JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

4f'3 5d2 Configuration of Singly Ionized Ytterbium (Yb ii)tNISSAN SPECTOR

Israel Atomic Energy Commission, Soreq Nuclear Researclh Center, Yavne, Israel(Received 25 November 1967)

Sixty high odd levels of Yb II are interpreted as 4f 35d

2 levels. A least-squares fit with 10 parametersresults in a rms error of 239 cm-' (1.1% of the total width). Good agreement is also obtained betweencalculated and observed g factors. Designations in JjLii coupling are given.INDEX HEADINGS: Spectra; Ytterbium; Atomic spectra.

IN 1960 Racahl published a preliminary interpreta-tion of some singly ionized ytterbium levels pre-

viously established by Meggers. Racah showed sche-matically that J1 Jir coupling prevailed in the 4f"3 6s6pconfiguration, and JILI, in the 4f'3 5d6s configuration.Of the odd levels given in Table II of Ref. 1, all but twobelong to 4f"3 6s2 and 4f'3 5d6s. Racah 2 also made a com-plete calculation of these two configurations, includingthe interaction between them. He did not have, how-ever, a detailed interpretation of the 4f' 1 5d2 configura-tion, since the energy matrices of this configuration hadnot been calculated. Also, very few levels of this con-figuration were known. We thought it worthwhile,therefore, to complete the theoretical interpretation ofthe odd levels by calculating the missing matrices andobtaining predictions for the missing energy levels.

At the same time material based on the observationsand analysis by W. F. Meggers was available for inter-pretation. The results are included in a recent mono-graph on Yb ii by Meggers,1 published posthumouslyby C. E. Moore. It turns out that the levels from4f'3 5d2 have very little overlap with the rest of theconfigurations of the same parity. They lie completely

above the 4f"3 5d6s levels. This justifies an attempt togive a preliminary interpretation of 4f'3 5d2 , neglectingconfiguration interaction. We present here the resultsof this attempt, which proved very useful in the processof establishing the positions of the new odd levels, as wellas their J values and g factors.

I. CALCULATION OF THE ENERGY MATRICES

There are ten adjustable radial parameters in the4f' 3 5d2 configuration: A-the additive constant; F2 , F4 ,G1 , G3, Gs-Slater parameters for the electrostatic inter-action between the 4f and 5d electrons; B, C-the elec-trostatic interaction between the two equivalent 5delectrons; rf and Pd-two spin-orbit interactions. Ganz4

has calculated the f-d interaction and the two spin-orbitmatrices for the case of fd 2 . Since f 3

cd2 is the almost-

closed-shell conjugate of fd2 , Ganz's td matrices couldbe used without a change, while the signs of tf and theF2, F4 matrices had to be reversed according to Eq. (74)of Racah.5 According to the same equation, usingRef. 4, we obtained the following formulas for thecoefficients of the Gk parameters of f' 3 d2

:

F21 9G:- -3+3±V70(,uf(i).U('))-1 5V6(uf(2 )-U(2 ))+-V115(uf(3)-U(3))--9V55(,uf4). U' 4))+6(sf- S)

2 2

-120\/70(vf)l )- V'l))+60V6(vf( 1 2 ). V 12 D)-42-/15(vf(1 3 )- V' 3))+18\/55(vfi°4) V(14))]

9 55G3: - -12+--\ 70(uf(1)- U('))+-V6(Uf (2). U(2))- 28Vl15(uf(1 ) . U(3)) - 18V55(uf(4 .*U(4))+24(sf *S)

2 2

-18V/70(vf(1l) . V(")-110V6(vf(l2) *V(12))+112V/15(vf('l)* V(13))+72V55(Vfl14). V(14) ]- 99 275 77 9G5: - -66---V70(uf(')- U(1))- - V6(Uf(2). U(2)) -- 15(Uf(3)* U(3))--V-\55(Uf(4 ). U(4 ))+132(sf.S)

L 2 2 2 2

+ 198v\7 0(vf (l) . V(11))+550V6(vf(12). V( 12 ))+ 154V15(,f(13). V(13))+ 18\/55(7Jf(1 4 ). V(14))]

t Work supported in part by the National Bureau of Standards, Washington, D. C.'G. Racah, J. Opt. Soc. Am. 50, 396 (1960).2 G. Racah, private communication (1962).3 W. F. Meggers, J. Res. Natl. Bur. Std. (U.S.) 71A, 396 (1967).

J. B. Ganz, Bull. Res. Counc. Israel 9F, 31 (1960).G. Racah, Phys. Rev. 62, 438 (1942).

837

VOLUME 58, NUMBER 6 JUNE 1968

The complete matrices of the Gk's are given in the appendix. It is interesting to note that each Gk appears only inthose SL terms for which S= 2 (doublets only) and L satisfies the triangular condition with k and 2. For the quartetstherefore, all Gk's vanish. This means that G, is zero for all the terms of f I3d2 except 2PDF. G3 is nonzero only for2PDFGH, and G5 vanishes for all but 2FGHIK. It is also easy to see that all Gk matrices are of rank 1; that is, all

their rows (or columns) are proportional. These facts considerably simplify the calculation of the matrices.Racah6 commented that the Talmi check (Ref. 6, p. 9 "check B") could not be applied to fd' "because the

matrices of d~f were not calculated." With the matrices of f '3d2 (which are identical' to those of d~f) now available,we applied check B to these conjugate configurations. The parameters we used for the check were

A =6930

F2== 165

F4=45

G,=-G,=77G3 =-G 3 = 297

G5=- G5= 25,

where bars indicate almost closed shells. With these parameters, the complete electrostatic-energy matrix vanishes.The check indicates that in the electrostatic matrices given in Ref. 4

(d2(3P) 21I El fd 2( 3p)21)

(fd2(IG)2DlEJ fd2('G) 2D)

(fd 2('D)2F I EI fd2(3p)2F)

(fd2(IF)2F I EI fd2 (l'D)2F)

(f(12(G)2F I EI fd 2 (3 p) 2F)

(fd 2Q(G)2FI El fd12(3 F)2F)

should read A-8B+5F 2-9F 4+15G1/2+35G3/2+4G5

should read A+4B+2C+(165F 2-143F 4+5G,/2+95G 3 /2-1562G,/7)

should read +(15G1-495G5/2)(3/7) 1

should read - (15G,/2+ 165G5)(6/7) I

should read - (3G,+42G3+24G5)(11/14)i

should read +(3G,/2+21G3-114G5)(11/7)!.

The coefficients of B and C were taken from Eq. (78) of Ref. 5.

II. ESTIMATES OF THE INITIAL VALUESOF THE RADIAL PARAMETERS

It is well known that reliable initial values of theradial parameters are essential if a meaningful finalagreement is to be obtained in the least-squares fitbetween the calculated and observed energy levels. Theinitial values of the 4f-5d interaction parameters inYb II, as well as ad were estimated from the correspond-ing parameters calculated previously by us for theneighboring Tm ii, allowing for the slight increase inthe nuclear charge. The initial values for B and C wereestimated on the basis of known data for Hf i and

TABLE I. Parameters for 4 f 3 5d'.

Value used in last Value obtained bydiagonalization least-squares fit

Parameter (cm-') (cm-)

A 55 500 55 636 ±74B 370 376 :±9C 1460 1431 ±65F2 160 159 ±4F4 11 11 ±41G. 144 143 44G3 21 22 43G 3.3 3.0±t 0.6

2910 2897 i27830 813 ±34

rms error in cm-' 239rms error in 0/% configuration width 1.1%

6 G. Racali, Bull. Res. Counc. Israel 8F, 1 (1959).E. U. Condon and G. H. Shortley, Theory o/ Atlomnic Spectra

(Cambridge University Press, 1950), p. 298.

La II. For Pf the value of 2899 cm-l was obtaineddirectly from the two 4f"36s2 levels of Yb iI.I To deter-mine the initial value of A, the additive constant, wetook advantage of the fact that the energy matrix ofJ =7 is a 2>X 2 array and both energy levels are known.We added up the traces of all the parameters andequated the sum to the sum of the observed levels, thusavoiding even the easy job of diagonalizing the J=7matrix. Since the trace is invariant under a unitarytransformation, the equation obtained, namely,

2A-4B+2C-25F2+3F 4+21OG5-3, 2+ 2Dd= 102246,

is independent of the coupling scheme. We now solvefor A, assuming for the other parameters their pre-viously estimated values.

III. INTERPRETATION OF THE LEVELS

The configuration 4f135d2 comprises 81 energy levels.In Ref. 3 more than a hundred high odd levels arelisted, 73 of which are attributed to 4f"5d 2 . This con-figuration starts around 45 000 cm-l. The configuration4fI3 5d6s starts at 26 759 cm-l, as given by Racah.'There are 39 possible levels, out of which 20 were givenassignments by Racah.' Since the same structure asgiven in Fig. 7 of Ref. 1 is repeated 10 000 cm-l higher,based on the f13 2F21 parent level, it was easy to separatemost of the levels of 4f "35d6s from those of 4f "S5d'. Onlyin one case was there some doubt as to the assignment,namely in the case of the two J = 0° levels, at 46 902cnml and 47 229 cm-l. Owing to their closeness and therms error of the predicted positions of the 4f"35d2 levels,

838 NISSAN SPECTOR Vol. 58

4f 135d2 CONFIGURATION OF SINGLY IONIZED Yb

TABLE II. Energy levels, g factors, and percentage composition of 4fl35d2 of Yb ii.

Observed Calc. level Obs.-Calc.Designation J Comp. level (cm-') (cm-,) (cm-,) g (obs.) g (calc.)

3F[21] 34 30% 47 759 47 877 - 118 1.08 1.17

3F[32 3

'F[14]

3F[61]

3F[51

3F[02 I

3F[42 ]

'G[04]G[74]

IG[24]IG[34]'G[64]3'G[4]23'G[5 2]

P[31 ]

'P[42

3P[2E]

'D[32 ]2D[54]

'D[4]2

'D[14][223

'2[34]'F'[54]

'F'[24]

'F'[02]

F'[42-]

3F2[12]

212

12

412342120402

12

22526272425262

6214

3 24252032

223262

425 2L4-2

3422324252312221 2

345242112

21X32425262,

2.2

3120 1

12

3 24125X2

212

3 2412

ox

12

2 21

50%30%68%42%18%56%49%43%81%86%96%56%51%83%63%53%44%49%56%66%96%37%72%83%74%82%42%71%38%53%71%71%70%52%70%37%79%43%78%45%38%95%95%90%45%12%61%90%35%48%45%90%57%57%49%64%70%31%

45 01350 83346 35545 27351 24946 90245 73748 27346 54847 68048 92347 66348 50449 72749 41948 02549 00849 91650 46852 06853 32353 71657 10357 56258 48459 75352 88152 92253 12155 46256 08856 62156 50057 79958 16253 64554 19354 64055 22156 057

56 48058 05159 632?56 84059 43962 04656 97760 91057 53459 04661 44258 66159 62560 64958 67259 77762 164

45 25950 96546 44445 39951 20147 28245 59048 16146 60647 59248 66047 66948 51249 65049 83348 16849 12450 03050 31552 17053 17453 69657 32257 68858 84360 02352 28552 87553 11255 63856 15456 55556 58157 67258 29953 02053 69153 62354 94955 90169 03956 36957 99959 85956 7945936964 46357 16460 68657 61959 30761 63058 40759 44060 57758 66359 51162 031

- 246- 132- 89- 126

48- 380

147112

- 5888

263-6-8

77-414- 143- 116- 114

153- 102

14920

-219

- 126- 359-270

59647

9- 176- 66

65- 81

127- 137

625502

1018)272156

11152

(- 227)4670

(-2417)- 187

224- 85

(-260)

- 188253185

( 72)9

266133

1.2810.9561.291.181.0341.301.3331.140.971.121.14?0.981.101.201.411.381.031.0901.2130.541.041.1831.101.05?1.021.131.261.150.9440.8781.111.121.3521.1510.941.181.111.131.111.12

0.760.95

0.900.9941.142.191.000.871.08

0.86

1.100.5381.63

1.300.971.291.241.071.381.471.210.971.121.201.021.141.201.371.361.011.051.190.531.071.141.071.051.051.081.211.190.960.881.111.221.361.160.911.231.111.181.121.111.130.760.971.090.931.041.122.171.080.861.071.130.840.901.110.541.601.06

June 1968 839

NISSAN SPECTOR

ObservedJ Comp. level (cm-1 )Designation

'P'[2-4J

'P'[14.]

G'016-']'01114]1

IG'C!24]I'GE2-11'G'[3-x]'G'[54]'G'[42]'D'[4-]'D'[3X]D'[O1)-0]'D'[2-4]D'[1 x]

'S[24]

1 422-43:224?24314-x2-1,

1 -,s

6-2

142

24-325-x444-243-201-224-1 I

2-,t

Calc. level(cm-,)

65%44%61%44%81%72%66%88%72%92%81%70%58%98%94%81%52%70%56%79%66%

no clear-cut discrimination was possible. But the ex-perimental g factors of the two levels are 1.30 and 0.74,respectively, which singles out 46 902 cm-' as the4f"15d2 level in view of the predicted g value, 1.38(given in Table II).

In order to improve the initial values of the parame-ters, we made a least-squares fit using only the lowerlevels of 4fl3 5d2 . These results gave us predictions forthe missing levels, some of which were found later bythe process of searching and were used in furtheriterations. In Table I, we give the values of the parame-ters used in the final diagonalization, and also theirbest values and sharpness obtained by least squares.Since the parameters in the two columns are the same,within their rms errors, the problem has reached con-vergence. The rms error is 239 cm-l, which is only1.1% of the total width of the configuration. It is re-markable that the final rf is 2897 cm"l, almost exactlythe same value as is obtained in the estimate in Sec. II.

In Table II we give the predicted levels and g factors,together with a comparison with the observed levelsand Meggers's observed g values, as given in Ref. 3.(In the column "Obs.-Calc.," parentheses indicate thatthe'observed level was not included in the calculation).We also give the percentage of the major component ofeach level in the JiLri coupling scheme. In this scheme,the d

2 electrons give Russell-Saunders terms whose re-sultant orbital angular momentum Lir then couples tothe Jr of the f"3 core to give a certain K. The spin SInthen adds to K. Since Sir is an integer, we get, insteadof pairs, groups of three levels or single levels, accordingto whether Sri is 1 or 0. Then, following Racah,l thenotation will be that usual for Jil coupling, with the

addition of the name of the parent d2 term, written tothe left of the square brackets containing K. For ex-ample a level with J=6, which results from 3F of d2coupling to f"l 2F3, and has an intermediate K of 54, willbe denoted by 'FLS4]64. Since in our case J, has onlytwo values, namely 24 and 32, we prime the d2 term ofthe levels based on f13 F2X,. For example, the same levelas above could be based on 2F,; in this case it would bedenoted by 'F'[55]61. (See Table I in Ref. 1.) FromTable II we see that the JIL17 coupling scheme indeedenables us to give satisfactory names to the levels,since many of the major components are greater than50%. These names agree in all but four cases with theones given by Meggers.'

In Fig. 1 we plot the observed levels based on JI = 32and group them according to the names in Table II. Itbecomes evident immediately that, unlike the case of4f"15d6s, it would have been very difficult to connectthe groups meaningfully without the aid of the cal-culated percentage compositions. The observed levelsbased on J,=24 are omitted from the figure in orderto make it clearer.

IV. DISCUSSION OF THE VARIOUSINTERACTIONS

As already mentioned, we consider the introductionof configuration interaction (which is outside the scopeof the present paper) a second step in the process ofinterpreting the odd configurations of Yb ii. Beforethis step is taken it is essential to have a full under-standing of the structure of each odd configurationseparately, and to have reliable values for the inter-

840

TABLE II (continued)

Vol. 58

63 02863 41764 16965 59466 796

67 93467 790?

64 19265 739

64 3656497165 14266 20666 189

62 75463 32164 46364 46266 58567 70868 08967 68567 87164 09164 61665 71868 30668 96070 05363 73263 81965 04666 29265 61078 605

Obs.-Calc.(cm-,)

27496

(- 294)( 1132)

211

(- 155)( 105)

- 42421

( 633)( 1162)( 93)

- 86( 579)

g (obs.)

0.511.01

1.19?

1.17

1.1930.879

0.91

g (calc.)

0.571.021.120.910.871.121.181.070.200.971.110.700.990.960.950.961.060.471.060.860.86

4f"35d2 CONFIGURATION OF SINGLY IONIZED Yb

action parameters involved. Only then can additionalinteractions-if deemed necessary-be introduced safelyand meaningfully. Racah,' as has been indicated,followed the same principle when he interpreted the4f1 35d6s configuration without taking into account anypossible interaction of 4f "S5d'. In the light of our resultsand those given in Table II of Ref. 1, we make the fol-lowing comments concerning configuration interactions.

There are three odd configurations below 4f"5d',namely: 4f"6s', 4f ' 46p, and 4f"5d6s. There is noevidence that the first two interact very strongly with4f "35d'. On the contrary, they are too small and too farremoved from 4f"5d2 to exert a significant perturba-tion on it. Furthermore, their experimental g factorsare almost identical with the theoretical ones; for4f J"6S2 the observed values are g3,l= 1.143 and g2l = 0.861as compared with the theoretical gq = 1.143 andg2l=0.857, while for 4f' 46p both theoretical and ob-served values are g&n==1.333 and ewL=0.667. (Thetheoretschemecloseneate leewhich (interacduced,eight robtainsstrictly

On tthat thcan briIn the4f 5d2*understthat thform s

60

`. 55

leU

us 50

45

,9

I?7

1 5

ENERGY

in (1000cm )

13

5 6

I

I

8 9 10

J3-

FIG. 2. Observed energy levels of 4 f"5d6s2 ofEr I grouped in JiLil coupling.

ical values a i5d6s-5d2 interaction to be important. This interactionica vauesare independent of the coupling cabeahvdonytruhhemxgofheI'

). In addition, we have already noted the striking can be achieved only through the mixing of the 'D'sss between the value of If obtained after elabor- in both even subconfigurations2 Indeed, the two 'D'sist-squares calculations for 4f "35d2 and that of 4f' 35d' seem to be pushed somewhat upwards relative-an be obtained directly from 4f"6s'. Finally, if to their calculated position. However, as pointed oution of these configurationswith 4f"5d is intro- by Camus,5 there is no meaning to introducing the

it will involve the addition of four new levels and interaction with f l'-lds without simultaneously in-Lew parameters! Any improvement in the fit cluding fop as well. It is clear, therefore, that any.d under such conditions should be considered further theoretical treatment of the odd configurationsfortuitous. of Yb ii will have to include all of the four odd con-

he other hand, there seems to be an indication figurations 4f"3(5d+6s)'+4f' 4 6p. This will increase the.e introduction of the interaction with 4f"5d6s number of fitted levels over the number fitted in theng about a significant improvement of the fit. present work by 70%, and their range by 80%, but thefirst place, this configuration is much closer to number of adjustable parameters will increase by 200%.* Also, which is more to the point, our present It is clear that such a problem will require great care. Weanding of the 4f"m5d configuration indicates hope that our independent treatment of 4f"S5d2 pres-e thirteen 4f electrons and the two 5d electrons ented here paves the way for this subsequent problem.eparate groups, so that we may expect the A striking feature of Fig. 1 is the highly perturbed

form of three of the triplet groups based on IF of d .This is due to an interaction within the configurationitself. It is a characteristic feature of the f-d inter-

3P[23' r \ action which destroys the pure JrL11, aspect of thecoupling. A similar situation exists in 4f"5d6s' of neutral

3P 42 \erbium (Fig. 2), when it is interpreted in J1 L1I1 coupling. 9

A departure from the conditions which favor JI1TL1 iscaused by the strong exchange interaction in f-d.

- ,Taking the almost-closed-shells case the major contribu-P [32 G tion to this exchange energy in fl~d, as can be seen from

Ref. 7, p. 299, is 70 Gi, which appears in the energy ex-m3IF 4 1] pression of 'P only. For Yb iI, where Gl= 150 cm-' this

\ 3F [5 1] means an addition of 10 500 cm-' to the electrostatic]F[il L2J 'F[64] energy of 'P. For comparision, G3 appears in the energy

L2

J 'F[2 expression of 'F only. While its coefficient is 1.7 timesX [3 greater than for G1, its value is -} as much, resulting

in an addition of only 4200 cm-' to the electrostatic°2 jl 1 34 44 54 e 72energy of 'F. In the case of 4f"ScdI a detailed unitary

FIG. 1. Observed energy levels of 4f "S5d' ofYb II grouped in JILII coupling.

8 P. Camus, private communication (1967).9 N. Spector, J. Opt. Soc. Am. 56, 341 (1966), Fig. 3.

June 1968

I

- I II I

I A

I %11

11 I

NISSAN SPECTOR

decoupling transformation is needed in order to tracethe influence of this additional electrostatic ener-N, tothe lP "Parent." From Fig. 1 we see, however, how theaffected levels are brought up.

Meggers' also used calculations made by the Racahgroup on some odd as well as even configurations ofYb ii. The good agreement which we obtained between

calculated and observed levels indicates that theinterpretation of the levels in JiLii coupling and theprediction of the g factors are certainly not impaired byconfining our attention to 4f' 3 5d' only.

The writer wishes to acknowledge his indebtednessto the late W. F. Meggers for the use of the extensiveobservational data on this complex rare-earth spectrum.

APPENDIX

Exchange Electrostatic Energy Matrices for f 13 d2.

GI

'D03 F 'G 'D 3F1~/Ii 3~f ~289 72 -1 24,F3P 1 1/_1 347 7 ' f7

21 917 4 1817

277

3P 0 3F 'G

LY3V7

0 63 3r7D 65

189 9 -55

3-6r2 6' 377

24 "1 I r 6r/54F7 7

72 I8142 6 v6-7 7 7

27 - 977

73

3p ID 3F 'G

324 54(7-0 378 54175 -- 35 5 7

68 9 17_0 9Vi0

i5 -9745

77

27 3 33D7 3F I

12-F 1121 2417w 12"777 7

7

P '0 _3F,4- 'G4

14 7 7

-7-

'D F'D 3F I50 755F2 5 5 %2b31-

774 7

75 7

77

2K

IS 3P ID 3F i

66 99\(2 55\(4 3 3VT7 33

297 T1 9917 9V1547 2 14

275 5 5 A- 5 V6767 1I4 1

33 3r7_72 1-4

3

3P ID 3F IG

231 -331/S_ 63(22 2=3 0

2 2189 N310i2~ 2-i

57-

ID 3F 'G

104 -26\(- 2/S39-

273 -3/-453

3F IG441 631-7

63

3F210

842 Vol. 58

G5