Characterisation of the b 3 Σ + ,v=0 State and Its Interaction with the A 1 Π State in Aluminium Monofluoride
Maximilian Doppelbauer, Nicole Walter, Simon Hofsäss, Silvio Marx, H. Christian Schewe, Sebastian Kray, Jesús Pérez-Ríos, Boris G. Sartakov, Stefan Truppe, Gerard Meijer
aa r X i v : . [ phy s i c s . a t o m - ph ] A ug ARTICLE
Characterisation of the b Σ + , v = 0 State and Its Interaction withthe A Π State in Aluminium Monofluoride
M. Doppelbauer a , N. Walter a , S. Hofs¨ass a , S. Marx a , H. C. Schewe a , S. Kray a ,J. P´erez-R´ıos a , B. G. Sartakov b , S. Truppe a , G. Meijer a a Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany b General Physics Institute, Russian Academy of Sciences, Vavilovstreet 38, 119991 Moscow,Russia
ARTICLE HISTORY
Compiled August 13, 2020
ABSTRACT
Recently, we determined the detailed energy level structure of the X Σ + , A Πand a Π states of AlF that are relevant to laser cooling and trapping experi-ments [1]. Here, we investigate the b Σ + , v = 0 state of the AlF molecule. Arotationally-resolved (1+2)-REMPI spectrum of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0band is presented and the lifetime of the b Σ + , v = 0 state is measured to be190(2) ns. Hyperfine-resolved, laser-induced fluorescence spectra of the b Σ + , v ′ =0 ← X Σ + , v ′′ = 1 and the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 bands are recorded todetermine fine- and hyperfine structure parameters. The interaction between the b Σ + , v = 0 and the nearby A Π state is studied and the magnitude of the spin-orbit coupling between the two electronic states is derived using three independentmethods to give a consistent value of 10(1) cm − . The triplet character of the A state causes an A → a loss from the main A − X laser cooling cycle below the 10 − level. KEYWORDS
Cold molecules, laser cooling, hyperfine-resolved spectroscopy, spin-orbit coupling
1. Introduction
Polar molecules, cooled to low temperatures, have wide-ranging applications in physicsand chemistry [2, 3]. Recently, we identified the AlF molecule as an excellent candidatefor direct laser cooling to low temperatures and with a high density [1]. AlF hasone of the strongest chemical bonds known (6.9 eV), can be produced efficiently,and captured and cooled in a magneto-optical trap using any Q-line of its strong A Π , v ′ = 0 ← X Σ + , v ′′ = 0 band near 227.5 nm. Subsequent to trapping andcooling on the strong A − X transition, the molecules may be cooled to a few µ Kon any (narrow) Q-line of the spin-forbidden a Π , v ′ = 0 ← X Σ + , v ′′ = 0 band. Tolaser-cool AlF successfully, it is essential to study the detailed energy level structure ofthe molecule, to measure the radiative lifetime of the states involved and to investigateand quantify the potential decay channels to states that are not coupled by the coolinglaser, i.e. losses from the optical cycle. CONTACT S. Truppe. Email: [email protected] G. Meijer. Email: [email protected] or nearly a century, spectroscopists have been interested in the AlF molecule, inpart due to its similarity to the much studied CO and N molecules. AlF is also im-portant to the astrophysics community and it has been detected in sunspots, stellaratmospheres, circumstellar envelopes and proto-planetary nebulae [4, 5]. Moreover, AlF was the first radioactive molecule to be discovered in space [6]. AlF has beenthe subject of theoretical studies using ab initio quantum chemistry to determine ra-diative lifetimes, dipole moments and potential energy curves for its electronic states[7–10]. Precise spectroscopic parameters for AlF are useful for future astrophysical ob-servations and new spectroscopic studies of electronic states can serve as a benchmarkfor quantum chemistry calculations.In this paper, we present rotationally-resolved optical spectra of the b Σ + , v ′ =0 ← a Π , v ′′ = 0 band and demonstrate state-selective ionization of a Π moleculesvia (1+2)-resonance-enhanced multi-photon ionization (REMPI). Pulsed excitationfollowed by delayed ionization with a KrF excimer laser is used to determine theradiative lifetime of the b Σ + , v = 0 state. Next, high-resolution, cw laser-inducedfluorescence (LIF) spectra of the b Σ + , v ′ = 0 ← X Σ + , v ′′ = 1 band are presented.The hyperfine structure in the b Σ + , v = 0 state is resolved and precise spectroscopicconstants for the b state are determined. Laser-induced fluorescence spectra, recordedusing a cw laser to drive rotational lines of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 band, allowus to reduce the uncertainty in the spin-orbit constant A and the spin-spin interactionconstant λ of the a Π state by nearly two orders of magnitude. A fraction of theexcited state molecules decays on the b Σ + , v ′ = 0 → X Σ + , v ′′ bands to the groundstate. This is caused by spin-orbit coupling of the b Σ + , v = 0 state with the nearby A Π state. This perturbation of the A Π state and the b Σ + state is analysed in detailas it induces also a small loss channel for the strong A − X cooling transition.
2. Previous Work
In 1953, Rowlinson and Barrow reported the first observation of transitions betweentriplet states in AlF by recording emission spectra from a hollow-cathode discharge[11]. In the following decades, new triplet states were identified and characterised inmore detail [12–16]. Kopp and Barrow analysed the interaction of the A Π state withthe nearby b Σ + state and thereby determined the term energy of the triplet statesrelative to the singlet states [16]. A comprehensive study of the electronic states andthe interaction between the singlet and triplet states followed [17]. In 1976, Rosenwaks et al. observed the a Π → X Σ + transition directly in emission [18] at the same time asKopp et al. did in absorption [19]. Both studies confirmed the singlet-triplet separationdetermined from the earlier perturbation analysis.The hyperfine structure of the rotational states in b Σ + was partly resolved andanalysed by Barrow et al. [17]. Their low-resolution spectra (typical linewidth of0.05 cm − ) showed a triplet structure, which they ascribed to magnetic hyperfineeffects of the Al nucleus, whose nuclear spin is I Al = 5 /
2. A more detailed analysis ofboth the fine and hyperfine structure in the triplet states was presented by Brown et al. in 1978 [20]. The hyperfine structure was partially resolved and they determined valuesfor the Fermi contact parameter b F (Al), the electron spin-spin interaction parameter λ , and the spin-rotation parameter γ .We have recently reported on the detailed energy level structure of the X Σ + , A Π and a Π states in AlF [1]. That study resolves the complete hyperfine structure,including the contribution due to fluorine nuclear spin, in all three states. The2nergy levels in X Σ + and in the three Ω manifolds of the metastable a Π statewere measured with kHz resolution; this allowed us to determine the spectroscopicconstants precisely and study subtle effects, such as a spin-orbit correction to thenuclear quadrupole interaction of the Al nucleus.
3. Rotational Structure of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 Band and theRadiative Lifetime of the b Σ + , v = 0 State To study the b − a transition, the molecules must first be prepared in the a Π state.Previously, we described this state and the a − X transition in detail [1]. The hyperfinestructure in X Σ + is small and for the purpose of this study, the ground state iscomprised of single rotational levels with energies given by BJ ( J + 1).The angular momentum coupling in the a Π state of AlF is well-described by Hund’scase (a). The energy levels are labelled by the total angular momentum (excluding hy-perfine interaction) J = R + L + S , with R the rotational angular momentum of therigid nuclear framework, and L and S the total orbital and spin angular momenta ofthe electrons, respectively. Both, L and S are not well defined. However, their projec-tion onto the internuclear axis Λ , Σ and the total electronic angular momentum alongthe internuclear axis Ω = Λ + Σ are well defined. The spin-orbit interaction leads tothree fine-structure states with Ω = | Λ + Σ | , labelled with F , F and F , in orderof increasing energy, corresponding to the states a Π , a Π and a Π . The splittingbetween the Ω manifolds is determined by A , the electron spin-orbit coupling constantand the spin-spin interaction coupling constant λ . Each Ω has its own rotational man-ifold with energies BJ ( J + 1), where the rotational constant B is slightly different foreach Ω manifold.The b Σ + state is well-described by Hund’s case (b), for which L couples to therotation R to form N = R + L . The rotational energy levels follow BN ( N + 1).For Σ electronic states N = R and the total angular momentum (without hyperfineinteraction) is J = N + S . The combined effect of the spin-rotation interaction γ ( N · S )and the spin-spin interaction λ (3 S z − S ) splits each rotational level N > b Σ + into three J-levels; these components are labelled F , F and F with quantum numbers J = N + 1, J = N and J = N −
1, respectively.Following the convention of Brown et al. [21] the parity states can be labelled by e and f , where e levels have parity +( − J and f labels have parity − ( − J . Allrotational levels of the X Σ + state are e -levels, while in the b Σ + state, all F levelsare e -levels, while F and F are f -levels. In the a Π state, each J-level has an e andan f component due to Λ-doubling.Figure 1a shows the energy level diagram of the electronic states relevant to thisstudy, together with a rotationally resolved spectrum of the b Σ + , v ′ = 0 ← a Π , v ′′ =0 band in 1b, and a sketch of the experimental setup in 1c, which is similar tothe one reported previously [1]. The molecules are produced by laser-ablating analuminium rod in a supersonic expansion of 2% SF seeded in Ne. After passingthrough a skimmer, the ground-state molecules are optically pumped to the metastable a Π , v = 0 state by a frequency-doubled pulsed dye laser using the Q-branch of the a Π , v ′ = 0 ← X Σ + , v ′′ = 0 band. For this, 367 nm radiation with a bandwidthof 0.1 cm − and a pulse energy of 6 mJ in a beam with a e − waist radius of about2 mm is used. The Q-branch of this transition falls within this bandwidth. Therefore,many rotational levels in the metastable a Π , v = 0 state are populated simultane-3usly. Via the Q-branch, only the f -levels in a Π are populated. Alternatively, if themolecules are optically pumped to the metastable state using spectrally isolated R orP lines, only the e -levels are populated. Further downstream, at z = 55 cm from thesource, the molecular beam is intersected with light from a second pulsed dye lasertuned to the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 transition near 569 nm. For pulse energiesexceeding 6 mJ (unfocused, with an e − waist radius of about 5 mm), this laser trans-fers population to the b Σ + , v = 0 state and subsequently ionises the molecules byhaving them absorb two more photons from the same laser. Such a one-colour (1+2)-REMPI scheme using an unfocused laser beam is very uncommon. However, AlF hasnumerous electronically excited states that lie one photon-energy above the b stateenergy, strongly enhancing the non-resonant two-photon ionisation probability. Theions are mass-selected in a short time-of-flight mass spectrometer (TOF-MS) and de-tected using micro-channel plates. The TOF-MS voltages are switched on shortly afterthe ionisation laser fires; this way ionisation occurs under field-free conditions and thestates have a well-defined parity. The (1+2)-REMPI scheme uses low-energy photonsand an unfocused laser beam, which has the benefit of producing a mass spectrumwith only a single peak, corresponding to AlF. The spectrum, displayed in Figure 1b,shows the ion signal as a function of the REMPI laser frequency. The spectral linesare labelled by ∆( N J ) F ′′ ( J ′′ ), where ∆( N J ) = N ′ − J ′′ , as the quantum number J isnot well-defined in the b state, vide infra . Since only the f -levels of the a Π state arepopulated, the spectrum consists of ∆(
N J ) = − , , +2, i.e., O, Q and S branches.To determine the radiative lifetime of the b Σ + , v = 0 state, we reduce the pulseenergy of the b ← a excitation laser to about 1 mJ. At pulse energies below 2 mJ,the a Π molecules are excited to the b state, but are not ionised via (1+2)-REMPI.Instead, a KrF excimer laser is used to ionise the b state molecules with a single248 nm photon and a pulse energy of about 3 mJ. The radiative lifetime of the b Σ + , v = 0 state is measured by varying the time-delay between excitation andionisation [22]. The determination of radiative lifetimes in the range of 20 ns to 1 µ s isrelatively straightforward, because it is longer than the laser pulse duration, but shortenough so that the molecules do not leave the detection region. Figure 2 shows theion signal (black dots) as a function of the time delay with 1 − σ standard error bars.The blue line is a fit to the data using the model S ( t ) = Ce − t/τ b erfc[( t − t ) / ( √ σ )],where erfc is the complementary error function, C and t are fit parameters, τ b isthe lifetime of the b Σ + , v = 0 state and σ is the measured, combined pulse-width ofthe excitation and ionisation laser. In this model the lifetime τ b = 190(2) ns is fixedto the value determined from a linear fit to the semi-log plot shown in the inset. In1988, Langhoff et al. calculated an approximate radiative lifetime of the b Σ + stateof 135 ns [8], considerably shorter than the measured lifetime.
4. The Fine and Hyperfine Structure of the b Σ + State
Aluminium and fluorine have a nuclear spin of I Al = 5 / I F = 1 /
2, respectively.Following the description of Brown et al. [20] and including the magnetic interactionof the fluorine nuclear spin, the effective Hamiltonian reads4 igure 1. a) Electronic energy level scheme of the relevant states of AlF. The transitions used for laserexcitation are shown as solid arrows. The laser-induced fluorescence used for detecting the molecules is indicatedby downward wavy arrows. The indicated energies are the gravity centres of the respective states in absenceof hyperfine structure. b) (1+2)-REMPI spectrum of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 band. The a Π , v = 0state is populated via the Q-branch of the a Π , v ′ = 0 ← X Σ + , v ′′ = 0 band using a frequency-doubledpulsed dye laser. c) Schematic of the experimental setup used for the determination of the lifetime of the b Σ + , v = 0 state. H eff = 23 λ (cid:0) S z − S (cid:1) + γ ( N · S ) (1)+ b F (Al) I Al · S + 13 c (Al)(3 I Al ,z S z − I Al · S )+ b F (F) I F · S + 13 c (F)(3 I F ,z S z − I F · S )+ eq Q I Al (2 I Al −
1) (3 I ,z − I ) , where λ is the spin-spin interaction constant and γ the spin-rotation interactionconstant. The parameters b F (Al) and b F (F) describe the Fermi contact interactionfor the aluminium and fluorine nucleus, respectively, c (Al) and c (F) the dipolarinteraction, and eq Q the electric quadrupole interaction of the Al nucleus.In the b Σ + state of AlF the Fermi contact interaction b F (Al) ( I Al · S ) betweenthe nuclear spin of aluminium and the electronic spin angular momentum is strongcompared to the spin-rotation interaction [20]. The coupling case approximates ( b βS ).The N = 0 level has only one spin-component for which J = N + S = 1; this J = 1 levelis split into three components due to the aluminium nuclear spin, each of which is againsplit by the nuclear spin of fluorine. This results in a total of six F levels. For N > J is not well defined and it is useful to introduce an intermediate angular momentum5 igure 2. Measurement of the radiative lifetime of the b Σ + , v = 0 state. First, the ground-state moleculesare optically pumped to the b Σ + , v = 0 state via the two-colour excitation scheme described in the text. Thepopulation in the b Σ + , v = 0 state is probed by single-photon ionisation using a KrF excimer laser, followedby TOF-MS detection. The black dots show the AlF + ion signal as a function of the time delay betweenexcitation and ionisation. The blue line is a fit to the data using the model described in the text. The insetshows a semi-log plot of a second measurement for eight specific time delays. A linear fit to the data gives a b state lifetime of 190(2) ns. G = I Al + S (see Figure 4). G then couples to N which results in three sets of sub-levels, with quantum numbers G = 3 / , / N +3 / , ...N − / N +5 / , ..., N − / N +7 / , N +5 / ..., N − / F = N + G + I F . For N = 1 , F levels, respectively. For N > F levels reaches its limit of 36.To determine the hyperfine energy levels of the b Σ + state, we drive the b Σ + , v ′ =0 ← X Σ + , v ′′ = 1 band near 227.2 nm with a cw laser and detect the b − a fluorescenceat 569 nm. The spectrum of this transition directly reflects the energy level structurein the b Σ + state, because the hyperfine structure of the X Σ + state is smaller thanthe residual Doppler broadening in the molecular beam. The wavelength required todrive this transition is close to that of the A Π , v ′ = 0 − X Σ , v ′′ = 0 band near227.5 nm, for which we have a powerful UV laser system installed.The b Σ + , v ′ = 0 ← X Σ + , v ′′ = 1 transition is spin-forbidden and the overlapof the vibrational wave functions is poor, with a calculated Franck-Condon factor of0.02. However, the transition becomes weakly allowed due to the spin-orbit interactionbetween the b Σ + and the nearby A Π state. In section 7 this interaction will bediscussed in more detail. To compensate for the weak transition dipole moment, weuse a high excitation laser intensity of up to 75 mW in a laser beam with a e − waist radius of 0.6 mm. The resulting laser-induced fluorescence occurs mainly on thedipole-allowed b − a transition near 569 nm. The far off-resonant fluorescence allowsus to block scattered laser light with a bandpass filter and to record background-freespectra. To increase the number of molecules in X Σ + , v = 1, we use a cryogenichelium buffer gas source, instead of the supersonic molecular beam introduced in theprevious section. Figure 3 shows a sketch of the experimental setup that is used forthis measurement.The design of this source is similar to the one described in [23–25]. A pulsedNd:YAG laser ablates a solid aluminium target in the presence of a continuous flow of0.01 sccm room-temperature SF gas, which is mixed with 1 sccm of cryogenic helium6as (2.7 K) inside a buffer gas cell. The hot Al atoms react with the SF and form hotAlF molecules, which are subsequently cooled through collisions with the cold heliumatoms. Compared to the supersonic molecular beam, this source delivers over 100 timesmore AlF molecules per pulse in the ro-vibronic ground-state to the detection region.The forward velocity of the molecules is four times lower. To increase the number ofmolecules in X Σ + , v = 1, we increase the pulse energy and repetition rate of theablation laser, increase the temperature of the SF gas to 350 K and increase its massflow rate to 0.1 sccm. At z = 35 cm the molecules interact with cw UV laser light todrive the b Σ + , v ′ = 0 ← X Σ + , v ′′ = 1 transition near 227.2 nm. The laser light isproduced by frequency-doubling the output of a cw titanium sapphire laser twice. Thelaser-induced fluorescence passes through an optical filter to block scattered light fromthe excitation laser, and is imaged onto a photomultiplier tube (PMT). The photo-current is amplified and acquired by a computer. The wavelength of the excitationlaser is recorded with an absolute accuracy of 120 MHz using a calibrated wavemeter(HighFinesse WS8-10).Figure 5 shows the recorded spectra reaching the three lowest N levels in the b state. The panels demonstrate the increasing complexity of the hyperfine structurewith increasing N . The three spectra allow us to measure the rotational constant ofthe b state. Gaussian lineshapes are fitted to the experimental spectra to determine theline-centres. We then fit the eigenvalues of the hyperfine Hamiltonian to the measuredenergy levels with the spectroscopic parameters as fit parameters. We assign a totalof 48 lines and the standard deviation of the fit is 11 MHz. The best fit parameterstogether with their standard deviations are summarised in Table 1. E is the pure vi-bronic energy of b Σ + , v = 0, i.e. the energy of the N = 0 level in absence of spin, fineand hyperfine splitting. This is referenced to the J = 0 level of the X Σ + , v = 0 stateby using the precise infrared emission lines of [26] to determine the energy differencebetween the v = 0 and v = 1 level in the X state. The inverted spectra in Figure 5 aresimulated spectra using the spectroscopic parameters presented in Table 1 and repro-duce the measured spectra well. The Fermi contact parameter b F (F) for fluorine andthe two hyperfine parameters c (Al) and c (F) for the aluminium and fluorine nucleus,respectively, are determined for the first time. In previous, low-resolution studies, theinteraction of the fluorine nuclear spin has been neglected. However, we conclude thatthe magnitude of the interaction parameter for the two nuclei is comparable. Thisindicates that there is a significant electron density at both nuclei which is in starkcontrast to the situation in the a Π state. In the latter state, the Fermi contact termfor the F nucleus is about seven times smaller than for the Al nucleus [1]. TheFermi contact parameter for the aluminium nucleus b F (Al) and the spin-spin interac-tion parameter λ presented here are consistent with the previously determined values,but their uncertainty is reduced significantly.
5. The b Σ + , v ′ = 0 ← a Π , v ′′ = 0 Transition The b − a bands have a diagonal Franck-Condon matrix and the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 band has a Franck-Condon factor of 0.994. Its natural linewidth is 100times smaller than the natural linewidth of the strong A Π , v ′ = 0 ← X Σ + , v ′′ = 0transition near 227.5 nm. Laser cooling AlF molecules on the b − a transition couldtherefore reach temperatures far below the Doppler limit of the strong A − X tran-sition. The vibrational branching to a Π , v = 1 is small and, if addressed with arepump laser, a molecule could scatter about 1000 photons before being pumped into7 igure 3. Schematic of the experimental setup used to measure the fine and hyperfine structure of the b Σ + , v = 0 state. AlF molecules are produced in a cryogenic helium buffer gas source. The molecular beamis intersected with UV laser light from a frequency-quadrupled cw titanium sapphire laser. The molecules areexcited on the weak, spin-forbidden b Σ + , v ′ = 0 ← X Σ + , v ′′ = 1 transition. The laser-induced fluorescenceoccurs mainly on the b Σ + , v ′ = 0 → a Π , v ′′ = 0 transition near 569 nm and is imaged onto a PMT. Table 1.
Experimentally determined spectroscopic constants of b Σ + , v = 0. E and its uncertainty is givenin cm − , all other parameters are given in MHz. SD gives the standard deviation in the absence of correlationsand SD · √ Q gives the standard deviation of the parameter including the correlations between the parameters[27]. The previous best values are taken from [17] and [20] and given with the reported standard error (SE). Parameter This Work SD SD ·√ Q Refs. [17] & [20] SE E B λ -919 15 18 -750 300 γ -9 7 13 0 9 b F (Al) 1311 2 3 1469 90 c (Al) 73 12 18 eq Q (Al) -62 69 99 b F (F) 870 10 11 c (F) 305 50 53 a Π , v = 2. However, the radiative decay from the b state to multiple J levels in allthree spin-orbit manifolds of the a state is allowed. This results in a large number ofrotational branches that must be addressed to close the optical cycle [28]. In addition,the hyperfine structure in a is large compared to the linewidth of the transition. Botheffects make laser-cooling of AlF, using an optical transition in the triplet manifoldvery challenging. This is in stark contrast to the strong A − X transition, for which allQ-lines are rotationally closed and for which all hyperfine levels of a given rotationallevel in the X state lie within the natural linewidth.Here, we demonstrate that laser-induced fluorescence spectroscopy of the b Σ + , v ′ =0 ← a Π , v ′′ = 0 transition can be used to efficiently detect a Π molecules withhyperfine resolution. In this section, we show that this method works well to detectmolecules in all three Ω manifolds to improve two important spectroscopic constantsof the a Π state: the spin-orbit ( A ) and spin-spin ( λ ) interaction parameter, whichdetermine the relative spacing of the three Ω manifolds in a Π. The effective finestructure Hamiltonian is 8 able 2.
Energies, E , of the hyperfine levels in b Σ + , v = 0, relative to the X Σ + , v = 0 , J = 0 level,magnetic g-factors g F , rotational quantum number N , total angular momentum F and parity p . Since theassignment of the quantum numbers N, F, p , is not unique, we use n to index the levels that share the sameset of quantum numbers F and p , with increasing energy. The final state of the transition used to investigatethe singlet contribution to the b Σ + state wave function in section 5 is highlighted in red. E (cm − ) g F N F p n E (cm − ) g F N F p n − .
661 0 2 +1 1 44807.7613 − .
298 2 4 +1 244804.4305 − .
000 0 1 +1 1 44807.7630 − .
517 2 1 +1 244804.5235 0 .
327 0 2 +1 2 44807.7685 0 .
000 2 0 +1 144804.5268 0 .
170 0 3 +1 1 44807.7838 − .
228 2 2 +1 444804.6622 0 .
663 0 3 +1 2 44807.7868 − .
291 2 3 +1 444804.6928 0 .
500 0 4 +1 1 44807.7883 0 .
261 2 1 +1 344805.5139 − .
617 1 2 − .
000 2 0 +1 244805.5217 − .
411 1 3 − .
517 2 1 +1 444805.5282 − .
991 1 1 − .
551 2 1 +1 544805.5454 − .
712 1 1 − .
140 2 4 +1 344805.5479 − .
461 1 2 − .
090 2 5 +1 144805.5540 0 .
000 1 0 − .
308 2 2 +1 544805.6297 0 .
703 1 1 − .
323 2 2 +1 644805.6344 0 .
292 1 2 − .
179 2 3 +1 544805.6373 0 .
196 1 3 − .
153 2 3 +1 644805.6396 0 .
117 1 4 − .
088 2 4 +1 444805.6548 0 .
341 1 2 − .
454 2 4 +1 544805.6615 0 .
172 1 3 − .
470 2 3 +1 744805.7708 0 .
632 1 3 − .
397 2 5 +1 244805.7845 0 .
497 1 4 − .
569 2 2 +1 744805.7958 0 .
779 1 2 − .
188 2 1 +1 644805.8053 0 .
486 1 4 − .
380 2 5 +1 344805.8142 0 .
400 1 5 − .
415 2 4 +1 644805.8216 0 .
578 1 3 − .
333 2 6 +1 144807.7541 − .
342 2 3 +1 3 44808.0530 0 .
499 2 3 +1 844807.7565 − .
416 2 2 +1 3 44808.0597 0 .
777 2 2 +1 89 igure 4.
Relative weights W G and W J of the | N, G, F i and | N, J, F i basis wave functions in the eigenfunc-tions | E i of the energy levels in the three lowest rotational states of the b Σ + , v = 0 state. The weights areobtained by projecting the eigenstate of the Hamiltonian onto the basis wave function | N, G, F i and | N, J, F i ,respectively. G can take the values 3 / , / / J can be N − , N or N + 1. The energy levels aredescribed uniquely by the quantum numbers F , p and the index n . This analysis shows that G is a goodquantum number for the description of any rotational level of the b Σ + , v = 0 state, while J only characterisesthe levels well for N = 0. H = AL z S z + 23 λ (3 S z − S ) . (2)The interval between the Ω = 0 and Ω = 1 manifolds is approximately A − λ , whilethe Ω = 1 and Ω = 2 manifolds are about A + 2 λ apart.For this experiment we use the supersonic molecular beam setup introduced insection 3, with an additional LIF detector installed between the excitation regionand the TOF-MS. After passing through the skimmer, the molecules are excited on aspecific rotational line of the a Π , v ′ = 0 ← X Σ + , v ′′ = 0 band to one of the threespin-orbit manifolds, using a frequency-doubled pulsed dye amplifier (PDA), which isseeded by a cw titanium sapphire laser. About 30 cm further downstream, a cw ringdye laser intersects the molecular beam orthogonally and is scanned over a rotationalline of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 band. The laser frequency is stabilised andscanned with respect to a frequency-stabilised and calibrated HeNe reference laser(SIOS SL 03), using a scanning transfer cavity. The wavelength is recorded with anabsolute accuracy of 10 MHz using the wavemeter, calibrated by the same HeNe laser.Figure 6 shows hyperfine-resolved LIF excitation spectra of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 band. Figure 6a shows the hyperfine spectrum of the b Σ + , v ′ = 0 , N ′ =1 ← a Π , v ′′ = 0 , J ′′ = 0 lines. The positive parity Λ-doublet level of a Π , J = 010 igure 5. Direct measurement of the fine and hyperfine structure of the three lowest rotational levels in the b state, i.e. for N ′ = 0 in panel c), N ′ = 1, in panel b) and N ′ = 2 in panel a). The experimental data are shownin black, pointing up and the simulated spectra are shown in blue, pointing down. The spectra are centred atthe gravity centre, i.e., the line position in absence of hyperfine, spin-spin and spin-rotation interaction. has only two hyperfine components with total angular momentum quantum numbers F = 2 and F = 3 that are only split by about 3 MHz, much smaller than the residualDoppler broadening in the molecular beam. Therefore, this b − a spectrum directlyreflects the energy level structure in the b state and is identical to the one of the b − X spectrum shown in Figure 5b.The spectrum of the b Σ + , v ′ = 0 , N ′ = 2 ← a Π , v ′′ = 0 , J ′′ = 1 transition,presented in Figure 6b, shows a richer structure, and no longer directly reflects theenergy level structure in the b state. The total span of the hyperfine splitting in the a Π , v = 0 , J = 1 level is about 500 MHz [1], and contributes to the complexity ofthis spectrum.Optical pumping to the a Π , v = 0 state is challenging because the R (1) transitionof the a Π , v ′ = 0 ← X Σ + , v ′′ = 0 band is about 1000 times weaker than the11orresponding R (1) line to the a Π state. Only a small fraction of the ground-statemolecules produced in the source is transferred to the a Π , v = 0 level, even with aPDA pulse energy of about 10 mJ in beam with a e − waist radius of 1.5 mm. Figure 6cshows the R (1) line of the a Π , v ′ = 0 ← X Σ + , v ′′ = 0 band, recorded by scanningthe seed laser of the frequency-doubled PDA, followed by (1+2)-REMPI and TOF-MSdetection. The hyperfine structure in a Π , v = 0 is very large and it is possible toresolve six of the ten hyperfine levels using a narrow-band pulsed laser. This enablesus to populate a specific hyperfine component of the a Π , v = 0 , J = 2 level, whichis then followed by cw excitation to the b state and LIF detection. Figure 6d showstwo LIF spectra that originate from two different hyperfine states in a Π , indicatedby the two colours. The green spectrum originates from the a Π , v = 0 , J = 2 , F = 4level and the grey spectrum from the a Π , v = 0 , J = 2 , F = 5 level. The line-centresare determined by fitting Gaussians to the spectral lines.The hyperfine structure and Λ-doubling of the a Π state has been investigatedextensively in our previous study and is known to kHz precision [1]. However, theconstants A and λ of the a Π state could only be determined with an accuracy ofabout 200 MHz, due to the finite bandwidth of the pulsed laser and the lower accuracyof the wavemeter that was used in that study to measure the relative energy of the Ωmanifolds. Here, we take advantage of the reduced linewidth in the cw LIF spectra ofthe b − a transition, in combination with the increased accuracy of our new wavemeter,to improve this measurement and therefore the spectroscopic constants A and λ . Byusing the hyperfine constants for the a Π state from reference [1] in combination withthe parameters for the b state, listed in Table 1, the b − a spectra can be simulated, andnew values for A and λ are derived. The simulated spectra, using the full Hamiltonian,including the hyperfine interactions in both states are shown in Figure 6 as blue,inverted curves. The uncertainty of the improved spectroscopic constants, listed inTable 3, is reduced by nearly two orders of magnitude compared to the ones presentedpreviously [1]. E is the term energy of the a state in the absence of rotation, fineand hyperfine structure, calculated by using the term energy of the b state determinedin the previous section. This means that the gravity centre, i.e, the position of the a Π , v = 0 , J = 1 level in the absence of hyperfine structure, relative to the X Σ + , J =0 level is at 27255.1737(5) cm − .It is also possible to record the LIF excitation spectra by detecting the weak UVfluorescence, on the b Σ + , v ′ = 0 → X Σ + , v ′′ bands which occurs predominantly at223.1 nm. Part of the spectrum displayed in 6b has also been recorded this way andis shown by the inset labelled ‘UV’. To measure the ratio of the emission occurringin the UV, relative to the emission occurring in the visible, we lock the excitationlaser to the resonance indicated by the arrow in 6b and average the signal over 1000shots. To distinguish the two wavelengths we use two different PMTs: a UV sensitiveone with a specified quantum efficiency of 0 . ± .
05 at 223.1 nm and negligiblesensitivity for wavelengths >
350 nm and a second PMT with a specified quantumefficiency of 0 . ± .
015 at 569 nm that is sensitive in the range of 200 −
800 nm.The quantum efficiencies of the PMTs are taken from the data sheet and are notcalibrated further. However, three identical UV PMTs give the same photon countrate to within 15% which we take as the systematic uncertainty. Both PMTs areoperated in photon-counting mode, counting the number of UV and visible photons, n uv and n vis , respectively. The visible PMT is combined with a band-pass interferencefilter to block the UV fluorescence and a small amount of phosphorescence on the a Π , v ′ = 0 → X Σ + , v ′′ = 0 transition near 367 nm. The measured transmission ofthe filter at 569 nm is 0.57. We combine the slightly different transmission through12he imaging optics, the different detector efficiencies and the filter transmission intothe total detection efficiencies η uv and η vis . Including these values, we measure a ratioof R b = n uv n vis η vis η uv = (4 . ± . × − . (3)This is the result of multiple experimental runs with a statistical error that is sig-nificantly smaller than the uncertainty in the quantum efficiencies of the PMTs. Anadditional 20% systematic uncertainty is added to the total error budget because wedo not correct for the spherical and chromatic aberration of the fluorescence detector.However, simulations using ray-tracing software show a typical difference in imagingefficiency for the two wavelengths of about 10%. The value for R b is identical to theratio of the Einstein A -coefficients for b → X and b → a emission, i.e. R b = A b,X /A b,a .The specific level in the b Σ + , v = 0 state for which this ratio of Einstein A -coefficientsis determined, is the positive parity level with N = 2, F = 4, n = 4 which is almosta pure f -level, with 42.3% F and 57.4% F character, and only 0.3% F character.The level is highlighted in red in table 2 and figure 4. Table 3.
Spectroscopic constants for the a Π , v = 0 state of AlF, determined from a fit to the spectra. E is the energy of the electronic state in the absence of rotation, fine- and hyperfine structure relative tothe X Σ + , v = 0 , J = 0 level (in cm − ). The spin-orbit coupling constant, A , and the spin-spin interactionconstant, λ , are given in MHz. Their respective values from our earlier publication [1] are also reproduced forcomparison. Parameter Value SD Value [1] SD E A λ
6. The A Π → a Π Bands
To probe the amount of triplet wave function that is mixed into the A Π state directly,we measure the ratio R A of the number of fluorescence photons emitted on the A Π → a Π and on the A Π → X Σ + transition. The value for R A is identical to the ratio ofthe Einstein A -coefficients for A → a and A → X emission, i.e., R A = A A,a /A A,X . Thevalue for R A gives the loss from the main laser cooling cycle due to electronic branchingto the a Π state. Previously, we measured this electronic branching ratio on the A → a bands indirectly to be at the 10 − level, by comparing the absorption-strength ofthe A Π , v = 0 ← a Π , v ′ = 0 transition relative to the absorption-strength of the A Π , v = 0 ← X Σ + , v ′′ = 0 transition [1]. In that analysis we only took into accountthe amount of singlet character in the wave function of the a state to deduce thestrength of the A ← a transition. We did not account for the (then unknown) amountof triplet character in the wave function of the A state due to the interaction of the A and b states, which, as we will see here, turns out to be the dominant effect.To measure such a small branching ratio directly, we use the buffer gas molecu-lar beam source in combination with optical cycling on the Q(1) line of the A − X transition. The small amount of visible fluorescence is isolated from the strong UVfluorescence by a high-reflectivity UV mirror, which is transparent in the visible, incombination with a long-pass and two bandpass filters in front of the PMT. The13 igure 6. LIF excitation spectra of the b − a transition. Panels a) and b) show spectra of the b Σ + , v ′ =0 , N ′ = 1 ← a Π , v ′′ = 0 , J ′′ = 0 and b Σ + , v ′ = 0 , N ′ = 2 ← a Π , v ′′ = 0 , J ′′ = 1 transitions, respectively.The spectrum in the inset labelled UV is detected by recording the emission on the b Σ + , v = 0 , N = 2 → X Σ + , v ′′ bands as a function of the excitation frequency. c) Hyperfine resolved spectrum of the R (1) line ofthe a Π , v ′ = 0 ← X Σ + , v ′′ = 0 band, showing the large hyperfine splitting in a Π . d) Part of the hyperfineresolved spectrum of the b Σ + , v ′ = 0 , N ′ = 1 ← a Π , v ′′ = 0 , J ′′ = 2 transition. The low (high) frequencypart of this spectrum was recorded after excitation to the F = 4 ( F = 5) component of the R (1) line of the a Π ← X Σ + transition. a) - d): The relevant energy level scheme is displayed next to each spectrum. Since J is not well-defined in the b state, the b Σ + ← a Π transitions are labelled with ∆( NJ ) F ′ ( J ′ ). The measuredspectra are printed in black, while the simulated spectra are shown in blue and inverted. transmission band of the filters is chosen such that only wavelengths that cover the A Π , v ′ = 0 → a Π , v ′′ = 0 transition are detected by the PMT. The mirror reducesthe UV fluorescence that is incident on the spectral filters to the 10 − level, suppress-ing their broadband phosphorescence when irradiated with UV light. The transmissionof each optical element is measured individually at 227.5 nm and 599 nm, using laserlight and a calibrated photo diode. The total detection efficiency, accounting for thetransmittances and PMT responses becomes η uv = 0 . ± .
05 for UV photons and η vis = 0 . ± .
01 for visible photons in the range of 596 −
604 nm. The ratio ofEinstein coefficients becomes R A = n vis n uv η uv η vis = (6 ± × − , (4)14here n vis and n uv are the number of photons detected in the visible and UV, respec-tively. A typical measurement is presented in Figure 7. The molecules are opticallypumped on the Q(1) line of the 0 − A − X transition and the LIF is im-aged and detected by two different PMTs. The majority of the fluorescence is emittedin the UV and imaged onto the UV sensitive PMT. The PMT is operated in currentmode, which is converted into a voltage, amplified and read into the computer. Wecalibrate this PMT output voltage against the output of the PMT in photon-countingmode for low incident light intensities. The small fraction of the LIF that is emittedin the visible is shown as red dots. The two time of flight profiles are very similar,with the detected signal in the visible being (1 . ± . × − of the emission in theUV. By accounting for the different detection efficiencies for the two wavelengths themeasured ratio is as given in equation 4. The total uncertainty in this measurement isdominated by the systematic uncertainty in the quantum efficiency of the two PMTsand by the uncertainty in the imaging efficiency for the two wavelengths as describedin section 5.The measurement of R A is significantly more challenging than the measurement of R b . This is because n uv is 10 times larger than n vis and the phosphorescence of theoptical elements typically occurs red-shifted, i.e., in the wavelength range of the weakfluorescence in the visible. This is in contrast to the measurement of R b , for which n uv is about 200 times smaller than n vis , and for which any background caused byphosphorescence of the optical elements is absent.We also measure the ratio of the visible to the UV fluorescence subsequent to exci-tation on the Q(1) line of the 1 − A − X transition. Since the A Π , v = 1level is energetically much closer to the b Σ + , v = 0 level, one might expect a signifi-cantly larger fraction of visible fluorescence. However, we find that the two ratios R A are equal to within the 15% uncertainty of the measurement. Figure 7.
The LIF emitted in the UV (227.5 nm, black) and VIS (599 nm, red) as a function of time whenthe UV laser frequency is locked to the Q(1) line of the A Π , v ′ = 0 ← X Σ + , v ′′ = 0 transition. The insetshows the configuration of the fluorescence detector used for the VIS experiment.
7. Spin-orbit Interaction Between the A Π and b Σ + States
The observation of the b Σ + , v ′ = 0 ↔ X Σ + , v ′′ and of the A Π , v ′ = 0 → a Π , v ′′ intersystem bands reported here indicates that the wave function of the b state ofAlF contains a fraction of singlet character and that the A state contains a fractionof triplet character. This is due to the spin-orbit interaction of the b state with the15earby A Π state. Figure 8 shows the potential energy curves for the lowest singletand triplet electronic states of AlF. The inset shows a more detailed view of the A Πand b Σ + states with their vibrational levels indicated. For low vibrational quantumnumbers, the vibrational level v b in the b state lies slightly above the vibrational level v A = v b + 1 in the A state. The energy difference between v b and v A = v b + 1 decreaseswith increasing vibrational quantum numbers until the levels have just crossed and arenearly degenerate at v A = 6 ( v b = 5), leading to a large perturbation of the rotationalenergy levels. This interaction has already been analysed by Barrow et al. in 1974, whointroduced the parameter H ( v A ) to describe the effect of the spin-orbit interactionof specific pairs of vibrational levels, i.e., of v A = 4 , v b = 3 , − .In this section we give a more general description of the spin-orbit interaction be-tween the A Π , v A state and the b Σ + , v b state. We deduce the expressions for R b and R A in terms of the spin-orbit coupling constant A so between the A and the b state, toexperimentally determine the value for A so . This is then compared to the value of A so that can be deduced from the H ( v A ) parameters as given by Barrow and co-workers[17]. The effect of the interaction on the rotational energy levels in the A Π , v = 0 and b Σ + , v = 0 states is also discussed.For the spin-orbit interaction of a Π state with a Σ + state, the interaction termsbetween the e - and f -levels belonging to a given J are given by [29]: H ( Π , f | Σ + , F ) = r J + 12 J + 1 ξ √ H ( Π , e | Σ + , F ) = − ξ √ H ( Π , f | Σ + , F ) = r J J + 1 ξ √ . (7)In the Born-Oppenheimer approximation, the parameter ξ only depends on the vi-brational wave functions of the coupled states and can be written in terms of thespin-orbit operator H so as ξ ( v A , v b ) = h Ψ A,v A | H so | Ψ b,v b i = A so Z φ ∗ b,v b ( ρ ) φ A,v A ( ρ ) dρ = A so √ q v A v b . (8)The expressions φ b,v b ( ρ ) and φ A,v A ( ρ ) are the vibrational wave functions for the b and A state, respectively, and ρ is the inter-nuclear distance between the Al and F atoms.The square of the expression for the integral is the Franck-Condon factor q v A v b betweenthe A, v A and b, v b levels. The Franck-Condon matrix between the A and b states isvery diagonal. The value of q is very close to one and even though the v = 0 levelsof both states are about ( E b, − E A, ) = 855 cm − apart, the interaction betweenthese levels dominates. The value of q is much smaller than one, but as the v b = 0and v A = 1 levels are only ( E b, − E A, ) = 63 cm − apart, this interaction also hasto be taken into account. The interaction with all the other vibrational levels can beneglected. 16 .1. Intensities of the Intersystem Bands The spin-orbit interaction between the A state and the b state mixes their wave func-tions and causes a shift of the rotational levels. If the interacting levels are muchfurther apart than the magnitude of the interaction terms, we can calculate the effectof the interaction using first order perturbation theory. In this case, the contributionto the wave function of the A Π , v = 0 ( b Σ + , v = 0) state due to the interaction withthe b ( A ) state is given by the interaction term divided by the energy separation ofthe interacting levels. The wave function of the A state can be written as: (cid:12)(cid:12) Ψ ′ A, (cid:11) = C sA, | Ψ A, i + C tA, b, | Ψ b, i , (9)where | C sA, | ≈
1, is the total amount of singlet character and where | C tA, b, | ≡ | C tA, | is the total amount of triplet character in the wave function of the A state. It is readilyseen from the interaction terms given above that the triplet contribution to the wavefunctions of the e - and f -levels is equal, does not depend on J , and is given by | C tA, | = A so √ q √ E b, − E A, ) . (10)The wave function of the b state can be written as: (cid:12)(cid:12) Ψ ′ b, (cid:11) = C tb, | Ψ b, i + C sb, A, ( J, F i ) | Ψ A, i + C sb, A, ( J, F i ) | Ψ A, i , (11)where | C tb, | ≈ q | C sb, A, ( J, F i ) | + | C sb, A, ( J, F i ) | ≡ | C sb, ( J, F i ) | is the total amount of singlet char-acter in the wave function of the b state. The singlet contribution to the wave functionof the b state depends on the ( J, F i ) ( i = 1 , ,
3) level. For the positive parity, N = 2, F = 4, n = 4 level that we used for the measurement of R b we find, using the weights W J as indicated in figure 4 ( W J = 42 . W J = 0 .
3% and W J = 57 . | C sb, | = 0 . A so √ r q ( E b, − E A, ) + q ( E b, − E A, ) . (12)The expression for R A can now be rewritten as: R A = A A,a A A,X = |h Ψ ′ A, | µ ( r ) | Ψ a, i| λ A,X |h Ψ ′ A, | µ ( r ) | Ψ X, i| λ A,a = | C tA, | A b,a λ b,a A A,X λ A,a = | C tA, | , (13)where µ ( r ) is the electronic transition dipole moment, λ b,a = 569 nm, λ A,a = 599 nmare the wavelengths of the b − a and A − a transitions, respectively. The ratio of theEinstein A -coefficients is calculated from the experimentally known lifetimes of the A state (1.90 ns, [1]) and of the b state (190 ns, Section 3). The expression for R b cannow be rewritten as: R b = A b,X A b,a = P i =0 |h Ψ ′ b, | µ ( r ) | Ψ X,i i| λ b,a |h Ψ ′ b, | µ ( r ) | Ψ a, i| λ b,X = | C sb, | A A,X λ A,X A b,a λ b,X = 106 | C sb, | , (14)17here λ A,X = 227.5 nm and λ b,X = 223 nm. The sum in the numerator results in thetwo terms given in equation 12, because the Franck-Condon matrix between the A and X states is highly diagonal.The parameter H ( v A ), introduced by Barrow and co-workers, describes the effectof the spin-orbit interaction between two specific vibrational levels and is equivalentto H ( v A ) = ξ ( v A , v b = v A − √ A so r q v A v A − . (15)Barrow and co-workers did not discuss the dependence of the values for H ( v A ) on thesquare root of the Franck-Condon factors, and they therefore did not extract a singlevalue for the spin-orbit interaction parameter A so from the three values of H ( v A ) thatthey reported [17].In the following, we determine the Franck-Condon factors q v A v b and A so from themeasured H ( v A ) values [17], and then use these Franck-Condon factors to extract avalue for A so from our measurements of R A and R b . For this, a Morse potential is fittedto the term-values of the vibrational levels in the A and b state, as listed in [17]. Theparameters of the Morse potential are optimised to reproduce the measured vibrationallevels to better than 0.3 cm − . This optimization is independent of the equilibriumdistance r e . Next, the difference between the equilibrium distances of the A and b state, ∆ r e = r e ( A ) − r e ( b ), is optimised such that the vibrational level dependence of √ q v A v A − agrees with the observed vibrational level dependence of H ( v A ). We findthe best agreement for ∆ r e = 0 . r e extracted fromthe reported values for B e in the A and b state [17] of 0.0094 ˚A. Finally, we fit to theexperimentally determined rotational constants reported in [17] using only r e ( A ) as afit parameter. The data is reproduced to within 1% for r e ( A ) = 1.63098 ˚A. Consideringthe simple model for the potentials, this is an excellent agreement. The value for A so determined from this fitting procedure is A so = 8 . − and the correspondingFranck-Condon matrix is shown in Table 4. The uncertainty in the value of A so isdifficult to determine, as the main contribution to the total uncertainty stems fromthe assumption that the potentials can be approximated by Morse potentials. Weestimate this uncertainty to be at least 2.5 cm − . Table 4.
Calculated Franck-Condon factors between the A Π state and the b Σ + state. The elements high-lighted in red are used to predict the three measured values for H ( v A ) given in [17] v A \ v b R A = (6 ± × − implies a value for A so = (10 ±
2) cm − . Thevalue of R b = (4 . ± . × − implies a value for A so = (10 . ± .
5) cm − . These twovalues for A so , as well as the value for A so extracted from the H ( v A ) values, all overlap18ithin their error bars, yielding a final experimental value for A so = (10 . ± .
1) cm − .As mentioned in the previous section, we observe that the ratio of the visible to theUV fluorescence that is emitted from the A Π , v = 1 level is equal to that from the A Π , v = 0 level. Based on the derivation presented in this section we expect that thefractional emission in the visible from the A Π , v = 1 level is larger than the emissionfrom the A Π , v = 0 level by a factor of (cid:18) E b, − E A, E b, − E A, (cid:19) q q + (cid:18) E b, − E A, E b, − E A, (cid:19) q q = 1 . , (16)where the energy difference ( E b, − E A, ) = 834 cm − , and where the values for theFranck-Condon factors are taken from Table 4. The predicted 23% increase in visiblefluorescence is consistent with the experimental observation, within the experimentaluncertainty of 15%. This highlights that the spin-orbit mixing is dominated by theinteraction between vibrational levels that have the same vibrational quantum number. b Σ + , v = 0 State
In first order perturbation theory, the shift of the energy levels is given by the squareof the interaction matrix elements, divided by the energy separation of the interactinglevels. The interaction is repulsive, and for the A Π , v = 0 state all rotational levelsshift downwards by A so q / [2( E b, − E A, )]. Such an overall shift is difficult to de-termine, and is absorbed in the value for the term-energy. In the b Σ + , v = 0 state,the e -levels will be shifted upward by about the same amount, but the f -levels willhave a lower, J -dependent shift.‘Curiously’, Hebb wrote originally in 1936, the shiftof the levels in a Σ + state due to the spin-orbit interaction with a Π state has thesame J -dependence as the shift due to the spin-spin and spin-rotation interaction, andboth effects cannot be distinguished [30]. The origin of the spin-spin and spin-rotationinteraction in a Σ state has first been described in a classic paper by Kramers [31],and more general expressions have been given soon after that by Schlapp [32]. Nor-mally, the spin-spin and spin-rotation interactions are expected to be the dominanteffects and the spin-orbit interaction with a nearby Π state is expected to be only asecond order correction. Both effects add up, and this means that the values for λ and γ as found from fitting the energy levels in the b Σ + , v = 0 state should actually beinterpreted as λ = λ ss + A so q E b, − E A, ) (17) γ = γ sr + BA so q E b, − E A, ) (18)where λ ss and γ sr describe the contribution due to the spin-spin and spin-rotationinteraction in the b Σ + state, respectively, and where the additional terms describethe contribution due to the spin-orbit interaction with the nearby A Π state.When we take the final experimental value for A so , then the spin-orbit contributionto λ amounts to about +900 MHz. We conclude therefore that the value for λ ss in the b Σ + , v = 0 state is about -1800 MHz, and that about half of this value is cancelledby the spin-orbit interaction with the nearby A Π state. The spin-orbit contribution19o γ amounts only to +1.1 MHz, and this contribution can be neglected. For highervibrational levels in the b Σ + state a slightly different behaviour is expected. The spin-orbit contribution to λ remains about +900 MHz for v b = 0 −
3, but increases to about+1200 MHz for v b = 4 due to the near-resonant interaction with the v A = 5 level.For the v b = 6 level, the near-resonant contribution to λ due to spin-orbit interactionwith the v A = 7 level is negative, reducing the total spin-orbit contribution to λ toabout +600 MHz. The v b = 5 level is special, as the F levels of the b -state and the f -levels of A, v A = 6 cross, between J = 1 and J = 2, making an interpretation interms of a contribution to λ less meaningful. It is interesting to note that this crossingcauses the J -levels that belong to low N -values in the b -state to be split considerablyfurther than for the b Σ + , v = 0 state, making J a good quantum number. This stronginteraction opens a ‘doorway’ to efficiently drive transitions between the singlet andtriplet manifolds [33, 34] and when J is a good quantum number, this can be donehighly rotationally selective. Figure 8.
Potential energy curves for the lowest singlet and triplet electronic states of AlF, using preciseExpanded Morse Oscillator (EMO) functions. We obtain these EMO potentials by fitting to the point-wiseRKR potentials generated by LeRoy’s program [35] and adjust the parameters to predict the vibrational levelswith a high accuracy of 0.05 cm − (using the non-perturbed or de-perturbed values from the appendix from[17]) and from [36]. These potentials are much more precise than the simple Morse model we use in section 7,but do not allow us to treat the two electronic states independently to extract the spin-orbit interaction. Abinitio calculations indicate that the A Π state has a barrier in the region marked with a flash, which cannot bereproduced with our EMO potentials. The inset shows a more detailed view of the A Π and b Σ + potentials,with the vibrational levels indicated.
8. Conclusion
We investigated the b Σ + state of AlF and the spin-orbit interaction between thislowest electronically excited state in the triplet system and the first electronicallyexcited singlet state, the A Π state. First, we presented a low-resolution rotationalspectrum of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 transition and determined the radiative20ifetime of the b Σ + , v = 0 state to be 190(2) ns. Molecules in the a Π , v = 0 state canbe efficiently detected using a (1+2)-REMPI scheme via the b Σ + , v = 0 state even atrelatively low laser intensity. Then, we used cw laser induced fluorescence excitationspectroscopy of the b Σ + , v ′ = 0 ← X Σ + , v ′′ = 1 transition to determine the fine andhyperfine structure of the b Σ + , v = 0 state with a precision of about 10 MHz. Theeigenvalues of the hyperfine Hamiltonian have been fitted to the experimentally deter-mined line positions and all relevant spectroscopic constants have been determined.Hyperfine-resolved LIF spectra of the b Σ + , v ′ = 0 ← a Π , v ′′ = 0 band, originatingfrom all three spin-orbit manifolds in a Π, were used to improve the spin-orbit ( A )and spin-spin ( λ ) interaction parameters that determine the relative spacing of thethree Ω manifolds in the a state. Despite the highly-diagonal Franck-Condon matrixof the b − a transition, laser cooling of AlF in the triplet manifold is challenging,due to the large number of rotational branches. This is further complicated by thespin-orbit interaction between the A Π state and the b Σ + state, which is concludedto be governed by an interaction parameter A so of about 10 cm − . The spin-orbitinteraction mixes up to about 1% of the wave function of the A Π state into thewave function of the hyperfine levels in the v = 0 level of the triplet state; the exactamount of mixing depends on the J, F i ( i = 1 , ,
3) character of the hyperfine levelsin the b Σ + state. By the same mechanism, about 1% of the wave function of the b Σ + state is mixed into the wave functions of the A Π , v = 0 state; the amount oftriplet character is the same for all levels in the A state. The triplet character of theA Π , v = 0 state, causes an A Π , v = 0 → a Π , v = 0 loss below the 10 − level fromthe main A Π − X Σ + laser cooling transition. This article has been accepted for publication in Molecular Physics, published byTaylor & Francis.
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