THE 4 F-5 D INTERACTIONS in SAMARIUM, GADOLINIUM and TERBIUM N

THE 4 f-5 d INTERACTIONS IN SAMARIUM, GADOLINIUM AND TERBIUM N. Spector

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N. Spector. THE 4 f-5 d INTERACTIONS IN SAMARIUM, GADOLINIUM AND TERBIUM. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-173-C4-188. 10.1051/jphyscol:1970429. jpa- 00213884

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THE 4 f-5 d INTERACTIONS IN SAMARIUM, GADOLINIUM AND TERBIUM

N. SPECTOR

Israel Atomic Energy Commission, Soreq Nuclear Research Center Yavne, Israel.

Rksumk. - Des niveaux d'energie recemment etablis qui appartiennent aux sous-configurations 4 f6(7F) 5 d du samarium ionise une fois (Sm 11), 4 fs(7F) 5 d du gadolinium ionis6 une fois (Gd 11) et 4 fg(7F) 5 d 6 s' du terbium neutre ont ete utilises pour obtenir des valeurs pour les six para- mktres electrostatiques (Slater) et les deux de spin orbite autour de la demi-couche 4 f7. I1 y a 57 niveaux dans chaque atonle ; 54 de Smi, 45 de Gd-+et 27 de Tb sont optimises avec une erreur de 168 cm-1 (1,24 % du domaine CtudiC) 239 cm-1 (1,76 %) et 152 cm-1 (1,76 %) respectivement. Les niveaux peuvent @tredesignes en utilisant les notations du couplage L-S, malgre quelques melanges. Des vecteurs propres notablement differents de ceux publies precedemment ainsi que des previsions pour les niveaux manquants sont presentes. Environ 70 raies de Gd I1 sont classtes.

Abstract. - The newly established levels of the subconfigurations 4 f6(7F) 5 d of singly ionized samarium (Sm II), 4 fs(7F) 5 d of singly ionized gadoliniun~(Gd TI) and 4 fs(7F) 5 d 6 sZ of neutral terbium (Tb I) are used to obtain reliable values for the six electrostatic (Slater) and two spin- orbit interaction parameters around the half filled 4 f shell. Out of 57 possible levels in each case, 54 of Sm.-,45 of Gdr and 27 of Tb are fitted with an vms error of 168 ~111-1 (1.24 % of total width), 239 cm-1 (1.76 "/,) and 152 cn-1 (1.76 %) respectively. L-S coupling notations can be used for level designations despite some heavy admixtures. Eigenvectors differring significantly from previously given ones, as well as predictions for missing levels are tabulated. About 70 Gd 11 lines are classified.

I. Introduction. - Our knowledge of the 4 f-5 d tation. The high number of fully observed terms interactions around the middle of the 4 f shell has, warrants the convergence of a least squares calcula- until now, been limited to cases based on a single tions involving the radial electrostatic parameters. state of a 4 f electron. The reason is that only confi- In Gd I1 and Sm II we were able to obtain reliable gurations of the type 4 f7('S,+) IIIIZ'I' have been values for each of the G,'s separately as well as for observed in the spectra europium and gadolinium. the F,'s. The latter appear for the first time as inde- The high symmetry of the 4 f7 core configuration pendent variables in least squares calculations around results in the vanishing of the coefficients of the the middle of the 4 f shell. We present here the results direct Slater parameters F, and F, for these confi- of our optimization process in Smf, Gdf and Tb. gurations. Thus, observed term splittings in Eu I, Eu 11, Gd I and Gd II depend only on the single 11. Theoretical Calculations. - The assumption exchange integral G = GI -t- 4 G, + 22 G,. Values underlying our theoretical treatment of the observed for the individual G,'s as well as for the F,'s needed levels is the purity of the 'F term in L-S coupling. for a full understanding of the 4 f-5 d interaction This assumption has been carefully established both around the half-filled 4 f shell could not, thus, be theoretically and experimentally in various instances derived. and is now commonly accepted. The theoretical Recently we [I] have observed levels belonging to predictions of the 4 f6('F) 5 d and 4 f8(7F)5 d levels the 4 f8(7F) 5 d s~lbconfig~~rationof Gd TI. Similar and their g-values involve diagonalizing the energy observations, including Zeeman effect data, were matrix that is a linear combination of electrostatic made by Blaise and Van Kleef [2]. Blaise et al. [3] and spin orbit interactions. In this section we describe also observed levels of 4 f6(7F) 5 d in Sm 11. Klin- the calculation of the angular coefficients (matrix kenberg et al. [4] have improved on some of their elements) of this linear combination. previously published levels of 4 f8(7F) 5 d 6 s2 of Tb I. This recent accumulation of observational material A. THECALCULATION OF THE ELECTROSTATIC MATRIX seemed to us well worthy of a theoretical interpre- ELEMENTS. - The 4 f6(7F) 5 d subconfiguration has 10 terms : five sextets and five octets, 6p8P~FGH. Index headings : ~~~~i~ spectl.a, theory : samarium ; gado- Since theji's (coeficients of F,'s) are functions of the hiurn ; terbium. orbital angular momentum only, they are the same for

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970429 C1- 174 N. SPECTOR

the two multiplicities of each L. For the 7Fparent all accomplished by a change of sign. In order to obtain spins are parallel. There is one electron missing from 2'" we use the following relation by Judd [7] : the half-filled 4 f shell, which in the orbital space, can be considered as a closed shell. Thus the f6(7F)d is the almost closed shell conjugate of fd. According to Racah [5] (eq. (74)). therefore, the ,h's for the terms which gives g'"' in terms of the ~(~''sobtained from of f6(7F)5 d have simply the same magnitude but the eq. (2). opposite sign of the corresponding ones in fdt6). For the gk's w: used the procedure outlined in ref. 1, equations (I) to (6). Modifying eq. (6) of ref. 1 to fit our case we obtain : The coefficients of the Slater exchange integrals for all octet terms of fai7F) d are independent of the term's L. It is thus seen that the observation of terms with next to higher multiplicity is an essential condi- 5 1 Since 2(Sf.sd) = S(S + 1) - 7 we can write (1) tion for the independent determination of all three Slater exchange parameters GI, G, and G,, in fN d configuration when N > 7. As we shall see in Section 111 this condition is satisfied in Gd I1 but not in Tb I. For N < 7 the coiidition is not essential since even terms with highest multiplicity contain different We denote with bars the corresponding expressions linear combinations of all Gkls. for the almost closed shell conjugate of f6(7F) d In Table I we give the f (k', g'A)and g(') for the namely f8('F) d. The transition from f(k' to f(k' is 6,8PDFGH of f6(7F) d and f8(7F) d.

TABLEI Electrostatic matrix elementb for f6('F) d and f8(7~)d

-- B. THE CALCULATION OF THE SPIN-ORBIT MATRIX x14,/(2s+1)(2s1+1)(2L+1)(2L'+1) ELEMENTS. - We calculated the coefficients of Cf and C, according to the following formulas : For lf : (3tS32L.l (sl) 3+S132L'J = I i 1 For id system. All are accomp;inied by L-S designations and g-factors.

IV. Calculations and results. -A. Sm 11.4 l"('F) 5 d. - We selected 54 levcls out of the 56 ones designated as The matrices fi)r these coefficients arc given in the belongirlg to 4fb(7F) 5d in rel'. 3, and fitted them to Appendix. the predicted cigenvalves. A preliminary set of predicted positions was obtained using values for the C. - It should be noted that if we assign to the radial parameters that were derived by interpolation F,'s and G,'s the set of values given in Table I1 the from other spectra. In Table 111 we give the values complete electrostatic energy expression used in the final diagonalization and those obtained

vanishes for each term separately. This may serve as Paratneters for 4 f6(7~)5 d of Sm I1 a check for the angular coefficients. Diagonalization Least squares Name (in cm- ') (in cm- I) - d - TABLEI1 A 16 600 16 597 + 36 Vahres for checkirlg the electrostatic rnatrix elements f-2 138 138 + 3 F4 9 8.6 + 0.7 GI 183 182 + 5 G3 I I 10 + 2 G 5 2.3 2.4 + 0.2 if 1 150 1139 k26 kd 400 403 + 42 rt11.~error 168 in % of total width 1.24 % 111. The Observed Levels. - The recent observational material used in the present calculations includes after the optimization process (least squares). Since the following : the rr?ls error is only 168 cm-' (1.24 ';;: of the total 1. Sm 11.-In addition to thegroup of 284f6(('~)5d width) and the paranieters were invariant -. within levels given by Albertson [8] we had a group their definition - to the least squares adjustment we of 28 new levels, 8 in the range of Albertson's obser- conclude that convergence has been reached. vations and 20 above them, given in rel'. 3. All but 4 An interesting outcome is the small value of <, of these SINI1 levels \vcrc accompanied by g-values. that was also anticipated by Eremin et al. [I I]. Russell Sa~~ndersdesignations \vcrc i~.;ed throughout In Table IV we give the comparison between calcu- the list. lated and observed levels and g-factors, as well as L-S coupling notations. While tlie "1 and 'D 2. Gd If. - A11 4 FR('F) 5d !evi.ls are the results ol' arc purc. Ilea\ y admixtures arc indicated between recent investigations (ref. 1. 2). .I'he preiious work 'G-'F. 9'-"1'. :111d "F-"11-"G. Still. L-S notations on this atom by Russell [9] contained confipi~rotions haw a m;!jor component that is higher than 50 :',: with 4 f7 ;IS the onl!. corc conIigt~~.atio~i.As ~~icntioncd in 90 ",, ol'the cases and therefore Russcll Saunders in ref. I. the lower levcls of 4 fH('F) 5d ;ire wcll scheme is suitable for level designations ill this case. isolated fro111 tlie rest of the c\cn Ic\els. and the .Thcr.e is quite a sntisl';~ctory agreement bctween the higher ones overlap those belonging to 4 I"(%S) dp obser~edand c:~lculatcd levels and g-fl~ctors.except and 4 f7(%S) sp. This overlap. \vhich i~lcrcases tlie Isor the two Ic\cls with J = 0 i where the observed configuration interaction between tlie group ol' even to c:~lculatcd corrcsponda~icedictated by tlie posi- levels results in the breaking ol' the strict selection tions co~itradictsthe one indicated by tlie g-f'actors rules observed in ref: 1 to go\,ern thc pure predictions. At this stage we decided to follow the theoretical predictions for the level positions rather than the g-factor considerations for the following transitions i~ivolvi~igthe isolated part of' 4 f8('F) 5d. reasons : I) The interchange would involve a devia- The 16 levels of 4 f8(7F) 5 d reported in ref. 1 were tion of more than 1 000 cm-' fi)r the 6~,,!.We believe supplemented by 29 other even levels later selected our calculations arc better than such a deviation. from Table 1 of ref. 2. 2) A change of the parameters causing only a second

3. Tb I. -- The most recent compilation by Klin- order change in the prediction of level position will kenberg ct al. [lo] gives 27 corrected 4 f8(7F) 5d 6s' c:~use a lirst order correction to the g-l';~ctors. This levels. 01' these all but one arc members of the octet sensitivity of the S-fr~ctors to sm;~llchanges in thc N. SPECTOR

Calculated positions, g-values and percentages in L-S coupling for the 4 f6('~)5 d levels of Sm I1

Observed Observed Calculated designa- J Percentage level level 0-C tion composition (in cm- ') (in cm-') (in cm- ') - - - 'H 13 100 % 'H 'H 24 100 % 'H H 3 3 100 % 'H H 4% 100 % 'H H 5 3 100 % 'H 'H 6t 100 % 'H H 7 + 100 % 'H H 8 t 100 % 'H D 1 3 85 % 'D 'D 23 88 % 'D 'D 3 f 86 % 'D 'D 4f 81 % 'D D 5 3 79 % 'D 'G 0 3 90 % 'F G 1 3 72 % 'F + 22 % 'G G 2 4 59 %'F + 34 %'G 'G 3 3 50 % 'F + 41 % 'G G 4 f 41 % 'F + 46 % 'G 'G 5 f 49 % 'G + 35 % 'F G 6 $ 56 % 'F + 42 % 'G G 7 4 96 % 'G 'F 0 3 90 % 'G 'F 1 3 69 %'G + 20 %'F F 2t 55 % 'G + 31 % 'F F 3 3 56 % 'C + 37 % 'F F 4t 52 %'G + 44 %'F F 5 f 46 %'F + 49 % 'G 'F 6 3 55 % 'G + 44 % 'F

6P 1 + 77 % 6~ P 2 4 42 % 6~ + 40 % 'P 6P 3 t 63 x6P + 31 %'P

'P 2 t 52 % 'P + 41 % 6~ P 3 f 62 % 'P + 30 % 6P 'P 4 3 94 % 'P

F 03 86 % 6~ F 1 f 64 % 6~ + 31 % 6F 6F 2 3 49 % 6D + 45 % 6~ 6~ 3% 52 % 6F + 36 % 6~ F 4 3 55 Z6F + 34 X6D 5 3 72 % 6~ D 04 86 % 6F 6~ 13 55 z6F + 31 x6D 6D 23 30 % 6F + 42 % 6~ D 3 f 46 % 6~ + 34 % 6~ 6~ 4 3 53 % 6~ + 34 % 6~ TABLEIV (contd.) Observed Observed Calculated designa- Pzrcen tage level level 0-C tion cornposit ion (in crn-') (in crn- ') (in crn- ') - . - - - - H 100 O/, 6H 14 193 14 431 - 238 H 97 O/, 6H 14 668 14 772 - 104 H 97 % 6H 15 243 15 232 11 6H 96 "/, 6~ 15 897 15 828 69 H 96 % "H 16 615 16 575 40 H 98 % "H 17 392 17 484 - 92

Eigenoectors in L-S scheme for 4f6('~)5 d of. Srn 11 Level - 7 272 7 570 7 990 8 532 9 200 9 994 10 917 11 971 9 043 9 503 10 043 10 747 11 710 10 372 10 546 10 855 11 305 11 904 13 500 13 406 10 790 11 083 11 533 12 083 I2 760 12 645 14 202 14 765 11 038 11 450 14 069 12 641 12 623 14 240 17 080 17 444 17 966 16913 17 613 N. SPECTOR

TABLEV (contd.) Level - 19 100 16 063 16 155 16 421 18 598 19 241 14 222 14 768 I5 233 15 836 16 592 17 511 18 498 19 063 19 739 20 403 20 873 21 184 parameters makes position calculations a better While the new 6~ indeed falls close to its predicted criterion for the fitting of observed levels. position its spread of about 2 300 cm-' is far from The small 0-C values for both 'F and 6~ indi- what can be expected from the same prediction. On cates the weakness of configuration interaction with the other hand its structure agrees with the predic- 4 f6('F) 6 S. tions of our calculation. Furthermore, this 6D make: In Table V we give the eigenvectors in L-S transitions to the ground configuration which arc coupling for the 57 levels of f6(7F) 5 d of Sm 11. very similar in character to those made by the neigh- B. Gd It 4 f8(7F) 5 d. - From the available lists boring sextet belonging to 4 f8(7~)5 d (see Table IX) of even levels of Gd TI established by Russell [9], We see, therefore, no reason to exclude it at thi! Spector [I] and Blaise and Van Kleef [2] we picked stage from the least squares calculation. 45 for the least squares calculations. From its first The final parameters obtained after the optimization member, 'G7+ at around 18 400 cm- ' up to appro- process are given in Table VI. In Table VII we givc ximately 25 000 cm-' the 4 f8(7F) 5d group of levels our predictions for the positions and g-values of the is well isolated from other groups. In the region above levels of 4 f8('~)5d and compare these to the avai- 25 000 cm-' where about half of its levels lie, it lable observed levels. The agreement is quite satis- overlaps levels of 4 f7('S) 5d 6p and 4 f7('S) 6s 6p. Accurate predictions for both positions and g-factors are essential for the correct selection of experimental Paratweters for 4 f8(7F) 5 d of Gd I1 levels. In the case of 6D, our predictions fit well, Diagonalization Least squares both in positions and g-factors, a certain 6~ given Name (in cm - ') (in cm- ') in ref. 2 the designation 4f8 6s. Such designation - - - seems, at present to need further corroboration. The A 27 500 27 527 87 theoretical prediction for the location of the center + F2 147 144 + 4 of gravity of the 5~ of 4 f8 above the 7F is given by F4 12 10 4 1 Elliott [I21 et al. as 51.8 F2 where. G I 144 129 + 9 G3 15 15 & 2 This puts 5~ about 19 300 cm-' above 7~.Also in G5 2.5 2.5 + 0.5 ref. 12 Table 3 we get values for the splitting factors if 1 240 1219 f 39 that enable us to estimate the total splitting of id 550 607 4 73 each multiplet. For 'D we get an estimated spread rms error 239 of' about 9 000 cm-'. Since the 6,4~of 4f8(5~) in % of total width 1.76 % Gs are obtained by adding 6s electron to the 5D, we expect the two terms to follow closely the factory. Again there is no evident perturbation of structure of 5D, as do the 'r6F of 4f8(7F) 6s. Also either 6F or 'F by their analogues in 4 f8(7F) 6 S. the transitions made from a 6~ belonging to 4 f8('D) 6 s The rms error is 239 cm-' which is 1.76 % of the to the ground configuration 4f7('S) 5d 6s should total width of this configuration. be quite different from those made from a 6D which Table VIII presents the eigenvectors in L-S belongs to 4 f8(7F) 5d that overlaps 4 f7(*s) 6s 6p. coupling of the levels of this subconfiguration. THE 4 f-5 d INTERACTIONS IN SAMARIUM, GADOLLNlUhl AND TEKBIUM

TABLEVII Calctrlafed positions, g factors and L-S percentages for the 4f8('F) 5 d levels of Gd I1

Observed Observed Calculated designa- J Percentage level level 0-C gobs. tion composition (in cm - ') (in cm-') (in cm-') - - - - 7 C4-180 N. SPECTOR

TABLE VII (contd.) Observed Observed Calculated designa J Percentage level level O-C oobs. ocalc. tion composition (in cm-1) (in cm-1) (in cm_i)

6G 6± 94 %6G 25 075 1.380 5i 84 %6G 26 640 26 343 297 1.320 1.338 4i 77 %6G 27 346 1.287 3i 79 %6G 28 220 28 151 69 1.205 1.164 2i 88 %6G 28 770 0.868 U 96 %6G 29 196 0.034 6H 90%6H 25 545 1.339 n 6 6i 86 % H 26 921 1.293 5i 84 %6H 28 030 1.221 4i 86 %6H 28 916 1.098 3i 88 %6H 29 605 0.362 6 2i 94 % H 30 118 29 820 - 298 0.345 0.325

6 6p 3i 86 % P 30 145 30 252 - 107 1.655 1.698 6 2* 92 % P 31 238 31 510 - 272 1.835 1.870 6 li 98 % P 31915 32 368 - 453 2.32 2.386

TABLE VIII Eigenvectors components in L-S scheme for 4 f8 (7F) 5 d of Gd II Calc. Level / 6p sp «D »D 6p sp «G »G «H m (in cm-1) 18 687 74 .986 6 — .053 5 .154 1 18 618 64 .478 4 — .1010 .859 8 —.040 4 .141 5 18 852 54 .329 9 — .052 3 .556 9 — .089 2 .745 6 — .027 3 .117 5 19 379 44 .071 2 — .004 0 .333 3 — .038 4 .550 4 — .076 7 .749 1 —.020 1 .109 6 20 010 34 .005 9 .045 3 .005 8 .256 5 — .013 8 .505 7 — .065 2 .812 7 — .014 4 .106 0 20 596 24 .005 4 .018 7 .012 3 .160 6 .012 0 .422 3 —.050 9 .885 3 — .008 1 .093 8 21 060 14 — .002 9 — .012 8 — .071 7 — .034 1 — .303 3 .032 7 — .946 6 — .066 8 21 356 04 .008 0 .048 9 .158 2 .986 2 20 381 54 .746 9 — .100 9 .325 6 .043 3 —.555 6 .027 2 — .1209 20 849 44 .200 7 — .032 1 .706 2 — .088 7 .340 5 .042 2 — .566 5 .020 5 — .1145 21 500 34 — .016 6 — .153 9 .016 6 —.689 6 .086 3 — .474 3 — .030 1 .507 8 — .012 6 .094 5 22 103 24 — .022 2 — .092 4 — .004 7 — .639 2 — .078 3 — .634 2 — .017 2 .411 6 — .005 6 .065 1 22 597 1 4 .021 2 .028 5 .510 2 — .061 8 .805 0 .007 1 — .292 3 — .032 7 20 845 64 — .874 9 — .001 2 .461 5 — .037 7 .141 9 22 389 54 — .573 9 —.065 9 .750 1 — .001 4 — .288 0 .031 0 — .140 3 22 912 44 — .292 8 .059 4 — .5183 — .088 2 .738 3 .002 0 — .268 4 .023 1 — .129 5 23 485 3 4 .016 6 .217 7 —.059 9 .620 8 .077 7 — .706 2 — .001 8 .219 6 — .013 5 .1004 23 862 2 4 .026 9 .144 6 — .056 6 .730 5 .056 8 — .639 1 — .001 5 .1627 — .005 1 .058 6 24 067 1 4 — .038 6 .045 1 — .855 6 — .031 3 .504 1 .000 8 — .094 6 — .020 5 22 926 04 .052 2 — .036 1 .985 6 — .1568 22 615 84 1 23 048 74 — .1630 — .303 8 .938 7 23 604 61 .061 0 .051 4 — .195 2 — .273 8 .938 4 24 107 54 .031 2 .0310 — .083 2 — .064 0 .199 6 .230 9 — .945 5 24 524 44 .042 1 — .005 6 .017 1 .023 1 — .075 4 — .072 0 .1828 .1862 — .958 3 24 862 34 .000 8 .014 8 — .004 0 .024 5 .016 1 — .064 3 — .074 0 .155 1 .140 5 — .972 5 25 119 24 .000 6 .004 7 — .002 7 .016 5 .009 4 — .040 1 — .073 1 .1170 .090 7 — .985 3 25 300 1 4 .000 2 — .001 1 .005 7 .004 4 — .016 7 — .071 3 .072 5 — .994 6 24 580 54 — .052 2 — .959 8 — .1192 — .246 0 .015 9 — .030 1 — .010 0 25 521 4 4 .161 4 .213 1 — .021 2 .917 1 .1304 .261 I — .025 8 .033 7 — .000 6 26 373 3 1 — .037 9 .021 1 — .244 3 — .029 7 — .928 7 — .090 5 — .255 1 .028 3 — .030 6 .010 6 27 109 2 4 — .027 4 .004 5 — .2100 — .013 5 — .947 3 — .069 8 —.225 4 .037 0 — .020 4 .013 4 THE 1 f-5 d INTERACTIONS IN SAMARIUIM, CiADOLlNltiM AND TEKBlUM C4-181

TABLEVIll (contd.) Calc. Level (in cm-1) J 6P CP 6D SD 6F s F 6G SG 6H nH ------27 677 14 .012 9 .I54 1 .005 2 .972 1 ,046 9 .I62 9 - .046 1 - .Oll 7 28 038 O? .082 2 .994 9 .023 6 - .053 8 25 443 4 1 .9110 -.I234 -.3472 -.I447 .I051 -.0362 -.0242 -.0087 .0192 27 293 3 1 - .002 7 - .959 1 .058 1 .265 7 - .0340 - ,067 9 - .016 4 .009 8 - .002 1 - .001 7 28 623 2 1 - .002 6 - .983 7 .039 5 .I72 1 - .010 3 - .029 6 - ,013 4 .002 5 - .002 3 .000 3 27 993 44 .I000 .919 5 .013 8 - .2868 -.049 1 .208 5 .033 7 ,122 1 .0107 28 805 34 .3488 .0656 ,8571 .0133 -.2892 -.0569 .I940 ,0311 .I184 .0043 29 639 2 ? .264 2 .042 0 .923 8 .016 7 - .232 9 - .058 7 .099 8 .019 3 .082 4 .OOO 8 30 259 14 ,1640 .9704 .0152 -.I605 P.0571 ,0458 .0111 - .003 2 30 636 0 6 .995 2 -- ,080 7 - .054 9 ,004 8 25 075 61 .043 8 ,966 7 ,094 8 2 6 .031 5 26 343 56 - .015 1 - ,242 2 ,001 5 96I ,105 3 .298 6 ,024 6 27 346 4 4 .033 5 ,2970 .013 1 .I849 - .002 2 - ,881 3 -.0940 -.301 2 - ,005 6 28 151 3? .081 0 .0270 .2698 .008 1 .I790 - ,001 2 - ,894 5 - .081 8 - .284 1 ,016 6 28 770 24 .0343 ,0180 .I602 .0035 ,1892 .0027 -.9374 -.0677 -.2279 .0420 29 196 I 4 ,008 9 ,0709 .0018 .I518 .0024 P.9824 -.0461 ,067 7 25 545 7? .003 4 .951 2 .308 5 26 921 61 .007 6 .229 5 .025 0 - .931 9 - ,279 7 28 030 5 1 - .002 4 - .042 9 9 .292 0 .035 4 - .924 2 - .239 7 28 916 4 1 .0033 .0333 -.0010 -.0619 ---.0032 .3122 .0355 -.9258 -.I979 29 605 3 1 .0245 .001 9 .0327 -.0004 -.059 6 -.0045 .2934 .0284 --.9402 -.I544 29 820 2 1 .0155 .@I16 .0452 .0005 p.0444 -.0044 .2282 .0172 u.9657 -.I041 30 252 3 1 .932 5 - .035 8 - .352 8 - ,031 0 .056 5 .018 6 - .012 9 - ,004 2 .003 9 .001 4 31 510 2 1 ,962 7 - .021 0 - .264 4 - .040 7 .031 1 .014 9 - .004 2 - .002 1 .000 5 .000 3 32 368 14 .985 4 -.I61 3 -.0458 .0126 .0095 P.0009 -.0008 .000 1

TABLE1X Classified lines of Gd I I

Intensity f.7 Odd level Even level in arc (in cm- ') (in cm-') (in cm- ') ------50 11 626.39 19 223 30 849 500 12 554.23 19 750 32 304 6 13 267.23 19 223 32 490 5 13 845.85 19 750 33 596 150 13 882.34 9 142 23 025 80 14 140.51 8 884 23 025 15 14 474.28 8 551 23 025 60 16 71 1.01 19 750 36 461 10 16 939.66 12 776 29 715 150 18 095.83 19 750 37 846 100 18 279.63 19 750 33 029 15 18 307.72 19 750 33 057 200 18 470.13 10 091 28 561 10 18 622.75 19 223 37 846 100 18 803.07 19 750 38 553 20 18 806.63 19 223 38 029 8 18 913.24 10 802 29 715 30 19 324.06 10 391 29 715 60 19 51 1.89 10 633 30 145 3 19 518.45 18 955 38 473 100 19 753.19 10 391 30 145 60 19 786.94 19 750 39 537 50 19 801.22 I9 223 39 024 100 20 053.37 I0 091 30 145 N. SPECTOR

TABLEIX (contd.)

Intensity 0- Odd level Even level in arc (in cm- ') (in cm-') (in cm- ') - -- - - 20 20 264.12 9 451 29 715 15 20 322.79 18 150 28 473 30 20 572.92 9 142 29 715 25 20 604.88 10 633 31 238 2 21 002.13 9 142 30 145 4 21 035.01 19 750 40 785 8 21 260.19 8 884 30 145 8 21 701.71 19 223 40 924 10 21 786.26 9 451 31 238 1 21 873.94 19 223 41 097 4 21 909.12 9 328 31 238 5 22 036.95 19 223 41 260 2 22 569.57 18 319 40 888 10 24 077.84 4 483 28 561 15 24 348.96 4 212 28 561 15 24 589.61 3 972 28 561 30 25 23 1.97 4 483 29 715 1 25 292.65 4 852 30 145 4 25 479.78 3 082 28 561 6 25 503.05 4 212 29 715 4 25 688.56 4 027 29 715 40 26 288.60 3 427 29 715 10 26 633.81 3 082 29 715 4 26 717.72 3 427 30 145 80 27 063.00 3 082 30 145 15 27 670.86 10 802 38 473 1 27 793.76 3 444 31 238 6 27 810.73 3 427 31 238 10 27 973.79 10 599 38 473 8 28 155.97 3 082 31 238 50 28 381.25 2 856 31 238 30 28 561.77 0 28 561 3 29 021.75 9 451 38 473 50 29 303.61 19 750 49 053 40 29 453.99 26 1 29 715 4 29 511.73 633 30 145 500 29.715.85 0 29 715 40 29 819.81 0 29 819 3 29 821.85 11 066 40 888 100 29 883.09 261 30 145 4 30 067.80 19 223 49 291 20 30 108.94 19 223 49 332 100 30 145.03 0 30 145 6 30 323.87 19 223 49 547 20 30 403.34 0 30 403 25 30 797.32 10 091 40 888 6 30 976.12 26 1 31 238 50 31 237.93 0 31 238 15 31 915.76 0 31 915 100 32 004.10 8 884 40 888 2 34 446.29 4 027 38 473 150 35 029.31 3 444 38 473 6 35 391.55 3 082 38 473 THE 3 f-5 d INTERACTIONS IN SAMARIUM, GADOLINIUM AND TERUlUM CJ-18.1

C. Tb I 4 f'(7F) 5 d 6 s'. - The 4 f8(7F) 5 d 6 s2 give reliable \.slues to the F2 and F,, but not to subconfiguration of neutral terbium has been the G,, G,. G,. We follo\ved the gradual unravelling of subject of several theoretical calculations. If it turns the 4 f8t7F)5 d 6 S2 of Tb I in a series of articles by out to be the ground configuration of neutral terbium Klinkenberg et al. Each time we inserted more levels it will provide a glaring exception to both Hund's to the least squares calculations. But it was clear that rule and Land6 interval rule. Its lowest term is not before some sextet levels are found, not all G,' s of highest L value, as required by Hund's rule. Its would be usable in such calculations. At a certain lowest level is not of highest J in this term as required point, a value was given to a 'H,: 1151. Its inclusion in by Lande's rule. This situation is quite well known an optimization process resulted in negative values to in configurations where a 5 d electron coexists with a most of the electrostatic parameters and extremely 4 fN core. Our theoretical calculations accurately large rnis errors for tlie two spin orbit parameters. reproduce this situation. It was later concluded in ref. 10 that this level belonged Even if it is not the ground configuratio~iit is low to another configuration : tlie 4 f8(7~)5 d2 6 S. enough as to be a significant contributor to states When the final list of observed 4 f8('F) 5 d 6 s2 appearing in atomic beam experiments. Such expe- levels was published [lo] we attempted another least riments determine magnetic dipole and electric qua- squares calculation. This time very reasonable values drupole moments, and a set of good eigenvectors is for all parameters were obtained when G, was made essential for their determinations. The first published equal to 2 cni-' in the diagonalization and then held attempt to predict levels of this subconfiguration was fixed during the least squares process. The results of made 5 years ago by Arnor~ltand Gerstenkorn [13]. this calculations are given in Tables X and XI. The They selected initial radial parameters for the electrostatic and spin orbit parameters and diagonalized its energy matrix. Their predictions fitted will the then available observed levels, established by Klinkenberg. But difficulties arose when attempts were made to Paranieters for 4 f8('F) 5 d 6 s2 of Tb I establish further levels of 4 fs(7F) 5 d 6 s2 of Tb I. Diagonalization Least squares Several newly found ones did not retain tlie good Name (in cm-') (in cm-') agreement to their predicted positions as did the first levels. A calculation of the magnetic and quadrupole 10 334 + 376 moment of Tb I by Childs and Goodman [14] using 147+ 5 the eigenvectors of ref. 11 did not produce satisfactory 16 + 1 results. 124 + 19 Ref. 11 did not mention a possibility for improving 26 + 9 the initial values of the radial parameters by optimizing fixed the theoretical prediction using the available observed 1618 + 31 levels. It also did not give a value for if.But it was r& 793 + 53 clear that any attempt at performing a least squares rms error 152 calculation based on the levels used in ref. 13 would in % of total width 1.76 %

Calculated positions, g-factors and L-S coupling percentages for the 4 f'(7F) 5 d 6 s2 levels of Tb I

Observed Observed Calculated designa- J Percentage level level 0-C gobs. gca~c. tion composition (cm- l) (cm - ') (cm- ') N. SPECTOR

TABLEXI (contd.)

Observed Observed Calculated designa- J Percentage level level gobs. tion composition (cm - ') (cm - ') THE 4 f-5 d LNTERACTIONS 1N SAMARIUM. GADOLINIUM AND TtKBLUM c‘4- 1 85

Eigenvectors con?ponents in L-S schet~lrfor 4 fs('~)5 d 6 s2 of Tb I Calc. Level (in cm-1) J - - 512 6 0 653 7 4 721 5 4 1 421 4 1 2 342 3 f 3 203 2 i? 3 884 13 4 318 0 4 2 440 5 i? 3 038 4 1 3 944 31 4 789 2 1 5 519 I f 3 292 6 1 5 081 5t 5 617 4t 6 429 34 6 918 2 1 7 129 1 f. 6 086 0 * 4 672 8 f 5 130 7? 5 969 6 5 6 736 53 7 289 4 4 7 770 34 8 122 2 * 8 369 I f 6 645 5 f 7 772 4 1- 8 916 3 & 9 916 23 10 706 I g 11 221 0 4 8 038 4 4 10 285 34 11 963 2 f 10 340 44 10 945 3% 12 357 2 1 13 125 I I 13 604 01 7 090 6 1 8 765 5? 9 907 41 11 359 3 * 11 890 2 4 I2 517 14 7 429 7t 9 213 6 1 10 646 5 4 11 783 4; 12 649 3 & 13 278 21 I2 892 3; 14 547 2 4 15 676 I! N. SPECTOR

Spin orbit matrices for the f and d electrons of f6(7F)d THE J f-5 d INTERACTIONS 1N SAMARLUM, GADOLINIUM AND TERBIUM CJ-187 C4- I XX N. SPECTOR former gives the adjusted values for the radial para- parameters to the presence or absence of even a meters resulting in an rn7s error of 152 cm-', which single level or parameter. To demonstrate this sensi- is 1.76 '7,of the total width. The latter gives the tivity we give, in Tables XIII, the results of four least predictions for level position and g-values as well squares on the same diagonalization parameters. as a comparison between them and the available Ln 1. s. 1 G, was free for adjustment and the level experimental values. L-S designations can be used 8 647 cm-' with J = 5 1 was included. All G,'s throughout the Table, despite some heavy admixtures. have unreasonable values. The situation does not Marked differences between our designation and the improve upon eliminating this level, nor yet upon ones used in ref. 10 and 13 are noticed for 'D,~ fixing G, while the level is still out. Only when G, and 'GAk.Also the level at 8 647 cm-' whose observed is held fixed and the level is restored to its place do designation is 'F,: fits well to our 6G,L.When made we get acceptable values for all the parameters. It tocorrespond to the predicted value of 'F,: (1 700cm-' is seen that exactly then the rt~iserror becomes the below) unreasonable values for the radial parameters biggest. But it is also evident that the other rms are obtained in the least squares. The most noticeable errors, though smaller, are totally meaningless. variation from ref. 13 is provided by our Table XI1 Contrary to the previous two sections the results where eigenvectors for this low subconfiguration are reported in the present one for Tbt should be consi- given in L-S coupling (as was done in ref. 13). dered merely as preliminary. Only when the sextets Differences in magnitude of various components and, of this subconfiguration are established could we in particular, prominent changes of phases are note- hope to have a more significant optimization for the worthy. parameters. Meanwhile the calculated positions for With the scanty experimental material yet established the missing levels given in Table XI should serve as there is a remarkable sensitivity of the optimized guidelines for searching the latter.

TABLEXI11 Various least squnres in Tb 1

Diagona- G, free, G5 free, G, fixed, G, fixed, Name lization 8 647 in 8 647 out 8 647 in 8 647 out ------A 10 000 6 505 + 1 373 8 089 + 1 758 10 333 + 376 9 234 + 463 F2 144 147+ 4 147f 4 147f 5 148 + 4 F4 12 16 f 1 16 + 1 16 + 1 17 + 1 G, 140 - 46 + 62 - 30 + 62 124 f 19 Of 42 G3 15 -145+_ 60 -26+ 104 26 + 9 44 + 9 G5 2 33 + 10 14 + 17 fixed fixed 1 620 1 654 f 30 1 632 + 33 1618 f 31 1619 f 27 4d 750 830 + 47 800 f 51 793 + 53 783 + 43 rms error 131 128 152 126

V. Conclusion. - New values for the 4 f-5 d inter- subconfiguration. The reliability of the parameters is action parameters around the half-filled 4 f shell manifest by their regular behaviour from Sm to Tb, have been obtained. The theoretical prediction are by the accuracy of their reproducing observed level capable of reproducing quite accurately recent obser- position and g-values, and by their being determined vational data in the 4 f6('F) 5 d and 4 f8(7~)5 d by 126 low levels observed in these atoms.

References

[I] SPECTOR(Nissan), J. Opt. Soc. Am., 1970, 60, 763. [9] RUSSELL(H. N.), J. Opt. SOC. Am., 1950, 40, 550. [2] BLAISE(J.) and VANKLEEF (Th. A. M.), C. R. Acad. [lo] KLINKENBERG(P. F. A.) and MEINDERS(E.), Physics, Sci., 1969, 268, 792. 1969, 42, 213. [3] BLAISE(J.), MARILLON(C.), SCHWEIGHOFER(Marie- [Ill EREMIN(M. V.) and MARYAKH~NA(0. I.), O~I~CS Gabrielle) et VERGES(J.), Spectrochim. Actu, nnd Spectroscopy, 1969, 26, 479. 1969, 24B, 405. [I21 ELLIOTT(J. P.), JUDD(B. R.) and RUNCIMAN(W. A.), [4] KLINKENBERG(P. F. A.), Physica, 1967, 37, 197. Proc. Roy. Soc., 1957, 240, 509. [5] RACAH(G.), Phys. Rev., 1942, 62, 138. [13] ARNOULT(C.) and GERSTENKORN(S.), J. Opt. SOC. [6] CONDON(E. U.) and SHORTLEY(G. H.), Theory of Am., 1966, 56, 177. Atomic Spectra, Cambridge University Press, [14] CHILDS(W. J.) and GOODMAN(L. S.), J. Opt. SOC. 1963. Am., 1969, 59, 875. [7] JUDD(B. R.), Pl~ys.Rev., 1962, 125, 613. [IS] MEINDERS(E.) and KLINKENBERG(P. F. A.), Physica, [8] ALBERTSON(W.), ASII.OP/~.J., 1936, 84, 26. 1968, 38, 253.