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Semiclassical calculation of the probabilities for collision-induced vibrational transitions in Ar*2
18 March 1994
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 2 19 ( 1994) 360-365
Semiclassical calculation of the probabilities for collision-induced vibrational transitions in Ar T L. Goubert a, G.D. Billing b, E. Desoppere a, W. Wieme a nDepartment ofApplied Physics, Universiteit Gent, Rozier 44, 9000 Gent, Belgium b Chemistry Department, H.C. Orsted Institute, Universitetsparken 5, DK-2100 Knbenhavn 0, Denmark
Received 1 July 1993; in final form 20 December 1993
Abstract Although the VUV continua emitted by rare gas excimers have been extensively investigated, no theoretical calculations are available for their intensity distribution. In this Letter a semiclassical method, based on the split operator technique, has been used to describe the collisions between Ar2 ( 3Zz ) excimers and ground state Ar atoms. The experimentally observed intensity distribution in the so-called first and second WV continuum has been qualitatively reproduced. The calculations yield a value of u = (4.10 + 0.27 ) x 10”~ (s- ’ ) for the global vibrational relaxation process frequency, where p is the pressure in Torr.
1. Theory
Excimers in the Ar2 (‘C,+ ) state can decay electronically from any of the 35 possible vibrational levels to the dissociative Ar, ( ‘X: ) ground state. Hence, the observed VUV spectrum is a weighted average of the individual contributions of each vibrational level, the weight factors being determined by the corresponding vibrational transition probabilities. We propose to evaluate the probabilities LZ?~(E) for transitions between the vibrational levels i, energy Eipand j, energyEj,inArt(3ZC:) (i,j=l,...,35),causedbycollisions with a ground-state atom,
Ar( *So) atom will be indicated as A and the At$ (‘C,’ ) excimer as BC. In the semiclassical (classical path) theory [ 21, the total energy E of system ( 1) is related to the classical energy (translational + rotational) U through E-Ei=U+fAE+(AE)2/16U,
(2)
with AE=E,-Ep The Hamiltonian of system ( 1) is expressed in the center-of-mass reference frame of the excimer (Fig. l),
A= g + V(R, CY, 8,
r) +
$ - $$
+u(r) , (3)
(1) Following ref. [ I] we have applied a semiclassical method to solve this problem, i.e. the translational and rotational degrees of freedom have been treated classically and the vibrational degree of freedom quantum mechanically. To simplify notation, the
where m is the reduced mass of the relative motion, m=mA (mB+mc)/(m,+mB+m,-); p is the reduced mass of BC; P= (P,, Py,P,), is the momentum vector of A; R is the distance vector between A and the center-of-mass of BC; r is the internuclear distance of BC, I’, 21:the inter- and intramolecular exci-
0009-2614/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)00100-S
L. Goubert et al. /Chemical Physics Letters 219 (1994) 360-365
able r is determined by the time-dependent dinger equation (TDSE ) ,
1 7.
fiY(r,t)=iAT,
Y”
*
Y’
Fig. 1. Configuration
ofthe system Ar(‘S,,) +A$(%,+).
mer potential respectively; (Y, /3 are angles defining the orientation of BC; and Jis the rotational angular momentum of BC. For further details the reader is referred to ref. [ 2 1. In (3), the translational and rotational degrees of freedom are treated classically, therefore the expectation value of Z? is calculated to obtain the efictive Hamiltonian [ 2 1, H,,=(~(~(r,t)Ifilyl(r,t)),
361
(4)
where Y( r, t) denotes the vibrational wavefunction of the excimer. The time propagation of the classical variables is then determined by Hamilton’s equations,
Schr&
alu(r, 0
(6)
withfigivenby (3). Instead of expanding !P( r, t ) as a linear combination of vibrational eigenfunctions with time-dependent coeffkients, we have chosen a non-expansive method to integrate ( 6 ) [ 3 1. A one-dimensional grid in r is fixed, consisting of K equidistant points, r,i, and r,, being the minimum and maximum r values. The time evolution of the vibrational wavefunction is determined in each point of the grid by means of the split operator technique [ 41, using a fast Fourier transform method (see, e.g., ref. [ 5 ] ) for the evaluation of the second-order derivative in r in the TDSE (6). At t= - cx), we let Yequal the eigenfunction belonging to a given vibrational level i. At the end of the trajectory, the square of the overlap of the final wavefunction and the eigenfunction belonging to vibrational levelj yields the probability for a transition from level i to levelj, P&S a,B, Y, &f) = ( 7 y*(r, --oo
+~MrJ
df, (7)
with (Y,B, y and 6 as defined in Fig. 1 andfe [ 0, 1 ] a parameter defining the partitioning of the rotational energy of BC between the components J, and J,. A Monte Carlo technique is used to evaluate the integral *j(E)=
I...SPo(E,a,B,y,6,ndadpdydsdf. (8)
In the center-of-mass system, one can express HeR as a function of the Cartesian coordinates x, y and z of A and of the angles (Yand /3 and their canonically conjugate variables P,, P,,, P,, .I, and JB (Fig. 1). One then obtains ten coupled differential equations [ 21. The vibrational motion has to be treated quantum mechanically, therefore in ( 3 ) the corresponding kinetic energy is represented by its operator. The time evolution of the wavefunction of the quantum vari-
As potential energy hypersurface, an additive two-body potential V(R cr,8, r) +4r)
we have chosen
= VAB(L)
+ V~‘,,(&,) + VK(&)
3
(9)
where *(r) E VBC(dB,) and & is the distance between the nuclei k and I( k, I= A, B or C). The first two terms on the right-hand side of (9 ) represent Morse potentials, for the last term a BomMayer potential (for dAC< 0.27 nm) has been joined
362
L. Goubert et al. /Chemical Physics Letters 219 (1994) 360-365
to a Lennard-Jones potential (for & > 0.29 nm) using a cubic spline polynomial fit. The appropriate potential parameters have been taken from Nowak et al. [6]. To obtain the global VUV spectrum, allowing comparison with experimental data, a weighted sum of the individual spectra emanating from each of the 35 vibrational levels of the excimer AI$ is calculated. Fig. 2 shows the Franck-Condon elements for electric dipole transition from the vibrational levels 0, 10 and 30 of the electronically excited state A$ ( 3C,+) to the ground state Ar2 ( ‘Cl ) , calculated as +CO &(a)=
I
dE
0
110
120
s
130
wavelength
+a, @i(r)p(r)ll/(E,
r) dr
9
(10)
0
where q$(r) is the eigenfunction of vibrational level i; p(r) is the electric dipole moment of the excimer and y/(E, r) is the eigenfunction of the ground state belonging to the continuum level E. The potential energy curves (PECs) of the ground state and the excited state, used to obtain the eigenfunctions, are again taken from Nowak et al. [ 6 1. The sensitivity of the vibrational spectra to the PEC parameters was tested by slightly modifying those parameters in their analytical expressions. A 1% variation had little effect on the position of the maxima of the spectra (less than 1% deviation) while the height and the (approximate) width (fwhm) changed at most by 7% and 1 l%, respectively. The relative weight of the spectrum from level i is proportional to the number of excimers which decay radiatively from this level to the ground state. This number is obtained as follows: consider N “just formed” excimers (N for instance 106): all will eventually decay, each one while being in one of the 35 possible vibrational states. If qd_. is the number of excimers which have decayed at t = + cmfrom vibrational level i, then clearly 7 ni,decaY=N.
100
140
150
[nml
0.25 I
7 ? 2
0.20 -
‘-
0.15 -
vibrational
level
10
is E : ..z Lo 2
L 0.10 -
0.05 -
2
IL 0.00
, 100
I
I 110
120
wavelength 0.75
130
1 150
140
lnml
1
vibrational
100
110
120
wavelength
(11)
level
130
30
140
150
(nm)
Fig. 2. Vibrational spectra of the levels 0, 10 and 30. TO
evaluate n/decay a Monte Carlo procedure is applied consisting of five steps: (a) Random generation of one velocity, say v, for the N excimers from a Maxwell-Boltzmann velocity distribution. (b) Random generation of one time-of-flight for
the N excimers; the time-of-flight distributed exponentially r(t) dt= f exp
(
- f t dt , >
is assumed
to be
(12)
L. Goubertet al. / Chemical PhysicsLetters219 (1994) 360-365
1being the mean free path of an excimer given at absolute temperature T and at low gas pressures p by
20
I+/_,
15 -
(13)
where k is the Boltzmann constant and c= 4 (r,+ rBC); r, and rBc being the hard sphere diameter of the Ar atom and excimer respectively, for which we have taken 0.363 and 0.910 nm. T was taken to be 300 K. (c) Determination of that fraction of the number of excimers in level i which will decay during the free flight, according to: %,dmayed
=
&
em
( -
l/
ri ) F
+
ni,before co1 -
c
j#i
C ~i(E)nj,befmeco* j#i
-
initial
level
= 10
l
.
.
2 .z : In
10 -
: L? 0 S-
. O.
.
I
I
,
I
I
4
6
8
10
12
25
30
vibrational 6
.
n
-
(14)
where ni is the number of excimers presently in vibrational level i; t is the time-of-flight of the excimers; and rj is the natural lifetime of an excimer in vibrational level i [ 7 1. Subsequently nj,d_y& is added to the aCtUa1 ni,dsay (d) The populations of the vibrational levels are changed by each collision. The resulting populations are calculated from R,afterco1=
363
1
initial
level
.
20
q
level
%j.(mbefmol
-
(15) 15
(e) A test is run to verify whether more than 99% of the excimers have already decayed, if this is the case the procedure is stopped, otherwise it is repeated from step (b) on. Steps (a)-(d) are repeated and the average value is calculated until a predefined statistical of n&decay accuracy is obtained. Finally, using appropriate Franck-Condon elements such as shown in Fig. 2, the wavelength dependence of the VUV spectrum can be calculated as (16)
20
vibrational 3.5 3.0
initial
level
= 30
‘-
1 z .f! ;;
.
2.0
2
. .
.
. 0.5
.
_I
. . , 15
n
-
-
.
’
.
.
.
-
1
I
I
20
30
25
vibrational
Fig. 3 shows the cross sections tag- which yield the desired %j(E) after normalization - obtained for i=lO, 20 and 30, with E-Eic0.026 eV. For each profile 100 trajectories have been calculated. It is clear
.
2.5
0.0
2. Results and discussion
level
35
level
Fig. 3. Cross sections (in relative units) a, for i= lo,20 and 30 and E-&=0.026 eV.
from Fig. 3 that the probability for further relaxation decreases the more the excimer has relaxed. However, this effect is more or less compensated
L. Goubertet al. /Chemical PhysicsLetters219 (1994) 360-365
364
by the strong increase of the natural lifetime for dipole radiation of the excimer with decreasing vibrational level [ 7 1. This explains the appearance of two clearly separated continua in the spectra evident from our final result ( 16), plotted in Fig. 4, and corresponding with experimental observations [ 8 1. The asymmetric, narrow continuum around 110 nm known as the first continuum - originates at higher (i=20-35) vibrational levels, while the more symmetric and broader continuum around 130 nm - the second continuum - is due to the decay from lower levels, say i= O-5. Fig. 4 also shows the characteristic increase of the second continuum with increasing gas pressure, also in accordance with ref. [ 81. Our calculations only include collisional de-excitation processes such as ( 1 ), i.e. electron collisions are not considered. This corresponds to the conditions prevailing during the late afterglow of low-pressure discharges. In kinetic models of such afterglows the process of vibrational relaxation has up to now always been simply modelled as relaxation from a single high level to a single low level [ 9 1. From the ratio of the integrated intensity Z2 of the second continuum to the integrated intensity I1 of the first continuum a global process frequency for vibrational relaxation can be obtained as
0.10 p = 0.5 Torr
- 0.06 ,x ‘g 0.04 w 2 0.02
T* being the natural lifetime of excimers in the vibrational states contributing to the first continuum (i.e. the highest levels) and t a provisionally unknown factor. In practice it is difficult to differentiate these two continua unequivocally. As the first (second) continuum reaches its maximum intensity at 108 nm ( 130.7 nm) we have taken, somewhat pragmatically, for II (Z2) the intensity in the interval 105-l 11 nm (127.7-133.7 nm). From a plot ofZ,/Z, versusp such as shown in Fig. 5 one then obtains Zz/Z1 =0.68p
(pin Torr)
With r*=O.166+0.011 and (15) v=ep=
.
us [7], we find using (14)
(4.10f0.27)~
~=ep=5.6x
10”~
(s-l)
.
(19)
10’~
(s-l)
.
(20)
We found however ref. [ 111 does not contain any direct reference to such value and we feel the authors
0.05 2 5 0.04
p = 1 Torr
‘;
wavelength (nml
O.Ooo 100 4. Calculated
(18)
It should be possible, at least in principle, to compare this with experimental results obtained from afterglows. In ref. [ 10 ] the authors quote Firestone [ 111 as proposing the value
wavelength (nml
Fig.
(17)
0.06
I!,
3 0.08
z*/z,= cppr* )
110 120 130 140 wavelength (nml
WV spectra at pressures p= 0.5, 1,20 and 60 Torr.
150
L. Goubertet al. /Chemical PhysicsLetters219 (1994) 360-365 75 ‘
60 -
..r . L.4
45
-
30
-
/
365
106p s-l, where p is the pressure in Torr. We are planning our own afterglow experiments to obtain further confirmation of this value.
Acknowledgement
0
20
40
pressure
60
80
1 100
This work was made possible by a grant of the National Science Foundation of Belgium (NFWO). The help of Dr. C. Henriet (CRAY RESEARCH FRANCE) in granting computer time for this research is gratefully acknowledged.
[Torrl
Fig. 5. Linear dependence of X2/I,upon the gas pressure. References provide no information from which such value could be deduced. A further extensive search in the literature has not revealed a single experimental or theoretical value for v. While no meaningful comparison can therefore be made we note ( 19) is at least of the expected order of magnitude.
3. Conclusion
In this paper we present a first theoretical calculation of a typical intensity pattern for an inert gas WV spectrum. Using a semiclassical method, both wavelength and pressure dependence can be reproduced qualitatively. The vibrational relaxation process frequency has been calculated as v= (4.10 f 0.27) x
[ 1 ] G.D. Billing, Computer Phys. Rept. 1 ( 1984) 237. G.D. Billing, Chem. Phys. 30 (1978) 387.
[2]
[3] G.D. Billing, Computer Phys. Rept. 12 (1990) 383. [4] M.D. Feit, J.A. Flick Jr. and A. Steiger, J. Comput. Phys. 47 (1982) 412. [5] R. Kosloff, J. Phys. Chem. 92 (1988) 2087. [6] G.Nowak,L. FreyandJ.Fricke, J.Phys.B 18 (1985) 2851. [7]A.A. Madej, P.R. Herman and B.P. Stoicheff, Phys. Rev. Letters 57 (1986) 1574. [8] Y. Tanaka, J. Opt. Sot. Am. 48 (1955) 710; N. Thonnard and G.S. Hurst, Phys. Rev. A 5 (1972) 1110. [9] H. Janssens, M. Vanmarcke, E. Desoppere, J. Lenaerts, R. BouciquC and W. Wieme, J. Chem. Phys. 86 (1986) 4925;
H. Janssens, M. Vanmarcke,E. Desoppere,R. Bouciqu6and W. Wieme, J. Chem. Phys. 86 (1986) 4935. [lo] P. Millet, A. Birot, H. Bnmet, H. Dijols, J. Galy and Y. Salamero, J. Phys. B 15 (1982) 2935. [ 111 R.F. Firestone, J. Chem. Phys. 70 (1979) 123.
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 2 19 ( 1994) 360-365
Semiclassical calculation of the probabilities for collision-induced vibrational transitions in Ar T L. Goubert a, G.D. Billing b, E. Desoppere a, W. Wieme a nDepartment ofApplied Physics, Universiteit Gent, Rozier 44, 9000 Gent, Belgium b Chemistry Department, H.C. Orsted Institute, Universitetsparken 5, DK-2100 Knbenhavn 0, Denmark
Received 1 July 1993; in final form 20 December 1993
Abstract Although the VUV continua emitted by rare gas excimers have been extensively investigated, no theoretical calculations are available for their intensity distribution. In this Letter a semiclassical method, based on the split operator technique, has been used to describe the collisions between Ar2 ( 3Zz ) excimers and ground state Ar atoms. The experimentally observed intensity distribution in the so-called first and second WV continuum has been qualitatively reproduced. The calculations yield a value of u = (4.10 + 0.27 ) x 10”~ (s- ’ ) for the global vibrational relaxation process frequency, where p is the pressure in Torr.
1. Theory
Excimers in the Ar2 (‘C,+ ) state can decay electronically from any of the 35 possible vibrational levels to the dissociative Ar, ( ‘X: ) ground state. Hence, the observed VUV spectrum is a weighted average of the individual contributions of each vibrational level, the weight factors being determined by the corresponding vibrational transition probabilities. We propose to evaluate the probabilities LZ?~(E) for transitions between the vibrational levels i, energy Eipand j, energyEj,inArt(3ZC:) (i,j=l,...,35),causedbycollisions with a ground-state atom,
Ar( *So) atom will be indicated as A and the At$ (‘C,’ ) excimer as BC. In the semiclassical (classical path) theory [ 21, the total energy E of system ( 1) is related to the classical energy (translational + rotational) U through E-Ei=U+fAE+(AE)2/16U,
(2)
with AE=E,-Ep The Hamiltonian of system ( 1) is expressed in the center-of-mass reference frame of the excimer (Fig. l),
A= g + V(R, CY, 8,
r) +
$ - $$
+u(r) , (3)
(1) Following ref. [ I] we have applied a semiclassical method to solve this problem, i.e. the translational and rotational degrees of freedom have been treated classically and the vibrational degree of freedom quantum mechanically. To simplify notation, the
where m is the reduced mass of the relative motion, m=mA (mB+mc)/(m,+mB+m,-); p is the reduced mass of BC; P= (P,, Py,P,), is the momentum vector of A; R is the distance vector between A and the center-of-mass of BC; r is the internuclear distance of BC, I’, 21:the inter- and intramolecular exci-
0009-2614/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)00100-S
L. Goubert et al. /Chemical Physics Letters 219 (1994) 360-365
able r is determined by the time-dependent dinger equation (TDSE ) ,
1 7.
fiY(r,t)=iAT,
Y”
*
Y’
Fig. 1. Configuration
ofthe system Ar(‘S,,) +A$(%,+).
mer potential respectively; (Y, /3 are angles defining the orientation of BC; and Jis the rotational angular momentum of BC. For further details the reader is referred to ref. [ 2 1. In (3), the translational and rotational degrees of freedom are treated classically, therefore the expectation value of Z? is calculated to obtain the efictive Hamiltonian [ 2 1, H,,=(~(~(r,t)Ifilyl(r,t)),
361
(4)
where Y( r, t) denotes the vibrational wavefunction of the excimer. The time propagation of the classical variables is then determined by Hamilton’s equations,
Schr&
alu(r, 0
(6)
withfigivenby (3). Instead of expanding !P( r, t ) as a linear combination of vibrational eigenfunctions with time-dependent coeffkients, we have chosen a non-expansive method to integrate ( 6 ) [ 3 1. A one-dimensional grid in r is fixed, consisting of K equidistant points, r,i, and r,, being the minimum and maximum r values. The time evolution of the vibrational wavefunction is determined in each point of the grid by means of the split operator technique [ 41, using a fast Fourier transform method (see, e.g., ref. [ 5 ] ) for the evaluation of the second-order derivative in r in the TDSE (6). At t= - cx), we let Yequal the eigenfunction belonging to a given vibrational level i. At the end of the trajectory, the square of the overlap of the final wavefunction and the eigenfunction belonging to vibrational levelj yields the probability for a transition from level i to levelj, P&S a,B, Y, &f) = ( 7 y*(r, --oo
+~MrJ
df, (7)
with (Y,B, y and 6 as defined in Fig. 1 andfe [ 0, 1 ] a parameter defining the partitioning of the rotational energy of BC between the components J, and J,. A Monte Carlo technique is used to evaluate the integral *j(E)=
I...SPo(E,a,B,y,6,ndadpdydsdf. (8)
In the center-of-mass system, one can express HeR as a function of the Cartesian coordinates x, y and z of A and of the angles (Yand /3 and their canonically conjugate variables P,, P,,, P,, .I, and JB (Fig. 1). One then obtains ten coupled differential equations [ 21. The vibrational motion has to be treated quantum mechanically, therefore in ( 3 ) the corresponding kinetic energy is represented by its operator. The time evolution of the wavefunction of the quantum vari-
As potential energy hypersurface, an additive two-body potential V(R cr,8, r) +4r)
we have chosen
= VAB(L)
+ V~‘,,(&,) + VK(&)
3
(9)
where *(r) E VBC(dB,) and & is the distance between the nuclei k and I( k, I= A, B or C). The first two terms on the right-hand side of (9 ) represent Morse potentials, for the last term a BomMayer potential (for dAC< 0.27 nm) has been joined
362
L. Goubert et al. /Chemical Physics Letters 219 (1994) 360-365
to a Lennard-Jones potential (for & > 0.29 nm) using a cubic spline polynomial fit. The appropriate potential parameters have been taken from Nowak et al. [6]. To obtain the global VUV spectrum, allowing comparison with experimental data, a weighted sum of the individual spectra emanating from each of the 35 vibrational levels of the excimer AI$ is calculated. Fig. 2 shows the Franck-Condon elements for electric dipole transition from the vibrational levels 0, 10 and 30 of the electronically excited state A$ ( 3C,+) to the ground state Ar2 ( ‘Cl ) , calculated as +CO &(a)=
I
dE
0
110
120
s
130
wavelength
+a, @i(r)p(r)ll/(E,
r) dr
9
(10)
0
where q$(r) is the eigenfunction of vibrational level i; p(r) is the electric dipole moment of the excimer and y/(E, r) is the eigenfunction of the ground state belonging to the continuum level E. The potential energy curves (PECs) of the ground state and the excited state, used to obtain the eigenfunctions, are again taken from Nowak et al. [ 6 1. The sensitivity of the vibrational spectra to the PEC parameters was tested by slightly modifying those parameters in their analytical expressions. A 1% variation had little effect on the position of the maxima of the spectra (less than 1% deviation) while the height and the (approximate) width (fwhm) changed at most by 7% and 1 l%, respectively. The relative weight of the spectrum from level i is proportional to the number of excimers which decay radiatively from this level to the ground state. This number is obtained as follows: consider N “just formed” excimers (N for instance 106): all will eventually decay, each one while being in one of the 35 possible vibrational states. If qd_. is the number of excimers which have decayed at t = + cmfrom vibrational level i, then clearly 7 ni,decaY=N.
100
140
150
[nml
0.25 I
7 ? 2
0.20 -
‘-
0.15 -
vibrational
level
10
is E : ..z Lo 2
L 0.10 -
0.05 -
2
IL 0.00
, 100
I
I 110
120
wavelength 0.75
130
1 150
140
lnml
1
vibrational
100
110
120
wavelength
(11)
level
130
30
140
150
(nm)
Fig. 2. Vibrational spectra of the levels 0, 10 and 30. TO
evaluate n/decay a Monte Carlo procedure is applied consisting of five steps: (a) Random generation of one velocity, say v, for the N excimers from a Maxwell-Boltzmann velocity distribution. (b) Random generation of one time-of-flight for
the N excimers; the time-of-flight distributed exponentially r(t) dt= f exp
(
- f t dt , >
is assumed
to be
(12)
L. Goubertet al. / Chemical PhysicsLetters219 (1994) 360-365
1being the mean free path of an excimer given at absolute temperature T and at low gas pressures p by
20
I+/_,
15 -
(13)
where k is the Boltzmann constant and c= 4 (r,+ rBC); r, and rBc being the hard sphere diameter of the Ar atom and excimer respectively, for which we have taken 0.363 and 0.910 nm. T was taken to be 300 K. (c) Determination of that fraction of the number of excimers in level i which will decay during the free flight, according to: %,dmayed
=
&
em
( -
l/
ri ) F
+
ni,before co1 -
c
j#i
C ~i(E)nj,befmeco* j#i
-
initial
level
= 10
l
.
.
2 .z : In
10 -
: L? 0 S-
. O.
.
I
I
,
I
I
4
6
8
10
12
25
30
vibrational 6
.
n
-
(14)
where ni is the number of excimers presently in vibrational level i; t is the time-of-flight of the excimers; and rj is the natural lifetime of an excimer in vibrational level i [ 7 1. Subsequently nj,d_y& is added to the aCtUa1 ni,dsay (d) The populations of the vibrational levels are changed by each collision. The resulting populations are calculated from R,afterco1=
363
1
initial
level
.
20
q
level
%j.(mbefmol
-
(15) 15
(e) A test is run to verify whether more than 99% of the excimers have already decayed, if this is the case the procedure is stopped, otherwise it is repeated from step (b) on. Steps (a)-(d) are repeated and the average value is calculated until a predefined statistical of n&decay accuracy is obtained. Finally, using appropriate Franck-Condon elements such as shown in Fig. 2, the wavelength dependence of the VUV spectrum can be calculated as (16)
20
vibrational 3.5 3.0
initial
level
= 30
‘-
1 z .f! ;;
.
2.0
2
. .
.
. 0.5
.
_I
. . , 15
n
-
-
.
’
.
.
.
-
1
I
I
20
30
25
vibrational
Fig. 3 shows the cross sections tag- which yield the desired %j(E) after normalization - obtained for i=lO, 20 and 30, with E-Eic0.026 eV. For each profile 100 trajectories have been calculated. It is clear
.
2.5
0.0
2. Results and discussion
level
35
level
Fig. 3. Cross sections (in relative units) a, for i= lo,20 and 30 and E-&=0.026 eV.
from Fig. 3 that the probability for further relaxation decreases the more the excimer has relaxed. However, this effect is more or less compensated
L. Goubertet al. /Chemical PhysicsLetters219 (1994) 360-365
364
by the strong increase of the natural lifetime for dipole radiation of the excimer with decreasing vibrational level [ 7 1. This explains the appearance of two clearly separated continua in the spectra evident from our final result ( 16), plotted in Fig. 4, and corresponding with experimental observations [ 8 1. The asymmetric, narrow continuum around 110 nm known as the first continuum - originates at higher (i=20-35) vibrational levels, while the more symmetric and broader continuum around 130 nm - the second continuum - is due to the decay from lower levels, say i= O-5. Fig. 4 also shows the characteristic increase of the second continuum with increasing gas pressure, also in accordance with ref. [ 81. Our calculations only include collisional de-excitation processes such as ( 1 ), i.e. electron collisions are not considered. This corresponds to the conditions prevailing during the late afterglow of low-pressure discharges. In kinetic models of such afterglows the process of vibrational relaxation has up to now always been simply modelled as relaxation from a single high level to a single low level [ 9 1. From the ratio of the integrated intensity Z2 of the second continuum to the integrated intensity I1 of the first continuum a global process frequency for vibrational relaxation can be obtained as
0.10 p = 0.5 Torr
- 0.06 ,x ‘g 0.04 w 2 0.02
T* being the natural lifetime of excimers in the vibrational states contributing to the first continuum (i.e. the highest levels) and t a provisionally unknown factor. In practice it is difficult to differentiate these two continua unequivocally. As the first (second) continuum reaches its maximum intensity at 108 nm ( 130.7 nm) we have taken, somewhat pragmatically, for II (Z2) the intensity in the interval 105-l 11 nm (127.7-133.7 nm). From a plot ofZ,/Z, versusp such as shown in Fig. 5 one then obtains Zz/Z1 =0.68p
(pin Torr)
With r*=O.166+0.011 and (15) v=ep=
.
us [7], we find using (14)
(4.10f0.27)~
~=ep=5.6x
10”~
(s-l)
.
(19)
10’~
(s-l)
.
(20)
We found however ref. [ 111 does not contain any direct reference to such value and we feel the authors
0.05 2 5 0.04
p = 1 Torr
‘;
wavelength (nml
O.Ooo 100 4. Calculated
(18)
It should be possible, at least in principle, to compare this with experimental results obtained from afterglows. In ref. [ 10 ] the authors quote Firestone [ 111 as proposing the value
wavelength (nml
Fig.
(17)
0.06
I!,
3 0.08
z*/z,= cppr* )
110 120 130 140 wavelength (nml
WV spectra at pressures p= 0.5, 1,20 and 60 Torr.
150
L. Goubertet al. /Chemical PhysicsLetters219 (1994) 360-365 75 ‘
60 -
..r . L.4
45
-
30
-
/
365
106p s-l, where p is the pressure in Torr. We are planning our own afterglow experiments to obtain further confirmation of this value.
Acknowledgement
0
20
40
pressure
60
80
1 100
This work was made possible by a grant of the National Science Foundation of Belgium (NFWO). The help of Dr. C. Henriet (CRAY RESEARCH FRANCE) in granting computer time for this research is gratefully acknowledged.
[Torrl
Fig. 5. Linear dependence of X2/I,upon the gas pressure. References provide no information from which such value could be deduced. A further extensive search in the literature has not revealed a single experimental or theoretical value for v. While no meaningful comparison can therefore be made we note ( 19) is at least of the expected order of magnitude.
3. Conclusion
In this paper we present a first theoretical calculation of a typical intensity pattern for an inert gas WV spectrum. Using a semiclassical method, both wavelength and pressure dependence can be reproduced qualitatively. The vibrational relaxation process frequency has been calculated as v= (4.10 f 0.27) x
[ 1 ] G.D. Billing, Computer Phys. Rept. 1 ( 1984) 237. G.D. Billing, Chem. Phys. 30 (1978) 387.
[2]
[3] G.D. Billing, Computer Phys. Rept. 12 (1990) 383. [4] M.D. Feit, J.A. Flick Jr. and A. Steiger, J. Comput. Phys. 47 (1982) 412. [5] R. Kosloff, J. Phys. Chem. 92 (1988) 2087. [6] G.Nowak,L. FreyandJ.Fricke, J.Phys.B 18 (1985) 2851. [7]A.A. Madej, P.R. Herman and B.P. Stoicheff, Phys. Rev. Letters 57 (1986) 1574. [8] Y. Tanaka, J. Opt. Sot. Am. 48 (1955) 710; N. Thonnard and G.S. Hurst, Phys. Rev. A 5 (1972) 1110. [9] H. Janssens, M. Vanmarcke, E. Desoppere, J. Lenaerts, R. BouciquC and W. Wieme, J. Chem. Phys. 86 (1986) 4925;
H. Janssens, M. Vanmarcke,E. Desoppere,R. Bouciqu6and W. Wieme, J. Chem. Phys. 86 (1986) 4935. [lo] P. Millet, A. Birot, H. Bnmet, H. Dijols, J. Galy and Y. Salamero, J. Phys. B 15 (1982) 2935. [ 111 R.F. Firestone, J. Chem. Phys. 70 (1979) 123.