum-’ mechanical IX results,‘even in the case of higher initial excitation, in which the second term of eq. (8) is of great infIu&e. Furthermore table I shows a remarkably good agreement bepveen Zz” and Z, (this has dso been found for dther systems and other energies). The quantum-mechanical werage energy transfer in the case of backward simple

scattering

seems

to follow the

equation (8). Th& simplicity asks for an ex-.

.. :

;.

295

._ Vo!&e

31,number

2 :

CHEMIC.iL

PHYSICS LETTERS

plannation irk a classical or. semiclassical model but we. .did tiot *succeed in doi& this. so we propose eq. (8) as an empirica! formula for the average enerpj;.tians,fey in the case of threedimensional b&&d scatter-’ ing;: .’ :.’ : : :

.,. .:

.

4. Conolusions

I.

‘From our calculations

we conclude,

that the co]-

linear model is unable to describe the three-dimen-. sional backward scatteritig. We found serious differ-

en+

in the structu:e of the energy loss spectra and considerable disagreement even in the average energy tia.n’sfer.‘ne results given in this letter are of course all derived for the special model of hard-sphere colbsions, but nevertheless we suppose.that our conclusions for this mode1 remain valid also in the case of more realistic interactionpotentials.

‘.,.

..

1 March 1975

[4] M.H. Cheng, M.H: Chiang, E.A. Gislason, B.H. hlehan, C.W. Tsao and A.S. Werner, J. Chem.‘Phys. 52 (1970)

6’1.50. ’[S] P.C. Cosby- andT.F. Moiarj J. C’hem. Phys. 52 (1970) 6:!57. [6] H. van Dop, AJ.H. Boerboom’and J. Los, Physica 54 (1971) 223. [ 71 P.F. Dittner and S. Data, J. Chem. Phys’ 54 (1971) 4228. [8] J.D. KelJey and M. Wolfsberg; J. Chem: Phys 53 (1970) 296 7. [9] H. van:Dop, kJ.H. Boerboom and J. Los, Physica 61 (1972) 616. [lo] hj: Faubel and J.P. Toennies, Chem. Phys. 4 (1974) 36. [ l,l] D. &crest md B.R Johnson, J. Chem. Phys 45 (1966) 4!356. [12] W.A. Lester Jr. and J. Schaefer, J. Chem. Phys. 59 (1973) 3676. (131 P. Eckelt and FLJ. Korsch, Chem. Phys. Letters 11

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[ 161 P. Eckelt, H.J. Korsch and V. Philipp, J. Phys. B7 (1974) 1649.

RefererA ,. [I] D. Rapp r&T. K&al, Chem. Rev. 69 (1969) 61. [2] J. Schlittler and J.P. Toermies, 2. Physik 214 (1968) 472. [?I W.D. Held, J. SchGttler and J.P. Tpcmr.ies, Chem. Phys. Letters 6 (1970) 304.

[ 171 K.E. Shuler and R. Zwanzig, J. Chem. Pbys 33 (1960) 1778.

[IS] J. SchattJer and.J.P. Toeties,

Chem. Phys 2. (1973) 137;4 (1974) 24. [19] H. Udseth, CF. Gieseand W.R. Gentry, J. Chem. Phys 511(1971) 3642. [20] W. Eastes and J.D. DoU, J.