JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Extended identifications of odd energy levels of Si i: Lu-Fano graphical analysis

Charles M. Brown and S. G. Tilford

E. 0. Hulburt Center for Space Research, U. S. Naval Research Laboratory, Washington, D. C. 20375

Marshall L. Ginter

Institute for Molecular Physics, University of Maryland, College Park, Maryland 20740

(Received 14 December 1974)

The Lu-Fano graphical technique has been applied to the analysis of the strongly interacting J = 2

and J = 3 Rydberg levels of SiI which arise from the 3pnd configurations. Approximately 20 new

J = 3 and 40 new J = 2 energy levels, are reported. A periodic intensity variation in the

'D 2-3

P n d J = 3 transitions is discussed. Twenty-three new 3p n d 1Po' levels are reported.

Index Headings: Spectra; Silicon.

Recently, we reported' the results of an investigationof the absorption spectrum of Sii in the 1900-1500 Aregion. In that work, silicon was heated to 1800-2200 'Cin a King furnace, 2 and absorption spectra of Si I wereobtained in the third and fourth orders of a 6. 6 m vac-uum spectrograph. Transitions were observed fromthe ground-state terms 3s 23p 2 3 P and 3s23p2 'D to alllevels of odd parity with 05J<3 associated with the

3s23pns and 3s23pnd electronic configurations. Energy-level assignments were based on regularities of the ef-fective quantum numbers for the various Rydberg se-ries and on the manner in which the excited levels com-bine with the levels of the 3s23p2 3

p and 'D states. Incases where some of the 3pnd J= 2 and 3pnd J= 3 levelseries interact strongly, the previous analysis 1 waslimited to assignments that (a) could be justified by useof the simple approach of approximating the interactionas a pair-wise repulsion between neighboring levels ofthe same total angular momentum J, and (b) involvedlevels whose J values could be determined spectroscop-ically.

In the present work, we have extended the J-value

identification of strongly interacting levels in Sii by ap-

plying the techniques developed by Lu and co-workers 3-5

to the previously reported transitions.' Briefly, thismethod of level assignments takes maximum advantageof the periodic nature of the interactions between Ryd-berg series (channels) that converge to different serieslimits. In the present case, there are two series limitsthat correspond to the two levels of the ground state3s23p 2po of Siii. For each energy level, T, two ef-fective quantum numbers can be calculated,

ni= [(Ti.- T)-Rs]-12 i= 1. 2 (1)

where Ti. = 65747. 55 cm- (2 p 1/ 2 Si II limit),T2.. = 66 035. 00 cm-' (2 P,/ 2 Si II limit), andRsi = 109 735. 18 cm-. If the fractional parts of thesetwo effective quantum numbers, A, (=-ny modulo 1) and,112 (= n modulo 1) are then plotted as abscissa andordinate, respectively, for all known levels of inter-acting Rydberg series, the resulting curve(s) establishempirically the shape of the ,1,(ju2) curve(s) that de-scribe the periodic nature of the series interactions.When the curve l 1 (A 2), which is an implicit function

FIG. 1. Plot of p1J=3 levels of Sii.determining 1.11 (12)-

1.0

~1(=nt modulo 1) vs 112 (=n4 modulo 1) for theSolid circles represent data points used in

F11FIG. 2. Plot of pi (=nnt modulo 1) VS J12 (=4 modulo 1) for theJ=2 levels of Sii. Asterisks mark levels of low n. The opencircles mark levels perturbed by the 3s3p3 3P, level (see text).Solid circles represent data points used in determining thecurves for PI(92)-

385

VOLUME 65, NUMBER 4 APRIL 1975

BROWN, TILFORD, AND GINTER

4518 4518

40 40

35 17 35 17

30 30

16 16

25 25

24 2415 15

23 2-3

22 22

21 21

,14 140.0 0.5 1.0 0.0 0.5 1.0

FIG. 3. Plots of Al (nt) for several periods of n2* together withthe function ,u (nt) obtained from Eq. (2). Figures 3 (a) and 3 (b)apply to the J=2 and J=3 levels of Sii, respectively. The in-tersection points predict the nt values of the extended Rydberglevels; solid circles indicate observed values (see Tables I andII).

of n2*, is extended over all values of n*, the points ofintersection between the gu (n?) curve(s) and the curves,u',(n ) derived from

ncr (ne)p=ond t /[1o -t no 2(T2 n- Tl)/Res f 12 (2)

correspond to the possible energy levels of the inter-

acting series. 3 The ji 1(n * ) curve provides a methodof assigning energy levels in this scheme, particularlyin the region near the series limits where spectral linesare closely packed, often weak, and frequently blendedwith or obscured by other lines.

The methods discussed make no distinction between theindividual series; hence, all reference to term symbols,series classifications, and specific correlations of in-dividual series with specific ionization limits must beabandoned. Discarding conventionallevel orseries des-ignations should be regarded as a realistic step whenlevels or channels mix because of strong interactions.In such situations the n I and J values of the energy levelscontain all of the information that we determined spec-troscopically.

The AJ= 0 selection rule for level interactions is valid'for Sii; thus series with different J values can be con-sidered separately. From Ref. 1 it is obvious that allseries arising from 3pns configurations and the J= 0and J= 1 level series arising from the 3pnd configurationscan be handled adequately by the constant quantum-de-fect methods described. Hence, only the J= 2 and J= 3level series associated with the 3pnd configurations re-quire additional attention here. Of these, applicationof the Lu-Fano method is simpler in the case of the3pnd J= 3 levels. In the LS notation there are threeJ= 3 levels associated with each 3pnd configuration,'F°, 3 F°, and 3D°. Of these, only the two F levels needbe considered further because the 3D° levels do not in-teract strongly with any of the other J= 3 levels and con-verge regularly on the 2P°/ 2 Siii ionization limit (seeTable XIV, Ref. 1). The remaining series, formerlydesignated 'F° and 3F°, provide an example of a two-channel system; one channel belongs to the 2P 1/2 andthe other to the 2P°3/2 state of the ion. Data for all ofthe levels designated 'F' and 3F' in Ref. 1 were usedto construct the ,u I vs 11 2 curve in Fig. 1. The manywell-distributed points on the curve allowed the shapeof gj(i,1), and hence j.L1(n*), to be accurately deter-mined empirically.

The J= 2 levels present a slightly more complex prob-

FIG. 4. Spectrum of Si i observed near 1690 A in the third order of the grating. Absorption lines are black, whereas white linesare emission from the second order. High n-value members of the 3p

2 1D2 - 3pnd 3D3 series and the 3p 2 1D2 -nt J = 3 perturbed series

are indicated. Note the periodic intensity fluctuations in the perturbed series (see text). The open triangles mark transitions fromthe 3p2 3p levels.

386 Vol. 6 5

ODD ENERGY LEVELS OF SiI

lem. In this case there are four interacting channels,which would correspond to the 3pnd 3P', 3DW, 3F',and 'D' level series in LS notation. Two of the fourseries converge to each of the 3p 2po limits in Si i;hence, two ,u 1 vs p,1 curves exist in this case. As canbe seen in Fig. 2, sufficient data were available fromprevious work ' on the J= 2 levels to determine empiri-cally the shapes of both of the ,u (. 2) curves. The firstfive members of the 3pnd 3P' series were strongly per-turbed by the 3s3p3 3P2 level and were not used in ourdeterminations.

TABLE I. J=2 levels of Si I

Energy levelsPre-dictednt19.54219. 90820.17420. 57720. 92521.35521. 67322. 10022. 55422. 98023. 31523. 63724. 04024. 54124. 82325. 28525. 64726. 09126. 56527. 00527.46427. 79628.15028.58728. 98229. 53429. 84530.33530.69531.16031. 60332. 06332. 56133. 00633. 51733.92034.29034.7035. 1235. 5936. 0036. 5636. 9337. 5237.8238.36

aShoulder on a stronger line.bBlended line."Obscured line.

Observed transitions

Pi-T2m p2-T2) mD2-T)2

(CM-') (CM-,) (CM-,)

Ob-servednt19. 54319. 91020.1920. 57820. 91121. 35021. 68822. 08822. 56522. 98323.29223. 63624. 0024. 52024. 82825.28

26. 08726. 57127. 00327.4527. 800

28. 97529. 5229. 83630. 33130.6731.1631.5232. 0532. 55333. 03433. 5333.92334.27734. 68

35. 5935. 98

36.9537. 5737. 8438. 36

65 237.0665 247. 5 6b65 255. 1 7b65265.2065273.39

65 291 1165 299.3,65 308. 8765 316. 7 0 b65 322.1465 327. 9765 333. 76 b65341.8765 346.2d

65 363. 1465 368. 96

b

65382.38

65 393. 56bc

65 401. 27b

65407. 72 bb

65 413. 9b

65 420. 87

65 426.7965 4293 0065 431. 0065 433. le

59 189. 58e . .

59 207. 96c

59 266. 32

59 277. 06.. *.

T2(cm-,)

65460.2265 470. 7265 478.3,65 488.4065 496. 5965506.8165 514.2565522.6365 532.0365 539.8165545.2865551.1365557.0765565.0365 569.5365 575. 9065 580. 7d65586.3065 592.1265 597. 0565 601.965 605. 5665 609. ld65 613. 3d65 616. 8465 621. 665624.2865 628.2765630.965634.565637.165640.765644.0365 646. 9965 649. 9565 652.1965 654.1565656.3465 658. 6d

65 660. 9,65662.806 5 66 5.4d65667.265669.7965 670.9165 672. 98

dEstimated.'Not reported in Ref. 1.

TABLE II. J=3 levels of SiI.

Energy levels ObservedPre-dictednt25 66226. 39727. 06827. 88828.82829.78030. 70331. 63232.35633.10433. 93334. 88035.84036. 79937.77338. 72439. 67740. 63641.47042.3143.1644. 0544.9645. 9046. 8847. 8648. 8449.82

Ob-servednt

26. 39727. 06227. 89828.83729. 79430.67

32. 36533.10633. 9334. 90835.81536. 86437. 790

39. 721

42.4143.1944.08

46.0246. 82

48.8249. 7

'Predicted.bObscured by a strong line.

'Obscured by an emission line.dBlended line.

65 383. 40

65 393. 59P6 5 401. 27b

65 419. 51

65 437. 1 b65 445. 5165 454. 8365 462. 6965468.1465 474. 31 b

65 479. 95

65 492.41

65 509. 08b

65519.9365 524. 77b65 528.46

cC

65 539.7265 544. 5e65547.16

65 557. 39P

65 563. 6 1b

65569.87

65 575. 0 7 b

65 579. 2 2 b

65 583. 7 8 b65 585. 68"

65 590.08 ...655932.67964e 7.565 593.79 65 447.51P

T°3(cm-l)

65 580. 965 590.0665 597.7365 606. 5665 615.6164623.9365 630.8865 637. 9"65642.7965647.4365652.2465 657.5065 662.0065 666.865 670.7165 674. 4a65 678.065 681. la65 683. 7a65686.5465 688.7265 691.0765 693. 2a65 695.7465 697. 565 699. 6"65701.565 703.1

3p,-T3'(cm'l)

65 366. 9065 374. 5565 383.4065392.4465 401. 27 d65 407. 7 2 d

65419. 5 1 d65 424.2465 429. 0 8 d65 434.3365 438. 82 d65 443. 5465447.55

65 454. 8 3d

65 465. 51

bb

65 474. 3 1 d

65 478. 3 3d

65 479. 95 d

transitions

lD2-T3(cm1l)

b59291.2259 298.90

c59 316.7759 325.08

59 343.9459 348.6159 353. 69 d59 358.6559 363. 17 d59 368.05

59 387.6959 389.9259 392.22

b59 396.89

59 402. 7 9dc

59 329.42

59 374.13

The ,u ,(n 2 ) curves were generated by repeating theA 1(A 2) curves in Figs. 1 and 2 over several periods ofn*. The intersections of these curves with the On*)curves from Eq. (2) then were used to predict additionalenergy levels. Figures 3(a) and 3(b) are examples of

TABLE m. 3pnd lPlevels of SiI.

Tnb T b

n nt (cm) n nt (cm1l)

20 19.958 65 472. 02 33 32.97 65 646. 6221 20.966 65 497.91 34 33.96 65 652.4,22 21.957 65 519.93 35 35. 00 65 658.0023 22.978 65 539.72 36 35. 96e 65662.7e24 23.916 65 555.69 37 36.97 65 667.2625 24.956 65 571.35 38 37.95 65 671. 36e26 25.969 6 5 5 8 4 . 8 , d 39 38.98 65 675.3027 26.928 65 596.21 40 40.01 65 678. 9928 28.094 6 5 6 0 8 . d 41 40.99 65682.24e29 28.92 65 616.3e 42 4 1 .9 6 e 65 685.2e30 29.94 65 625.150 43 42.98 65 688.1531 30.97 65633.15 44 44.02 65690.93032 31.97 65 640.16 -0 65 747. 55

"See Ref. 1, Table VII for lower members of series.bObserved from 3 Po-3pnd 'P' transitions only.'Blended line.dObscured line.eEstimated.

387Apr. 1975

BROWN, TILFORD, AND GINTER

such plots for the 3pnd J= 2 and J= 3 levels of SLI. Thepredicted values of n 1 produced in this manner aretabulated in Tables I and II, along with the observedspectroscopic data and extended energy-level assign-ments. Because the method does not distinguish indivi-dual series, the energy levels in Tables I and II aredesignated only by their n t and J values. It should berecalled, however, that the newly assigned J= 3 andJ= 2 levels have as their ancestors at low n the3pnd 1"3F' and the 3pnd 3P', 3D', 3 Fo, and 'Do levelseries, respectively.

As discussed by Lu 4 and Starace, 6 the oscillatorstrengths of the discrete portion of a spectrum can berelated to the shape of the ju i(n 2 ) curve and thus shouldreflect its periodic nature in n . Figure 4 illustratesthis feature in Sii near 1690 A. This spectral regioncontains transitions from the 3p2 'D2 level to J= 3 levelsnear the series limits. The 3p2 lD2 -3pnd 3D' transitionsare not of special interest because the 3DO levels arenot involved in the interaction; however, these transi-tions are useful markers because successive membersof this series differ by almost exactly one unit of n 2.Itis apparent from Fig. 4 that the interacting J= 3 levelscombine strongly with 3p2 'D2 if their n* value is ap-proximately integral and weakly if their n2 value is ap-proximately half integral. Such an intensity variationmay account for the unobserved 9d 3F' and 1ld 3F'levels, since the nt values predicted for these levelsare 8. 635 and 10. 607, respectively.

Since the quantum defects for the J= 0 and J= 1 levelsare quite regular, I an attempt was made to extend theseseries once all possible J= 2 and J= 3 levels were iden-tified. With the exception of the 3pnd 'Po series, noextensions were possible. Additional members of the3pnd 'Po series were identified through n = 44 from theremaining unassigned spectral lines. The new 3pnd 'Polevel assignments are summarized in Table III.

The analyses reported here and in Ref. 1 result ina virtually complete accounting for all the observed dis-crete absorption spectrum listed in Table I of Ref. 1with the exception of a few weak features of doubtfulorigin. The present work also illustrates the powerof the Lu-Fano technique in untangling interacting Ryd-berg series, especially at high n values.

The authors wish to thank R. Naber for his assistancein collecting the experimental data and one of the ref-erees of Ref. 1 for suggesting the use of the Lu-Fanotechnique.

'C. M. Brown, S. G. Tilford, R. Tousey, and M. L. Ginter,J. Opt. Soc. Am. 64, 1663 (1974).

2 C. M. Brown, R. H. Naber, S. G. Tilford, and M. L. Ginter,Appl. Opt. 12, 1858 (1973).

3K. T. Lu and U. Fano, Phys. Rev. A 2, 81 (1970).4K. T. Lu, Phys. Rev. A 4, 579 (1971).5 Chia-Ming Lee and K. T. Lu, Phys. Rev. A 8, 1241 (1973).6A. F. Starace, J. Phys. B 6, 76 (1973).

388 Vol. 6 5