1039

Review/Synthèse

Energy transfer between the 2S and2D states in alkalis: experiments andtheory

A. Ekers, M. Głódź, V. Grushevsky, J. Klavins, and J. Szonert

Abstract: Collisional energy transfer (ET) between the2S and2D states of alkali atoms isreviewed. Recent experimental results on the ET in homonuclear collisions in Na, K, andRb vapours upon cw (continuous wave) and pulsed laser excitation, as well as theoreticalcalculations of the respective cross sections are discussed. The mechanisms of these processesare considered in an attempt to clarify the role of the dipole–dipole interaction.

PACS No.: 34.50Fa

Résumé: Nous passons en revue les transferts d’énergie par collision (ET) entre les états2S et2D dans les atomes alcalins. Nous analysons des résultats expérimentaux ET récents obtenus parcollisions homonucléaires dans des vapeurs de Na, K et Rb excitées par laser en mode continuet en mode pulsé, ainsi que des résultats de calculs théoriques des sections efficaces impliquées.Nous étudions les mécanismes pour ces transferts dans le but d’éclaircir le rôle de l’interactiondipôle–dipôle.

[Traduit par la Rédaction]

1. Introduction

Excitation energy transfer (ET) in collisions of excited atoms and molecules plays an importantrole in many phenomena, including processes occurring in planetary atmospheres, gas lasers, andplasmas. Studies of low-energy collisions have contributed significantly to the understanding of gas-phase chemical reactions and laser physics. As a classical example, one can mention here the resonantET process from the metastable He(2s1S0) state to the Ne(3s) state, which is used to achieve generation

Received June 22, 2000. Accepted May 18, 2001. Published on the NRC Research Press Web site on August 24,2001.

A. Ekers. Institute of Atomic Physics and Spectroscopy, University of Latvia, Rai¸na bulv. 19, LV-1586 Riga,Latvia and Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland.Telephone: +371-7229758; FAX: +371-7820113; e-mail: aekers@latnet.lvM. Głódź1 and J. Szonert.Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668Warsaw, Poland.V. Grushevsky and J. Klavins.Institute ofAtomic Physics and Spectroscopy, University of Latvia, Rai¸na bulv.19, LV-1586 Riga, Latvia.

1 Corresponding author Telephone: + 48-22-8437001 ext. 3331; FAX: +48-22-8430926;e-mail: glodz@ifpan.edu.pl

Can. J. Phys.79: 1039–1053 (2001) DOI: 10.1139/cjp-79-8-1039 © 2001 NRC Canada

1040 Can. J. Phys. Vol. 79, 2001

of the 633 nm radiation in a He–Ne laser. The large variety of elementary processes, which take place inlow-temperature plasmas, as well as the experimental and theoretical methods that are used to investigatethem, have been reviewed, e.g., in the monograph ref. 1.

Until about the mid 1970s, most of the ET experiments were performed on mercury and alkalimetals, the resonance states of which are easy to access for optical excitation (see ref. 2 and referencestherein). Most of these studies concerned ET between resonance states in homonuclear and heteronuclearcollisions, and the quenching of resonance radiation. To interpret these processes, nonadiabatic collisiontheory was developed (see refs. 3 and 4 and references therein).

Starting in the 1970s, the so-called energy-pooling (EP) processes were studied where two elec-tronically excited atoms collide to produce one highly excited atom and one atom in a low-lying state(typically the ground state). In the first studies of the homonuclear collisions of two alkali resonanceatoms (n21P) the formation of a highly excited atom (n

22S orn

22D) and an atom in the ground state (n

20S)

was observed [5,6]

M∗ (n1P) + M∗ (n1P) → M∗∗ (n2L) + M (n0S) + 1E (1)Numerous studies concerning this and other kinds of ET processes have been published since thesefirst observations, including processes like EP in heteronuclear collisions (see, e.g., refs. 7–9) or reverseenergy pooling (REP) in which an atom prepared in the highly excited state collides with an atom inthe ground state and in the exit channel two excited atoms appear [8,10]. The theoretical approach tosuch processes was developed for type (1) process in Cs [11], and later extended and then applied toEP in other alkalis by various authors [12–17]. The task consists of

(i) determination of the adiabatic quasimolecular terms for the given atomic pair, and

(ii) description of the dynamics of motion in the system of these terms, including determination ofnonadiabatic couplings (due to relative motion of the nuclei) in the regions of curve crossingsand calculation of the resultant transition probabilities and cross sections.

The theoretical studies show that the high efficiency of type (1) EP processes is related to the strongdipole–dipole interaction between the initial and final atomic states [11–13]. The vast experimental datathat have been gathered up to the present day allow one to establish a common empirical relation betweenthe energy defects and the cross sections of different EP processes [7,8]. Anticipating the considerationsof Sect. 3, it can be noted that other ET processes, in which the dipole–dipole interaction seems to bereplaced by higher multipoles (e.g., ET from the initial2P+2P to the final2P+2S states of separatedatoms, or from2F+2S to2D+2S, and the like), follow the same tendency as the dipole–dipole processescf. Fig. 1. Thus, on the one hand, the theory envisages that ET is efficient if induced by the dipole–dipoleinteraction. On the other hand, the experimental results depicted in Fig. 1 seemingly suggest that it isthe energy defect and not the type of interaction that plays the decisive role in the ET. To obtain someinsight into this question, we will consider the ET between the atomic2S and2D states in alkalis, whichare not directly coupled by the dipole–dipole interaction (Sect. 4). By summarizing the results of aseries of our previous experimental [18–21] and theoretical [22] work and comparing the mechanismsof such processes in various alkalis, we shall attempt to explain the features of the dependence of theET cross sections upon the energy defect seen in Fig. 1. We shall first, however, sketch in Sect. 2 someelements of the theory of ET, with an emphasis on the case of strong dipole–dipole coupling betweenthe initial and final atomic states.

2. Elements of the theory of energy transfer

One can introduce the so-called global Massey parameterξaa′ = 1Eaa′R∗/~v, where1Eaa′ denotesthe energy difference between the initial and final states|a〉 and∣∣a′〉 of a pair of colliding atoms,v istheir relative velocity, andR∗ is the characteristic size of the region where the mutual interaction of the

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Fig. 1. Experimental cross sections of different exothermic energy transfer processes in alkali atom collisions asa function of energy defect. Data points are labelled consecutively by small letters given above the upper axis ofthe figure (symbols:◦ and�) or by arrows accompanied by brief description in the plot (symbol:N). ◦, energypooling (and reverse energy pooling) processes, in which the initial and final atomic configurations are coupledby the dipole–dipole interaction: (d) K(4P)+ Rb(5P)→ K(4S) + Rb(5D) [7], (e) K(5D)+ Na(3S)→ K(4P) +Na(3P) [8], (f) Cs(6P1/2) + Cs(6P3/2) → Cs(6D)+ Cs(6S) [27], (g) K(7S)+ Na(3S)→ K(4P)+ Na(3P) [8], (i)Na(3P)+ Rb(5P)→ Na(3D)+ Rb(5S) [7], (j) Na(3P)+ Rb(5P)→ Na(3S)+ Rb(8S) [7], (l) Na(3P)+ Na(3P)→ Na(5S)+ Na(3S) [29], (m) Na(3P)+ Na(3P)→ Na(5S)+ Na(3S) [28], (n) K(4P)+ Na(3P)→ K(4S) +Na(3D) [8,9], (o) Na(3P)+ Rb(5P)→ Na(3S)+ Rb(6D) [7], (p) Cs(6P3/2) + Cs(6P3/2) → Cs(6D)+ Cs(6S)[26]; �, processes (energy pooling and other) in which the dipole–dipole interaction between the initial and finalconfigurations is not obvious: (a)7Li(2S) + 6Li(3S) → 7Li(3S) + 6Li(2S) [45], (b) Li(4F)+ Li(2S) → Li(4D) +Li(2S) [33], (c) Cs(5D)+ Cs(5D)→ Cs(7F)+ Cs(6S) [25], (h) Rb(8S)+ Rb(5S)→ Rb(6D)+ Rb(5S) [35], (k)Cs(6P1/2) + Cs(6P1/2) → Cs(7P)+ Cs(6S) [26], (q) K(4P1/2) + K(4P1/2) → K(5P)+ K(4S) [43], (r) K(4P3/2) +K(4P3/2) → K(5P)+ K(4S) [43], (s) and (t) Cs(6P3/2) + Cs(6P3/2) → Cs(7P)+ Cs(6S) [26] and [46], respectively;N, S-D transfer discussed in the present paper: K(6S)+ K(4S) → K(4D) + K(4S), process (13) [19]; Rb(7S)+Rb(5S)→ Rb(5D)+ Rb(5S), process (14) [20]; Rb(8S)+ Rb(5S)→ Rb(6D)+ Rb(5S), process (15) [21]; andan estimate for the D-S transfer: Na(4D)+ Na(3S)→ Na(5S)+ Na(3S), process (12) [18].

atoms is significant [23]. The probability of a transition between the atomic states|a〉 and∣∣a′〉 can beroughly estimated as

P ∼ exp(−ξaa′) (2)which suggests an exponential dependence of the reaction cross section on the energy defect1Eaa′ . Theabove approach is, however, oversimplified. It ignores the fact that a quasimolecule is formed duringthe collision. Therefore, instead of transitions between the atomic states|a〉 and∣∣a′〉 one should ratherconsider transitions between the states|ϕ〉 and∣∣ϕ′〉 of the quasimolecule. The local Massey parameterξϕϕ′ (R) is then introduced [23]; it depends on the interatomic potentials of the quasimolecule, andhence, on the dynamic change in the distanceR between the colliding atoms.

A number of models have been proposed to simplify the calculations of ET cross sections [4].Typical for these models is the semiclassical treatment of the relative motion of colliding atoms, andrestriction to a small number of strongly coupled electronic states with well-localized regions of strongnonadiabatic coupling. A relatively simple yet efficient method to describe atomic collisions at thermal

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energies uses the expression of quasimolecular wave functions through the wave functions of collidingatoms. These molecular functions are usually constructed as linear combinations of antisymmetrizedproducts of wave functions of isolated atoms [4]. The regions of coupling between the quasimolecularstates correspond to the regions of crossing or proximity of the potential curves. Accounting for themotion of atoms introduces the so-called nonadiabatic interactions between the molecular states. Afurther simplification is reached by using the approximation of two-states (i.e., only the initial andfinal states of separated atoms are used as basis sets for the quasimolecular terms) configurations, withconfinement to the long-range interactions in an asymptotic form. The latter is dependent upon anassumption that the main contribution to the ET is due to nonadiabatic transitions at large internucleardistancesR > 20 a.u.

Let us consider as an example the EP process in rubidium [12].

Rb(5P) + Rb(5P) → Rb(5D) + Rb(5S) + 607 cm−1 (3)First, the quasimolecular terms that correlate with the 5P+ 5P and 5D+ 5S states of separated atomsare constructed. Both groups of terms are coupled by the dipole–dipole interaction (allowed opticaldipole 5P→5S and 5P→5D transitions), which couples the terms of the initial 5P+ 5P and final5D+ 5S configurations. At large internuclear distances, the interaction of atoms weakens and becomescomparable with the splitting of the resonance doublet due to the spin–orbit interaction. It is thereforeconvenient to choose the adiabatic-basis (Hund’s case (c)) in which the spin-axis interaction isaccounted for in the molecular wave functions, and the molecular statesσw are characterized by theprojection () of the total electronic angular momentum on the internuclear axis, by the even–odd parity(w = g, u) of reflection of the electronic wave function at the centre of the molecule, and (for termswith = 0) by the symmetry (σ = +, −) on reflection through any plane containing the internuclearaxis.

On neglecting the nonadiabatic coupling operator, the effective Hamiltonian of the colliding atomscan be written as

H = HA + HB + VCoul + Vexc + Vdisp (4)whereHA andHB are the Hamiltonians of isolated atoms, while the remaining three terms denoteCoulomb, exchange, and dispersion interactions, respectively. Note, that usually only the dominatingdipole–dipole interaction term is retained in the multipole expansion of the Coulomb interaction. Afterthe matrix elements of (4) have been calculated (see, e.g., refs. 11 and 13 for the calculation procedure),the adiabatic quasimolecular termsσw are obtained by diagonalizing (at different internuclear distances)the HamiltonianH

(σw

), which is expressed in the form

H(σw

) =(

EDS + Edisp + αY6 (DS) VddVdd EPP

)(5)

whereEDS andEPPdenote the diagonal matrices with eigenvalues corresponding to energies of isolatedatoms in the 5D+ 5S and 5P+ 5P states;Edisp,αY6 (DS), andVdd denote the matrices of the dispersion,exchange, and dipole–dipole interaction, respectively.

Nonadiabatic transitions between the adiabatic terms are caused by the motion of nuclei.The strength

of the nonadiabatic coupling between the quasimolecular terms∣∣σw〉 = |ϕ〉 and

∣∣∣′σ ′w′〉

= ∣∣ϕ′〉 ischaracterized by the local Massey parameter [23],ξϕϕ′ (R) = ωϕϕ′ (R) τϕϕ′ (R), whereωϕϕ′ (R) is theelectronic transition frequency between the terms|ϕ〉 and∣∣ϕ′〉, and 1/τϕϕ′(R) = 〈ϕ |∂/∂t | ϕ′〉 denotesthe rate of change of the adiabatic wave functions|ϕ〉 and∣∣ϕ′〉. The parameterξϕϕ′ (R) characterizesthe ability of the fast subsystem (electrons) to follow adiabatically the motion of the slow subsystem(nuclei). In the regions ofR whereξϕϕ′ (R) � 1 the dynamic coupling between the electrons and nucleibecomes strong, which leads to efficient transitions between the states|ϕ〉 and∣∣ϕ′〉.

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Review/Synthèse 1043

The operator of the nonadiabatic coupling consists of two separate contributions — radial andtangential

i

〈σw

∣∣∣∣ ∂∂t∣∣∣∣ ′σ ′w′

〉= vR

〈σw

∣∣∣∣i ∂∂R∣∣∣∣ ′σ ′w′

〉+ ω

〈σw |ij⊥| ′σ

′w′

〉(6)

wherevR is the radial relative velocity of the nuclei;ω is the angular velocity of the internuclear axis;j⊥ is the component of the electronic angular momentum operator of a quasimolecule, perpendicularto the rotational plane of the internuclear axis. Strictly speaking, the derivatives in (6) should be takenalong the curved classical trajectories of the colliding atoms. More often, however, the straight trajectoryapproximation is used, which is independent of the molecular states|ϕ〉 and∣∣ϕ′〉. The selection rulesfor the operator (6) are

(a) for perturbations due to the radial motion

1 = 0, g ↔ g, u ↔ u, 0+ ↔ 0+, 0− ↔ 0−

(b) for perturbations due to the rotation of the internuclear axis (Coriolis interaction)

1 = ±1, g ↔ g, u ↔ u

Since the nonadiabatic regions are usually localized at large internuclear distances, the internuclearaxis rotates very slowly at thermal collision energies. Therefore, the transitions induced by Coriolisinteraction (second term on the right-hand side (RHS) of (6)) can be neglected in most cases, and thecalculations can be performed independently for each molecular symmetryp = σw. The probabilityPmn of the transition between two molecular statesm andn, induced by the radial motion of nuclei(first term on RHS of (6)), is often calculated using the well-known Landau–Zener (LZ) formula [24](for derivation see also ref. 4, Chap. 8)

Pmn (Rk, Ei, ρ) = exp −2πH 2mn(

~vRk

∣∣∣ ∂Hmm∂R − ∂Hnn∂R∣∣∣R=Rk

) (7)

whereRk denotes the coordinate of the centre of the avoided crossing of the adiabatic termsm andn;|∂Hmm/∂R − ∂Hnn/∂R|R=Rk is the difference in slopes of diabatic potential curves atR = Rk; Hmnis the nondiagonal interaction matrix element;Ei is the initial kinetic energy;vRk is the radial velocityof nuclei atR = Rk; andρ is the impact parameter. The LZ model relies on the usual conditions ofthe semiclassical approximation and is derived assuming that in the vicinity of the crossing pointRkthe radial velocity and the nondiagonal interaction matrix elements are constant and diabatic statesdepend linearly onR. These assumptions restrict the nonadiabatic transitions to a small region aroundthe crossing point.

By taking into account that the avoided crossing is passed twice in the course of a collision (asthe atoms approach each other and separate), in the idealized case of two isolated states the resultingtransition probability is

Pmn = 2Pmn (1 − Pmn) (8)which peaks atPmn = 0.5, and approaches zero forPmn → 0 andPmn → 1. In real situations, however,even for one selected symmetryp, it is necessary to deal with entire systems of terms, among whichseveral nonadiabatic transitions are possible in a single collision. One should account for all the possibleentrance channels and all avoided crossings present within this symmetry.Application of the LZ formula

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(7) is justified, when these avoided crossings are well separated from each other. If this is the case, theresulting transition probabilityPmn is obtained by solving a system of equations that accounts for theredistribution of the population among many terms applying (7) successively to each avoided crossing[16,22]. A complication encountered in these calculations is related to the fact that the parametersentering (7) can be correctly determined only for the two-level case, when it is possible to rationallydefine the diabatic states. In the multichannel case, the diabatic states cannot be unambiguously definedfor all values ofR, which leads to the uncertainties in the values of these parameters. This problemcan be overcome by applying numerical procedures that allow determination of the parameters of (7)directly from the term diagrams (see, e.g., ref. 22).

To allow comparison with the cross section measured at a temperatureT , the resulting probabilityhas to be integrated over the impact parameterρ and the relative velocity distributionf (v, T ), summedover all symmetriesp and all the statesm andn correlating with the initial and final configurations ofthe atoms, and account should be made for the statistical weightsgp,n of the final statesn

σ (T ) =∑

p,m,n

gp,n

∫v

f (v, T ) dv∫ρ

Pmn2πρ dρ =∫v

σ (v) f (v, T ) dv (9)

Note that the Stueckelberg phases are not included in the above calculations because their effect isusually completely suppressed after integration over the impact parameterρ.

Asymptotic calculations [11–13] tend to give cross sections that are systematically smaller than theexperimental values [12,25–29]. This may be explained by the fact that they do not account for possibletransitions at shorter internuclear distances. It has also been noted [15] that besides the dipole–dipoleinteraction higher terms of the multipole expansion may play a role. However, these higher terms in theexpansion of a Coulomb interaction fall off much more rapidly with increasing internuclear distance thanthe dipole–dipole interaction. As a result, the higher order interactions can cause noticeable couplingonly at small internuclear distances, which imposes a geometrical limit on their contribution to the totalET cross section. For the cross sections of the orders of magnitude considered in this work, the higherorder terms may only contribute as small corrections (see the analysis in ref. 22). Until now, exactcalculations based on accurate potential curves for a broad range of internuclear distances [30,31] havebeen possible only for the process (1) in sodium [16,17].

3. Empirical relation between the efficiency of the ET and the energydefect

The effective ET cross sectionsσ eff(T ) are usually considered in experiments. They are obtainedfrom measured rate constantsk (T ) = 〈σ (v) v〉 = ∫ σ (v) f (v, T ) v dv asσ eff(T ) = 〈σ (v) v〉 / 〈v〉.Such determined experimental cross sectionsσ eff(T ) are compared with the theoretical cross sectionsσ(T ) calculated according to (9), assuming an approximate relationσ eff(T ) ≈ σ(T ). For the sake ofsimplicity, index “eff” for the measured cross sections will be omitted in the following discussion.

In the case of Maxwellian velocity distribution of colliding atoms (experiments in vapours), themean relative velocity is given as〈v〉 = √8kBT/πµ, whereµ is the reduced mass of the collidingatoms. Hereafter, we shall restrict ourselves to experiments in thermal cells, for which the most extensiveinformation is available. Note that cross sections obtained under different experimental conditions, suchas in thermal cells, and alternatively with single or crossed beams, are not directly comparable becauseof very different collision velocity distributionsf (v, T ) [32].

Since the early ET studies, it was noticed that the cross sections tend to increase as the energy defectdecreases [2] and attempts were made to establish empirical formulas describing such a tendency (e.g.,in refs. 7 and 8 for EP processes). Considerations summarized in the previous section suggest that thecross section of each individual process is determined by the relevant molecular terms and dominatinginteractions. We shall, therefore, consider the variation of cross sections as a function of energy defect,

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Review/Synthèse 1045

classifying the processes according to the presence or absence of the dipole–dipole interaction betweenthe initial and final states of the colliding atoms.

A logarithmic plot of the thermally averaged experimental cross sections for different exothermicET processes against the energy defect1E is shown in Fig. 1. The tendency of the cross sections toincrease as1E decreases is obvious. Interestingly, no systematic differences are observed between theprocesses in which the initial and final configurations are coupled by the dipole–dipole interaction, andthe processes in which these configurations are not coupled by this type of interaction. One might expectthat the cross sections for the latter processes should deviate towards smaller values from the tendencyof dipole–dipole processes. It is natural to suppose that the efficiency of the ET would decrease whenthe nondiagonal dipole–dipole interactionsVdd in Hamiltonian (5) are replaced by terms describing aweaker, higher order interaction. In an attempt to clarify the role of interactions involved in ET, we shallconsider a class of processes in which the dipole transitions between the initial and final atomic statesare forbidden, namely, the ET between the S and D states (hereafter referred to as S-Dtransfer)

M∗∗ (n1S) + M (n0S) ↔ M∗∗ (n2D) + M (n0S) + 1E (10)For these processes one can expect that the nondiagonal term in Hamiltonian (5) is nearly zero.

4. S-D transfer processes

Until recently very little information was available in the literature on S-D transfer processes. Forlithium, the rate constant for the process

Li (4S) + Li (2S) → Li (4D) + Li (2S) − 1611 cm−1 (11)determined in thel-mixing experiment [33] allows one to estimate the cross sectionσ Li−Li4S→4D ≈ 1.7 ×10−16 cm2. However, this cross section cannot be directly compared to the values shown in Fig. 1,because process (11) is strongly endothermic.

In rubidium, for the ET between the states 8S and 6D, the available information was contradictory.An unexpectedly large cross sectionσRb−Rb8S→6D = (6.7 ± 3) × 10−13 cm2 was reported in ref. 34. At thesame time, in ref. 35 the very same cross section was determined as (6.1± 2.1) × 10−14 cm2, which issmaller in magnitude by more than an order of the former value. It suggested that this process shouldbe reconsidered with care.

In cesium, cross sections have been measured for the ET between some Rydberg states: from the 9Dand 10D states to the 10S, 11S, and 12S states, and they are all of an order of 10−15 cm2 [36]. It shouldbe noted, however, that collisions involving Rydberg atoms present a separate category. Due to theirspecific properties, Rydberg atoms are treated by theoretical methods different from those describedin Sect. 2. Such collisions are considered in terms of two separate processes: scattering of the partneratom on (i) the highly excited electron and (ii ) the ionic core of the Rydberg atom [37]. For this reasonwe do not include these processes in our discussion.

We shall concentrate hereafter on the recent experiments [18–21] on S-D transfer in Na, K, and Rb,in which the cross sections for the following processes were determined:

Na(4D) + Na(3S) → Na(5S) + Na(3S) + 1348 cm−1 (12)

K(6S) + K(4S) → K(4D) + K(4S) + 53 cm−1 (13)

Rb(7S) + Rb(5S) → Rb(5D) + Rb(5S) + 607 cm−1 (14)

Rb(8S) + Rb(5S) → Rb(6D) + Rb(5S) + 358 cm−1 (15)

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Fig. 2. Energy levels and excitation schemes for (a) process (12) in Na,1 denotes the difference between theenergy of the laser photonhν3P−4D and that of the 3P3/2 state; (b) process (13) in K, two lasers with photon energieshν4S−4P andhν4P−6S are used for excitation; (c) processes (14) and (15) in Rb,1 denotes the difference betweenthe energy of the laser photonhν5P−7S and that of the 5P3/2 state on stepwise excitation of the 7S state, whereashνmarks the photon energy on two-photon excitation of the 8S state. Broken lines depict the energies of the (PJ +PJ ′ )states.

We shall review briefly the experimental methods and the main results of these studies and the theoreticalinterpretation [22] for process (14). On the basis of the latter, we shall also attempt to explain qualitativelythe results for the remaining three processes.

4.1. Experimental methods and resultsProcesses (12)–(15) were studied in vapour, using different schemes of optical excitation (see Fig. 2).

The Na(4D), K(6S), and Rb(7S) states were excited by cw (continuous wave) laser radiation. Forexcitation of the K(6S) state the radiation from two lasers was used resonantly in a two-step excitation

scheme 4Shν1→ 4P hν2→ 6S (Fig. 2b) [19]. The Na(4D) and Rb(7S) states were excited by using a single

laser resonant with the second step transition 3Phν→ 4D or 5P hν→ 7S (Figs. 2a and 2c) [18,20]. In the

first step transition, the resonance states were excited by means of the optical collision induced excitation[10,38]

Na(3S) + Na(3S) + hν3P−4D → Na(3P) + Na(3S)

Rb(5S) + Rb(5S) + hν5P−7S → Rb(5P) + Rb(5S)©2001 NRC Canada

Review/Synthèse 1047

Fig. 3. Dependence of the intensity ratioI4D→4P / I6S→4P onN0 〈v〉 for process (13) in K:•, experiment; continuousline, least-squares fit of (18) to the experimental points withσ6S→4D = (1.8 ± 0.6) × 10−15 cm2 and σ q4D =(2.7 ± 1.1) × 10−14 cm2. (Adopted from ref. 19.)

Although such a method allows one to simplify the experiment considerably, it is restricted to thecases when the energy of the laser photon is not much different from the energy of the resonance state.Moreover, since the excitation is collision-induced, it is efficient only at sufficiently high densities ofatoms.

For the Rb(8S) state, the two-photon excitation by pulsed laser radiation was used (Fig. 2c) [21]. Anadvantage of pulsed excitation is that it allows for time-resolved detection, which gives direct access totemporal development of excited state populations and thus the mechanisms of population transfer. Italso allows determination of the excited state lifetime and quenching cross section from the populationdecay of the laser-excited state.

Solution of the rate equations developed to model the experimental situation allows one to extractthe corresponding ET cross sections from the experimental data. The cases of continuous and pulsedexcitation schemes will be considered separately.

4.1.1. Continuous excitation

If the S state is excited by laser radiation, and excitation is collisionally transferred to the D state,the rate of population changes in the D state is

dNDdt

= σS→D〈v〉N0NS − 1τD

ND − σ qD〈v〉N0ND (16)

whereNS, ND, andN0 are the number densities of atoms in the excited S and D, and ground states,respectively;τD is the lifetime of the D state, andσS→D andσ qD are the cross sections for S-D transferand the D-state quenching in collisions with the parent ground-state atoms, respectively. If the intensityIi→P of fluorescence from an excited state|i〉 to the resonance state P is expressed as the number ofemitted photons per unit of time, then

Ii→P = Ai→PNi (17)

whereNi denotes population of|i〉 andAi→P the appropriate Einstein coefficient. Under steady-stateconditions of cw excitation applied in refs. 18–20, dND

/dt = 0 and the ratio of collisionally induced

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Fig. 4. Possible processes leading to population transfer from the 8S state to the 6D state in Rb. Apart from thedirect collisional 8S→6D transfer (ET), the other contributing processes are (i) absorption of thermal radiation(BBR) to 8P, followed by spontaneous emission to 6D, (ii ) spontaneous emission to 7P followed by absorption ofthermal radiation. (Adopted from ref. 21.)

(ID→P) and optically excited (IS→P) fluorescence intensities is given as

η = ID→PIS→P

= AD→PAS→P

σS→Dσ

qD + 1

/(τDN0〈v〉)

(18)

whereAD→P andAS→P are the relevant Einstein coefficients for the transitions to the resonance statecorresponding to the observed fluorescence. Thus, the S-D transfer and quenching cross sections canbe obtained by fitting (18) to the experimental dependence of intensity ratio onN0〈v〉. Figure 3 showssuch dependence for the ET from 6S to 4D in potassium (process (13)) [19]. A similar approach wasused to obtain the cross sections for 4D→5S transfer in sodium (process (12)) and for 7S→5D transferin rubidium (process (14)) [18,20].) Radiation trapping [39,40], which should be accounted for in theenergy-pooling measurements, where fluorescence of resonance transitions is detected (see, e.g., refs.1, 26, and 41 and references therein), does not affect the S-D transfer measurements because of thelow number densities of the atoms in the resonance states constituting lower states of the fluorescencetransitions and the relatively small probabilitiesAD→P andAS→P of these transitions.

4.1.2. Pulsed excitation

In the study [21] of process (15), pulsed two-photon excitation of the 8S state and time-resolvedfluorescence detection were used. To reproduce the time dependence of the fluorescence signal fromthe 6D state, it was necessary to account for other transitions, apart from the collisional S-D transfer, bywhich this state is populated. Contribution to the population transfer due to thermal radiation-inducedstepwise transitions through the neighbouring 7P and 8P levels (Fig. 4) appeared to be significant at lowconcentrations in the experiment in ref. 21 (5.5 × 1011 cm−3 ≤ N0 ≤ 1.5 × 1013 cm−3).2 They werenot accounted for in earlier studies [34,35] of process (15), though the conditions of these experiments

2 In the case of cw experiments [18–20] with processes (12)–(14), the thermal radiation induced transfer can be disregarded,since in the atomic density range (1015 cm−3 6 N0 6 1016 cm−3) of those experiments the collisional S-D transfer rateexceeds significantly any rate of the thermal radiation-induced transitions:σS→DN0 〈v〉 � Ta→b.

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Fig. 5. Time-resolved fluorescence signal from the 6D state of Rb following two-photon pulsed excitation of the8S state: •, experiment; continuous line, least-squares fit according to (20b). (Adopted from ref. 21.)

were similar to the conditions of [21]. The ET from 8S to 5F followed by the spontaneous emission5F→6D can be neglected due to the small 5F→6D branching ratio. Evolution of population of thestates concerned is described by equations

dN8Sdt

= −08SN8S (19a)dN6D

dt= R8S→6DN8S − 06DN6D +

∑k

Tk→6DNk (19b)

dNkdt

= −0kNk + T8S→kN8S (19c)

Here,R8S→6D = σ8S→6D〈v〉N0 is the rate of collisional S-D transfer;0i is the total decay rate, due tospontaneous emission and collisional quenching of a state|i〉; k denotes the intermediate states 7P and8P. The rate of absorption of thermal radiation is expressed byTa→b = Ab→a (ga/gb) 〈nba〉, and ofemission (both spontaneous and that stimulated by thermal radiation) byTb→a = Ab→a (1 + 〈nba〉),where〈nba〉 =

[exp(Eba/kBT ) − 1

]−1 is the number of photons of thermal radiation per energy modeEba [42], andga ,gb are the statistical weights of the states. On solving system (19) with initial conditionsN8S(t = 0) = N08S, Ni 6=8S(t = 0) = 0, wheret = 0 corresponds to the moment of incidence of thelaser pulse (the laser pulse is much shorter than the lifetime of the excited state,1tL � τi), one obtains,with the help of (17)

I8S(t) = ξA8S→5PN08Se−08St (20a)I6D (t) = ξA6D→5PN08S

{R8S→6D08S−06D

(e−06Dt − e−08St) + ∑

k

T8S→kTk→6D

×[

e−06Dt(08S−06D)(0k−06D) − e

−0kt(08S−0k)(0k−06D) + e

−08St(08S−0k)(08S−06D)

]} (20b)

Here the factorξ accounts for the detection efficiency. Note, that (20b) contains only one fitting param-eter,R8S→6D, because0i are known from experiment and (or) theory,Ta→b andTb→a are calculated,and ξN08S is determined from the direct fluorescence signal according to (20a). Figure 5 shows theexperimental 6D→5P fluorescence signal and a result of fitting of (20b) to it. From the fitted value ofR8S→6D, the cross section for the process (15) was determined.

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1050 Can. J. Phys. Vol. 79, 2001

Table 1. Recently obtained cross sections for S-D transfer in homonuclear alkali atomcollisions.

1E Cross section (cm2)

Process (cm−1) Experimental Theoretical

Na(4D) + Na(3S)→Na(5S) + Na(3S) 1348 ∼10−15aK(6S) + K(4S)→ K(4D) + K(4S) 53 (1.8 ± 0.6) × 10−15bRb(7S) + Rb(5S)→ Rb(5D) + Rb(5S) 607 (8 ± 4) × 10−15c 7.0 × 10−15eRb(8S) + Rb(5S)→ Rb(6D) + Rb(5S) 358 (3.4 ± 1.2) × 10−14da Ekers, ref. 18.b Ekers and Alnis, ref. 19.c Caiyan et al., ref. 20.d Ekers et al., ref. 21.e Grushevsky et al., Orlovsky et al., ref. 22.

The shape ofI6D (t) is determined by the decay rates0i of the states involved in the populationtransfer process. If some of the processes were disregarded, the corresponding equation would fail toreproduce the experimental signal of sensitized fluorescence. This gives the possibility of verifyingthe assumed population transfer mechanism, which constitutes the main advantage of using pulsedexcitation and time-resolved fluorescence analysis vs. detection of time-averaged fluorescence signalsin cw experiments.

The cross sections for S-D transfer processes (12)–(15) measured in refs. 18–21 are summarized inTable 1. As can be seen in Fig. 1, the results for sodium and rubidium are in approximate agreementwith the cross sections for other ET processes with similar energy defects. Only for potassium does thecross section deviate considerably from the general trend towards smaller values.

4.2. Theoretical interpretation of S-D transferAmong the S-D transfer processes, only process (14) in rubidium has been considered theoretically

[22]. In that work, ref. 22, it was shown that besides the molecular terms correlating with the initial 7S+ 5S and final 5D+ 5S states, one has to also consider the terms correlating with the 5P+ 5P state,which lies only 68 cm−1 below 5D+ 5S. It is, therefore, necessary to extend the Hamiltonian (5) toaccount for the interaction with the terms of the 5P+ 5P configuration

H(σw

) = ESS+ αY6 (SS) 0 Vdd (SS− PP)0 EDS + βY6 (DS) Vdd (DS− PP)

Vdd (SS− PP) Vdd (DS− PP) EPP

(21)

whereVdd (SS− PP) andVdd (DS− PP) are the matrices of dipole–dipole interactions between theterms of the 7S+ 5S and 5P+ 5P configurations, and the 5D+ 5S and 5P+ 5P configurations, respec-tively. Figure 6 shows the system of the 0+g terms, resulting from Hamiltonian (21). The nonadiabatictransitions within these terms give the main contribution to the S-D transfer [22]. The role of the 5P+ 5P configuration is decisive in this case. It introduces a strong dipole–dipole coupling among theterms correlating with the 7S+ 5S and 5D+ 5S states, thus opening efficient channels that lead to thecollisional population of the 5D state.

The mechanism of the S-D transfer process (12) in sodium should be similar, because here the 3P+ 3P state lies between the 4D+ 3S and 5S+ 3S states (Fig. 2a), and the system can be describedby a Hamiltonian similar to (21). On generalizing the results of ref. 22, it is because of the P+ Pconfiguration that the energy transfer in both cases (12) and (14) is determined by the dipole–dipoleinteraction. In the transfer between the 8S and 6D states of Rb (process (15)), a possible perturbing roleis played by the terms correlating with the neighbouring 7P+ 5S and 8P+ 5S states. In view of these

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Fig. 6. Rb2 terms of 0+g (continuous lines) and 1g (broken lines) symmetry, correlating with the 7S+ 5S, 5D+ 5Sand 5P+ 5P states of separated atoms. (Adopted from ref. 22.)

arguments, it is not surprising that cross sections for processes (12), (14), and perhaps also (15), canbe found in Fig. 1 at a place consistent with the common trend for the cross sections of dipole–dipoleprocesses.

The energy transfer between the 6S and 4D states of potassium (13) is noteworthy because it hasthe smallest energy defect among the S-D transfer processes considered. If the cross section for process(13) were to follow the tendency seen in Fig. 1, a value some 30 or 40 times bigger would be expected.Process (13) differs, however, from the other S-D transfer processes considered because the initial 6S+ 4S and the final 4D+ 4S configurations are isolated from the other states (Fig. 2b). Dipole–dipoleinteraction with the 4P+ 4P configuration can be practically neglected because of the large energyseparation (−1365 cm−1). It is evidenced also by EP experiments [43], in which no fluorescence fromthe 4D state was observed when the 4P state was optically excited. One can exclude also the influenceof the 6P+ 4S state, which lies 1548 cm−1 above 6S+ 4S. The contribution of the higher order termsin the expansion of the Coulomb interaction may be neglected (see the remark at the end of Sect. 2). It ispossible that the process proceeds owing to interaction of the 6S+ 4S and 4D+ 4S configurations withthe ionic K+ + K− state at large internuclear distancesR > 60 a.u. (see the potential curves calculatedin ref. 44). However, more certain conclusions about the mechanism would require detailed theoreticalcalculations.

5. Summary

As a starting point, we have taken an empirical fact noticed earlier by various authors [2,7,8]concerning the cross sections for collisional ET between various states of alkali metal atoms. Whensuch values are plotted as a function of energy defects they tend to cluster along a common line, onaverage decreasing with increasing energy defects. On first glance, such an approximate relation holds,without any discernible systematic differences, both for processes in which the initial and final statesof colliding atoms are coupled by the dipole–dipole interaction, and for those in which the states arecoupled by higher order interactions (Fig. 1). This leads to a superficial, surprising conclusion that it isthe energy defect and not the type of interaction that determines efficiency of the ET.

In this paper, we have examined the ET between the excited S and D states induced by collisionswith parent ground-state (S) atoms, on the basis of recent experimental results for processes (12)–(15),and the theoretical treatment of the process (14). We argue that this example demonstrates that the typeof interaction does play an important role for the efficiency of these processes.

For the S-D transfer, the dipole–dipole interaction between the initial S+ S and final D+ S atomic©2001 NRC Canada

1052 Can. J. Phys. Vol. 79, 2001

states seems to be excluded. However, in most cases the ET between the initial and final atomic statescan be influenced by other atomic states lying nearby. For process (12), Na(4D)+ Na(3S)→Na(5S)+ Na(3S), and (14), Rb(7S)+ Rb(5S)→Rb(5D) + Rb(5S), efficient nonadiabatic transitions occurbetween the terms of the S+ S and D+ S configurations due to their dipole–dipole interaction with thenearby P+ P configuration. The strong dipole–dipole coupling introduced in this “indirect” way causesefficient S-D transfer. It is possible to explain process (15), Rb(8S)+ Rb(5S)→Rb(6D)+ Rb(5S), ina similar way, by accounting for two nearby P+ S configurations.

The case of the process (13), K(6S)+ K(4S)→K(4D) + K(4S), is different. Here, the initial andfinal states are actually well isolated from the other states, which allows the dipole–dipole couplingto be excluded. The cross section for this process deviates strongly from the common trend of Fig. 1towards smaller values and this could be expected from the theory sketched in Sect. 2 if applied for anon dipole–dipole process. One should, therefore, be careful in attempts to predict an unknown crosssection for an ET process merely from its energy defect and an empirical relation like the one suggestedby Fig. 1.

It is outside the scope of this paper to explain why the cross sections for the other non dipole–dipoleprocesses, marked by open squares in Fig. 1, follow the same trend as the dipole–dipole ones. It isanticipated, however, that an explanation can be found along the same lines as that for processes (12),(14), and (15). The existence of isolated terms, as in the case of process (13) in potassium, is ratherrare. The higher the states are excited, the more abundant are the neighbouring terms and the higher thechance that perturbations similar to those found for processes (12) and (14) lead to strong dipole–dipolecoupling.

Acknowledgements

This work was partially supported by the Polish Committee for Scientific Research (grant No. 2P03B 065 11) and by the Latvian Science Council (grant No. 96.0609).

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