Charles B. Ross, David R. Wood, and Pamela S. SchollWright State University, Dayton, Ohio 45431
(Received 15 August 1975)
The first six members of the 6 s 2n g series of Pbii were fitted to a polarization formula, from which the serieslimit was determined to be 121 245.14 4 0.05 cm-1. This value is in good agreement with that determined fromthe same series by an extended Ritz formula, as is also the case with the nf, ng, and nh series of Mgii, thenf series of Csi, and the ng series of Pbk. The polarization parameters empirically determined from the ngseries of Pbn were used to predict level values for the nh series, six members of which were subsequentlyfound.
INTRODUCTION
Most of the experimentally determined levels of singlyionized lead belong to configurations consisting of asingle electron outside of a core composed of closedshells, with the 6W2 subshell outermost. The resulting
36 J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
energy structure therefore consists of what are essen-tially one-electron series of levels, characterized byquantum defects of about 0. 02 for the hydrogenlike6 s2 g series of levels, and quantum defects of morethan one for the penetrating orbits 6s2 ns, np, wad, and nf.The first six members of the ng series were accurately
determined by interferometric measurements reportedearlier. 1 These levels are relatively unperturbed byother configurations.
While all unperturbed series can be approximated byan empirical extended Ritz formula, hydrogenlikeseries can also be described in terms of parametersempirically determined to fit a polarization formula.Either of these formulas can be used to establish aseries limit from a hydrogenlike series. The polar-ization formula, however, has the advantage that theseparameters can be used to predict level values forother hydrogenlike levels of the same atom or ion.
Hydrogenlike series of terms in the lighter elementsMgii and Civ have been accurately represented by po-larization parameters. 2 Kleiman3 was able to fit the2F series of Csi to the polarization formula with an rmsdeviation of 0. 04 cm-'. In Pbi, the value for the serieslimit was derived by applying the extended Ritz formulato four different series while the good fit of the 6s2 6P ngseries to the polarization formula was noted. 4 This fitimplied a value for the series limit in good agreementwith the result from the extended Ritz formula and withcomparable uncertainty.
With these indications of the applicability of the po-larization formula to hydrogenlike series in heavierelements, we fit the 6s 2 ng series of Pbii to the polar-
ization formula and derived a value for the series limitof Pbii. The resulting parameters were then used topredict level values for members of the previously un-reported 6s2 nh series, six members of which we weresubsequently able to locate experimentally.
THE POLARIZATION FORMULA
The difference between hydrogenlike term values,Tn, in a multielectron atom and the corresponding hy-drogenic values,
Z2 R F a 2 z! ( n 3
T = n2[1 + n +kl 4 4)]
is primarily due to dipole and quadrupole polarizabilityof the core, if the effects of penetration are negligible.Bockasten2 was able to fit hydrogenlike series of Mgiiand Civ to the polarization formula:
Tn= TO+ AO(n, 1) + Bb(n, 1).
In this expression
p (n4 1) = a4 (r- 4)/Z 4 ,
where ao is the radius of the first Bohr orbit of hydro-gen, and r is the distance from the nucleus to the va-lence electron, and0(n, I) = ao ( /
Therefore (P and 4 depend only on Z and the quantumnumbers n and 1 of the valence electron. A table ofvalues is given by Bockasten. The parameters A and Bare proportional to the dipole polarizability ad of theremaining core of electrons and its quadrupole polar-izability 0a as
A = R a,,Z4a3, B = RcaQZ6/asa
and are independent of the particular hydrogen-likeseries chosen.
Edlen7 points out that the experimentally derivedvalue of the parameter B may depend upon penetrationand other effects as well as quadrupole polarizability.The polarization formula still provides a useful repre-sentation of hydrogenlike terms in these instances,though B no longer has a simple interpretation.
To graphically illustrate fitting of data to the polar-ization formula and in addition its sensitivity in obtain-ing a series limit, the polarization formula will be re-written in the form
(T.- T0)/'P =A+B(P/'p),
which should give a straight line when (T, - T )/'P isplotted as a function of the corresponding values of p/'Pas n is varied. Since the term values are dependentupon the chosen series limit Loo as well as the levelvalues Ln, a value of the series limit may be derived byadjusting this parameter until a straight line is obtainedfor the data.
Figure 1 shows the sensitivity of this method in de-termining the series limit of Pbii from the 6s2 ng seriesof levels. As the formula neglects the fine-structuresplitting, the centers of gravity have been used. Thestraight line corresponds to a choice of 121 245. 15 cm-'as the series limit. If a series limit 0. 15 cm-1 greateris chosen, the graph is clearly concave upward, whilea series limit 0. 15 cm-' less gives a graph curvingconcave downward.
CALCULATION OF POLARIZATION PARAMETERS
By digital computer, the three parameters LX, A,and Bcan be determined which give a best least-squaresfit of the experimental level values to the polarizationformula. All level values are equally weighted. Todetermine the uncertainty in each of these parameters,we hold this parameter fixed at a chosen value, performa least-squares fit with the remaining two parameters,and then increment the value of the fixed parameteruntil the value of the mean square deviation is twicethat for the best least-squares fit. The values of theparameter above and below the best least-squares valuefor which this condition holds true give us the uncer-tainty range.
A least-squares fit of the centers of gravity of theexperimental 6s2 ng levels of Pbii to the polarizationformula yields a series limit of 121 245.14 ± 0.05 cm-.The value of 2. 349 ± 0. 004X 107 cm-' for A correspondsto a dipole polarizability of 1. 982 A., while B= 2. 55± 0. 08X 108 cm'1 corresponds to a quadrupole polariz-ability of 1. 51 Ai. The standard deviation of 0. 018 cm'
C. B. Ross et al. 37
TABLE I. Series limits derived from the polarization formulaand from the extended Ritz formula.
Polarization limit Ritz limitElement Series (cm"') (cm')
Mgx' inf (n = 4- 10)a 121267.619 ± 0.017 121267.61'ng (=i5-9) a 121267. 612 ± 0.013nhi (i = 77-10) a 121267.87 ± 0.13
'P. Risberg, Ark. Fys. 9, 483 (1955).bH. Kleiman, J. Opt. Soc. Am. 52, 441 (1962).t D. R. Wood and K. L. Andrew, J. Opt. Soc. Am. 58, 818(1968).
in the best fit is less than the uncertainties assignedto any of the g levels, which vary from 0. 02 to 0. 06cm-1.
The series limit of Pbii was also determined fromthe ng levels using the extended Ritz formula with twofree parameters in addition to the series limit. Forthe ng 207/2 series, this method gave a series limit of121 245.12 cm-'. For the ng 2G092 series, the derivedseries limit was 121 245.02 cm-'. Using the centersof gravity of the ng series, the limit was 121 245. 04cm- 1. The 0. 027 cm-' standard deviation for the ex-tended Ritz fit was higher than the 0. 018 cm-1 standarddeviation when these same centers of gravity were fitwith the polarization formula. The uncertainty in theseries limit, calculated by varying this parameter inthe Ritz formula until the mean square deviation dou-bled, was 0. 07 cm-1 compared to the value of 0. 05 cm t
derived from the polarization formula.
To demonstrate that limit values calculated using po-larization parameters agree closely with Ritz values,we also calculated limits from several other hydrogen-like series, again using the centers of gravity of theenergy levels. Table I summarizes our results andcompares our limits with those derived from averagesof calculations based on the extended Ritz formula.
In Kleiman's3 fit of the centers of gravity of the nfseries of Csi to the polarization formula, he used afixed limit value of 31406. 450 cm-1 and calculated Aand B from the values of the 4f and 5f levels. He foundthat the remaining levels were predicted with an rmsdeviation of 0. 04 cm"', which was an order of magni-tude greater than his experimental accuracy. By omit-ting the 4f level and allowing three parameters, L>., A,and B, to vary, we were able to fit the nf levels of Csi
2.60
I
Predicted Predicted Measured Derivedlevel value wavenumber wavenumber level value Difference
TABLE II. nh level values ofPbn and transitions to 5g.
38 J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
La I] 12145.3 0T
t 2.55 - <hi, 9 log0
2.50 -
2.4 5'-5 X 10- 6 7 8 9
FIG. 1. Graphical determination of the series limit of Pbnfrom the ng series. The correct series limit should give astraight line according to the polarization formula.
(n= 5-11) with an rms deviation of 0.002 cm-1 . Most ofthis rms deviation came from the 0. 004 and 0.003 cm-1deviations of the n = 10 and 11 levels. The deviations ofthe well determined n = 5-9 levels were within 0. 0015cmnf. The calculated position of the 4f level is 0. 3 cm1
higher than its experimental value. Its fine structuresplitting and quantum defect also deviate from the reg-ularities present in the other nf levels.
THE hI SERIES OF Pbil
The polarization parameters derived from the ngseries of Pbii were used to predict term values formembers of the previously unreported 6s 2 nh series.The transitions most likely to be observed are fromthese h levels to the 5g pair. The 6h-5g transitionsare predicted to lie in the lead sulfide region of the in-frared which we are not equipped to investigate, buttransitions from higher members of the h series to 5glie in the photographic infrared. Spectral lines corre-sponding to transitions from 8h, 9h, 10h, and 1h werefound close to their predicted positions on plates takenon the 9. 2-m Paschen-Runge spectrograph at PurdueUniversity, and additional transitions from 7h and 12kthe 2-m Czerny-Turner spectrograph. For our hollowcathode source, these lines are all weak and diffuse,with widths ranging from a few tenths to over 2 cm t .These widths mask the fine-structure splittings, dueprimarily to the 5g separation.
C. B. Ross et aL 38
FIG. 2. The polarizationformula applied to hydrogen-like terms of Pbix. Thestraight line is the leastsquares fit to the ng levels.
2 X 10-3
3 4 5 6 7 8
In Table II, the predicted and measured wavenumbersfor these transitions are compared. As shown, theagreement is close. All the experimental level valuesare larger than the predicted levels. The nearly con-stant difference of about 0. 1 cm-' between the experi-mental h levels and those predicted from the g seriesparameters could conceivably be the result of either oftwo effects. The first possibility is a Stark shift frominteratomic electric fields. The observed lines showedconsiderable broadening, as would be expected withStark effect, but no asymmetry was observed. Nor didthese differences become greater as the series limitwas approached. The second possibility is that theempirical values of the g series parameters were af-fected by penetration of the g electrons into the core.Since this penetration would lower the g levels, theresulting parameters, when used to calculate levelscorresponding to less penetrating h orbitals, wouldpredict level values lower than expected. As this wasobserved, we believe that these differences arise frompenetration effects, rather than Stark shifts.
We performed a least-squares fit of the h series ofPbri to the polarization formula which yielded a serieslimit of 121245.12± 0.15 cm-', with A = 2. 4± 0. 1X 107
cm f, and B = -±1 5X 108 cm-'. The large uncertaintiesassociated with these parameters are due to the smallvalues of 4 and 4 for the h series as well as the largelevel uncertainties. Each of the three parametersderived from the h levels agrees with the correspondingparameter determined from the g levels, within theestimated uncertainties.
In Fig. 2, experimental values of (T, - T°)/I for theng and nh series are plotted as a function of 4/1 on thesame graph. The straight line represents the leastsquares fit of the ng levels to the polarization formula.This graph indicates that the terms of both hydrogen-like series can be represented by means of the same
39 J. Opt. Soc. Am., Vol. 66. No. 1. January 1976
polarization parameters. It is similar to the graph ob-tained by Edlen' in plotting the nf and ng series of Mg"i.Little significance can be attached to the small negativeslopes shown by the higher angular momentum serieson both graphs because of experimental uncertainty.Note, however, that the members of these series tendto lie slightly below the straight lines determined bythe lower angular momentum series. These devia-tions are large enough that the lines do not intersect theupper error bars for most data points. In fact, thedeviations for the three lowest members of the niseries in Pbii are more than twice the experimentaluncertainties. We believe that these deviations can beascribed to the slightly differing penetration energiesof the two series, which have been neglected in the for-mulation. However, the difference does not appear togreatly affect the calculated value of the series limitparameter. In Table I, it can be seen that the serieslimits derived from the nf and ng series of Mgii are invery close agreement. The same is true for the limitsderived from the ng and nh series of Pbii.
ACKNOWLEDGMENTWe are grateful to K. L. Andrew who made available
to us the Purdue University Paschen-Runge spectro-graph, and who also provided useful information withregard to the Csi data.
'D. R. Wood, C. B. Ross, P. S. Scholl, and M. L. Hoke, J.Opt. Soc. Am. 64, 1159 (1974).
2K. Bockasten, Ark. Fys. 10, 567 (1956).3H. Kleiman, J. Opt. Soc. Am. 52, 441 (1962).4D. R. Wood, Ph.D. Thesis, Purdue University (1967).5I. Waller, Z. Phys. 38, 635 (1926).6J. H. Van Vleck, Proc. R. Soc. London, Ser. A 143, 679
(1934).7B. Edl6n, HandbuchderPhysik, Vol. 27, edited by S. Fligge
(Springer, Berlin, 1964) p. 187.8B. Edlen, in Ref. 7, p. 127.
