JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

4fnl Configurations of Doubly Ionized Cerium (Ce III)

NIssAN SPECTOR

National Bureau of Standards, Washington, D. C. 20234(Received 19 November 1964)

A theoretical interpretation is given for the configurations of the type 4fnl of Ce III. New values for radialparameters are obtained by least-squares. Percentage compositions in the appropriate coupling schemes aregiven for the energy levels. The rms error is less than 0.7% for most cases.

INTRODUCTION

THE first report on the energy levels of doublyionized cerium was published by Russell, King,

and Lang 30 years ago.1 Recently Sugar2 revised theanalysis of this spectrum, and found more than 100 newlevels and about ten new configurations, including thelowest one of Ce in, namely 4f2. Some of these config-urations, like 4f5f and 4f5g, were the first of their kindto have been reported in the spectroscopic literature.This ion, therefore, poses important and interestingtheoretical problems. This is a first attempt to interprettheoretically some of the configurations of Ce iii. In thisfirst attempt we did not include interaction betweenconfigurations. In some cases this leaves much to bedesired, especially in the case of 4f6p in which the rmserror could be greatly reduced by taking into accountits interaction with the three mutually interacting con-figurations 5d'+5d6s+6s2 . Also in the case of 4f5d theexperimental data strongly indicate an interaction with4f6s (the 4f 2 levels were found mainly by transitionsfrom the ',IF of 4f6s). Work is still in progress on thisphase. In what follows we consider the interactionparameters: spin-orbit, designated by Al; and theSlater parameters, designated by Fk and Gk. In the caseof 4f' we consider also ca, the L(L+ 1) correction, and Ekwhich are linear combinations of the Fk given by Racah.'Fo is always the additive parameter.

THE EVEN CONFIGURATIONS

1. The configuration 4f2

This configuration was not located by Russell, King,and Lang; therefore the question of the ground state forCe III was left open. The center of gravity of 4f2 wasestimated by Racah4 to be 60004A3000 cm-' below thatof 4f5d. Experimentally, the 'H4 of 4f 2 was observed at3277 cm-' below the 'G of 4f5d. There are 7 possibleterms in this configuration, which split into 13 levels.All but one of these terms, 3P, were found by Sugar. Thefirst estimate for the parameters of this configurationwas obtained from papers by Trees5 and Runciman and

' H. N. Russell, R. B. King, and R. J. Lang, Phys. Rev. 52, 456(1937).

2 J. Sugar, J. Opt. Soc. Am. 55, 33 (1965).G. Racah, Phys. Rev. 76, 1352 (1949), Sec. 6.G. Racah (private communication, 1963).R. E. Trees, J. Opt. Soc. Am. 54, 651 (1964).

Wybourne.6 We give in Table I the parameters used inthe final diagonalization, and their best values andsharpness obtained by three different least-squarescalculations. In the first least-squares we included allthe energy levels given by Sugar. This resulted in anrms error of 488 cm-l. By repeating the least-squares(l.s.) without the 'So level, which was based on onlytwo lines, the rms dropped to 34 cm-l. But this does nothave too much significance because the number ofelectrostatic parameters now equals the number ofterms. So we performed another l.s. holding the twoelectrostatic parameters E2 and E3 fixed to their valuesin the diagonalization. This brought the ratio of electro-static parameters to terms down to 4:6, and increasedthe rms error only to 75 cm-l, which is about 0.5% ofthe width of this configuration.

Table II gives the energy levels predicted by each ofthe l.s. calculations. In the Russell-Saunders (L-S)coupling scheme the levels are almost pure, except forthe 'G4 and 'F4 which are strongly mixed. The groundstate is almost pure IH4, with only 4% 1G4. Thus thisconfiguration is an example of good L-S coupling, obey-ing Hund's rule.

The main problem is the 'So level. Table I shows thatits introduction increases the total rms error from 34 to488 cm-l-that is, by a factor of 14. The rms errors inthe parameters increase by the same factor. When thislevel is discarded, its predicted position deviates by5000 cm-l from the observed value.7

2. The configuration 4f6p

This configuration consists of 6 terms which splitinto 12 levels. The level diagram shows that it is over-lapped by 5d2 and is not far from 5d6s. We thereforewould not expect a very good fit by treating it withoutthe possible interactions. Indeed, the rms error is394cm-1 which is about 6.2% of the width of the con-

' W. A. Runciman and B. G. Wybourne, J. Chem. Phys. 31,1149 (1959).

7 The same situation, however, occurs in the isoelectronic spec-trum La ni. This was pointed out to us by Racah,3 who-by takinginto account the perturbation from 'So of p2 and the relative widthof the corresponding configurations-predicted that the calculatedposition of 'So of f' in La II would be off by 5000 cm--. We tend toaccept W. C. Martin's suggestion at this point, that the theory isnot capable of an extrapolation from 17 421 (the nearest observedlevel) to 33 000 cm-'. This justifies our discarding of this level fromour later calculations.

492

VOLUME 55, NUMBER 5 MAY 1965

THE 4fnl CONFIGURATION OF Ce iii

TABLE I. Parameters for 4f2.

Diagonalization l.s. 1 I.s. 2 ('So discarded) l.s. 3 ('So discarded,Parameter (cm-') (cm-) (cm-') E,, E3 fixed) (cmn"l

E0 4320 5087 ±t437 4720 ±t33 4692=t59Ei 3166 3070 ±t 79 3642 i21 3694±33E2 17.9 20.8± 1.6 18.5± 0.1 fixedE3 414 452 ±33 417 ±3 fixeda 50 33 15 28 1 27±2Pf 630 607 ± 83 554 ± 6 547±t13

rms error (in cm-1 ) 488 34 75in percent of configuration width 1.5% 0.2% 0.4%

TABLE II. Energy levels and percentage composition of 4f2.

l.s. 2 l.s. 3 ('So dis-l.s. 1 ('So discarded) carded E2, E3 fixed)

Exp. Obs. Caic. O-C CaIc. O-C Caic. O-CJT name Percentage composition (cm-') (cm7') (cm-') (cm-) (cm-') (cm-') (cm--)

4 3H 95% 3H+4% 'G 0 -80 80 -20 20 -1 15 100% 3H 1527 1706 -179 1523 4 1507 206 99% 3H 3132 3487 -355 3154 -22 3119 132 3F 99% 3F 3763 4004 -241 3758 5 3740 233 100% 3F 4765 5179 -414 4777 -12 4744 214 52% 3F+44% 1G 5006 4451 555 5038 -32 5151 -1454 IG 56%' G+43% 3F 7120 6876 244 7082 38 7142 -222 'D 94% 1D 12 836 12 474 362 12 836 0 12 767 696 1I 99% 1I 17 421 17 275 146 17 421 0 17 401 200 100% 'P 19 139 17 838 17 7251 100% 3P 19 765 18 269 18 1352 96% 3P 20 571 19 051 18 9070 iS 100% iS 32 839 33 036 -197 37 652 38 073

figuration. The parameters used in the diagonalization TABLE III. Parameters for 4f6p.

and also their final values, obtained in the l.s. calcula-tion, are given in Table III. Except for G2 and G4, whose Diagonalization l.s.

Parameter (cm-') (cm-')rms errors are greater than their values, all the param-eters remained, within their rms errors, approximately Fo 51 790 51 606±+i145the same. This means that G2 and G4 do not play signifi- F2 65 644 24

G212 -6±4 6cant roles in this calculation. Table IV gives the pre- G4 6 4± 11

dicted energy levels and their percentage compositions 645 593± 71in j-j coupling. The purity is more than 94% for all but 2200 2060±t217two levels. Clearly, the conditions for good j-j coupling rms error (in cm-') 394(f, in percent of configuration width 6.2%

TABLE IV. Energy levels and percentage composition of 4f6p.

Observed Calculated O-CJ Exp. name Percentage composition in j-j coupling (cm'l) (cm-l) (cm71)

3 (2F2,)poi 99% ('F21)pOJ 48 267 48 465 -1982 ('F2,1)PO 94%/0 (F2'F)Poi+5% (2F2j)plj 48 405 48 060 3454 ('F3,)poj 94% (eF3,)po,+3%o ('F21 )p11 50 058 50 504 -4463 (7F3J)Po, 92% ('F3,,)po,+4% (2F2J)pll+4% (2Fs-)p1q 50375 50062 3133 ('F2,1)pI 94%,c (2F2,)p,1 +4% (2F31)POJ 51 262 51 093 1694 (7F2J)p 1 i 88% ('F21)p,1 -+-9% (2F3,)pjj 51 289 51 619 -3302 (2F2,)pj 94% ('F2,)p 1 +5% (eFN0)po1 51 641 51 467 1741 (2F2j)pj 100% ('F2f)plj 51 932 52 090 -1584 (2F31)pI 88% ('F3,)pjj+7% (2F2,)p 1j+4% (2F31)poi 52 441 52 843 -4023 ('F3j)pjj 94% ('F3,)pj+4% (YF3,)pol 53 616 53 462 1545 (fF31 )pq 100% (2F3j)p,1 54 194 54 115 792 (7F3J)p11 100% ('F3D)p11 54 556 54 256 300

493May 1965

NISSAN SPECTOR

105

104 1-

(03 I-

102 -

101 ~-E,

000

W2

100 I-

54 -

531-

521-

5'

50 (2F/2~

49'-

48- ( F5 /%)p

_l l/

1 2 3J -4.

IF3 -4 6p|

I F

4 5

FIG. 1. Energy level diagram for 4f6p and 4f7p (see Ref. 2).

3. The configuration 4f7p

As expected for the second member of the 4fnp series,this configuration exhibits an even purer j-j coupling

TABLE V. Parameters for 4f7p

Diagonalization I.s.Parameter (cm-) (cm-)

Fo 103 018 103 028 -411F2 30 32 + 2G2 5 6.2d1 0.5G4 4 4.94 0.8

654 658 i 5915 868 417

rms error (in cm-l) 31in percent of configuration width 0.4%

than the first. Table V gives the parameters used in thediagonalization and those resulting from the l.s. calcu-lation. We see that the problem converged, because theparameters remained the same, within their rms errors.The rms error is 31 cm-', which is only 0.4% of thewidth of the configuration. Table VI gives the predictedenergy levels, and their percentage composition in j-jcoupling. It is interesting to note that, although F2 hereis about half its value in 4f6p, and the contribution ofthe Gk's is not bigger, still there are some mixturesbetween levels of the same J. This may be partly be-cause A, is reduced by a factor of 3 from its value in4f6p, so that levels based on the same 2F parent are nowcloser, and therefore can interact more strongly.

In Fig. 1 we give the level diagram of both 4f6p and4f7p. This shows that the reduction of Up from about2100 cm-' to about 870 cm'I (while Pf remained aboutthe same) caused a different grouping of the levels, al-though their coupling is always j-j.

4. The configuration 4f5f

This configuration was reported in the spectroscopicliterature for the first time by Sugar2 ; therefore, onlynow can we have an idea about the values of Slater'sintegrals Fk and Gk (k= 0, 2, 4, 6) for the f-f' interaction.This configuration has 14 possible terms, which splitinto 26 levels; 25 of them are given by Sugar. He

TABLE VI. Energy levels and percentage composition of 4f7p.

Percentage composition in Observed Calculated O-CJ Exp. name j-j coupling (cm-l) (cm ') (cm-)

3 ('F21)PO, 98% ('F2,)Pj 100 663 100 609 542 (2F21)POI 94% (2F2,)Po+6% (2F2j)P1, 100 734 100 768 -343 (2F2j)P1 j 98% (2F2,)P1, 101 822 101 793 292 (2F2,)P1j 94% (2F2j)p1j+6% ('F21 )POI 102 174 102 162 124 (2F1 2)PI1 98% (eF2,)pj1 102 222 102 205 171 (2F,2)pIJ 100% (

2 F"2)P11 102 369 102 340 293 (2F3 j)p,1 94% (2F3j)po,+4% ('F31 )pq 102 961 102 971 -104 (eFJ,)p°o 92% (2F,3)po0 +5% (2F3I)p 1i 103 080 103 082 -24 (2F3,)pjj 94% (2F3,)pjj+5% (2F31 )po, 104 289 104 327 -383 (2Fj)p1 1 94% (2F3j)pj+6% (2F3j)poj 104293 104319 -265 ('F O)P1m 100% (2Fr,)p,1 104351 104343 -82 (7F31)pi 100% (2F3,)P,1 104841 104850 -10

494 Vol. 55

|. Ce2+-4f u ,

C 1, -" `1_1��

THE 4fnl CONFIGURATION OF Ce iii

TABLE VII. Parameters for 4f5f.

Diagonalization l.s.Parameter (cm-') (cm-')

Fo 101500 101544 ±412F2 14 14 i 0.4F4 1.6 1.3±4 0.2F6 0.3 0.1±4 0.0Go 1190 1175 414G2 8 6.84 0.5G4 0 0.8±4 0.3G6 0 0.1±t 0.0l4f 654 639 i 7r5f 27 26 ± 8

rms error (in cm-') 39in percent of configuration width 0.7%

assigned L-S coupling names for the 4f5f configurationdespite its being more than 100 000 cm-l above theground level, and although it overlaps the 4f7p, whichis in pure j-j coupling. The bases for these assignmentswere transitions to the 4f5d configuration, which mani-fests a good L-S coupling. The level diagram given bySugar2 shows that the multiplets are widely split. Theparameters, given in Table VII, do not satisfy the condi-tions for good j-j coupling (.4f, ¢5f>>Fk, Gk) nor do

they satisfy the j-l coupling conditions (¢4f> F2>>5»,

Fk>,, Gk). The condition is not ideal for good L-Scoupling either, because Fk, Gk>>»4f, t5f is not satisfied.The big value of Go causes the L-S character of thisconfiguration, and makes it possible to assign names inL-S coupling to the energy levels. We show, however, inTable VIII, where percentage compositions for theenergy levels are given, that the j-l coupling scheme ismore adequate for assignments in this case. There isonly one level whose major component is less than 55%in j-l. But there are eight such levels in L-S, along withmany interchanges in the designations, notably the ISwhich is a strong mixture, the 'I5 which is mainly a 1H,and the 'P which is mainly a 'S. All these levels arestrong mixtures in both coupling schemes, but in j-1they have a definite major component, which enablesus to name them. The most remarkable deviation inassignments is exhibited in the lowest level of thisconfiguration. Sugar called it 'G3, indicating a dis-crepancy from Hund's rule. According to our percentagecomposition, its major component is 3G, but we had toassign to it a IF name, in order to be able to assign allthe rest of the J= 3 levels their major components. Thishappened twice more: in the 'P case and in the ID case(in the latter the designation agrees with the experi-mental name). When we consider the j-l names, there is

TABLE VIII. Energy levels and percentage composition of 4f5f.

Exp. Percentage composition in Observed Calculated 0-CJ name L-S couplinga Percentage composition in j-l coupling (cm-') (cm-') (cm-')

3 3G 36% (IF)t+59%3G 79%21[31]+21%20[21] 98914 98918 -45 'H 67%o 31+27% 'H 80%o 21[5.0+20% 21[41] 99 178 99 140 381 3D 52% 3D+45% 'P 85% 2DL1U+15% 21 [0] 99 248 99 285 -374 3G 83% 3G 77% 21[31L]+13% 21[4-] 99 577 99 582 -55 3I 40% '1H+27% 3I 69% 22[41]+20% 21E5J]+10% 3t[51] 99 604 99 599 53 'F 29% (3G)t+30% 3D+23% 'F+18% 3F 67% 212E2]+19% 22[34]+12% 31[31] 99 708 99 698 102 3D 86% 3D 56% 2.K1XJ+27% 24[24] 99 894 99 870 246 31 88% 31 83% 2 H51] 100 016 99 987 291 3S 23% (1P)t+33% 3D+32%3S+12%3P 62%21[012+25% 3i1EI] 100190 100302 -1124 3H 77% 3H+21% 1G 86% 21[4,]+14% 2.K31] 100 814 100 814 05 3G 77% 3G+23% 1H 83% 31241]+16% 31[5A] 101 178 101 193 -153 3D 55% 3D+38% 'F 66% 3l[21]+34% 31[31] 101344 101350 -62 3F 69% 3F+25% 'D 72% 21L21]+23% 2C[11] 101 354 101 323 31

7 3I 100% 3I 100% 31[61] 101 565 101 551 141 'P 50% 35+31% lP+13% 3D+6% 3P 46% 31[11]+41% 31[01] 101 647 101 589 585 3H 83% 3H 72% 3X[51]+16% 31E4fl+10% 2!E4J6 102 409 102 410 -10 'P 81%'P±19%1S 94% 2 202] 102 502 102 500 24 'F 42% 1G+24% 3F+17%3G 74% 314 ]+14% 31C31] 102 566 102 577 -113 3F 83% 3F 551% 3AE3!]+32% 3121]+1l% 2121] 102 649 102 665 -166 3H 83% 3H 77% 31[51]+16% 3 [6l] 102 898 102 904 -62 'D 35% ('D) t+36% 3P+19% 3F+10% 3D 62% 34[21]+20% 21[11] 103 231 103 247 -164 1G 67% F+30% 'G 83% 313I]+17% 3 [4] 103 351 103 343 81 3P 85% 3P 58% 31iI0]J+29% 32[11] 103 613 103 595 186 1I 81% 1+14% 3H 67% 3ME621+22% 3t:5§] 103 676 103 674 22 3P 55% 3P+41% ID 74% 3t[1E]+25% 31[21] 104 177 104 189 -120 81% 'S+19% 3P 94% 3X[01] 105 383

3 ('**)t designation was assigned to term other than the largest component.

495May 1965

496 ~~NISSAN SPECTORVo.5

TABLE IX. Parameters for 4f6f.

Diagonalization l.s.Parameter (cm-') (cm-')

FO 124170 124187 ±5F2 8 7.5±t0.2F4 1.3 0.6±t0.1F6 0.1 0.140.0Go 650 653 ±t6G2 3 4 ±40.2G4 0 0.5±0.1Go 0 0.1±0.0P4f 635 636 ±42PGf 20 18 ±t2

rms error (in cm-') 13in percent of configuration width 0.3%,

no doubt about the assignment of the lowest level, orany other level, and we prefer, therefore, to adopt thiscoupling scheme for the 4f5f configuration.

Table VII shows that the rms error is 39 cm-'. Thiserror is only 0.7% of the width of this configuration.We consider this an indication that the configurations4f5f and 4f7p do not interact, although they overlap.

5. The configuration 4f6f

Only 21 of the 26 possible levels of this configurationwere given by Sugar. Here the j-l character is muchmore pronounced than in its analog the 4f5f configura-tion. Table IX gives the parameters of this configura-tion; the only drastic change is in Go, which dropped toalmost half the value it had in 4f5f. The rms error is13 cm-1 (0.3% of the total width). Table X gives thecalculated energy levels and their percentage composi-tion in the j-l coupling scheme. The fit is good in thistable, where only one level deviates by more than twicethe average rms error.

ODD CONFIGURATIONS

6. Configuration 4f5d

This configuration consists of 10 terms, which splitinto 20 levels, all of which have been found experi-mentally. Initial values for the radial parameters weregiven by Gotthelf.8 In Table XI we give the parametersused in the diagonalization and those resulting from thel.s. calculation. Although the l.s. values of the param-eters remained unchanged, within their rms errors,indicating the convergence of the calculations, the fit,

TABLE X. Energy levels and percentage composition of 4f6f.

Exp. Percentage composition Observed Calculated O-CJ name in j-l coupling (cm-') (cm-') (cm-')

3 (IF,2)[3'1 79% 24![3f]+21% 241[2)L] 122 160 122 164 -45 [54] 75% 21[54]+25% 24[41] 122 289 122 288 11 88%, 2[11] 122 3415 [4D] 70% 21[41]+25% 24[51] 122 611 122 619 -84 [3.] 80% 2 [31] 122 629 122 630 -13 [241] 74% 21 C21]+20% 21[3!] 122 689 122 691 -22 [i2] 59% 22[12]+32% 22[22] 122 808 122 804 46 [54] 92% 21[54] 122 870 122 871 -11 C[0] 83% 2X[04] 122 980 122 982 -24 [44] 83% 21 [4] 123 202 123 214 -122 [21] 67% 2[21-]+31%12[L14] 123 555 123 527 280 99% 22[042 ] 124 1685 (2F31)[41] 80% 31[41]+20% 34[51] 124 433 124 427 63 E241] 64% 31[24]+36% 3AC34] 124 510 124 509 17 64t] 100% 31464] 124 610 124 603 71 66% 32E[1]+31% 31[01] 124 6615 [54] 80% 3[54]+20%/ 34[4l] 125 006 125 009 -34 [44] 74% 314E4]+20% 31[34] 125 091 125 082 93 [34] 60% 34[34]+35% 34[24] 125 132 125 131 16 [54] 62% 34[51]+32% 341[64] 125 301 125 311 -102 [24] 74% 34[24]+18% 34[14] 125 400 125 410 -101 67% 34C04]+27% 3[E10] 125 5504 [34] 80% 34[3]+20% 3.[44] 125 616 125 605 116 [61] 59% 31[66]+36% 34[541] 125 710 125 724 -142 [14] 77% 31E41]+21% 34:24] 126 053 126 054 -10 99% 3120-42] 126 735

.thesis, Hebrew University of Jerusalem (1960).

496 Vol. 55

I U. Gotthelf, M.Sc.

THE 4fnl CONFIGURATION OF Ce ii I

TABLE XI. Parameters for 4f5d.

Diagonalization i.s.Parameter (cm-,) (cm-')

Fo 9406 9408 ± 56F2 177 176 4 5F4 21 21 ±t 1GI 300 297 ±t 7G3 38 39 ± 3G5 5 4.8±4 0.4Pf 690 669 ±t 51ad 700 777 -i102

rms error (in cm-') 226in percent of configuration width 1.5%

with a level rms error of 226 cm-l, is an unsatisfactory1.5% of the width of the configuration. Table XII givesthe calculated energy levels and their percentage compo-

to name it after the slightly smaller component 'G inorder to save the preferred name for the higher 'H 4.Also, these names agree with the experimental assign-ments. They also agree with the assignments in La ii(Ref. 9) 4f5d and Ce i (Ref. 10) 4f5d6s2, where theyare supported by experimental gi values. Apart fromthis strong mixture, the L-S coupling scheme is adequatefor designations in this configuration. The rather big rmserror cannot be explained offhand by configurationinteraction. The only plausible configuration for 4f5dto interact with is 4f6s, which starts only 2000 cm-'above it. (The next odd configuration is 70 000 cm-lhigher.) The levels which would be expected to beprimarily affected by this interaction are the 'F3 and'F2,3, 4. But Table XII shows that these particular levelsdo not seem to undergo a marked repulsion by the 4f6slevels. Also, there are levels with higher deviation (O- C)than the ''F, e.g., 1D2 (which deviates by twice therms error), 'H5, and 'G3. It seems that interaction be-

TABLE XII. Energy levels and percentage composition of 4f5d.

Exp. Percentage composi- ObservedJ name tion in L-S couplinga (cm-,)

46% (IG)t+49% 31-

77% 3F+22% ID49% 3H+40% 'G86%/ 'F+11% 3G

85% 3G+12% 3F100% 3H

69% 'D+22% 3F

64% 3F+30% 3G67% 3G+20% 3F

100% 3H

3277382251275502626563616571715078378350

Calcu-lated

(cm-')

3452391050955581594560197030733977348159

O-C(cm-l)

-175-88

32-79

320342

-459-189

103191

Exp. Percentage composi- ObservedJ name tion in L-S couplinga (cm-')

1523013251

3D 98% 3D

3G 98% 3G

3D 92% 3D74% 3D+25% 'F

3P 100% 3P94% 3P

IF 71%' F+26% 3D3P 96% 3P

'H 98% 'H'P 94% ',P

892293259900

10 12711 57711 61312 50112 64216 15218 444

a (.*.*)t designation was assigned to term other than the largest component.

sitions in L-S coupling. The outstanding feature of thedesignations is that Hund's rule is not obeyed. Thelowest level is a strong mixture of 'G and 'H 4. We had

TABLE XIII. Parameters for 4f6d.

Diagonalization l.s.Parameter (cm-,) (cm-')

Fo 91 500 91 509 ±36F2 31 31 ± 3F4 2.5 2.54 1GI 38 38 ± 5G3 2.5 1.7± 2.2G5 1.3 1.3± 0.4rf 642 642 ±20ad 243 239 ±36

rms error (in cm-') 139in percent of configuration width 3.2%

tween configurations here, although present, is a pertur-bation of only secondary importance. It is interestingto note that by eliminating the 1D2 from the calculationsthe rms error drops from 226 cm-' down to 112 cm-'.But this alone is not enough reason to discard this level.

7. The configuration 4f6d

In analogy with 4f5d this configuration has 20 levels,all of which were found experimentally. Table XIIIgives the parameters for the configuration and TableXIV the calculated energy levels with their percentagecompositions. Sugar gave assignments to the levels ofthis configuration in j-l coupling. By considering theparameters we see that conditions are not favorable forthis type of coupling because G, is not smaller than F2.On the other hand Pf, rd>>Fk, Gk; therefore, we should

I H. N. Russell and W. F. Meggers, J. Res. Natl. Bur. Std. 9, 625(1932).

10 W. C. Martin, J. Opt. Soc. Am. 53, 1047 (1963).

4243352446

'G3F3H3F3G

3HID

3F3G311

Calcu-lated(cm-l)

885191879948

10 08911 60411 62212 63212 69916 17418 397

O-C(cm-l)

71138

-4838

-27-9

-131-57-22

47

497May 1965

NISSAN SPECTOR

TABLE XIV. Energy levels and percentage composition of 4f6d.

Calcu-Percentage composition in Percentage composition in Percentage composition in Observed lated O-C

J Exp. name i-i coupling j-1 coupling L-S couplinga (cm') (cm') (cm ')

2 (F2FJ)[21] 92% (2F12)dj1 86% (2CF2)[21]+13% (2F23)[14J 77% 3F+20% ID 89 350 89 529 -179

4 [44] 94% (F 1F2)di1 71% (2F12)[44J+28% (2F2i)[34] 64% 3H+30% 'G 89 652 89 476 1763 [34] 98% (' 1F2)dij 86% (2F2s) [34) +14% (' 1`2)[21] 80% 3G+17% 3F 89 744 89 735 9

4 94% (2F2j)d2s 70% (' 1r2)[31]+30% ('21F)[44J 25% ('G)t+36% 3G+30% 3H 90 045 90 044 13 [2-i] 97% (2 1F2)d2, 85% (F2,1 ) [24] +14% ('2Ps) E34] 67% 3F+14% 'F 90 087 90 151 -641 [1i] 71% (21F2)d 1j+27% (2 1i2j)d2j 90% (2F21 ) [1E) 88% 3D 90 145 90 159 -142 92% (2F2j)d2j 86% (2F21 )[14]+13% (21F2)[2)] 57% 3D+21% 'D 90224 90337 -113

5 [44] 92% (2'F12)d2s 92% (2F23)E[4] 76% 3H+18% 'H 90 659 90 543 1161 [0o] 72% (2F2j)d2s+26% (2F21 )dij 90% (2F 12)[04] 81% 3P 90 879 90 859 200 100% ('21F)d2i 100% (' 1F2) [04] 100% 3P 90 902 90 855 474 (2Fsa) [44] 98% (2 1rq)dij 78% (23F1) [41] +20% (2F'I) [34) 59% 3G+21% 'G+14% 3F 91 736 91 699 37

3 [34] 98% (2F13)d,1 55% (2F31) [34] +43% ('1F)[21] 44% 3D+29% 3F+20% 'F 91 955 91 901 542 [24) 86% (2F 13)d 1j+12% (2rF3s)d2j 71% ('F31 )[21]+28% (2F3I)[14) 26% (1D)t+40% 3D+25% 3P 92 019 92 190 -1714 [34] 99% (2F3j)d2j 80% (2 1,3i)[34]3+20% (23F1) [44] 74% 3F+24% 'G 92 081 92 127 -46

5 [44] 64% (2F3j)dij+34% (2Fi3)d2i 69% ('F31)[41]+29% ('F31) [5i] 71% 3G+16% 3H 92 180 92 201 -216 [54] 100% (2F3j)d2j 100% (2F31 )[51] 100% 3H 92 527 92 440 873 [24] 99% (2F13)d2j 56% (QF31 )[24]+43% (F31)[3E3] 48% 'F+45% 3D 92 705 92 710 -52 [1)] 86% (2F3p)d2 1 +12% (2F31)dij 71% (2F3)E[14]+28% (F3I) [24-] 64% 3P+31% ID 92 795 92 849 -54

5 [54] 65% (2F,3)d2j+30% ('F31 )di 63% (F31)[5-i]+31% (2F31) [44) 69% 'H+24% 3G 93 227 92 963 2641 [14] 96% (2 1F3)d2j 96% ('FI)E[14] 74% 'P+18% 3P 93 603 93 745 -142

a (- *)t designation was assigned to term other than largest component.

expect good j-j names for the levels. This is actually tions in this coupling scheme are recorded also inthe case. The composition of the levels is much purer in Table XIV for comparison. Unlike 4f5d, this configura-the j-j coupling scheme than it is in the j-l scheme. tion overlaps completely the competing 4f7s, but theTherefore we give in Table XIV the percentage compo- interaction between them is much smaller. This can be

seen by considering the structure of the 4f7s configura-TABLE XV. Parameters for 5d6p. tion, which seems to be unperturbed. We shall return

to this later. The rms error is 139 cm-', which is aboutDia-onalization l.S. 3.2% of the width of this configuration. This relatively

Parameter (cm-) (cm-) big rms error can probably be attributed to the inter-action with 5d6p, which partly overlaps 4f6d. We have

Fo 98 675 98 676429 calculated the levels of 5d6p. The next step would be toF2 377 377± 6 take into account all the interactions among 4f6d, 4f7s,G1 413 412± 6 5d6p, 6s6p. The matrix elements of these interactions

31 2 12±t-1Gd 1085 1069±t27 have been calculated,8 but it is outside the scope of thisvp 2200 2203±t58 paper to give the calculations.

rms error (in cm-') 58in percent of configuration width 0.6% 8. The configuration 5d6p

Although not included under the title of this work,we give here the results of our calculations of this con-

sition in both j-j and j-l coupling. It is remarkable that, figuration because it will serve to clarify the situationexcept for a few cases, we are able to give good names in 4f6d. There are six terms in this configuration whichto the levels even in L-S coupling. Percentage composi- split into 12 levels. Table XV gives the parameters of

TABLE XVI. Energy levels and percentage composition of 5d6p.

Exp. Percentage composi- Observed Calculated O-C Exp. Percentage composi- Observed Calculated O-CJ name tion in L-S coupling (cm') (cm-') (cm-') J name tion in L-S coupling (cm-') (cm-') (cm-)

2 3F 58%F+37%'D 92 635 92645 -10 4 3F 100% 3F 99 169 99 095 741 3D 88% 3D 94 509 94 447 62 1 3P 79% 3P+13% 'P 99 288 99 287 12 1D 53%', 'D+41%' 3F 95 827 95 874 -47 0 100% 3P 99 6543 3F 92% 3F 96 022 96 048 -26 2 3P 88% 3P 100 968 100 984 -162 3D 94% 3D 96 376 96 378 -2 3 1F 90% IF+10% 3D 102 369 102 374 -53 3D 83%' 3D+10% 1F 97 964 97 994 -30 1 79% 'P+18% 3P 103 780

498 Vol. 55

THE 4fnl CONFIGURATION OF Ce Iii

TABLE XVII. Parameters for 4f7d.

Diagonalization l.s.Parameter (cm:-) (cm-,)

Fo 119 912 119 910 +2F2 17.5 17.4±0.2F4 2.2 2.140.1GI 17.5 17.2±0.4G0 3 3.1±t0.1G5 0.5 0.5=1=0.0A! 644 642 ±1Vd 90 87 ±3

rms error (in cm-') 9in percent of configuration width 0.3%

the configuration, and Table XVI the calculated levelswith their percentage composition. The rms error is

and only 4f terms (one of the 3P levels is missing). Theagreement, therefore, may not be as good as it seems,and taking into account the configuration interactionmentioned in Sec. 7 might give a better prediction forthe two missing levels.

9. The configuration 4f7d

Unlike the 4f6d configuration, Sugar gave assign-ments inj-l coupling for the levels of 4f7d. ConsideringTable XVII, which gives the parameters of this con-figuration, we see that this is one of the cases in whichthere is not much difference between the assignments inj-j or j-l coupling. So, for comparison, we give inTable XVIII the calculated energy levels with theirpercentage composition in both coupling schemes. Onthe whole the f-t coupling scheme gives slightly betternames, but j-j is also good, and in J= 4 it gives higher

TABLE XVIII. Energy levels and percentage composition of 4f7d.

Percentage composition in Percentage composition in Observed Calculated O-CJ Exp. name j-1 coupling j-j coupling (cm-') (cm-') (cm-,)

2 (2F2 1 )[24] 83% ('F 2 1 )[2fl+17% ('F2,)[114] 95% (21'21)dij 118291 118286 54 [44] 50% ('F21)[44]+50% ('F2)O[32] 81% (2F2O)d,+±18% (2F21)d2j 118312 118314 -23 [34] 97% (2F2j)[31] 88% (2F2O)dq+12% (2F21)d2j 118318 118313 54 50% f ('F2)[34]+50% ('F21)[42] 81% (2F2O)d2i+18% (2F2,O)di 118477 118478 -13 [24] 97% (2F2O)[21] 88% (2F2j)d21 +11% (2 F2O)dii 118 588 118 584 42 [1i] 83% (2F2,O[11]+17% ('F2KE2'] 95% (2F2')d2 118666 118659 71 98% eF2)2[14] 56% ('F2,)d,+43% (2F2O)d2i 118683 118683 05 [44] 98% ('F2) [42] 98% ('F2J)d2 118794 118805 -110 E04] 100% (R2 0[)E04] 100% ('F2O)d2 119 0101 98% ('F21)[01] 57% (2F2j)d2,+42% ('211 )dij 119 043 119 049 -64 [441 86% ('F 3 02)[44]+l4% (2F3 1)[31] 97% (f2f3)di 120468 120469 -1

5 84% (2Fn[424]+15% (QF31 )[5] 52% (2Fa1 )d2j+48% (2F31)dl 120 646 120 647 -13 j34] 55% (

2F3,O[34]+45% ('F,)O[24] 99% (Faj)dni 120652 120653 -1

4 86% (2F3i)s[3]+14% ('FO)[42] 97% (2Fsj)d2J 120 685 120 684 12 [24] 95% ('F3[24M]+15% (2F3,[5] 58% (2F31)d,,+42% (2F3j)d 2l 120 740 120 733 7

6 [54] 100% ('F3 l) [51] 100% (2F'm)d2, 120 846 120 852 -63 [21] 55% (F 3 O)[21]+45% ('F31)[31] 99% (2F,3)d21 121001 121001 05 154] 83% OF3 O[51]+15% (2F30 E44] 50% (2F31)d,+47% (2F3O)d2l 121 057 121 076 -19

2 114] 95% (2F3aO[1E1] 58% (2F31)d2,+42% M3F,1)dij 121 096 121 088 81 98% (2F31)[14] 98% (2F8 N)d21 121 559 121 546 13

58 cm-l, which is 0.6% of the width of the configuration.The l.s. parameters are the same as those introducedfor diagonalization, and the problem has, therefore,converged. However, we should not be misled by thegood fit, because there are 4 electrostatic parameters

TABLE XIX. Parameters for 4f6s, 4f7s, 4f8s.

Gf,Configuration (cm')

Vf(car')

R (= 1.333theor.)

percentages to major components than does j-l. Themean error is 10 cm-1, which is less than 0.3% of thewidth of the configuration.

10. The configurations 4f6s, 4f7s, 4f8s

Each of these configurations gives rise to four energylevels. The parameters are Fo, Gf, and gf. The ?f canbe obtained by using the formula Pf= 2[J(4)-J(2)]/7.Gf, is the arithmetic mean of the two differencesJ(3)-J(2) and J'(3)-J(4), where J' is the upper J=3level; i.e., G18= ffIJ(3)-J(2)+J' (3)-J(4)]. It is easyto see that both formulas hold independently of thecoupling scheme. The ratio R= (1F3-3F2)/(F 4-'F3)

4f6s 300 640 1.6364f7s 75 641 1.2314f8s 44 643 1.358

499May 1965

NISSAN SPECTOR

TABLE XX. Parameters for 4f5g.

Diagonali- l.s. 2 (F 4 , F6,Param- zation l.s. 1 (all free) GL. eliminated)

eter (cm-') (cm-') (cm-')

Fo 124 242 124 244 1:2 124 242 t2F2 0.35 0.36 140.02 0.3440.01F 4 0.0 0.01 -40.01 eliminatedF6 0.0 0.0 410.0 eliminatedGI 0.0 -0.16 4:0.18 eliminatedG3 0.0 -0.1 1t 0.1 eliminatedG5 0.0 0.0 410.0 eliminatedG7 0.0 0.0064:0.002 eliminated

643 643 i:1 643 I11 1.06 :1:0.97 1 I1

rms error (in cm-') 9 10in percent of config- 0.3% 0.4%

uration width

is 4 theoretically and the amount of deviation from itindicates the distortion due to interaction with otherconfigurations." In Table XIX we give the values of

Gf8, ¢, and R for the three configurations. We seethat in both 4f6s and 4f7s there are some perturbationswhich distort the structure of these configurations soas to cause a marked deviation from the theoreticalvalue of R. Only in 4f8s do we have overlapping evenconfigurations which do not interact.

11. The configuration 4f5g

This configuration gives rise to 28 levels, all reportedby Sugar. Since the 4f5g configuration has not pre-viously been reported, the first problem was to findinitial values for the parameters. Consideration of theenergy-level diagram indicates that the configurationexhibits pure j-l coupling, with Pj>>F2>>Gk. By calculat-ing the coefficients of F2 in the j-l coupling scheme" andcomparing with the experimental data, we obtained afirst estimate of F2. It was so small that we set all therest of the Fk's and Gk's equal to zero. In Table XX wegive the parameters of this configuration resultingfrom two different l.s. calculations. In the first, all were

TABLE XXI. Energy levels of 4f5g.

l.s. 1 l.s. 2(all parameters free) (F4, F6, G7, eliminated)

Observed Calculated O-C Calculated O-CJ Exp. name (cm-') (crrr) (cm-') (cm',) (cm-')

4 (IF,21)[4] 122 906 122 909 -3 122 907 -15 122 909 122 908 1 122 909 06 [5-l] 122 920 122 917 3 122 919 15 122 922 122 916 6 122 916 64 [31] 122 932 122 938 -6 122 940 -83 122 933 122 934 -1 122 937 -42 [2-1] 122 976 122 988 -12 122 985 -93 122 978 122 976 2 122 988 -106 [64] 123 010 123 005 5 122 994 167 123 017 123 003 14 122 997 202 [14] 123 028 123 029 -1 123 036 -71 123 029 123 036 -7 123 033 -36 ( 2F,1 )[51] 125 156 125 162 -6 125 158 -25 125 159 125 155 4 125 155 44 [44] 125 165 125 170 -5 125 169 -45 125 168 125 170 -1 125 171 -36 2[6] 125 181 125 175 6 125 177 47 125 187 125 182 5 125 180 74 [31] 125 194 125 201 -7 125 203 -93 125 196 125 198 -2 125 201 -52 [21] 125 231 125 243 -12 125 240 -93 125 233 125 230 3 125 242 -92 [112] 125 268 125 276 -8 125 276 -81 125 269 125 266 3 125 274 -58 [74] 125 271 125 258 13 125 252 197 125 280 125 270 10 125 255 250 [01] 125 295 125 287 8 125 297 -21 125 297 125 307 -10 125 300 -3

11 The formulas in this section can be found in B. Edldn, AtomicVerlag, Heidelberg, 1964), Vol. 27.

12 G. Racah, Phys. Rev. 61, 537 (1942).

Spectra, Encyclopedia of Physics, edited by S. FlUgge (Springer-

500 Vol. 55

THE 4fnl CONFIGURATION OF Ce fii

free. The rms error is 9 cm-l, which is 0.3% of theconfiguration width. We see that F4 , F6, G,, G3, G5 arenot physically significant, because they have negativevalues, and their rms errors are bigger than their values.It is remarkable that only G7 is positive; although small,it is sharp. However, guided by the relations among theGk'S, 13,14 we made another l.s. calculation eliminatingall the Gk's as well as F6 and F4. This made the totalrms error 10 cm-', still only 0.4% of the total width. Inboth least-squares the parameters used in the diagonali-zation resulted in the same values, within their rmserrors, indicating the convergence of the problem.Table XXI gives the predicted energy levels. The levelshave all pure (100%) j-l names, which agree with theexperimental names. Therefore we omit the column"percentage composition" in this case.

13 E. U. Condon and G. H. Shortley, Theory of Atomic Spectra(Cambridge University Press, New York, 1935), 46.

14 G. Racah, Phys. Rev. 62, 438 (1942), Sec. 10.

CONCLUSION

A theoretical interpretation of all the configurationsof the type 4fnl of doubly ionized cerium has been made.The interpretation consists of giving percentage compo-sition for all the energy levels in different types ofcoupling schemes, and of tabulating best values, ob-tained by least-squares calculations, of the various two-body electrostatic interaction parameters, as well as-spin-orbit interaction parameters. On the whole, thefit is satisfactory. In some cases (of pure coupling andunperturbed configurations) the rms error is as low asone-half of one percent of the total width. There areother cases, however, where good fits were not obtained,owing to strong perturbations by nearby or overlappingconfigurations. It was beyond the scope of this paper totake into account these interactions, but improve-ments can be expected for 4f6p+ (5d+6s)' and for4f6d+4f7s+5d6p+6s6p.

May 1965 501