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On the statistics of an ensemble of oscillators under excitation. IV. Evaluation of the time for kinetic excitation into the quasicontinuum based on Kramers' theory
Chemical Physics 76 (1983) 2 5 - 2 9 North-HoUand P u b h s h m g Conapany
ON THE STATISTICS OF AN ENSEMBLE OF OSCILLATORS UNDER EXCITATION. IV. EVALUATION OF THE TIME FOR KINETIC EXCITATION INTO THE QUASICONTINUUM BASED ON KRAMERS" THEORY V . N . S A Z O N O V a n d I.E. K H R O M O V Department o f Theoretical Physics. P N. Lebedev PhyMcal Institute, Leninsk) Prospect 53, 3losco*~ I 1 7924, L'SSR Received 6 Jul~r 1982, in final i c r m 2 December 1982
We consider the model with kineuc excltatton into tile quasicontanuum (KEQ) t'or resonant p o l y a t o m i c molecules which a b s o r b laser radmtion and are s u r r o u n d e d by b u f f e r molecules KEQ takes place w h e n the resonant molecules m the loxser part o f the energ3 s p e c t r u m interact weakly with the laser r a d m n o n , b u t the molecules m the q u a s i c o n t i n u u m are rapidly excited to still higher energy and dissociate. Under these conditions the colhsions ol the r e s o n a n t and buffer molecules lead to excitation o f resonant molecules i n t o the q u a s i c o n t i n u u m because the p o p u l a t i o n o f the quasicontinuunl is mucla less Illdn its llleImod)heroical equilibrium value It is found~ xiaal the smaller ihe V - - T relaxation time u'y-1-, the larger the rate o f KEQ and the dissociation rate (if onl~ ~'VT is n o t too small) T h u s if we change the e x p e r i m e m a l c a n d y n o n s and decrease r V T (tar instance, by passing from the heavy buffer gas Xe to tile hght buffer gas tte). for some resonant molecules we may observe that the probability o f dlssocmtion increases
1. I n t r o d u c t i o n a n d d e s c r i p t i o n o f p h y s i c a l s i t u a t i o n 1.1. I n t h i s p a p e r w e c o n s i d e r t h e case w h e n p o l y a t o m m m o l e c u l e s w i t h w b r a t t o n a l energ~ E a b o v e s o m e t h r e s h o l d e n e r g y E c are r a p i d l y e x c i t e d b y l a s e r r a d i a t i o n t o still h i g h e r e n e r ~ , . S u c h a ph2¢sical s i t u a t i o n arises w h e n t h e a b s o r p t i o n c r o s s s e c t i o n o i n c r e a s e s w i t h i n c r e a s i n g w b r a t i o n a l q u a n t u m n u m b e r ( o r e n e r g y E ) L e t t h e laser freq u e n c y co t b e s m a l l e r t h a n co01, t h e f r e q u e n c y o f t h e m a x i m u m o f t h e h n e a r a b s o r p n o n s p e c t r u m I f co01 -- co t = ( 1 0 - 3 0 ) c m - 1 . t h e n co e is set o u t s i d e t h e v i b r a t i o n a l - r o t a t i o n a l b a n d . In t l u s case t'or s m a l l v t h e c r o s s s e c n o n ot,(coe) is v e r y s m a l l t o o ( n m c h s m a l l e r t h a n 1 0 - 1 9 _ 10 - 2 0 c n t 2 ) . W h e n o i n c r e a s e s , t h e t r a n s i t x o n f r e q u e n c y ¢°o.v+ 1 d e c r e a s e s d u e t o n o n - h n e a r l t y , a n d t h e b a n d o f r e s o n a n t f r e q u e n c i e s m o v e s t o t h e l e f t . S o . f o r h i g h e r levels, one must e x p e c t that o o increases up to 1 0 - 1 7 - 1 0 - 1 8 cm z b o t h for two-atomic and p o l y a t o m i c molecules. B u t f o r t w o - a t o m i c m o l e c u l e s o o a g a i n b e c o m e s v e r y s m a l l , ff o is large e n o u g h a n d coo.v+ 1 -- co~ < - - ( 1 0 - - 3 0 ) c m - 1 . F o r p o l y a t o m i e m o l e c u l e s w h e n o i n c r e a s e s w e get i n t o t h e q u a s i c o n t m u u m , a n d t h e c r o s s s e c t i o n o o c o u l d n o t b e c o m e v e r y s m a l l . P r o b a b l y 0 o d e c r e a s e s in t h e q u a s i c o n t i n u u m t o s o m e e x t e n t (see fig. 1). b u t t y p i c a l v a l u e s o f o o i n t h e q u a s i c o n t i n u u m are 1 0 - 1 9 - 1 0 - 2 0 c m 2, i.e. m u c h l a r g e r t h a n 0 u f o r s m a l l o. R e l a x a t m n c a u s e d b y c o l h s i o n s o f r e s o n a n t m o l e c u l e s w l t h b u f f e r m o l e c u l e s in a gas m i x t u r e , h i n d e r s t h e exci110-~
0
"~
0301-0104/83/0000-0000/$
l~
"~
03.00 © 1983 North-Holland
Fig 1 Dots schemaucall.x shot, tile relaxation rate v v. The dashed line schematicalb show s tile exaltation rate **v = °v J, x~here J is the p h o t o n flux for a Its o-atomic molecule or for an individual ~ibrationai mode in a pol.x atomic molecule. The solid line schematicall~ shox~ s i,t, for a pol.x atomic molecule x, ith quasicontinuum at E > Eqc. Sohd and dashed lines coincide for E < Eqc.
V.N. ~azono),, LE. Khromov/Stattstics o f an ensemble o f o%cdlator% under excitation I V
26
tation caused b y laser radiation. We assume that the relaxation rate uo is larger than the excitation rate w o f f o < oe (E < Ec) , and t,u < w o ff o > oc (E > Ec). Generally speaking, the threshold o f the q u a s i c o n t i n u u m Eqc differs from E~ But E e depends on the laser frequency we, the intensity I, and the buffer gas pressure p. We imply that u n d e r the g~ven conditions E c ~ Eqc and treat the quas~continuum as a resonant part o f the energy spectrum 1.2. Under the action o f radiation, the molecules in the q u a s i c o n t i n u u m are rapidly excited and dissocmte or enter some other chemical reactions [ 1,2]. Because o f this, the q u a s i c o n t i n u u m is quickly depleted. The molecules in the lower part o f the energy spectrum interact weakly w~th the radmtion These molecules are excited into the quas~continuum kinetlcally, i.e. because o f collisions with buffer molecules. Owmg to these colhsions the population o f the q u a s i c o n t m u u m tries to increase to its t h e r m o d y n a m i c a l e q u d l b r m m value. In the present paper we consider the mare features o f this kinetic excitation into the q u a s i c o n t m u u m (KEQ). KEQ has n o t been m e n t i o n e d previously in the literature, though the physical situation is absolutely realistic KEQ must take place for m a n y polyatomic molecules u n d e r proper conditions, i.e. ~01 -- w~ = (10--30) cm - 1 , I >~ 109 W/era 2, p ~ 300 Torr (see section 3 below). 1.3. To deduce the ttrne o f KEQ, we use a theoretical scheme which is sinular to Kramers" theory of chemical kinetics [3] (see, for example, s o m e r e c e n t papers [ 4 - 8 ] ) . We make an a t t e m p t to use Kramers' ideas m the case o f a discrete variable q u a n t u m numbfir o We could n o t find any a t t e m p t o f this kind m the hterature.
2. Model and mathematics
2.1. To clear up the mare features o f KEQ we consider a sLrnple model, namely an ensemble of q u a n t u m nonlinear one-dimensional oscillators; the oscdlators are placed m a heat bath and are non-coherently excited b y laser radiation (in previous papers [9--11 ] we have considered coherent excitation). Let Pv be the p o p u l a t i o n o f level o. The dynamics o f p o is defined b y the well-known rate equations (o = 0, I, ...) IJo = - w o ( P o - Po+ l ) + Vo+l(Po+l -- e v P , ) + W o - l ( P , _ 1 - P , ) -- uo(p,-- % _ l P v - 1 ) - - ~[vPv ,
(1)
where w v = J o o is the rate o f excitation from o to o + 1 , J =l/hcoe is the p h o t o n f l u x ; vo Is the rate of V--T r e l a x a t i o n from v to v -- l , e o = exp [--(Et,+l - Ev)]T], T i s the temperature o f the buffer gas, which is the heat bath, ~'o is the rate o f dissociation from level o. In eqs. ( 1 ) , p , = )vz = 0 l f i = - 1 , vi = 0 lfz = 0. If at t = 0 we have Y'oPv = 1, t h e n f ( t ) = 1 - Z o P o ( O Is the probability o f dissociation at time t T o describe in our model the physical sltuatlon, we assume ~[v = 0
if o < od ,
ov = 0
If o < o c ,
w v = J ° o >~ ~'v+ 1
ff o >/v c .
(2)
The dissociation threshold od exceeds oc. Rate equations were successfully applied to the descripuon o f excitation and relaxation o f p o l y a t o m l c molecules b o t h at lower levels [ 12,13] and in q u a s i c o n t i n u u m [14] (see also references m r e f . [15]). 2.2. Let the action o f the laser radiation begin at t = 0, i.e. at t < 0 we have J = w o = 70 = 0 and Pv obeys the Gibbs d i s t r i b u t i o n Pu = A exp ( - E o / T ) , where A is a constant. If eo '< 1 (v = 0, 1,2 .... ), then almost all the population is in ground level o = 0. At t > 0 we assume the dissociation rates 7t, to be very large 0 f o >/Od, then 70 ~ wo, vo) and c o n s e q u e n t l y we p u t Po ~ 0 if o/> 0 d. There ts n o stationary solution o f (1) with this condition The population o f the lower levels will be transferred to the q u a s i c o n t i n u u m and to the dissociation region o > / o d. We want to esttmate the decrease ttme ~- o f the p o p u l a t i o n P0 in the ground level.
V.N. Sazonov, LE. Khromov[Statistics o f an ensemble o f oscillators under exctmtion I V 2.3.
27
We rewrite e q . ( I ) for v < v d in t h e f o l l o w i n g f o r m
Po =
-/~+t
(3)
+/o,
where ]o Is equal to iv = B ' v - 1 0 9 o - I -- P v ) - - P v ( P v - - e v - l P v -
(4)
l) "
],, must be treated as a current According to (4) we have
IV Pv- I
Wv-- l + IJv
Vu
W o - I -I- eo_ l
Wv-
Pu = bv + auPv -
l + e o - l Vv
A f t e r K -- I r e c u r r e n c e s w e get
Pv-K = bv-K+l
N o w we p u t v
P0
=
h
x-,
Vd
Vlff0
~-I v ~ az + P v ~ a, . i=v-K+t i=v-K+l
]Q
~-
1
~Vc
1
Wt _
17I
- + z_J
. . . . . vc -
]I
b~
(5)
0 , K = v d and eq. (5) gives
=
l fw~=0(i=0.1
P0
£=v-K+2
= Od,Pvd
w0+e0v
=--
+
~ = 2 w £ _ 1 + v l t e ~ _ 1 z=t
1
wi_ 1 +v/e/_ 1
l) andiv i>~v~+l(i=v c ..... vd]£
+ ~__25
°d
=_ ~lte0ffl...eQ_ l
+
+ Vt
(6)
l).then
]e
E
Q=uc+l W ~ _ l
(7)
A c c o r d i n g to Kramers" w o r k [3] b o t h m t h e case o f c o n t m u o u s and discrete variables the p o p u l a t i o n f l o w d e p e n d s w e a k l y o n t h e variable o u t s i d e t h e b o t t o m o f a p o t e n t i a l well, w h e r e a h n o s t all the p o p u l a n o n is s i t u a t e d ( m o u r case levels w~th v = 0, 1 . . . . . v~ - 1 m a y be t r e a t e d as t h e p o t e n n a ! well. and v = 0 is the b o t t o m o f the well) I n d e e d , t h e r e is n o source for c u r r e n t af t h e p o p u l a t m n is small, g o : - e c a n p u t in eq_ (7) i e ~ ] (~ = 1 . 2 . . . . ). I f t h e relaxan o n rate % increases w i t h v so rapidly that vi e l - 1 >~ v~_ 1 •
(8)
then a c c o r d i n g to ( 7 ) P O ~']/Vl CO. I f the r e l a x a t i o n rate % increases n o t ve W rapidb" and v~ez_t < v z - t
,
(9)
then ]
P0 ~
VVceO e l ---euc - l
] exp [(E c --
=
Eo)IT ]
vv c
When v = 0, eq. (3) gives/J0 = - ] I = - ] - N o w in the case (8) we have ~" = --Po/tSO = ( I / V l ) e x p [ ( E 1 - E o ) / T ] .
(10)
In the case (9) we have "r = -Po]l~ 0 = ( l / w e ) e x p [(Eve - E o ) / T ] .
(1 1)
We b e h e v e t h e case (9), ( 1 1 ) to b e m o r e reahstic and c o n s i d e r this case b e l o w . E x p r e s s i o n (1 1) c a n b e o b t a i n e d b y simpler c o n s i d e r a t i o n s . But t h e range o f v a h d i t y o f (1 1) can o n l y b e o b t a i n e d b y e x a c t c a l c u l a t i o n s [ c o m p a r e (8), ( 1 0 ) w l t h (9), ( 1 1 ) ] .
28
V.N. Sazonov, I.E. Khrornov/Statlstics o f an ensemble o f oscillators under excztation. I V
2.4. Our calculations are valid only i f ~ ' i s so large t h a t rvv >> 1 ( v = ] . . . . . Vd) and rwt, >> 1 ( o = o e . . . . . v d -- 1). Thus the ensemble has no stationary state, b u t has a statistical distribution f u n c t i o n , whose tmae variation is c o m p a r a t i v e l y slow. PO ~ e - t / r
P l = e - t i t wO + eOVl -- l / z •
w 0 + l,, l
Po = e-rl'~{ °~I1 "i + e,"i+l z=O
Pv "~0
l"dz + i)z+l
1
T
]
¢v_l+Vv
,
+
=
I-]
Wi+Vz+lK=t
. . . . .
W K + V K + 1 /11
(o = 2 .....
od -
1),
(V>lVd).
We neglect e o everywhere• ff possible. In this a p p r o x i m a t i o n P0 (t = 0) = 1.
3. Discussion The relaxation rate v u equals v o = T~V)r(V), where rxrr(O ) is t h e V - - T relaxation time o f resonant molecules in level o in the buffer gas. F o r molecules w~thm the laser b e a m the d:ssociatton p r o b a b i l i t y f d u n n g the laser pulse time te is equal to f = t J r = [te/ZVT(Ve)] e x p [ - - ( E c -- E o ) / T ] ,
(12)
see eq. (11). Expression (12) is valid i f the relative p o p u l a t i o n o f the q u a s i c o n t i n u u m before irradiation is m u c h smaller than f In our ease e x c i t a t i o n b y radiation in the q u a s i e o n t i n u u m is large enough to efficiently excite the resonant molecules, t h o u g h these molecules heat the buffer gas in the laser b e a m due to V - - T relaxation [see c o n d i t i o n ( 2 ) ] . The b u f f e r gas is n o t needed for e x c i t a t i o n b y radiation, b u t is n e e d e d for dissociation because initially there is only a very small a m o u n t o f molecules in the q u a s i c o n t i n u u m , which can dissociate wathout b u f f e r gas. O t h e r molecules can dtssoeiate o n l y due to kinetic e x c i t a t i o n into the d e p l e t e d q u a s i c o n t i n u u m ; this e x c i t a t i o n , caused b y the buffer gas, deereases the energy o f the latter. Molecules in the q u a s i c o n t i n u u m absorb laser energy and transmit part o f this energy to the b u f f e r gas because o f V - - T relaxation. I f the laser radiation intensity is large enough (see below), the l a t t e r process is m o r e effective than the former, i.e. the buffer gas is h e a t e d within the laser b e a m . F o r subsequent estimations we p u t T = 500 K (see ref. [16]). This value can be achieved also b y simple thermal heating o f t h e cuvette containing the gas m i x t u r e . The quasicontinuum o f the molecule C18H12 is at an energy E > ~ E q c = (1--2) X 103 cm - 1 [17], where the state density increases rapidly. F o r subsequent estimations we p u t E c = Eqc = 2000 e r a - I . The V - - T relaxation time rVT for e x c i t e d molecules can be e s t i m a t e d as 3 X 10 - 8 s, when the pressure o f the b u f f e r gas is 300 Torr. N o w , if t~ = 10 - 7 s, then according t o eq. (12) we g ~ t f ~ 9 X 10 - 3 . Since f v a l u e s o f ~ 1 0 - 3 are detectable• it should be possible to observe this effect e x p e r i m e n t a l l y . Our simple m o d e l ignores degeneracy o f v i b r a u o n a l levels. It is possible to show that this degeneracy will considerably increase the rate o f KEQ. I n d e e d , let lo be equal to ]o = w o - 1 [t7o-1 - ( g o - 1 / g o ) Po] -- z'o [Po -- e o - l ( g o / g v - 1 ) P v - 1 ] • where go is t h e effective ztatistical weight o f level v. In this ease we get the faetor guclgO in the rhg o f expression (12). It ~s difficult t o obtain a reliable estimate o f this factor; b u t it is clear t h a t it m u s t b e m u c h larger than u n i t y . The c o n d i t i o n (2) o f strong excitation in the q u a s i c o n t i n u u m can be r e w r i t t e n as Pexc > Pth- T h e e x c i t a t i o n p o w e r Pex e is equal t o P e x e =1o. I f / = 109 W]cm 2 and o ( E > ~ E ¢ ) = 2 X 10 - 2 o c m 2, t h e n Pex e = 1012 c m - 1 / s (we
V.N. Sazonov, I.E. Khromov/Statisrics o f an ensemble o f oscdlarors under e_xeltatzon I V
29
m e a s u r e e n e r g y in u n i t s o f c m - 1 ) . T h e t h e r m a h z a t i o n p o w e r Pth ~s e q u a l to Pflt = A E 7 - 1 w h e r e A E is t h e m e a n e n e r g y passing i n t o t r a n s l a t i o n a l degrees o f f r e e d o m p e r collision, r is t h e t i m e b e t w e e n t h e s e collisions. A c c o r d m g to ref. [ 18] for s t r o n g l y e x c i t e d p o l y a t o m i c m o l e c u l e s A E rs n o m o r e t h a n 3 k c a l / n l o l ; let A E = 2 k c a l / m o l = 7 0 0 c m - 1 . T h e t i m e ~- rs n o less t h a n 3 X 10 - 9 s, t h e n Pill ~ 2 X 1011 c m - 1 / s . S i n c e Pexc > Pflt- the c o n d l t t o n (2) is satisfied T h e specific f e a t u r e s o f K E Q m a y be revealed in t h e f o l l o w i n g w a y . Let us carry o u t a sertes o f e x p e r i m e n t s c l l a n g m g the c h e n d c a l c o m p o s i t i o n o f the b u f f e r gas b u t m a i n t a i n i n g t h e b u f f e r gas pressure and k e e p i n g all o t h e r e x p e r i m e n t a l c o n d i t i o n s f i x e d . In g o i n g f r o m h e a v y b u f f e r gases to light o n e s ( f o r e x a m p l e , m a set Xe, Kr, Ar, N e , He) the t i m e ~'VT is usually d e c r e a s i n g [ 1 9 , 2 0 ] . A c c o r d i n g to t h e K E Q m o d e l [see (12)] x~e e x p e c t an increase o f the d~ssoeiatton rate. T h i s increase c a n n o t be e x p l a i n e d b y r o t a t i o n a l r e l a x a t i o n b e c a u s e the r e l a x a u o n t i m e o f t h e p o p u l a t l o n in r o t a t i o n a l sublevels usually increases o n g o m g f r o m h e a v y to h g h t b u f f e r gases [ 1 9 , 2 0 ] .
References
[ 11 v N. Sazonov. ZhETI- 79 (1980) 39 (English transl So~iet Phys JETP 52 (1980) 19] [2] W. Fuss and K L Kompa, The hnportance of Spectroscopy for Infrared Muir(photon Excitation preprint No 30. MaxPlanek Ge~ellschaft, Garehing (1980) [3l tt.A. Kramers, Phystea 7 (1940) 284. [4] W. Brenlg, H Muller and R. Sedlmeier, Phys Letters S4A (1975) 109 [5] R J D o n n e l y a n d P H Roberts, Proe. Roy. Soc 312A (1969) 519. [6] R Landauer a n d J S~anson, Ph3s Rev. 121 (1961)1668. [7] M Mangel, J. Chem. Phys 72 (1980) 6 6 0 6 . 7 4 ( 1 9 8 1 ) 3 6 3 5 . [ 8] g Reshetnyak. g M Kharehev and L.A. Shelepm, Teor. Mat. l'iz. 49 (1981 ) 13 I. 19l v N. Sazonov and S V. Zatsepin, Chem Phys 52 (1980) 305 [ 10] V N Sazonov and A A. Stuchebrukhov, Chem. Phys 56 (1981) 391. [ 11] V.N. Sazonov. A.A. Stuchebrukhov and S V. Zatsepin, Chem. Phx s 69 (1982) 459 [ 121 M. Tamlr and R.D. Levine, Chem Phys Letters 46 (1977) 208 [13] W Fuss, Chem. Phys 36 (1979) 135. [ 14] E R Grant, M.L Coggiola, Y.T. Lee, P A. Schulz, A S Sudbo and ~ R Shen, Chem Phx ~ Letters 52 (1977) 595 [ 15] V.S. Letokhov and A.A. Makarov, Usp. riz. Nauk 134 (1981) 45 [ 16| R.N. Zitter. D.V. Koster, A. Cantom and A Ringx~elski, Chem. Ph~s 57 {1981) 11. [ 17] A. Amlrav. U. Even and J Jortner, Opt Commun. 32 (1980) 266. [ 18] M Quak and J Troe, m. Gas kmeucs and energx transfer, Vol 2, eds P G A,hmore and R.J. Donox~an {Chem. Soc . London 1977) [ 191 E E Nlkitin. Teoriya elemenrarnyikh atomnomolekulyarn3 ikh protsessox ~ gazakh (Klfimi.x a. Xlo-~cox~. 1970) [20] B F. Gordiets, A I O~ipov and L A Shelepin, Kineticheskie protsessyi ; gazakh i mol~kuly arn.x e lazed e (Nauka Moscoxx, 1980)
ON THE STATISTICS OF AN ENSEMBLE OF OSCILLATORS UNDER EXCITATION. IV. EVALUATION OF THE TIME FOR KINETIC EXCITATION INTO THE QUASICONTINUUM BASED ON KRAMERS" THEORY V . N . S A Z O N O V a n d I.E. K H R O M O V Department o f Theoretical Physics. P N. Lebedev PhyMcal Institute, Leninsk) Prospect 53, 3losco*~ I 1 7924, L'SSR Received 6 Jul~r 1982, in final i c r m 2 December 1982
We consider the model with kineuc excltatton into tile quasicontanuum (KEQ) t'or resonant p o l y a t o m i c molecules which a b s o r b laser radmtion and are s u r r o u n d e d by b u f f e r molecules KEQ takes place w h e n the resonant molecules m the loxser part o f the energ3 s p e c t r u m interact weakly with the laser r a d m n o n , b u t the molecules m the q u a s i c o n t i n u u m are rapidly excited to still higher energy and dissociate. Under these conditions the colhsions ol the r e s o n a n t and buffer molecules lead to excitation o f resonant molecules i n t o the q u a s i c o n t i n u u m because the p o p u l a t i o n o f the quasicontinuunl is mucla less Illdn its llleImod)heroical equilibrium value It is found~ xiaal the smaller ihe V - - T relaxation time u'y-1-, the larger the rate o f KEQ and the dissociation rate (if onl~ ~'VT is n o t too small) T h u s if we change the e x p e r i m e m a l c a n d y n o n s and decrease r V T (tar instance, by passing from the heavy buffer gas Xe to tile hght buffer gas tte). for some resonant molecules we may observe that the probability o f dlssocmtion increases
1. I n t r o d u c t i o n a n d d e s c r i p t i o n o f p h y s i c a l s i t u a t i o n 1.1. I n t h i s p a p e r w e c o n s i d e r t h e case w h e n p o l y a t o m m m o l e c u l e s w i t h w b r a t t o n a l energ~ E a b o v e s o m e t h r e s h o l d e n e r g y E c are r a p i d l y e x c i t e d b y l a s e r r a d i a t i o n t o still h i g h e r e n e r ~ , . S u c h a ph2¢sical s i t u a t i o n arises w h e n t h e a b s o r p t i o n c r o s s s e c t i o n o i n c r e a s e s w i t h i n c r e a s i n g w b r a t i o n a l q u a n t u m n u m b e r ( o r e n e r g y E ) L e t t h e laser freq u e n c y co t b e s m a l l e r t h a n co01, t h e f r e q u e n c y o f t h e m a x i m u m o f t h e h n e a r a b s o r p n o n s p e c t r u m I f co01 -- co t = ( 1 0 - 3 0 ) c m - 1 . t h e n co e is set o u t s i d e t h e v i b r a t i o n a l - r o t a t i o n a l b a n d . In t l u s case t'or s m a l l v t h e c r o s s s e c n o n ot,(coe) is v e r y s m a l l t o o ( n m c h s m a l l e r t h a n 1 0 - 1 9 _ 10 - 2 0 c n t 2 ) . W h e n o i n c r e a s e s , t h e t r a n s i t x o n f r e q u e n c y ¢°o.v+ 1 d e c r e a s e s d u e t o n o n - h n e a r l t y , a n d t h e b a n d o f r e s o n a n t f r e q u e n c i e s m o v e s t o t h e l e f t . S o . f o r h i g h e r levels, one must e x p e c t that o o increases up to 1 0 - 1 7 - 1 0 - 1 8 cm z b o t h for two-atomic and p o l y a t o m i c molecules. B u t f o r t w o - a t o m i c m o l e c u l e s o o a g a i n b e c o m e s v e r y s m a l l , ff o is large e n o u g h a n d coo.v+ 1 -- co~ < - - ( 1 0 - - 3 0 ) c m - 1 . F o r p o l y a t o m i e m o l e c u l e s w h e n o i n c r e a s e s w e get i n t o t h e q u a s i c o n t m u u m , a n d t h e c r o s s s e c t i o n o o c o u l d n o t b e c o m e v e r y s m a l l . P r o b a b l y 0 o d e c r e a s e s in t h e q u a s i c o n t i n u u m t o s o m e e x t e n t (see fig. 1). b u t t y p i c a l v a l u e s o f o o i n t h e q u a s i c o n t i n u u m are 1 0 - 1 9 - 1 0 - 2 0 c m 2, i.e. m u c h l a r g e r t h a n 0 u f o r s m a l l o. R e l a x a t m n c a u s e d b y c o l h s i o n s o f r e s o n a n t m o l e c u l e s w l t h b u f f e r m o l e c u l e s in a gas m i x t u r e , h i n d e r s t h e exci110-~
0
"~
0301-0104/83/0000-0000/$
l~
"~
03.00 © 1983 North-Holland
Fig 1 Dots schemaucall.x shot, tile relaxation rate v v. The dashed line schematicalb show s tile exaltation rate **v = °v J, x~here J is the p h o t o n flux for a Its o-atomic molecule or for an individual ~ibrationai mode in a pol.x atomic molecule. The solid line schematicall~ shox~ s i,t, for a pol.x atomic molecule x, ith quasicontinuum at E > Eqc. Sohd and dashed lines coincide for E < Eqc.
V.N. ~azono),, LE. Khromov/Stattstics o f an ensemble o f o%cdlator% under excitation I V
26
tation caused b y laser radiation. We assume that the relaxation rate uo is larger than the excitation rate w o f f o < oe (E < Ec) , and t,u < w o ff o > oc (E > Ec). Generally speaking, the threshold o f the q u a s i c o n t i n u u m Eqc differs from E~ But E e depends on the laser frequency we, the intensity I, and the buffer gas pressure p. We imply that u n d e r the g~ven conditions E c ~ Eqc and treat the quas~continuum as a resonant part o f the energy spectrum 1.2. Under the action o f radiation, the molecules in the q u a s i c o n t i n u u m are rapidly excited and dissocmte or enter some other chemical reactions [ 1,2]. Because o f this, the q u a s i c o n t i n u u m is quickly depleted. The molecules in the lower part o f the energy spectrum interact weakly w~th the radmtion These molecules are excited into the quas~continuum kinetlcally, i.e. because o f collisions with buffer molecules. Owmg to these colhsions the population o f the q u a s i c o n t m u u m tries to increase to its t h e r m o d y n a m i c a l e q u d l b r m m value. In the present paper we consider the mare features o f this kinetic excitation into the q u a s i c o n t m u u m (KEQ). KEQ has n o t been m e n t i o n e d previously in the literature, though the physical situation is absolutely realistic KEQ must take place for m a n y polyatomic molecules u n d e r proper conditions, i.e. ~01 -- w~ = (10--30) cm - 1 , I >~ 109 W/era 2, p ~ 300 Torr (see section 3 below). 1.3. To deduce the ttrne o f KEQ, we use a theoretical scheme which is sinular to Kramers" theory of chemical kinetics [3] (see, for example, s o m e r e c e n t papers [ 4 - 8 ] ) . We make an a t t e m p t to use Kramers' ideas m the case o f a discrete variable q u a n t u m numbfir o We could n o t find any a t t e m p t o f this kind m the hterature.
2. Model and mathematics
2.1. To clear up the mare features o f KEQ we consider a sLrnple model, namely an ensemble of q u a n t u m nonlinear one-dimensional oscillators; the oscdlators are placed m a heat bath and are non-coherently excited b y laser radiation (in previous papers [9--11 ] we have considered coherent excitation). Let Pv be the p o p u l a t i o n o f level o. The dynamics o f p o is defined b y the well-known rate equations (o = 0, I, ...) IJo = - w o ( P o - Po+ l ) + Vo+l(Po+l -- e v P , ) + W o - l ( P , _ 1 - P , ) -- uo(p,-- % _ l P v - 1 ) - - ~[vPv ,
(1)
where w v = J o o is the rate o f excitation from o to o + 1 , J =l/hcoe is the p h o t o n f l u x ; vo Is the rate of V--T r e l a x a t i o n from v to v -- l , e o = exp [--(Et,+l - Ev)]T], T i s the temperature o f the buffer gas, which is the heat bath, ~'o is the rate o f dissociation from level o. In eqs. ( 1 ) , p , = )vz = 0 l f i = - 1 , vi = 0 lfz = 0. If at t = 0 we have Y'oPv = 1, t h e n f ( t ) = 1 - Z o P o ( O Is the probability o f dissociation at time t T o describe in our model the physical sltuatlon, we assume ~[v = 0
if o < od ,
ov = 0
If o < o c ,
w v = J ° o >~ ~'v+ 1
ff o >/v c .
(2)
The dissociation threshold od exceeds oc. Rate equations were successfully applied to the descripuon o f excitation and relaxation o f p o l y a t o m l c molecules b o t h at lower levels [ 12,13] and in q u a s i c o n t i n u u m [14] (see also references m r e f . [15]). 2.2. Let the action o f the laser radiation begin at t = 0, i.e. at t < 0 we have J = w o = 70 = 0 and Pv obeys the Gibbs d i s t r i b u t i o n Pu = A exp ( - E o / T ) , where A is a constant. If eo '< 1 (v = 0, 1,2 .... ), then almost all the population is in ground level o = 0. At t > 0 we assume the dissociation rates 7t, to be very large 0 f o >/Od, then 70 ~ wo, vo) and c o n s e q u e n t l y we p u t Po ~ 0 if o/> 0 d. There ts n o stationary solution o f (1) with this condition The population o f the lower levels will be transferred to the q u a s i c o n t i n u u m and to the dissociation region o > / o d. We want to esttmate the decrease ttme ~- o f the p o p u l a t i o n P0 in the ground level.
V.N. Sazonov, LE. Khromov[Statistics o f an ensemble o f oscillators under exctmtion I V 2.3.
27
We rewrite e q . ( I ) for v < v d in t h e f o l l o w i n g f o r m
Po =
-/~+t
(3)
+/o,
where ]o Is equal to iv = B ' v - 1 0 9 o - I -- P v ) - - P v ( P v - - e v - l P v -
(4)
l) "
],, must be treated as a current According to (4) we have
IV Pv- I
Wv-- l + IJv
Vu
W o - I -I- eo_ l
Wv-
Pu = bv + auPv -
l + e o - l Vv
A f t e r K -- I r e c u r r e n c e s w e get
Pv-K = bv-K+l
N o w we p u t v
P0
=
h
x-,
Vd
Vlff0
~-I v ~ az + P v ~ a, . i=v-K+t i=v-K+l
]Q
~-
1
~Vc
1
Wt _
17I
- + z_J
. . . . . vc -
]I
b~
(5)
0 , K = v d and eq. (5) gives
=
l fw~=0(i=0.1
P0
£=v-K+2
= Od,Pvd
w0+e0v
=--
+
~ = 2 w £ _ 1 + v l t e ~ _ 1 z=t
1
wi_ 1 +v/e/_ 1
l) andiv i>~v~+l(i=v c ..... vd]£
+ ~__25
°d
=_ ~lte0ffl...eQ_ l
+
+ Vt
(6)
l).then
]e
E
Q=uc+l W ~ _ l
(7)
A c c o r d i n g to Kramers" w o r k [3] b o t h m t h e case o f c o n t m u o u s and discrete variables the p o p u l a t i o n f l o w d e p e n d s w e a k l y o n t h e variable o u t s i d e t h e b o t t o m o f a p o t e n t i a l well, w h e r e a h n o s t all the p o p u l a n o n is s i t u a t e d ( m o u r case levels w~th v = 0, 1 . . . . . v~ - 1 m a y be t r e a t e d as t h e p o t e n n a ! well. and v = 0 is the b o t t o m o f the well) I n d e e d , t h e r e is n o source for c u r r e n t af t h e p o p u l a t m n is small, g o : - e c a n p u t in eq_ (7) i e ~ ] (~ = 1 . 2 . . . . ). I f t h e relaxan o n rate % increases w i t h v so rapidly that vi e l - 1 >~ v~_ 1 •
(8)
then a c c o r d i n g to ( 7 ) P O ~']/Vl CO. I f the r e l a x a t i o n rate % increases n o t ve W rapidb" and v~ez_t < v z - t
,
(9)
then ]
P0 ~
VVceO e l ---euc - l
] exp [(E c --
=
Eo)IT ]
vv c
When v = 0, eq. (3) gives/J0 = - ] I = - ] - N o w in the case (8) we have ~" = --Po/tSO = ( I / V l ) e x p [ ( E 1 - E o ) / T ] .
(10)
In the case (9) we have "r = -Po]l~ 0 = ( l / w e ) e x p [(Eve - E o ) / T ] .
(1 1)
We b e h e v e t h e case (9), ( 1 1 ) to b e m o r e reahstic and c o n s i d e r this case b e l o w . E x p r e s s i o n (1 1) c a n b e o b t a i n e d b y simpler c o n s i d e r a t i o n s . But t h e range o f v a h d i t y o f (1 1) can o n l y b e o b t a i n e d b y e x a c t c a l c u l a t i o n s [ c o m p a r e (8), ( 1 0 ) w l t h (9), ( 1 1 ) ] .
28
V.N. Sazonov, I.E. Khrornov/Statlstics o f an ensemble o f oscillators under excztation. I V
2.4. Our calculations are valid only i f ~ ' i s so large t h a t rvv >> 1 ( v = ] . . . . . Vd) and rwt, >> 1 ( o = o e . . . . . v d -- 1). Thus the ensemble has no stationary state, b u t has a statistical distribution f u n c t i o n , whose tmae variation is c o m p a r a t i v e l y slow. PO ~ e - t / r
P l = e - t i t wO + eOVl -- l / z •
w 0 + l,, l
Po = e-rl'~{ °~I1 "i + e,"i+l z=O
Pv "~0
l"dz + i)z+l
1
T
]
¢v_l+Vv
,
+
=
I-]
Wi+Vz+lK=t
. . . . .
W K + V K + 1 /11
(o = 2 .....
od -
1),
(V>lVd).
We neglect e o everywhere• ff possible. In this a p p r o x i m a t i o n P0 (t = 0) = 1.
3. Discussion The relaxation rate v u equals v o = T~V)r(V), where rxrr(O ) is t h e V - - T relaxation time o f resonant molecules in level o in the buffer gas. F o r molecules w~thm the laser b e a m the d:ssociatton p r o b a b i l i t y f d u n n g the laser pulse time te is equal to f = t J r = [te/ZVT(Ve)] e x p [ - - ( E c -- E o ) / T ] ,
(12)
see eq. (11). Expression (12) is valid i f the relative p o p u l a t i o n o f the q u a s i c o n t i n u u m before irradiation is m u c h smaller than f In our ease e x c i t a t i o n b y radiation in the q u a s i e o n t i n u u m is large enough to efficiently excite the resonant molecules, t h o u g h these molecules heat the buffer gas in the laser b e a m due to V - - T relaxation [see c o n d i t i o n ( 2 ) ] . The b u f f e r gas is n o t needed for e x c i t a t i o n b y radiation, b u t is n e e d e d for dissociation because initially there is only a very small a m o u n t o f molecules in the q u a s i c o n t i n u u m , which can dissociate wathout b u f f e r gas. O t h e r molecules can dtssoeiate o n l y due to kinetic e x c i t a t i o n into the d e p l e t e d q u a s i c o n t i n u u m ; this e x c i t a t i o n , caused b y the buffer gas, deereases the energy o f the latter. Molecules in the q u a s i c o n t i n u u m absorb laser energy and transmit part o f this energy to the b u f f e r gas because o f V - - T relaxation. I f the laser radiation intensity is large enough (see below), the l a t t e r process is m o r e effective than the former, i.e. the buffer gas is h e a t e d within the laser b e a m . F o r subsequent estimations we p u t T = 500 K (see ref. [16]). This value can be achieved also b y simple thermal heating o f t h e cuvette containing the gas m i x t u r e . The quasicontinuum o f the molecule C18H12 is at an energy E > ~ E q c = (1--2) X 103 cm - 1 [17], where the state density increases rapidly. F o r subsequent estimations we p u t E c = Eqc = 2000 e r a - I . The V - - T relaxation time rVT for e x c i t e d molecules can be e s t i m a t e d as 3 X 10 - 8 s, when the pressure o f the b u f f e r gas is 300 Torr. N o w , if t~ = 10 - 7 s, then according t o eq. (12) we g ~ t f ~ 9 X 10 - 3 . Since f v a l u e s o f ~ 1 0 - 3 are detectable• it should be possible to observe this effect e x p e r i m e n t a l l y . Our simple m o d e l ignores degeneracy o f v i b r a u o n a l levels. It is possible to show that this degeneracy will considerably increase the rate o f KEQ. I n d e e d , let lo be equal to ]o = w o - 1 [t7o-1 - ( g o - 1 / g o ) Po] -- z'o [Po -- e o - l ( g o / g v - 1 ) P v - 1 ] • where go is t h e effective ztatistical weight o f level v. In this ease we get the faetor guclgO in the rhg o f expression (12). It ~s difficult t o obtain a reliable estimate o f this factor; b u t it is clear t h a t it m u s t b e m u c h larger than u n i t y . The c o n d i t i o n (2) o f strong excitation in the q u a s i c o n t i n u u m can be r e w r i t t e n as Pexc > Pth- T h e e x c i t a t i o n p o w e r Pex e is equal t o P e x e =1o. I f / = 109 W]cm 2 and o ( E > ~ E ¢ ) = 2 X 10 - 2 o c m 2, t h e n Pex e = 1012 c m - 1 / s (we
V.N. Sazonov, I.E. Khromov/Statisrics o f an ensemble o f oscdlarors under e_xeltatzon I V
29
m e a s u r e e n e r g y in u n i t s o f c m - 1 ) . T h e t h e r m a h z a t i o n p o w e r Pth ~s e q u a l to Pflt = A E 7 - 1 w h e r e A E is t h e m e a n e n e r g y passing i n t o t r a n s l a t i o n a l degrees o f f r e e d o m p e r collision, r is t h e t i m e b e t w e e n t h e s e collisions. A c c o r d m g to ref. [ 18] for s t r o n g l y e x c i t e d p o l y a t o m i c m o l e c u l e s A E rs n o m o r e t h a n 3 k c a l / n l o l ; let A E = 2 k c a l / m o l = 7 0 0 c m - 1 . T h e t i m e ~- rs n o less t h a n 3 X 10 - 9 s, t h e n Pill ~ 2 X 1011 c m - 1 / s . S i n c e Pexc > Pflt- the c o n d l t t o n (2) is satisfied T h e specific f e a t u r e s o f K E Q m a y be revealed in t h e f o l l o w i n g w a y . Let us carry o u t a sertes o f e x p e r i m e n t s c l l a n g m g the c h e n d c a l c o m p o s i t i o n o f the b u f f e r gas b u t m a i n t a i n i n g t h e b u f f e r gas pressure and k e e p i n g all o t h e r e x p e r i m e n t a l c o n d i t i o n s f i x e d . In g o i n g f r o m h e a v y b u f f e r gases to light o n e s ( f o r e x a m p l e , m a set Xe, Kr, Ar, N e , He) the t i m e ~'VT is usually d e c r e a s i n g [ 1 9 , 2 0 ] . A c c o r d i n g to t h e K E Q m o d e l [see (12)] x~e e x p e c t an increase o f the d~ssoeiatton rate. T h i s increase c a n n o t be e x p l a i n e d b y r o t a t i o n a l r e l a x a t i o n b e c a u s e the r e l a x a u o n t i m e o f t h e p o p u l a t l o n in r o t a t i o n a l sublevels usually increases o n g o m g f r o m h e a v y to h g h t b u f f e r gases [ 1 9 , 2 0 ] .
References
[ 11 v N. Sazonov. ZhETI- 79 (1980) 39 (English transl So~iet Phys JETP 52 (1980) 19] [2] W. Fuss and K L Kompa, The hnportance of Spectroscopy for Infrared Muir(photon Excitation preprint No 30. MaxPlanek Ge~ellschaft, Garehing (1980) [3l tt.A. Kramers, Phystea 7 (1940) 284. [4] W. Brenlg, H Muller and R. Sedlmeier, Phys Letters S4A (1975) 109 [5] R J D o n n e l y a n d P H Roberts, Proe. Roy. Soc 312A (1969) 519. [6] R Landauer a n d J S~anson, Ph3s Rev. 121 (1961)1668. [7] M Mangel, J. Chem. Phys 72 (1980) 6 6 0 6 . 7 4 ( 1 9 8 1 ) 3 6 3 5 . [ 8] g Reshetnyak. g M Kharehev and L.A. Shelepm, Teor. Mat. l'iz. 49 (1981 ) 13 I. 19l v N. Sazonov and S V. Zatsepin, Chem Phys 52 (1980) 305 [ 10] V N Sazonov and A A. Stuchebrukhov, Chem. Phys 56 (1981) 391. [ 11] V.N. Sazonov. A.A. Stuchebrukhov and S V. Zatsepin, Chem. Phx s 69 (1982) 459 [ 121 M. Tamlr and R.D. Levine, Chem Phys Letters 46 (1977) 208 [13] W Fuss, Chem. Phys 36 (1979) 135. [ 14] E R Grant, M.L Coggiola, Y.T. Lee, P A. Schulz, A S Sudbo and ~ R Shen, Chem Phx ~ Letters 52 (1977) 595 [ 15] V.S. Letokhov and A.A. Makarov, Usp. riz. Nauk 134 (1981) 45 [ 16| R.N. Zitter. D.V. Koster, A. Cantom and A Ringx~elski, Chem. Ph~s 57 {1981) 11. [ 17] A. Amlrav. U. Even and J Jortner, Opt Commun. 32 (1980) 266. [ 18] M Quak and J Troe, m. Gas kmeucs and energx transfer, Vol 2, eds P G A,hmore and R.J. Donox~an {Chem. Soc . London 1977) [ 191 E E Nlkitin. Teoriya elemenrarnyikh atomnomolekulyarn3 ikh protsessox ~ gazakh (Klfimi.x a. Xlo-~cox~. 1970) [20] B F. Gordiets, A I O~ipov and L A Shelepin, Kineticheskie protsessyi ; gazakh i mol~kuly arn.x e lazed e (Nauka Moscoxx, 1980)