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Page 1: singlet-triplet anticrossings 3He : precise determination ...He by means of Electric Field Induced Singlet-Triplet Anticrossings [5]. The coupling between 1 D and 3D states occurs

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Forbidden singlet-triplet anticrossings in 3He : precise determination ofn1D-n3D (n = 3-6) intervals

J. Derouard, M. Lombardi and R. Jost

Laboratoire de Spectrométrie Physique (*),Université Scientifique et Médicale de Grenoble, B.P. 53X, 38041 Grenoble Cedex, France

(Reçu le 21 décembre 1979, accepté le 11 avril 1980)

Résumé. 2014 Nous avons étudié sur 3He des signaux d’anticroisement de faible largeur; dus à la conjonction desinteractions de structure fine et hyperfine, ils n’ont pas d’équivalent dans 4He. Des mesures de grande précisionsont rendues possibles grâce à l’utilisation d’une bobine de Bitter de grande homogénéité pilotée par RMN. Lessignaux expérimentaux sont en parfait accord avec les prédictions déduites d’une diagonalisation complète duhamiltonien de Breit à l’intérieur de chaque configuration (Is, nd) de 3He; les valeurs utilisées pour les constantesradiales sont déduites de résultats expérimentaux antérieurs portant sur les intervalles de structure fine et hyperfinede 4He et 3He obtenus par différents auteurs; on trouve que ces valeurs diffèrent de moins de 1 % de celles calculéesà l’approximation hydrogénoïde. Nous avons déterminé : d’abord la constante de couplage spin-orbite singulet-triplet dans le cas des états 3D : a(3D) = 650 ± 1 MHz. Ensuite, les écarts singulet-triplet pour les étatsnD (n = 3 à 6) avec une précision allant jusqu’à 5 x 10-5 en valeur relative; nos valeurs sont plus petites, d’en-viron 1 % que les quantités correspondantes mesurées dans le cas de 4He ; ce déplacement isotopique, dont lagrandeur est inattendue, est, pensons-nous, dû à une faible interaction de configuration induite par l’interactionhyperfine.

Abstract. 2014 We have studied narrow anticrossing signals in 3He which have no equivalent in 4He because theyare due to the conjonction of fine and hyperfine interactions. High precision results are obtained by the use of a highhomogeneity Bitter coil driven by NMR. The experimental signals obtained are in perfect agreement with thepredictions deduced from an entire diagonalization of the 3He Breit hamiltonian restricted to the (Is, nd) configu-ration. The values used for the radial constants are deduced from an analysis of previous experimental resultsobtained by various authors on 4He and 3He. These values are found to differ from the hydrogenic ones by lessthan 1 %. We have determined : first the singlet-triplet spin-orbit coupling constant for the 31-3D states :a(3D) = 650 ± 1 MHz and secondly the singlet-triplet separation of nD states (n = 3 to 6) with a precision ofup to 5 x 10-5 in relative value. An unexpectedly large ( ~ 10- 3), negative isotope shift is found compared to theequivalent 4He values, presumably due to a slight configuration interaction induced by a hyperfine interaction.

J. Physique 41 (1980) 819-830 AOÛT 1980,

Classification

Physics Abstracts32.80B - 32.60

1. Introduction. - Anticrossing phenomena [1]have been extensively used in the past to measureintervals between states of different multiplicity(see Refs [2, 3] and references therein). As it is well

known, this occurs when two levels (e.g. singlet andtriplet), weakly coupled by a perturbation, (e.g.spin-orbit) are tuned near degeneracy by the magneticfield; the resonant mixing of the wave functionswhich results is observed as a variation of the intensityof the light emitted by each level. This variation has aLorentzian shape as a function of the magnetic field,the width of which is fixed by the magnitude of the,

(*) Laboratoire associé au Centre National de la Recherche

Scientifique.

coupling matrix element. That width is the main limit’for the accuracy of the method.Hence it is very interesting to search for cases where

the effective coupling responsible for the anticrossingis very weak. That has been achieved in the D states of’He by means of Electric Field Induced Singlet-TripletAnticrossings [5]. The coupling between 1 D and 3Dstates occurs via mixing with F states and has providedexperimental values for the singlet-triplet intervalswhich are more accurate, typically a few 10- 4 insteadof a few 10-3 than the ones derived from conventionalsinglet-triplet anticrossings. Another situation hasbeen reported in the case of H2 [4].

In 3He, the existence of nuclear spin induces somenarrow singlet-triplet anticrossings [cf. Figs. 5-7] as a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004108081900

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result of mixed spin-orbit-hyperfine 2nd order coupl-ings. The object of this paper is to study them.

In contrast to ’He [5-15], very few spectroscopicmeasurements already exist on D levels of ’He. Apartfrom some determinations of the hyperfine structure ofnID states [6], and one interval in the n 3D struc-ture [6], an accurate measurement of some intervalsin the 3 3D states [16, 17] was achieved only veryrecently.However, when we started our experiments, our

aim was to improve the experimental values of thesinglet-triplet intervals of D levels, the exchangeenergy in He. In fact the comparison with otheraccurate measurements already existing for 5D and6D states in ’He showed a systematic difference.There exists a relatively large isotope shift (about10-3). Apart from a very small mass effect ( 10-4)such an effect was unexpected. A tentative explana-tion is developed in the last sections.

2. Experiment. - 2.1 ANTICROSSING SET-UP. -

The principle of the experiment is similar to that ofreference [15]; the intensity and polarization of linesemitted by anticrossing states, singlet or triplet arerecorded as a function of the magnetic field.As in reference [15], the observation of n 1- 3D

anticrossings for n 5 requires magnetic fields ofseveral tesla which cannot be attained by classicalelectromagnets. Thus the experiments were carriedout in a Bitter coil at the Service National des ChampsIntenses (C.N.R.S. Grenoble).

3He is contained in a sealed Pyrex cell and isexcited by a triod system [30]. The measurements weremade at two pressures, 10 mtorr and 100 mtorr. Theelectrons were produced by an indirectly heatedcathode (S - 1 cm’) and accelerated parallel to themagnetic field at 30 to 40 eV energy by a grid placed atabout 2 mm from the cathode ; the resulting currentcollected at the anode was 15 to 25 mA. The anodewas at the same potential as the grid and 3He emissionspectra were taken in the grid anode space which isabout 1 cm long. Some space charge electric fieldsshould however exist and induce slight Stark shifts ;however, as specified in section 5, several runs wereperformed in different conditions of pressure, currentintensity and voltage, and the corresponding dis-

persion of results was found small in face of the othersources of uncertainty.

Light emitted by the He atoms perpendicularlyto the magnetic field, reflected by a small mirrorplaced at 450 was collected by a 2 m fused silica lightpipe which was fed into a polychromator built in ourlaboratory [31] from a Jobin-Yvon HRS 2 monochro-mator. Singlet and triplet lines corresponding to thesame (ls, nd) configuration could be then recordedsimultaneously. The light was detected by thermoelec-trically cooled and magnetically shielded HamamatsuR 268, R 269 and R 374 photomultipliers. The signalwas digitized by a DANA 160 000 points voltmeter

and fed into a multichannel analyser also built inour laboratory. A laboratory-made multiplexer wasused to store simultaneously the two signals, singletand triplet, on adjacent channels of the multichannelanalyser and restitute them during readout. An expe-rimental curve consisted of 200 points accumulatedin 10 to 30 passes of 40 s each.

2.2 FIELD CONTROL AND SWEEP. - For large (andnot too precise) magnetic sweeps, a (also laboratory-made) step generator was used to drive the field andtrigger the multichannel analyser.The magnetic field was produced by a 5 MW Bitter

coil which provided a field of up to 13 tesla with ahomogeneity of about ± 10-5 in a sphere of 8 mmdiameter. Of equal importance in setting the possibleaccuracy of the experiments is the time stability ofthe field. Several kinds of instability éxist. First a jitterof the order of 10-5 in relative units. Second a driftof the current of 10-5 in 20 min. Third a drift of upto 10-4 in relative units which is due to the non-

independence of two Bitter coils which are operatedsimultaneously ; the coupling comes from the factthat the cold part of the cooling water circuits arecommon. When the second Bitter magnet goes fromzero to full current, the temperature of the coolingwater in our coil increases by 7 °C, which correspondsto a thermal dilatation of 10-’ and thence a 10-’decrease of the field for constant current.To remedy that, in high precision measurements,

the field was locked onto a NMR magnetometerwhich corrected for the drift (but nor for the jitter)so that the final accuracy was of the order of 10-5in relative units.The principle of the NMR system is the same as the

one used by R. S. Freund and T. A. Miller at the BitterMagnet Facility of MIT [15]. We have only changedthe way of making thé probes and made some modi-fications in order to use it to lock the field on the NMR

signal at any NMR frequency, conveniently, withouttuning or adjustment.The heart of the system is the magic tee 6 (Fig. 1) :

When the two proton probes 7 and 8 are carefullymatched as explained below, the output of the teeis more than 40 dB below the input. When the protonsin the probe resound under the influence of the appliedR.F. 50 mW power, they induce an e.m.f. (withcomponents in phase and quadrature with the appliedR.F.) which propagates back to the magic tee, whereit is not balanced by a signal coming from 8. It is thenamplified by 9 and R.F. phase detected by the mixer 12.If the length of cable 11 in the reference channel is

adjusted so that the electrical lengths

and 5 - 10 --+ 11 - 12 are equal, the mixer detects atany frequency the part of the resonance signal whichis in phase with the applied R.F. power, which is

absorption shaped when one varies the R.F. frequencyat constant field (or vice versa).

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Fig. 1. - NMR set-up. 1) Adret 6100 0.4-600 MHz frequency synthetizer, FM modulated rate 1 kHz, depth 100 kHz for search, 20 kHzfor lock by L.F. generator 3 ; 2) Laboratory made controller : sets and sweeps the frequency of the synthetizer 1 by mean of its extemaldigital frequency control. Sends start and channel advance pulses to the multichannel analyser. Activate 14 to hold the voltage at thé outputof the NMR system during frequency switch to eliminate associated transients; 3) L.F. generator for FM modulation; 4) Wideband(5-500 MHz) amplifier ENI 510-300 mW output; 5) Two ways power divider ; 6) Magic Tee ; 7) Field probe ; 8) Reference probe ; 9) Lownoise, wideband 30 dB gain, 5-500 MHz amplifier; 10) R.F. attenuator; 11) Coaxial cable of length calibrated so that the electric delayduring 5-6-7-6-9-12 and 5-10-11-12 paths are equal ; 12) High level (10 dBm) frequency mixer ; 13) L.F. low noise 1 kHz-10 kHz amplifier ;14) Sample and hold activated by 2 to avoid frequency switch transients ; 15) PAR 124 lock-in detector. The frequency reference is 3 timesthe FM rate of the synthetizer for reasons explained in the text ; 16) Frequency tripler 1 kHz - 3 kHz ; 17) Integrator contained in PAR 124 ;18) SNCI made field control which adds a function of the output of 17 to a manually settable reference voltage ; 19) 1 kHz notch filter ;20) Scope ; 21 ) Box shielded against H. F.

To sweep the resonance one can modulate the field

by a small modulation coil, or sweep the R.F. fre-quency by frequency modulation of the synthetizer.We have preferred the FM because this does not

perturb the anticrossing experiment.For the field lock, one needs an NMR signal disper-

sion shaped as a function of the magnetic field. Theprobe signal is absorption shaped. Modulating theR.F. frequency sinusoidally and detecting at any oddharmonic thus gives a dispersion shaped signal.However at the fundamental frequency there is a bigparasitic signal : Owing to the propagation in the 3 mprobe cable, the residual (non-resonant) signal, notrejected by the magic tee due to imperfections inbalance of the two probes, varies rather rapidly as afunction of frequency ; since this residual is much big-ger than the resonance signal, when one sinusoidallymodulates the R.F. fréquency, the resonance is

superimposed on a sinusoid much bigger than itself.We have thus selected the third harmonic. The result-

ing 3 kHz (Frequency modulation being 1 kHz) lock-in detected signal is amplified by an integrator in sucha way as to cancel the D.C. error of the loop and is thenadded to a reference voltage which drives the 5 MW

field power supply. For the scope observation

however, we just looked at the 1 kHz fundamental

signal rejected by a 1 kHz notch filter 19.,

The probe (Fig. 2) consists of a 4 mm diameter cellfilled with water doped to 0.1 moles/1 of MnS04to give the protons a resonance linewidth of 0.1 to

0.2 G. It is transmitted to the R.F. power by meansof a few turns of copper wire wrapped around it.However this small coil has in itself a cut-off fre-

quency of roughly 100 MHz. To extend this frequencyrange to more than 500 MHz, one tries to transformthe coil to a standard low pass filter (Fig. 2b) closedonto its iterative impedance L/C. For that we use asmall brass 4 mm inner diameter cylinder cut along onegeneratrix to avoid current loops. It is covered by a0.1 mm thick autoadhesive sheet of mylar and thenwrapped with four turns of flat, 0.25 mm wide,0.1 mm thick copper wire. The brass cylinder is

grounded and the copper wire is directly linked to theheart of the R.F. cable. The iterative impedance ofsuch a coil is 75 Q and the impedance of the probe israther flat on the 0-600 MHz frequency range. Thesmall cell is of course inserted inside the brass cylinder.With careful construction, two identical probes may

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Fig. 2. - a) NMR proton probe ; b) Equivalent discrete elementcircuit.

be found to give better than 40 dB insulation of themagic tee without any adjustments. In order to

maintain the symmetry the probes are then embeddedin an epoxy resin so as to maintain their characteris-tics. When more than 40 dB insulation is achievedthe most important noise in our set-up comes from theamplifier 9 (which has a 2.5 dBnF), which is of theorder of the noise of the synthetizer, so that it is

pointless to further increase this insulation.

3. Qualitative analysis. - 3.1 In a magnetic fieldthe eigenenergies of the ’He atom are obtained bydiagonalizing the hamiltonian inside each

subspace. The Zeeman diagram which results isshown in figure 3 for two of them, Mj = 1 (conti-nuous curve) and Mj = 2 (dashed curve).

Notice the two anticrossings (indicated by arrows)occurring one in each Mj subspace which can beinterpreted as due to the spin-orbit coupling

(see Sect. 4.1) between the pairs of states

and

of the decoupled basis (which are ordinarily nearlyeigenstates in high magnetic field). Such anticrossingshave been observed some time ago [15].

Fig. 3. - Energy level diagram of Mj = 1 (full curve) and Mj = 2(dashed curve) of D states of 4He as a function of the magneticfield. Notice the crossings in A and B, and the anticrossings (markedby arrows) induced by the spin-orbit interaction.

Notice also the two crossing points A and B. Aand B can’however be transformed into anticrossingsby applying an external transverse electric field.The symmetry of the system is then destroyed, Mjis no longer a good quantum number and indeèdstates of different Mj become weakly coupled bymixing with F states so that we have the narrowElectric Field Induced Singlet-Triplet Anticrossings [5]mentioned in the introduction.

3.2 In the case of 3He, the existence of a nuclearspin I = 1/2 introduces some new features when

compared with 4He.First of course Mj is no longer a good quantum

number but MF is : MF = Mj + MI, withMI = ± 1/2. Then each state of the decoupled basis ,1 L, ML ; S, MS > becomes a two fold L, ML ; S, MS ;I = 1/2, MI = ± 1/2 ). That enhances the dimensionof the irreducible subspaces of the hamiltonian andmakes possible the occurrence of new kinds of anti-crossing : as can be seen in figure 4, the previousMJ = 1 and Mj = 2 subspaces of 4He are nowsomewhat brought together in the same subspaceMF = + 3/2 ; we see two new anticrossings appear-ing in A and B in place of the previous crossingsbecause two states of the same symmetry cannotcross.

This can also be explained by considering the newterms in the hamiltonian which correspond to the

hyperfine interaction. Of them, only the Fermicontact term is of importance (see Sect. 4.1) andcan be written

with

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Fig. 4. - Energy level diagram of MF = 3/2 D states of 3He as afunction of the magnetic field, with the corresponding anticrossingsignals observed on the fluorescence lines emitted from triplet andsinglet. Three kinds of anticrossings are present :- hyperfine interaction induced anticrossings at about 2 tesla

6n A) ; .

- spin-orbit interaction induced anticrossings marked by uparrows ;- mixed hyperfine-spin-orbit interactions induced 2nd order

anticrossings marked by one down arrow (in B).

We notice a the presence of a singlet-triplet matrixelement. The large magnitude of this matrix elementis responsible for the huge width of the anticrossingsignal observed in A as shown at the bottom of

figure 4. It is also responsible [20] for the relativelylarge hyperfine structure of the 1 D states, of about100 MHz [6].The symmetric part l(Sl + S2) yields the hyper-

’ fine structure of the 3D states. As a consequence,notice the splitting of the spin-orbit induced anti-crossings into two humps corresponding to thesubstates MI = 1/2 and MI = - 1/2, with a magne-tic field interval approximately equal to

The « B type » singlet-triplet , anticrossing is verynarrow. Indeed it can be considered as a 2nd ordercorrection to the energy levels calculated in the

decoupled basis, as there is no matrix element of

the hamiltonian which directly connects the twostates involved

of the decoupled basis.However there exists a 2nd order effective coupling

a Veff P > between ce > and fi > which can beunderstood as [26]

where HJ and HF are the fine and hyperfine interac-tions respectively (see Sect. 4.1) and where i >represents all the states with which fi > and ex > arecoupled : an example of such a state is

for which we have

In the vicinity of the anticrossing we have Ea # Ep

so that we expect to have an effective coupling of theorder of

for any n. An exact diagonalization of the hamil-tonian will indeed show (see Sect. 4.3 and table II)that the eigenlevels considered repell each other froma similar quantity. Also figures 5 to 7 show that theexperimentally observed signals have a width ofabout 4 x 30/,YB - 100 G. Owing to the good signalto noise ratio, a precision of a few MHz on the

intervals is thus expected.Of course a > is not restricted to ML = 1 ;

S = 0, Ms = 0 ; MI = + 1/2 > but four forbiddenanticrossings are expected between the pairs ofsubstates

and

with ML = 1, 0, - 1, - 2, corresponding to the

subspaces MF = 3/2, 1/2, - 1/2, - 3/2 respectively.That is experimentally confirmed as seen in figure 5.

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Fig. 5. - Forbidden anticrossing signals observed on 3D statesof 3He. The experimental curve has been numerically fitted (conti-nuous line) with four lorentzians, whose position and relative

intensity are marked by the vertical bars.

Fig. 6. - Forbidden anticrossing signals observed on 4D statesof 3He ; enlargement of figure 4. As for figure 5 the experimentalcurve has been numerically fitted, but by three lorentzians only.

However the spacing between these four anti-

crossings is fixed essentially by the magnitude of thefine structure interaction, and their width by theeffective coupling Veff. As the latter is roughly inde-pendent of n while the former decreases like n-3, thedifferent components tend to overlap as n increases.

Fig. 7. - Forbidden anticrossing signals observed on 5D statesof 3He. Fit by two lorentzians.

That is completely realized for n = 6 where we cannotresolve any component. It greatly reduces the accuracyand the power of the method and limits its range of

applicability to low excited states.In the next section we discuss the quantitative

analysis of the signals. Due to the large singlet-tripletmixing created by HF, a simplified analysis as wasperformed in the case of ’He is not sufficient for highprecision measurements : entire diagonalization ofthe hamiltonian is needed, with great care given tothe choice of the radial constants involved.

4. Quantitative analysis. - 4. 1 THE 3He HAMIL- TONIAN. - To within a good approximation, thehamiltonian of the He atom can be written as [19,20] :

HLs contains the electrostatic interactions and givesrise to the L singlet and L triplet terms. HJ, the Breithamiltonian gives rise to the magnetic fine structureand contains the interactions such as spin-orbit,spin-other orbit and spin-spin. HF contains the

hyperfine interactions and is present only in the caseof ’He. HB describes the coupling with an appliedstatic magnetic field and contains the ordinary linearZeeman interaction plus the diamagnetic term (some-times called quadratic Zeeman).From previous measurements [15, 5] the singlet-

triplet separation due to HLS is already fairly wellknown in ’He, with an accuracy of 10-3 to 10-4.By use of the Wigner-Eckart theorem, the matrix

elements of Hj and HF restricted to the nD subspaceare expressed as the products of well known angularparts and reduced matrix elements [20].

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Table 1. - Numerical values (in MHz) for the radial integral involved in Breit hamiltonian. Notice the goodagreement between the experimental values and the hydrogenic ones (labelled by the upperscript « th »). In thecomputation of section 4.3 the values used are Aexp, bexp, ath and jËh (except for 3D states where we tooka(dtand X(fx)P).

e) Values deduced from a compilation of references [6-12].(6) Values deduced from Avexp taking into account the singlet-triplet mixing.(C) Deduced from anticrossing signal width (Ref. [15]).(4) This work (see text).(e) Values computed using hydrogen-like wave functions (Ref. [20]). The uncertainties are due to the rounding off of the numerical

values plotted in reference [20].(f ) Deduced from hyperfine structure measurements of 3He (Ref. [16]).(9) Hyperfine structure constant of the 3He+ ground state (Ref. [18]).

4.1.1 Fine structure interaction HJ. - We have [2,19]

with S = S, + S2 the total spin of the atom, andS = 81 - S2.A and b are the fine structure constants which are

usually used to describe the spin-orbit, spin-otherorbit and spin-spin interactions. Experimentally wecan determine these constants from the zero fieldune structure measurements on ’He. If we neglectsinglet-triplet mixing byaL 8, we have [ 15] for the Dstates :

The experimental values are then found in fairlygood agreement with the ones computed assuming thewave functions to be linear combinations of Slaterdeterminants built on the single configuration (ls, nd)where 1 s and nd are hydrogen-like functions withrespectively Z = 2 and Z = 1 for the nuclear

charge [20] (1).a describes the spin-orbit singlet-triplet coupling

and has been previously evaluated experimentally

(1) The theoretical values quoted in table 1 of reference [15]are slightly in error due to errors in equations (4) and (6). In

fact equations (4) and (6) of reference [15] must be replaced byA = zo/2 - 3zJ2 - 3 z2/5 and a = zo/2 + zl/2, respectively.

from the anticrossing signal width [15], and is alsofound to be in agreement with the hydrogen-likevalues of reference [20].

In fact, at the degree of accuracy now reached, thesinglet-triplet mixing (of about 1 % for D states ofHe [15]) is not negligible. In order to take it into

account, we have fitted the fine structure constantsso that by an entire diagonalization of the zero fieldhamiltonian restricted to the nD subspace we recoverexactly the experimental fine structure intervals. Ofcourse, as we have only two intervals, we can fit

only A and b ; a was fixed at the hydrogenic value ofreference [20] given in table I. On the other hand thesinglet-triplet separation A was fixed at the alreadyfairly well known experimental value. It must benoted anyway, that these two latter constants et and Aact in second order perturbation, so that the finestructure constants A and b can be very well knowneven when using relatively approximate values for aand d.

Results are shown in table I. We see that, takinginto account the singlet-triplet mixing, the agreementbetween the experimental and hydrogen-like valuesis extremely good, better than 1 %. Notice that thecorrelation between A and b is considerably strongerthan one would think seeing the error bars, thatis because AV23, the small interval is very sensitive tothe relative values of A and b. If we take one value forA inside the error bar, say Ao + ôA (where Ao marksthe centre of the error bar interval), the correspondingvalue for b must be bo + l5b, with l5b - 8 bA/3(cf. Eq. (5)). If we take a value slightly different for b,the AV23 which would result would be very different.As we will show in the section 4.3, we were able to

determine a (3D) = 650 ± 1 MHz in perfect agree-ment with the hydrogenic value.

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4.1.2 Hyperfine structure interaction HF. - Wehave [2, 19]

Two parts can be distinguished in HF :

. The first term is the dipole-dipole term whichis itself composed of two parts, the short range Fermicontact interaction, and the long range one, herenoted DD.

. The second term is the dipole-orbit term.

Of them the most important is obviously the Fermiterm. Due to the short range interaction of the core 1 selectron with the nuclear dipole, its magnitude is

nearly independent of the quantum number n andshould be very near the hyperfine interaction constantof the 3 He + ground state, which is [18] :

The outer nd electron introduces a slight screening ofthe nuclear charge which might somewhat decreasethat value. However, in the case of 3D levels, a recentexperimental determination [16] of hyperfine structureintervals in 3 3D allows us to give an accurate value (2)for

and justifies our feeling that the 3He+ value should beexact for higher excited nD states of 3He.The long range part DD of the dipole-dipole term

is very cumbersome to calculate ; indeed the angularpart involves products of 2nd order rank tensors

mixed in spin and orbit [21] and the matrix elementscannot be expressed in a straightforward and compactway. In fact they were evaluated with the aid of acomputer using Racah’s formula for the nJ symbols.Our procedure was found exact by predicting valuesfor hyperfine intervals of ’He ’P states in agreementwith those of Hambro [22]. In any case the effect ofthis term is weak, yielding to energy shifts of the orderof a few tenths of MHz, below the experimentaluncertainties. Note in passing that its effect on the

singlet states is exactly zero.The dipole-orbit term is easy to evaluate ; the

radial part ? is directly related to the mean valueof r-3 of the outer nd electron

where gs MB and gN MN are the gyromagnetic ratios

(2) We have derived that value by fitting the zero magnetic fieldenergy levels deduced from an entire diagonalization of

with the experimental ones of reference [16]. The radial constantsinvolved in HLs and Hj were taken as equal to those which fit theaccurate measurements of 4He 3 l,3D levels (see table 1).

of the electron and nucleus respectively. For nD levelsof 3He, we have [19]

4. 1 .3 Interaction with the magnetic field. - Theinteraction with the static magnetic field B can bewritten as

where HL and HQ are respectively linear and quadraticwith respect to B.At our level of accuracy relativistic terms of the

order of (X2 JlB B must be included in HL, which thenslightly differs from the familiar form used previously.The exhaustive list of these additional terms and theirtensorial decomposition is rather cumbersome andcan be found in reference [23]. Fortunately, as nand L increase, it can be easily shown that all theseterms tend to zero except one, which tends to [19, 23]

and is essentially due to the relativistic motion of thecore Is electron. For 3D levels HL is several tens oftimes larger than the other additional terms, so thatwe can finally write :

with gL = 1 - m = 0.999 82 ; where M is the massof the ’He nuclei, and m that of the electron

The diamagnetic term [19, 23]

can be rewritten in the decoupled basis set as

the values for the radial constants D,,(n, L) werecalculated using hydrogenic wave functions so that

and Dzq(n, 2) = 7. 10 x10-12 n2(5 n2 -17) MHz/G 2 fornD states.

C(n, L) is only responsible for a global energy shiftof the entire nL subspace and has not to be takeninto account.

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Table 11. - Experimental results concerning the forbidden anticrossing signals. Deconvolution of the signal shapeand comparison with computation. Only the relative positions of the anticrossings are predicted. The values for thesinglet-triplet separations were adjusted in the computation in such a way that the predicted absolute positionsquoted in that table were in agreement with the experimental ones, thus providing the values listed in table III.Veff is given by half the computed energy difference at the anticrossing point between the levels which anticross ;theoretically the full width at half height of the corresponding anticrossing signal should be 4 VeffpB however for3D states there is overlapping of « forbidden » and « allowed » first order spin-orbit anticrossings, thus the expectedwidth is enhanced (see text).

Although there exists no direct expérimental veri-fication, we have good confidence in these valuessince fine structure constants which involve r-3 meanvalues differ very little from the hydrogenic ones;it must be even better for r’ mean values which areinvolved in Dzq, since these are less sensitive to theshort range deviations which stem from the presenceof the ls core electron.We are now in a position to calculate the energy

levels in a high magnetic field with an accuracy of10-5 relative to the singlet-triplet interval, and thento analyse our experimental signals.

4.2 DATA ANALYSIS. - 4.2.1 The experimentalcurves were treated numerically by fitting their shapewith a sum of several Lorentzian profiles. Indeed, theforbidden anticrossing signals consist of four compo-nents, the shape of each should be very near a Lorent-zian ; several facts however should affect the shape ofthe experimental curves :

. If there were a coherence in the excitation betweenthe two substates of each pair which anticross, thisshould add a dispersion shape to the absorption,

LE JOURNAL DE PHYSIQUE. - T. 41, N° 8, AOÛT 1980

one [1]. But as the two substates of each pair are ofdifferent nuclear spin MI, this cannot be realized byelectron bombardment excitation [27].

. A dispersion part is also present for anticrossingsdue to 2nd order couplings [28]. Physically this stemsfrom the fact that the intermediate states ,i > (secSect. 3) involved in the effective coupling introduce acoherence between the two levels of each anticrossingpair in an effective excitation matrix. But in our casewe generally have (see below and table II) :

where T is the deexcitation rate of the nD levels,going from 6 x 107s-1 for 3D t0 9 x 106 s -1 for 6D.Then the effect should be weak [28].

. On the other hand, three of the four singletsubstates which undergo a forbidden anticrossingalso undergo an allowed first order spin-orbit induced

’ anticrossing with a third triplet substate. For nDstate of n > 4 the centres of the forbidden and allow-ed anticrossings are very well sepàrated, so that we mayneglect the existence of the latter when studying theformer. But that is not the case for 3D states where

55

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they overlap strongly (this is because in that case a isnot small in front of A), and then interfere.For each MF we have three substates :

where ML may have the values 1, 0, - 1.

1 A ) is the singlet substate strongly coupled by thespin-orbit couplinga L 8 to P ), which is a triplet.The 1 B) triplet substate is coupled to 1 A ) by the2nd order effective coupling Veff and undergoes onlya forbidden anticrossing.

If we neglect Veff we have three eigenlevels E+, E_,EB and three corresponding eigenstates :

where 0 varies with the magnetic field B. It is zero or noutside the spin-orbit induced anticrossing region,then the eigenstates are pure triplet or singlet and n/2at the centre, where eigenstates are mixed singlet-triplet.EB always varies linearly with the magnetic field

(linear Zeeman effect) but this is absolutely not thecase for E+ and E- when 0 - n/2 (they follow arcs ofhyperbola). Then, as we treat the forbidden anti-

crossing as occurring between B > and + >, wemust now introduce a new effective couplingV’eff - Veff cos 0/2 which depends on B. Also the

energy difference E+ - EB obviously does not varylinearly with B. These are two causes of distortionfor the corresponding forbidden anticrossing signal.However if Veff a, which is the case, we may

neglect the variation of 0 in the range where

E+ - EB ’" Veff, where the forbidden anticrossingoccurs ; in that limited range Veff may then be consi-dered as constant (V’eff ’" Veff cos 0/2) and

where Bo is the value of the magnetic field whichmarks the centre of the forbidden anticrossing, maybe considered as varying linearly with B. Thén thesignal remains Lorentzian. It should be noted that itsfull width at half height is then given by

4.2.2 For n > 4, some components of the expe-rimental signals strongly overlap (see Figs. 6, 7)and the least square fit procedure was not able torecover the position of each component. Thus thedeconvolution was partially performed by fitting ourexperimental curves with less than 4 Lorentzians.Results are shown in table II. From the width of theindividual components it might be possible to recoverthe experimental values of Veff, the effective coupling

responsible for the forbidden anticrossing. Since inour case we have generally V,,ff » hr, Veff shouldbe given by the relation :

where AB is the full width at half height. Howeverrelaxation processes [29] should affect this relationand anomalously broaden the signal. Also for 3D,this relation should be modified as is explained insection 4.2.1, (Eq. (17)).The very large différences observed in the relative

intensities of the various components seem to be toolarge to be explained by differences in the emissiondiagrams in the direction of observation. In particularthe MF = - 1/2 component of 3D anticrossings. isvery weak (see Fig. 5). Probably there are strongalignments created by the excitation.

4. 3 COMPARISON WITH THEORY AND DERIVATION OFTHE SINGLET-TRIPLET INTERVALS IN ZERO MAGNETIC

FIELD. - We have diagonalized the entire hamiltonianin each MF subspace as a function of the magneticfield. Anticrossing positions are detected on the

components of the eigenstates in the decoupled basisset. At the centre of the anticrossing, two componentsof each of the eigenstates which anticross are equal.This procedure is more sensitive than detecting theminimum distance of the eigenenergies since the varia-tion of the latter is very flat around the anticrossingpoint. On the contrary, at the same time, the compo-nents of the eigenstates vary linearly with the magneticfield. On the other hand, the value of half the distanceof the eigenenergies at the anticrossing point gives Veff(or V’eff, see Sect. 4.2.1), and thus we can predictthe width of the anticrossing signal.We found excellent agreement with the observed

relative positions (magnetic field intervals betweenthe various components) of the different forbiddenanticrossing signals. Strictly speaking, it is true onlyfor the 3D cases, since it is the only one where all thefour components are resolved : nevertheless, for theother cases the partial deconvolution has yieldedresults which although partial are also in perfectagreement with the calculations. See table II.We then adjusted the value of the parameter which

describes the singlet-triplet interval in the hamiltonianso as to recover in the computation exactly theabsolute positions of the observed signals whichprovided the exchange energies in ’He nD states

quoted in table III. It must be recalled that the singlet-triplet intervals were already fairly well known [15, 5](- 1 0/00 to 1 %) so that a variation in that range doesnot affect the relative positions of the forbiddenanticrossings. On the other hand a variation of thefine structure parameters (A, b) inside their error

bars does not yield deviations significantly larger thanthe experimental uncertainties of the relative positionsof the experimental signals. Also the calculated

positions were hardly sensitive to a variation of a

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829

Table III. - Values (in MHz) of the « electrostaticfine structure » in D states of He. By this we mean theinterval between n 1D and n 3D states in the hypotheticalabsence of magnetic fine and hyperfine interactions.In fact configuration interactions induced by the lattermight contribute to these « electrostatic » intervals

(see Sect. 6).

(D) Reference [15] spin-orbit induced singlet-triplet anticrossingsexperiment.

(6) Reference [5] electric field induced singlet-triplet anticrossingexperiment.

(c) Reference [14] microwave spectroscopy measurements.

(singlet-triplet spin-orbit coupling) in a range ofseveral MHz around the hydrogenic values of table I.However, there is an exception for the 3D states. Inthat case, and only in that case, due to the overlappingof forbidden’ and Ist order spin-orbit induced anti-crossings, the relative positions of the former dependon a, and we can even fit the value for a to the experi-mental intervals. We obtained :

which is once again very near the hydrogenic value,and gives us confidence that use of hydrogenic valueswill yield very good results for the higher excitedstates.

In conclusion, our experimental results in highmagnetic field seem to be well explained by thecalculation. Taking B = 0, we are now able to givethe zero field levels quoted in table IV.

5. Error analysis. - Various causes are responsiblefor the uncertainties quoted :

. First there is the magnetic field homogeneityand calibration. We estimate at 10- 5 the correspond-ing relative uncertainty which results.

. Secondly there is some dispersion in the experi-mental signals obtained. Several runs were performedat different conditions of pressure and excitation

(intensity current and electron energy in the electrongun) showing a total dispersion range going from 1to 5 G, depending on the case.

. Finally there is the deconvolution of the signalshape in order to recover the position of the individualcomponents. Here there are two limiting cases. For3D states the components are all well resolved with

Table IV. - Derived zero magnetic field energy levels(in MHz) of 3He nD states deduced from a diagonaliza-tion of Breit hamiltonian within each nD subspaceusing the same parameters as for the computation ofthe position of the forbidden anticrossing signals.In each case, the zero energy level is that of the 3Dterm in the hypothetical absence of fine and hyperfineinteractions.

Estimated uncertainties :

a signal to noise ratio of several tenths (see Fig. 5)so that signal processing yields the position of thecomponents with a mean standard deviation ofabout 1 G. Also the approximative nature of Breithamiltonian as well as the neglect of some verysmall terms (such as hyperfine DD, Stark effect, ...)introduce additional errors which we estimate as beingsome tenths of MHz.On the other hand, for 6D states, no components

can be resolved so that uncertainty about theirabsolute positions is given essentially by their relativeintervals as they are theoretically predicted.

6. Discussion and conclusions. - Obviously as

shown in table III, the precision of our method isbetter than conventional spin-orbit induced anticross-ing and yields improved values for the singlet-tripletinterval in He 3D and 4D states. However, whencompared with other high precision measurementsperformed on ’He 5D and 6D states, we see a syste-matic deviation conceming the electrostatic exchangeenergy. Hence there is a relatively large isotope shift(- 10- 3).

This was first unexpected since exchange energy,responsible for the singlet-triplet interval, involves

exchange integrals essentially to the first and secondorder [24], (terms of higher order contribute onlyabout 1 %) respectively proportional to llai and

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830

1 /a¿ 2 ; a’ is the Bohr radius taking into account thefinite mass of the nucleus and ai rr ao(1 + m/M)where m is the electron mass, M that of the nucleus.The expected mass effect was thus about 1 to 2 times

in relative value, hence 0.5 to 1 x 10-4.However, as was pointed out by Chang and Poe [25]

hyperfine interactions (probably as well as fine struc-ture interactions, although in a slighter proportion)should affect the energy splittings between differentelectronic terms. And in fact, as hyperfine interactionsare not very small in front of the electrostatic finestructure interactions, it is not surprising that the for-mer could produce some configuration interactionswhich should manifest themselves as noticeableadditional shifts in the energy splittings between

singlet and triplet terms.In conclusion the influence of hyperfine interconfi-

guration interactions in a two electron system, ’He,has been experimentally demonstrated. Of coursethat could be more clearly elucidated by having more

data. For that microwave spectroscopy for examplecould be applied to 3He as was done [14] in the caseof ’He Rydberg states of quantum number n > 6.Another interesting result is the measurement of

the singlet-triplet spin-orbit coupling constant of 3Dstates with an accuracy not previously achievedfor such a non-diagonal parameter. The result

(a (3D) = 650 ± 1 MHz) is in agreement with the

hydrogenic value. More extensive results concerningthese coupling constants can be experimentally obtain-ed by magnetic resonance experiments performedin the vicinity of the spin-orbit induced anticross-ing [32, 33].

Acknowledgments. - This work has been performedat the Bitter magnet facility of the Service Nationaldes Champs Intenses at Grenoble.We are greatly indebted to Drs. T. A. Miller and

R. S. Freund for the communications of the detailsof their NMR systems, to M. Molé who built themultichannel analyser, to M. Aventurier who builtthe sweep generator and the multiplexer, to M. Sim-plice who built the synthetizer controller and to

M. Castejon who realized the triode system cell.

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