Energy levels of Ho III

Physica 85C (1977) 386 392 © North-tlolland Publishing Company

ENERGY LEVELS OF Ho 111 J. F. WYART Laboratoire Aim~ Cotton, C N. R.S. I1, Bat. 50.5, 91405 Orsay, France

and H. M. CROSSWHITE* and RIAZ HUSSAINt The Johns Hopkins University, Baltimore, Mao'land 21218, USA

Received 21 September 1976

The early analysis of Ho 111 has been extended, including parametric studies for complete configurations of 4f I 1,4f105d ' 4f106s and 4flO6p, with 78 new energy levels found, including the lowest even level at 18033.4 cm-1. After optimization of real and effective parameters, the mean deviations on the energies are 32 cm-I for 72 even levels, 48 cm-1 for twelve levels of 4f 11 and 6 cm-1 for 37 levels of 4f106p.

The present state of the analysis has therefore been reached in three stages: (1) McElaney [2] was able to determine 42 low levels of these four configurations, making use o f rudimentary calculation o f the energy level positions and hyperfine structure patterns; (2) Hussain [3] extended the 4f106s and 4f106p analysis by making a detailed fit of the individual hyperfine components to a theoretical model of both the 6s and 6p splittings; (3) the most recent work led to classify 665 transitions among 121 levels. In this, parametric studies o f all four configurations used matrices containing all theoretical possible basis states.

1. Introduction The parametric analysis of the spectrum o f Ho Ill reported here is based primarily on J. H. McElaney's measurements on a holmium metal sliding spark [1 ], supplemented by electrodeless discharge Fourier transform spectra taken at Aired Cotton and spectrograms supplied by the Argonne National Laboratory. The electrodeless discharge source does not produce Ho 111 spectra but they have nevertheless proved very useful in re-interpreting some of McElaney's assignments of the degree of ionization. Holmium is blessed (plagued) by the presence of pronounced hyperfine structure, especially for the 4f 106s levels. The empirical energy level analysis therefore falls naturally into two parts: (1) assignment of the 4f106s-4f106p transitions, which are recognizable because of their hyperfine splitting; and (2) analysis of the 4f105d-4f106p and 4 f l l - 4 f 1 0 5 d arrays by constant difference searches. These latter are made more difficult by their incipient hyperfine structure which serves to modify the line centers in obscure ways without revealing the structure.

2. Analysis of data A detailed description of the experimental material which was used has been given by McElaney [1], but some corrections have been made in the interpretations given in the early line list. Four pairs of very strong lines at 4570.2 A, 4494.5 A, 4467.7 A and 4080.1 A are now interpreted as both sides of self-reversed resonance transitions. Recent extension o f the observations from 2500 A to 4 ~m at the Laboratoire Aim6 Cotton by J. Verg6s have led to a better separation of the Ho II and Ho Ill spectra: comparison is now possible with spectra emitted by electrodeless discharge

* Present address: Chemistry Division, Argonne National Laboratory, Argonne, IlL 60439, U.S.A. + Present address: Physics Department, University of Scranton, PA, 18510, U.S.A. 386

J. F. Wyart et al./Energy levels o f Ho III

tubes in which only Ho I and Ho II lines are excited. Between 7000 and 11000 A many lines previously attributed to Ho III are now classified as 4fl15d 4f 116p transitions of Ho II [4]. Initially considered as Ho II the line at 4580.8 A (which does not appear in the electrodeless discharge spectrum) is interpreted as a transition in Ho III between the lowest level of 4fl°6s and the ground state 4f ll 4I°5/2, which is in agreement with the hyperfine structure of the 4f106s level. As a result of the parametric interpretation of the even levels, configuration mixing induces in the eigenfunction of the 4f106s level a small component 4f105d 4K (0.6% of the total) which makes the transition possible. A critical examination of the early energy diagram [1, 2] by means of parametric studies brought out some irregularities in the J-value assignments. The interpretation of the even levels at 19010, 25699, 26081 and 28809 cm -1 could only be made consistent with the theoretical energy diagram of 4f105d by lowering their J-value by one unit. For odd parity levels, it was found that the energy difference (4f 10 5Ij, 6Pl/2)j + ~ - ( 4 f 10 5Ij, 6Pl/2)j_ [. is positive for every J, which led us to permute the J values of the levels 661280 and 664930 . These modifications facilitated the extension of the analysis. New energy levels have been found in classical ways. Strong lines showing wide hyperfine structures around 35000 and 40000 cm -1 are considered as 6Sl/2-6Pl/2 and 6Sl/2-6P3/2 transitions, respectively. Small systems built on the same level of the 4f 10 core were found by searching for constant differences among their wavenumbers. Owing to the purity of the 4fl°6s and 4f106p configurations in the J - ] coupling scheme, interconnections between states based on different 4f 10 core states were difficult to find by means of 6 s - 6 p transitions only. As the coupling conditions in 4f105d deviate from the J - ] limit above 25000 cm -1, selection rules for a pure coupling are not obeyed and numerous 4f106p-4flO5d transitions are observed between 29000 and 45000 cm -1 by means of which the needed interconnections could be established. In this way energies relative to the ground level 4 -¢11 410 H5/2 were found for all the new levels. Somewhat independently the system 4 f l l - 4 f l ° 5 d was investigated. As a result of the parametric study of 4f105d, even levels involving quartet terms in their eigenfunction were located by means of very strong

387

transitions to the ground term 4I and led to the new levels of 4f 11. In this extension of the energy level diagram, the early 4f 11 4F9/2 level at 13804 cm -1 has been replaced with a level at 13329 cm - I , which is now confirmed by eleven transitions to 4fl°5d levels. Some of the new levels have been found from one or two lines only. The levels 4f 114F5/2 is calculated as 86% pure in LS coupling and combines only with two levels of 4fl°(5I)5d: 36622.07 (60% 4G7/2) and 38680.01 (16% 4G7/2). Their energy difference only provided this one level in a search between 19000 and 20000 cm -1 with lines in the expected intensity ratio. The two levels 6G3/2 and 6L21/2 of the sub-configuration 4f10(5I)Sd are expected to combine only by single transitions from the corresponding subconfiguration 4f10(5I)6p. At the end of the present analysis, the strongest unclassified line in the energy range 40680-40872 cm-1 (the range represented by three times the mean deviation given by the parametric study) lies at 40758.84 cm -1 and places the 6L21/2 level at 22153.79 cm - I , only 17 cm - I above the calculated energy. On the other hand it was impossible to locate the transition 4f105d 6G3/2-4f106 p 6H5/2, calculated to be near 40980 cm -1. In order to fit the experimental energies, we have used preferentially 4 f - 5 d and 5 d - 6 p transitions. Energies for 4f106s have been determined afterwards by considering the third diagonal hyperfine component of resolved hfs patterns, which leads to a fair approximation for the nuclear spin I = 7/2 and for mean J-values. A more detailed discussion of the 4f106s and 4fl°6p hyperfine structures will be given separately [5]. With a tolerance of 0.2 cm -1 on the wavenumbers, 665 lines are now classified, but discrepancies are smaller than 0.1 cm -1 for 87% of these lines. Twelve lines are doubly classified.

3. Parametric interpretation of the levels As described in earlier papers [6, 7], the hamiltonian operator presently used comprises the spinorbit interaction and all spin-independent two-particle operators, which leads to 23 real and "effective" parameters for fl°d + fl°s and 14 parameters for flOp. Matrices of angular coefficients had been previously set up for describing spectra of Ho I and Dy II [8]. All the basic states of these configurations have been taken into account.

J. F. Wyart et al./Energy levels of rio III

388 Table 1 Energy levels of Ho III

Odd Levels Configuration

J

Energy (cm -1 )

Leading component

4fl 1

15/2 13/2 11/2 9/2 9/2 11/2 7/2 5/2 9/2 9/2 7/2 5/2 15/2 17/2 13/2 17/2 15/2 19/2 15/2 13/2 11/2 1312 15/2 17/2 13/2 11/2 9/2 11/2 7/2 9/2 13/2 11/2 15/2 9/2 9/2 11/2 9/2 1112 7/2 13/2 5/2 7/2

.00 5438.53 8644.59 10770.40 13329.42 16891.20 17868.42 19375.33 21533.75 23884.67 24648.07 28960.42 57497.72 57853.14 62484.48 62804.69 62844.11 62912.63 63468.62 64115.86 66128.33 66493.62 67896.30 68104.87 68266.03 68755.93 68788.48 69148.27 70979.23 71052.23 71499.30 71611.84 71784.40 72090.65 72978.94 73420.46 74140.80 74209.17 74280.00 74530.99 75867.06 76237.80

41 97 4I 99 41 85 4I 61 4F 65 2H(21) 48 4F 93 4F 86 4F 23 4G 80 4G 46 4G 93 518 1/2 93 5I 8 1/2 92 5I 7 1/2 84 5I 8 3/2 92 5I 7 1/2 77 518 3/2 94 518 3/2 75 518 3/2 80 5I 6 1/2 89 5I 6 1/2 92 517 3/2 95 5I 7 3/2 96 5I 7 3/2 93 517 3/2 69 5I s 1/2 85 5I s 1/2 68 514 1/2 85 5I 4 1/2 88 516 3/292 5I 6 3/2 90 5I 6 3/2 94 5I 6 3/2 86 5F s 1/2 78 5F s 1/2 78 5I s 3/2 84 51s 3/2 86 5I s 3/2 83 5I 5 3/2 88 514 3/2 88 5I 4 3/2 66

4fl°6p

Eexp, Ecalc (cm T M )

Comment

29 2

A A

1

A

5 -54 -34 -12 49 -16 -20 55 36 4 -2 2 -3 1 2 -3 -3

A C A C C C B C C A A A A A B A A

1

A

3 1 4 0 -4 4 4 10 -4 -1 -3

A A B A A B B C C B C

1

B

1

B,C C C C B,C C C C C

-15 -2 -2 -1 -1 0 -1 -1

389

J. F. Wyart et aL /Energy levels of rio III

Table I (cont.) Odd Levels

Configuration 4f106p

9/2 11/2 13/2 11/2 9/2 11/2 13/2

Energy (cm-1)

Leading component

76355.20 76810.88 78048.95 78290.61 79047.71 80190.52 80360.21

5I 4 3/2 514 3/2 5F 5 3/2 5F 5 3/2 5F 5 3/2 5G 6 1/2 5G6 1/2

87 88 78 57 72 51 79

EexpsEcale

Comment

(cmTM) -11 2 -3 8 17 2 -4

C C C C C C C

Even Levels

Configuration

4f105d

4f106s 4f105d

4f106s 4f105d

4f106s 4f105d 4f106s 4f105d

15/2 17/2 13/2 19/2 17/2 21/2 17/2 13/2 19/2 15/2 15/2 11/2 17/2 13/2 11/2 15/2 9/2 15/2 15/2 17/2 19/2 13/2 13/2 9/2 11/2 11/2 13/2 7/2 15/2 11/2

Energy (cm-1)

Leading component

18033.40 18099.14 19010.09 19940.47 21824.15 22153.79 22243.06 22431.10 22504.08 22637.87 22993.98 23212.29 25557.75 25699.23 25973.40 26081.47 26603.07 27074.39 27180.12 27260.26 27666.77 27735.94 27913.71 28683.36 28724.89 28809.29 28867.24 29148.88 29162.57 30618.22

(Sl) 6H 51 (51) 6I 72 (51) 6G 63 (51) 6K 63 (Sl) 61 92 (5I) 6L 93 (Sl) 4K 41 (SI) 6I 35 (sI) 4L 51 (Sl) 6H 32 (51) 41 66 (sI) 6G 45 (51) 6L 35 (Sl) 6I 26 (Sl) 6I 45 (Sl) 6K 40 (Sl) 6H 41 (Sl) 4I 31 (Sl) 6I 52 (sI) 6K 42 (si) 6L 66 (Sl) 6I 50 (Sl) 4H 48 (sI) 6I 45 (sI) 6H 30 (sI) 4G 63 (sI) 6K 41 (sI) 6H 41 (sI) 6L 57 (SI) 6K 53

Perturbing conf. (%)

1 1

3 4

26 26

1 1

1

Eexp-Eealc

Comment

-52 -10 16 -49 59 17 47 3 20 16 30 -16 -12 23 8 -11 -41 16 -10 -11 -8 9 -30 -43 7 -22 23 -32 18 5

C C A,C C A C A A C A A A A A,C A A, C C A A C C A A C A A,C C C C C

390

J.F. Wyart et at/Energy levels o f rio III

Table I (cont.) Even Levels

Configuration

J

Energy (cm -1)

Leading component

Perturbing conf. (%)

4fl°5d 4fl°6s 4fl°5d 4fl°6s 4fl°5d 4fl°6s 4fl°5d 4fl°6s 4flO5d 4flO6s 4fl°5d 4fl°6s 4f105d 4f106s 4fl°5d 4fl°6s

7/2 11/2 13/2 9/2 15/2 13/2 5/2 17/2 13/2 9/2 11/2 7/2 11/2 9/2 9/2 9/2 ll/2 11/2 5/2 13/2 15/2 13/2 7/2 11/2 15/2 9/2 11/2 9/2 11/2 7/2 13/2 11/2 9/2 9/2 13/2 7/2 13/2 5/2 5/2 11/2 13/2 13/2

30887.77 30953.20 30997.66 31046.42 31150.22 31313.51 31666.95 31707.90 31903.06 32157.15 32349.13 32712.80 33046.50 33146.66 33277.20 33609.78 33644.25 33875.59 33923.10 34066.99 34666.32 34759.72 35067.71 35203.80 35639.95 35822.86 35876.89 36234.06 36607.82 36622.07 37300.35 37450.31 37746.04 38529.09 38551.49 38680.01 39388.26 39399.24 39633.40 42095.85 42626.94 44242.39

(5I) 6I 49 (51) 61 70 (51) 41 48 (51) 6H 32 (5I) 4K37 (51) 6L 63 (51) 6H62 (51) 4L 51 (5I) 4I 44 (5I) 6K65 (5I) 4H41 (5I) 6G 45 (51) 6L 77 (51) 4G26 (51) 61 70 (51) 4G 42 (SF) 6F 40 (51) 41 66 (5I) 6G71 (5I) 4K 24 (5I) 4L 54 (SF) 6H 26 (51) 61 89 (51) 41 33 (SF) 6H 65 (51) 4H39 (SF) 6H 32 (51) 41 79 (51) 4K 44 (51) 4G59 (5I) 4L 76 (SF) 6F 79 (51) 41 46 (SF) 4F 53 (SF) 4H 23 (5I) 4H43 (SF) 6G 48 (SF) 6H 46 (51) 4G 64 (SG) 6G 25 (SG) 6G 38 (SG) 6G 84

1

3

Eexp-Ecalc

Comment

56

C

-2

A

13

A

-

-6

C

-

-29

A

1

27

C

2

C

-

-21

C

-37

A

-

--33

C

-

12

A

2

-

-17

C

-

22

C

13 14

4 13

A,C

1

17

C

1

58

2

B C

0

B

4

C

-

8

C

-

15

A

-

8

C

-7

A

41

C

-

-17

-

1 -

C

7

1

C

-15

C

-9

C

-

-5

C

-

-19

-

C 0

-

-78 1

C

18

-

-31

-

-46

C

C C C

-

46

-

38

C

-

16

C

5

C

-

-30

-

C

41

1

C

3

C

C

J. F. Wyart et aL/Energy levels of rio III

391

Table II Fitted parameter values in Ho III, with associated standard-errors (all values in cm-1) 4fl 1 A

15784

/zq E2 E3

4fl°6p (61)

5764 (49) 27.8 (0,6) 574 (3)

81201

4f1°6s (14)

6436 (10) 31.5 636.2 (.7)

42783

4f 1°5 d (76)

46228

6398 33.5 635.7

6398 (29) 33.5 (1) 635.7 (3.7)

F2(4f5d) F4 G1 G3 G5 -

189.4 20.4 225 28.8 3.35

F2(4f6p) G2 G4 G3(4f6s)

(84)

(2) (0.4) (4) (1) (0.16)

79.5 (.5) 10.50 (.05) 8.08 (.09) 312.8 (9) 2012.2 (14)

~'4f ~'sd ~'6p

2157.5

(.8)

3505.1

(1.8)

2167.5 (5)

R(2)(4f5d, 4f6s) R(3)(4f5d, 6s4f) 18.0 (3) -780 2500

3'

15.5 -620 2000

(.5)

(3) (8)

566 a 2265

(564)

19.8 -620 2000

19.8 -620 2000

9.7 (0.2)

AL(L + l) X3(4f, 5d) Y2Y4-

2157.8 1039

(2.7)

9.5 (0.6) 39 (19) -41.7 (4) -206 (53)

1


47.5 \

N-p

I

6.1

32.1

a held in a fixed ratio to R(3)(4f5d, 6s4f). As experimental energy levels arise from seven levels belonging to three SL-terms o f the 4 f 10 core, n o t all electrostatic parameters E k, and those of the correction a L ( L + 1) + / 3 G ( G 2 ) + 7 G(R7) could be varied i n d e p e n d e n t l y . The effective parameter Y3(4f, 6p) was eliminated, and/3 and ~/were fixed to a mean value for the lanthanides [9]. For the even parity

states, corresponding electrostatic parameters have been assumed to be equal for 4 f l ° 5 d and 4 f l ° 6 s . As b o t h ~'4f parameters converge to well-defined, b u t slightly different, values, they were considered as i n d e p e n d e n t in these configurations. The effective parameter X3(4f, 5d), though poorly defined, has been kept in the final optimization because o f agree-

392

J. F. Wyart et al./Energy levels o f rio III

ment with values derived from Ce 1II [6], Pr III [7] and Yb III [6, 7]. As is observed in other overlapping 4fN5d and 4fN6s configurations, the "configuration interaction" leads to small mixings and many levels are calculated as pure. Consequently the R(2)(4f5d, 4f6s) and R(3)(4f5d, 6s4f) Slater integrals have been constrained to retain the same ratio as in Tm III and Yb III. Concerning 4fl°6p we want to emphasize the agreement between theoretical and experimental energies: the mean deviation represents 1/3000 of the interpreted range. The present study involves 37 experimental levels and 11 parameters. With the same hamiltonian and only 16 experimental levels of Er III, similar agreement was previously obtained in 4f 116p, for which the present analysis provides the needed confirmation [10]. At present Er III appears as quite compatible with the neighbouring Ho IIl and Tm Ill, with regard to the fitted parameters. For all the levels presently known in Ho III, the following quantities have been reported in Table I: configuration assignment, J quantum number, energy (in cm-1), leading component of the eigenfunction, the squared amplitude (in percentage of the total eigenfunction) and (for even levels only) the percentage of the perturbing configuration; in the last two columns, the discrepancy between experimental and theoretical energies and a comment which refers to the three successive steps in the analysis (A, B for levels found at Johns Hopkins, C for levels found or newly interpreted at Laboratoire Aim6 Cotton). Parameters used in the last diagonalization are reported in table II.

4. Conclusion At present the lowest levels of 4fl0nl, with nl = 6s, 6p and 5d are experimentally known, and an attempt to determine either upper levels of these configurations or the lowest levels of 4f107s and 4fl°6d was not conclusive. The extension of the spectral analysis in this system seems to involve lines of moderate or weak intensity. Most of the unclassified lines can be attributed to the extensive system 4f95d 2 + 4f95d6s +~ 4f95d6p + 4f96s6p and to transitions between upper levels of 4f 11 and 4f105d. This work confirms the

regular behaviour of various interactions at the end of the 4f-subshell and will be developed in two further steps: a systematic investigation of effective magnetic operators in various configurations (Crosswhite), and a global description of the whole sequence of lanthahides by means of generalized least-squares (Wyart). Taking advantage of analogies with the present descriptions of Dy II [8] and Ho II1, a similar extension of the analysis of Er IV [11 ] is also projected.

5. Acknowledgements Theoretical results reported here have been obtained by using four programs written by Y. Bordarier and A. Bachelier-Carlier (Laboratoire Aim~ Cotton), who are gratefully thanked. One of the authors (J. F. W.) wishes to thank M. Fred of Argonne National Laboratory who has taken spectral plates for Ho I and Ho II, and J. Verg6s (Laboratoire Aim6 Cotton) who has recorded Fourier Transform spectrograms of the same spectra in the infrared. The work at Johns Hopkins was carried out with the partial support of the National Science Foundation. 6. References [1 ] J. H. McElaney, Ph.D. Thesis, The Johns Hopkins University (1966). [2] J. H. McElaney, J. Opt. Soc. Am. 57 (1967) 870. [3] R. Hussain, Ph.D. Thesis, The Johns Hopkins University (1973). [4] J. F. Wyart, J. Blaise and P. Camus, Physica Scripta 9 (1974) 323. N. Spector, Physica Scrip ta 13 ( 1976) 181. [5] R. Hussain and H. M. Crosswhite, to be published. [6] H. M. Crosswhite, Phys. Rev. A4 (1971) 485. [7} J. F. Wyart, J. Blaise and P. Camus, Physica Scripta 9 (1974) 325. [8] J. F. Wyart, Physica 83C (1976) 361. [9] H. M. Crosswhite, presented at the C.N.R.S. International Colloquium, Spectroscopic des Elements de Transition et des Elements Lourds dans les Solides, Lyon, France, June 1976. [10] J. F. Wyart, J. J. A. Koot and T. A. M. van Kleef, Physica 77 (1974) 159. [11 ] W. J. Carter, Ph.D. Thesis, The Johns Hopkins University (1966).