BRANDI, HS

KOILLER, B

LINSDEBARROS, HGP

MIRANDA, LCM

The Journal of Chemical Physics

Repositorio da Producao Cientifica e Intelectual da Unicamp

Inert gas atom-molecule collisions in a laser field: 
Vibrational excitations in homopolar molecular ions 
H. S. Brandi and Belita Koiller 
Departamento de Fisica, Ponti/tcia Universidade Cat6lica, CX.P. 38071, Rio de Janeiro, RJ, Brasil 
H. G. P. Lins de Barros 
Centro Brasileiro de Pesquisas Fisicas 20.000, Rio de Janeiro, RJ, Brasil 
L. C. M. Miranda 
Instituto de Fisica, Universidade Estadual de Campinas, 13.000, Campinas, SP, Brasil 
(Received 5 July 1977) 
We investigate the effect of an electromagnetic field in the vibrational excitations of diatomic molecules, 
considering its collisions with noble gas atoms. Considering colinear collisions we obtain an expression for 
the vibrational transition amplitUdes, and we apply the formalism to the specific collision He + Hi in CO2 
laser field, in the high energy approximation. The general behavior of the multiphoton transition 
amplitude as a function of the relevant parameters of the system is discussed for the 0---> I vibrational 
excitation. 
I. INTRODUCTION 
The interaction of matter with electromagnetic fields 
(EMF) has been the subject of great interest in the last 
few years. Comprehension of processes involving par-
ticle scattering by atoms and atomic transitions in the 
presence of laser fields are very important in the study 
plasma heating by electromagnetic waves, resonantly 
excited atoms, molecular dissociation (gas breakdown), 
etc. 1- 6 Important applications are related with the use 
of intense laser fields to study selective excitation of 
molecules and molecular dissociation by many photon 
processes. 7,8 
In the present work we examine the mechanism for 
the vibrational excitation of diatomic molecules in the 
presence of a strong EMF taking into account the pos-
sibility of collisions with inert-gas atoms. The gas 
concentration is such that the time r between collisions 
is shorter than the laser pulse, i. e., r < r pu1.... The 
EMF can be strong enough so that it can not be treated 
in a simple manner by perturbation theory. 9,10 It is 
well known that the theoretical treatment of collisions 
involving atoms and molecules requires several approx-
imations and usually simple models are assumed to 
represent the effect of the interactions. 11-13 
The theory for the colinear collisions of an inert gas 
atom with a diatomic molecule is presented in Sec. II, 
and we apply it to homopolar molecules in Sec. III. In 
Sec. IV we present the numerical calculations and dis-
cussions for the specific collision process He + Hi in the 
presence of a CO2 laser field. 
II. FORMULATION 
For sake of simplicity, let us consider the one-di-
mensional collision of a diatomic molecule BC with an 
inert gas atom A: the colinear collision is the most 
favorable orientation for vibrational transition in this 
system. 11 The inert gas atom is treated as a structure-
less neutral particle, and its interaction with the mole-
cule is apprOXimated by an effective potential. The laser 
field is treated as a classical plane electromagnetic wave 
in the dipole approximation, i. e., one considers the 
wavelength of the field long as compared with the size 
of the bound system BC(>' »10-8 cm), so that the vector 
potential A (t) = Ao cosw t is spacially homogeneous. The 
polarization is linear and parallel to the collision direc-
tion. 
We call x the deviation of the B-C bond length from 
equilibrium (do is the equilibrium distance) and z the 
position of A with respect to the center of mass of the 
molecule BC. If rA , r B , and re are the positions of A, 
B, and C along the collision direction, and mA, ~, and 
me their masses, then 
(1) 
(2) 
The Hamiltonian for the problem in terms of these ca-
nonical variables is 
H=HM + V(x,z)+(1/2J,L)[P.-(q/c)A(t))2 , (3) 
where 
J,L = mA(mB + me) 
mA+mB+mC 
_ _ m--'lA,-,q~M!.-_ q= 
m A +mB +me 
and qM = qB + q e is the excess charge in the molecule; 
(4) 
(5) 
qB and qc are the net charges of atoms Band C, and of 
course qA = O. 
In (3), V(x, z) is the effective atom-molecule inter-
action potential, therefore lim, .,_ ~ V(x, z) = 0; The 
specific form of V(x,z) is discussed in Sec. III. The 
part of the Hamiltonian describing the molecular vibra-
tions in the relative position variable x is also approxi-
mated by an effective potential U(x), so that 
HM = (l/2m) [Px - (q' /c)A(t)]2 + U(x) 
with 
(6) 
1096 J. Chern. Phys. 69(3), 1 Aug. 1978 0021-9606/78/6903-1096'$01.00 © 1978 American Institute of Physics 
Brandi, Koiller, de Barros, and Miranda: Atom-molecule collisions in a laser field 1097 
and 
q' = m B qe - me qB 
mB+me 
(7) 
(8) 
The molecule vibrational potential U(x) is presented in 
Sec. III. 
We can immediately differenciate the homopolar and 
heteropolar molecule cases: for an homopolar mole-
cule) q' = 0 and the field can not directly induce vibronic 
transitions in the dipole approximation, as experimental-
ly observed, 14 while an heteropolar molecule behaves as 
a charged oscillator in the presence of the field, inter-
acting significantly with it. During the colliSion, there 
may exist some distortion of the molecular charge dis-
tribution causing a change in q'. This is a very small 
effect, and difficult to be included from first principles, 
therefore we neglect it here. 
A possible solution for the molecular problem in the 
presence of an EMF is obtained by introducing a space 
translation transformation6• 1S : 
(9) 
where 
Ii(t) = - L A(t')dt' = - ~sinwt= - 00sinwt I f 'A 
mc mwc 
and (10) 
Under this transformation the Schrodinger equation 
associated with Eq. (3) is 
{HM + (l/2J..1.)[P. - (q/ c) A (t)]2 + V [X+ o(t), z ]}w(x, z, t) 
= i1i[aw(x, z, t)/at] , 
where 
HMr/>(x, t) = {(P~/2m) + U[X+ o(t)] }r/>(x, t) 
= i1i[ar/>(x, t)/at] • 
(11) 
(12) 
No exact solution can be obtained for the states r/>(x, t); 
the so-called space translation approximation (STA) 
consists of neglecting completely o(t) in Eq. (12), and 
therefore the bound states r/>(x, t) are taken as the solu-
tion of the field free problem r/>n(x, t). This approxima-
tion is justifiable6 only when the characteristic range 
of the molecular interaction (a-1) is large as compared 
with 00' i. e. , aoo« 1. 
Once established the approximations for the problem, 
we obtain the scattering amplitude using the Green's 
function formalism. The approximate asymptotic solu-
tion of the Schrodinger equation 
- . B4>"" 
Ho4>",,= f1i -at 
obtained from (11) with I z I - 00 {V[x+ o(t), z] - o} is 
(13) 
4>",,(x,z,t)=4>II(x,tH,,(z,t) , (14) 
where 
~,,(z, t) = exp{ik[z - a(t) ]}exp( - i€"t./1i) , 
€" = (n2~ /2J..1.) 
(15) 
is the analytic asymptotic solution of the z dependent 
part of the Hamiltonian, 16 with 
a(t) = .!L jA(t') dt' = qAo sinwt 
J..I.c J..I.wc 
(16) 
and, in the STA, 
r/>n(x, t) = r/>n(x) exp [- i( €n/1f)t , (17) 
where r/>n is the eigenfunction and €n the corresponding 
eigenvalue of the field free molecular vibration Hamilto-
nian. 
Therefore the Green's function for this problem is 
G(x,z,t,x',z',t') 
= i.. L: f dk iP!,,,(x', z', t') iPn,,,(x, z, t) 8(t - t ' ), (18) 
21T n' 
where 8(t) = 1 for t> 0, and vanishes for t< 0, and the 
solution of Eq. (15) is 
lP"o.n(x,z,t) = r/>"o.n - f dx' dz' dt' G(x,z,t,x',z',t') 
x Vex' + o(t'), z,] l/J"o.n(x' ,z' ,t). (19) 
Following the procedure described in the Appendix A, 
it is possible to write Eq. (19) in the form 
where 
In Eq. (21), the symbol (II) means integration over 
coordinates z and x, and 
(22) 
The sign of k(lI) is chosen according to the asymptotic 
region of interest: k(lI) is positive for forward scatter-
ing (z - (0) and negative for the backward scattering 
case (z - _ (0). 
It is clear from Eqs. (19)-(22) that a~n' represents 
the probability amplitude for the vibrational transition 
from state n to state 11' due to the collision with simul-
taneous absorption (II >0) or emission (11< 0) of 1111 pho-
tons. 
Assuming high-energy collisions (Born approxima-
tion), we may replace lP"o,n by iP"o.n in Eq. (21); from 
Appendix B we have 
-iJ..l. - [ ] a~'n(Ao) = (21T)Z 1izk(lI) V2 K(II) 
x I~ dsFn'n(s) V1(s)J~[aoK(II>+00S] (23) 
In the application presented below, we restrict our-
J. Chern. Phys., Vol. 69, No.3, 1 August 1978 
1098 Brandi, Koiller, de Barros, and Miranda: Atom-molecule collisions in a laser field 
selves to excitations caused by the forward scattering 
of the atom [k(v) >0]. 
III. COLLISIONS INVOLVING HOMOPOLAR 
MOLECULAR ION 
In this section we apply the previously described for-
malism to study multiphoton vibrational excitations of 
homopolar molecules in the presence of an EMF. As 
previously discussed, for homopolar molecules, q' = 0, 
and since we are considering its collision with a charge-
less particle, the effect of the field appears only for 
homopolar molecular ions, i. e., for q"* 0, Of course 
we can not expect the effect of the EMF to be as impor-
tant as for polar molecules: here the action of the field 
arises only via the collision process, 
For q'=O(oo=O), Eq. (23) reduces to 
a~'n(Ao) = (21f)Z ~~k(V) V2 [K(v)] Jv [aof{(v)) 
x L~ ds Fn'n(s) V1(s) , (24) 
For homopolar molecules, the Morse potential17 is 
known to represent well the molecule vibrations. It is 
defined as: 
(25) 
where D is the molecule dissociation energy and a-I 
represents the range parameter. The eigenfunctions 
and eigenvalues for the Hamiltonian (6) with the 
Morse potential (25) and q' = 0 are given in Ref. 17. 
It is convenient to define the ratio of the transition 
probability amplitude with and without the radiation 
field 
(26) 
For the energy range we are considering (high ener-
gy) the dominant multiphoton processes are associated 
with a relatively small number of photons, and there-
fore k(v) ~ k(O) [see Eq. (22)]. From (26) it is clear 
that R~_ n' is not strongly dependent on the particular 
choice of the interaction potential. To perform calcu-
lations we chose a model potential of the form 11.12 
(27) 
so that 
(28) 
IV. RESULTS AND DISCUSSION 
In this section we present numerical results for 
RO- 1 considering the particular collision process be-
tween the He atom and the Hi molecule ion. For this 
molecule the Morse potential parameters IS are D= 2.79 
eV and a = 0,72 aii1 (ao = Bohr radius). These parame-
ters yield .6.E = El - Eo = O. 28 eV, For the EMF we chose 
initially typical values of a CO2 laser, that is, field 
strength 80 = 1. 4 x 107 V / cm and frequency W C02 = 103 
cm-1• 20 The Born approximation requires the collision 
energy Eo = 1i2 kV2/J. to be much larger than .6.E, i. e. , 
Eo» O. 28 eV, As we discussed previously the results 
are not expected to be strongly dependent on the value 
of the collision parameter {3 in Eq. (21); we assume 
{3 = lOS cm-1 which is a typical value for this parameter ,12 
To understand the relevance of multiphoton processes 
we investigate the dependence of RO_1 with the field 
strength and frequency, and with the relative collision 
energy. 
In Table I we present the values of RO- 1 for field 
strengths increasing from 8 0 to 1000 , The colliding 
energy is 3 eV and w = W C02 ' The increase in the field 
strength causes a significant increase in the probability 
amplitude for processes involving absorbed photons, and 
a slight decrease in the zero photon processes, The im-
portance of multiphoton processes is clear from Table 
II, where the EMF frequency is decreased from weo2 
to weo /10, keeping 0 0 = 1. 4x 107 V/cm and Eo= 3 eV: 
for low enough frequencies, one or more photon pro-
cesses become dominant over that involving no photons. 
In Table III the collision energy Eo is increased up to 
10 eV and the values of the field strength and frequency 
are those of a typical CO2 laser. As expected, the in-
crease in Eo causes the multiphoton transition ampli-
tudes to decrease. 
We have also tested the sensitivity of our results with 
the mass mA , which is the only parameter related to 
the inert gas involved in the collision. In Eq. (26), 
TABLE 1. Relative multiphoton transition amplitude probability RO->I for the He-H; collision 
with collision energy Eo =3 eV in the presence of a CO2 laser field (Weo2= 10
3 em-I) of variable 
strength (So=1.4 x 101 V/cm). 
~ 
S/So 
v 1 2 4 6 8 10 
0 1 1 0.99 0.99 0.98 0.96 
1 0.24x10-1 O. 48X 10-1 O. 97x 10-1 0.15 0.19 0.24 
2 O. 94X 10-5 O. 38X 10-4 0.15 x 10-3 0.34 X 10-3 0.60 X 10-3 O. 94x 10-3 
3 0.16x 10-6 0.13X 10-5 0.10X 10-4 O. 34x 10-4 0.80X10-4 0.16X 10-3 
J. Chern. Phys., Vol. 69, No.3, 1 August 1978 
Brandi, Koiller, de Barros, and Miranda: Atom-molecule collisions in a laser field 1099 
TABLE II. Relative multiphoton transition amplitude probability RO-t for the He-H; collision 
with collision energy Eo =3 eV, in the presence of a laser field of strength So =1. 4x 107 V Icm 
for different values of the frequency w (Wc02 = 103 cm-i ). 
~ 
Wco2/w 
1 2 4 6 
0 1 0.99 0.90 0.56 
1 O. 25X 10-t O. 89x 10-t 0.32 0.59 
2 O. 98± 10-5 O. 21x 10-2 0.42x 10-t 0.20 
3 a 0.94 x 10-5 O. 26X 10-2 O. 36x 10-t 
4 a a O. 80x 10-4 O. 39X 10-2 
5 a a O.l1x 10-5 0.25 X 10-3 
6 a a a O. 99x 10-5 
a R~_t < 10-6• 
since q/ Ii = qM/(mB + me), the only dependence in mA 
comes from the various wave vectors involved in the 
energy balance equation. For this reason, comparing 
our results for the standard values of the parameters 
(W eo2 , go and Eo = 3 eV) for m A = mHe and for m A - "", 
the values of Ro~ 1 for II up to 3 vary (increase) by less 
than a factor of 2. The general behavior of the proba-
bility amplitude for any inert gas atom is the same as 
described for He. 
As argued previously, our results should not be very 
sensitive to the collision potential parameter f3. For 
the same standard values of the parameters weo2 , go 
and Eo, a decrease in f3 by an order of magnitude 
causes a very slight increase in Ro~ l' except for II = 2, 
when it increases also by an order of magnitude. This 
is easy to understand, since the II = 2 process is the 
closest one to resonance, so that the transferred mo-
mentum K(II) is minimum, and from (28), the depen-
dence in f3 is enhanced. 
8 10 
0.49x 10-t 0.40 
0.58 0.45 x 10-t 
0.48 0.52 
0.19 0.46 
0.44x10-t 0.21 
0.68 X 10-2 0.65 X lO- t 
O. 71x 10-2 0.13 X 10-t 
The results presented in Tables I-III confirm the 
qualitative behavior one expects from simple arguments, 
and since the calculations are restricted to the Born 
apprOXimation, the main contribution to the excitation 
process is due to the energy exchange between the col-
liding particles; energy balance is always garanteed by 
the collision process. 
The role of the EMF is much more significative for 
colliSions involving polar molecules, since there exists, 
even in the dipole approximation, a direct contribution 
to the vibrational excitations from the field. Calcula-
tions for collisions of inert gases with heteropolar mole-
cules in the present formalism are computationally more 
involved, and are in progress. 
APPENDIX A. THE SCATTERING AMPLITUDE 
In this Appendix we obtain an expression for the tran-
sition amplitude. Using Eqs. (14)-(18) in (19), this yields 
to 
x exp[ikO!o (sinwf - sinwt)) exp[(i/n) (fi,. + fin' - fiko - fin)!'] exp[(i/n)(fi,.o Hn)!'] v[x' + oCr), z'] l/!ko,n(X' ,z' ,t'). 
Using the expansion of the periodiC function1,2 (AI) 
00 
exp(ikO!osinwt') + exp[(i/n) (fi,.o + fin)t'] 1P"!l'n(x' ,z' ,t') '" l/!k,ko,n(X' ,z', t') = L 1]Jtk
o
,n(X', z') exp(ijwt') 
f:-CIO 
(A2) 
and 
where p" is the momentum operator, the integration in t' 
yields 2i(Ii/n-2){1![k2 - k2(v)]}where v=j-landk(v)isde-
fined by Eq. (22). Integration in k is readily performed 
and after taking the asymptotic limit I z I - "" we have 
I 
TABLE III. Relative multiphoton transition amplitude proba-
bility RO_ t ' for the He-H; collision in the presence of a CO2 
laser field (wC02=10s em-t , So=1.4 x 107 V/cm) for different 
values of the collision energy Eo. 
S Eo(eV) 3 5 7 9 10 
0 
O. 24X 10-1 0.16X io-1 0.12X 10-1 O.lOX 10-1 0.94xlO-2 
2 O. 94x 10-5 O. 39X 10-5 O. 22X 10-5 0.15X 10-' 0.12x 10-' 
J. Chern. Phys., Vol. 69, No.3, 1 August 1978 
1100 Brandi, Koiller, de Barros, and Miranda: Atom-molecule collisions in a laser field 
X {J1 [(liol1I)Px] V(x, z)h~~~(.).,,(x, z) 
From (A2) we get 
(A4) 
(A5) 
Substituting (A5) in (A4) and using (A3) we get Eq. (21). 
APPENDIX B. THE BORN APPROXIMATION 
In Eq. (21) we replace IPkO'''(x,z) by ko'''(x,z) and get 
a~.,,(Ao) = - 21T!iz:(V) t too J/[aof{(v») 
x(".(x)exp[ -iK(v)z]{J!(lio/1I)Px] V(x,z)}1 ,,(x» 
r2r1w x L exp{i/1i[ E:k(.) + E:". - E:ko - E:" + (1 - s)1fw] t} . 
o 
After integration (B1) reduces to 
a~.,,(Ao)=- '/k'll( ) t IN[aoK(v)] 
v .9=_'" 
x (<t>".(x) exp[ - iK(v)z) Js [(li o/1I)P x) I V(x, z) I ,,(x) , 
where K(v)=k(v) -ko' 
Using Ref. 19, we get 
a~.,,(Ao) = - [ill/1f 2k(v») (<1>". (x) exp[ - iK(v)z] 
(B1) 
(B2) 
x{ J. [aoK(v) + (lio/1I)Px) V(x, z)} I ,,(x» (B3) 
For simplicity one usually takes V(x, z) as a separable 
potential V(x, z) = Vl(x) V2(Z)12 and define the Fourier 
transform of Vi and V2 by 
V1(Y)= {"YM)e'OYdq 
and the bound state form factor by 
F".,,(q) = f dxeIQX<t>:(x) ,,(x) • 
USing (B4) and (B5) in Eq. (B3) we get 
(B4) 
(B5) 
• ill - [ ] 
a".,,(Ao) = - (21T)z1fzk(v) V2 K( v) 
x iooOOdSF".,,(S)Yl(S)J.[aof{(v)+lioS] (B6) 
IN. M. Kroll and K. M. Watson, Phys. Rev. A 8, 804 (1973). 
2B. S. Bhakar and B. J. Choudhury, J. Phys. B 7, 1866 
(1974). 
3M. H. Mittleman, Phys. Rev. A 14, 586 (1976). 
4N. D. Bhaskar, B. Jaduszliwer, and B. Bederson, Phys. 
Rev. Lett. 38, 14 (1977). 
sJ. Macek and 1. V. Hertel, J. Phys. B 7, 2173 (1974). 
6F. H. M. Faisal, J. Phys. B 6, L 89, and L 312, (1973). 
7S. 1. Vetchinkin, V. L. Bakhrakh, and I. M. Umanskii, Opt. 
Spectrosc. 40, 28 (1976). 
8N. G. Basov, E. M. Belenov, E. P. Markin, A. N. Draevskii, 
and A. V. Pankratov, Sov. Phys. JETP 37, 247(1973); L. 
C. M. Miranda, Phys. Lett. A 56, 374 (1976); N. M. Kroll 
and K. M. Watson, Phys. Rev. A 13, 1018 (1976); A. M. F. 
Lau, Phys. Rev. A 13,139 (1976),A. M. F. Lau and C. K. 
Rhodes, Phys. Rev. A 15, 1570 (1977); J. M. Yuan, J. R. 
Laing, and T. F. George, J. Chern. Phys. 66, 1107 (1977); 
I. H. Zimmerman, J. M. Yuan, and T. F. George, J. 
Chern. Phys. 66, 2638 (1977); T. F. George, J. M. Yuan, 
1. H. Zimmerman, and J. R. Laing, Disc. Faraday Soc. 62, 
246 (1977). 
9H. Reiss, Phys. Rev. A 1, 803 (1970). 
tbH• Reiss, Phys. Rev. D 4, 3533 (1971). 
I1H. K. Shin, J. Chern. Phys. 49, 3964 (1968). 
t2H. K. Shin, J. Phys. Chern. 77, 1394 (1973). 
t3L. Dube and A. Herzenberg, Phys. Rev. Lett. 38, 820 
(1977). 
14F. V. Bunkin, R. V. Karapetyan, and A. M. Prokhorov, 
Soviet Phys. JETP 20, 145 (1965). 
lSW. C. Henneberger, Phys. Rev. Lett. 21, 838 (1968). 
16J • J. Gersten and M. H. Mittleman, Phys. Rev. A 13, 123 
(1976). 
17L. D. Landau and E. M. Lifshitz, Mecanique Quantique, 
Editions Mir, Moscou (1966), p. 92. 
18F • V. Bunkin and 1. 1. Tugov, Phys. Rev. A 8, 601 (1973). 
191. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals. 
Series and Products, (Academic, New York, 1965), p. 979. 
20Notice that lio = O. 27 X 10-9 cm for these parameters. For the 
Morse potential u(x + liol = U(x) +L;;= 1 (lioa)"V,,(x), where V,,(x) 
are of the same order of U(x). Since lioa ~ 3 x10-2, the STA 
is completely satisfactory in the present example. 
J. Chern. Phys., Vol. 69, No.3, 1 August 1978

INERT-GAS ATOM-MOLECULE COLLISIONS IN A LASER FIELD - VIBRATIONAL EXCITATIONS IN HOMOPOLAR MOLECULAR-IONS

http://repositorio.unicamp.br/jspui/bitstream/REPOSIP/60124/1/WOSA1978FN93500021.pdf

INERT-GAS ATOM-MOLECULE COLLISIONS IN A LASER FIELD - VIBRATIONAL EXCITATIONS IN HOMOPOLAR MOLECULAR-IONS

Abstract

Similar works

Full text

Available Versions

Repositorio da Producao Cientifica e Intelectual da Unicamp