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Forbidden singlet-triplet anticrossings in 3He : precise determination of n1D-n3D (n = 3-6) intervals
J. Derouard, M. Lombardi, R. Jost
To cite this version:
J. Derouard, M. Lombardi, R. Jost. Forbidden singlet-triplet anticrossings in 3He : precise de- termination of n1D-n3D (n = 3-6) intervals. Journal de Physique, 1980, 41 (8), pp.819-830.
�10.1051/jphys:01980004108081900�. �jpa-00209304�
Forbidden singlet-triplet anticrossings in 3He : precise determination of
n1D-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 des
interactions de structure fine et hyperfine, ils n’ont pas d’équivalent dans 4He. Des mesures de grande précision
sont rendues possibles grâce à l’utilisation d’une bobine de Bitter de grande homogénéité pilotée par RMN. Les
signaux expérimentaux sont en parfait accord avec les prédictions déduites d’une diagonalisation complète du
hamiltonien de Breit à l’intérieur de chaque configuration (Is, nd) de 3He; les valeurs utilisées pour les constantes radiales sont déduites de résultats expérimentaux antérieurs portant sur les intervalles de structure fine et hyperfine
de 4He et 3He obtenus par différents auteurs; on trouve que ces valeurs diffèrent de moins de 1 % de celles calculées à 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 états nD (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 la grandeur est inattendue, est, pensons-nous, dû à une faible interaction de configuration induite par l’interaction
hyperfine.
Abstract. 2014 We have studied narrow anticrossing signals in 3He which have no equivalent in 4He because they
are due to the conjonction of fine and hyperfine interactions. High precision results are obtained by the use of a high homogeneity Bitter coil driven by NMR. The experimental signals obtained are in perfect agreement with the predictions 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 results
obtained by various authors on 4He and 3He. These values are found to differ from the hydrogenic ones by less
than 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 of
up to 5 x 10-5 in relative value. An unexpectedly large ( ~ 10- 3), negative isotope shift is found compared to the equivalent 4He values, presumably due to a slight configuration interaction induced by a hyperfine interaction.
Classification
Physics Abstracts
32.80B - 32.60
1. Introduction. - Anticrossing phenomena [1]
have been extensively used in the past to measure intervals 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 and triplet), weakly coupled by a perturbation, (e.g.
spin-orbit) are tuned near degeneracy by the magnetic field; the resonant mixing of the wave functions
which results is observed as a variation of the intensity
of the light emitted by each level. This variation has a
Lorentzian 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 anticrossing
is very weak. That has been achieved in the D states of
’He by means of Electric Field Induced Singlet-Triplet Anticrossings [5]. The coupling between 1 D and 3D
states occurs via mixing with F states and has provided experimental values for the singlet-triplet intervals
which are more accurate, typically a few 10- 4 instead
of a few 10-3 than the ones derived from conventional
singlet-triplet anticrossings. Another situation has been reported in the case of H2 [4].
In 3He, the existence of nuclear spin induces some
narrow 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
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 spectroscopic
measurements already exist on D levels of ’He. Apart
from some determinations of the hyperfine structure of nID states [6], and one interval in the n 3D struc- ture [6], an accurate measurement of some intervals in the 3 3D states [16, 17] was achieved only very recently.
However, when we started our experiments, our
aim was to improve the experimental values of the
singlet-triplet intervals of D levels, the exchange
energy in He. In fact the comparison with other accurate measurements already existing for 5D and
6D states in ’He showed a systematic difference.
There exists a relatively large isotope shift (about 10-3). 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 of
reference [15]; the intensity and polarization of lines
emitted by anticrossing states, singlet or triplet are
recorded as a function of the magnetic field.
As in reference [15], the observation of n 1- 3D
anticrossings for n 5 requires magnetic fields of
several tesla which cannot be attained by classical electromagnets. Thus the experiments were carried
out in a Bitter coil at the Service National des Champs
Intenses (C.N.R.S. Grenoble).
3He is contained in a sealed Pyrex cell and is
excited by a triod system [30]. The measurements were
made at two pressures, 10 mtorr and 100 mtorr. The
electrons were produced by an indirectly heated
cathode (S - 1 cm’) and accelerated parallel to the magnetic field at 30 to 40 eV energy by a grid placed at
about 2 mm from the cathode ; the resulting current
collected at the anode was 15 to 25 mA. The anode
was at the same potential as the grid and 3He emission spectra were taken in the grid anode space which is about 1 cm long. Some space charge electric fields
should however exist and induce slight Stark shifts ; however, as specified in section 5, several runs were performed in different conditions of pressure, current
intensity and voltage, and the corresponding dis- persion of results was found small in face of the other
sources of uncertainty.
Light emitted by the He atoms perpendicularly
to the magnetic field, reflected by a small mirror
placed at 450 was collected by a 2 m fused silica light pipe which was fed into a polychromator built in our laboratory [31] from a Jobin-Yvon HRS 2 monochro- mator. Singlet and triplet lines corresponding to the
same (ls, nd) configuration could be then recorded
simultaneously. The light was detected by thermoelec-
trically cooled and magnetically shielded Hamamatsu R 268, R 269 and R 374 photomultipliers. The signal
was digitized by a DANA 160 000 points voltmeter
and fed into a multichannel analyser also built in
our laboratory. A laboratory-made multiplexer was
used to store simultaneously the two signals, singlet
and triplet, on adjacent channels of the multichannel
analyser and restitute them during readout. An expe- rimental curve consisted of 200 points accumulated in 10 to 30 passes of 40 s each.
2.2 FIELD CONTROL AND SWEEP. - For large (and
not too precise) magnetic sweeps, a (also laboratory- made) step generator was used to drive the field and
trigger the multichannel analyser.
The magnetic field was produced by a 5 MW Bitter
coil which provided a field of up to 13 tesla with a
homogeneity of about ± 10-5 in a sphere of 8 mm
diameter. Of equal importance in setting the possible
accuracy of the experiments is the time stability of
the field. Several kinds of instability éxist. First a jitter
of the order of 10-5 in relative units. Second a drift of the current of 10-5 in 20 min. Third a drift of up to 10-4 in relative units which is due to the non-
independence of two Bitter coils which are operated simultaneously ; the coupling comes from the fact
that the cold part of the cooling water circuits are
common. When the second Bitter magnet goes from
zero to full current, the temperature of the cooling
water in our coil increases by 7 °C, which corresponds
to 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 magnetometer which corrected for the drift (but nor for the jitter)
so that the final accuracy was of the order of 10-5 in 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 Bitter
Magnet Facility of MIT [15]. We have only changed
the 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, without tuning or adjustment.
The heart of the system is the magic tee 6 (Fig. 1) :
When the two proton probes 7 and 8 are carefully
matched as explained below, the output of the tee
is more than 40 dB below the input. When the protons
in the probe resound under the influence of the applied
R.F. 50 mW power, they induce an e.m.f. (with components in phase and quadrature with the applied R.F.) which propagates back to the magic tee, where it is not balanced by a signal coming from 8. It is then amplified 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 at any frequency the part of the resonance signal which
is in phase with the applied R.F. power, which is
absorption shaped when one varies the R.F. frequency
at constant field (or vice versa).
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 kHz for lock by L.F. generator 3 ; 2) Laboratory made controller : sets and sweeps the frequency of the synthetizer 1 by mean of its extemal
digital frequency control. Sends start and channel advance pulses to the multichannel analyser. Activate 14 to hold the voltage at thé output of 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) Low noise, wideband 30 dB gain, 5-500 MHz amplifier; 10) R.F. attenuator; 11) Coaxial cable of length calibrated so that the electric delay during 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 times the 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. The
probe signal is absorption shaped. Modulating the
R.F. frequency sinusoidally and detecting at any odd harmonic thus gives a dispersion shaped signal.
However at the fundamental frequency there is a big parasitic signal : Owing to the propagation in the 3 m probe cable, the residual (non-resonant) signal, not rejected by the magic tee due to imperfections in
balance of the two probes, varies rather rapidly as a
function of frequency ; since this residual is much big-
ger than the resonance signal, when one sinusoidally
modulates 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 such
a way as to cancel the D.C. error of the loop and is then
added 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 cell filled with water doped to 0.1 moles/1 of MnS04
to give the protons a resonance linewidth of 0.1 to 0.2 G. It is transmitted to the R.F. power by means
of 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 frequency
range to more than 500 MHz, one tries to transform the coil to a standard low pass filter (Fig. 2b) closed
onto its iterative impedance L/C. For that we use a
small brass 4 mm inner diameter cylinder cut along one generatrix to avoid current loops. It is covered by a
0.1 mm thick autoadhesive sheet of mylar and then wrapped 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 the
heart of the R.F. cable. The iterative impedance of
such a coil is 75 Q and the impedance of the probe is
rather flat on the 0-600 MHz frequency range. The small cell is of course inserted inside the brass cylinder.
With careful construction, two identical probes may
Fig. 2. - a) NMR proton probe ; b) Equivalent discrete element circuit.
be found to give better than 40 dB insulation of the
magic tee without any adjustments. In order to
maintain the symmetry the probes are then embedded in an epoxy resin so as to maintain their characteris- tics. When more than 40 dB insulation is achieved the most important noise in our set-up comes from the
amplifier 9 (which has a 2.5 dBnF), which is of the order of the noise of the synthetizer, so that it is pointless to further increase this insulation.
3. Qualitative analysis. - 3.1 In a magnetic field
the eigenenergies of the ’He atom are obtained by diagonalizing the hamiltonian inside each
subspace. The Zeeman diagram which results is shown 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 be interpreted as due to the spin-orbit coupling
(see Sect. 4.1) between the pairs of states
and
of the decoupled basis (which are ordinarily nearly eigenstates in high magnetic field). Such anticrossings
have 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 magnetic field. Notice the crossings in A and B, and the anticrossings (marked by arrows) induced by the spin-orbit interaction.
Notice also the two crossing points A and B. A
and B can’however be transformed into anticrossings by applying an external transverse electric field.
The symmetry of the system is then destroyed, Mj
is no longer a good quantum number and indeèd states of different Mj become weakly coupled by mixing with F states so that we have the narrow
Electric Field Induced Singlet-Triplet Anticrossings [5]
mentioned in the introduction.
3.2 In the case of 3He, the existence of a nuclear
spin 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, with
MI = ± 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 dimension of the irreducible subspaces of the hamiltonian and makes possible the occurrence of new kinds of anti-
crossing : as can be seen in figure 4, the previous MJ = 1 and Mj = 2 subspaces of 4He are now
somewhat brought together in the same subspace MF = + 3/2 ; we see two new anticrossings appear-
ing in A and B in place of the previous crossings
because two states of the same symmetry cannot
cross.
This can also be explained by considering the new
terms in the hamiltonian which correspond to the hyperfine interaction. Of them, only the Fermi
contact term is of importance (see Sect. 4.1) and
can be written
with
Fig. 4. - Energy level diagram of MF = 3/2 D states of 3He as a function of the magnetic field, with the corresponding anticrossing signals observed on the fluorescence lines emitted from triplet and singlet. Three kinds of anticrossings are present :
- hyperfine interaction induced anticrossings at about 2 tesla
6n A) ; .
- spin-orbit interaction induced anticrossings marked by up arrows ;
- 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 matrix
element. The large magnitude of this matrix element is responsible for the huge width of the anticrossing signal observed in A as shown at the bottom of
figure 4. It is also responsible [20] for the relatively large hyperfine structure of the 1 D states, of about 100 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 the
substates MI = 1/2 and MI = - 1/2, with a magne- tic field interval approximately equal to
The « B type » singlet-triplet , anticrossing is very
narrow. Indeed it can be considered as a 2nd order correction to the energy levels calculated in the
decoupled basis, as there is no matrix element of
the hamiltonian which directly connects the two
states involved
of the decoupled basis.
However there exists a 2nd order effective coupling a Veff P > between ce > and fi > which can be
understood 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 > are coupled : 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 the
order 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 from
a similar quantity. Also figures 5 to 7 show that the
experimentally observed signals have a width of
about 4 x 30/,YB - 100 G. Owing to the good signal
to 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 forbidden
anticrossings are expected between the pairs of
substates
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.
