Excitation of oscillations in gas by light-induced drift

29 August 1994 PHYSICS LETTERS A

ELSEVIER

Physics Letters A 192 (1994) 22-26

Excitation of oscillations in gas by light-induced drift F.Kh. Gel'muldaanov a, G. Nienhuis b, T.I. Privalov a a Institute of Automation and Electrometry, 630090 Novosibirsk, Russian Federation b Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

Received 5 November 1993; revised manuscript received 6 June 1994; accepted for publication 22 June 1994 Communicated by A.P. Fordy

Abstract

It is shown that the anomalous temperature dependence of the drift velocity of light-induced drift (LID) causes oscillations of the temperature and concentration of the absorbing gas.

The phenomenon of light-induced drift ( L I D ) [ 1 ] is one of the strongest manifestations of the interaction of light and matter. LID can occur when optically active atoms or molecules immersed in a buffer gas are excited in a velocity-selective fashion, so that a flux of excited-state particles and an opposite flux of ground-state particles will arise. These two fluxes are exposed to different resistances, and do not cancel, due to different collision cross sections with buffer-gas particles. The result is a net flux or lightinduced drift of absorbing particles [ 1 ]. By now the LID effect has been well enough studied both theoretically and experimentally [ 2,3 ]. By numerical investigations of LID in alkali-noble gas mixtures the effect of anomalous temperature dependence of the drift velocity was predicted [ 4 ]. It means that the drift velocity U changes its sign in some area near a so-called critical temperature To, U=ot(T-To),

ot=(OU/OT)r=ro.

(1)

Recently the existence of the critical temperature To was proved in experiments on LID of molecular gases

[51. In this paper we discuss a new and nontrivial effect in L I D , caused by the anomalous temperature depenElsevierScienceB.V. SSDI0375-9601 (94)00522-Q

dence of the drift velocity ( 1 ). We will show that in some area near the critical temperature LID can lead to oscillations of the temperature and concentration of the absorbing component of the gas mixture. The qualitative picture of the oscillations is the following. Let us consider the mixture of buffer and absorbing gases in the field of resonant radiation. We suppose also that the initial temperature of the gas mixture is equal to the critical temperature To. So, due to Eq. ( I ), the drift is absent ( U= 0) and the concentration of the absorbing component is equal to equilibrium value no. However, in some conditions, the equilibrium state of the gas is unstable. Let us imagine a small perturbation of the gas near the left end of the cell, 6n (x) = n ( x ) - no (Fig. 1 ), where x is the longitudinal coordinate in the cell. The signchanging character of 6n(x) is caused by the particle conservation law in the closed cell. The temperature of the region A (Fig. 1 ) increases due to photoabsorption and dissipation of the energy of optical excitation into heat. In its turn, the region B is cooled off. So the drift velocity is now nonzero, precisely U> 0 in the region A and U< 0 in the region B, if a > 0. The characteristic time of heat exchange between the gas and surroundings is tex and the charac-

F.Kh. Gel'mukhanov et al. / Physics Letters A 192 (1994) 22-26

x

r-r,

than the radius of its cross section. The non-linear equations of the LID theory for concentration of absorbing gas n, the gas temperature T, the cell temperature Tc and the intensity of radiation I (in a linear approximation in I) are the following, On O 02n 0--7 + °t ~x [ ( T - T o ) n ] = D ~ - ~ ,

OT

0 Fig. 1. Qualitativepicture of the origin of oscillations. teristic time of LID is t~ = l / U ~- D~ U 2. Here the l is a typical length of LID (l~-D~ U), and D is the coefficient of diffusion. It is evident that in the case when tex << t~ the "driftless state" of the gas is stable. But when the heat exchange is not so fast and te~ >-tu , because of the rapid movement (with the drift velocity) of the temperature nonhomogeneity along the cell, the gas's temperature has no time to relax to the equilibrium one To. In other words, in the case te~>~tu the temperature distribution is retarded with respect to the distribution of the concentration. Thus the region A, where U> 0, will drift to the fight with some deformation but without fading, in its turn the region B will drift to the left (Fig. 1, dashed curve). The drift of particles from the region B into the left direction will cause the accumulation of absorbing gas near the left end of the cell. Because of this the positive part will appear on the curve 8n(x,t) near the left end of the cell, hence, the drift velocity will be positive ( U > 0) in that region. So the new region with increased density will appear near the left end of the cell and it will drift to the right. The qualitative arguments discussed above show the origin of oscillations of the concentration and temperature of the absorbing gas in a finite cell. Now let us consider the non-linear dynamics of LID, taking into account the anomalous temperature dependence of the drift velocity ( 1 ) and the heat exchange between the gas and its surroundings. It is possible to use a one-dimensional description of the problem, when the length of the cell L is much larger

23

O2T

(2a)

p c - ~ = x ~ - ~ - b ( T - Tc) + naKI,

(2b)

OTc 02Tc pccc-~-=~T-~-b'(~c-Ta)-b(r~-T),

(2c)

OI = -ntrl. OX

(2d)

Here p, c, x and Pc, co, xc are the mass density, specific heat capacity and heat conductivity of the gas and the material of the cell, respectively; b and b' are the coefficients of heat exchange between the gas mixture and the cell wall, the cell walls and the surroundings (with temperature Ta), respectively; tr is the cross section of photoabsorption. It is necessary to use some coefficient K in the last term of Eq. (2b), because only a Kth part of the absorbed energy of the radiation can be converted into heat in the gas. Eq. (2b) can be simplified, taking into account the smallness of the density and heat conductivity of the gas in comparison with those of the cell. So we may substitute Eq. (2b) by the following: T= Tc + ntrKI/ b. For more simplicity let us make two additional approximations. ( 1 ) We analyze only the case of an optically thin gas (Lano << 1 ). (2) Also we consider only the case when the heat exchange between the cell walls and the surroundings is negligible in comparison with that between the walls of the cell and the inner gas, eb'/b << 1, where the dimensionless parameter E is I2 a ( Io ) no aKtex ¢= Iop¢cc(D/tex)1/2"

(3)

In Eq. (3) we use a linear approximation for the Idependence of the parameter a ( I ) : a = a ( I ) = a ( I o ) l / I o ( I ) , where Io is some fixed value of the intensity of the radiation. It is reasonable to use dimensionless notations,

F.Kh. Gel'rnukhanovet al. ~PhysicsLettersA 192 (1994)22-26

24 z=t/tex,

z=x/(Dtex) 1/2, N = n / n o ,

(4)

O= e t ( T ¢ - T o ) (D/tex)l/2 "

0

NOW Eqs. (2a) and (2c) read ON + 0 OZN ~z Oz (NO)= Oz 2 ,

O0

(5a)

~020 =A ~ z 2 - O + ~N+ O~ ,

(5b)

where A=z/D,

X=x¢/p¢c¢,

Fig. 2. Dependenceof the instabilityincrementO" on the wave numberk.

t~x=p~cdb',

Oa =ot( T a - Tc) / ( bD ) 1/2 .

Eqs. (5a) and (5b) have the steady state space homogeneous solution N=I,

O=Oa+e.

(6)

When we substitute a small deviation from the stationary solution (6) proportional to exp ( - il2z+ i k z ) . into the linearized equations (5a) and (5b) we obtain the dispersion relation between I2 and k, g22+ LQ[ ( l + A)kZ + l ] - k Z ( l + A k 2 ) - i e k = O .

(7)

For real values of the wave number k Eq. (7) has two complex solutions I2=t2'+i,Q", one is stable (12"< 0) and the other is unstable (g2">0). When k<< 1 the unstable solution of Eq. (7) is t2'=ek[l-(l+d)k:l,

(8)

Q"= - e 2 ( 3 / 1 + 2 ) k 2 ( k 2 - k 2) ,

(9)

tionary solution (6) increases at every point in the cell. The absolute instability takes place when the complex solution k(O) of the dispersion equation (7) has a branch point for values oft20 of I2 with positive imaginary part g2~ [ 6 ]. Eq. (7) can be rewritten as A ( k - k l )2(k-k3) ( k - k a ) = 0 ,

( 11)

if two roots of this equation coincide. Here k~ = k2, k3 and k4 are the roots of the dispersion equation. Comparison of Eqs. (7) and ( 11 ) yields a value of g2o in a branch point. The region of absolute instability g2"> 0 is characterized by its border line I2" = 0 in the e-A 2 plane. This line termed the neutral curve can be put in the parametric form x 3 ( A - I )4+ 2x2 (2A_ 5) ( A - 1 )2

+ x [ 5 - 4A + ~2(9 - ~ ( I +/I) 2) ] + ~e2( 1 -

~ d e 2) = 0 ,

x2(A-- 1)3(5--A)

and we introduce a characteristic wave number kc,

+ x [ 2 ( A - 1 ) (3/1-5) + ¼e2(A+ 1 ) ( 1 -34A+A 2) l

~2--1 K~= 3A+2"

- I+~E2(llA- I)=0.

(10)

The stationary solution (6) is unstable when ~2" is positive, so the condition of instability is Ikl
e2>l.

The behavior of g2" as a function of wave number k is presented in Fig. 2. Let us find now the region of absolute instability, which occurs when an initial perturbation of the sta-

The relation between e and d describing the neutral curve is determined after elimination of x = t2 z from this pair of equations. The result in the e2-A plane is sketched in Fig. 3. This is a good upper level estimation for the area of instability in the ez-A plain for a finite, but sufficiently long cell. The approximate criterion of instability tex>/tu is in qualitative agreement with the numerical results of Fig. 3. The neutral curve E2(A) (see Fig. 3 ) determines the threshold Io for the

25

F.Kh. Gel"mukhanov et aLI Physics Letters A 192 (1994) 22-26 8.0

I

I

I

i

s2[A) ,o,

<'>

7.5 II 7.0

I

6.5 0

I

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

9

10

Fig. 3. The neutral curve, (I) region of stability and (II) area of absolute instability.

intensity of radiation/. In the case I > Io the solution (6) is unstable in an infinite cell. The threshold condition on the radiation intensity (I > Io) is not a sufficient condition of instability. For creation of instability in a finite cell its length must be larger than some critical value Lc(¢ 2, Lf) ( L > L c ) . With the help of a numerical solution of the Eqs. (5a), (5b) we obtain a good semi-empirical formula for L¢,

L>L,,

( 3 A + 2 ~ 1/2

L~=6n(Dtex)t/2~-T-~_l]

\--

~

(12)

Unfortunately all attempts to solve the Eqs. (5a), (5b) analytically failed. So we obtained numerical solutions of these equations with various types of boundary conditions. The solution ofEq. (5a), (5b) for boundary conditions O=0, aN/az=O on both ends of the cell are represented in Figs. 4a, 4b. The initial perturbation dN of the space homogeneous solution (6) was proportional to cos[ ( 2 ~ x / L ) Y ] , here X is the number of waves of the perturbation. In the region of instability (Fig. 3 and Eq. (12) ) we have found the oscillations of concentration and temperature of the absorbing gas as a result of the rather complex evolution of the initial perturbation. The space distributions of N and O for different times are shown in Figs. 4a, 4b. In this oscillatory regime the peaks of the concentration of the absorbing component of the gas mixture move from the left end of the cell to the fight one. The phase curve of that solution (Fig. 4c) is closed, and the time dependence is truly one-periodical. Note that the phase curve in Fig. 4c is the limit cycle of the oscillatory solution

N

um(seo) N-I

.

/

(C)

/ Fig. 4. (a) Space distributions of the dimensionless concentration N (4) at different times. (b) Space distribution of the dimensionless temperature O (4) at different times, x is the longitudinal coordinate in the cell, ~2=20, ,4=1, L = 6 0 cm (the dimensionless length of the cell is 13.4). (e) Phase curve of the oscillatory solution of Eq. ( 5 ) in the middle of the cell. The period of oscillations is 6.48 s.

(Figs. 4a, 4b). Similar solutions were obtained with other values of the parameters ¢, A and L in the re# o n of instability (Fig. 3 together with Eq. (12) ). The character of the solutions to a very large extent depends on the length of the cell. The evolution of

26

F.Kh. Gel'mukhanov et aL / Physics Letters A 192 (1994)22-26

solutions in cells with larger lengths is more complicated and we obtained oscillatory solutions multi-periodical in time. In conclusion let us estimate the real parameters for L I D o f Li in a buffer mixture Ne-CEH4. A small admixture of C2H4 is necessary for the collisional transformation o f the absorbed radiation into the heat [ 7 ]. The numerical estimations for the critical temperature in a L i - N e mixture give Tc i> 1000 K. Estimations for L I D in a copper cell are Ic-~ l0 W / c m 2 and L¢--- 10 cm. It is necessary to mention that oscillations were observed in experiments on L I D o f Na vapors in a sapphire capillary [ 3 ]. But the nature o f these oscillations is not quite clear, and we speculate that the interaction o f Na atoms with the surface o f the cell plays a dominant role. The authors thank S.N. Atutov, A.A. Chernykh and

A.M. Shalagin for comments and valuable discussions. The work was supported by the Netherlands Organization for Scientific Research ( N W O ) , the Program Universities o f Russia, the International Science Foundation (ISF) and the Russian Foundation for Fundamental Research.

References [ 1] F.Kh. Gel'mukhanov and A.M. Shalagin, Pis'ma Zh. Eksp. Teor. Fiz. 29 (1979) 773 [JETP Len. 29 (1979) 711 ]. [2] S.G. Rautian and A.M. Shalagin, Kinetic problems of nonlinear spectroscopy(North-Holland, Amsterdam, 1991). [3] H.G.C. Weryand J.P. Woerdam, Phys. Rep. 169 (1988) 145. [4] A.I. Parkhomenko, Opt. Spektrosk. 67 (1989) 26. [ 5 ] G.J. van der Meer, Molecular collision processes studied by light-induced kinetic effects, Ph.D. thesis, Leiden, 1992. [6]E.M. Lifshitz and L.P. Pitaevskii, Physical kinetics (Pergamon, Oxford, 1981). [7] M.C. de Lignie and J.P. Woerdman, J. Phys. B 23 (1990) 417.