PHYSICAL REVIEW A VOLUME 44, NUMBER 9 1 NOVEMBER 1991

J off-diagonal effects in the hyperfine-structure splitting in the Ku I term e D of 4f 6s 6d

H.-D. Kronfeldt and D.-J. Weber*Optisches Institut der Technischen Uni Uersitat Berlin, Strasse des 17. Juni 135, 1000 Berlin 12, Germany

J. Dembczynski and E. StachowskaInstytut Fizyki, Politechnika Poznanska ulica Piotromo 3, 60-965 Poznan, Poland

(Received 1 April 1991)

In the analysis of high-resolution Doppler-free two-photon hyperfine-structure (hfs) measurements oflevels in the Eui term e D of the configuration 4f 6s6d of "'Eu and "Eu the necessity occurs totake into account also hfs perturbations. For the magnetic hfs interactions of "'Eu, the followingoff-diagonal matrix elements "'A&(JJ') could be determined from the high-resolved two-photonline profiles: '"A &(1/2, 3/2) =5.70(3) GHz, "'A &(3/2, 5/2) =4.3(2) GHz, '"3 &(5/2, 7/2) =2.9(2) GHz,"'A, (7/2, 9/2) =2.2(2) GHz. The high quality of the calculations is represented both in the reproducibil-ity of the absolute hfs level energies and also in the hfs anomaly where a mean value deduced from thecorrected magnetic hfs constants of —0.66(8)% can be stated.

PACS number(s): 32.30.Jc, 31.30.Gs, 35.10.Fk

I. INTRODUCTION

In the past few years, we published several papers re-porting on investigations of hyperfine-structure (hfs)splittings and higher-order eff'ects in the isotope shift (IS)between the two stable isotopes 151 and 153 of thelanthanide Eu (nuclear spin of both isotopes I =5/2); see,e.g. , Refs. [1] and [2]. During these investigations we re-cently analyzed by means of Doppler-free two-photonspectroscopy the fine structure, hfs and IS, in the energyregion between 34400 and 36700 cm ' connecting 27odd fine-structure levels of ' " Eu to the ground level4f 6s a S7&2 [3]. In the e D term of the Eulconfiguration 4f 6s6d we found significant deviations be-tween the ratio of the electric hyperfine-structure con-stants B of ' 'Eu and ' Eu deduced from the experimen-tal line profiles and the ratio of the corresponding quad-rupole constants. To our present knowledge no unambi-guous quadrupole anomaly has been found, neither withoptical methods nor with radio-frequency methods.Every deviation from this constant quadrupole ratio orB-factor ratio is an indicator for possible J off-diagonaleffects in the hyperfine-structure splitting presupposingthat any inAuence of external parameters, e.g., electricand magnetic stray fields, is excluded carefully. To checkthis assumption we calculated these hyperfine correctionsfor the e D term of 4f 6s 6d analytically.

II. EXPERIMENTS AND INVESTIGATIONS

The Doppler-free two-photon setup employing a ther-mionic diode filled with europium in its natural composi-tion was used in order to determine the isotope shift andhyperfine structure of the e D levels with J =3/2 —9/2of 4f 6s 6d directly from the Eu ground state4f 6s a S7/Q As a small-band tunable light source weused a coherent ring-dye-laser system (CR 699-29 auto-scan) pumped by an Ar-ion laser. A cavity marker per-

mits the frequency calibration, so that after data process-ing a linearity of typical under 1 MHz in a 16-6Hz scanwas achieved [2]. As detector for the two-photon absorp-tion we used a thermionic diode with a high-ion sensitivi-ty [4]. The thermionic diode works in an usual two-photon setup, i.e., after passing an optical isolator thelaser light was focused into the active region of the diode.The focus of the counterpropagating beam, refiected by aconcave mirror, was carefully adjusted to coincide withthe beam waist of the incoming wave. The backreAectedlight was chopped and the two-photon signal was detect-ed by a lock-in amplifier system —for further details seeRefs. [3] and [5].

Spectra of the Doppler-free two-photon transitions tothe levels of the e D term are shown in Fig. 1 and illus-trate the expected high resolution. Due to restriction inthe selection rules the level with J= 1/2 could not beconnected to the ground state with a two-photon transi-tion. Nevertheless, this level was connected via the535.0-nm line to the well-known level z I'3&2 of 4f 6s6pby means of optical interference spectroscopy [3,6].

From the two-photon transitions we extracted the hfsconstants and the isotope shift values in the followingway. As depicted in Fig. 1 one group of closely spacedhfs components corresponds to each upper value F of thetotal angular momentum quantum number. In the begin-ning of the analysis each of these hfs complexes was fittedseparately with its theoretical line profiles utilizing thewell-known hfs splitting constants of the Eu ground stategiven by Ref. [7] with the aim of extracting one center ofgravity for each upper F value. As an example in Fig. 2 asection from the hfs two-photon transition to the e D3/2level with F=F'=4 is shown together with the corre-sponding hfs transition scheme. Good agreement be-tween experimental and theoretical line intensities wasachieved. From all these centers of gravity we obtain onecenter of gravity for the whole hfs complex of ' 'Eu and

Eu, respectively. This way with the fine-structure en-

5737 1991 The American Physical Society

5738 KRONFELDT, WEBER, DEMBCZ~NSKy AND STACHO

5ection in I"ig. 2

I4

e D6

5 5~ tH

~ - 6 6' 4' 3'f 2' 1'i, 4

0)(

Dsz26

6

6'

( i I i i I l

2 4 6

6

&O 16

4D9f'2 2

63

A) ) I I I I l

18 20 v (GHZ)

6 ofthe round state s a 7/24 76 2 8g ) to the levels e L 3/2 —9/2o- hoton transitions in Eu from the g153E

Y p o o-p o omon an ular momenta F o t e upper4f '6s6d. Numbers represent the common a g

M

CD

CD

153E

4f 6s6d(e D3~z)

O00OO

l5lE s

bC

e

b

4f 6s ( S7p)

l

—200

v (MHz)l

200

FIG. 2. Section of the two-photonton transition to the Eu levele 2

'h F=F'=4 and the hfs transition sc..erne with theire D 2wit

f. ~8&~). The europium was used in its na-e eu' ' nat-relative intensities (Ref. ~ &

~. e eu' ' na-

ural abundance, i.e., Eu: 47.8%,o and '"Eu: 52. o.

bl t determine absolute hfs energy posi-ergies we were a e o eiled in Table I for both Eu isotopes.tions, compi e in

'ns we calculated in aI th b ginning of our evaluations wen e e

'

s fit from these energies the hfs constanstants andleast-squares t romarameters accord-the IS with the following free running parame

'533 for thein to Kopfermanns formulasISg ', ' 'B ' B for the electric hfs, ATfor the Ima netic hfs, B,

Eu. The resulting values are com-between Eu an u.mentionedi ed in Table II, columns 2 and 3. As already men

'

B constants are here un-in the Introduction the resulting con'

ldh D level the B constants yierealistic, e.g. , for t e e 3/pell-B = —8.3, which is far off the we-

known quadrupole ratio given in Ref. [9:' 'Q/' Q =+0.3928(20) .

(2)

Here the a and p are coefficients depending on the quan-irnir 11 . In our case the split-tum numbers due to Casimir

s of the lower level, the Eu groun sta eting constants o eE . (2) reduces to4f 6s a 57&2, are known (Ref. [7]) and q. re

differences of the hfs constants of the upper level:

g~red (au u)A u+(Pu Pu)Bu (3)

ribed bEquation (3) ea s o a1 d t linear dependence describ ythis formula

To exclude any external effects the following experimen-s stematically: the element oftal parameters were varie sys em

diode the u er-fft eno e uh bl b ffer gas in the thermionic dima neticre the laser power, and electric and ggas p ess

fields. In these experiments no sp i ingue to Zeeman or Stark split-h erfine components, e.g. , ue to eyper

shifts reater than 1 MHz were observable.For a better visualization of possible in uence

lied Krebs-Winkler plot [10]. The energythe so-ca e re s-zr between two hfs transitions andi erence e w

I stand for upperpressed by the following formula (u and l stanand lower hyperfine levels, respectively):

u u I I I l5E, z =a", A "+P",B"—a', A P,B—1 I l I—(a"A "+P"B"—az A PzB ) . —

JOFF-DIAGONAL EFFECTS IN THE HYPERFINE-. . . 5739

TABLE I. Absolute hfs-level energies in the ""'Eu I e D term of 4f 6s6d and energy differencesbetween experimental and recalculated energy values.

hfs level

(upper E)

e Dl/2(2(3)

Experimentalhfs-level energy

151E

(MHz)

1 098 162 510.71 098 149 812.1

AEof "'Eu(MHz)

—0.20.4

Experimentalhfs-level energy

of '"Eu(MHz)

1 098 155 454.61 098 149 797.6

hEof ' Eu(MHz)

—0.1—0.1

e D3/p(1)(2)(3)(4)

1 097 909 433.01 097 906 295.71 097 901 705.91 097 895 797.0

1.8—2.0—2.8—0.1

1 097 902 115.61 097 900 783.71 097 898 769.1

1 097 896 060.1

—0.80.9

—1.2—0 8

e Dg/2(0)(1)(2)(3)(4)(5)

1 097 504 258.51 097 503 113.81 097 500 813.51 097 497 382.01 097 492 854. 1

1 097 487 242.7

—4.03.60.2

—3.91 ~ 8

—1.0

1 097 495 882.41 097 495 376.71 097 494 368.1

1 097 492 859.71 097 490 843.1

1 097 488 315.4

1.9—0.6—2.2

0.31.7

—0.2

e D, /~{1){2)(3)(4)(5)(6)

1 096 931 827.91 096 929 811.61 096 926 777.31 096 922 753.31 096 917 728.61 096 911727.9

—2.63.0

—2.42.5

—2.60.6

1 096 922 710.71 096 921 799.61 096 920 454.21 096 918 657.41 096 916420.01 096 913753.7

2.9—5 1

1.11.2

—1.1—0.2

e D9/2(2)(3)(4)(5)(6)(7)

1 096 254 998.51 096 252 102.21 096 248 241.21 096 243 433.41 096 237 685.61 096 231 008.2

—1.92.2

—0.6—0.6

0.1

0.0

1 096 244 840.01 096 243 523.21 096 241 782.41 096 239 642.21 096 237 083.51 096 234 161.9

1.7—1.4—3.2

7.0—4.4

1.0

(5E';"z/a) = A "+(13/a)8", (4)

i.e., the slope of the line characterizes the electric hfssplitting constant B"and the intercept with the ordinateaxis yields the magnetic splitting constant A". In thisway all possible hfs energy differences can be examinedgraphically. As an example in Fig. 3 for the e D3/2 termof 4f 6s6d all possible six hfs differences between F =1and 4 are shown, in the upper part for ' 'Eu and in thelower part for ' Eu. In case (a) the laser power wasvaried using the buffer gas Ne, in (b) again the laserpower was varied but using Ar as buffer gas, and in (c)the buffer-gas pressure was changed. In all cases nosignificant deviations from a linearity were observed. Allsymbols for the various experimental conditions nearlycoincide in the plot, i.e., the experimental deviations inmost cases are less than 0.5 MHz. The resulting slopes,i.e., the electric hfs constants B, lead to the above-statedunrealistic B ratio, i.e., ' 'B/' B =+151.6 MHz/ —18.2MHz = —8.3.

For the determination of the inAuence due to pressurebroadening we varied the pressure from about 60 to 985Pa: see Fig. 4(a). Although this variation of the buffer-gas pressure resulted in large pressure broadenings of the

line profiles, see Fig. 4(b), no signification pressure shift isobservable. The natural linewidth was deduced from Fig.4(b), and amounts to a value of about 10 MHz —for de-tails see Ref. [5].

Since experimental influences could fully be excluded,deviations of the hyperfine-level energy positions fromthe standard first-order theory involving perturbationeffects of the hfs had to be taken into account.

III. DISCUSSION

It is to be expected that for the mixed configurationnl n'l'n "I",where nl is a half-filled shell, term splittingdue to spin-orbit interaction is small. This is the case ob-served for the term 4f 6s6d e D of Eut, for which theenergy intervals between the fine-structure levels are onlya few cm '. As both isotopes ' 'Eu and ' Eu have anonzero nuclear spin (I =5/2), a mixing of electronicwave functions via hyperfine-structure interaction is pos-sible. This effect is called as "breakdown in J as a goodquantum number. " In such a case secular equationsshould be constructed for an atomic structure, whereonly one quantum number F, representing a total angular

5740 KRONFELDT, WEBER, DEMBCZYNSKI, AND STACHOWSKA

TABLE II. hfs and IS parameter values for "'Eu and "Eu without and with off-diagonal hfs correc-tions. The asterisk denotes values calculated with the fixed ratio according to Eq. (9).

hfs and ISparameters

Without off-diagonalhfs corrections (MHz)

151E 153E

With off'-diagonalhfs corrections (MHz)

151E 153E

A e D1/26

3/2

5/2

7/2

9/2

—4232.9(9)—1507.4(0.4)—1132.0(0.3 )—1004.5(0.2)

959.0(0.2)

—1885.7(9.4)—673.8(0.5)—504.6(0.3)—447.3(0.2)—426.1(0.2)

—4255.3(1.5)—1523.2(0.5 )—1137.2(0.3 )—1006.4(0.2)—959.6(0.2)

—1889.7(1.8)—676.8(0.6)—505.4(0.3 )—447.6(0.2)—426.2(0.2)

B e D1/2

3/2

5/2

7/2

9/2

2 1( 1/2, 3/2)A1(3/2, 5/2)A1(5/2, 7/2)A 1(7/2, 9/2)

151.6(3.3)84.8(3.5)45.2(4.2)76.4(4.5 )

—18.2(4.0)—5.5(4.4)31.6(5.2)

120.0(5.7)

—19.2( 3.6)—9.2(4.0)

9.6(4.8)43.2( 5.2)

5.70(3) GHz4.3(2) GHz2.9(2) GHz2.2(2) GHz

—50.8(4.0)—23.6(4.4)

24.0(5 ~ 2)113.2( 6.4)

2.5 GHz*1.9 GHz*1.3 GHz*1.0 GHz*

AT e D1/23/2

5/2

7/2

9/2

—2967.9(4.5)—2821.0(3.4)—2826.8(3.9)—2845.9(2.8)—2853.3(2.7)

—2802.3(5.0)—2830.0(3.7)—2843.4(3.2)—2850.2(3.1)—2864.8(2.7)

momentum of the atom, is a good quantum number. Theconcept of atomic secular equation was first proposed byCasimir [11]. The energy matrix consists of submatricesconstructed for each F value. In this way for I=5/2 and

J = 1/2 to 9/2 an energy matrix is composed out of eightsubmatrices of rank: 1,3,5,5,4,3,2, 1 for F =0 to 7, respec-tively. As an illustration, the submatrix for F = 1 is givenbelow:

-0.SI

151

-0.3I

-0.1I

0.1I

0.3I

153

(b)

(a)

(c)

(b)

{a)

(c)

~153B- -18.2 MHz

-700—

-700—

700 —P@kor/~

15l1)

Hz Av)(g(MHz)

200

160

120

00

10 MHz

530(14

—1 600I I I

-000 0 000

985(30) Pa

0(30) Pa

)pI I

v (MHz)

FICx. 3. Krebs-Winkler plot, according to Ref. [10], for thee D3/2 term with the six possible hfs differences between F =1and 4 for different experimental conditions. (a) Laser power inmW: ~ 320(20), o 200(20), A 140(20); const: Ne at 120(30) Pa.(b) Laser power in mW: & 310(20), A 130(20); const. Ar at140(30) Pa. (c) ~ Ar at 150(30) Pa, A Ar at 60(30) Pa; const:laser power 290(20) mW.

200 400 600p (Pa)

FIG. 4. (a) Dependence of the half-width of the two photoncomplex e D3/2 with F=F'=4, same section as in Fig. 2, dueto various pressures of the buffer gas Ar. (b) Half-width Av, /2

of the five complexes from (a) against the Ar buffer-gas pressure.

JOFF-DIAGONAL EFFECTS IN THE HYPERFINE-. . . 5741

TABLE III. The ratio ' 'B /" B calculated without and with hfs off-diagonal corrections.

Fine-structurelevel

e D3/26

5/2

7/2

9/2

Without off-diagonalhfs corrections

—8.3—15.4

1.40.6

With off-diagonalhfs corrections

0.380.390.400.38

(3 7)3/7

A, ( —,', —,' )+ A2( —'„—,'

)

(5 7) A (5 7)3&a ' ' 53/10

A2( —'„—,')5

—A1(-', -') — — 2( —', -')2 T 53/1O 2 2~T

. Ak(J, J')I E I—I 0 Imin

Jmin Jmin

The matrix elements have been calculated using the following formula from Childs [12,13]:

J J' Kg (

1)J'+I+F I I F(aJIFMIHhf, la'J'IFM ) =

J K J'

The oft'-diagonal matrix elements involving A2(J, J') es-timated theoretically by us are much smaller than thoseof A, (J,J') and have been neglected in our calculations.Finally two hfs fits to 24 experimental hfs sublevels foreach isotope were performed simultaneously. TheA1(J,J), A1(J J'), and A2(J J) were used as adjustableparameters and if JWJ' the following ratio between pa-rameters was assumed [9]:

A 151(JJ~) (151F=2.265,

A 153(J,J' ) (15'E )

(9)

in which J in is the minimum of J and J . The hfs con-stants Ak(J, J') for J =J' are related to familiar hfs con-stants 3 and B by

A, (J,J)=IJA,A2(J, J)= ,'B . —

in Table III. The corrected ratios are close to the value' 'Q/' Q from Ref. [9],given in Eq. (1).

The hfs constants 3 are expressible as linear combina-tions of the relativistic one-electron parameters a 4f, a 6d,a 6d a 6d, and a 6, introduced in Ref, [14]. They can easilybe estimated in pure SL coupling if the values of the pa-rameters a„I are known from independent investigations[3]. A comparison between the off-diagonal hfs constantsA, (J,J') calculated from one-electron parameters a„&,and those obtained from the hfs fit is given in Table IV.The A, (J,J') values estimated from a„& and calculatedfrom the hfs fit are in very good agreement. Therefore weconclude that the anomaly B ratios ' 'B/' B=—8. 3was caused by the neglection of J off-diagonal hfs interac-tion in our preliminary calculation. The consideration ofa mixing of electronic wave functions via hfs interactionleads to the plausible B ratio, which is in agreement withthe quadrupole ratio of Ref. [9]. The hfs analysis present-

where pl denotes a nuclear magnetic moment.The values of the hfs-sublevel energies and the

differences between the experimental and calculated ener-

gy values AE are given in Table I. In general thedifferences are very small and within the limit of the ex-perimental uncertainties. Hence one can assume that theset of adjustable parameters in the hfs fit is sufficient toexplain the available experimental results. The hfs and ISparameter values for ' 'Eu and ' Eu both without andwith off-diagonal hfs corrections are given in Table II.The values of the ratios of the B constants obtainedwithout and with off-diagonal contributions are collected

(1/2, 3/2)(3/2, 5/2)(5/2, 7/2)(7/2, 9/2)

5.64.43.32.2

5.74.32.92.2

TABLE IV. Comparison of the off-diagonal constants"'A& (J,J') calculated from one-electron hfs parameters and ob-

tained from the hfs fit.

"'A, (J,J') In GHzCalculated From the hfs fit

5742 KRONFELDT, WEBER, DEMBCZYNSKI, AND STACHOWSKA

TABLE V. hfs anomalies of e D Eu1 levels in 4f 6s6d with statistical errors from the hfs A con-stants given in Table II.

Fine-structurelevel

hfs anomaly ' '4' (%)From uncorrected A From corrected A

e D, /26

3/2

5/2

7/2

9/2

mean:

—0.897—1.231—0.958—0.854—0.636

—0.91(21)

—0.594—0.666—0.697—0.775—0.584

—0.66(8)

ed here satisfactorily explains the hfs splitting observedfor the 4f 6s6d D term. (The results for the e D termcalculated here are already taken into account by us inRef. [3].)

As a further test for the J off-diagonal hfs inhuence wecalculated the hfs anomalies ' '5' from the magneticconstants ' 'A and ' A listed in Table II both for the un-corrected and for the corrected hfs constant values. InTable V the resulting hfs anomalies are compiled. Theanomalies of all corrected A values are very close to thewell-known value of —0.65(3)% state, e.g., by us in Ref.[15]. It follows here a mean hfs anomaly value of

151'1 53 () 66( 8 ) o/

Furthermore, we would like to point out that the atomicwave functions obtained in IJ-coupling basis predict thepossibility to observe two-photon transition with hJ =3.This assumption can be derived from the composition ofthe atomic wave function, which is, e.g. , given for the hfslevel with F = 3 (three main components):

~

' 'Eu 4f 6s 6d e D, i2 F = 3 )

=0.9996~e D J=1/2 F=3)+0.0290~e D J=3/2 F =3)+0.0003

~

e D J =5 /2 F = 3 ) .

The leading component is connected with state J =1/2,but solely the nonzero amplitude of state J =3/2 givesrise to the possibility of the following two-photon transi-tion:

4f 6s6d e D, ~2,F'=2, 3~4f 6s S7r2, F =1,2, 3,4, 5 .

The intensity of the above transition is expected to be

about three orders of magnitude smaller (i.e., 1188 timessmaller) than that of the transition with b,J=2:

4f 6s6d e D3&2,F'=1,2, 3,4

~4f 6s S7)2,F =1,2, 3,4, 5, 6 .

A similar situation is expected for the isotope ' Eu,where the intensity ratio of the corresponding transitionsis about 1.6X10

IV. CONCLUSION

In this work we demonstrated that, for the correctionof an obviously misleading quadrupole ratio, determinedthrough high-resolution two-photon absorption spectros-copy, it was an absolute necessity to take into account Joff-diagonal hfs elements. This introduction of off-diagonal magnetic hfs constants A (J,J') proved to be asensible interpretation of the measurements in that thequadrupole ratio was corrected and that the recalculatedhfs-level energies show a strong agreement with the ex-perimental energies.

It should be interesting to extend the experimentalresearch to further terms in europium, which alsopossesses close-lying energy levels, e.g., the termsf ' ' D and h D in the configuration 4f 6s7d or toterms in other lanthanide elements. Furthermore, thetheoretical possibility of observing two-photon transi-tions with hJ = 3 is presented.

ACKNOWLEDGMENTS

We are indebted to D. Ashkenasi and G. Klemz for as-sistance in the preparation of the manuscript.

*Present address: B. Halle Nachfolger GmbH, Huber-tusstrasse 10-11, 1000 Berlin 41, Germany.

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JOFF-DIAGONAL EFFECTS IN THE HYPERFINE-. . . 5743

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