Vol. 72, No. 4/April 1982/J. Opt. Soc. Am. 493

Effective-operator interpretation of the hyperfine structurein the 5p 2 6p configuration of 52Te II

L. Augustyniak and K. Werel

Institute of Physics, University of Gdahsk, Wita Stwosza 57, 80-952 Gdahisk, Poland

Nissan Spector

Soreq Research Center, Yavne 70600, Israel

Received August 10, 1981

The magnetic-dipole hyperfine-structure (hfs) constants for 15 levels of the 5p 2 6p configuration of Te II were re-cently observed. To carry out the theoretical analysis of these experimental data, the intermediate coupling coeffi-cients for all 21 levels belonging to 5p 26p were calculated. The theoretical gj factors show good agreement withthe experimental values known from Zeeman-effect data. The hyperfine-structure analysis was performed by theeffective-operator formalism. On the basis of the experimentally determined A hfs constants the values of the ef-fective radial parameters akskl have been obtained in two versions: in version 1 we have taken into considerationall 15 known A values; in version 2 the A constant for the 11l/2 level was eliminated. The following values for thehfs parameters akSil in these two versions were obtained: a = -107.7 and -108.8 mK, a12 = -170.3 and -151.9mK, a0 = 1.1 and 1.9 mK, atj = -21.8 and -10.9 mK, al = -6.2 and -29.5 mK, a = 1.5 and -6.7 mK, respec-tively.

1. INTRODUCTION

The spectrum of Te II is one of the least-known spectra. Thespectroscopic identification of Te II energy levels was madeby Handrup and Mack.' So far, only the ground configura-tion 5s

25p

3 of Te II is fully described. Most of the excitedlevels have been designated only in terms of their J values andenergy. Many of the lines listed in Handrup and Mack's workremain unclassified.

In this paper we limit our attention to the theoreticalanalysis of the hyperfine structure (hfs) of the 5p26p excitedconfiguration. This configuration consists of 21 levels. Thehfs of 15 states has been determined by Ross and Murakawa 2

and Werel and Augustyniak. 3 Based on these data, ouranalysis was performed by means of an effective Hamiltonianformalism. This method, introduced by Sandars and Beckin 1965,4 shows that hfs calculations can be performed con-veniently for any type of configuration and that all relativisticeffects predicted in the classical jj-coupling scheme can bereproduced if one uses the effective operators that act betweenthe nonrelativistic LS-basis states. Many works confirm theusefulness of this method in many electron atoms (see, e.g.,Refs. 5-12).

To interpret the hfs data it was necessary to perform thefollowing operations:

(1) Calculate the intermediate coupling coefficients forall levels of 5p 26p configuration of Te II;

(2) Calculate the matrix elements of the effective operatorbetween the LS-basis states;

(3) Derive, by using the results of operations (1) and (2),the parametrized expressions for the hfs interaction constantsas functions of the effective radial parameters akski. Theseexpressions enabled us to determine the ahski parameters for

the 5p 26p configuration of Te If on the basis of experimentallydetermined A hfs constants.

2. SHORT EXPERIMENTAL REVIEW

The experimentally determined hfs splittings and A hfsconstants for the levels of the 5p 26p configuration of Te II aregiven in Table 1. The level designations are due to Handrupand Mack.1

The results of Ross and Murakawa, as well as our results, 3

were obtained by using the classical experimental setup con-sisting of a Fabry-Perot interferometer crossed with a spec-trograph with photographic registration.

The A constant for the 971/2 level is presented here for thefirst time to our knowledge; therefore the intensity distribu-tion for the X 629.4-nm transition, which involves the 97s/2level, is given in Fig. 1. The numerical analysis of the ex-perimentally obtained hfs pattern of this transition was car-ried out in a way described in our recent paper.3

3. THEORETICAL FINE-STRUCTURECALCULATIONS

The configuration p2p has 10 terms that split into 21 theo-retical energy levels. In Ref. 13, all except the two high 2 P's,namely, (1S)2P and ('D)2P, were given. We calculated these21 energy levels by diagonalizing the Hamiltonian that con-sists of the following interactions: the electrostatic interac-tion of the two 5p core electrons that is expressed by two radialparameters: Slater's F2 (5p, 5p) and Trees's correction a; theelectrostatic interaction between the core electrons and theexternal 6p electron, expressed by the three radial Slaterparameters F2, Go, and G2 (5p, 6p); the two-spin-orbit in-

0030-3941/82/040493-06$01.00 © 1982 Optical Society of America

Augustyniak et al.

494 J. Opt. Soc. Am./Vol. 72, No. 4/April 1982

teraction expressed by the two radial parameters a5p and t6p;and the additive constant A.

We obtained initial values for these parameters by usinginterpolations between neighboring atoms as well as otherwell-known tellurium levels. These initial parameters werethen fitted to the 17 experimentally known levels.

In Table 2 we give the system of radial parameters that re-sulted from this least-squares calculation. It is seen from thistable that convergence was reached. The total rms error isonly 36 cm- 1 , which is 0.2% of the total configuration width.It should be noted that, despite the rather low sharpness ofa, a is important in its contribution to the eigenvalues. Asecond least-squares calculation, in which a was held fixed,was also performed. The other parameter values stayedpractically unchanged, and the rms error went down to only34 cm-1 . We therefore felt that adjusting a was not justi-fied.

In Table 3 we compare the predicted and observed energylevels and g factors for the 5p 26p configuration of Te II. Inthe first column we give the J value of the level and in thesecond column we give the percentage composition in LScoupling. It is seen that, despite several admixtures, most ofthe levels have their major components about 50%, and thusthe LS coupling scheme can still be used to describe the levelstructure of this configuration. The agreement between theobserved and calculated g factors is quite good. No inter-change is indicated on this basis.

The extended list of levels by Handrup and Mack gives thetwo missing doublets. Their (lS)2 P levels fall quite far fromour predictions and cannot be included because of their largeO-C values as well as their g factors. The other pair, however,fits both calculated positions and g factors well and serves asa confirmation of the validity of our calculations.

In Table 4 we give the eigenvectors obtained in the diago-nalization that used the parameters given in Table 2. We give

Table 1. Hyperfine-Structure Splittings and AConstants for the Energy Levels of the 5s25p2 6p

Configuration of Te iI

Level

931/2963/2971/2996/

100,2101/321011/21025X2

1083/2103;/2105,21055/21061,2

1115/2112,/21126/31127/21141/2

1163/2125,12125;/2

Hfs Splitting (mK)Werel and Ross and

Augustyniak Murakawa

-50 I 13 -53- -20

+54 + 6 -

-61 I 2 -64- 0

- -61+75 I 4 +78

-256 + 3 -256- -246- -193- -109

-222 I 2 -221-252 + 3 --169 I 3 -

-196 ± 5

A (mK)

-50 + 13-10+54 ± 6-30 + 1

0-30+75 + 4-85 + 1-61-96-54-74 + 1

-252 ± 3-56 ± 3

-49 ± 2

A 629.4 nm

0 200

WAVE NUMBER (m /Fig. 1. Intensity distribution of the X 629.4-nm (813/2-97'1/2) tran-sition. Open circles, experimental profile linear in intensity; solidline, theoretical profile; dashed lines, theoretical profiles of single hfscomponents.

Table 2. Radial Parameters for 5p26p Configurationof Te II

Initial Value Final ValueName (cm'1) (cm'1)

A 110356 110983 + 366F2 1295 1364 ± 41F2 ' 335 335 + 4Go 1561 1551 + 17G2 50 48±3a -120 -128 ± 75t5p 5250 5290 i 44t6p 780 785 + 16rms - 36 (0.2%)

for each J in the first column the observed energy level (in casethe observed value is missing we give the theoretical value)followed by its corresponding eigenvector in LS coupling.These eigenvectors were then used in the following work toevaluate the theoretical expressions for the relativistic hfscoefficients.

4. HYPERFINE-STRUCTURE THEORY IN ANEFFECTIVE-OPERATOR TREATMENT

In this work we confine our attention to the magnetic-dipole(MI) interaction. The electric-quadrupole constant vanishes

Augustyniak et al.

Vol. 72, No. 4/April 1982/J. Opt. Soc. Am. 495

Table 3. Observed and Calculated Energy Levels and g Factors of the 5p26p Configuration in Te [I

Percentage Composition Observed Level Calculated Level O-C

J in LS Coupling (cm'1) (cm'1) (cm-1 ) gobs gcal

1/2 51% 4 D + 21% 2 S 93979 93959 21 0.79 0.78

90% (1S)2P - 125798 - - 0.67

72% ( 3p)

2p 106119 106101 18 0.86 0.88

90% (1D)2 p - 114320 - - 0.7677% 4P 101371 101368 3 2.32 2.33

52% 2 S + 38% 4D 97780 97798 -18 1.26 1.26

3/2 12% 4 S + 42% 4 D + 23% (3 P) 2D 99585 99592 -7 1.26 1.24

90% (lS) 2P - 126504 - - 1.3364% (3

p)2

p + 25% (ID)2 D 105006 105006 0 1.21 1.21

53% ('D)2P + 22% (3P) 2P - 116279 - - 1.2842% 4P + 37% 4S 103936 103883 53 1.74 1.73

46% (3p) 2 p + 31% 4S 101221 101217 4 1.31 1.32

48% (1 D)2 D + 24% ('D) 2 P 112272 112303 -31 1.09 1.09

51% 4 D + 20% 4 p 96145 96165 -20 1.32 1.32

5/2 31% 4 P + 28% (3 P)2D + 18% ('D) 2D 102324 102300 24 1.31 1.30

41% (3 P)2 D + 34% 4 P 105585 105608 -23 1.29 1.28

53% (1D)2D + 24% ('D) 2F 111947 111894 53 1.19 1.19

81% 4D 100112 100149 -37 1.40 1.39

45% 2 F + 22% (3 P)2 D + 21% (2 D)2 D 112549 112574 -25 1.06 1.06

7/2 81% 4 D 103106 103102 4 1.38 1.37

81% 2 F 112788 112802 -14 1.19 1.20

in the case of 125Te because I = 1/2. For magnetic-dipole where a01 represents the interaction between the nuclear

hyperfine interaction in a lNl' configuration the Hamiltonian magnetic-dipole moment Au and the magnetic field producedmay be written, after Sandars and Beck,4 in the form by purely orbital motion of the electrons, a12 represents the

N interaction between Al and the magnetic moment of theHhfc(Ml) = ; [a'li -101/ 2aF2(s X C(2))(1) + a' 0 si]I electron, and a10 represents the relativistic and configuration

i=1 interaction (core polarization). C(2) is proportional to the+ [a~"alN+1-101 12alNs X C(2)))1Y + a'OsN+11I, (1) spherical harmonic of order 2.

Table 4. Eigenvectors a,,, of the 5p26p Configuration in Te II

LevelJ value (cm-1 ) Eigenvector Component

(3P)2S (lS) 2p (3p)2p (ID)2P (3p)4 p (3 P)4DJ= 1/2 93979 -0.4651 0.2692 0.3335 -0.0938 -0.2887 0.7127

125798 -0.0458 -0.9462 0.2102 -0.1390 -0.1000 0.1700106119 0.3667 0.1202 0.8534 0.1650 -0.1753 -0.2547114069 -0.2241 -0.1307 -0.0185 0.9538 0.1227 0.0871101371 0.2879 -0.0161 -0.3312 0.1711 -0.8819 0.0145

97780 -0.7170 -0.0186 0.0795 -0.0753 -0.2885 -0.6248

(3 P)4S (lS) 2p (3p)2p ('D)2p (3p)4p (3P)2D ('D)2D (3P)4DJ= 3/2 99585 0.3367 0.0630 -0.2684 0.0502 -0.3232 0.4838 0.2289 -0.6459

126504 -0.0704 0.9548 0.0207 0.1400 0.1579 -0.1500 0.0263 -0.1233

105006 0.1225 0.0116 0.7924 -0.2598 -0.1331 -0.0841 0.5043 -0.1022

116102 0.2458 -0.0740 0.4665 0.7290 0.0860 0.2104 -0.3639 -0.0306

103936 0.6149 0.0345 -0.0131 -0.3753 0.6543 0.1872 -0.1036 0.0763

101221 -0.5638 -0.0011 0.0818 0.0260 0.3836 0.6768 0.2518 0.0784

112272 0.2030 -0.1223 -0.2711 0.4864 0.2442 -0.2957 0.6941 0.1916

96145 0.2628 0.2502 -0.0407 -0.1297 -0.4590 0.3786 0.0655 0.7124

(3P)4P (3 P)2D (1D)2D (3 P)4D (2D)2F

J= 5/2 102324 -0.5608 0.5347 0.4327 -0.3179 0.3336105585 -0.5811 -0.6394 0.2577 -0.1319 -0.4120111947 -0.4237 -0.2077 -0.7289 -0.0800 0.4895100112 0.3707 -0.2050 0.0055 -0.9001 0.0932112549 0.1758 -0.4674 0.4638 0.2548 0.6860

(3P)4D ('D)2FJ = 7/2 103106 -0.8991 0.4376

112788 -0.4376 -0.8991

Augustyniak et al.

496 J. Opt. Soc. Am./Vol. 72, No. 4/April 1982

Table 5. Nonvanishing Matrix Elements Diagonal in J for Magnetic-Dipole hfs Interaction in the 5p26pConfiguration of Te ii

F (,y, y,)a (ry IHhfs (M1) 1y)

(3 P)4 D7 /2 , (3P)4D7 /24 (3P)4D7 /2 , (1D)2F7 /2

(1D)2F7 /2, (1D)2 F7/2

(3p)4 P5 /2 , (3P)4P5/2(3p)4P5/2 , (3 P)2 D5 /2

3 (3p) 4Pr/2, (1D)2D5/2(3 P)4 P5 /2 , (3 P)4 Ds/2(3 P)4 P5 /2 , (lD)2 F5 /2

(3 p)2 D5 /2 , (3 p)2 D5 /2(3 p)2 D5 /2 , ('D)2 D5 /2(3P)2D5 /2, (3 P)4 D5 /2(3 p)2 D5 /2 , (lD)2 F5 /2('D)2D5 /2 , (lD)2 D5 /2

3 (lD)2 D5/2, (3p) 4D5/2('D)2D5 1 2 , ('D) 2F5 /2(3 P)4 D5/2 , (3 P)4D5 /2(3 P)4D5 /2 , (1D)2 F5 /2('D)2F5 /2 , ('D) 2F5 /2

(3P)4 S3 /2 , (3P)4 S3 /2(3P)4S 3 /2, (3 P)4 P3 /2(3P)4 S3 /2 , (3 P)2 D3 /2(3P)4 S3 /2 , ('D)2 D3 /2(3 P)4 S3/2 , (3 P)4 D3 /2(lS) 2P3/ 2 , (1S)2 P3 /2(3p)2 p3 /2 , (3p) 2P3 /2

(3p)2 P3 /2 , (1 D)2P3 /22 (

3P)

2 P312 , (3 P)4P31 2(3 p)2P3 /2 , (3 P)2 D3 /2(3 p)2P3 /2 , ('D) 2D3 /2(3 p)2 P3 / 2 , (3p)4 D3 /2

('D)2 P3 /2, (1 D)2P3 /2

( pD)2 P3 /2 , (3 P)4 P3 /2('D)2 P3X2 , (3p) 2D31 2

('D)2 P3 /2 , (lD)2 D3 /2(3p)4P3 /2 , (3p)4P3 /2

(3p)4p3 /2, (3P)2 D3 /2(3p)4p3 /2 , ('D) 2D3 /2(3 P)4 P3 /2 , (3 P)4D3 /2(3 P)2 D3 /2, (3P)2D3 /2

2 (3 P)2 D3 /2 , ('D) 2D 3 /2(3 p)2 D3 /2 , (3 P)4D3 /2(ID)2D3 /2, ('D) 2 D3 /2('D)2D3 /2 , (3P)4D3 /2(3 p)4 D3 /2, (3 P)4 D3 /2

(3 p) 2S 1/2, (3 p)2S 1/2(3p)2 S1/2 , (3p)2P 1 /2

(3p) 2P1/ 2 , (3p)4D)2P

(lS)2p,/2, (lS)2P,/2(3P) 2 p,/ 2 , ('D) 2 P1 /2(3p) 2 p,/2, (3 P)2P1 /2

1 (3P)

2P 1 /2 , (PD)2pD1 2

(3p)2p,/2, (3p)4 p1 /2(3p)2p,/2, (3p)4Dl/2

('D)2p,/2, (3p)4p i/2('D) 2P,/ 2, (3 P)4D1l 2

(3p) 4p 1 /2 , (3 P)4P1 /2(3p)4p 1 /2, (3 P)4 D1 /2(3 P)4 D1 /2, (3 P)4 D1 /2

0.5a°, + 0.5a8' + O.1a12 - 0.1a12 + 0.5a5° + 0.25al,-0.212a52ag51 + 0.5a8,1- O.la12 + 0.25al,

0.25ag5, + 0.25al - 0.05a02 + 0.05a12, + 0.5a5° + 0.25a',0.061a2 - 0.122a 2,-0.061a5p0.164a' - 0.164ao' + 0.164a 2 + 0.164a'2-0.065alp

0.5a0 + 0.5a~l + 0.067a12, + 0.033a 2, + 0.333a5° - 0.083al,-O.1al,-O.O76a5 p- O.151al,-0.178a5° + O.178a60.214a5p

0.833a5° + 0.167al, + O.1a'2 + 0.25al,-0.227ac540.089a5l - 0.089ao, + 0.214a120.393a°5 + 0.393aop - 0.088a p + 0.088a 2 + 0.309a° + 0.155a'°-0.028alp0.952a°5l + 0.476ao' + 0.171a, -0.178al,

0.5a5° + 0.25al°0.316a5l - 0.315a'l0.032a5p + 0.063a 6-0.095a52p0.126a5 2p- 0.126ao,0.5ap - 0.1a12 + 0.25a'°0.25ao, + 0.25ao, - 0.033a~ -0.017al, + 0.333a5°p- O.83a 6-0.067a5p0.048a12p + 0.097a2, - 0.149a5° + 0.149al,0.112a°p - 0.112aol + 0.134a2 - 0.067a12p0.067a5kp-O0.1Ola2 + 0.201a12,0.75a°5 - 0.25agl - 0.01a 2, + O.25al°-0.195a5 2

-0.09a5p0.15a°5p - 0.15a, - 0.09a12,O.lag5p + 0.Aa°l + 0.1i3a2, - 0.113a 2, + 0.367a5° + 0.183al,

-0.015a02 + 0.03a12,0.015a5p0.2a°5l - 0.2a°l0.45agp + 0.45aop - 0.14a12, - 0.07a2, - 0.2a5° + 0.05a6°0.21alp-0.35ap - 0.07a p - 0.2ap + 0.2a'p0.75a°5p + 0.15aol - 0.21a12 -0.15a',

-0.105a5lp0.3a°5 + 0.3a8l - 0.14a5 + 0.14a6. + 0.1a, + 0.05a6°

0.333a5p - 0.083a'6p0.236a°p - 0.236a~p-0.074a52 - 0.149a 2,0.333aop + 0.333a2, - 0.083a lo

0.333ap + 0.333a6p - 0.083al6p0.167a°5p + 0.167ap + 0.11la5p + 0.055a6p - 0.111ao, + 0.028a6lp0.224a5'p-0.02a2, - 0.04a2 - 0.157a5° + 0.157al,-0.079a5p + 0.158a62p0.079a5p-0.106a6p-0.083a°p - 0.083a~l - 0.028a12, + 0.028a12, + 0.278a5° + 0.139al°0.186a°5 - 0.186aol - 0.112a, - 0.112a6,0.26a°5 + 0.25a~o - 0.117a'p + 0.117a12p - 0.167alp - 0.083alp

a y = (2S1+lLI)2S+lLj.

Augustyniak et al.

Vol. 72, No. 4/April 1982/J. Opt. Soc. Am. 497

The matrix elements of the magnetic-dipole hfs interactionof Eq. (1) evaluated between any two levels of the same J onan LS basis are given by Childs.6 Combining these matrixelements and LS eigenvectors of the atomic states of the5p 26p configuration of Te II in the following way:

2A (B) = -E aOaT,5aOf (SLJIFIHhfs(M1)IS'L'JIF), (2)

C 7tT

where r stands for the set of quantum numbers (SLJIF) ofthe LS state, fl denotes the set of quantum numbers of the realstate,-and C = [F(F + 1) - J(J + 1) - I(I + 1)], one can obtainin the first-order perturbation theory the expressions for themagnetic-dipole hyperfine-interaction constants as a linearcombination of the effective radial parameters akskl for allstates of the 5p26p configuration.

It was shown by Sandars and Beck4 that in the absence ofconfiguration interaction the akskI parameters may be eval-uated by taking the proper combinations of several relativisticradial integrals. If the radial integrals are not available, ak-kimay be approximated with the help of Casimir factors'4" 5 inthe following way:

al' = a,1(21 + 1)-2 [21(1 + 1)F,(l + 1/2, Zef)+ 21(1 + 1)Fr(l - 1/2, Zef) + Gr(l, Zef)],

a12 = ai(21 + 1)-2 1/3[_41(l + 1)(21- 1)Fr(l + 1/2, Zef)

+ 41(1 + 1)(21 + 3)F,(l - 1/2, Zef)- (21 + 3)(21 - 1)Gr(, Zef)],

alo = 4/3ahl(l + 1)(21 + 1)-2[(l + 1)F,(l + 1/2, Zef)

-IF,(I - 1/2, Zef) - G,(l, Zef)] (3)The quantities ao1 and a12 approach a.l =(2ABAI1Ih)(r-3).1,and a 10 vanishes in the nonrelativistic limit if the configura-tion interaction is ignored. (r- 3 ),j is given by

(r-3)n= R.o 2a8ZefHr(p, Zef)

where tp is the s-o parameter, R- is the Rydberg constant, Zefis the effective atomic number, and Hr (p, Zejf) is the relativ-istic correction factor.15

5. RESULTS

To compare the experimental magnetic-dipole hfs constantsof all known levels of the 5p26p configuration of Te II with thetheoretical ones it was necessary to express these constantsin terms of one-electron parameters akski. For this purposetwo computer programs have been written to calculate thematrix elements of the Hamiltonian [Eq.(1)] in the LS schemeby using the expression given by Childs6 and to combine theseelements with the eigenvectors (see Table 4) according to Eq.(2). All nonvanishing matrix elements are given in Table5.

Taking into account all experimentally determined A hfsconstants, the following relations were obtained:

0.34039agl + 0.28568a8l + 0.14158ap, - 0.05714a p+ 0.23096a', + 0.14284a3° =-61 mK (103;/2),

0.51664a01 + 0.28568a~l - 0.08444a12 - 0.05713a p

+ 0.05471al° + 0.14284a', = -49 mK (112;/2),

0.48589adi + 0.20912aol + 0.15566a12, + 0.25358a'2

+ 0.27551a5l + 0.02944a~l = -85 mK (102;/2),

0.41507agp + 0.30518a°l + 0.16927ap - 0.05527a 1

+ 0.22435a5p + 0.05551al, = -74 miK (105;/2),

0.54994a5°p + 0.25927al - 0.07679a1 - 0.50307a 1

+ 0.08016a'p + 0.11059a', = -56 miK (111;/2),

0.21892a°5 + 0.39020aol - 0.18268a - 0.05265a~,+ 0.21360alp + 0.17727al2, = 0 mK (1006,2),

0.32995a°5p + 0.42775al - 0.13996a 1 + 0.29357a'2+0.28404a5° - 0.04175a', = -30 mK (996/2),

0.45950a° + 0.33093a°6 + 0.00688a p + 0.02846a 1

+ 0.33458ap - 0.12494a', = -54 mK (105;/2),

0.59948a5p - 0.33093a6p + 0.22331a5p - 0.11483a6p-0.44866ap + 0.28280a'p = -96 mK (1033/2),

0.20799a5°p + 0.46683a°l + 0.01858ap - 0.12732a 1+ 0.12472ap + 0.20045a'° = -30 miK (101'/2),

0.12853da5p + 0.55424al - 0.04155a1 - 0.07610a'2+ 0.02795ap + 0.28930alp = -10 mK (966,2),

-0.02730a5° + 1.24426a°6p + 0.06752a0p + 0.97838a'2

+ 0.11401ap - 0.33087a'p = -50 mK (931/2),

1.25123ap - 0.11539aol + 0.73597a"p + 0.33688a 1.03456a5 - 0.17040alp = -252 miK (1061/2),

-0.32621a° - 0.00618a°l - 0.27058a", + 0.02984a'2+ 0.55859a5°p + 0.77389a'° = 75 miK (1011/2),

0.53941a5 + 0.20230a1 - 0.61881a 1 - 0.49422a 1

+ 0.54371alp - 0.28537a', = 54 mK (971/2)- (4)

By treating akskl in the above set of equations as adjustableparameters, one can obtain the values of akshl that includeboth relativistic and configuration interaction effects. InTable 6 the effective radial parameters akskl determined intwo versions are given. In version 1 the fit to all known 15values of A was made. However, for this version the differ-ences between Afit and Aexp in many cases exceed the exper-imental uncertainty (see Table 7). It turned out that the fitis much better when the A constant for 1115/2 state wasomitted (version 2). In this case Afit - Aexp does not exceed±3 mK. Taking into account the uncertainty of the measuredA values, the fit in version 2 is satisfactory. The theoreticalvalues of akskl were obtained in Casimir approximation fromEq. (3). Casimir factors were taken from Kopfermann,15

nuclear magnetic moment /ui from Dharmatti and Weaver,16

and tp values from Table 2.

Table 6. Experimental and Theoretical (Calculatedin Casimir Approximation) a kgki Values for the 5p 26p

Configuration of Te II (in mK)

a~p alp al° a~p al2 al°

ExperimentalVersion 1 -107.7 -170.3 1.1 -21.8 -6.2 1.5Version 2 -108.8 -151.9 1.9 -10.9 -29.5 -6.7

Theoretical -113.9 -136.9 6.8 -17.1 -20.5 1.0

Augustyniak et al.

498 J. Opt. Soc. Am./Vol. 72, No. 4/April 1982

Table 7. Experimental and Calculated A hfsConstants for the Energy Levels of 5p 26p

Configuration of Te II (in mK)

Version 1 Version 2Level Aexp Aft Afit-Aexp Art Afjt-Aexp

931/2 -50 + 13 -42 +8 -47 +3963/2 -10 -18 +8 -13 -397;/2 +54 4 6 +46 -8 +51 -3991/2 -30 + 1 -23 +7 -27 +3

1005/2 0 -2 -2 0 01013/2 -30 -34 -4 -28 +21011/2 +75 i 4 +83 +8 +78 +3102;/2 -85 -84 +1 -86 -1103;/2 -61 -66 -5 -60 +1103;/2 -96 -94 +2 -93 +3105/2 -54 -58 -4 -54 0105,3 -74 4 1 -79 -5 -72 +2106,/2 -252 ± 3 -260 -8 -255 -31115/2 -56 3 -48 +8 - -1127X2 -44 ' 2 -47 +2 -46 +3

The experimentally determined values of a~l parameter arealmost identical for both versions. The same relation can beseen for the al parameter. The values of those parametersare close to the theoretically calculated ones (the differencesbetween experimentally determined and calculated values donot exceed 7.7 mK, which is of the order of the experimentalerror in determining the A hfs constants). The values of thea12 parameter are slightly different in versions 1 and 2. Bothvalues differ also from theoretically calculated a5p (the dif-ferences are 33.4 and 15 MiK, respectively, for versions 1 and2).

The values of a 01 and a12 parameters are 1 order of mag-nitude smaller than those of a~l and a", a'° is of the sameorder as that of al° The differences between the experi-mental and theoretical a86 and a' parameters do not exceed7.7 mK, in analogy with agi and a0% For the alp parameterthe differences between the experimental and theoreticalvalues are equal to 14.3 and 9.5 mK for versions 1 and 2, re-spectively.

6. CONCLUSIONS

The experimental values of the aksks parameters obtained inversion 2 (with better fit) are closer to the theoretically cal-culated ones than those obtained in version 1. The dis-crepancies exceed the experimental error only in the case ofa12 parameters. The facts that a2 (exp) and a12(exp) are

larger than the values of a 5 and a 6 calculated on the basisof relativity, and that both ratios a12/a~i and a12/aM are largerthan expected, suggest that configuration interaction may playan important role.

To carry out more-detailed analysis of the hyperfinestructure in the 5p 2 6p configuration of Te II it would be nec-essary to perform a high-accuracy experimental determinationof all hfs A constants and to obtain values of the akskl pa-rameters from ab initio theoretical calculations.

REFERENCES

1. M. B. Handrup and J. E. Mack, "On the spectrum of ionizedtellurium," Physica 30, 1245-1275 (1964).

2. J. S. Ross and K. Murakawa, "Hyperfine structure and isotopeshift in the spectrum of tellurium," Phys. Rev. 85, 559-563(1952).

3. K. Werel and L. Augustiniak, "Hyperfine structure analysis ofsome tellurium II lines," Phys. Scr. 23, 856-858 (1981).

4. P. G. H. Sandars and J. E. Beck, "Relativistic effects in manyelectron hyperfine structure," Proc. R. Soc. London Sect. A 289,97-107 (1965).

5. A. Ros6n, "Analysis of the hyperfine structure of the ground statemultiplet of the samarium atom," J. Phys. B 2, 1257-1260(1969).

6. W. J. Childs, "Hyperfine structure of many atomic levels of Tb159

and the Tb'59 nuclear electric quadrupole moment," Phys. Rev.A 2, 316-336 (1970).

7. W. J. Childs and L. S. Goodman, "Hyperfine and Zeeman studiesof low lying atomic levels of La13 9 and the nuclear electricquadrupole moment," Phys. Rev. A 3, 25-45 (1971).

8. J. Dembczyfiski and M. Frackoviak, "Hyperfine structure in in-termediate coupling of the first excited electron configuration6p2 7s of 839Bi (I = 9/2)," Acta Phys. Pol. 48, 139-155 (1975).

9. J. Dembczyfiski, "Determination of the quadrupole moment of5 5

1Sb (I = 5/2) nucleus from hyperfine structure analysis," ActaPhys. Pol. 49, 541-554 (1976).

10. B. Arcimowicz and J. Dembczynski, "Relativistic effects in thehyperfine structure of the second spectrum of the Bi II ion," ActaPhys. Pol. 56, 661-671 (1979).

11. S. Buttgenbach, R. Dicke, H. Gebauer, R. Kuhnen, and F. Traber,"Hyperfine structure of seven atomic levels of 9 1Zn and the 9 1Znnuclear electric quadrupole moment," Z. Phys. A 286, 125-131(1978).

12. K. H. Burger, S. Buittgenbach, R. Dicke, H. Gebauer, R. Kuhnen,and F. Traber, "Hyperfine and Zeeman studies of low lying atomiclevels of 181Ta and the nuclear electric quadrupole moment," Z.Phys. A 298, 159-165 (1980).

13. C. E. Moore, Atomic Energy Levels (U.S. Government PrintingOffice, Washington, D.C., 1958).

14. L. Armstrong Jr., Theory of the Hyperfine Structure of FreeAtoms (Wiley, Interscience, New York, 1971).

15. H. Kopfermann, Kernmomente (Frankfurt am Main, Germany,1956).

16. S. S. Dharmatti and H. E. Weaver, Jr., "On the magnetic momentof Te123,125 and Si2 9 '," Phys. Rev. 84, 843-844 (1951).

Augustyniak et al.