1. INTRODUCTIONThe fundamental three-body Coulomb problem is of long-standing interest in atomic physics. The main reason forthis is that the Hamiltonian of the system is in generalnonseparable. However, for atomic states with twohighly and asymmetrically excited electrons Nl1nl (N! n), several simplifications that are due to their distin-guishability in space and their negligible exchange effectmake the problem theoretically tractable. Such asym-metric double-Rydberg (ADR) states have been investi-gated experimentally in Ba,1,2 in Sr,3 and more recentlyin Yb (Ref. 4) because these atoms possess low double-ionization limits compatible with current pulsed laser se-quential excitations of the two valence electrons. Prima-rily long-range electronic correlations in highly excitedlow-l momentum ADR states5 have been observed. Ithas been shown that the recorded data closely fit a gen-eralized core excitation model in which the Ba1Nl1 core isno longer isolated for N ' 0.6n and is under the influenceof the outer nl Rydberg electron. The core is in fact po-larized by the electric field induced by the presence of theouter electron. Similar results were observed whenhighly excited Ba1-ion Rydberg states were recorded inthe presence of an appropriate static field simulating thisouter-electron-induced polarization effect.3 To fit the de-convoluted line profile, the field strength value is approxi-mately 70% of the calculated one in a classical picture inwhich the outer electron is frozen at its outer turningpoint. Obviously a detailed and quantitative analysis ofsuch adiabatic effects including the long-range Coulombelectron–electron (e –e) interaction appears to be of im-mediate interest. To investigate this interaction, ADRstates with a high-l nonpenetrating electrons have beensuccessfully produced by the Stark switching technique6,7
and are still under study for two crucial reasons. First,because the centrifugal barrier experienced by the outerelectron prevents its penetration to the inner one as wellas to the residual doubly charged core, autoionization
widths8 are usually small enough to permit full benefit ofthe pulsed laser instrumental resolution. Second, in ad-dition to the usual ADR spectroscopic investigations byvarying N, l1 , and n values, l is introduced as a fourthadjustable parameter9 to change the strength of the Cou-lomb repulsion between the two excited electrons. Untilnow in Ba, this l-dependence study has allowed for l1Þ 0 the observation of subtle long-range correlation ef-fects inducing K splitting of the excited high-l Npnl andNdnl ADR states.10,11 This splitting has been shown tobe essentially due to the dipole term of the multipole Cou-lomb e –e interaction expansion. In Sr (Ref. 12) long-range configuration mixing between the nominally pre-pared 5fnl states and the 6gn8l8 series has also beenobserved and briefly explained within a dipole polariza-tion model as being due to high Sr1 5f state dipole polar-izability. More recently an energy-level shift analysis ofthe nonpenetrating high-l orbitals in Ba Nsnl ADR states(N 5 7 –10, n5 12–15, l > 7) (Ref. 13) has revealed theincreasing strength of dynamic effects on the Ns coreelectron caused by the motion of the outer nl electron thatcannot be reproduced in a pure static polarization model.In a different experiment, similar energy-level shifts ofhigh-l Rydberg states of berylliumlike O41 ions14 havealso proved to be in disagreement with ab initio calcula-tions using the core polarization model. For Ba 8snlADR states with l 5 6 and l 5 7, this strong l-dependentconfiguration mixing with the 5fn8l8 ADR series was re-vealed in a preliminary study.15 It was roughly under-stood as been due to higher-order multipole polarizationeffects, in contrast to the long-range dipole mixture ob-served in Sr.12 Here we present a detailed analysis ofour preliminary study completed by observations coveringa more extended energy range. Of course, a full andquantitative understanding of such spectra needs ad-vanced theories that take in account the e –e Coulomb re-pulsion on a large radial area. Recent calculations16 us-ing the R-matrix procedure in combination with the
1997 Optical Society of America
P. Camus and S. Cohen Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2431
multichannel quantum defect theory (MQDT) and includ-ing the adiabatic effects of long-range multipole interac-tions are a first step in this direction. However, asshown below, many details of the photoionization spec-trum are not yet accurately reproduced by this method.Some of the discrepancies, which were attributed in Ref.16 to the presence of Stark mixing and lack of l selectivityin our experiment, have obliged us to examine the re-corded data by using a parametric MQDT model.
This paper is divided into four sections: In Section 2we briefly describe our experimental procedure and theresults of Stark calculations that we performed to ensurethat the initial high-l 6snl Rydberg states for l 5 6 andl 5 7 are free of l mixing induced by the small parasiticresidual field (<1 V/cm) that is present in our setup atthe end of the Stark switching procedure. This parasiticl-mixing effect, which increases in importance as n in-creases, was observed previously in a study of 10sn5 14l > 10 ADR states.17 It was undoubtedly verifiedby theoretical calculations that the lack of l selectivity forhigher-l states (l > 8) was due to an l admixture causedby a weak Stark field. In Section 3 we present all the re-corded 6s → 8s two-photon excitation spectra; in Section4 we use parametric MQDT fits to identify and interpretenergy-level positions and line profiles between the two5fj1 ionic limits. Then a comparison between the photo-ionization spectra calculated by the parametric MQDTmodel and those calculated with the combined R-matrixand MQDT method16 is given. Finally, in Section 5 wecomment on the spectra below the 5f5/2 limit.
2. PRINCIPLES OF THE EXPERIMENTALPROCEDUREOur apparatus and experimental procedure were de-scribed previously,10 so here we give a brief summary andfocus only on details relevant to the present study.Ground-state Ba atoms emerging from an effusive atomicbeam interact with two dye laser beams in the presence ofa static electric field F (1500–800 V/cm). The first laserexcites the atoms to the 6s6p 1P1 level (l1 5 553.5 nm)and the second (l2 ' 430 nm), to a selected 6snk Starkstate (n 5 12–15 and k 5 n 2 1 –5). The field is thenturned off slowly enough that the Stark state can evolveadiabatically to the corresponding 6snl 5 k Rydberg levelin zero field. Then a third laser drives the 6s → 8s two-photon excitation (l3 ' 344.6 nm) of the remaining va-lence electron to produce a final Nsnl ADR state. Theselevels are detected by a time-of-flight analyzer and a tan-dem of microchannel plates and by monitoring the Ba21
signal recorded as a function of the third laser’s fre-quency. The detected Ba21 ions are created in either oftwo ways: photoionization of excited Ba1 levels producedby autoionization of the Nsnl ADR states or direct doublephotoionization from those states. Evidence for the lat-ter mechanism, frequently considered to have negligibleprobability, was seen in an analysis of Ba21 yield versuslaser intensity of the 8sn 5 13, 14l (l > 10) ADRstates.18 The third laser also photoionizes parasiticground-state Ba1 ions in a two-photon-resonant one-photon-ionization scheme through the 8s ionic level. Itprovides a reference line for standard frequency calibra-
tion of the Fabry–Perot etalon’s transmitted fringes,which are recorded simultaneously with the Ba21 ionspectrum. This fortuitous Ba1 8s resonance is also auseful marker pointing to the expected 8snl energy-levelposition in the extreme case of uncorrelated ADR states.
In a previous publication17 we performed a detailedstudy of residual stray field effects present in our appara-tus (<1 V/cm) at the same time that the third laser isfired. For l . 7, in addition to the main 8snl excitationline, which corresponds to the principal l 5 k componentof the selected 6snl 5 k initial level, the spectra exhibitsome contamination of neighboring weaker 8snl 5 k6 1 transitions that correspond to the initial adjacent l5 k 6 1 weak-field Stark-state degeneracy (see, in Fig. 1of Ref. 15, the spectrum k 5 9). To identify the newlyobserved data without ambiguity it is crucial to estimateproperly this parasitic contamination effect. Using astandard numerical code19 for the Stark effect, we calcu-lated the percent of l admixture in the initial 6snl Ryd-berg states in the presence of a 1-V/cm static electric field.Some results are presented in Table 1. By referring toFig. 1 of Ref. 15 we can see for the spectrum k 5 8 thatthe weak 8sn 5 13l 5 k 1 1 5 9 transition comparedwith the main l 5 8 transition fits approximately the0.23% calculated Stark admixture in the initial 6sn5 13k 5 8 Rydberg state. Using a similar approach, wesee immediately that the sudden appearance in the k5 7 spectrum of a strong resonance in the expected posi-tion of the 8sn 5 13l 5 k 1 1 5 8 peak cannot be fullyexplained by the very small (0.06%) calculated admixturein the initial 6sn 5 13k 5 7 Rydberg state. So we be-lieve that the strong intensity of this line, which cannotbe entirely due to the initial Stark level admixture, is duemainly to a configuration mixing in the final state in-duced by the outer electron. Even more important, theweak peak at '2 cm21 on the blue side of this strong line,which does not fit any of the expected 8sn 5 13l reso-nances, can be explained only by a final state admixture.It is for this reason that in Fig. 1 of Ref. 15 we were ableto distinguish weak l 6 1 resonances that were due tostray electric field contamination for k > 8 from newlines (and possibly strong l 6 1, 6 2 resonances) mainlyas a result of final state multipole mixing in the k < 7
Table 1. Calculated Stark l Admixture (%)in Each Initial 6snl Rydberg State
for a 1-V/cm Static Field
n k 5 l %ul 1 1& %ul 2 1&
15 9 3.43 1.218 1.13 0.307 0.29 0.056 0.05 0.01
13 9 0.07 0.238 0.23 0.067 0.06 0.016 0.01 0.00
12 9 0.24 0.098 0.09 0.027 0.02 0.006 0.00 0.00
2432 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 P. Camus and S. Cohen
spectra. Obviously, for the n 5 12, 13 k 5 6 spectra pre-sented in Section 3, the calculated percent of Stark ad-mixtures for the l 5 7, 5 neighboring states, which is al-ways less than 0.01, cannot be at the origin of the stronglymixed observed spectra. This argument contradicts theassumption formulated in Ref. 15 in which the complexityin the experimental k 5 7 spectrum was tentatively at-tributed to mixing of the l 5 7 and l 5 6 final states bystray electric fields. Compared with the much strongere –e interaction caused by the extension of the inner-electron spatial distribution, this '1-V/cm stray field rep-resents a small perturbation for the final ADR state andevidently has no effect on it. To prevent future erroneousinterpretations of our data we note that spectra that aredominated by weak Stark mixing in the initial statesshould present peak and profile similarities when theyare recorded from two adjacent k selected values in ourStark-switching setup. Normally in a rigorous adiabaticStark switching technique we would expect only the l5 k initial Rydberg state from the k selected value in themanifold. By the way, because of a small residual para-sitic field at the end of the adiabatic decrease, we ob-served and identified resonances from the l 1 1 and thel 2 1 initial Rydberg states corresponding to the weak-field Stark admixture. Of course those resonancesshould also appear in the recorded spectra from the twoadjacent k 1 1 and k 2 1 selected values in the mani-fold. Conversely, any structure that is not reproducedsimilarly by recording of two adjacent k spectra should bedue mainly to multipole polarization effects and mixing inthe final ADR state. For example, the broad structuresnear 20 cm21 on the red side of the Ba1 8s ionic transi-tion, which appear in the k 5 7 and k 5 6 8sn 5 13photoionization spectra [traces (a) and (b) in Fig. 2 of Ref.15] are not the same, as was speculated in Ref. 16. Wecan easily see that the shape of the envelope of this com-plex structure (located in place of the expected 8sn 5 13l 5 7 resonance) is symmetrical when it is recorded fromk 5 7 and asymmetrical when it is recorded from k 5 6.Moreover, the measured energy peak positions of thelines that appear in place of the expected 8sn 5 13l5 7 broad resonance are different from those measuredon the asymmetrical profile in the l 5 6 spectrum [seeFig. 5 below for a better comparison, spectrum (a) for l5 6 and spectrum (b) for l 5 7]. It is only because ofthe large-scale reduction of our published spectra in Ref.15 that this broad resonance has been attributed by Woodand Greene16 to Stark mixing in the electric field switch-ing procedure. The above discussion shows that for l5 6 the percent admixture that is due to the Stark mix-ing in the initial state is negligible for the 8sn 5 12–n5 15 excitation spectra presented in Section 3.
3. OBSERVED 6snl ˜ 8snl TWO-PHOTONEXCITATION SPECTRABefore proceeding to the analysis of the experimentalspectra it is useful to examine the energy-level diagram ofFig. 1, where the 8snl ADR states in the range99 222–99 670 cm21 above the Ba 6s2 ground level areshown. The principal closed and open Nl1nl channels in-
volved in the following discussion are represented on thisdiagram by a continuous line corresponding to an ob-served level and by a dashed line corresponding to a pre-dicted (or calculated) one. As we can see, the closed8snl 5 6 channel extends from n 5 12 to n 5 17 in theenergy range investigated. There are two regimes in thetwo-photon excitation spectra distinguished by the outerelectron l momentum in the 6snl initial Rydberg state.The first one for, l > 8, is characterized by simple single-line spectra,13,15 which correspond to the 6snl → 8snltwo-photon electric dipole excitation. The l momentumof the outer electron in the final ADR state can be identi-fied without ambiguity because it is conserved during the6s → 8s inner-electron two-photon transition. Typicall-dependent energy-level positions of these high-l 8snlstates are drawn only for n 5 13 and l 5 8 –10 in Fig. 1.They are not strongly mixed with the other closed Nl1nlchannels and are weakly autoionizing18 through the long-range dipole interaction with the nearest available7pj1el618 open channels (see the dashed arrow to the con-tinua in Fig. 1). The magnitude of the shifts relative tothe Ba1 6s → 8s ionic transition frequency reveals the ef-fects of the mainly dipole long-range interaction betweenthe two electrons. These shifts can be described by use ofa calculated effective quantum number n* relative to theBa1 8s limit that reflects the changes that affect theouter nl nonpenetrating (planetary) electron in the pres-ence of the excited Ba1 8s core state. All these situationshave been extensively studied and satisfactorily analyzedin an effective core-polarization model.13
The second regime, for l < 7, is the subject of thepresent paper. It is characterized by an extended andcomplex line density, in complete contrast with the simplel > 8 spectra in the same energy range and under thesame excitation conditions. This complex structure is
Fig. 1. Energy-level diagram of the observed 8sn 5 12–17l5 6 autoionizing ADR states (bold lines). Mixing with openchannels is indicated by dashed arrows; solid arrows indicate in-teraction with the 5fn8l8 series.
P. Camus and S. Cohen Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2433
the result of a strong configuration mixing with the5f5/2,7/2n8l8 closed channels15 (see in Fig. 1 thecontinuous-line arrows from 8sn 5 13l 5 6 toward5f5/2n8l8 and from 8sn 5 14l 5 6 toward 5f7/2n9l9). Forclarity, in Fig. 1 we show only the spectra recorded fromthe 8snl 5 6 initial states and only one of the two identi-fied series that belong to each 5fj1 channel. The5f5/2,7/2n8l8 states are denoted by bars whose sizes areroughly proportional to the observed line intensities inthe corresponding excitation spectra. It is evident thatthe mixing with the 5fj1 channels extends well outsidethe 8snl 5 6 energy-level positions in the vicinity ofwhich the resonances are roughly more intense. We canalso see that for n > 14 the 5f5/2el8 channel is open. Ob-viously the principal anomaly of these excitation spectracompared with those observed from the 6snl > 8 initialRydberg states consists in the fact that a direct two-photon dipole excitation of the 5f5/2,7/2n8l8 states isstrictly forbidden in a pure isolated core-excitation model.Because this mixing implies an inner-electron angularmomentum change of three units (8s → 5f ), the octopolenature of the observed 8snl–5fn8l8 configuration interac-tion can be understood from different possible types ofcoupling: a direct octopole polarization effect induced bythe third term of the multipole expansion of the aniso-tropic part of the e –e interaction15 or a quadrupolar plusdipolar mixing that is due to the presence of nearby7dj1n-g and 5gj1n-d perturbers, which have been theo-retically predicted16 at the positions drawn in Fig. 1. InRef. 15 none of these perturbers was directly identified in
the spectra. Because of the strong l dependence of theobserved configuration interaction [no mixing for l > 8,weak mixing for l 5 7, and strong mixing for l 5 6 (andeven l 5 5, which is not reported in this paper)], the l de-pendence was initially attributed15 to short-range multi-pole effects, the importance of which increases as the in-ner turning point of the outer electron decreases. Theseclassically calculated n 5 13l 5 6 –12 inner turningpoints are shown in Fig. 2 for comparison with the outerturning points of the Ba1 8s and 5f orbitals. From aclassical point of view it appears evident that short-rangeeffects should be stronger for l 5 6 –8snl ADR than for l> 8 because of the close proximity of the two electron or-bitals.
By symmetry considerations and using the ( j1l)K cou-pling scheme, we can calculate the number of 5f7/2,5/2n8l8
Fig. 2. Classical radial inner turning points of the n 5 13, l5 6 –12 orbitals compared with the outer turning points for theBa1 8s and 5f core states.
Fig. 3. Observed ADR 8snl two-photon excitation spectra from the 6snl 5 6 Rydberg state: (a) n 5 12, (b) n 5 13, (c) n 5 14, (d)n 5 15. (e) Some of the 4-cm21 free-spectral-range Fabry–Perot fringes. The four spectra are presented on an absolute energy scaleto show resonance lines that are due to the same upper level [the vertical dashed line through spectra (b), (c), and (d)]. The resonanceintensity [spectrum (d)] shows the characteristic isolated core excitation modulation: a central lobe centered on the position of the Ba1
6s → 8s parasitic two-photon ionic transition with a secondary lobe on each side. From a selected 6snl 5 6 Rydberg state we observethe n 2 1 to n 1 2 members of the 8snl 5 6 series. Because of an '0.5 quantum-defect difference between the 6snl and the 8snlstates, the n and the n 1 1 resonances are under the principal lobe. For a given 8snl spectrum the two first zeros, one on either sideof the central lobe, correspond to the positions of the Ba1 8s ionic transitions in the contiguous 8sn 6 1l spectra [the vertical solid linethrough spectra (c) and (d)]. Calculated positions of the 8snl members are indicated below spectrum (c). The positions of the highermembers of the 5f5/2n8l8 and 5f7/2n9l9 series are indicated below spectra (b) and (d), respectively. The spectra surrounded by the boxedrule are repeated in Fig. 4 on a larger scale to show more details.
2434 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 P. Camus and S. Cohen
Table 2. 6s ˜ 8s Excitation Spectra from 6snk with k 5 6
Measured Wave Number of the Line (cm21)
Energy Level (cm21)
n* a
n 5 12 n 5 13 n 5 14 n 5 15 5f 2F5/2 5f 2F7/2 8s 2S1/2
a Effective quantum numbers in roman type correspond to the assigned series limits for the levels; italic type denotes calculated numbers on the otherchannels.
double-Rydberg series that are in principle allowed to in-teract with the 8snl 5 6 K 5 11/2, 13/2 states and arenormally expected to appear in the excitation spectra(here K denotes the coupled angular momentum betweenthe total angular momentum j1 of the inner electron with
the l momentum of the outer electron). Applying parityand K conservations, we predict a total of six 5f5/2n8l8 Kseries and eight 5f7/2n8l8 K series, with l8 5 9, 7, 5, 3 forthe outer electron. This large number of available5fj1n8l8 closed channels is not confirmed by the recorded
2436 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 P. Camus and S. Cohen
spectra, for which only two series converging to 5f5/2 andtwo series converging to 5f7/2 are well resolved at our'3.5-GHz (FWHM) laser linewidth.
The two-photon excitation spectra obtained from the6snl 5 6 initial Rydberg states with n ranging from n5 12 to n 5 15 are assembled in Fig. 3. To show thecorrespondence between the transitions that reach thesame upper level, the 6snl n-different recorded spectraare appropriately shifted in energy to fit an absolute en-ergy scale representation. As an example, the upperlevel 99 458.3 cm21 given in Table 1 was observed threetimes from different 6snl 5 6 initial Rydberg states:trace (b) for n 5 13, trace (c) for n 5 14, and trace (d) forn 5 15. It is indicated in Fig. 3 by a vertical dashed line.The total observed energy range in Fig. 3 covers99 240–99 650 cm21 above the 6s2 ground level. Be-cause of this absolute energy scale representation, the in-dicated Ba1 8s lines that correspond to the Ba II 6s → 8stwo-photon transition are obviously shifted from one toanother spectrum by the energy difference of the initiallyinvolved 6snl 5 6 Rydberg states. All the recorded spec-
Fig. 4. Slow-scan two-photon excitation spectra in the vicinityof the 8sn 5 13l ADR from the 6snl 5 6 Rydberg state: (a) n5 12, (b) n 5 13, (c) n 5 14. It can be seen clearly that the se-ries line profile shown within the boxed rule in Fig. 3 is in fact adouble-peaked structure that is just resolved at our instrumentallaser resolution. The two peaks are interpreted as members oftwo 5f5/2n8l8 series with different l8 values. The last four digitsof the measured level wave numbers are indicated on spectrum(a) for the lower series and on spectrum (b) for the upper series.Trace (d) shows the 4-cm21 free-spectral-range fringes. Thewave numbers of the '99 370-cm21 lines are listed in Table 2.
tra [see particularly spectrum (d) in Fig. 3] exhibit therecognizable overlap integral intensity envelope20 thattraduces the change experienced by the outer Rydbergelectron in the two-photon excitation of the inner valenceelectron to reach the final 8snl 5 6 ADR state. This en-velope, which is composed of a principal lobe centered onthe Ba1 8s transition, has secondary a weaker lobe on ei-ther side. Because of an '0.5 quantum-defect differencebetween the initial 6snl 5 6 Rydberg state and the final8snl 5 6 ADR state, there are two broad profiles that cor-respond to the 8snl 5 6 and 8s(n 1 1)l 5 6 members in-side the principal lobe and two weaker 8s(n 2 1)l 5 6and 8s(n 1 2)l 5 6 members in the secondary lobes onthe two sides of the principal lobe. Without any mixingwith the 5fn8l8 configuration, the n 5 13–16 8snl 5 6series members should appear as simple broad autoioniz-ing resonances such as the n 1 2 5 17 member in spec-trum (d). Below spectrum (c) we have indicated the cal-culated positions of these 8sn 5 13–17l 5 6 seriesmembers calculated with a polarization quantum-defectmodel in which the dipole and the quadrupole terms areincluded. So it can be seen clearly from spectrum (d)that the stronger narrow peaks, which are identified withthe highest members of the 5f7/2n9l9 series, appearmostly in the vicinity where the 8sn 5 14l 5 6 –8sn5 16l 5 6 broad resonances are expected. A similarfeature involving the highest members of the 5f5/2n8l8 se-ries indicated below spectrum (b) can be seen also in spec-tra (a) and (c) for 8sn 5 13l 5 6. In fact, the recordedspectra present a better laser resolution than those pre-sented in Fig. 3, which have been compressed for presen-tation purposes. In Fig. 4 we present a high-resolutionscan of the vicinity of the 8sn 5 13l 5 6 member, whichexhibits mixing with two 5f5/2n8l8 series and two 5f7/2n9l9
Fig. 5. Slow-scan two-photon excitation spectra from the 6sn5 13l Rydberg state with (a) l 5 6 and (b) l 5 7. These twospectra are presented to show that the l 5 7 modulated profile[at '99389.21 cm21 in spectrum (b)], which corresponds to mix-ing with higher 5f5/2n8l8 series members, is different in energyfrom the broad blue wing [at 99384.70 cm21 in spectrum (a)] ofthe same perturbed series recorded for l 5 6. In Ref. 15 thesetwo profiles were considered identical and were attributed toStark l mixing in our initial 6snl Rydberg populating procedurefor low-l values (for further comments see Section 2).
P. Camus and S. Cohen Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2437
members. Although the two series are easily observable,their line separation is close to our instrumental laserwidth. We note that the intensity of these perturbed se-ries is well reproduced from one to another 6snl initialstate. As noted above, there are four possible values forl8 in the final 5fj1n8l8 ADR state, namely, 3, 5, 7, and 9.Assuming that the observed series correspond to differentl8 values for the outer electron, it is not possible from theexperimental data alone to specify them. Therefore inour spectral analysis the remaining ambiguity that con-cerns the assignment of the l8 momentum does not permita complete identification of each of the four observed5f5/2,7/2n8l8 double-Rydberg series. The high-energy sideof this double-line structure is dominated by a 5f7/2n9l9perturber close to 99 384.70 cm21, which cancels the os-cillator strength of the higher 5f5/2n8l8 series member forE . 99 384.70 cm21. We remind the reader that this ob-served asymmetrical line shape cannot be connected to aresidual Stark mixing in the 6sn 5 13l 5 6 initial state.To probe this problem we compare in Fig. 5 the recorded
Fig. 6. Two-photon excitation spectra 6sn 5 12l → 8sn 5 12lrecorded from an initially selected 6sn 5 12k Stark state with kvarying from 7 to 11. Spectra refer to the 6s → 8s two-photonBa1 transition at s3 5 29 012.6 cm21 for the third laser, and theFabry–Perot fringe spacing is 4 cm21. On each k spectrum anarrow indicates the calculated position of the corresponding un-perturbed 8sn 5 12l 5 k resonance determined from a polariza-tion quantum-defect formula including the dipole and the quad-rupole terms and our previously fitted polarizabilities.13
spectra from 6sn 5 13l with l 5 6 and l 5 7 with alonger energy scale than the one used in Fig. 2 of Ref. 15.First, it appears immediately that the peak positions in-dicated by arrows (or points) in spectrum (b) of Fig. 5 near99 389.21 cm21, which correspond to 5f5/2n8l- seriesmixed with the 8sn 5 13l 5 7 state, do not fit those ofspectrum (a) on the red side of the 99 384.70-cm21 asym-metrical profile and obviously with the similar higherresolution recording [spectrum (b) of Fig. 4]. Second, thetwo perturbed profiles of Fig. 5, which have different ori-gins [5f7/2n9l9 for spectrum (a), l 5 6, and 8sn 5 13l5 7 for spectrum (b), l 5 7] are characterized by differ-ent shapes. So it is clear that the main features of thesetwo spectra recorded from different l 5 6 and l 5 7 Ryd-berg states are due not to a residual Stark mixing in theinitial zero-field 6snl5 k populating procedure but to apolarization quantum defect in the final state. To sup-port this conclusion we show in Fig. 6 the two-photon ex-citation spectra observed from 6sn 5 12k, with k varyingfrom 7 to 11. Normally, for n 5 12 the mixing of the8sn 5 12l ADR with the 5fn8l- closed channel should bemore pronounced than for n 5 13 because the two elec-trons approach each other and the e –e interaction isstronger. For n 5 13 this mixing, as we saw above, ispresent in the analyzed spectra mainly for l 5 6, and itstarts to disappear for l 5 7 (remember that all the spec-tra recorded with l > 8 present the simple single-line di-pole excitation structure that characterizes the planetaryADR13). As expected, for l5 5 (not presented here), thismixing is considerably stronger. For n 5 12 it appearsimmediately from Fig. 6 that, even for the spectrum k5 10, the mixing of the 8sn 5 12l 5 k ADR with the5fn8l8 closed channels starts to occur because the ob-served two-photon excitation spectrum is different fromthe expected single-line dipole planetary spectrum. Thisdifference should normally be represented by a singlestrong 8sn 5 12l 5 k peak at the position indicated bythe arrow, which corresponds to a polarization quantum-defect calculation25 that includes the dipole and the quad-rupole terms. The arrow’s position error is of the order of
Fig. 7. Two-photon excitation spectra of the 8snl 5 6 highmembers from the 6sn 5 15l 5 6 Rydberg state. The n 5 17member is the same as that in spectrum (d) of Fig. 3. The n5 18 member is blended by the strong Ba1 5d3/2 → 5g two-photon resonance. The spectrum is here quite noisy because ofthe fast and unusually long scan-recording conditions. The in-set shows the measured m8s quantum defects versus n.
2438 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 P. Camus and S. Cohen
0.6 cm21 for a relative shift to the ionic line of 10 cm21.So we have at this place two small resonances that do notfit the general regularity observed for the Nsnl dipoleplanetary ADR.13 The same conditions obtain for the k5 9 spectrum, where several peaks appear on both sidesof the calculated position of the 8sn 5 12l 5 9 ADR. Fork 5 8 the two-photon excitation spectrum seems to revertto a more-conventional dipole spectrum because of astrong resonance in place of the calculated 8sn 5 12l5 8 position, even if many weaker peaks are alwayspresent. We can see immediately that this strong reso-nance in the k5 8 spectrum does not appear in the k5 9 spectrum, a fact that confirms that we have no re-sidual field Stark contamination in the selected 6sn5 12k Rydberg state (even if, in this case, Stark calcula-tions given in Table 1 predict a 0.09% uk 5 8& admixturein uk 5 9&). So the apparent similarity among various kspectra, particularly for the small resonances, is due prin-cipally to e –e interaction l mixing in the final state. Forthe k 5 7 spectrum the e –e interaction coupling with theclosed channels is probably at the origin of the resonances(one is particularly strong) that are superimposed upon asmaller and wider profile, which can be identified as the8sn 5 12l 5 7 resonance (even if its calculated positionis a little further to the red). All these 8snl spectra withl < 10 need more-elaborate calculations to be interpretedsatisfactorily, but we can keep in mind that only the 8sn5 12l 5 11 ADR seems free of coupling with the 5fn8l8closed channels. Finally, a longer scan of the two-photon6sn 5 15l 5 6 → 8snl 5 6 excitation spectra toward theseries limit is given in Fig. 7. Looking at the regularityof the measured m8s quantum defect (see the inset of Fig.7), we can claim that there are no local perturbationsclose to the end of the series limit.
4. PARAMETRIC MULTIPLE QUANTUMDEFECT THEORY INTERPRETATIONThe observed spectra for E . I5f5/2
are analyzed by thephase-shifted MQDT21,22 model shown in Fig. 8. Themodel consists of two Rydberg series converging to the
Fig. 8. Five-channel phase-shifted MQDT model used to ana-lyze the states with E . I5f5/2
.
I5f7/2ionization limit (channels 1 and 2) coupled to one se-
ries converging to the I8s limit (channel 3). These threeclosed channels may autoionize to two continua. Be-cause of the high excitation energy of the two valenceelectrons the true number of available continua is verylarge, an attribute that is taken into account in an effec-tive way22 by introduction of two open channels. Never-theless, if it is desirable to label these two effective openchannels, we expect one of them (channel 5) to character-ize mostly the 7pel 6 1 continuum (strongly coupled tothe 8snl series by the dipole interaction) and the other tocharacterize mostly the 5f5/2el channel (channel 4), inwhich the 5f7/2nl series are expected to autoionize pre-dominantly. Having in mind previous observations23 inwhich a considerable number of Ba1 5fj1 ions are pro-duced through autoionization of low-l 8snl states, wehave also introduced coupling between channels 3 and 4.
The basic matrix MQDT equation to be solved is
@R8 1 T#a 5 0, (1)
where the phase-shifted reaction matrix R8 has zero di-agonal elements and its off-diagonal ones account for thecoupling among all the channels. The elements of the di-agonal matrix T are given by Tii 5 tan(pni8), with n i85 n i 1 m i . The parameters m i can be considered thezero-coupling quantum defects of each channel, i.e., whenR8 5 0. The vector a is related to the vector Z of thechannel admixture coefficients by ai 5 Zi cos(pni8). Thecompatibility condition for Eq. (1) to have a nontrivial so-lution is
det@R8 1 T# 5 0. (2)
In a first step, ignoring the open channels, we determinethe energy levels by satisfying simultaneously Eq. (2) andthe effective quantum-number relation I5f7/2
–1/2n12
5 I8s –1/2n32 in atomic units. Then a fit of these pre-
dicted energy levels to the observed values determinesthe parameters m1 , m2 , m3 , uR138u, and uR238u. The qual-ity of this fit is probed by inspection of the @1 2 n1 (mod1), n3] Lu–Fano plot of Eq. (2). Now, introducing in asecond step the open channels, we write the photoioniza-tion cross section within the isolated core excitationapproximation7 in our case as
s~E ! } d6s→8s~2 ! n3
3O2~n* ; n3!($nO%
~Z3~r!!2, (3)
where d6s→8s(2) is the two-photon dipole matrix element for
the inner electron transition and O(n* ; n3) stands for theoverlap integral between the initial and final wave func-tions of the nl outer electron, given by
O~n* ; n3! 5 2~n* n3!1/2sin@p~n* 2 n3!#
p~n* 2 2 n32!
, (4)
where n* is the effective quantum number of the 6snl5 6 level with respect to the limit I6s . The index r inEq. (3) runs over the number no of open channels (in thepresent model no 5 2), which is the number of indepen-dent solutions of Eq. (1). For each solution we require acommon phase shift for all the open channels so tan(pn48)5 tan(pn58) 5 tan(2d (r)), where we find tan(2d (r)) for r5 1, 2 by solving Eq. (2). The admixture coefficients Zi
(r)
P. Camus and S. Cohen Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2439
are then determined by Eq. (1) and the normalization con-dition (Z4
(r))2 1 (Z5(r))2 5 1. In Eq. (3), only excitation of
the 8s inner electron is considered, so only the 8snl char-acter of the final-state wave function is probed. The two-photon dipole 6s → 5f core transition is forbidden, andthe two-photon 6s → 7d transition (not considered ex-plicitly by the model) is suppressed by the overlap inte-gral of Eq. (4) because the effective quantum number withrespect to the 7d limit is far smaller than n* . Addition-ally, the absence of structure in the neighborhood of thefirst photon (no ionic states are present at this energy) al-lows us to ignore any double-resonance effects.24
If the MQDT model above is sufficiently comprehen-sive, with all the relevant closed channels included, theparameters m i and Rij8 should be constant or almost con-stant with energy. Then the Lu–Fano plot should ex-hibit periodic perturbations of the 5f7/2nl series by the8snl series with the same pattern repeated every unitrange (m, m 1 1) of n3 (m is a positive integer). How-ever, as can be seen from Fig. 9, the experimental Lu–Fano plot shows a drastic nonperiodicity. The quantumdefects, which initially have (modulo 1) values of ;0.45and ;0.7 near n8s ' 13, approach unity (or zero) close tothe I5f7/2
limit (n8s ' 16.5). Moreover, the evolution ofthe branch slopes indicates a decrease of the channel cou-pling with increasing energy. There are two possiblecomplementary reasons for this behavior that have thesame origin (the two-excited-electron interaction). First,we have shown for that, high-l ADR (l . 7 for 8snl), corepolarization effects induce energy dependences for thequantum defects.25 Second, there may be additional per-turbers that belong to series that converge to higher lim-its than I8s . Although the overlap between the two ex-cited electrons cannot be excluded for l , 8 (see Fig. 2),the outer electron does not penetrate the residual doublycharged core, and this mixing should also be due to short-range interaction between the two valence electrons.However, this interaction can be of much higher strengththan that of a simple core polarization picture. Woodand Greene16 have indeed predicted four perturbers inthis region: 5gj1nd (n8s ' 13.49) and 7dj1ng (n8s' 15.25) for the K 5 11/2 symmetry and 7dj1ng (n8s' 13.38) and 5gj1nd (n8s ' 15.16) for the K 5 13/2 sym-metry. The two perturbers that lie just above the I5f5/2
limit (n8s 5 13.154) can be responsible for the nonperiod-icity of the Lu–Fano plot. In contrast, the almost unper-turbed part of the plot does not seem to be influenced bythe last two higher-energy perturbers. It is obvious thatneither core polarization nor additional closed-channel in-teractions can be taken into account by a constant-parameter MQDT model as described above. Conse-quently, to simulate these perturbations we attempt tointroduce energy-dependent parameters. Because thequantum defect that is due to the dipole e –e interactionterm is expected to vary linearly with energy,25 a lineardependence is assumed for both the m i (i 5 1, 2) and theclosed-channel coupling parameters Rij8. However, it isimpossible (especially for the quantum defects) to fit thewhole range of interest by this kind of linear parameter-ization. Tests with higher-order polynomials do not pro-vide better results. Thus it becomes clear that core po-larization effects alone cannot explain the strong
nonperiodicity of the experimental Lu–Fano plot, and theintroduction of perturbers is a necessity. It is wellknown that near an isolated perturber the phase shiftjumps by p. In a multichannel problem this jump is notnecessarily equal to p, so we adopt the following form forthe quantum defects m1 and m2 :
m i~e! 5 Ai tan21S e 2 e0~i !
G i/2D 1 Bi , i 5 1, 2, (5)
where e 5 21/2n12. In this case four parameters are re-
quired for each quantum defect in the fit, but we can re-duce this number to three by setting m i(0) 5 1 close tothe I5f7/2
limit e 5 0 because the experimental quantumdefects approach unity there. For the closed-channelcoupling parameters R138 and R238 the linear parameter-ization
Ri38 5 eRi38~1 !, i 5 1, 2 (6)
has proved to be sufficient. We obtained the fitted Lu–Fano plot of Fig. 9(a) by using a set of nine parameters,the values of which as follows: For channels 1 and 2 (i5 1, 2), 5f7/2nl8, Ii 5 99666.54 cm21. m1 : A1 5 0.1156 0.009, e0
Fig. 9. (a) Experimental and fitted Lu–Fano plots of the energylevels observed for E . I5f5/2
. (b) m i(e) quantum-defect func-tions, (c) their derivatives 2]m i /]e versus n8s , showing the po-sitions (n8s ' 13.4 and n8s ' 13.9) and the widths of the two fit-ted Lorentzian profiles. One of the Lorentzians is located closeto the 7djng (n8s ' 13.38) K 5 13/2 perturber predicted by theR-matrix calculation of Ref. 16; the other is blue shifted with re-spect to the calculated 5gjnd (n8s ' 13.49) K 5 11/2 perturberby ;35 cm21.
2440 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 P. Camus and S. Cohen
5 100060.08 cm21, m3 5 0.40 6 0.02. The phase-shifted reaction matrix is
The behavior of the experimental points is then satisfac-torily reproduced. The larger discrepancies occur justnear the I5f5/2
limit in a range where the widths of thelines are quite large [see spectrum (b) of Fig. 3]. In Fig.9(b) we show the resulting m i(e) quantum-defect func-tions and their derivatives 2]m i /]e that representLorentzians centered at e0 with a width G (FWHM). Theposition of one of these two fitted Lorentzians at n8s' 13.4 is very close to the 7dj1ng K 5 13/2 perturbermentioned above and predicted by Wood and Greene.16
The other Lorentzian, however, located at n8s ' 13.9, is
Fig. 10. Comparison between observed and calculated spectra:(a) experimental data15 [part of spectrum (c) of Fig. 3], (b) para-metric MQDT (this work), (c) R-matrix 1 MQDT calculations.16
somewhat blue shifted with respect to the second 5gj1ndK 5 11/2 perturber also predicted at this region. Addi-tionally, for n8s . 14 the experimental points are veryclose to our fitted curve without introducing the two otherperturbers, 5gj1nd K 5 13/2 (n8s ' 15.16) and 7dj1ngK 5 11/2 (n8s ' 15.25), predicted by them in this energyrange. To check whether the perturber characteristicsextracted from the fit are sensitive to the specific param-eterization used [Eqs. (5) and (6)], we performed varioustrial-and-error tests with simpler as well as more-complicated functional forms. The positions of these twoperturbers did not change appreciably (less than 10%)and can be considered the most reliable output of the fit.The energy levels of the series as well as the spectra arehighly sensitive to these fitted positions of the perturbers.The quality of our values can be appreciated not only bycomparison of those values with the fitted Lu–Fano plotbut also by the close agreement between our observed andfitted spectra, which are presented in Figs. 10(a) and10(b). In Fig. 10(c) for comparison we have reproducedthe spectrum calculated by the R-matrix method.16 Wecan see several features that we believe are better repro-duced by our fit. First, a small shift toward higher ener-gies of the R-matrix spectrum [Fig. 10(c)] with respect tothe experimental data [Fig. 10(a)] is evident that is ab-sent in the fitted data [Fig. 10(b)]. Second, we can seethat in the spectrum of Fig. 10(b) near 99470 cm21 more-intense lines are wider than the corresponding ones inFig. 10(c). This narrowing of the profiles was attributedby Wood and Greene16 to the presence of the 5gj1nd K5 11/2 perturber. The absence of line narrowing shouldthen be probably due to the difference between fitted andR-matrix calculated positions ('35 cm21) of this per-turber. Finally, the relative intensity of the doublet se-ries between 99 430 and 99 540 cm21 [Fig. 10(b)] is wellreproduced, whereas in Fig. 10(c) the broad profiles arenot consistent with the observed spectrum [Fig. 10(a)].
The parametric MQDT model confirms the existence oftwo perturbers that were not taken into account in ourprevious analysis.15 However, the position of one of theperturbers does not seem to agree with the theoreticalpredictions of Wood and Greene.16 This fact may explainthe discrepancies between the calculated R matrix andexperimental spectra. On the contrary, the MQDTmodel well reproduces the positions, widths, and relativeintensities of the observed doublet 5f7/2n8l8 series nearthe 8snl 5 6, n 5 14–15 perturbers.
5. LU–FANO PLOT ANALYSIS BELOW THE5f5/2 LIMITIn Fig. 11 we show the Lu–Fano plot of the energy-levelpositions that correspond to spectrum (a) in Fig. 3. Wedid not attempt to model this part of the spectrum be-cause the required number of parameters is large. Nev-ertheless, the plot clearly shows two easily distinguish-able 5f5/2n8l8 series periodically perturbed by 5f7/2n8l8members (n5f7/2
5 17–20) as well as the 8sn 5 13l 5 6level near n8s ' 12.5. We can see that this relativelysimple picture of the Lu–Fano plot changes abruptly forn5f7/2
< 17, where the number of observed resonances is
P. Camus and S. Cohen Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2441
considerably increased. We remind the reader that thepredicted number of 5fj1n8l8 series that can be coupled to8snl K 5 11/2,13/2 is six for j1 5 5/2 and eight for j15 7/2.15 Moreover, in this energy region the level sepa-ration is larger than our laser resolution, and it may benot surprising to observe more members there than closeto the 5f5/2 limit. Unfortunately the line intensities areweak because of the absence of any perturbing effects intheir vicinity by an 8snl 5 6 member. We believe thatan R-matrix analysis for this part of the spectrum couldsolve a remaining problem in this study, namely, theidentification of the outer-electron angular momentum.
6. CONCLUSIONWe have studied the 8snl 5 6 ADR series for n5 12–31 and l 5 7 for n 5 13 from selectively prepared6snl Rydberg states. One outcome of this study was theintroduction of l (the angular momentum of the outerelectron) as a chosen parameter that controls the strengthof the Coulomb repulsion between two excited electrons.For l5 6 and n 5 12–16, strong interaction configura-tion mixing with the high members of the neighboring5f5/2,7/2n8l8 series was observed. The l 5 6 data contrastwith the higher, l . 7, simpler, essentially dipole spectrathat present a planetary character and are free of short-range e –e interaction. At our laser instrumental resolu-tion the spectra present only two 5f5/2n8l8 and two5f7/2n8l8 resolved series of a total number of 14 seriesthat are allowed to be mixed with the nominally excited8snl 5 6 K 5 11/2 and 13/2 ADR states. That the l8 ofthese series is always undetermined makes difficult a pre-cise analysis of this strong configuration interaction.Nevertheless, a parametrical MQDT model permits us toanalyze the 8snl 5 6 1 5f7/2n8l8 mixing and to interpretthe irregularities of the Lu–Fano plots just above the 5f5/2limit as being due to the presence of two perturbers. Oneof the perturbers is near the position predicted by Woodand Greene16; the second is shifted by 35 cm21 to an en-ergy higher than the position calculated by Wood andGreene. Additionally, two other perturbers present intheir R-matrix 1 MQDT calculations do not seem to af-fect either the energy-level positions or their observed
Fig. 11. Experimental Lu–Fano plot for the energy levels belowthe 5f5/2 limits. These levels correspond to observed spectra(a)–(c) in Fig. 4.
line profiles. This result is confirmed by the good repro-duction above the 5f5/2 limit of the observed spectra whenthe parametric MQDT model is used. Finally, this para-metric study demonstrates that to achieve more-preciseseries identification requires a more accurate R-matrix1 MQDT calculation.
ACKNOWLEDGMENTSIt is a pleasure to thank A. Bolovinos for his help duringthe experiment and G. Hubbard for her technical assis-tance. The Laboratoire Aime Cotton is associated withthe Universite Paris Sud.
*Present address, Institute of Accelerating Systemsand Applications, P.O. Box 17214, Athens 10024, Greece.
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