depolarization mechanisms in alkali-noble gas half collisions

Z. Phys. D - Atoms, Molecules and Clusters 2, 193-205 (1986) Atoms, Molecules and Clusters ~r Phys~k D

© Springer-Verlag 1986

Depolarization Mechanisms in Alkali-Noble Gas Half Collisions*

F. Schuller Laboratoire des Interactions Mol~culaires et des Hautes Pressions C.N.R.S., Centre Universitaire Paris-Nord, Villetaneuse, France

G. Nienhuis Fysisch Laboratorium, Rijksuniversiteit, Utrecht, The Netherlands

W. Behmenburg Physikalisches Institut I, UniversitM Dtisseldorf, Federal Republic of Germany

Received March 17, 1986

The polarization properties of excited alkali atoms having suffered half collisions with noble gas atoms following excitation by nonresonant photon absorption during a strong collision, are investigated. On the basis of a semiclassical perturbation treatment of the density matrix up to second order in the driving field a general expression for the polarization dependent intensity of the emitted alkali fluorescence is derived. Assuming quasistatic excitation of the collision complex and considering reorientation of the orbital and spin electronic angular momenta within a cut off radius, explicit results for the polarization degrees are presented and discussed, depending on the initial molecular excited state and coupling case.

PACS: 32.70. Jz; 34.25.+t

0. Introduction

The investigation of half collisions, initiated several years ago in connection with photodissociation [1], has recently received increasing attention as a general tool to study collision processes in excited states with spectroscopic methods [2-4]. In principle the experiment consists in preparing a collision complex in one or more molecular excited states by absorption of a nonresonant photon (far wing absorption) from the ground state at small internuclear separation during a strong collision. During the half collision following excitation (dissociation of the complex) mixing between the molecular states takes

* Dedicated to Professor Dr. W. Hanle on the occasion of his 85 birthday

place, and the resulting distribution among the asymptotic levels may be monitored via fluorescence emitted from them. Its characteristics (frequency, intensity, polarization) are thus directly related to the dynamics of the collision.

In the following we will restrict to alkali noble gas systems, since here plenty of potential data are available, both from temperature dependent fluorescence line wings [5] as well as from model potential calculations [6, 7], facilitating a quantitative com- parison with experiment. In these systems the D1/D 2 ratio of the fluorescence intensities is determined by nonadiabatic transitions and change in coupling case; the polarization of the fluorescence, on the other hand, is governed by coupling of the radiating alkali dipole to the internuclear axis.

Recently, optical collision cross sections have

194 F. Schuller et al.: Alkali-Noble Gas Half Collisions

been obtained for the Na-Ar case in the framework of a nonadiabatic theory in a quantum coupled channels (CC) calculation [8]. Fluorescence ratios for the NaD 1 and D 2 lines calculated using poten- tials derived by Diiren [7] reproduced correctly experimental data under single collision conditions [9], in particular the rather different behaviour of the DI/Dz ratios on both sides of the resonance lines. A detailed analysis of the fluorescence polarization following from the same calculation [10] led to qualitative agreement with available experimental data [4] in the near wing regions up to 50 cm-1 detunings from the resonance lines. CC-calculations performed also for other alkali noble gas systems [11] predict in the extreme blue wings a consider- able residual linear polarization of the D z line, its value being independent of the system; in the extreme red wings, on the other hand, the polarization is found to decrease to zero with increasing molecular mass.

The purpose of the paper presented is a quantitative understanding of these features in terms of the different mechanisms contributing to the polarization of the alkali fine structure levels produced in half collisions. For this purpose a semiclassical treatment is chosen which is based on the model of quasistatic excitation into molecular states by far wing absorption and reorientation of the orbital and spin electronic angular momenta L and S during the half collision following excitation. Classical trajectories for nuclear motion are assumed throughout this treatment.

We take as the starting point of our derivation the full expression for the polarization-dependent spectrum of the redistributed light, which is the La- place transform of the dipole correlation function. In the special case of excitation in the quasistatic wing, and at sufficiently low perturber density as to make subsequent depolarizing collisions during the excited-state lifetime negligible, the simple picture arises where the polarization of the lines reflects the distribution over the excited substates as it emerges from the collision that induced the absorption. This simple picture of photon absorption during an atom- perturber collision with depolarization of the atom during the separation of the collision system was taken as the starting point by Lewis et al. [12].

Using a similar model, an early consideration of the case L = I , S=0, i.e. J = l , assuming complete adiabatic reorientation [13] predicted zero J =1 polarization in the far wings of the fluorescence line following broad-band excitation. However, a subsequent more detailed analysis E14] accounting for coherences between degenerate substates of the molecular A # 0 levels led to nonzero residual J = l

polarization for the case of //-excitation. On the contrary excitation of the non degenerate Z state, which cannot support coherences, led again to zero polarization.

In the case of alkali atoms, considered in this paper, particular attention has to be paid to the behaviour of the spin with respect to the interatomic motion during the half-collision.

1. The Excited-State Density Matrix

We consider an optically active atom perturbed by collisions with inert perturber atoms. We define a density matrix p depending on both the internal state of the optical atom and the translational degrees of freedom of the perturber bath. The time evolution of the density matrix is described by the Liouville equation. Introducing the Laplace-transformed quantity.

oo

p(z)= ~f e-~t P(t) dt 0

the Liouville equation takes the form [15]

-p(t =0) + zp + iHp -1- i l~eff p -~/~p ~-0 (1.1)

Here the quantities /1 a n d [~eff are commutators corresponding respectively to the hamiltonian H of the entire system of atom and perturber bath and to Vef f, the interaction potential of the system with an incident monochromatic driving field, f is an operator representing spontaneous decay.

We assume that the atom can be modeled by taking into account the ground state IJiMi) and only one excited multiplet with components IJeMe) the separations between the latter being small com- pared to their distance from the ground state. The interaction hamiltonian with the incident laserfield of frequency COL and polarization e L can then be written as follows: [16]

i i Vef f = --~ PeES-eLEo +~ P~pPe-e~Eo (1.2)

where P/, Pe are projectors defined as

P~=lJiMi)(J~Mit Pe=ILMe)(LMet (1.3)

_E and E 0 are respectively the electronic dipole operator and the amplitude of the exciting field.

The liouvillian /4 can be split into a term /4a depending only on the state of the atom and a sum /4p+/4i containing respectively the translation energy of perturbers and the atom-perturber interaction. /Id, the "dressed-atom" liouvillian [t6], has

F. Schuller et al.: Alkali-Noble Gas HaIf Collisions 195

eigenvalues given by

I~[ elJ~Mi) ( J~M ~[ = A j~IJiM~) ( JeM ~l (1.4)

Iid IJ~M~) ( JiM~l -- - A j~ IJ~M~) ( JiM, I

where Aj=caz-cojo designates the frequency difference between the incident radiation and the atomic transition frequency for the upper level Jr.

For the purpose of our problem of calculating polarization properties of fluorescence radiation we need p up to second order in E o. We thus write the expansion

P =Po + Pl + P~ (1.5)

with

1 Po =P~P~ .......... (1.6) 2J i + 1 =P~a°

where PB is the density matrix of the perturber bath. From Eq. (1.1) we then get for the first-order term p~ the equation

zPl + i H p l + f P l + i geffPo =0 (1.7)

and hence

1 Pl = - i z ~ _ i / ~ + f l~effPo (1.8)

In the following we shall always assume that excitation takes place in the far wings of the lines. In order to separate in (1.8) spontaneous decay from the collision-induced part, we use the operator iden- tity

i 1 1 ~ 1

z + i H + F z + i / ? 4-z+i~il-t-Y~z+iI~+f'' " (1.9)

Now, I~effp 0 couples the excited levels to the ground level. Hence, according to (1.4), the part /ld contained in /~ can be considered as being of the order of A. Thus, by far-wing excitation, the term /~d will dominate in the denominators and we may simplify the second term in (1.9) and write

i ^ . A p i = - i zO+~-~ - I/~ffp o - , ~ - pBo-' (1.10)

where the quantity a' depends only on /4a i.e. on A

atomic states. ~- (with A the Einstein coefficient) is

the relevant matrix element of F. We now discuss the first term on the right of

(1.10) assuming, as stated above, that excitation takes place in the far wings. Furthermore we in-

troduce the binary collision approximation by considering only one perturber at one time. According to the first assumption we may expect that the pair a tom+per turber is in a quasi molecular state 193l~) at the moment of excitation. This state can be con- nected to the atomic states as follows:

~01e

The atom-perturber interaction hamiltonian H i, which now reduces to a one-particle quantity, is related to the binary interatomic potential; the following relations hold:

ni [JgMi) (gJ~,l = - V d [J~M,) (99l~1 (1.12)

/t,199le> (J~M~I = Ve lg)le) (J~Mi]

I/d being the difference potential between the excited molecular state and the ground state of the atom- perturber pair. In the present case of far-wing excitation the quasi-static theory should be valid, which allows us to write

1 1 z+iH 193t~)(J,M,I = z + i ( - A a + V~)l!lJt~) (J,M,I

1 1 z-+ill IJiMi) (9)l el - z + i(A~- E) ]J~M~) (9~1.

(1.13)

Here ,~ labels the atomic state that is correlated to the molecular state 1~32~). For the calculation of integrated line intensities we are only interested in stationary values of the density matrix; this amounts to considering in all our expressions z as an infinitesimal positive number. In this limit the Eqs. (1.13) take the following form

1 z ÷iI~ I~)(J~M~I

= { - i ~ _ A a + v a l ~_n6(Aa_V~)}I93t~)<j~M~I (1.14)

1 z + i / 4 IJgMi) (gJ~l

= { - i ~ 1 Vd)} [J~M~)(99l~I. ~ K + n ~ ( A 3 - 3 - - d

In the first term of these expressions one may neglect V d with respect to A~and then evaluate in Eq. (1.10) the corresponding contribution to the value of Pl. Similarly, the contribution of the second term can be evaluated in a straightforward way. We leave these calculations to the appendix and consider now the second-order term of the density matrix which is

196 F. Schuller et al•: Alkali-Noble Gas Half Collisions

relevant for the study of depolarization. The problem there consists in deriving an evolution equation for the reduced quantity

0"2 =Tr~ P2 (1.15)

representing the average over variables of the perturber bath. To derive such an equation we chose a projection operator method together with the binary-collision approximation. This procedure is presented in detail in the appendix. Here we indicate only the result which can be written in the form

A (~ -}- A)0" 2 -[- \ ] (2-- At- 2- t Neff 0-'--[-i<l~ ~'~eff P])-~--0 (1.16)

valid for the excited-state parts of the density matrix. In this equation the most important term is the last one, where p~ designates the contribution to p~ that stems from the delta-functions in (1.14). p~ describes the state of the system after quasi-static excitation• The operator /~ takes the system from the state at the moment of excitation to the final state after the binary collision is over. In a quasi-classical picture/~ corresponds to a "half-collision" and describes es- sentially a rotation in space followed by a transition from a molecular to an atomic state• In this picture the brackets indicate an average over all trajectories and initial positions. In the first two terms of (1.16) the operator ~ stands for the effect of relaxation of the excited state density matrix by subsequent col-

lisions. It is remarkable that the term ~- effO', which

1 has its origin in the functions !13 of Eq. +_A3+V~ (1.14), involves that same quantity 0"' that appears in the spontaneous decay part of Eq. (1.10); this will explicitly be demonstrated in the appendix.

For the calculation of the polarization properties of fluorescence radiation studied in this paper it is convenient to introduce in (1.16) expansions in terms of irreducible tensor operators Tkq. So we write

0"2 = 2 ~(3)T(~) (1.17a) Vkq *kq kq3

VeffG = 2 "(~)T(~) (1.17b) ~kq *kq kq.~

• ~ ^ c I(R Veffpl) = 2 R(5) T(3) (i.17c) Vkq kq kq3

where in the case of alkali atoms ~ takes the values 1/2 and 3/2. Note that according to the remark following Eq. (1.10) a' and hence -(-~) are independent ~kq of collisions.

By definition ~ is a collisional relaxation operator and its action on Tk~ ) is expressed by the re-

lation

~ Tk(~) = fk(3) Tk(~ ) , (1.18)

involving the collisional relaxation constant j~(3) for a given tensorial rank k. Substituting these relations into Eq. (1.16) and solving for a 2 we obtain the expression

:A ¢(3) t ~* .a_ : k ~ . ( 3 ) _ R ( 3 )

=(~)_ ~2 -- 2 ] ~kq Ykq ekqt~('~) Vkq - -

_ 1 .(3) . (1.19) fk('~) + A - - ~ f~('~) + A

The first term on the right is independent of collisions and yields the Rayleigh part in the emission spectrum [17]. This part can thus be ignored if, as will be the case in the present work, only intensities of the fluorescence components are of interest. So the quantity that has to be calculated is the following:

fl (3) kq (1.20) G(3) --

kq -- f(3) + A"

In the next paragraph we shall derive expressions for this quantity on the basis of specific collision models.

2. Express ions for the Coll is ional Part of the Dens i ty Matr ix

The quantities needed for our problem of collisional depolarization are the coefficients figq introduced in Eq. (1.17c). In order to determine these coefficients we proceed step by step, calculating first Pl, then ^ c * * V~ffp ~ and finally i(RVeffpCl). The quantity p{ corresponds to the first term on the right of Eq. (1.10) and more precisely to the part involving the delta-function introduced in (1.14) (see Eq. (1.17) of the appendix). Hence by using Eq. (1.6) for P0 we can write

p~ = - i z + ~ - I?eff PBP~ (2.1) 24+ 1

where it is understood that in the stationary limit, which is of interest here, z is an infinitesimal positive quantity.

According to Eq. (t.2), the operator Vet f is given in terms of atomic states IJiMi), IJeMe) defined with respect to a spacefixed "laboratory" frame. Be-

1 fore considering the action of in (2.1), these z+i~- states (at least the excited ones) have to be transformed into a molecular frame. We first introduce

F. Schuller et al.: Alkali-Noble Gas Half Collisions 197

atomic states 1J~57/~) with respect to the internuclear axis by applying the transformation

IJeM~) = Z ~" ~ D ~M~ IJ~Me). (2.2)

From these we obtain molecular states by assuming that, in the case of alkali atoms, at the moment of excitation the atom-perturber interaction is so strong that it decouples the spin from the orbital angular momentum. Thus we write

lg)le)__ll~L ) t - I~Ms) or shorter If~ILl/Is) (2.3)

and

IJ~)e) = ~ C(1½J~; MLMsM~)IfdLMs). (2.4) M L M s

Now, if excitation takes place in the far wings of the line, we may assume that the quasi-static theory is

1 valid and then the action of z---~- on the molecular

states is particularly simple as it is expressed by the &function part of Eq. (1.14). We thus write

1 I~/LMs> (J~Mil = ~ a IMaMs> (JiMil (2.5) z+ i /~

where 7 ~ is the quasi-static linewidth which is related to the difference potential K -d - according M L M s

to the expression [14]

7A# Ms = r c N ( 6 ( A 3 - V-~L~s) ). (2.6)

Here the average is over intermolecular distances, N being the number density of perturbers. The exciting frequency ~o L determines the quantity A.~ defined as the difference between co L and the transition frequency c%, ,3 labeling the doublet component to which the state IMLMs) is correlated.

With these definitions a straightforward calculation leads to the result

1 Eo P ] = 2 2 j i + 1 ~ D};.Mo(LM~I_f2IJ, M~)-eL

Je ./l~/'e, .,, etc

+ adjoint operator. (2.7)

geffp 1 by As the next step we calculate the quantity ~ noticing that according to (1.2) Vaf is invariant with respect to the change from IJ~Me) to IJ~f/le). We then obtain the expression

D~ Mo(L el#lJiM~) I/offpl - 4 2J~ + 1 _ j,jo Me, *.. etc

x_Q~(JiMil_tztLMe)_e~. C(1½ge; 57/zMsM ~)

x ? ~ ILa~) (MLA?sl. (2.s)

We rewrite this expression by introducing the trans- formations

(J~Ynel =Z D~.m.(J~m~l me

and

~e/~/e) = 2 C(1½jfi f f t L r ~ S t ~ e ) l ~ t L r ~ s ) r~Lffcs

and we express furthermore the dipole matrix elements by means of the Wigner-Eckart theorem, noticing that in our case reduced matrix elements may be replaced as follows:

(l'~l[#llJi)(dilr#llde)=-~(-1) j°-J' (2Je+ltl/2 (2.9) \ 2 J , + 11 "

The result of these operations is then given by

i E~ VeffP~ --4 2 J ~ 1 Z DL*~ D J°

- J e j e MeMe--Vaerae

Me, ... etc

× (2Je+ 1] "2 \ 2 ~ + t ! L~),,,~L)M,

x C(J i lje; MiMme) C(J~ 1Ji; MeM'Mi)

x ~ [ r~Lr~s) (hT/LMs[. (2.10)

Further we have

p~ Vef f ~-(gef f p~) 't. (2.11)

In the following we shall make the simp_Iifyi_ng assumption that in the operator Ir~Lff~s)(MLMs[ the ket and bra correspond to states belonging to the same potential curve. According to this secular approximation we thus retain only coherences between states which are degenerate in energy and neglect all coherences between states of different energy. This approximation is justified by the fact that in the latter case phase integrals would occur which, after averaging, would almost annihilate the corresponding terms. With the secular approximation the following relation holds:


~-+

111/2

j=_3 2

a = ! 2

Fig. 1. Schematic adiabatic potential curves for alkali-noble gas systems A 2 Pj - X 1 S o

and hence

c _ c c Veff=2Veffpcl. (2.12) Veff Pl - Veff P1-1Ol

We now proceed to the study of the action of /~. According to the arguments developed in the appendix, iR describes the change in atomic states between the moment after excitation and the final state when the collision is over i.e. /~ represents the effect of a half-collision. We assume that when the collision terminates, the molecular states go over into atomic states of angular momentum .~ and we express the resulting operator in terms of irreducible tensor components Tk(~ ). So we set

f i I f f lLn'ls) ( I ~ L M s I = 2 R(3)T(3) 2k + 1 (2.13) "'k~ *kq kq 2,~ + 1

Further we express the tensor components with respect to the initial space-fixed " laboratory" frame by using the transformation

Tk(3)-V nk* T(-~) (2.14) q - - / ~ g l q * k q • q

With these definitions we obtain from (2.10) the following result:

/~12effp 1 -- ^ __2R Veefp 1 i E 2

-~2 2 f ~- 1 E E D~ D~, m Dk* 5 _Jejek eMe "5 e glq Me, - - , e t c

× 1( __ 1)j~- Ji (2Je -I- 1~ 1/2 \2J i + 1 ] ~-*)M~L)M'

x C(J~IL; m~mme)CUe 14; mem'm~)

× C(1½Jo; ~ i - ~ . ) C(1½je; mLm~mO

XyM~;, ~ 2 k + l (~'T(~) (2.15) a~ ~ Rkc~''kq .

This expression has now to be averaged over all initial orientations and over all trajectories of half- collision. The latter average will at the moment be formally expressed by writing /W.~)\ instead of R(.~) \*-kq / ~'kq • For the average over initial orientations we use the relation [18]

( D(* D j~ D e ) MeMe r~erne 7:1q-

( ~( 1 ) k - j ~ - M e \(2je + 1)(2k + 1)!

x C(Jekje ; l~ieglffZe) C(JeJek ; -Memeq ). (2.16)

The resulting expression of the average of Eq. (2.16) can be simplified by introducing a summation over some of the quantum numbers referring to the "laboratory" frame. This summation is expressed by the relation

(--1)-M~C(Jj~k; --Memeq) MiMeme

x CGlje; M~Mme)C(Jel4; MeM'M~) =( -- 1)J"-k((2je + 1)(2Ji + 1)) 1/2

× W(1Jelje ; Jik)C(llk; M'Mq). (2.17)

This leads to the final result

geffD1) : --~-L.az..B(3)F'LTk~q ) k kq (2.18) V kq

where the quantity B~k "~) is given by

B~ 3)= Z ( -1 ) j " - J ' [ ( 2 J~+ l ) ( zk+ l ) ] l / 2 _ aoj~ 3(2J i + 1) ML, ... etc

x W(1Jelje ; Jik) 2~ ~ C(1½Je; M L M s J ~ e )

x C ( l l j e ; fflLffIsffle) C ( J e k J e ; J~/le~tI~le)

7 ~2M s (2.19)

and where we introduced the definition

C( l l k; M' M q)(~*)M(~JM, ----F£q.L (2.20) MM'

By comparing Eq. (2.18) with the expansion (1.17c) of the previous paragraph, we note that the coefficients fl(k] ) are given by

2 g(.~) = _ E~ B(3) FL (2.21) I~kq 2 k J kq"

In order to exploit the expressions derived so far, it is necessary to introduce a model describing the effect of the half-collision.

F. Schuller et al. : Alkali-Noble Gas Half Collisions 199

RI.

Fig. 2. Semiclassical picture of a half-collision in an atom-perturber system

We shall assume that the orbital angular momentum rotates adiabatically by following the motion of the internuclear axis until the perturber has reached a distance which corresponds to some cut- off radius R~ (Lewis model [-19]). We call fl the angle between the final orientation of the internuclear axis and the initial one, Fig. 2. We shall also consider the particular case of complete reorientation, where R~ becomes infinity. Regarding the spin, we shall discuss two cases assuming either that the spin also follows the motion of the internuclear axis or, on the contrary, that it is decoupled from that axis and stays fixed in space. In both cases we assume that when the final position is reached the molecular states are transformed according to the correlations X1/2, 113/2--+ P3/2 and 111/2--+ P1/2" Note that for the X,/2 component only the second case is physically meaningful.

In the first case of moving spin we write

2 k + l ~" M~Qm~) T£~ (2.22) = ~ ] - c(~1¢4 , " - - (s~

where the quantities Re, ~7/e refer to the final position of the internuclear axis. But since these quantities are conserved during the motion, we also have

l~e =ITIL-~tTFIS )~ e =3)L + Ms" (2.23)

Naturally, we assume that at the moment of recou- pling these quantities are conserved as well. We also

need tensor components refering to the initial position which are obtained by the transformation

- ~ d0s(fl) ks" (2.24t q

Equations (2.22), (2.24), and (2.13) then yield the result

e ( S ) _ V C ( 3 k 3 ; MeOfne)d~s. (2.25) kS --z. .

The case of the non-moving spin is slightly more complicated. As a first step we rotate the orbital angular momentum and express at the same time the space-fixed spin in terms of components with respect to the final position of the internuclear axis. This amounts to replacing

by

1/2 ~ (2.26) IrhL> ~ d ~/2,~ I rSs> <~/LI F, d ~ < M s l . ms ~/s

Note that in the last line r~c, r~ s and -~/L, ~/s refer to the final position.

At this stage an additional complication arises in the case of/ /-excitat ion where mixing of the molecular states //3/2 and //~/2 has to be accounted for. But let us first consider the Z~/2-state. Then we have r~L=~/L=0 and we may assume that after recou- pling of the spin the molecular states go over into atomic states as follows:

10>I r~s> <01<Msl ~ 13'~s> <3M~I with ,3=3/2.

(2.27)

Expansion into irreducible tensor components followed by the transformation (2.24) then yields

2 k + l

×d~/2 d~/2 d~-Tk(~). (2.28) r~sr~s &tsMs tgq q

Hence we obtain

R(3) . . . . A~t /~t~ ~ni'l/2 ,41/2 dk (2.29) O_#,s~s

with

3=3/2 .

In the case of the //-states the ,3-value of the atomic state that results from one given molecular

200 F. Schul ler et al. : A l k a l i - N o b l e Gas H a l f Col l i s ions

state is either 1/2 or 3/2 according to the total magnetic quantum number of the considered state. With the secular approximation the transition from molecular to atomic states is now the following:

Ir~L~s> <~/r~/sl ~ l'3r~e> <'3~/el (2.30)

with

and

3=1 kel =lA4~l,

By replacing in (2.28) molecular states by the corresponding atomic states and by expressing afterwards the result in terms of irreducible tensor components, we obtain the relation

=" 2 dl/2 dl/2 --2k+l ~ ~. ~/t tSn] ~d ~ g(-~) _ = ~s,~s v ~ 2 , 3 + 1 C(~5k~5 . . . . ~ e~ 0,o ~c~.

3 ~ s M s

kQ~ (2.31)

Because of the additional summation over ,3 which appears in this expression, it is necessary to modify Eq. (2.13) by writing more generally

/~ l ff~Lrhs) <ML*Qsl = 2 R(3) T'('~) 2 k + 1 3k~ kq kO 2 , 3 + i "

(2.32)

With this definition Eq. (2.19) can be generalized as follows:

B~ "~)= ~ ( -1 ) jo-J ,[(2Je+l)(2k+l)]~/z _ j,j~ 3(23"/+ 1) ML, ... ere

1 x W(1J~lje; J,k)~-~--+-( C(1½Je; ~L~]ls;Ie)

x C(1½je; mLmsFne)C(Jekje; Meglr~e)

/m-~)', 7 ~ s (2.33) x \~'kq/ A3"

where we have me=mL-}-ms, ]~Ie=-J~/IL"~~/Is and '3' =lr~el =lAY/el and where ,3 equals 1/2 for the II1/2 - and 3/2 for the H3/a component. The quantity R(k~ ) is defined by Eq. (2.29). Equation (2.33) contains in the sum on the right - hand side, in general, terms

:6 =IML + Ms[ and belonging to both values ~' ~ =½ '3' = 3/2.

The results of explicit calculations of the constants B~ -~) in terms of rotation matrices will be presented in the next paragraph.

3. The Polarization Degrees

Let us first consider the total intensity of emitted radiation with polarization direction _~. This is given by the general expression [16]

I F =Tr {add*} (3.1)

with a the stationary density matrix and

d =~(+)~ =Pe_IjPN; d* =e* _/j (-) =_e* P~ _/zP e. (3.2)

Labeling an excited state belonging to one given multiplet component by the angular momentum quantum number ,3 we define Pe and Pi as

Pe= ~j~, [`3M~>(`3Me[ and P~=~ IJ~M~>(J~M~[. 3M~ M~

(3.3)

We disregard coherences between different states of the multiplet and write the density operator in the form

a = 2 %M;, .~.I3M'e>(3MeI ' (3.4) 3MeM'~

With these definitions we obtain the expression

T r a d d * = ~ c%M;,.~M~(MeI~IMi)_~(MiIuIMe)_~*. 3M~M'~

(3.5)

In the case of well-separated lines we may assume that the intensity of each component of the multiplet is represented by the terms on the right of Eq. (3.5) which correspond to one given ,3-value. We then introduce multipole components defined by the relation

C%~;,.~M~ = 2 a(k~ ) C ('3k `3 ; Meq M'~) kq

and consider the intensity of one given multiplet component for which we have

1('3) 2 (3) . . . . = akq C(~3k;3, MeqMe)(Me[~lMi) MoM'~

kq

x_e ( Mil ~ l M'e) ~_*. (3.6)

This expression is transformed by use of the Wigner- Eckart theorem and summation over M e, M;. The result is

-- ~ /2Ji + l "~ 1/2 G "~) =(`3 II~ll 4)(4 II,ttl ̀3) )2 (2;5 +t ) (2-~-1)

kq

~ - I / ~ ,,-(3) F, x W(1;31~, (3.7) ~i '~]~kq kq"

Here the following abreviation analogous to (3.20) has been introduced

F. Schu l le r et al.: A l k a l i - N o b l e G a s H a l f Col l i s ions 201

Fkq= ~ C( l lk ; M'Mq)e~teu,. (3.8) M M '

Furthermore, in o- only the collisional part has to be taken into account since the collision-independent term must be attributed to the Rayleigh component not measured in the type of experiment that we are considering here.

From Eqs. (1.19) and (2.21) the quantity -~('~) is ~ k q

given by

E 2 B(,~) ~q'~('~)- L°2 fk (3) + A F~, (3.9)

This yields for I(~ ) the final expression

E 2 (2J i + 1 ]l/z t(F 3) =(3 [IPt It Ji)(Ji [t/~ I13) ~ ~ (23 + 1) 12 Tfl

B(.~) x W(1313; J ~ k ) ~ P ~ (3.10)

Jd +A

with

(3.11) q

It is this latter quantity that depends on the polarization state of the incident and the emitted radiation.

We define a polarization degree ~, for linear polarization and !13~ tbr circular one by the following relations

III _ I z ~z --it I +i_~ (3.12a)

I r - - I t

~3 ~ =F + iz (3.12b)

where III and I ± are intensity components respectively parallel and perpendicular to the incident polarization and where I" and I ~ are the intensities corresponding to right and left-handed polarization. The quantities ~31 and ~c can be calculated by substituting into (3.10) for a given 3 the proper values of Pk defined by (3.11). These values are

1) Linear polarization:

k = 0 1 2

1 0 2 pl[ .g

p± .t 0 1 3 3

2) Circular )olarization"

k = 0 1 2

p, ± ± ! 3 2 6

pl _1 3

1_ 6

The following expressions for the polarization degrees of the components of a 3 = t/2, 3 = 3/2 doublet are then obtained:

3 = ½ : ~ = 0 t B~/2

~c = - - ~ A (f11/2 + A)B~/2 (3.13)

3I/~B 3/2

3 =-~: !!3x = A (4t/~B3o/2 +]//~B~/2) A + 4t/-~B3o/Zf23/2

1 B~/2 ~ = - - ~ A (f?/2 + A)Bg/2"

We have considered linear polarization and we have calculated the constants B(0 "~) and B(2 -~) for both the model of non-rotating and rotating spin. The results are presented in Table 1. The calculation, al- though straightforward, is a little lengthy due to the large nomber of summations involved. The notations in Table 1 are such that we have the obvious relations

B(kl/2) __ R F I 1 / 2 . __ I ~ 2 1 / z - - ~ k , B(3/2) ~- R/'/3/z (3.14)

In general, the model of non-rotating spin leads to mixing of the two H-components. The reason that this does not appear in the particular case of Bz n ~ is, that in (2.31) the terms with 3=lMel =1 r~el-± - - 2 '

are eliminated because of the factor C(~sk~,

We finally note that Zeeman coherences do not contribute to the alignment; this is because of the relation ~ " - ~ , _ 25=[Me[=[ r~] -3" ( ~ = + 3 which elim- inates the case k =2.

To obtain numerical values of the B-coefficients computer calculations are necessary. The averages represented by the brackets are over all values of the total rotation angle fl and it should be noticed that they depend on the initial position of the perturber if a finite cut-off radius is used. On the other hand, according to the quasi-static picture, this initial position is related to the exciting frequency so that in the end these averages are frequency-dependent quantities. This is no longer true in the special case of complete reorientation with rectilinear trajectories, where we have


Table 1. Coefficients z~/~ n~/~ n3/~ B k , B k , B k for the calculation of the linear polarization degree: Eq's (3.13) and (3.14)

k = 0 2

36 d~/2 180 aa/2

non-rotating spin

36 \ \ 2 / A~/2 2 / A~/2] 360 2 °°/ A~l z

i8 \ \ 2 / A1~ z \ 2 / A3z21

36 z12/2 360 (d°°)7]~72 rotating spin

H*/2 18 AZjz

H

4 . . . > y . . . . 0

However, in this special case all B2-values vanish and consequently the linear polarization degree becomes zero. According to this result, remaining linear polarization can only be attributed to incom- plete reorientation and to account for this effect models using a cut-off radius, and, possibly, non- linear trajectories should be considered.

4. Conclusions

We have considered a mechanism consisting in quasistatic excitation followed by reorientation of the atomic angular momenta through a half-collision. The effect of subsequent relaxing collisions is also incorporated in our formalism. We assume that right after excitation the states of the atom-perturber pair are molecular states Z1/z, H1/2, H3/2. The orbital angular momentum rotates adiabatically, its projection on the internuclear axis being conserved during the half-collision. For the spin S two limiting cases are distinguished. In the first case S is un- coupled from the internuclear axis, its orientation in space remaining fixed. This corresponds to Hund's coupling case (b) and will be a good model for S- excitation of all diatomics considered and for H- excitation of the low mass systems. In the second case S is coupled strongly to the internuclear axis and together with L reorients adiabatically with respect to this axis. This corresponds to Hund's case (a), being a good approximation for H-excitation of the large mass systems. After dissociation the molecular states are assumed to transform adiabatically

slow into atomic states according to the correlations

21 /2 , [13/2-~P3/2 a n d H1/2 ~ P1/2" The assumption of adiabatic L-reorientation is

strictly valid only at small R <R e, within an interaction Radius Re, where the molecular H - Z splitting equals the nuclear rotational energy. Thus the model of complete adiabatic L-reorientation will apply only in the limiting case of extreme wing excitation R L ~ R ~ (RL=internuclear separation at the moment of excitation). For the more general case R L ~ R c one should consider the L-decoupling from the internuclear axis at Re. In this refined model one would assume, that at R c molecular states are transformed into atomic states with the same correlations as before and the resulting atomic J = L + S remains space fixed in the region R > R~.

A further refinement would consist in considering non-adiabatic effects especially those leading to H - S transitions which are known to be important for fine-structure transfer. Possibly such transitions might also have some influence on polarization properties.

We have derived formulae for the linear polarization degree involving some quantities B k which are expressed in terms of averages over functions of the total rotation angle /? of the internuclear axis during the half collision. In general, fi will depend on both the initial distance R L and the final position of the perturber at the moment when the cut-off radius R~ is reached. Because of the correlation between R c and the exciting frequency, the averaged angular functions introduce an additional frequency dependence to that already contained in the line broadening functions ,/(co 0.

So far we have not treated numerically the case of finite R e and curved trajectories, where a remain-

F. Schuller et al.: Alkali-Noble Gas Haif Collisions 203

ing frequency dependent polarization is to be expect- ed. In the much simpler case of linear trajectories and complete reorientation the resulting alignment is zero despite the fact, that at all stages of the calculation Zeeman coherences are considered. This makes the case of alkali atoms different from that of atoms with d = 1, where even with complete reorientation a nonzero result due to these Zeeman coherences has been obtained [14].

A p p e n d i x

The expansion of the density matrix with respect to the incident field leads to Eq. (1.7) of Sect. 1 for the first-order term and to a similar equation for the second-order term. Thus we have

zp~ + iI~p~ +Fp~ + i Ig~ffp o =0 (A.1)

zP2 q- i/tfl2 q- f P 2 q- i ~effPl =0. (A.2)

The quantity of interest to us is the reduced density matrix

a 2 =TrBp z (A.3)

obtained by taking the trace of P2 over variables of the perturber bath. In order to derive an evolution equation for this latter quantity, we define a projection operator [20] by the relation

Pa =PB TrB a (A.4)

(where o" is an arbitrary operator) and its comple- mentary

Q = 1 - P. (A.5)

We have from the definitions

p2 = p (A.6a)

Q2 =Q. (A.6b)

Acting on Eq. (A.2) first with P and then with (2 we obtain the following equations:

zPP2+iPIt(P+Q)p2+FPp2+iVeffPp~ =0, (A.7a)

zQpz+iQI4(P+Q)pz+FQp2+iveffQp~=O. (A.7 b)

From the second equation we can formally express Qp2 as a function of Pp2 and by substituting the expression obtained into (A.7a) eliminate QP2 from this equation. The result is given by

1 (z + inIIn + F + inflQ z + iQI~Q + F (-i)Q I~P)P P2

1 +il~ffPp~ +iPt?IQ z+iQI4Q+F ( - i ) I?~trQP~ =0.

(A.8)

The quantity Q.pl at the end of the last term in (A.8) can be obtained by acting on Eq. (A.1) with Q and solving for QPl; this yields

1 QP' - z + iQItQ + f ( - i)QfilPP p~. (A.9)

To obtain the evolution equation for a 2 =TrBp 2 it is now sufficient to take in (A.8) the trace over perturber states. By introducing the abbreviation

(.--} =TrB {(...)PB} (A.10)

we obtain from (A.8) an equation of the form

(z+i/?d +/~+~)o-2

{ 1 } +TrB iVef tPPl+P/?Q z+iQlqQ 17effQP~ =0 (A.11)

where ~ is a collision operator which is formally represented by the expression

q~=I PI~IQ z+iQI4Q1 Qi~1P)" (1.12)

Here / ] i represents the part containing perturber variables i.e. the part designated in Sect. 1 as /~p +/I~; /~d is the perturber-independent "dressed- atom" hamiltonian introduced also in Sect. 1. Fur- thermore, in (A.11) and (A.12) use has been made of the well-known substitutions

PI4P -..', t t d

and

PI2IQ ~ PI4xQ;

(A.13)

Q IIP ~ QI~IP. (A. 14)

One notices also that in (A.11) and (A.12) the quantity/~ has been dropped in the denominators.

We now proceed to the explicit calculation of the second term in Eq. (A.11). First we consider the first-order quantity Pl as given by Eq. (1.8). By using the decomposition (1.9) and the approximations discussed right afterwards, we write

. ( 1 ~ A 1 (A.15)

Expressing Vafpo in terms of molecular states t~J~} we have

I~afp0=~ C~),o[gJle} ( diMil + C* ]d~Mi} (?JJlel (A.16) ~0te

and by using (1.14) we obtain for the first term in the brackets of (A.15) the relation

204 F. Schuller et at.: Alkali-Noble Gas Half Collisions

t ^ z + i / t Veff P°

=z( i , )

+ 7~(5(A 3 -- Vd) l~eff P0 i

= --Hd V~ffP° + r~a(Aa - Vd) 12~ffp0, (A.17)

where the definitions (1.4) have been considered. Hence we have

t .A 1 ) = r e f f p o + p i (A.18) , , , ^ ,

with

p] = -ir~b(A ~ - Vd) l?orfp 0. (A.19)

Let us now consider the expression within the brackets in the evolution Eq. (A.11). Regarding the term i I?offP&, it can be shown by using explicit expressions for l,~ff and Po, that we have

( 1 )[~eff/)O= 0 (1.20) iVeff --Hdd

if only the excited-state contribution is considered. On the other hand, since Po factorizes, it is obvious that we must have

Q ( _ i A 1 5~2e2) l?offp o =0. (A.21)

Finally before substituting the part of Pl represented 1

by - ~ l?~frp o into the second term within the brack-

ets, we apply first Eq. (A.9). We then obtain

1 ¢ = iVeffPPl + P/4 'Q z + iQ/ iQ I~eff Q Pl

Q A 1 =P ~. ~ ~ G:o

1 Veff 1 + P I i , Q z+iQ.IqQ z+iQI~Q ( - i )

1 +ilToffPp{+P/~rQ . ..-,a.-, ^ VeffQpl. (A.22)

z+l~gr1~g

In order to exploit this formal expression it is now essential to make the binary collision approximation, assuming that only one perturber at a time acts on the optical atom. Mathematically this amounts to

replacing in expressions like (A.22) the P and Q operators by unity as has been shown by previous authors [21]. The resulting expression is

A ^ 1

1 _ _ ~ 1 ~ 1 ^ +/{ z+--i/~ ¢"~ H, ~ Vo.po

1 • ^ c A

Voff&, (A.23) +lVeffpl +Hx z_~l~l H ^ c

where now /~i represents a one-perturber contribution.

It can be shown by a straightforward, but a little lengthy calculation, that in the second term of (A.23) /t~ can be shifted to a different position according to the relation

1 iT f f 1 ^ 1 ^

1 ^ t , ^ ^ 1 = ~ H~ z + i 1t H ~ Voff ~ ~:ff P o- ( A . 2 4 )

If on the right-hand side of this equation we take the trace over bath variables, we obtain an expression containing the operator

¢ : / / 4 , z ~ i H / 4 i ) (A.25)

which is the binary version of the coIlision operator defined by (A.t2). After these manipulations the evolution Eq. (A. 11) for the stationary excited-state density matrix is reduced to the form

(C-[-(ib) o" 2-'~ ('2-+-~- ) eff ~2d2 VeffO" 0

{ 1 } ^ ~ =0. (A.26) +Tr, iG:~ + / { ~ Vo~:~

Finally, the significance of the term containing p] must be made apparent. To do this we use the operator equation

1 1 / t , ~ =i(z + i/4d) ~ - i

with

=/~+/~.

Then we have

1 iz ilPeffP; + / t ' z+i------H VeffPl z+ i /q

F. Schuller et al.: Alkali-Noble Gas Half Collisions 205

(since I t d in the numerator has no effect). Now, the 1

operator z + ~ - describes in Laplace-Fourier space

the evolution of the quantity ~ c V~ffp 1 and letting z go Z ~ c

to zero, we can interpret ~ l/effp 1 as the corre-

sponding stationary value. Hence we can write

1 ^ c i l~efep ~ + n I ~ Ve¢fp 1

z _ " ^ c * A ~ c

--1 ~ Veffp 1 - - t R Veffp 1 (A.27)

and we see that iff has the significance of an operator representing the evolution of the quantity Veffp ] from the moment of excitation up to the stationary value at the end of the binary collision process.

To obtain from (A.26) the final form of the evolution equation presented in Sect. 1, it is now sufficient to make the identification

A

cr' = ~ V~f f ao (A.28)

and to replace f by the relevant matrix element A.

References

1. Vigu6, J , Beswick, J.A., Broyer, M.: J. Phys. (Paris) 44, 1225 (1983)

2. Foth, H J , Polany, J.C., Telle, H.H.: In: Spectral line shapes. Burnett, K. (ed.), Vol. 2, p. 813, Berlin: Walter de Gruiter 1982

3. Burnett, K.: In: Electronic and atomic collisions. Eichler, J,, Hertel, I.V., Stolterfoht, N. (eds.). Amsterdam, Oxford, New York: Elsevier 1984

4. Kroop, V.: Dissertation Diisseldorf 1983 5. Hedges, R.M., Drummond, D.L., Gallagher, A.: Phys. Rev.

A6, 1519 (1972)

6. Pascale, J., Vandeplanque, J.: J. Chem. Phys. 60, 2278 (1974) 7. Diiren, R., Hasselbrinck, E., Moritz, G.: Z. Phys. A - Atoms

and Nuclei 307, 1 (1982) 8. Kulander, K.C., Rebentrost, F.: J. Chem. Phys. 80, 5623 (1984) 9. Havey, M.D., Copeland, G.E., Wang, W.J.: Phys. Rev. Lett.

50, 1767 (1983) 10. Behmenburg, W., Kroop, V., Rebentrost, F.: J. Phys. B 18,

2693 (1985) t 1, Rebentrost, F.: Private communication 12. Lewis, E.L., Harris, M., Alford, W.J., Cooper, J., Burnett, K.:

J. Phys. B 16, 553 (1983) 13. Schuller, F., Behmenburg, W.: Z. Naturforsch. 30a, 442 (1975) 14. Nienhuis, G., Behmenburg, W., Schuller, F.: Z. Phys. A -

Atoms and Nuclei 320, 165 (1985) 15. Schuller, E., Nienhuis, G.: Can. J. Phys. 62, 183 (1984) 16. Nienhuis, G.: J. Phys. B 14, 3117 (1981); B 15, 535 (1982) 17. Nienhuis, G.: J. Phys. B 16, 1 (1983) 18. Rose, M.E.: Elementary theory of angular momentum.

New York: Wiley 1966 19. Lewis, E.L., McNamara, L.F.: Phys. Rev. A 5, 2643 (1972) 20. Zwanzig, R.: Lectures in Theoretical Physics. Vol. III.

New York: Wiley Interscience Inc. 196I 21. Smith, E.W., Cooper, J., Vidat, C.R.: Phys. Rev. 185, 140

(1969)

F. Schuller Laboratoire des Interactions Mol6culaires et des Hautes Pressions C.N.R.S. Centre Universitaire Paris-Nord Villetaneuse France

G. Nienhuis Fysisch Laboratorium Rijksuniversiteit Utrecht The Netherlands

W. Behmenburg Physikatisches Institut I Universitiit Dtisseldorf D-4000 Diisseldorf Federal Republic of Germany

depolarization mechanisms in alkali-noble gas half collisions

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