Fine structure of the lowest triplet states in He2Martin Kristensen and Nis Bjerre Citation: The Journal of Chemical Physics 93, 983 (1990); doi: 10.1063/1.459125 View online: http://dx.doi.org/10.1063/1.459125 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/93/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Potential energy surfaces for the lowlying 2 A‘ and 2 A’ States of HO2: Use of the diatomics in moleculesmodel to fit ab initio data J. Chem. Phys. 102, 1994 (1995); 10.1063/1.468765 Deexcitation of He(21 P) in collisions with diatomic molecules J. Chem. Phys. 97, 3180 (1992); 10.1063/1.463957 OODR fluorescence and polarization spectroscopy of K2: Rydberg states and the A1Σ+ u state AIP Conf. Proc. 191, 572 (1989); 10.1063/1.38632 Numerical multiconfiguration selfconsistentfield study of the hyperfine structure in the infrared spectrum of3He4He+ J. Chem. Phys. 90, 4392 (1989); 10.1063/1.456625 The electronic and vibrational energies of two doublewelled 3Σ+ u states of He2 J. Chem. Phys. 87, 4000 (1987); 10.1063/1.452903

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Fine structure of the lowest triplet states in He2

Martin Kristensen and Nis Bjerre Institute of Physics, University of Aarhus, DK-BOOO Aarhus C. Denmark

(Received 6 February 1990; accepted 6 April 1990)

Laser driven transitions from the metastable a 3~u+ state to the predissociating level v' = 4 of the electronic state c 3~t ofHe2 are investigated by fast beam laser spectroscopy. For transitions to the rotational levels N' = 8, 10, and 12 ofthe c 3~g+ state, the fine structure is resolved and the fine structure parameters of both electronic states can be determined. The fine structure in the rotational levels N" = 7, 9, and 11 of v" = 0 in the a 3~u+ state is measured to 20 kHz precision (30-) using laser-radio frequency double-resonance spectroscopy. These measurements are complementary tathe previous radio frequency (rf) measurements for the levels N" = 1, 3, and 5 by Lichten et al. The fine structure parameters are compared to theoretical calculations and simple models for the electronic interactions and the couplings between different electronic states.

INTRODUCTION

Since the first observation of optical transitions among excited states of He2 , 1,2 many papers have been published dealing with this molecule. The optical spectroscopy has been investigated extensively by Ginter and others. 3

-6 Very precise radio frequency spectroscopy of the fine structure in the three lowest rotational levels of the metastable a 3~ +

state has been performed by Lichten et al. 7-9 Extensive ref:r­ences to the previous literature on He2 can be found in Ref. 7.

The basis for the present experiment is the observation by fast neutral beam laser spectroscopy10 that the two high­est vibrational levels of the c 3~t state predissociate by tun­neling through a potential barrier11 as illustrated in Fig. 1. The lowest dissociating levels, N' = 8, 10, and 12 of v' = 4 have a lifetime sufficient for the fine structure to be resolved in fast neutral beam laser spectroscopy, whereas the higher levels are homogeneously broadened due to rapid barrier tunneling. The fine structure of the lower states a 3~u+ v" = 0, N" = 7, 9, and 11 is measured to 20 kHz precision (30-) using a fast neutral beam photofragment version of laser-radio frequency double-resonance spectroscopy.12-14 Our measurements extend the previous radio frequency' measurements of the low rotational levels by Lichten et al. When combined, the two data sets allow an exceedingly ac­curate determination of the fine-structure parameters for v" = 0 of the a 3~: state. The analysis requires the addition of higher-order centrifugal terms to the fine-structure Ham­iltonian.

For v' = 4 of the c 3~g+ state, less accurate fine-struc­ture parameters are derived from the laser spectra. Recently, Yarkony et al. 15

•16 made accurate theoretical calculations

for several electronic states of He2 , including the relativistic coupling matrix elements between the different states. These results are used in the present paper for making comparisons to the experimental fine-structure parameters of the estate.

EXPERIMENTAL

The experimental setup is shown schematically in Fig. 2. He/ ions were produced in a hot filament ion source from

a discharge in pure He gas at a pressure of a few Torr. The amount of beam, which could be extracted from the 0.3 mm hole in the anode of the ion source, was very dependent on the discharge conditions. The output was strongly favored by high pressure, and the limitation in beam current was determined by the leaks in the ion source and the pumping capacity of the system.

The He/ ions were accelerated to 1.2 keV, focused, mass analyzed, and finally charge exchanged in Cs vapor. Remaining ions were bent away by a static electric field. This part of the setup was described previously. 11,17 Typical neu­tral beam currents were about 2 X 108 He2 molecules per second.

After the charge exchange, the neutral He2 molecules passed into a 54 em long differentially pumped drift region, where the beam from a single frequency continuous-wave (cw) ring dye laser counterpropagating with the He2 beam could selectively deplete one specific fine structure level in the a 3~: state by exciting the population to a predissociat­ing level in the c 3~g+ , v' = 4 state. After the pumping zone, the molecules entered a 53 cm long rfsection where the pre-

25000

20000

"i E 15000 2 > c 3[g (!) a:: 10000 UJ z UJ

b3ng

5000

00 2 3 4 5

R (a.u.)

FIG. 1. Potential curves for the three lowest triplet states ofHe2 • Data from Ref. 16.

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984 M. Kristensen and N. Sjerre: Lowest triplet states in He2

ACCELERATION 1.2keV

I DEFLECTION PLATES

FIG. 2. Experimental setup used for rf-optical double-resonance spectroscopy.

viously depleted fine structure level could be repopulated in magnetic dipole transitions from the other fine structure lev­els of the same rovibrational state. Finally, the population in the fine structure level was probed by photodissociation in a 90 cm long URV probe region. Photofragments induced by the laser were deflected out of the beam due to the energy release in the fragmentation process and were detected by a ceramic channeltron detector covering from 7 to 16 mm off the beam axis in the horizontal plane.

The laser wavelengths were measured with a phase­locked wave meter with a Zeeman stabilized ReNe reference laser18 calibrated against saturated absorptions in 12.19 All wavelengths were corrected for the dispersion of air.20 The laboratory laser frequencies were measured with both co­(v _ ) and counterpropagating (v + ) laserlbeam configura­tion. This determines the rest frame frequency

vo = [v + v _ ] 112

and the beam velocity

(1)

(2)

The measured radio frequencies were Doppler corrected us­ing the Doppler factor determined from the laser measure­ments.

The rf section was constructed according to a design by Sen, Goodman, and Childs;21 a cross section is shown in Fig. 3. On the outside is a shield against external magnetic fields made from 1 mm thick annealed fl metal. Inside this shield is placed a tri-plate strip line conductor carefully matched to 50 n impedance. The microwaves were coupled in from the side via a slightly modified commercial SMA feedthrough. On the other side of the central conductor the molecular beam passed through collinearly with the microwaves. The radio frequency was generated in an Ailtech synthesizer and amplified to 1 W in a broadband amplifier. At the end ofthe strip line conductor, the rf was dumped in a 50 n termi­nation. The whole system was optimized to minimum reflec­tions and losses. The setup was tested from direct current (dc) to 4 GRz using directional couplers. The design en-

~-I4ETAL SHIELD

sures that the microwaves propagate in a pure transverse­magnetic (TEM) mode with small reflections. As the strip line was made from Cu with polished surfaces, for which the skin depth is very small at these frequencies, and since no dielectric material was present in the interaction region, the propagation velocity was equal to the velocity of light in

23

9,3

10 M

FIG. 3. Cross section of the rf section. All sizes are in millimeters.

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M. Kristensen and N. Sjerre: Lowest triplet states in He2 985

vacuum. Compared to similar rf sections designed as coaxial lines, the strip line construction has a much more homoge­neous magnetic field ( 1 % inhomogeneity over typical beam diameters), less reflections, and a higher magnetic field strength. With 1 W ofrfpower afield strength of92 mGwas achieved at the position of the beam. These improvements typically increase the signal with more than an order of mag­nitude for the same input power.

To avoid ac Stark shifts from the laser in the rf section, the laser beam was chopped by an acousto-optic modulator such that the laser was on while the molecules were travers­ing the pumping region, off during the passage through the rf section, and finally on again in the detection region. The signal from the ceramic channeltron detector was gated ac­cordingly using a "photon counter" from Stanford Research Systems (SRS). A dark period was included for background subtraction.

The He2 beam energy was chosen as low as possible for four reasons:

( 1 ) The low-energy fragments produced during the pre­dissociation of the c 3~t, v' = 4 state need a long time to separate so much from the beam that they can be detected.

(2) A low beam energy increases the interaction time in the rf section. This reduces the transit time limited linewidth and increases the transition probability.

(3) As the op.tical transitions used for pumping and probing the fine structure levels are of moderate strength (av = 4), the longer times spent in the pump and probe regions increases the efficiency.

( 4) The longer time the molecules spend traveling from the charge exchange cell to the observation region, the more ofthe long-lived Bing state population will have decayed. This reduces the background from spontaneous decay.22 This point turned out to be very important to achieve a good signal-to-noise ratio.

Before the double-resonance experiments were con­ducted, the relevant optical spectra were recorded. Only lev­els in the c 3~g+ , V = 4 state with N'> 8 dissociate sufficiently by barrier tunneling to be observed with the present tech­nique. However, for N' > 12, the upper state is lifetime broadened so much that the fine structure is no longer re­solved.

Figure 4 shows a 4 GHz laser scan of the P 11 branch of the a (v" = 0) -+C (v' = 4) transition. The figure directly illustrates the signal-to-noise ratio since no background is subtracted. We note that the strong t:.J = aN transitions are poorly resolved while the weak aJ = 0 transitions are well separated from the other lines. Therefore, the t:.J = 0 transi­tions are very important for the determination of the fine structure of both levels. The position of the five fine-struc­ture components were determined by Lorentzian fits to the laser spectra.

The relative intensities of the fine structure components depend on the polarization of the laser relative to the direc­tion of the off-axis detector. For large values of J the angular dependence of the intensities is approximately given by23

t:.J = aN: I( e) a: 1 + cos2 e, (3)

t:.J = 0: I( e) a: sin2 e,

where e is the angle between the E vector of the laser field and the direction of the photofragments. To enhance the weak t:.J = 0 transitions we used vertical laser polarization for the laser scans. Optical pumping was avoided by misa­ligning the laser less than 1 mrad which gives a negligible frequency error.

For the rf double-resonance experiments we used the strong t:.J = aN transitions for optical pumping. The signal was optimized by using horizontal polarization of the laser

4000r----------.----------r----------r--------~

lJ) I-

3000

~ 2000 o u

01 I

o

I

2

11-10 112-11

°O~----------~--------~2~--------~3--------~4

LASER FREQUENCY (GHz)

J. Chern. Phys., Vol. 93, No.2, 15 July 1990

FIG. 4. Laser spectrum of the a--c (v' = 0, r! = 4) P 11 transition. The curve is a Lor­entzian fit to the spectrum. The numbers above the curve indicate J' -J'. The num­bers below the curve are the labels used in the text and in Tables I and II.

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986 M. Kristensen and N. Sjerre: Lowest triplet states in He2

light. All double-resonance spectra were recorded with 1 W of rf power, which was calculated to give optimum signal. The resonance frequencies were determined from fits to the rf scans. The rf line shape is determined by the transit time t and the Rabi frequency OR according to the expression24

0 2

P(t a) = R sin2(1 t [a2 + 0 2 ] 112) (4) , a2 + O~ 2 R'

where a = 21T( v - vo ) is the detuning from the resonance frequency Vo' The transit time was fixed at the value 3.124 f-tS

derived from the Doppler shifts.

RESULTS AND DISCUSSION

Fine-structure Hamiltonian

The fine structure in the two 3l: states can be described by the rotational energy plus the spin-spin and spin-rotation interactions:7

H = BvR2 - DvR4 + HvR6 + Ev (3S; - S2) + rvS·N, (5)

where Bv is the rotational constant and Dv and Hv are the centrifugal constants. Ev is the spin-spin coupling constant and rv is the spin-rotation coupling constant. Ev is often substituted by A.v defined as

A.v = (3/2)Ev' (6)

The relevant angular momenta are: R the rotational an­gular momentum, S the total electronic spin angular mo­mentum, L the electronic orbital angular momentum, N the total angular momentum except for spin, and J the total angular momentum including spin.

These angular momenta are coupled according to

J=L+S+R=N +S,

N=J-S=L+R. (7)

The fine structure terms in the Hamiltonian are much smaller than the rotational splittings. Therefore, the fine structure is most clearly described in Hund's case (b). Fig­ure 5 gives a schematic representation of the fine structure of the a 3l:: state. For reasons of nuclear symmetry N assumes odd values only. Each rotational level characterized by N splits into three fine structure components with J = Nand J = N ± 1. A given odd value of J appears only once with J = N. Thus, the Hamiltonian has only diagonal matrix ele­ments in Hund's case (b) for odd values of J. This is the case for J = 0 as well. The even values of J: J = N ± 1 occur twice-the same value of J appears in two adjacent rota­tionallevels. Therefore, the Hamiltonian in Hund's case (b) has off-diagonal matrix elements that couple the rotational levels. These have been treated by Schlapp2s by including only the Bv term in the energy difference of the two rota­tional states. Here we use the full Hamiltonian [Eq. (5)] and obtain for the energies of the fine structure components relative to the odd J = N component:

EI (J = N + I) = (2N + 3)Blv - A.v

- [(2N+3)2Biv +A.~ -UvBlv]1I2+ (N+ I)rv'

E2(J=N-I) = - (2N-I)B2v -A.v

--- J=N+l --- J=N+3

--- J=N+2

---J=N-l ---J=N+l

N ---J=N

--- J=N-3 --- J=N-l

N-2 --- J=N-2

FIG. 5. Schematic diagram of the fine structure levels in the a 3~: state.

+ [(2N-l)2B~v +A.~ -UvB2V]1I2-Nrv' (8)

E3 (J=N) =0.

These formulas correspond to Schlapp's result2s except for two important details: (1) !rv is subtracted from the Bv con­stant. This has already been discussed by several authors.26

(2) The Bv constant has been replaced by two effective Bv constants B lv and B2v ' These constants are given by

B lv = Bv - 2Dv(N2 + 3N + 3)

+ Hv (3N 4 + 18N 3 + 49N2 + 66N + 36) - !rv,

B2v = Bv - 2Dv (N 2 - N + 1)

+Hv(3N4 -6N 3 + 13N2-lON+4) -!rv' (9)

Here the expression for B2v can be obtained by substituting (N - 2) for N in the expression for B I v' In order to describe the very accurate rf measurements it is necessary to include centrifugal terms for the A. and r constants. They can be included in the Hamiltonian by adding the terms

2/3(3S; - S2) [A.I R2 + A.2R4] + rl S·N R2 (10)

to the fine-structure Hamiltonian [Eq. (5)]. Here we only use the first-order contributions in Hund's case (b) by sub­stituting for A.v and rv:

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M. Kristensen and N. Sjerre: Lowest triplet states in He2 987

A(N) =Ao +AIN(N+ 1) +A2 N 2(N+ 1)2,

yeN) = Yo + yIN(N + 1). (11)

Then we can calculate A(N) and yeN) for each rotational level and subsequently fit the resulting values to Eq. (11).

The fine structure of the c 3~g+ state can be treated simi­larly except for the fact that now only even values of N are allowed by symmetry. For the less accurate laser measure­ments in the c state the second-order centrifugal constant A2 can be omitted.

Spectroscopic results

All the recorded laser spectra consist of five fine-struc­ture components. We label those by the numbers 1 through 5 starting at the component at lowest laser frequency. Using both the P and R branches we obtain the unambiguous as­signments of the five components given in Table I. Transi­tions 1 and 5 are weak t:J = 0 transitions. Transitions 2-4 are strong t:J = aN transitions. Transition 2 is almost re­solved whereas 3 and 4 overlap. The t:J = - aN transition is calculated to be two orders of magnitude weaker than the t:J = aN transitions. Furthermore, it is hidden below the strong transitions which explains why it is not observed in any of our spectra.

The assignments in Table I give the order of the fine­structure components of both electronic states and conse­quently also the signs of the fine structure constants A and y. For the a- state the present measurements confirm the signs determined by Vierima.8 The measured laser frequencies are listed in Table II.

In the laser-rf double-resonance experiments it is possi­ble to observe magnetic dipole transitions with aJ H = 1. For each N" level there are two rf transitions:

VI =EI: J" =NH~J" =N" + 1,

V 2 =Ez : J" =N"~J" =N" -1, (12)

where EI and E2 are the energies given in Eq. (8). For pumping and probing the rf transitions, the strong laser tran­sitions 2 and 4 were used. Transition 2 gives the better signal­to-noise ratio because the overlap with other laser transitions is small. Figure 6 shows one of the rf resonances pumped on a laser transition of type 2. Unfortunately, laser transition 2 can be used only for measuring V 2 because the lower level is J H = N" - 1 for both Pand R transitions. Therefore, the VI

transitions were measured with optical pumping on transi­tion 4, giving a considerably worse signal-to-noise ratio. Ac-

TABLE I. Assignments of the fine structure components in the laser spec· tra. The components are numbered in order of increasing laser frequency.

Pbranch R branch

Component J" AJ J" AJ

I N"-I 0 N"+I 0 2 N"-I -I N"-I I 3 N" -I N" I 4 N"+I -I N"+I I 5 N" 0 N" 0

cumulation times on the order of an hour were needed for these measurements.

Table III gives a list of all the measured rf frequencies together with the Rabi frequencies extracted from the fits. The Rabi frequencies can be calculated theoretically by anal­ogy with the case of magnetic dipole transitions between hy­perfine structure levels in an atom:24 Instead of coupling J and I to give F, Sand N should now be coupled to give J. This gives for the Rabi frequencies

.n~ = (1/3 )(IlBgsHo/li)z(2J' + 1)(2S + l)(S + 1)

{J 1 J'}2 xS S N S ' (13 )

where}' is the larger Jvalue,IlB the Bohrmagneton, andHo the magnetic field strength. gs is the equivalent of the atomic Lande g factor arising from the coupling of the electronic orbital angular momentum and spin. In a ~ state we expect gs ::::;2. For the V 2 rf transitions we find the predicted value for gs within the experimental accuracy. For the VI transi­tions, however, there are systematic deviations: The gs val­ues increase with N" reaching as high as 2.7 for N H = 11. The reason for this is presently unknown. Influence from the nuclear rotation is expected to be too small to affect the Rabi frequency to any measurable extend.

Fine structure parameters

The fine structure parameters for the a 3~: state listed in Table IV were determined from a fit to all our rf measure­ments plus those of Lichten et al. 7

•9 which are nearly three

times more precise. The fine structure parameters A (N) and yeN) were determined for the individual rotational levels using Eqs. (8) and (9) with rotational constants from Ref. 6. They were then weighted according to their individual precision in polynomium fits to Eqs. (11). In these fits, the

TABLE II. Transition wave numbers (cm -') for the a 3I.u+ -c 3I.g+, v" = O-v' = 4 transitions.·

R7 P9 R9 Pll Rll P13

v, 16041.265 15785.995 IS 982.461 15668.896 15902.714 15531.954

V 2 16041.290 15786.027 15982.486 15668.928 15902.740 15531.984

V3 16041.297 15786.034 15982.494 15668.934 15902.748 15531.991

v. 16041.299 15786.036 15982.496 15668.937 15902.753 15531.997

v, 16041.330 15786.069 15982.525 15668.970 15902.780 15532.030

"The standard deviation on the wave numbers is 0.001 em - '.

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988 M. Kristensen and N. Sjerre: Lowest triplet states in He2

7000

~ 6500 z ::> o u

55001149.5 1150.5 1151.5 1152.5

FIG. 6. Radio frequency-optical double­resonance spectrum ofthe V 2 rf transition for N H = II optically pumped and probed on the P II transition component 2. The curve is a fit to Eq. (4).

MICROWAVE FREQUENCY (MHz)

low rotational levels from Refs. 7 and 9 lead to an accurate determination of the zeroth-order parameters Ao and Yo while the present measurements for high rotational levels give precision to the centrifugal constants AI' A2 , and YI. Both data sets are represented by these parameters to within the experimental accuracy.

With the a state energy levels completely determined, we then used the laser spectra to derive the fine structure parameters of the c 3I.g+ state also given in Table IV. The necessary rotational and centrifugal parameters were taken from Ref. 3. The constants for the c state are not as accurate as those for the a state because of the Doppler limited resolu­tion ( - 80 MHz) of the laser spectra.

The fine-structure parameters represent the experimen­tal data for the individual states but their values may reflect second-order perturbations from other electronic states as well as first-order contributions from the state in question. Experimentally it is impossible to separate the various con­tributions to the molecular parameters but we can estimate their magnitudes theoretically.

For the a 3I. u+ state, the first-order contribution to Ao

TABLE III. Radio frequency transition frequencies in the a 31..+ state.

N Transition Vo (MHz) OR (MHz) K..

7 VI 1004.886 (21)' 0.88 (45") 1.9 (10)" 9 VI 1006.217 (18) \.18 (18) 2.5 (4)

11 VI 1002.167 (21) 1.27 (\5) 2.7 (4) 7 V2 1188.718 (18) 1.03 (27) 2.1 (5) 9 V2 1166.449 (9) 0.99 (15) 2.0 (3)

11 V2 1150.073 (12) 0.99 (21) 2.0 (4)

a Numbers in parentheses indicate three standard deviations on the last digit.

has been calculated ab initio to be - 1226 MHz27 which has some 10% discrepancy with our present experimental value of - 1099.170 MHz. For the spin-rotation parameter Yo, Tinkham and Strandberg26 estimate the first-order contri­bution by considering the effect on an electron from the mag­netic field of the rotating nuclei:

r' = 2 ZILBILN MR3 '

(14)

where Z and M are the charge and mass (in amu) of the nucleus, ILB andlLN are the Bohr magneton and the nuclear magneton, and R is the internuclear distance. With the pa­rameters for the a 3I.: state we find r' = 12.4 MHz. Lichten et af.7 give an estimate of r' = 5 MHz. Both of these esti­mates are too large in magnitude and have the wrong sign when compared to the experimental value Yo = - 2.4221 MHz.

The second-order contributions to the fine structure of a

TABLE IV. Fine structure parameters for He2 •

Parameter a 31.: a.b C 31.t C

Ao - 1099.170 (5)d MHz - 952 (57)d MHz

AI 197.55 (27) kHz -0.48 (48) MHz

A2 - 5.6 (21) Hz

Yo - 2.4221 (18) MHz 10.74 (45) MHz

YI 0.680 (20) kHz 6.9 (39) kHz

• Fit based on present rf measurements plus rf data from Refs. 7 and 9. b Rotational parameters were taken from Ref. 6. CRotationai parameters were taken from Ref. 3. d Numbers in parentheses indicate three standard deviations on the last

digit.

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M. Kristensen and N. Sjerre: Lowest triplet states in He2 989

molecular l: state comes mainly from spin-orbit coupling to II states. All the 3Ilu states in He2 are Rydberg states lying far above the a 3l:u+ state. Therefore we choose to estimate the second-order contribution by summing over the whole manifold of Rydberg states. Tinkham and Strandberg26 give the second-order contributions as

~ (OIALxln)(nIBLxIO) y" =4Re £..

n En -Eo

(15)

Assuming the usual Rydberg state behavior for the matrix elements and the energies:

En = Eion - Ryln*2; n* = n - 8; (Olin) a: n* - 3/2

(16)

we obtain

(17)

The sum is calculated using the ionization potential and the quantum defect for the 3Il u states derived from Ginters spec­troscopic data!·5 As an estimate of A2-the spin-orbit pa­rameter for n* = 2-we use the value of - 0.226 94 cm - I

determined for the b 3Ilg state.6 We find assuming L = 1:

A. " = 0.04 MHz; y" = - 10.4 MHz. (18)

Thus, the second-order contribution is negligible compared to the first-order contribution for the spin-spin parameter. For the spin-rotation parameter, however, the second-order contribution is of the opposite sign and the same magnitude as the first-order contribution. These simple estimates ex­plain why Yo can become small and even negative, but they are not accurate enough to predict the actual value of the constant.

For the c 3l:g+ state there is to our knowledge no ab initio calculations of the first-order contributions to the fine struc­ture parameters. Using Eq. (14) we estimate the first-order contribution to the spin-rotation parameter to be 1" = 10.7 MHz, which is very close to the experimental value Yo = 10.74 MHz. However, as for the a state, there is a sig­nificant second-order contribution to y, which in this case ruins the excellent agreement between theory and experi­ment.

The c 3l:g+ state differs from the a- state in that there is a nearby II state of the same parity that can contribute to the fine structure via the spin-orbit coupling. Therefore, it is appropriate to use the "unique perturber" approximation28

in calculating y" and..1. ". Yarkonyl6 has calculated the spin­orbit coupling matrix elements between the c 3l:g+ state and the b 3Ilg and B IIlg states. Reference 16 lists these matrix elements for the nonrotating molecule as a function of the internuclear distance. Reference 28 treats such couplings by

a Van Vleck transformation which has the energy difference between the electronic states in the denominator. Here this gives a singularity because the potential curves cross as can be seen from Fig. 1. However, for a rotating molecule, rota­tional couplingl7 ensures that the two states never cross but are separated by - 2BN which for He2 is always much larger than the spin-orbit matrix elements. Therefore, the singular­ity of the nonrotating molecule is converted into a derivative Lorentzian, which can readily be integrated numerically to­gether with the electronic matrix elements and the vibration­al wave function.

As before, the coupling between 3l: and 3Il states con­tributes to both y" and A. ". The spin-orbit interaction cou­ples the c 3l:g+ state to the B I Ilg state as well, which also contributes to A. ". From the matrix elements ofYarkonyl6 we obtain for the two second-order contributions to the spin-spin parameter:

c 3l:t - b 3Ilg: A." = - 0.15 MHz,

c 3l:g+ - B IIlg: A." = 0.58 MHz. (19)

Thus, the second-order contribution to the spin-spin cou­pling parameter is again very small. For the second-order contribution to the spin-rotation parameter, we note that there is no contribution from the singlet states because there is no rotational singlet-triplet coupling. Therefore, we use the unique perturber approach of Zare et al.28 who treat the second-order perturbations in terms of two parameters 0 and p related to each other and to our parameters by

o = pA 18B, A." = - 20, y" = - p. (20)

Inserting the values for the spin-orbit constant A, as deter­mined for the b state6 and the rotational constant B for v' = 4 of the c state, we obtain

y" = 13.4 MHz (21)

which again is of similar magnitude as the first-order contri­bution to the spin-rotation parameter.

CONCLUSIONS

The present rf-optical double-resonance experiments il­lustrate how the fast neutral beam laser technique can be extended into the radio frequency regime and provide very precise fine structure measurements in small metastable molecules. Also, the present work illustrates how a set of very precise fine structure measurements can be represented in terms of an effective fine structure Hamiltonian. Still, the interpretation of the parameters in the Hamiltonian in terms of the electronic structure of the molecule lacks far behind the experimental precision. The estimates presented in the last section of the present paper clearly indicate that the first­order contribution dominates the value ofthe spin-spin cou­pling constant, which should therefore be a valuable param­eter for testing the quality of theoretically calculated electronic wave functions. The spin-rotation constants are more difficult to account for because of large second-order contributions.

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990 M. Kristensen and N. Bjerre: Lowest triplet states in He2

ACKNOWLEDGMENTS

We gratefully acknowledge helpful discussions with Dr. Donald C. Lorents. One of us (M. K.) thanks the Carlsberg Foundation for financial support.

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