© Copyright 1996 by the American Chemical Society
VOLUME 100, NUMBER 51, DECEMBER 19, 1996
LETTERS Time and Frequency Resolved ZEKE Spectroscopy F. Remacle† De´ partement de Chimie, B6, UniVersite´ de Lie` ge, B 4000 Lie` ge, Belgium
U. Even School of Chemistry, Tel AViV UniVersity, Ramat AViV, Tel AViV 69978, Israel
R. D. Levine* The Fritz Haber Research Center for Molecular Dynamics, The Hebrew UniVersity, Jerusalem 91904, Israel, and Department of Chemistry and Biochemistry, UniVersity of California at Los Angeles, Los Angeles, California 90024-1569 ReceiVed: September 30, 1996; In Final Form: NoVember 4, 1996X
ZEKE spectroscopy is based on delayed detection by pulsed field ionization. It is thereby possible to monitor the time evolution at a given excitation frequency. Moreover, by varying the depth of detection, one can harvest different Rydberg series. The qualitative features expected for such a spectrum are discussed. The quantitative theory required to compute spectra is outlined and applied to the realistic example of Na2+. The computed spectrum is found to very accurately exhibit two time scales, just as has been observed, with the shorter decay time being faster for lower Rydberg states. Extensive interseries coupling is noted.
Introduction A ZEKE spectrum is different from an ordinary photoionization spectrum in the essential way that only such states that have not ionized after a finite time delay (often for more than a microsecond) are detected.1,2 A secondary point is that those states are detected that can be ionized by an applied external electrical field. It is the delayed application of this field (socalled pulsed field ionization1-3) that serves as the detector. By varying the delay until the ionizing field is applied, one can observe the spectrum as a function of the delay.4-10 In this Letter we discuss the features expected for such a time and frequency resolved spectrum, outline the required theory, and †
Chercheur Qualifie´, FNRS, Belgium. * Corresponding author. Fax, 972-2-6513742; e-mail, rafi@batata. fh.huji.ac.il. X Abstract published in AdVance ACS Abstracts, December 15, 1996.
S0022-3654(96)03005-5 CCC: $12.00
present computational results for a realistic model of Na2+. We shall show that the time evolution of such a spectrum exhibits two time scales, as has been observed for quite a number of polyatomic molecules.4-6,8,9 The faster decay for the case of Na2+ was first reported by Bordas et al.11 The main special feature of ZEKE spectroscopy requires that even if the total energy is above the lowest ionization threshold, the state(s) optically accessed must be bound. Of course, this is possible only if there are bound states isoenergetic with the continuum. In molecules there is a natural mechanism: the Rydberg electron in its highly extended orbit revolves about an ionic core. The lowest ionization threshold is when after ionization the core is left in its ground state. But each rovibrational state of the core can support its own series of high Rydberg states. Each such series has its own ionization threshold which is higher than the lowest one by the excitation energy of the core. It follows that at any energy © 1996 American Chemical Society
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Figure 1. Rydberg series built upon different rotational states of the ionic core on a common total energy scale. The onset of ionization and the depth of detection (the “detection window”, shaded) are shown for each series. A stronger field F will allow detection also of the j ) 10 and 12 series. As the frequency of the laser is varied, one can cross new thresholds but also one can cross thresholds for detection of additional series.
above the lowest threshold there are bound states isoenergetic with the continuum. The point about ZEKE spectroscopy is that the thresholds can be resolved as each state of the ionic core is characterized by its own ZEKE peak. One of the essential results that the theory must provide is, why is the ZEKE spectrum observed anew hugging every new threshold? In other words, why do we not see a ZEKE spectrum at all excitation energies; what is special about being just below a new threshold? The answer, as originally argued,4 is that states which energetically lie not immediately below the threshold can be excited but are quick to escape from the delayed detection. This idea of a “detection window”12 is the other point that governs the features of the ZEKE spectrum. The detection window is determined primarily by the temporal profile of the delayed electrical field applied to ionize the excited states. It has been recognized for some time3 that one can shape the field to advantage, but it is only more recently that this is being used.13-15 In this Letter we shall use a simple approximation, namely, that an external dc field can ionize bound Rydberg states only down to a given depth below the ionization limit, Figure 1. The depth of detection is determined by the magnitude, F, of the field, and we take it to be 4xF cm-1, for a field measured in volts per centimeter. In principle, even lower states can ionize by tunneling, but for weak fields the rate is negligible.16 Strictly, speaking, the yield of classically allowed ionization is not quite a step function in the field strength,16 and this does have observable implications.14 Here, however, we take it that depending on its energy with respect to the series ionization limit, Figure 1, a Rydberg state will either ionize or will not. Figure 1 emphasizes a key feature of the detection, namely, that not all nearly isoenergetic Rydberg states can equally be detected. If a state is or is not detected is determined by its
Letters binding energy (-Ry/n2, where Ry is the Rydberg constant) with respect to the ionization threshold of the series it belongs to. At the same total energy E, different series can have rather different values of the principal quantum number n, i.e., E ) IP + ∆ - Ry/n2, where ∆ is the excitation energy of the ionic core and IP is the lowest ionization potential. Note in particular that it requires a dc field F of the order of [(vibrational spacing in cm-1)2/16] V/cm to ionize states which belong to cores in different vibrational states. Hence the situation shown in Figure 1 is generic: At a given excitation energy it will not be all bound Rydberg states that are detected. One important aspect is not well represented by Figure 1. The states depicted therein are zero-order states. Interseries coupling, if it is important, will mix states, preferentially those which are nearly isoenergetic. Such a process will, however, alter the electronic energy of the state and can therefore move states into or out of the detection window. In this first publication we do not discuss this point in further detail, but it is important to recognize that in so far that the (zero-order) states which are optically accessed and the (zero-order) states which are detected need not be the same, there is considerable scope for otherwise unexpected time evolution of the spectrum. In particular, due to interseries coupling states which have not been optically accessed (e.g., due to Franck-Condon propensity rules17 or due to rotational selection rules18,19) can be detected. A computational example of this point is provided below. Readers of this paper will differ in their assessment as to how important is interseries coupling between bound states. Rather than take a position, we shall, below, compute the dynamics of the Hamiltonian. There is, however, clear experimental evidence that the ZEKE spectrum does evolve with time. For states above the lowest ionization threshold, this can be due to autoionization. As is clear from Figure 1, autoionization is a bound-free interseries coupling. Moreover, at the energies in question (ZEKE), the density of bound states is comparable to if not higher than that of the continuum.20-22 If autoionization is possible, bound-bound interseries coupling can also not be ignored. The states lying below the lowest IP can only “disappear” by exiting from the detection window which requires that their electronic energy decreases sufficiently. Since the balance of the energy must be in the core, such processes are often labeled “predissociation”. Of course, the coupling to the core may be mediated by Rydberg-valence excitation whereby the high Rydberg excited state is coupled to a doubly electronically excited state of the neutral molecule.23 The ZEKE spectrum in the absence of any interseries coupling is expected to be quite simple. The optical transition to bound high Rydberg states scales as24 n-3 and in any series there are n3 states per energy interval. Say the width of the excitation laser exceeds the spacing of states equal to 2 Ry/n3 cm-1 ≈ 0.06 cm-1 at n ) 150. Then, as a function of frequency, a given series contributes an essentially flat cross section extending all the way up to its ionization threshold. Instead, what is observed is a ZEKE “peak” hugging the threshold, as in Figure 2. Moreover, the overall peak intensity decreases with time with the early time decrease being much more noticeable on the red side of the peak. Figure 2 is drawn for a laser width of 0.2 cm-1 and for a single bound Rydberg series whose time evolution is mimicked by assigning to each bound state a width Γ0/n3, with Γ0 ) 100 cm-1. The line shape25 of each state is thereby spread, and because its area remains unity its maximum is reduced by n3/ Γ0, leading to a ZEKE-like peak hugging the threshold. If we were to take Γ0 f 0, this computation would recover the flat
Letters
J. Phys. Chem., Vol. 100, No. 51, 1996 19737
Figure 2. ZEKE peak vs frequency, for a broad laser excitation, for two delays in the detection. The plot shows the characteristic asymmetric shape and the faster loss of population at the red side. In an actual experiment, there is a thermal distribution of initial states which allows several series to be excited simultaneously resulting in a (inhomogeneously) broader peak.
spectrum discussed above, with discrete lines at lower ns, where the laser can resolve individual states. ZEKE spectroscopy is particularly useful because ZEKE peaks hug any new threshold. Figure 2 suggests that this is due to the rate of disappearance of states of lower ns from the detection window being faster. It is up to the full dynamical theory, as presented below, to capture this behavior in an abinitio fashion. Theory The detailed theory which enables us to compute a time and frequency resolved spectra will be presented elsewhere. The essence is that the theory mimics the sequence of experimental steps. First, the molecule is excited over a long time interval, which we take to be from -∞ to time zero. In this way the frequency of the excitation can be sharply defined. Only transitions into bound states are allowed. Then, at time zero the laser is turned off, and the system evolves under its own Hamiltonian plus any other external perturbations that are deemed to be part of a realistic description of the experiment. At a time t, t > 0, we apply a detection which counts all the states within the detection window. Explicitly, the detection operator is given as
Π ≡ θ((16F/-E2n) - 1) - θ((16Fs/-E2n) - 1) Here θ(x) is the unit step function which serves to count all Rydberg states that can be ionized, F (V/cm) is the external field, and FS is any electrical field, stray or otherwise, which is present throughout the time evolution and causes ionization of the very highest Rydberg states. The second term in Π accounts for these higher-most states which have been ionized due to the field FS. En ) -Ry/n2 is the energy (in cm-1) of the Rydberg states. The ZEKE cross section is determined by the expectation value of Π for the (quasi-bound) wave function at the time t, 〈Ψ(t)|Π|Ψ(t)〉. Note that the detection operator does not commute with the time evolution operator because, due to the interseries coupling, the latter can move zero-order states into or out of the detection window. Hence the spectrum can depend on time in a possibly intricate manner. In addition, autoionization, when energetically allowed, will lead to a decrease in the intensity. The relation of the present approach to the general theory of photoionization spectra26 will be presented elsewhere. The wave function Ψ(t) is computed as already discussed. The laser field is applied over the range -∞ < t e 0 and is then turned off. For t > 0 the time evolution is due to the
Figure 3. ZEKE spectrum vs laser frequency ω, where ω ) 0 is the (lowest) threshold for ionization, for different delays before ionization. The frequency range shown is above the threshold for ionization of j ) 4 and below that of j ) 6, as in Figure 1. The width of the spectrum is determined by the (broad, 0.2 cm-1) laser width and by the width of the Stark manifold of states due to the weak (0.1 V/cm) external electrical field which is included in the computation to mimic the field imposed to remove any promptly ionized electrons. Shown are results for optical excitation into the l e 2 states of the j ) 10 and 12 series. In the upper panel detection is limited to the j ) 6 and 8 series, while in the lower panel the ionizing field is strong enough to detect the series j ) 6-12; cf. Figure 1. For each spectrum the inset shows the decay of the integrated ZEKE intensity vs time (in µs) and a fit to a biexponential functional form. Using the peak height as a measure of intensity results in a very similar decay curve.
molecular Hamiltonian and the stray electrical field FS. The latter has the additional important role that above the InglisTeller limit it ensures that due to the Stark splittings there is a quasicontinuum of Rydberg states so that interseries coupling is facilitated.27-29 Note that above this limit individual n states cannot be resolved. In a full report on the computations we shall also discuss two additional effects. One is the role of external ions30,31 and the other is the depletion of Rydberg states due to “predissociation”, by which we mean all processes which require a particularly close approach of the electron to the core and are hence only effective at very low values of the orbital angular momentum l. The molecular Hamiltonian used here is as used before21 with rotational constant of the molecular core (0.11 cm-1) and quadrupole constant (-18 au) which are those of Na2+. For the results shown in Figure 3, the external field is FS ) 0.1 V/cm, which is sufficient to remove the promptly produced electrons. To mimic realistic conditions for Na2+, four bound series are retained in the zero-order basis: j ) 6 (with 158 e n e 166), j ) 8 (with 120 e n e 122), j ) 10 (with n ) 97), and j ) 12 (with n ) 81). For j ) 14 one is already well below the Inglis-Teller limit so that the coupling to it is not effective.27-29 This basis includes all l values and has 28 327 quantum states spanning an energy of 0.74 cm-1 about the mean energy of 0.44 cm-1 which is just above the (lowered by the external field) threshold of the j ) 4. The basis states are grouped into 1999 groups of states of given n, l, j, and M. (In
19738 J. Phys. Chem., Vol. 100, No. 51, 1996
Figure 4. Computed time-resolved ZEKE spectrum over a frequency range for a wide laser width. Details as in Figure 3 but for a very weak external field, FS ) 0.01 V/cm. Note that in both Figures 3 and 4 the upper panel shows detection confined to the j ) 6 and 8 series which are not directly accessed by the laser. Without the coupling induced by the core, there can be no signal. Yet after about 2 µs the broad resolution spectrum is essentially identical in the upper and lower panels. The buildup of population in the series which are not optically excited and the subsequent decay (see insets) are slower by a factor of 2 when the external field is weak.
other words, states of common values of n, l, and j but different mj and ml such that mj + ml ) M ) 0 are lumped together.27) The bound basis states span the energy width of the Stark manifold, 3n2(FS/2.34 × 104) cm-1. Only the j ) 6 series is directly coupled to the continuum by the quadrupolar anisotropy of the core. There are 166 open channels of the continuum (so that, as expected in general,20-22 the number of bound states is much higher than that of the continuum). The continuum is accounted for by using an effective Hamiltonian32,33 which is numerically diagonalized by a biorthogonal transformation. Thereby the wave function can be propagated in time. For Figure 4 which is computed for a much weaker external field, FS ) 0.01 V/cm, the j ) 4 series is bound and is included in the basis (with 248 e n e 252). We reiterate one technical point about the computation. The basis of states that we use, while quite large, is only sufficient to describe a limited energy range about the mean energy of 0.44 cm-1. The spectra that we show below are therefore but a slice of a complete frequency scan. The trade-off is that such a zero-order basis is very convenient for examining the role of interseries coupling and for specifying the detection window and the optical selection rules. Results Spectra are shown, at different values of the delay time before detection, vs the laser frequency for two different detection windows at two values of the external electrical field. The basis set used in the computation spans an energy >0.3 cm-1 which is the range shown. In a ZEKE experiment one performs a frequency scan over a wider range, and two-dimensional plots vs both frequency and time will be presented elsewhere. One
Letters result of such studies, which is not further discussed here, is that the decay of the lower ns is indeed faster, giving rise to the typical ZEKE profile shown in Figure 2. For each spectrum, an inset shows the integrated ZEKE intensity vs time and a fit to a biexponential decay as was used to fit the observed data.4-8,9 In each case, the biexponential fit is extremely tight (correlation coefficient >0.999) and the short and long lifetimes differ by an order of magnitude and are, roughly, 1 and 16 µs, respectively. Essentially the same decay curve is obtained when the peak height is plotted vs time. Figure 3 shows the results for an excitation laser with a broad width (0.2 cm-1). In both panels excitation is into the l e 2 states of the j ) 10 and 12 series. (There are rotational selection rules which, in combination with the intermediate state excited in the actual two-photon scheme, enable one to limit the range of initially optically accessed j states19). In the upper panel the detection is limited to the j ) 6 and 8 series. At very short times there is somewhat less signal because there is no direct optical excitation into those series which are detected. However, while the laser is on (i.e., for t < 0), the interseries coupling does shift the zero-order population around so that by t ) 0 there is already some population to be detected. Very soon thereafter the population in the bound series reaches a steady state,27,31 and the ZEKE signal diminishes with time. In the lower panel, all series can be detected. We have computed spectra, not shown, for identical other conditions but for a narrow laser width of 0.02 cm-1. Both time scales for the decay are found to be unaffected by the laser width. Figure 4 is similar to Figure 3 but for a very weak external electrical field, FS ) 0.01 V/cm. We emphasize that without interseries coupling there will not be any signal shown in the upper panels of Figures 3 and 4. The reason is that, by design, the optical excitation is into different series than the series which are detected. By superposing the two panels, it can also be seen that by about 5 µs the two broad laser spectra are quantitatively very similar. After about 1 µs (or 2 µs for Figure 4), there is already a steady state distribution over the bound series, independent of the initial optical excitation. A one-series computation at the same energy (optical excitation and detection of j ) 6 other bound series not included, not shown) and external field as in Figure 3 results in a spectrum where the submicrosecond scale decay is faster by a factor of 5. The time evolution of the spectrum, as shown in Figures 3 and 4, cannot therefore be explained in terms of a one-series l mixing.2,12,34,35 On the other hand, the Stark spreading of the states due to the stray field is beneficial to an enhanced interseries coupling.27-29 In Figure 4, where the external field is lower by an order of magnitude, the two decay time scales are slowed by a factor of 2. The interseries coupling is thus reduced, but because of the quasi-degeneracy of the states it is still effective. (In its absence there will be no signal in the top panel.) Concluding Comments A frequency and time resolved ZEKE spectrum can be computed for a realistic Hamiltonian, including a weak external electrical field. The spectra show peaks which hug the threshold for production of each state of the cation, with the red wing of the spectrum having a faster time decay. A significant (up to 40%) long (tens of microseconds) time component is evident in the spectrum. A spectrum is seen even when the optical excitation is into a Rydberg series which cannot be detected by delayed ionization, and this is taken as direct evidence for the role of interseries coupling.17
Letters In accord with time-resolved experiments which probe the submicrosecond and microsecond time scales4,8,9 the full dynamical computations of the spectrum reported in this Letter exhibit two quite distinct time scales. In a full report we shall relate these to the bottlenecks in the time evolution.31 A one bound series computation results is a significantly faster decay of the intensity. This, and the presence of a signal when excitation is exclusively into series which cannot be detected, points clearly to the role of interseries coupling due to the anisotropy of the ionic core, even for a weak external electrical field. The long time scale found in the present computations does not rquire that the time evolution be under the influence of external ions. The presence of ions will, however, enhance the extent of the long time decay and also will increase its decay constant. When we include the effect of predissociation, the intensity of the long time component is reduced but is still significant. The present computations are at an energy range above the lowest IP. Below the lowest IP, when autoionization is not possible, external perturbations can play a bigger role in the time evolution of the optically excited state. Acknowledgment. This work was supported by the GermanIsrael Binational Foundation (GIF) and by the James Franck program and used the computational facilities of SFB 377. We thank Prof. G. Gerber for detailed discussions of the unpublished17 ZEKE experiments on Na2 and Prof. E. W. Schlag for many discussions and for continuous interest and support. We thank Prof. U. Kaldor and Dr. F. Masnou-Seeuws for computing the quadrupole moment of Na2+. References and Notes (1) Mu¨ller-Dethlefs, K.; Schlag, E. W. Annu. ReV. Phys. Chem. 1991, 42, 109. (2) Schlag, E. W.; Levine, R. D. Comments At. Mol. Phys., in press. (3) Merkt, F.; Softley, T. P. Int. ReV. Phys. Chem. 1993, 12, 205. (4) Bahatt, D.; Even, U.; Levine, R. D. J. Chem. Phys. 1993, 98, 1744.
J. Phys. Chem., Vol. 100, No. 51, 1996 19739 (5) Zhang, X.; Smith, J. M.; Knee, J. L. J. Chem. Phys. 1993, 99, 3133. (6) Merkt, F.; Mackenzie, S. R.; Softley, T. P. J. Chem. Phys. 1995, 103, 4509. (7) Held, A.; Selzle, H. L.; Schlag, E. W. J. Phys. Chem., in press. (8) Even, U.; Ben-Nun, M.; Levine, R. D. Chem. Phys. Lett. 1993, 210, 416. (9) Even, U.; Levine, R. D.; Bersohn, R. J. Phys. Chem. 1994, 98, 3472. (10) Vrakking, M. J. J.; Lee, Y. T. J. Chem. Phys. 1995, 102, 8818. (11) Bordas, C.; Brevet, P.; Broyer, M.; Chevaleyre, J.; Labastie, P. Europhys. Lett. 1987, 3, 789. (12) Rabani, E.; Levine, R. D.; Mu¨hlpfordt, A.; Even, U. J. Chem. Phys. 1995, 102, 1619. (13) Dietrich, H.-J.; Lindner, R.; Mu¨ller-Dethlefs, K. J. Chem. Phys. 1994, 101, 3399. (14) Dietrich, H. J.; Mu¨ller-Dethlefs, K. M.; Baranov, L. Y. Phys. ReV. Lett. 1996, 76, 3530. (15) Schlag, E. W.; Held, A. Private communication. (16) Baranov, L. Y.; Kris, R.; Levine, R. D.; Even, U. J. Chem. Phys. 1994, 100, 186. (17) Nemeth, G. I.; Ungar, H.; Yeretzian, C.; Selzle, H. L.; Schlag, E. W. Chem. Phys. Lett. 1994, 228, 1. (18) Mu¨ller-Dethlefs, K. J. Chem. Phys. 1991, 95, 4809. (19) Buhler, B. Ph.D. Thesis; Wu¨rzburg University, Germany, 1990. (20) Remacle, F.; Levine, R. D. Phys. Lett. 1993, A173, 284. (21) Remacle, F.; Levine, R. D. J. Chem. Phys. 1996, 104, 1399. (22) Remacle, F.; Levine, R. D. J. Phys. Chem. 1996, 100, 7962. (23) Dulieu, O.; Masnou-Seeuws, S. M. Z. Phys. D 1994, 32, 229. (24) Bethe, H. A.; Salpeter, E. E. Quantum Mechanics of One- and TwoElectron Atoms; Plenum-Rosetta: New York, 1977. (25) Bixon, M.; Jortner, J. J. Chem. Phys. 1996, 105, 1363. (26) Wang, K.; McKoy, V. Annu. ReV. Phys. Chem. 1995, 46, 275. (27) Baranov, L. Y.; Remacle, F.; Levine, R. D. Phys. ReV. in press. (28) Mahon, C. R.; Janik, G. R.; Gallagher, T. F. Phys. ReV. A 1990, 41, 3746. (29) Merkt, F.; Fielding, H. H.; Softley, T. P. Chem. Phys. Lett. 1993, 202, 153. (30) Merkt, F.; Zare, R. N. J. Chem. Phys. 1994, 101, 3495. (31) Remacle, F.; Levine, R. D. J. Chem. Phys. in press. (32) Feshbach, H. Annu. ReV. 1962, 19, 287. (33) Levine, R. D. Quantum Mechanics of Molecular Rate Processes; Oxford University Press: Oxford, UK, 1969. (34) Chapka, W. A. J. Chem. Phys. 1993, 98, 4520. (35) Bixon, M.; Jortner, J. J. Phys. Chem. 1995, 99, 7466.
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