Production of KRb+: the relative efficiency of two competing channels

Journal of Molecular Structure (Theochem) 529 (2000) 15–19 www.elsevier.nl/locate/theochem

Production of KRb ⫹: the relative efficiency of two competing channels B.C. Saha a,*, A. Kumar b, S.K. Verma c a

Department of Physics, Florida A&M University, Tallahassee, FL 32307, USA b Department of Physics, J.P. University, Chapra 841301, India c Department of Physics, Jagdam College, Chapra 841301, India Received 5 November 1999; accepted 17 January 2000

Abstract At thermal energies, the Hornbeck Molnar ionization (HMI) cross-sections for the production of heteronuclear alkali dimer KRb ⫹ are calculated using the improved theoretical approach within the framework of Fermi’s free-electron model. For the first time we report the associative ionization cross-section for a heteronuclear system involving excited K and Rb alkali-metal atoms. The roles of two alternative reactions for the production of KRb ⫹ are discussed. Present cross-sections successfully account for the relative efficiency of the two competing channels that can produce the above molecular ion. At large n (i.e. n ! ∞† the present cross-sections obey the correct asymptotic behavior, in conformity with the previous homonuclear studies of Kumar et al. (see A. Kumar, B.C. Saha, C.A. Weatherford, S.K.Verma, J. Mol. Struct. (Theochem) 487 (1999) 1). 䉷 2000 Elsevier Science B.V. All rights reserved. Keywords: Associative ionization; Hornbeck Molnar cross-sections; Heteronuclear alkali dimer PACS: 34.60. ⫹ z

1. Introduction Associative ionization (AI) happens to be an important outcome of the thermal collision of an excited atom A ⴱ with another ground-state (or excited) atom B: Aⴱ ⫹ B ! …AB†ⴱ ! AB⫹ ⫹ e⫺

…1†

provided the minimum energy supplied in the collision is equal to the difference between the lowest atomic ionization and the molecular binding [1]. * Corresponding author. Tel.: ⫹1-850-599-3000; fax: ⫹1-850599-3953. E-mail address: [email protected] (B.C. Saha).

This process becomes the principal mechanism of quenching the excited-state populations in cold plasmas by creating molecular ions. This has also been observed in low-energy electron-beam-excited gases, shock waves, and hot gases [2]. Due to their wide application, such ionizing collisions have been studied in detail for two resonantly excited alkali atoms [3–8]. Hornbeck Molnar ionization (HMI) reaction is a special case of AI, where one partner is prepared in an excited Rydberg state while the other remains in its ground state. Attempts have been made to investigate the HMI reaction rate coefficients as a function of principal quantum number n of the excited target atom through gas-cell, crossed-beam, and single-beam experiments. But the majority of these

0166-1280/00/$ - see front matter 䉷 2000 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(00)00525-X

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B.C. Saha et al. / Journal of Molecular Structure (Theochem) 529 (2000) 15–19

experimental studies are confined to homonuclear systems [1,5,6,9–13]. Only Djerad et al. [14], however, have measured the rate coefficients for various mixture of Rb and K Rydberg atoms for the production of the molecular dimer KRb ⫹ via these reactions: Rbⴱ …nl† ⫹ K…4s† ! KRb⫹ ⫹ e⫺

…2a†

Kⴱ …nl† ⫹ Rb…5s† ! KRb⫹ ⫹ e⫺

…2b†

at a cell temperature of 443 K. On the theoretical front, the free-electron model of Fermi [15] is known to provide a convenient simplification for the study of collisions involving Rydberg atoms. Invoking this model, Kumar and Gounand [16] were able to account for the production of both molecular and atomic ions (Penning ionization, PI) in thermal collisions of low-lying Rydberg alkalis. In this prescription the Rydberg atom is virtually “split” into two entities: the weakly bound distant electron and the positively charged core that encompasses the target nucleus along with the tightly bound electrons of the inner shell/shells. This facilitates treating the electron–perturber (the other colliding atom) and the core–perturber interactions quite independently; one does not affect the other as the regions of interaction of each are drastically different. Taking advantage of this virtual separation of the two pairs of interactions, Kumar and Gounand have proposed that the electron– perturber collision accounts for the production of atomic ions (PI) while the core–perturber interaction is mainly responsible for the formation of the molecular dimers like KRb ⫹ in reactions (2a) and (2b). The above method, though successful in explaining both PI and HMI reactions in the case of low-Rydberg alkali atoms, is known to suffer from an inherent limitation: it predicts a cross-section that increases monotonically with the principal quantum number (n) of the Rydberg atom. With rise in n we go up in the energy level diagram where we encounter the crowding of quantum states. As a result, other neutral channels, such as state-changing, l-mixing, and total quenching, become energetically more efficient and take over the ionizing collisions. As these features are ignored in the simplified, but convenient model of Ref. [16], the estimated cross-sections attain a somewhat unphysical nature in the domain of high

n. In the present work we report a simple improvement to the above model by introducing an external multiplier; this significantly enhances its utility, specially at large n-values. We then apply the modified formula to calculate the HMI cross-sections for reactions (2a) and (2b) at thermal impact energy, for which no prior theoretical calculations exist. According to Kumar and Gounand [16] the HMI reaction cross-section (with their notation, see also Ref. [18]) is given by Z∞ s HM ˆ 32u2n …D=E†1=2 dv2 s c-p ‰Fn …v; tmax † 0

⫺ Fn …v; tmin †Š

…3†

where D ˆ ⫺ed is the energy of dissociation of the molecular ion; following Alekseev and Sobelman [17] the core–perturber elastic cross-section (s c-p) can be written as

s c-p ˆ 7:18‰a=…v=mR ⫹ v0 †Š2=3 : It has been derived on the basis of JWKB phase shifts for the polarization potential. 2. Theoretical method: a brief sketch Although the details of the present modification have been discussed in Ref. [18], we provide a brief sketch here. It is useful to look at the concept of a transient-step to derive the required energy conditions for molecular binding in an HMI reaction (see Ref. [18] for details): Aⴱ ⫹ B $ A ⫹ B⫹ ⫹ e⫺ $ …AB†⫹ ⫹ e⫺

…INITIAL† Step I

…TRANSIENT† Step II

…FINAL† Step III

…4†

Thus the probability of the formation of the molecular ion is primarily determined by the relative magnitudes of two quantities, the ionization potential of the Rydberg target Aⴱ …en † and the energy of dissociation of the formed ion AB⫹ …ed †. Binding of the collision fragments (i.e. A ⫹ and B) will readily take place, leading to the formation of the molecular dimer provided 兩ed 兩 ⬎ 兩en 兩

…5†

This fact enables us to define a lower limit of the principal quantum number …nl † upto which no

B.C. Saha et al. / Journal of Molecular Structure (Theochem) 529 (2000) 15–19

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Fig. 1. HMI cross-section for the production of KRb ⫹ …a20 † in reactions (2a) and (2b). Present calculations for these reactions are shown as P1 and P2, respectively, and those due to Kumar and Gounand [16] are K1 and K2. The experimental measurements are due to Djerad et al. [14]: solid rectangles and circles are for s and d states, respectively, for reaction (2a), and the open symbols are for the other reaction.

correction is needed in the model of Kumar and Gounand [16]. Considering the nature of dependence of en on n, we define this lower limit by an identical expression:

ed ˆ 1=‰2…nl ⫺ dl †2 Š

…6†

For n ⱖ nl ; a lesser amount of energy will be needed to strip off the Rydberg electron. As a result, the process of fragment binding will have to compete against the enhanced kinetic energy made available during the collision (see also Ref. [16]). In other words, the conceptual step II of Eq. (4) will have a definite possibility of being realized in practice. The probability of binding (step III of Eq. (4)) will thus go on decreasing as the principal quantum number of the

Rydberg target increases, and beyond a certain maximum value of n (say nu) the same should reduce to zero. Physically, this signifies that the production of a molecular target will be practically impossible (the cross-section attaining negligibly small magnitude) as other exit channels will be energetically favored due to crowding of the quantum states. We correlate this upper limit of n with the amount of impact energy made available to the colliding pairs at the cell temperature (T ); this is determined by nⴱu ˆ nu ⫺ dl ˆ 1=…2kT†1=2

…7†

Both the lower and upper limits for n, defined above, are obtained by adding the amount of quantum defect to the effective quantum number; if it is not an integer

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B.C. Saha et al. / Journal of Molecular Structure (Theochem) 529 (2000) 15–19

one must take the next integral number as representative of the principal quantum number. We then define a dimensionless quantity to account for the relative separation of the target’s n with respect to the above-defined limits (nl and nu): f …n† ˆ …n ⫺ nl †=…n ⫺ nu †

…8†

The external multiplier is finally defined as …9†

F…n† ˆ ‰1 ⫺ f …n†Š2 which modifies Eq. (3):

s HM ˆ 32u2n …D=E†1=2 F…n†

Z∞

⫺ Fn …v; tmin †Š

0

dv2 s c-p ‰Fn …v; tmax † …10†

3. Results and discussion To assess the relative efficiency of the two possible entrance channels (2a) and (2b) producing the molecular dimer KRb ⫹ at thermal energies, we use the modified Eq. (10) to estimate the HMI cross-sections for the first time. Our calculated cross-sections, along with the experimental measurements of Djerad et al. [14] and the original results of Ref. [16] are presented in Fig. 1. It may here be pointed out that the HMI cross-sections, in general, are independent of cell temperature in the thermal energy region. Therefore, the measured reaction rates can easily be transformed into the corresponding cross-sections by dividing them with the most probable relative velocity at the cell temperature. Djerad et al. [14] have performed these experiments at 443 K; Kumar and Gounand have also presented their results at the same temperature, and so have we, in order to visualize the true role of the proposed modification. The estimated cross-sections for both interactions are found to increase with rising principal quantum number, attain comparatively flat maxima around n ˆ 11, and start decreasing with further increase in n. They exhibit satisfactory agreement with the measured values of Djerad et al. [14]; the results for Rb ⴱ(ns) always lie within the experimental uncertainties (shown through error bars); for Rb ⴱ(nd), the measured values are smaller than our calculated cross-sections. Further, the original estimates of Ref.

[16] are larger in case of Rb(nd) colliding with ground-state K. But the most significant improvement is that our present calculations exhibit a physically plausible n-dependent shape, specially in the region of large n-value, where the estimates based on relation (3) possess diverging nature. To the best of our knowledge, for reaction (2b) there is only one set of experimental measurements [14] for both s and d states (7s and 5d). At these nvalues, no modification in the original model [16] is needed, and hence the true effect of the proposed external multiplier cannot be fully appreciated. The estimated HMI cross-sections for this system overestimate the experimental results. But the proposed modification certainly provides a better n-dependence for the calculated cross-sections. The relative efficiency of the two possible reactions (2a) and (2b), producing the heteronuclear molecular dimer KRb ⫹ can be easily estimated from the magnitudes of the calculated cross-sections. In both the original and modified prescriptions, the cross-sections are larger for the entrance channel (2a), in complete agreement with the experiment [14]. For a better assessment of the relative efficiency of the two channels, we consider comparable systems: Rb(8s/ 6d) ⫹ K(4s) and K(7s/5d) ⫹ Rb(5s), because the effective quantum numbers for Rb[8s(6d)] and K[7s(5d)] are 4.869(4.653) and 4.82(4.723), respectively. The experimentally measured cross-sections for channels (2a) and (2b) for the above comparable colliding pairs have a ratio of 1.56 and 1.19 for the s and d states, respectively. In the same region of effective quantum number, our study predicts that the HMI cross-section for reaction (2a) should be nearly 1.35 times larger than that for reaction (2b). These observations are supported by physical considerations, as well: the Rydberg ns states of Rb are slightly more hydrogenic than the corresponding states of K. Comparatively larger deviation from an ideal hydrogenic configuration, in the case of the d states of both Rydberg targets (K and Rb), is perhaps responsible for producing smaller HMI cross-sections. But as the principal quantum number of the target atom increases, the two processes tend to attain nearly equal efficiency. As we go up in the energy level diagram the difference between the excited Rb (and K) atom and a true hydrogenic state diminishes, and hence the two channels turn out to be equally efficient

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in producing the dimer KRb . This observation, however, awaits further experimental measurements at higher n, in the case of system (2b), for confirmation. Finally, it may also be pointed out that the present approach has recently been successfully used to investigate the formation of homonuclear alkali dimers in slow collisions of low-Rydberg alkalis with the respective ground-state atoms [18]. We have thus improved upon the earlier results of Kumar and Gounand by introducing a simple multiplying term, and calculate the HMI cross-sections for the production of the heteronuclear molecular dimer KRb ⫹ through two possible reactions. The relative efficiency of these two competing channels, over a long range of n, are revealed through the estimated cross-sections. Our calculated results, in contrast to the earlier work of Kumar and Gounand [16], also exhibit a better asymptotic behavior in the limit of large n. Agreements with the experimentally measured cross-sections are also found to be satisfactory. The proposed improvement significantly enhances the utility of the free-electron model for estimating the HMI cross-sections at thermal impact energies. Acknowledgements B.C.S. would like to acknowledge the support from the Research Corporation and from the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army

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Research Laboratory cooperative agreement number DAAH04-95-2003/contract number DAAH04-95-C008, the content of which does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred.

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