Department of Physics, University of Virginia, Charlottesville, Virginia 22901
Received February 13, 1989; accepted May 1, 1989
The npl/2 and ngj Rydberg series of the Ba+ ion have been observed, using a multistep laser-excitation scheme. Asingle photon excites either the npl/2 levels from an initially populated Ba+ 8s1/2 state or the ngj levels from the Ba+5fh states. The Ba+ 8s1/2 state is populated directly through two-step laser excitation of ground-state Ba+ ions, andthe Ba+ 5fj state is a decay product of neutral-Ba 8snd autoionizing states, which are produced by an analogous two-step excitation of bound 6snd states. We observe no excitation of the np3/2 levels, indicating a zero in the excitationcross section to these levels. Using the quantum defects of the ng series, we are able to extract values for the electricdipole and quadrupole polarizabilities of the Ba2+ core of 10.3(2)ao 3 and 51(10)ao 5.
INTRODUCTIONAlthough Rydberg states of neutral atoms have been studiedextensively, little attention has been focused on the Rydbergstates of ions, which can be expected to have qualitativelydifferent properties by virtue of their charge. To our knowl-edge the only systematic spectroscopic study is the study ofthe Ba+ nd and ns Rydberg series by Boulmer et al.1 Herewe report the excitation spectra of the Ba+ np and ng seriesusing a multistep laser-excitation approach. Specifically wehave observed these two series from the Ba+ 8s1/2 and 5fjstates, respectively. The Ba+ 8s1/2 state is produced by two-step laser excitation of the 6s1/2 state, and the Ba+ 5fj statesare produced as the products of autoionization of laser-excited Ba 8snd states.
In the following sections we describe our experimentalprocedure, present the measured energies of the two Ryd-berg series, explain the apparent anomaly that no nP3 /2 se-ries is observed, and derive the polarizabilities of the Ba2+core.
EXPERIMENTAL PROCEDURE
Our two excitation schemes involve the use of five pulsedtunable dye lasers, as shown in Fig. 1. Consider first theexcitation of the Ba+ ng states (path g of Fig. 1). The firstdye laser drives the Ba 6s2-6s6p Pi transition, and thesecond laser then excites a 6snd Rydberg state. A thirdtunable dye laser then excites a 6p 3/2nd autoionizing state.The frequency of the third laser is set at the Ba+ 6s1/2-6P3/2
transition frequency and is in essence the ion transition withthe Rydberg electron remaining as a spectator. 2 The fre-quency of the fourth dye laser is fixed at one half of the Ba+6p3/2-8sl'2 interval, and when the laser is frequency doubledit can excite the neutral 6p3/2nd autoionizing state to a8s,1 2nd autoionizing state. Atoms in the 8s,/ 2nd stateautoionize to produce a substantial population in the Ba+ 5fjstates. The fifth dye laser is then used to excite the Ba+ ngRydberg series.
The temporal positions of the five dye lasers are critical.The first two lasers precede the other three by approximate-ly 100 nsec. This time separation ensures that only Baatoms in the ground state or in a 6snd Rydberg level are in
the interaction region when the other lasers appear. Next,the laser whose frequency matches the Ba+ 6p 3/2-8s/2 ener-gy interval must appear slightly before the laser that excitesthe 6s,/2-6P3/2 transition. This overlap is needed to preventall the 6p 3/2nd states from autoionizing before the appear-ance of the laser that will create the 8snd autoionizing states.The temporal position of the final laser excitation is relative-ly unimportant as long as the majority of the pulse arrivesafter the 8snd states have been excited. The 8snd statesautoionize in a time that is short compared with the 10-nsecpulse width of the final laser. Thus population will be trans-ferred to the 5fj levels before the end of the fifth laser pulse.
The Ba+ np states may be excited in either of two ways.When the 8s,1 2nd states are excited some of the atomsautoionize to the Ba+ 6s1/2 state, which can absorb a secondphoton from the third laser, yielding a Ba+ 6
p3/2 ion, whichcan in turn absorb the frequency-doubled light from thefourth laser to produce a Ba+ 8s1/2 ion. The fifth laser thendrives the transition to the Ba+ np series. A more efficientmeans, path p, is shown in Fig. 1. Rather than produce a6snd Rydberg state, the second laser is tuned just above thefirst ionization limit to produce Ba+ 6s1/2 ions directly, whichthen sequentially absorb photons from the third, fourth, andfifth lasers. The timing of the lasers is not so critical for theexcitation of the Ba+ np states since in all cases the interme-diate states are relatively long lived, >5 nsec.
In both excitation schemes Ba2+ ions are detected afterthe Ba+ Rydberg ions are field ionized. The field-ionizationpulse appears approximately 200 nsec after the appearanceof the last laser pulse. This delay time ensures that there isno electric field in the interaction region during the laserpulses. Rapid expansion of the small volume of excited ionsby the space charge of the Ba ions precludes the use of longerdelay times between the lasers and the field-ionizationpulse.
A basic schematic of the laser apparatus is shown in Fig. 2.The first two dye lasers are pumped by an excimer laseroperating with XeCl at 308 nm. The first laser is tuned tothe Ba 6s2 -6s6p 'PI resonance at 18 060.26 cm .3 Thislaser is of the Hansch4 type and has a linewidth of _0.8cm-1 . The second, commercial dye laser with a frequency of£-24 000 cm'1 and a linewidth of 0.3 cm-' is tuned to excite a6snd Rydberg level or a 6sed continuum state. Both of these
Fig. 1. Diagram of the laser-excitation scheme showing important intermediate levels but not drawn to scale. Path g, Two pulsed dye lasersexcite Ba atoms to a high-n (n > 60) 6snd state via a 6s6p intermediate state and then to a 6p3/2nd autoionizing state. A fourth dye laser excitesthis autoionizing state to an 8snd level, which autoionizes, leaving a significant fraction of Ba+ ions in a 5fj level. A final laser drives the 5fj-ngtransition. Path p, Similar to path g except that the first two dye lasers photoionize one electron directly. The third and fourth lasers thendrive the bare ion transitions to move population into an Ba+ 8s,/ 2 level, which is excited by the fifth laser to an np Rydberg state.
dye lasers have pulse energies of 100 ,J and pulse widthsof 10 nsec. The last three dye lasers are of the Hansch type,are pumped by different harmonics of a Nd:YAG laser, andhave pulse widths of 10 nsec. The third dye laser is pumpedby the Nd:YAG third harmonic and has its frequency fixedat the Ba+ 6sl/2 -6p3 /2 transition at 21 952.42 cm-1.3 Thislaser has a linewidth of 1 cm-' and a pulse energy of 100,uJ. The fourth dye laser uses the Nd:YAG second harmonicas a pump source. The frequency of this laser is fixed at18 036.38 cm-', and the beam is amplified before being fre-quency doubled in a KDP crystal. The doubled light has afrequency that matches the Ba+ 6p3 /2-8sl/2 transition at36 072.76 cm-', 3 a linewidth of _2 cm-', and a pulse energyof c_50 ,J. The fifth dye laser is pumped by the Nd:YAGthird harmonic and is also amplified. Its frequency can betuned from 23 000 to 24 000 cm-I to excite different Rydbergstates of the Ba+ ion. This laser has a pulse energy of 1 mJand a linewidth of 0.7 cm-'. All five dye lasers are operatedat a 20-Hz repetition rate.
The five dye-laser beams interact with an effusive Babeam originating from a resistively heated oven in a vacuumchamber with a background pressure of 10-7 Torr. The fivelaser beams enter the vacuum chamber parallel to one an-other and at right angles to the atomic beam. A 50-cm focal-length quartz lens focuses the beams to a spot of diameterf0.5 mm centered between a pair of 8 cm X 8 cm Al plates,which are separated by 0.66 cm. After the lasers are fired avoltage pulse applied to the lower Al-plate field ionizes theBa+ Rydberg states and accelerates both singly and doublyionized Ba through a 0.6-cm-diameter hole i the center ofthe upper plate. The rapid rise time (<0.5, tsec) of the field
Fig. 2. Schematic of the basic experimental apparatus showing thelaser and detection equipment for the time of flight (TOP) ionsignal.
100 060.22 cm 1
99 666.68 cm'99 425.97 cm'
63 987.46 cm- I
42 035.04 cm-'
18 060.26 cm-'
0
I-
-- J i
path g
R. R. Jones and T. F. Gallagher
R. R. Jones and T. F. Gallagher Vol. 6, No. 8/August 1989/J. Opt. Soc. Am. B 1469
pulse ensures that the Rydberg ions can sample large electricfields (>10 kV/cm) before being expelled from the fieldregion. The hole in the upper plate is covered by a finestainless-steel mesh to preserve the electric-field homogene-ity in the interaction region. After leaving the interactionregion, the ions travel a distance of 10 cm before striking adual-microchannel plate (MCP) detector. The ion currentis then amplified and sent to a boxcar averager. For thesemeasurements we record the Ba2+ signal, which is clearlytime resolved from the Ba+ signal by the different flighttimes of the two ions. A measure of the efficiency of theexcitation is that the ratio of the Ba+ to Ba2+ signals is
1000:1.
C
U)
0
0
22350
Z
. _
CS
0
C
0
++Z
rn
EXPERIMENTAL RESULTS
Rydberg Energy LevelsFigure 3 shows observed Ba2 + signals as a function of thefifth laser frequency when the second laser is tuned to excitea 6snd Rydberg level with n 70. Under these conditionsboth the Ba+ 8s/ 2 and the 5fj state are populated. For thesescans the field-ionization pulse has a maximum amplitude ofapproximately 3000 V/cm. Figure 3(a) shows excitation ofthe Ba+ 8s1/2 state to the np series at the low-frequency endof the scan. Toward the center of the scan the fifth laserexcites the Ba+ 8s/2 states directly into the Ba2+ continuum.The high-frequency side of Fig. 3(a) shows excitation of the
22550 22750 22950
Laser Frequency (cm 1)
22800 23000 23200
Laser Frequency (cm 1)
Fig. 3. Frequency scans of the laser driving the final excitation from an ionic 5f or 8s level to a Rydberg state or a doubly ionized continuum.(a) Excitation of the np series and its associated continuum from an initial 8s1/2 Ba+ state at the low-frequency end of the scan. The high-fre-quency side of the scan shows the ng series originating from the 5f7/2 ion level. (b) The ng series originating from both fine-structurecomponents of the Ba+ 5f level. The separation between series members with the same n value is simply the fine-structure interval of the 5flevel. This scan also exhibits excitation of several members of the Ba+ nd series, which also originate from the 5f states.
1470 J. Opt. Soc. Am. B/Vol. 6, No. 8/August 1989
Ba+ ng series from the 5f7/2 state. Careful inspection of thisregion of the figure also shows slight excitation of the Ba+nd5 /2 series, which can also be reached from the 5f7/2 level.Figure 3(b) again shows excitation of the Ba+ ng series butfrom both the 5f5/2 and the 5f7 /2 initial states. Features onthe low-frequency end of the scan originate from the 5f7/2
state, and those on the high-frequency end originate fromthe 5f5/2 state. Note that the lower members of the Ba+ ndseries are clearly visible. The increase in the Ba 2+ back-ground near the center of the scan is due to direct Ba2+excitation from the 5f7/2 level.
The energies of the npl/2 and ng levels were measured byusing frequency scans similar to those shown in Fig. 3. Forthe npl/2 series the second laser was tuned to excite Ba+ 6s1/2ions directly to increase the signal-to-noise ratio, as dis-cussed above. The energies for the ng levels were deter-mined from the spectra from an initial 5f7/2 state of the Baion. As can be seen from Fig. 3(b), the series excited fromthe 5f5/2 state is obscured in places by small resonances,which we are unable to assign at this time. The experimen-tal energy levels of the two series are shown in Table 1, andthe uncertainty for all determinations is 0.2 cm-'.
Absolute frequency calibrations for our energy measure-ments are obtained as follows. Scanning the fifth laser overthe same frequency range used to excite the ion series allowsus to excite atoms in the 6s6p 'PI state to the lower membersof the Ba 6snd(s) series (approximately 6s12d to 6s14d) ifonly the first and fifth lasers are used.5 Therefore, by block-ing the second, third, and fourth lasers we obtain severalfrequency marker lines from well-known neutral-Ba levels.Also, as we have mentioned above, the Ba+ nd series is
Table 1. Measured Term Energies of Ba+ np 1 2 andng Seriesa
excited along with the ng series. This ion series was ob-served previously,' and therefore we obtain nearly 20 fre-quency markers by using previously published results forthis ion series. Relative frequencies are determined with adielectric-coated, solid-quartz 6talon with a nominal thick-ness of 1 mm.
The energy levels of the initial Ba+ 8s/2 and 5f7/2 stateswere taken as 58 025.18 and 57 631.64 cm'1, respectively.3
The energy used for the second ionization limit was80 686.25 cm'1, as reported by Boulmer et al.1 The energyinterval between the Ba+ 5f72 state and the second ioniza-tion limit determined from these previous measurementsagrees to within 0.04 cm-' with the interval measurementderived from our Ba+ nd series calibration. In order todetermine a value for the Ba+ 8s12 level energy experimen-tally, we used an independent calibration of the laser that isused to drive the Ba+ 6p3/2-8s,/2 transition. The frequencyof this laser (before doubling) was compared with that of thewell-known Ba 6s2-6s6p 'P, resonance line. This determi-nation gives us a value for the 8s/2 term energy that agreeswith Moore's value3 within our experimental uncertainty of<0.3 cm-'.
The largest errors in our energy-level values originate inthe determination of the line positions relative to a givenetalon fringe and in the determination of the free spectralrange of the 6talon at the frequencies of interest. We esti-mate our errors in finding the relative energy-level positionsto be 0.2 cm-'. The free spectral range of the etalon istreated as a free parameter to minimize the discrepanciesbetween the energy levels that we observe for the Ba+ ndseries and those reported previously. For this fit we use theBa 6snd marker lines mentioned above as an absolute cali-bration. Our fit yields a free spectral range of (3.523 0.003) cm-' for the etalon over the frequency range of inter-est.
The energy levels reported for the Ba+ np/ 2 and ng seriesare the average positions relative to the nearly 20 Ba+ ndreference lines used. The average is weighted according tothe number of etalon fringes between the experimental leveland the reference. An average quantum defect for eachseries was also determined. For the np,12 series we obtain
3PI/2 =3.198 0.002
and, for the ng series,
bg = 0.022 + 0.002.
These values were obtained by weighting the quantum de-fect from each member of a given series according to theuncertainty in its value.
The Ba+ ng series has minimal core penetration, and wecan therefore use the variation of the quantum defect with nto determine the dipole and quadrupole polarizabilities, a'dand a'q, of Ba2 +.6 The polarization energy Ap, the depres-sion of the energy of an nl state below the analogous He+state, is given by6
Ap = a'dR(r-4) + atqR(r6 ). (1)
Here Ap is given in inverse centimeters; R is the Rydbergconstant for Ba, 10 9737.3 cm 1; a'd and a'q are the dipoleand quadrupole polarizabilities, respectively; and (r- 4) and(r- 6 ) are the expectation values of r 4 and r 6, respectively,
-
R. R. Jones and T. F. Gallagher
Vol. 6, No. 8/August 1989/J. Opt. Soc. Am. B 1471
210
2001
C-
a:
190
180
170
160
150 1 2 3 4 5 6 7 8 9 10
q(n,4)x 03
Fig. 4. Plot of Ap(n, 4)/P(n, 4) versus q(n, 4). For determinationof the dipole and quadrupole polarizabilities, as in Eq. (4), a'd is they intercept divided by 16 and a''d is the slope divided by 64.
for the valence nl electron. We have used primes on thepolarizabilities to denote the fact that the values derived inthis way may not be perfectly accurate since the valenceelectron does not provide a static field. Discrepancies dueto the fact that the outer electron is not static are oftencalled nonadiabatic effects, and a clear quantum-mechani-cal discussion of their origin was first given by van Vleck andWhitelaw. 7 Following Edl6n,6 we introduce
P(n I) = R (r 4) (2)
and
q(n, ) (3)(r-4 )Z 2
where Z is the charge of the Ba 2+-ion core, 2 in this case.Using Eqs. (2) and (3) in Eq. (1) yields
Ap/P(n, 1) = a'dZ4 + a'qZ6q(n, 1). (4)
In Fig. 4 we show a plot of the observed values of Ap/P(n,4) as a function of q(n, 4). The points for the high-n mea-surements reported here all fall at q(n, 4) - 0.0099, and wehave therefore plotted all the measurements as a single pointcorresponding to a quantum defect of 0.022. Owing to the0.002 uncertainty in this quantum defect, this point has onlyone fifth of the weight of the points corresponding to the 5gthrough 10g states, which are taken from Moore.3 Fittingthe points in Fig. 4 to a line yields the values
a'd = 10.3(2)ao 3
and
a'q = 51(10)ao 5.
These values may be compared with values of the polariza-bilities calculated by Mahan,8 who reports values of 10.5 a0
3
and 56.9 ao5 when self-consistent field Hartree-Fock calcu-lations are used. Both values are in good agreement withour values. Such good agreement is surprising when we
consider the isoelectronic case of Cs. Analogous polariza-tion analyses9"0 of the Cs ng series, of quantum defect-0.007, yield dipole polarizabilities of 15.77(1) and15.82(3)ao 3, in good agreement with Mahan's 7 calculatedvalue of 15.9 ao3. On the other hand, the quadrupole polar-izabilities from the polarization analyses,9"10 34(7) and44.1(19)ao 5, are in substantial disagreement with the calcu-lated value. The fact that analysis" of the higher-i stategives a value in better agreement with the theoretical valuessuggests that the discrepancy between the theoretical valueand the one determined from analyses of ng-term energies isdue to penetration of the Cs+ core by the ng electron, not tononadiabatic effects. In contrast, the Ba+ ng states appearto be nonpenetrating. This is consistent with the fact thatfor the l < 2 series, for which core penetration is important,the Cs quantum defects are larger than the Ba+ quantumdefects by several tenths.
DISAPPEARANCE OF THE npll2 SERIES
Since the linewidth of the laser used for the final step in ourexcitation scheme was .7 cm'1, we did not expect toresolve the fine structure of the Ba+ ng levels. However, wedid expect that we would observe both fine-structure com-ponents of the Ba+ np levels. If we scale the fine-structuresplittings of the Ba+ 6p, 7p, and 8p levels listed in Moore'stable,3 we obtain a prediction for the splitting of the 30plevel of 1.8 cm'1, which should be well resolved with our laserlinewidth. However, we observe only single levels whenexciting the np series from the initial 8sl/2 state. We assignthe observed levels as np/2 states since the observed quan-tum defect, 3.20, matches almost exactly a linear bindingenergy extrapolation of the 6p1/2, 7Pi/2, and 8p1/2 quantumdefects,3 which extrapolate to 3.205 at n = -. The quantumdefect of the P3/2 states extrapolates to 3.165 at n = -. Oursignal-to-noise ratio in the excitation of the lowest np states,where the fine-structure doublets should be resolved, is >50,so it seems that the np3/2 states are missing.
Why only the npl/2 states are observed can be appreciatedas follows. The electric dipole matrix element for the 8s1/2-
where C1o is the spherical harmonic tensor operator for lin-early polarized light in the z direction. The ratio of theangular part of Eq. (5) for j = 3/2 to j = 1/2 can be found byusing standard methods'2 and has a value of V2. Therefore,if we take the value of the radial matrix elements to be thesame for both fine-structure components, the observed exci-tation spectra should show the nP3/2 series to have twice theintensity of the npl/2 series. However, the assumption thatthe radial matrix element does not depend on j is not neces-sarily valid, and, in fact, the excitation probability for thedifferent fine-structure components can differ by severalorders of magnitude. This effect has been observed in Cs,and the disappearance of the DI line of the np states hasbeen studied in detail.'3 "14 The same effect is also responsi-ble for the fact that alkali atoms exhibit nonzero minima, notzeros, in their photoionization cross sections.' 5
Using a numerical Numerov integration technique, wehave calculated the radial matrix elements for excitation ofthe Ba+ 8s,/2 state to an np level, using Coulomb wave
0
R. R. Jones and T. F. Gallagher
1472 J. Opt. Soc. Am. B/Vol. 6, No. 8/August 1989
X 0.01E
>- 0.001
asco00a000
0.00001 I ' I I I I I I I I01
EFFECTIVE QUANTUM NUMBER (Modl)
Fig. 5. Calculated transition probability for excitation of an nplevel from the Ba+ 8s1/2 state plotted versus effective quantumnumber (mod 1) for the Rydberg state. Note the zero in the proba-bility near a quantum defect (mod 1) of 0.20. The position of thezero is extremely insensitive to the value of the principal quantumnumber n for n > 20.
functions for various quantum defects of the np state.Some of the results of the calculation are plotted in Fig. 5,where excitation probability is plotted versus effectivequantum number for excitation to n = 23 and n = 43 levels.Figure 5 shows a definite zero in the excitation probabilitynear a quantum defect of 3.20, independent of the principalquantum number n. Although we would expect the zero at aquantum defect of 3.16 to match the missing np 1/2 series, thefact that such a simple calculation predicts a zero in approxi-mately the correct place supports our explanation.
CONCLUSION
We have measured the energy levels of the high-n membersof the Ba+ np1 /2, nd5/2, and ng Rydberg series. The energylevels of the nd5 /2 states are in good agreement with thosepreviously reported.1 The average quantum defect of the
npl/ 2 series is 3.198, and that of the ng series is 0.022. Bothof these values are in good agreement with extrapolations ofprevious results. From the quantum defects of the ng serieswe determine the Ba2+ dipole and quadrupole polariza-bilities. Furthermore, we note vanishingly small excitationof the P3/2 series, as is the case in Cs.
ACKNOWLEDGMENTS
It is a pleasure to acknowledge useful comments of M.Kutzner, V. Radojevic, and W. C. Martin. This research hasbeen supported by the National Science Foundation undergrant PHY-861 9056.
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3. C. E. Moore, Atomic Energy Levels, Natl. Bur. Stand. (U.S.)Circ. 467 (U.S. Government Printing Office, Washington, D.C.,1949), Vol. 3.
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Stroke, J. Phys. B 11, L765 (1978).15. U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968).
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