- Email: [email protected]
Mechanism of collision-induced intersystem crossing in CO
Volume 57, number I
CHEMICAL PHYSICS LJZKTERS
1 July 1978
MECHANISM OF COLLI!3ION-INDUCED INTERSYSTEM CROSSZNC CNCC D. GRIMBERT, M. LAVOLLEE, A. NITZAN * and A. TRAMER Lnimr~~oire de Photophysique Moitkuk~ire C_BXS. and LURE_ UnixrsitZ Paris-%$
91405-Qrsry. Frame
Received 6 February1978
Revisedmanuscriptreceived6 March 2978
We have studiedinert-gaspressureeffects on the fluorescencedecay in CO selectivelyexcited to the ti = 0 to 7 vx%ronic levelsof the A’n electronicstate. It is shown that the dependenceof the quenchingcross sectionok on the averagevahze of the S-T mixingcoefficient C$> has a qua&ogarithmic form. A simple two-levelmodel describingsemiquantitativeiy‘&is behavioris proposed.
1.Introduction The fluorescence quenching and phosphorescence induction by collisions with inert partners (e.g. rare gases) is a common feature of small (CO [ 1,2], glyoxal [3]) and intermediate {pyrazine [4], biacetyl [5], etc.) molecules, As shown first by Geibart and Freed [6,7], these processes may be described in terms of coiiision induced transitions between “mixed” molecular states:
where S and T are pure sp@ states. In the following we denote mixed states as S (for 02 > sz) anf T (for p2 > ixZ)_ In smaR moZecules, cross sections for the colhsioninduced intersystem crossing (Cf ISC), ois,., show a pronounced dependence on the properties of the collider [3,8] and may be correlated to its polarizability fi3] or to the energy-well depth of the collisional complex [9 1. This suggests the important role of the’at&active part of the intermotecular potential for CI ISC, a5 it is the case for rotational relaxation. * On leavefrom Tel-Aviv University.
A unified treatment of CI ISC and of the rotationa! relaxation was given by Freed [7]_ A principal conclusion of this work is a &rear dependence:
(where oR denotes an averaged rovibronic relaxation cross section involving a single electronic manifold). This relation deduced from the first-order perturbation treatment (i.e. strictly valid in the weak-coupling limit) has never been experimentally checked. In this work, we have studied the effect of collisions with rare gases (X = He, Ar, Kr) on the CO fluerescence decay under a selecitve excitation of single vibronic levels {u i 0 to 7) of the A 1i1 state, coupied to quasi-resonant levels of a’?P, e?B:- and d3A triplet states. Coupling constants being known from highresolution spectroscopic studies [lo], the mixing coefficients of individual rovibronic levels, cyn and &, have been evaluated and checked by a study of the collision-free fluorescence decay [l I] _ Under the present experimental conditions the S-T transition involves many initiai levels: if the rotational relaxation within a singlet vibronic state is fast as compared to Cf ISC, the initial state is a mixture of rotational Ievefs with Boh~unann populations: N(J) = (2J’ -t I) exp{-ncrZ’J*(.i’
+ l)/kZJ
.
C%
On the other hand if rotational relaxation is slow, the populations N(J) are determined by the initial excita45
tion process:
X ex$f-+tcB”J”(S’
+ l)/kTJ
(3b)
)
where SJJ’) is the rotational line strength [ 12]_ In either case the mixed character of a given vibronic level is roughly determined by the average:
The (p2) values obtained from eq. (4) using eqs. (32) or (3b) are listed in table 1. It is seen that both calcu-
lations yield about the same results. Complete results of this work will be published elsewhere, this paper contains only preliminary data concerning ukc for a few collision partners and a preliminary model treatment.
2. Experimental
1 Jnty 1978
CHE!+lJCAL PHYSICS L5lTERS
Volume 57. number 1
and results
The CO fluorescence was excited by synchrotron radiation of the Orsay Collision Ring (ACO) using a specially designed monochroma:or [ lS] with spectral slits of the order of 10-I 5 A ensuring a good separation of individual vibronic bands and a constant intensity distribution within a contour of a single band.
The CO pressure was maintained constant (S-20 mtorr) and that of added gases was varied between 0 and 200 torr. Correlated single-photon counting technique adapted to specific properties of the light source 1141 was used for recording the fluorescence decay curves. In collision-free
conditions, the CO fluorescence decays quasi-exponentially with lifetimes varying withuinthe9-12nsrange [ll].inthepresenceof a coliider the decay becomes strongly non-exponential and may be approximated by a b&exponential function (fig. 1). It has been shown [l&16] that the long component of the decay is due to the reversible character of Ci !SC and to the vibrational relaxation processes, while the short one practically unaffected by vibrational relaxation, characterizes the initially excited vibronic level and obeys the Stem-Volmer relation:
1/r@x)
= 1 /r(O) + kstpx ,
where kg = urscQisthe rate constant of the S-T CI KC. orsc, for different vibronic states and different collision partners, is given in table 1. It may be clearly seen that there is no regular variation of oirc with u’ but a net correlation between ursc and @2). We do not Observe, however, the linear correlation expected from the first-order perturbation treatment [cf. eq. (2)] _ If we suppose that no transitions between pure singlets and pure triplets can oc-
Table 1
Mixing coeffkients ($) and quenching cross section Oisc for vibronic levels of the A ‘IJ state. Relative error in a~
v
46
Perturbing Ieveis
QJ2)from
values is f 10%
eisc (A21
eq. (3b)
eq. (3a)
He,A =45cm”. I/a = 3.13 A
Ar. A = 150 em-‘. I/a = 3.56 A
Rr, A = 185 em”, l/a = 3.65 A
e3Z- (U = 1)
0.102
0.092
3.4
27
44
d3A(u=4) d’A (u = 5) a’3ZZ+ (G = 10)
9.062
0.080
2.7
25
43
e3Z- cu - 4)
0.0044
0.0032
0.9
d3A (v = 8)
0.003
0.003
0.45
a’3Z+ (u = 14) e3.z- (u = 8)
0.012 0.002
0.016 0.0019
1.35 0.55
10.1 9.1 17.5 12.4
32 29s 45.5 30.5
d'A(v= 12) a’3X+ (u = 17) e32-(u= 11)
0.102
0.084
3.6
23.2
44.5
0.001
0.0012
0.5
9.8
36
CHEMICAL PHYSICS LJZTTERS
Volume 57, number 1
1 July 1978
the centers of mass of the colliding particles. Following Freed [7] we &G assume that this interaction which induces the coupling between 13rJ>and I?‘.‘.‘) levels is proportional to the mixing coefficient @ [eq. (I)] and to the collisional coupling between roiationaI levels of the S (or T) state:
.
cc” tins) Fig 1. Experimentaldecay curves(correctedfor the blackcountingand stray-lightbackground)for u = 2 level of CO. PC-J= 12 mtorr_
3”
. .
.
cur i.e. that ai,
+ 0 for fS2+ 0 and if we neglect the possibility of heavy-atom effects, the variation of oisc withmay be described in the following way: (i) InWally, oisc increases rapidly and quasi-lineariy wiht
Note that this collisional coupling does nor conserve the molecular angular momentum so that levels characterized by different rotational quantum numbers J may be collisionally coupled_ With the objective of obtaining rough estimates we make some more drastic assumptions: (1) Level shifts are disregarded. (2) Two simple choices of interaction potentials are made: (a) A square perturbation, different from zero during 2 collision time rc rc = liav
(5)
and given during this time by V= a1/2fiAe-a2b2
,
(6)
where b is the impact parameter, a measures the range of the interaction potential and A its strength_ v is the relative velocity of the colliding particles. Eqs. (5) and (6) are obtained by fitting a gaussian interaction IJ--?(~) = @A,-a*.P*(r)
=fiAe-a*b*-a*v*t*
(7)
to 2 square potential taking 3. Model cakdation The experimental data demonstrate thar the linear dependence predicted by the fmt-order perturbation treatment breaks down for the observed range of coupling strengths. It is natural to associate this observation with the breakdown of the linear approximation. In order to test this viewpoint, we checked whether the observed phenomena may be fitted to calculations based on a nonperturbative solution of extremely simpiified two-level models with physically reasonable parameters_ The collision is treated semi-classically so that the coordinate of the perturbing atom is taken as a time-dependent classical function in the interaction hamiltonian. For simplicity the interaction is assumed to depend only on the separation between
7-c =
[s -0
dt VCR(t))] ‘I2 J -0II
dt t2 ‘/R(t))/
@)
and V= (l/rc)
J
V(R(r))dt
_
(3
--o
(b) A time-dependent V,(t)
perturbation
= C, sech(C$)
,
of the form (10)
where Cl and C-J were determined so that rc and V defmed by eqs. (8) and (9) are again given by eqs. (5) and (6). This leads to CI =ip1/2&4e-a2b2 c2=&W.
,
(11) (12) 47
CHEMICAL
VoIume 57, number 1
PHYSICS
LETl-ERS
1 J-dy 1978
The transition probability in a collision with a given impact parameter, b, is obtained by soking the time-dependent SchriZlinger equation. This gives in case (a): &@2e-2a2b2
4 =
f@,
2(1 +tiZ+32e--2a2b2)
1 _cos (1
x
+GTq3k”‘b2)‘12]
)
[
(132)
where x= 2A/6E and ir = 2iazv/i5Eand in case (b) (according to ref. [171_): fib, vj = sin2(~~/2~~e-a2b5fir) The
Sd'12(2/ij).
(I=)
total cross section for a given velocity v:
aa o*(v)
= 2a s
fib, vjbdb 0 will he, for caSe (a): o&y)
= (~.@u”){hr(l
- 2Ci[(l
-I-lrzf12)
!I0 + 2Ci(l/Vj
f ?tS&rWJ)
(14)
F&_ Z ok plotted versusIII@~>.Top: experimentalvalues. Sottom: cakuiatedvaluesfor case (a). dotted line, and case O), solid line, for merent collisionpartners.
and for case (b): u&v)
= (&!a2)
4. Comments
sech2(2/Sj
[y +ln(2n’Gi#irj
- Ci(2rrWG3,Gj]
,
(14b)
where y is tke Euler constant_ This resu!t may be numerically averaged for MaxwelL-Roltzrrann distributions corresponding to different colliding pairs and temperatures in order to evAate the effective cross section defied as:
These formulae are applied to the CO case vlitb _ the following choice of parameters: 6E = 10 cm-l (of thg order of the average gap between the closest lying S and T states for u’ = 0, 1 and 2 AlII CO ievek); bfie is taken from ref. [l g] while for other gases we suppose ACOSC/ACOJle = (e&~_#* where E is the krnard-Jones constant; I/a is equal to the averaged Lennard-Jones radius (ucO + u-J/2_ G and e pardmeters are taken from ref. 1191. Plots of ok versus hr@~) obtained in this way for cases (a) and (hj are compared with experimental data ir? fig. 2. 48
(I) It has been shown that with a reasonable choice of parameters we obtain a correct order of magnitude for ok and reproduce fundamental features of the observed behavior (saturation, dependence on the collision partner. (2) In the weak perturbation limit (A6 + 0), eqs.
(14a) and (14b) give a linear dependence of ok on @, deduced previously from the fust-order perturbation treatment [7] _ (3) The two model potend& [eqs. (5) and (6) for case (a), eq. (IO) for case (b)] were chosen because they lead to analyticahy sohrble problems. The close similarity between the results obtained in both cases suggests that at least in the range of parameters considered, the exact form of the perturbation is relatively unimportant. This is seen in the range i+ 1 where 0 = (@z2jP~2/4~~
f or case (a) and
o = (12/2a2jP~2/irz for case (b)_ For ? 5 1 the results of the two models disagree
Volume 57, number 1
CHEMICAL
with each other because the square potential cannot account correctly for the adiabatic behavior in this limit. (4) Our model neglecting the level shifts is complementary to that of Freed and Tric 1201 accounting for level shifts by the first-order perturbation treatment and for the level crossing in the spirit of the Landau-Zener approach.
Acknowhedgement The authors are highly indebted to the team of the LURE and of the Linear Accelerator Laboratory in &say whose work made possible our experiments. Thanks are also due to Drs. G. Bergeron, K.F. Freed and C_ Tric for discussions and remarks. One of us (AN) thanks the Commission for Basic Research of the Israel Academy of Science for supporting this work and the France-israel Scientists Exchange Program for a travel grant.
References 1l] T-G. Skmger and G. Black, J_ Chem. Phys. 58 (1973) 194,3121_ [ 21 E-H_ Fii and FJ. Comes, Chem Phys Letters 14 (1972) 433.
PHYSICS
LETTERS
1 July 1978
[3] R-k Beyer and WC
Lineberger, J_ Chem. Phys 62 (1975) 4029_ [4j A. Frad, F. Lahmar& A. Tramer and C. Tric. J. Chem. Phys. 60 (1974) 4419. [S] R. van der Werff and J. Kommandeur, Chem_ Phys. i6 (1976) 125. [6] W-M_ Gelbart and ICF. Freed, Chem. Phys. Letters 18 (1973) 470. [7] K-F. Treed, J. Chem. Phys. 64 (1976) 1604. [8] C-A_ Jhayer and J-T. Yard!ay, J. Chem_ Phys. 61(1974) 2487. [9] C-S. parmenter, Discussion at the Meeting on Radiationless Processes, Breukelen (i977). [lo] R.W. Field, B.G. Wick, J.D. Simmons and S.G. TiXord, L MoL Spectry. 44 (1972) 383. [ il] R.W. Field, M. Lavoll&, R. Lopez-DeIgado and A_ Tramer, to be published. j 12 j G. Henberg, hi 3lecular spectra and mo!ecuLu structure, Vat 1 (Van Nostrand, Princeton, 1962). 1131 hf. Lavollie and R. Lopea-Delgado, Rev. Sci Instr. 48 (1977) 816. [ 141 R Lopez-Delgado. A_ Tramer azrd J. Munro. Chem_ Phys. 5 (1974) 72. [IS] M. LavoBt?e and A. Tramer, Chem. Phys. Letters 47 (1977) 523. [ 161 M. Lavollde and A. Tramer, to be published. [ 171 N. Rosen and C. Zener, Phys. Rev_ 40 (1932) 502. [ 181 S. Green and P. Thaddeus, Astrophys I_ 205 (1976) 766. [ 191 J.O. Hirschfelder, CF. Curtiss and RB. Bird, MolecuIar theory of gases and liquids (WiIey, New York, 1954) (201 K.F. Freed and C. Tric, Chem Phys., to be published.
49
CHEMICAL PHYSICS LJZKTERS
1 July 1978
MECHANISM OF COLLI!3ION-INDUCED INTERSYSTEM CROSSZNC CNCC D. GRIMBERT, M. LAVOLLEE, A. NITZAN * and A. TRAMER Lnimr~~oire de Photophysique Moitkuk~ire C_BXS. and LURE_ UnixrsitZ Paris-%$
91405-Qrsry. Frame
Received 6 February1978
Revisedmanuscriptreceived6 March 2978
We have studiedinert-gaspressureeffects on the fluorescencedecay in CO selectivelyexcited to the ti = 0 to 7 vx%ronic levelsof the A’n electronicstate. It is shown that the dependenceof the quenchingcross sectionok on the averagevahze of the S-T mixingcoefficient C$> has a qua&ogarithmic form. A simple two-levelmodel describingsemiquantitativeiy‘&is behavioris proposed.
1.Introduction The fluorescence quenching and phosphorescence induction by collisions with inert partners (e.g. rare gases) is a common feature of small (CO [ 1,2], glyoxal [3]) and intermediate {pyrazine [4], biacetyl [5], etc.) molecules, As shown first by Geibart and Freed [6,7], these processes may be described in terms of coiiision induced transitions between “mixed” molecular states:
where S and T are pure sp@ states. In the following we denote mixed states as S (for 02 > sz) anf T (for p2 > ixZ)_ In smaR moZecules, cross sections for the colhsioninduced intersystem crossing (Cf ISC), ois,., show a pronounced dependence on the properties of the collider [3,8] and may be correlated to its polarizability fi3] or to the energy-well depth of the collisional complex [9 1. This suggests the important role of the’at&active part of the intermotecular potential for CI ISC, a5 it is the case for rotational relaxation. * On leavefrom Tel-Aviv University.
A unified treatment of CI ISC and of the rotationa! relaxation was given by Freed [7]_ A principal conclusion of this work is a &rear dependence:
(where oR denotes an averaged rovibronic relaxation cross section involving a single electronic manifold). This relation deduced from the first-order perturbation treatment (i.e. strictly valid in the weak-coupling limit) has never been experimentally checked. In this work, we have studied the effect of collisions with rare gases (X = He, Ar, Kr) on the CO fluerescence decay under a selecitve excitation of single vibronic levels {u i 0 to 7) of the A 1i1 state, coupied to quasi-resonant levels of a’?P, e?B:- and d3A triplet states. Coupling constants being known from highresolution spectroscopic studies [lo], the mixing coefficients of individual rovibronic levels, cyn and &, have been evaluated and checked by a study of the collision-free fluorescence decay [l I] _ Under the present experimental conditions the S-T transition involves many initiai levels: if the rotational relaxation within a singlet vibronic state is fast as compared to Cf ISC, the initial state is a mixture of rotational Ievefs with Boh~unann populations: N(J) = (2J’ -t I) exp{-ncrZ’J*(.i’
+ l)/kZJ
.
C%
On the other hand if rotational relaxation is slow, the populations N(J) are determined by the initial excita45
tion process:
X ex$f-+tcB”J”(S’
+ l)/kTJ
(3b)
)
where SJJ’) is the rotational line strength [ 12]_ In either case the mixed character of a given vibronic level is roughly determined by the average:
The (p2) values obtained from eq. (4) using eqs. (32) or (3b) are listed in table 1. It is seen that both calcu-
lations yield about the same results. Complete results of this work will be published elsewhere, this paper contains only preliminary data concerning ukc for a few collision partners and a preliminary model treatment.
2. Experimental
1 Jnty 1978
CHE!+lJCAL PHYSICS L5lTERS
Volume 57. number 1
and results
The CO fluorescence was excited by synchrotron radiation of the Orsay Collision Ring (ACO) using a specially designed monochroma:or [ lS] with spectral slits of the order of 10-I 5 A ensuring a good separation of individual vibronic bands and a constant intensity distribution within a contour of a single band.
The CO pressure was maintained constant (S-20 mtorr) and that of added gases was varied between 0 and 200 torr. Correlated single-photon counting technique adapted to specific properties of the light source 1141 was used for recording the fluorescence decay curves. In collision-free
conditions, the CO fluorescence decays quasi-exponentially with lifetimes varying withuinthe9-12nsrange [ll].inthepresenceof a coliider the decay becomes strongly non-exponential and may be approximated by a b&exponential function (fig. 1). It has been shown [l&16] that the long component of the decay is due to the reversible character of Ci !SC and to the vibrational relaxation processes, while the short one practically unaffected by vibrational relaxation, characterizes the initially excited vibronic level and obeys the Stem-Volmer relation:
1/r@x)
= 1 /r(O) + kstpx ,
where kg = urscQisthe rate constant of the S-T CI KC. orsc, for different vibronic states and different collision partners, is given in table 1. It may be clearly seen that there is no regular variation of oirc with u’ but a net correlation between ursc and @2). We do not Observe, however, the linear correlation expected from the first-order perturbation treatment [cf. eq. (2)] _ If we suppose that no transitions between pure singlets and pure triplets can oc-
Table 1
Mixing coeffkients ($) and quenching cross section Oisc for vibronic levels of the A ‘IJ state. Relative error in a~
v
46
Perturbing Ieveis
QJ2)from
values is f 10%
eisc (A21
eq. (3b)
eq. (3a)
He,A =45cm”. I/a = 3.13 A
Ar. A = 150 em-‘. I/a = 3.56 A
Rr, A = 185 em”, l/a = 3.65 A
e3Z- (U = 1)
0.102
0.092
3.4
27
44
d3A(u=4) d’A (u = 5) a’3ZZ+ (G = 10)
9.062
0.080
2.7
25
43
e3Z- cu - 4)
0.0044
0.0032
0.9
d3A (v = 8)
0.003
0.003
0.45
a’3Z+ (u = 14) e3.z- (u = 8)
0.012 0.002
0.016 0.0019
1.35 0.55
10.1 9.1 17.5 12.4
32 29s 45.5 30.5
d'A(v= 12) a’3X+ (u = 17) e32-(u= 11)
0.102
0.084
3.6
23.2
44.5
0.001
0.0012
0.5
9.8
36
CHEMICAL PHYSICS LJZTTERS
Volume 57, number 1
1 July 1978
the centers of mass of the colliding particles. Following Freed [7] we &G assume that this interaction which induces the coupling between 13rJ>and I?‘.‘.‘) levels is proportional to the mixing coefficient @ [eq. (I)] and to the collisional coupling between roiationaI levels of the S (or T) state:
.
cc” tins) Fig 1. Experimentaldecay curves(correctedfor the blackcountingand stray-lightbackground)for u = 2 level of CO. PC-J= 12 mtorr_
3”
. .
.
cur i.e. that ai,
+ 0 for fS2+ 0 and if we neglect the possibility of heavy-atom effects, the variation of oisc with
Note that this collisional coupling does nor conserve the molecular angular momentum so that levels characterized by different rotational quantum numbers J may be collisionally coupled_ With the objective of obtaining rough estimates we make some more drastic assumptions: (1) Level shifts are disregarded. (2) Two simple choices of interaction potentials are made: (a) A square perturbation, different from zero during 2 collision time rc rc = liav
(5)
and given during this time by V= a1/2fiAe-a2b2
,
(6)
where b is the impact parameter, a measures the range of the interaction potential and A its strength_ v is the relative velocity of the colliding particles. Eqs. (5) and (6) are obtained by fitting a gaussian interaction IJ--?(~) = @A,-a*.P*(r)
=fiAe-a*b*-a*v*t*
(7)
to 2 square potential taking 3. Model cakdation The experimental data demonstrate thar the linear dependence predicted by the fmt-order perturbation treatment breaks down for the observed range of coupling strengths. It is natural to associate this observation with the breakdown of the linear approximation. In order to test this viewpoint, we checked whether the observed phenomena may be fitted to calculations based on a nonperturbative solution of extremely simpiified two-level models with physically reasonable parameters_ The collision is treated semi-classically so that the coordinate of the perturbing atom is taken as a time-dependent classical function in the interaction hamiltonian. For simplicity the interaction is assumed to depend only on the separation between
7-c =
[s -0
dt VCR(t))] ‘I2 J -0II
dt t2 ‘/R(t))/
@)
and V= (l/rc)
J
V(R(r))dt
_
(3
--o
(b) A time-dependent V,(t)
perturbation
= C, sech(C$)
,
of the form (10)
where Cl and C-J were determined so that rc and V defmed by eqs. (8) and (9) are again given by eqs. (5) and (6). This leads to CI =ip1/2&4e-a2b2 c2=&W.
,
(11) (12) 47
CHEMICAL
VoIume 57, number 1
PHYSICS
LETl-ERS
1 J-dy 1978
The transition probability in a collision with a given impact parameter, b, is obtained by soking the time-dependent SchriZlinger equation. This gives in case (a): &@2e-2a2b2
4 =
f@,
2(1 +tiZ+32e--2a2b2)
1 _cos (1
x
+GTq3k”‘b2)‘12]
)
[
(132)
where x= 2A/6E and ir = 2iazv/i5Eand in case (b) (according to ref. [171_): fib, vj = sin2(~~/2~~e-a2b5fir) The
Sd'12(2/ij).
(I=)
total cross section for a given velocity v:
aa o*(v)
= 2a s
fib, vjbdb 0 will he, for caSe (a): o&y)
= (~.@u”){hr(l
- 2Ci[(l
-I-lrzf12)
!I0 + 2Ci(l/Vj
f ?tS&rWJ)
(14)
F&_ Z ok plotted versusIII@~>.Top: experimentalvalues. Sottom: cakuiatedvaluesfor case (a). dotted line, and case O), solid line, for merent collisionpartners.
and for case (b): u&v)
= (&!a2)
4. Comments
sech2(2/Sj
[y +ln(2n’Gi#irj
- Ci(2rrWG3,Gj]
,
(14b)
where y is tke Euler constant_ This resu!t may be numerically averaged for MaxwelL-Roltzrrann distributions corresponding to different colliding pairs and temperatures in order to evAate the effective cross section defied as:
These formulae are applied to the CO case vlitb _ the following choice of parameters: 6E = 10 cm-l (of thg order of the average gap between the closest lying S and T states for u’ = 0, 1 and 2 AlII CO ievek); bfie is taken from ref. [l g] while for other gases we suppose ACOSC/ACOJle = (e&~_#* where E is the krnard-Jones constant; I/a is equal to the averaged Lennard-Jones radius (ucO + u-J/2_ G and e pardmeters are taken from ref. 1191. Plots of ok versus hr@~) obtained in this way for cases (a) and (hj are compared with experimental data ir? fig. 2. 48
(I) It has been shown that with a reasonable choice of parameters we obtain a correct order of magnitude for ok and reproduce fundamental features of the observed behavior (saturation, dependence on the collision partner. (2) In the weak perturbation limit (A6 + 0), eqs.
(14a) and (14b) give a linear dependence of ok on @, deduced previously from the fust-order perturbation treatment [7] _ (3) The two model potend& [eqs. (5) and (6) for case (a), eq. (IO) for case (b)] were chosen because they lead to analyticahy sohrble problems. The close similarity between the results obtained in both cases suggests that at least in the range of parameters considered, the exact form of the perturbation is relatively unimportant. This is seen in the range i+ 1 where 0 = (@z2jP~2/4~~
f or case (a) and
o = (12/2a2jP~2/irz for case (b)_ For ? 5 1 the results of the two models disagree
Volume 57, number 1
CHEMICAL
with each other because the square potential cannot account correctly for the adiabatic behavior in this limit. (4) Our model neglecting the level shifts is complementary to that of Freed and Tric 1201 accounting for level shifts by the first-order perturbation treatment and for the level crossing in the spirit of the Landau-Zener approach.
Acknowhedgement The authors are highly indebted to the team of the LURE and of the Linear Accelerator Laboratory in &say whose work made possible our experiments. Thanks are also due to Drs. G. Bergeron, K.F. Freed and C_ Tric for discussions and remarks. One of us (AN) thanks the Commission for Basic Research of the Israel Academy of Science for supporting this work and the France-israel Scientists Exchange Program for a travel grant.
References 1l] T-G. Skmger and G. Black, J_ Chem. Phys. 58 (1973) 194,3121_ [ 21 E-H_ Fii and FJ. Comes, Chem Phys Letters 14 (1972) 433.
PHYSICS
LETTERS
1 July 1978
[3] R-k Beyer and WC
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