# Spectroscopic characterization of aluminum monofluoride with relevance to
laser cooling and trapping

S. Truppe<EMAIL_ADDRESS>Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  S. Marx  Fritz-Haber-
Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
S. Kray  Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6,
14195 Berlin, Germany  M. Doppelbauer  Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  S. Hofsäss  Fritz-Haber-
Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
H. C. Schewe  Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg
4-6, 14195 Berlin, Germany  N. Walter  Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  J. Pérez-Ríos  Fritz-
Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin,
Germany  B. G. Sartakov  General Physics Institute, Russian Academy of
Sciences, Vavilovstreet 38, 119991 Moscow, Russia  G. Meijer  meijer@fhi-
berlin.mpg.de Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg
4-6, 14195 Berlin, Germany

###### Abstract

Here we report on spectroscopic measurements of the aluminum monofluoride
molecule (AlF) that are relevant to laser cooling and trapping experiments. We
measure the detailed energy level structure of AlF in the X${}^{1}\Sigma^{+}$
electronic ground state, in the A${}^{1}\Pi$ state, and in the metastable
a${}^{3}\Pi$ state. We determine the rotational, vibrational and electronic
branching ratios from the A${}^{1}\Pi$ state. We also study how the rotational
levels split and shift in external electric and magnetic fields. We find that
AlF is an excellent candidate for laser cooling on any Q-line of the
A${}^{1}\Pi$ \- X${}^{1}\Sigma^{+}$ transition and for trapping at high
densities.

The energy levels in the X${}^{1}\Sigma^{+},v=0$ state and within each
$\Omega$-manifold in the a${}^{3}\Pi,v=0$ state are determined with a relative
accuracy of a few kHz, using laser-radio-frequency multiple resonance and
ionization detection schemes in a jet-cooled, pulsed molecular beam. To
determine the hyperfine and $\Lambda$-doubling parameters we measure
transitions throughout the 0.1 MHz – 66 GHz range, between rotational levels
in the X${}^{1}\Sigma^{+},v=0$ state and between rotational and
$\Lambda$-doublet levels in all three spin-orbit manifolds of the
a${}^{3}\Pi,v=0$ state. We measure the hyperfine splitting in the A${}^{1}\Pi$
state using continuous wave (CW) laser-induced fluorescence spectroscopy of
the A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. The resolution is limited by the short radiative lifetime of the
A${}^{1}\Pi,v=0$ state, which we experimentally determine to be $1.90\pm 0.03$
ns. The hyperfine mixing of the lowest rotational levels in the A${}^{1}\Pi$
state causes a small loss from the the main laser cooling transition of 10-5.
The off-diagonal vibrational branching from the A${}^{1}\Pi,v=0$ state is
measured to be $(5.60\pm 0.02)\times 10^{-3}$ in good agreement with
theoretical predictions. The strength of the spin-forbidden
A${}^{1}\Pi,v=0\rightarrow\textrm{a}^{3}\Pi,v^{\prime}=0$ transition is
measured to be seven orders of magnitude lower than the strength of the
A${}^{1}\Pi,v=0\rightarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
transition. We determine the electric dipole moments $\mu(\textrm{X})=1.515\pm
0.004$ Debye, $\mu(\textrm{a})=1.780\pm 0.003$ Debye and
$\mu(\textrm{A})=1.45\pm 0.02$ Debye in X${}^{1}\Sigma^{+}$, $v=0$,
a${}^{3}\Pi,v=0$ and A${}^{1}\Pi,v=0$, respectively, by recording CW laser
excitation spectra in electric fields up to 150 kV/cm.

##  I  Introduction and motivation

Over the last two decades, there has been great progress in cooling and
trapping neutral molecules in the gas phase [1, 2]. Ultracold molecules can be
used for studying collisions and chemistry at low temperature [3, 4], for
precision measurements to test fundamental symmetries [5, 6] and for quantum
information and simulation [7, 8, 9, 10]. The molecules can be associated at
ultralow temperatures from pre-cooled atoms [11, 12, 13] or they can be
produced in a molecular beam and subsequently cooled and trapped. Experimental
techniques involve buffer gas cooling [14], Stark deceleration [15] and Zeeman
deceleration [16, 17], Sisyphus cooling [18] and laser cooling. Laser cooling
has so far been demonstrated for four diatomic species SrF [19], YO [20], CaF
[21], and YbF [22] and one polyatomic species SrOH [23]. Magneto-optical traps
(MOTs) for SrF [24], CaF [25, 26] and YO [27] have been demonstrated and sub-
Doppler temperatures have been reached [28]. The laser cooling of BaF [29, 30,
31], BaH [32], TlF [33], MgF [34], CaOH [35] and YbOH [36] is also being
pursued. Magnetic [37, 38] and optical [39, 40] trapping of laser cooled
molecules has been demonstrated and sympathetic [41, 42] and evaporative
cooling [43] is being explored. Ultracold molecular samples produced this way
typically have a density many orders of magnitude lower compared to the
association methods [44]. New methods are steadily being developed to deliver
more molecules at low speeds to the trapping region [45, 46, 47, 48, 49].

For all the diatomic molecules that have been laser-cooled so far, a
${}^{2}\Pi_{1/2}$ $\leftarrow$ ${}^{2}\Sigma^{+}$ transition is used [50]. In
this case, rotational branching is suppressed on the P(1) line by angular
momentum selection rules [51]. The molecule must have an excited state that
decays at a high rate to just one or a few vibrational levels in the ground
state, and there should be no accessible intermediate electronic state.
Preferably, the hyperfine structure of the molecule should be simple.
Molecules with a ${}^{1}\Sigma$ ground state and a ${}^{1}\Pi$ excited state
are very attractive candidates for laser cooling [52, 53, 54]. However, none
have been laser-cooled so far. All Q-lines of a ${}^{1}\Pi$ $\leftarrow$
${}^{1}\Sigma$ transition are rotationally closed, the hyperfine splitting in
the ${}^{1}\Sigma$ state is typically within the natural linewidth of the
optical transition, and molecules in ${}^{1}\Sigma$ states are intrinsically
more stable than radicals.

The feasibility of laser cooling of aluminum monofluoride (AlF) has been
investigated using $\it{ab~{}initio}$ quantum chemistry [55]. The study
concludes that the
A${}^{1}\Pi,v=0-\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band of AlF
around 227.5 nm has a calculated Franck-Condon factor of 0.99992 and is an
excellent candidate for cooling with just a single laser. The spontaneous
decay rate of the excited state is very high with a calculated value of
$\Gamma=1/\tau=2\pi\times 84$ MHz, where $\tau=1.89$ ns is the radiative
lifetime of the A${}^{1}\Pi,v=0$ state [56]. The spin-forbidden
a${}^{3}\Pi,v^{\prime}=0-\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band is
highly diagonal as well; it has a calculated Franck-Condon factor of 0.9967.
Furthermore, the P${}_{1}(1)$ line and all Q-lines of the
a${}^{3}\Pi-\textrm{X}^{1}\Sigma^{+}$ transition are rotationally closed.
These transitions can be used for laser cooling to final temperatures in the
low $\mathrm{\SIUnitSymbolMicro K}$ range, after pre-cooling on the much
stronger A${}^{1}\Pi-\textrm{X}^{1}\Sigma^{+}$ band. Figure 1 shows the
potential energy curves for the three electronic states relevant to laser
cooling. The main transition wavelength and the calculated Franck-Condon
factors, based on experimental spectroscopic data, are shown.

AlF is a promising candidate to reach high densities of ultracold molecular
samples. It has a binding energy of almost 7 eV and forms as a stable
constituent of aluminum-fluorine systems at high temperatures via the reaction

$2\textrm{Al(l)}+\textrm{AlF}_{3}\textrm{(g)}\rightarrow
3\textrm{AlF}\textrm{(g)}.$ (1)

Vapour pressures of tens of mbar can be reached at temperatures around
1200-1350 K [57]. Therefore, a bright beam of AlF can be produced, either
pulsed or CW. Because of the high spontaneous decay rate on the
A${}^{1}\Pi-\textrm{X}^{1}\Sigma^{+}$ band, the distance needed for laser
slowing a molecular beam to rest is only several centimeters and the capture
velocity of a MOT will be exceptionally large. The bimolecular rearrangement
channel in which two AlF molecules react to form a fluorine molecule and an
aluminum dimer is strongly endothermic. This reaction channel will therefore
not limit the lifetime of AlF molecules in dense, cold samples.

The spectroscopy of AlF is intrinsically interesting, as it is similar to
carbon monoxide, one of the most-studied diatomic molecules. AlF is heavier
and has eight electrons more than CO, shifting its electronic transitions to
the more accessible region of the spectrum. The spin-forbidden
a${}^{3}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band, for example, is centered
at 206 nm for CO (Cameron band), whereas for AlF it appears around 367 nm. The
most abundant isotopomer of CO has no nuclear spin. Aluminum and fluorine have
a single stable isotope, with a nuclear spin of I${}_{\textrm{Al}}=5/2$ for
27Al and I${}_{\textrm{F}}=1/2$ for 19F. There are not many diatomics for
which the most abundant isotopomer has this nuclear spin combination and for
which the hyperfine structure has been completely resolved. The spectroscopic
data on AlF has until now been restricted to absorption and emission
measurements in samples at high ($\geq 900$ K) temperatures with at best
Doppler-limited resolution. This data is suitable to get accurate information
on the electronic potential curves and on the ro-vibrational energy level
structure from which Franck-Condon factors can be calculated precisely. The
fine and hyperfine structure, however, has barely been resolved [58, 59, 60,
61, 62, 63, 64].

Figure 1: Potential energy curves of the three electronic states of AlF
relevant for laser cooling. The inset on the right shows the transition
wavelengths and calculated Franck-Condon factors presented in Sec. IX.2.

Cooling and trapping experiments rely on measurements of the fine and
hyperfine structure of the lowest rotational levels in X${}^{1}\Sigma^{+}$,
A${}^{1}\Pi$ and a${}^{3}\Pi$. First, the working principle of a MOT is
determined by the hyperfine structure and the magnetic g-factors. Second, the
degree of mixing of rotational levels due to hyperfine interactions determines
how far branching to other rotational levels, i.e. loss from the optical
cycling transition, occurs [33]. The vibrational branching ratios can be
calculated, but typically not to the required accuracy. Therefore, it is
essential to compare these calculations to the vibrational branching ratios
determined by experiments. Molecules that decay on the spin-forbidden
A${}^{1}\Pi\rightarrow\textrm{a}^{3}\Pi$ transition are also lost from the
optical cooling cycle. We measure the strength of this transition to quantify
this loss channel. The predicted spontaneous decay rate on the
A${}^{1}\Pi,v=0-\textrm{X}^{1}\Sigma^{+},v=0$ band requires experimental
verification. Besides, it is useful to measure how the hyperfine levels split
and shift in external electric and magnetic fields. We use a pulsed beam of
jet-cooled AlF in combination with radio-frequency, microwave, and optical
fields to experimentally determine this data.

This paper is organized in eleven sections. The early spectroscopic
characterization of AlF is discussed in Sec. II, with a particular focus on
the electronic states relevant for this study. Then the experimental setup is
described, together with the various multiple resonance excitation and
detection schemes that are used. In Sec. IV, the Hamiltonian we use to
describe the energy level structure of AlF is given. In Sec. V, overview
excitation spectra of the
a${}^{3}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v=0$ band are presented.
Radio-frequency and microwave measurements in the a${}^{3}\Pi,v=0$ state
together with the spectroscopic parameters we determine from a fit to the
Hamiltonian are presented in Sec. VI. We first analyze the hyperfine structure
in the a${}^{3}\Pi,v=0$ state, because the long radiative lifetime of this
state allows for high spectral resolution. Moreover, the hyperfine structure
in the a${}^{3}\Pi_{1}$ state is expected to resemble the one in the
A${}^{1}\Pi$ state. As shown below, the detailed knowledge of the hyperfine
structure in the a${}^{3}\Pi,v=0$ state is essential to resolve the hyperfine
structure in the X${}^{1}\Sigma^{+},v=0$ state. This measurement is presented
in Sec. VII. Section VIII presents UV excitation spectra from which we infer
the hyperfine structure and radiative lifetime of the A${}^{1}\Pi$ state. In
Sec. IX we discuss the potential loss channels from the main optical cycling
transition. We measure the intensities of the
A${}^{1}\Pi,v=0\rightarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=1$ and the
spin-forbidden A${}^{1}\Pi,v=0\rightarrow\textrm{a}^{3}\Pi,v^{\prime}$
transitions relative to the
A${}^{1}\Pi,v=0\rightarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
transition to quantify these loss channels. In Sec. X, the measurements of the
electric dipole moments in the X${}^{1}\Sigma^{+}$, the a${}^{3}\Pi$ and the
A${}^{1}\Pi$ state are presented. The paper closes with Sec. XI which
summarizes the most important results of this study and discusses the
prospects for laser cooling and electric field manipulation of AlF. Throughout
the manuscript experimental data is shown as solid black curves or black open
circles, simulated spectra are shown as solid blue curves and a fit to the
experimental data as a solid red curve.

##  II Early characterization of AlF

Over 80 years ago, George Rochester observed the first band-spectrum of AlF
[65]. He observed a system of five strong absorption bands in the 220-235 nm
region by heating aluminum fluoride (AlF3) in a graphite furnace to
temperatures up to 1973 K. Rochester did not resolve any rotational structure
but concluded from the appearance of the bands that these must be vibrational
sequences between two electronic states with nearly equal vibrational and
rotational constants. He attributed these spectra to the
A${}^{1}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band of AlF, based on a
comparison to previously measured spectra of AlCl and AlBr.

The A${}^{1}\Pi\rightarrow\textrm{X}^{1}\Sigma^{+}$ band was first observed in
emission by Rowlinson and Barrow in 1953. They used a mixture of Al and AlF3
in a hollow-cathode discharge [66]. From a detailed analysis of the
separations of the band-heads, they obtained a first estimate of the
rotational constant in the X${}^{1}\Sigma^{+},v=0$ state of $B=0.54$ cm-1. The
relative magnitude of rotational constants could be determined more
accurately, and its value in the A${}^{1}\Pi,v=0$ state was concluded to be
only about 1$\%$ larger than in the electronic ground state. In the mid
1950’s, high resolution emission spectra of AlF were recorded by Naudé and
Hugo throughout the 220-870 nm range [67, 68, 69]. This work improved the
vibrational constants, but could not reveal the rotational structure of the
A${}^{1}\Pi\rightarrow\textrm{X}^{1}\Sigma^{+}$ band. They reported on several
new band systems of AlF, spanning from the blue to the near infrared. These
bands had the A${}^{1}\Pi$ state as the common lower state. One band, centered
around 725 nm, appeared particularly strong and could be rotationally
resolved. It was assigned to the
C${}^{1}\Sigma^{+}\rightarrow\textrm{A}^{1}\Pi$ band, and a first value for
the rotational constant in the A${}^{1}\Pi,v=0$ state [68] could be
determined.

Rowlinson and Barrow also reported the first observation of, what they then
called, the b${}^{3}\Sigma^{+}\rightarrow\textrm{a}^{3}\Pi$ band of AlF in the
345-372 nm spectral region [66]. In 1974 a detailed study of the electronic
spectrum of gaseous AlF revised the labelling of this excited
${}^{3}\Sigma^{+}$ state [58]. From then on it is referred to as the
c${}^{3}\Sigma^{+}\rightarrow\textrm{a}^{3}\Pi$ band. A first estimate of the
energy of the triplet states relative to the singlet states was inferred from
spectral perturbations that were attributed to the interactions of vibrational
levels in the singlet and triplet manifolds [58]. This estimate was confirmed
when the a${}^{3}\Pi\rightarrow\textrm{X}^{1}\Sigma^{+}$ intercombination
emission was observed by Rosenwaks et al. in 1976 [70]. They recorded low
resolution spectra from flames produced by reactions of Al with SF6, NF3 and
F2. Shortly after this, the first rotationally resolved absorption spectrum of
the spin-forbidden a${}^{3}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band in AlF
was recorded [71].

The thermodynamic properties of gaseous aluminum monofluoride were also
studied from early on. The heat of formation of AlF was derived [72] and a
first value for its dissociation limit of about 6.87 eV was deduced [73]. The
dissociation limit was also derived from spectroscopic data on the
A${}^{1}\Pi$ state [74] to 7.24 eV, significantly higher than the
thermodynamic value. More than four decades later, the latter value was
revised by Bernath’s group to 6.9 eV [75], using improved spectroscopic data
on the X${}^{1}\Sigma^{+}$ state.

Figure 2: Energy level diagram of the relevant electronic states of AlF using
accurate spectroscopic data. The optical transitions used for excitation and
ionization are indicated. The lowest rotational levels in the electronic
states are shown on an expanded scale. The $\Lambda$-doubling can only be seen
in the a${}^{3}\Pi_{0}$ manifold.

Lide reported the first microwave spectrum of AlF in 1963 [60]. He heated a
mixture of powdered Al and AlF3 up to 900-975 K inside a quartz tube that
contained a microwave waveguide and recorded the spectrum in absorption via
Stark modulation spectroscopy. He observed a triplet structure for the
$J=0\rightarrow 1$ transition in the $v=0$ and $v=1$ levels of the
X${}^{1}\Sigma^{+}$ electronic ground-state, characteristic of a molecule
containing a single quadrupolar nucleus of spin 5/2. He found a quadrupole
coupling constant of $eq_{0}Q=-37.6\pm 1.0$ MHz and determined the electric
dipole moment in the $v=0$ state to be $1.53\pm 0.10$ Debye [60, 61]. A more
extensive set of millimeter and sub-millimeter measurements, including
rotational levels up to $J=14$ and vibrational levels up to $v=4$, confirmed
and further constrained the value for $eq_{0}Q$ and allowed for a more precise
determination of the vibrational and rotational constants [62, 63]. In 1974,
the Zeeman effect of the $J=0\rightarrow 1$ transition in the
X${}^{1}\Sigma^{+},v=0$ state was measured in fields of up to 4.4 T, from
which values for the $g$-factors were determined [64]. The Doppler-limited
linewidth in these measurements was about 200 kHz and no splitting due to the
19F nuclear spin could be detected.

Barrow et al. and Brown et al. obtained limited information on the hyperfine
structure in the triplet states by analyzing the line shapes of twenty
unblended rotational lines in the
b${}^{3}\Sigma^{+}\rightarrow\textrm{a}^{3}\Pi$ and
c${}^{3}\Sigma^{+}\rightarrow\textrm{a}^{3}\Pi$ bands [58, 59]. These lines
showed partly resolved sub-structure, i.e. they appeared as doublets or
triplets. Assuming that the magnetic interaction of the nuclear spin of Al
with the electrons dominates the hyperfine structure, they simulated the
experimental spectra and determined values for the Fermi contact parameter
$b_{F}\textrm{(Al)}$ for each of the three electronic states. We briefly
revisit this work in Sec. VI, which presents the hyperfine structure in the
a${}^{3}\Pi$ state in more detail.

From the available spectroscopic data, the energies of the low-lying
rotational levels in the various electronic states are known to within 0.01
cm-1. Figure 2 shows the electronic states of AlF that are relevant to the
experiments presented here, together with the optical transitions between
them.

##  III Experimental setup

The experimental setup is shown schematically in Fig. 3. It consists of a
molecular beam machine with a source chamber, a differentially pumped
preparation chamber and two differentially pumped detection chambers. The AlF
molecules are produced by laser ablation and cooled in a supersonic expansion.
A solenoid valve (General Valve, Series 9) emits short pulses of carrier gas
(Ne or Ar; 3 bar backing pressure) mixed with SF6 (2%) into a short reaction
channel. A diode-pumped, pulsed Nd:YAG laser (1064 nm, 3 mJ/pulse, 10 ns pulse
duration) with a near-Gaussian beam profile is focused to a spot size of
approximately 0.5 mm to ablate an aluminum rod that rotates back and forth
inside the reaction channel. The reaction of aluminum atoms with SF6 produces
vibrationally excited AlF molecules in the a${}^{3}\Pi$ state [70]. These
highly excited molecules are efficiently quenched in the high pressure region
of the reaction channel. The AlF molecules are translationally and internally
cooled through collisions with the carrier gas, while expanding from the
reaction channel into the source chamber through a conical nozzle. The AlF
molecules thermalize to a rotational temperature of about 10 K and have a mean
velocity of 780 m/s, when Ne is used as carrier gas). The experiment is
operated at a repetition rate of 10 Hz. About 60 mm downstream from the laser
ablation point, i.e. at $y=60$ mm, the molecular beam passes through a 5 mm
diameter skimmer that separates the source chamber from the preparation
chamber.

Figure 3: Schematic of the experimental setup. A supersonic molecular beam
passes through a skimmer into the preparation chamber, where the internal
quantum state of the AlF molecules is prepared via optical pumping. The
optical pumping can be done under field-free conditions or in a high electric
field (c). Alternatively, radio-frequency or microwave transitions can be
driven using a transmission line (a) or microwave horn antennae (b). In the
first detection chamber, resonant excitation and ionization, followed by mass-
selective detection of the parent ions can be performed. In the second
detection chamber, laser-induced fluorescence of the molecules is imaged onto
a photomultiplier tube.

In the preparation chamber, centered 190 mm behind the skimmer, the state of
the molecules can be prepared by optical pumping or resonant excitation,
driving transitions in either the
A${}^{1}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ or the
a${}^{3}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band. For optical pumping on
the A${}^{1}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band near 227.5 nm we use a
CW Ti:Sa laser that is frequency-doubled twice using commercial, resonant bow-
tie cavities, to produce up to 200 mW of UV radiation (MSquared). For resonant
excitation to selected rotational levels in the metastable a${}^{3}\Pi$ state,
i.e. to drive transitions within the
a${}^{3}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band near 367 nm, we either use
a frequency-doubled pulsed dye laser ($\approx$ 10 mJ, 5 ns pulse duration,
about 0.1 cm-1 bandwidth), a frequency-doubled pulsed dye amplifier (PDA)
seeded by a CW Ti:Sa laser ($\approx$ 10 mJ, 5 ns pulse duration, about 250
MHz linewidth) or a frequency-doubled CW Ti:Sa laser (up to
$1\text{\,}\mathrm{W}$, sub-MHz linewidth). Both, the PDL and PDA are pumped
by an injection seeded Nd:YAG laser (350 mJ, 5 ns). The molecules can be
excited under field-free conditions or in static electric fields up to 150
kV/cm (Fig. 3 (c)); during operation, the pressure is maintained below 10-6
mbar in this region.

Shortly after optical pumping or resonant excitation with (near) UV radiation,
the molecules can be exposed to radio-frequency radiation to drive electric
dipole allowed transitions between $\Lambda$-doublet or rotational levels. The
rf and microwave radiation is generated by a synthesizer (Rohde & Schwarz;
SMR20) that is phase-locked to a stable GPS frequency reference (Quartzlock).
The output of the synthesizer is connected to a fast, high-isolation switch
and coupled to free space. For frequencies in the range of 0.1 to 500 MHz a
TEM (transverse electromagnetic) transmission line is used (Fig. 3 (a)). The
transmission line consists of two square copper plates with a side length of
57 mm that are separated along $z$ by 7.5 mm. The TEM mode propagates along
$x$, orthogonal to the direction of the molecular beam. To produce radiation
near 10 GHz the output of the synthesizer is coupled to free space through a
horn antenna (Fig. 3 (b)). To produce radiation near 66 GHz, the output of the
synthesizer is connected to a frequency quadrupler (Militech, AMC-15-RFH00), a
precision attenuator and coupled to free space through a standard pyramidal
gain horn (Fig. 3 (b)). The quadrupler delivers microwave powers of up to 10
dBm. The interaction time of the molecules with the radiation is set by the
switch to values ranging from 40 to 80 $\mathrm{\SIUnitSymbolMicro s}$. In the
preparation chamber, the ambient magnetic field is compensated by three pairs
of coils that are close to Helmholtz configuration. They reduce the magnetic
field in the interaction region to $3\text{\,}\mathrm{\SIUnitSymbolMicro T}$.

For state preparation in well-defined electric fields, two high voltage
electrodes are installed in the preparation chamber to form a parallel plate
capacitor. The electrodes are polished, stainless steel disks with rounded
edges, a diameter of 55 mm and a thickness of 5 mm. The distance between the
accurately spaced plates can be set to $2.020\pm 0.010$ mm, $3.020\pm 0.010$
mm or $4.005\pm 0.010$ mm. The disks are mounted such that a high voltage of
opposite polarity can be applied to each of them, thus creating a well-defined
static electric field along the $z$-axis. Two power supplies (Spellman SL1200)
are used to produce voltages of up to $\pm 15.5$ kV. The voltage is measured
with a calibrated high voltage probe (CPS 250-M-01) to an accuracy of better
than $5\times 10^{-4}$. The molecular beam and the excitation laser beam cross
between the circular high voltage electrodes, within 3 mm from their center. A
finite element simulation of the plate geometry (COMSOL) shows that electric
field variations due to finite size effects are suppressed to below 10-5. The
electric field can, therefore, be determined with a fractional accuracy that
is mainly determined by the uncertainty in the distance between the plates.

In the first detection chamber, centered 210 mm downstream from the center of
the preparation chamber, molecules in the metastable a${}^{3}\Pi$ state can be
single-photon ionized using 193 nm light from an ArF excimer laser ($\approx$
2 mJ, 10 ns pulse duration). The molecules in the a${}^{3}\Pi$ state can also
be state-selectively ionized via resonant excitation on the
c${}^{3}\Sigma^{+}\leftarrow\textrm{a}^{3}\Pi$ transition, followed by single-
photon ionization from the c-state, i.e. via a (1+1)-resonance enhanced multi-
photon ionization (REMPI) process, using a frequency-doubled pulsed dye laser
near 360 nm. A linear time-of-flight (ToF) setup extracts the ions
perpendicular to the molecular beam and accelerates them to a microchannel
plate (MCP) detector. The signal from the MCP detector is amplified and read
into a computer on a fast (5 ns/channel) digitizer card. The ToF electrodes
are grounded during ionization and switched to high voltage to extract the
ions. This ensures field-free excitation and ionization in the detection
region. The mass-resolution $m/\Delta m$ of the ToF mass spectrometer is about
100, sufficient to unambiguously identify the AlF+ parent cations at mass 46
amu. A grounded metal plate with a vertical slit is attached to the entrance
of the ToF setup. The width of the slit can be adjusted between 5 and 15 mm.
This shields the ionization region from stray electric fields and can reduce
the residual Doppler broadening of the transitions in the
a${}^{3}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band to about 30 MHz. The
field-free REMPI ionization is parity selective and permits detecting the
transitions between $\Lambda$-doublet levels or rotational levels in the
a${}^{3}\Pi$ state against zero background. This optical-radio-
frequency/microwave-optical triple resonance ionization detection scheme has
large similarities to schemes that have been used in the past to characterize
the metastable a${}^{3}\Pi$ state of CO, for instance [76].

In the second detection chamber, at $y=715$ mm from the source, the AlF
molecules can be detected via laser-induced fluorescence (LIF) on the
A${}^{1}\Pi,v=0-\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band. The UV
fluorescence is imaged onto a photomultiplier tube, and the photo-electron
pulses are either counted (in case of low fluorescence intensity) or the
photo-current is amplified and measured.

##  IV The Molecular Hamiltonian

The Hamiltonian to describe the energy level structure of a diatomic molecule
has been considered in many articles [77, 78, 79, 80, 81] and text books [82,
83]. It can be written in a general form as

${H}=H_{\textrm{ev}}+H_{\textrm{rot}}+H_{\textrm{fs}}+H_{\textrm{hfs}},$ (2)

where $H_{\textrm{ev}}$ contains the terms describing the electronic and
vibrational part, $H_{\textrm{rot}}$ the rotational, $H_{\textrm{fs}}$ the
fine-structure and $H_{\textrm{hfs}}$ the hyperfine structure. The electronic
and vibrational term of the Hamiltonian determines the vibrational energy
levels $E_{\textrm{v}}$ for a given electronic state and can be approximated
by

$E_{\textrm{v}}=T_{e}+\hbar\omega_{e}(v+1/2)-\hbar\omega_{e}x_{e}(v+1/2)^{2}+\ldots,$
(3)

where $T_{e}$ is the electronic term energy, i.e. the minimum of the potential
energy, $\omega_{e}$ the vibrational energy, with its first order correction
term $\omega_{e}x_{e}$. The rotational term can be written as

$H_{\textrm{rot}}=A_{v}\left(\mathbf{L}\cdot\mathbf{S}\right)+B_{v}\,(\mathbf{J}-\mathbf{L}-\mathbf{S})^{2}-D_{v}\,(\mathbf{J}-\mathbf{L}-\mathbf{S})^{4},$
(4)

where $A_{v}$ is the electron spin-orbit coupling constant,
$B_{v}=B_{e}+\alpha_{e}(v+1/2)$ the rotational constant with $B_{e}$ the
rotational constant at equilibrium, $\alpha_{e}$ the anharmonicity correction
and $D_{v}$ the centrifugal distortion constant, $\mathbf{J}$ the angular
momentum of the molecule, $\bf{L}$ the total orbital angular momentum of
electron motion and $\mathbf{S}$ the total spin of the electrons. The spin-
orbit interaction splits electronic states with non-zero values of $L$ and $S$
into $\Omega$ manifolds, where $\Omega=\Lambda+\Sigma$, with $\Lambda$ the
projection of $\mathbf{L}$ and $\Sigma$ the projection of $\mathbf{S}$ along
the internuclear axis. The fine-structure part of the Hamiltonian can be
expressed as [78, 79]

$H_{\textrm{fs}}=\gamma\left(\mathbf{N}\cdot\mathbf{S}\right)+2\lambda\left(S_{\textrm{Z}}^{\,2}-\,\frac{1}{3}\mathbf{S}^{\,2}\right)+H_{\Lambda},$
(5)

where the first term describes the interaction of the electron spin $\bf{S}$
with the angular momentum $\bf{N}=\bf{J}-\bf{S}$, i.e. with the angular
momentum of pure spatial rotation of the diatomic molecule. Note that for the
fine and hyperfine structure we omit the subscript $v$ for the interaction
parameters. The second term describes the electron spin-spin interaction,
resulting from the spin-orbit interaction between a given and other electronic
states. It appears as a second order perturbation to the spin-orbit
interaction. Both of these terms do affect the $\Omega$ sub-levels but they do
not affect the $\Lambda$-splitting of the energy levels. The term
$\,H_{\Lambda}$ can be written as

$\begin{split}H_{\Lambda}=&\frac{1}{2}\\{-o(\Lambda_{+}^{2}S_{-}^{2}+\Lambda_{-}^{2}S_{+}^{2})+p(\Lambda_{+}^{2}S_{-}N_{-}+\Lambda_{-}^{2}S_{+}N_{+})\\\
&-q(\Lambda_{+}^{2}N_{-}^{2}+\Lambda_{-}^{2}N_{+}^{2})\\},\end{split}$ (6)

and contains the terms describing the second order perturbation with respect
to the spin-orbit interaction that do cause the $\Lambda$-splitting [80]. The
$+$ and $-$ subscripts designate the appropriate axial components of the
corresponding vector.

The hyperfine structure part of the Hamiltonian, $H_{\textrm{hfs}}$, describes
the interaction of the nuclear spins with the electronic and rotational
degrees of freedom. The strongest hyperfine interactions are magnetic
interactions between nuclear spins and the electrons’ degrees of freedom, as
considered in detail by Frosch and Foley [77]. For each nucleus, the magnetic
interaction with the electrons can be described using four parameters $a$,
$b$, $c$ and $d$. For the hyperfine part of the Hamiltonian we follow the
notation used by Brown and co-workers, in which the explicit expressions for
each of the terms is given [59]. We also use the Fermi contact parameter
$b_{F}=b+(1/3)c$ and $c$, instead of $b$ and $c$. The interaction strength of
the electric quadrupole moment $Q$ of the Al-nucleus with the electric field
gradient at the nucleus is determined by two parameters $eq_{0}Q$ and
$eq_{2}Q$ [59], where $q_{0}$ is the electric field gradient in the direction
of the internuclear axis and $q_{2}$ the field gradient in the perpendicular
direction.

Both, the interaction between the nuclear magnetic moments and the interaction
between the pure rotational angular momentum of the molecule with the nuclear
magnetic moments are a factor $m_{p}/m_{e}$ smaller than the electronic
contribution to the hyperfine structure, where $m_{p}$ and $m_{e}$ are the
mass of the proton and electron, respectively. The magnitude of the hyperfine
interaction involving the electron magneton is in the 200 – 2000 MHz range
whereas the nuclear interactions are in the 10 – 100 kHz range. The
Hamiltonian for the nuclear magnetic hyperfine interactions is similar to the
one for the electron hyperfine interactions, but with the electrons’ degrees
of freedom replaced by $\bf{N}$. It is convention to use the parameters
$C_{I}$ and $C^{\prime}_{I}$ for each of the nuclei to describe the nuclear
spin-rotation interaction between $\Lambda^{\prime}=\Lambda$ and
$\Lambda^{\prime}=-\Lambda$ basis wave functions, respectively, and use the
parameter $D_{1}$ to describe the spin-spin interaction between the nuclei
[84, 85].

In view of the complexity of the set of basis wave functions the numerical
calculations are based on the tensor representation of the terms in the
Hamiltonian. This facilitates the spectroscopic analysis and numerical
calculations for many polyatomic molecules [85, 86, 87] because it is not
necessary to derive the explicit formulae for matrix elements. It also
simplifies the calculation of the spectra in external electric fields.

##  V The $a^{3}\Pi,v^{\prime}=0\leftarrow X^{1}\Sigma^{+},v^{\prime\prime}=0$
band

The overall rotational structure of the spin-forbidden
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band of AlF is investigated using resonant excitation with a pulsed dye laser
around 367 nm. The beam of the dye laser is spatially overlapped in the
ionization detection region with the beam from an ArF excimer laser, whose
pulse is delayed relative to the dye laser by about 50 ns. Both laser beams
have a diameter of approximately 3 mm. The current best value for the
ionization potential (IP) of AlF is $78,472\pm 11$ cm-1 ($9.729\pm 0.001$ eV),
obtained by extrapolating the energies of a series of Rydberg states [88].
Single-photon ionization from the a${}^{3}\Pi,v=0$ state with a 6.40 - 6.42 eV
photon from the ArF laser creates AlF+ cations with an internal energy of
maximally 0.05 - 0.07 eV. The cations are therefore produced only in the $v=0$
level of their ${}^{2}\Sigma^{+}$ electronic ground state [88]. Fig. 4 shows
the low-resolution excitation spectra obtained by detecting the signal of the
parent ions in the ToF setup.

Figure 4: Excitation spectrum of the
a${}^{3}\Pi_{0},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
(a) and
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band (b) of AlF. The molecules are excited with a pulsed dye laser, followed
by ionization with an ArF excimer laser. The signal of the parent ions is
shown as a function of the excitation laser frequency. The pulse energy of the
excitation laser used to record spectrum (a) is ten times higher than the
energy used to record spectrum (b).

Figure 4 (b) shows the spectrum of the spin-forbidden
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. The spin-orbit coupling admixes a small amount of A${}^{1}\Pi$ character
to the wave function of the a${}^{3}\Pi_{1}$ state, and the transition becomes
weakly allowed, similar to the Cameron band of CO. The spin-orbit coupling
mixes levels with the same $\Omega$ quantum number, and the resulting spectrum
is equivalent to that of a ${}^{1}\Pi$ $\leftarrow$ ${}^{1}\Sigma^{+}$
transition. The observed relative intensities of the rotational lines are
consistent with a rotational temperature of 10 K. Figure 4 (a) shows the
spectrum of the spin-forbidden
a${}^{3}\Pi_{0},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. This band is about ten times weaker than the
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band shown in Fig. 4 (b), and is shifted to lower frequency by about 50 cm-1.
It is missing a Q-branch and thus resembles a ${}^{1}\Sigma^{+}$ $\leftarrow$
${}^{1}\Sigma^{+}$ band. This means that the transition becomes weakly allowed
due to the mixing of the a${}^{3}\Pi$ state with ${}^{1}\Sigma^{+}$ states. In
contrast, for metastable CO, the
a${}^{3}\Pi_{0}\leftarrow\textrm{X}^{1}\Sigma^{+}$ band mainly gets its
intensity via the $\Omega=1$ character in the wave functions of the rotational
levels in the $\Omega=0$ manifold, due to deviations from a pure Hund’s case
(a) description. This effect is much less pronounced in AlF, because the $A/B$
ratio in the a${}^{3}\Pi$ state is about four times larger for AlF than for
CO.

Using the excitation and detection scheme described above, we could not find
the
a${}^{3}\Pi_{2},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band because it is expected to be almost three orders of magnitude weaker than
the
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. The detection of this weak band is further complicated by the occurrence
of the
a${}^{3}\Pi_{1},v^{\prime}=2\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=2$
hot band in the same spectral region.

When the molecules are ionized with the ArF excimer laser, all rotational
levels in the a${}^{3}\Pi$ state are detected with the same efficiency. Using
two independent ArF excimer lasers, we determine the absolute ionization
cross-section from the a${}^{3}\Pi,v=0$ state at 193 nm in a pump-probe
experiment to be ($3.6\pm 0.5)\times 10^{-17}$ cm2, or $36\pm 5$ Mbarn which
is almost identical to the photoionization cross-section of the Al atom in
this wavelength region [89]. The excitation spectra show a constant background
signal due to the ionization of metastable AlF molecules. These are produced
in the source and are still present when the molecular beam reaches the
detector. These triplet molecules mainly reside in the a${}^{3}\Pi_{0}$
manifold due to cooling in the expansion and due to the longer radiative
lifetime of the rotational levels in this manifold. The $J=1$ level in the
a${}^{3}\Pi_{1},v=0$ state has the shortest lifetime. Nevertheless, molecules
in this level live long enough to travel the 210 mm from the preparation
chamber to the first detection chamber without significant loss. This
indicates a radiative lifetime of at least 200 $\mathrm{\SIUnitSymbolMicro
s}$. The lifetime of the corresponding rotational level in CO has been
accurately determined in Stark deceleration and electrostatic trapping
experiments as $2.63\pm 0.03$ ms [90]. Early calculations on the radiative
lifetime of this level in CO, using perturbation theory, give a value of 2.93
ms [91]. We apply the same model to AlF, assuming diagonal Franck-Condon
factors, and find a value for the radiative lifetime of about 1 ms for the
$J=1$ level in the a${}^{3}\Pi_{1},v=0$ state.

To excite the AlF molecules on weak rotational lines of the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band, we use the frequency doubled PDA described in Sec. III. For state-
selective ionization and to reduce the background ion signal we use the
frequency doubled pulsed dye laser to ionize the AlF molecules from the
a${}^{3}\Pi$ state via the c${}^{3}\Sigma^{+},v=0$ state. With this excitation
and ionization detection scheme, we can observe even the weak low-$J$
transitions to the $\Omega=2$ manifold. In Fig. 5 the spectrum of the R3(3)
transition, reaching the $J=4$, $e$-level in the a${}^{3}\Pi_{2}$ manifold, is
shown. The partly resolved hyperfine structure in this spectrum spans over
almost 3 GHz and is well reproduced by the simulations (vide infra).

Figure 5: Excitation spectrum of the R3(3) transition of the
a${}^{3}\Pi_{2},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band near 27304 cm-1, using excitation with the PDA and (1+1)-REMPI detection
via the c${}^{3}\Sigma^{+},v=0\leftarrow\textrm{a}^{3}\Pi_{1},v=0$ band around
360 nm. The simulated spectrum (solid curve) is generated using the parameters
presented in Sec. VI and VII

.

The AlF molecules can also be excited on the spin-forbidden transition using a
frequency-doubled CW Ti:Sa laser. This laser is used to record the hyperfine
resolved excitation spectra of the two lowest rotational transitions in the
Q2-branch as well as on the R2(0) line, shown in Fig. 6. We can saturate the
rotational transitions of the
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band with about 300 mW of CW radiation at 367 nm and a laser beam diameter of
2.0 mm. With about $1\text{\,}\mathrm{W}$ of CW radiation we can also detect
molecules that have been excited on the weak P1(1) line.

Figure 6: (a) High-resolution excitation spectrum of the Q2(1) and Q2(2)
transitions of the
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band, using CW laser excitation and (1+1)-REMPI detection via the Q-branch of
the c${}^{3}\Sigma^{+},v=0\leftarrow\textrm{a}^{3}\Pi_{1},v=0$ band around 360
nm. (b) High-resolution excitation spectrum of the R2(0) line of the
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. The assignment of the $F$ quantum numbers in the $J=1$ level of the
metastable state is based on the measurements presented in Sec. VI.

##  VI Hyperfine measurements in the $a^{3}\Pi,v=0$ state

To determine the hyperfine structure in the metastable state with higher
accuracy, we measure rf transitions between $\Lambda$-doublet components. In
the preparation chamber, we prepare the AlF molecules in the $J=1$ level of
the a${}^{3}\Pi_{1},v=0$ state via excitation with the PDA on the R2(0) line.
The molecules are now distributed across the negative parity components of the
$J=1$ level, i.e. in the $J=1$, $e$-levels [92]. We then pulse the rf
radiation on for a short period of 40 - 80 $\mathrm{\SIUnitSymbolMicro s}$ at
the time when the molecular pulse is inside the rf transmission line. If the
rf is tuned to a resonance frequency, the population is transferred to a level
with positive parity. Provided the electric fields inside the molecular beam
machine are kept sufficiently low, parity remains well-defined, and parity-
selective detection can be used to probe the transferred population. In the
ionization detection chamber, the AlF molecules are resonantly excited from
the metastable a${}^{3}\Pi$ state to the c${}^{3}\Sigma^{+},v=0$ state and are
subsequently ionized by the same laser. By choosing a rotational level in the
c${}^{3}\Sigma^{+}$ state with an odd rotational quantum number $N$, i.e. with
a negative parity, only molecules in the level with positive parity in the
a${}^{3}\Pi$ state are detected. In this particular case, we ionize the
molecules via the $N=3$ level to record the radio-frequency spectra
background-free.

In the $J=1$ level, there are twelve hyperfine components in total with 26
possible, electric-dipole allowed transitions between them. We determine the
transition frequencies by recording the ion signal originating from the
positive parity components as a function of the rf frequency. The strengths of
these transitions vary by over four orders of magnitude, and the rf power
coupled into the transmission line is adjusted accordingly. Transitions
between levels with small differential $g_{F}$-factors have the expected Rabi
line shape, and a fit to the data determines the line center with an accuracy
of well below 1 kHz. This is shown in Fig. 7 for the transition around 21.8264
MHz. The residual magnetic field in the interaction region broadens the line,
which can cause a systematic frequency shift. We model this effect, place an
upper bound to the frequency shift and add a magnetic field related
uncertainty to the error budget. Transitions with a large differential
$g_{F}$-factor appear considerably broader, and with a Gaussian lineshape. In
this case, a systematic uncertainty associated with the unknown lineshape is
added to the statistical uncertainty obtained from the fit. For some of the
weaker transitions, i.e. when high rf powers are needed, ac Stark shifts of
several kHz are observed. To find the transition frequency of these lines, we
determine the line center at different rf powers and extrapolate to zero
power. The transition frequencies with their statistical and systematic
uncertainties are given in the Appendix.

Figure 7: Observed $F=4,+\leftarrow F=4,-$ transition in the $J=1$ level of
the a${}^{3}\Pi_{1},v^{\prime}=0$ state. The line shows a fit to the data
using the Rabi line shape, with the background, amplitude and central
frequency as free parameters. The inset shows the Zeeman splitting of the line
when the compensation coils along $x$ and $z$ are turned off.

All 26 transitions are observed and we uniquely assign the $F$ quantum numbers
using angular momentum selection rules. We verify this assignment by applying
a static magnetic field in the preparation chamber along $x$ and/or $y$ to
measure the Zeeman splitting of some of these transitions. The inset of Fig. 7
shows the Zeeman splitting of the $F=4,+\leftarrow F=4,-$ transition when the
compensation coils along $x$ and $z$ are turned off. The splitting in three
relatively narrow $\Delta M_{F}=0$, $\pm$1 components indicates that this
transition connects two $F$ levels with almost identical $g_{F}$-factors. The
ambient magnetic field has a strength of approximately 46 $\pm$ 2
$\mathrm{\SIUnitSymbolMicro T}$ with an inclination of 67 degrees between the
lines of magnetic flux and the ($x$, $y$) plane. The $\Delta M_{F}=\pm 1$
lines are split by +/- 77 kHz which gives a magnetic $g_{F}$-factor of 0.115
$\pm$ 0.005, consistent with the theoretically predicted value of 0.113 (see
Appendix). The experimentally determined hyperfine energy level diagram of the
$J=1$ level, together with all the observed transitions, is shown in Fig. 8.

Figure 8: Hyperfine energy level diagram of the $J=1$ level in the
a${}^{3}\Pi_{1},v=0$ state. All 26 allowed transitions are shown with their
transition frequencies in MHz. To aid the identification of the levels, the
intermediate quantum number $F_{1}$ is given as well.

The level diagram shows six pairs of opposite parity $F$-levels. To identify
the hyperfine components we introduce the intermediate quantum number $F_{1}$
that labels the vectorial coupling of the rotational angular momentum
$\mathbf{J}$ with the spin of the aluminum nucleus $\mathbf{I}_{\textrm{Al}}$,
i.e. $\mathbf{F}_{1}=\mathbf{I}_{\textrm{Al}}+\bf{J}$. The spin of the
fluorine nucleus, $\mathbf{I}_{\textrm{F}}$, is then vectorially added to
$\mathbf{F}_{1}$ to get the final vector $\mathbf{F}$, i.e.
$\mathbf{F}=\mathbf{F}_{1}+\mathbf{I}_{\textrm{F}}$. The three sets of four
$F$-levels labeled by different $F_{1}$ quantum numbers in Fig. 8 reflect the
expected triplet structure arising from the nuclear spin of
${}^{27}\textrm{Al}$. The hyperfine splitting due to the nuclear spin of
${}^{19}\textrm{F}$ is of the same order of magnitude.

We fit the eigenvalues of the Hamiltonian to the measured hyperfine energy
levels of the $J=1$ level to determine preliminary values for the
spectroscopic parameters. We then use these parameters to predict the
hyperfine levels, transition frequencies and transition dipole moments in the
$J=2$ and $J=3$ level of the $\Omega=1$ manifold. The experimentally
determined transition frequencies are typically within several MHz of the
predicted frequencies. We then use the measured values to improve the
parameters in the fit or if necessary we include higher order terms. This
procedure is followed for the $J=7$ level in the $\Omega=1$ manifold, as well
as for the $J=4$ and $J=7$ levels in the $\Omega=2$ manifold. All relevant
hyperfine parameters can be determined this way. To accurately determine the
$\Lambda$-doubling parameters, we measure the hyperfine resolved
$\Lambda$-doubling transitions in the $J=0$, 1 and 2 levels of the $\Omega=0$
manifold around 10 GHz. To accurately determine the rotational splitting in
the a${}^{3}\Pi,v=0$ state, we measure several hyperfine resolved lines of the
$J=2\leftarrow 1$ transitions in both the $\Omega=1$ and $\Omega=0$ manifolds
around 66 GHz. Selected hyperfine resolved transitions for the measurements
around 10 GHz and 66 GHz are shown in Fig. 9. Compared to the rf transmission
line, the electric field distribution of the microwave radiation in free space
is not very well defined which affects the observed line shape. All the
measured frequencies of the rf and microwave transitions, together with their
experimental uncertainties, as well as the calculated frequencies with their
assignments and calculated intensities are given in the Appendix.

Figure 9: Observed line shapes for the $F=3,+\leftarrow F=2,-$
$\Lambda$-doublet transition in the a${}^{3}\Pi_{0},v=0,J=1$ level (a) and for
the $F=3,+\leftarrow F=3,-$ component of the $J=2\leftarrow J=1$ rotational
transition in the a${}^{3}\Pi_{1},v=0$ state (b).

The parameter $A$ and its second order correction $\lambda$ can not be
determined accurately from the rf and microwave measurements. We therefore
measure the center frequencies of selected rotational transitions from the
X${}^{1}\Sigma^{+},v^{\prime\prime}=0$ state to each of the $\Omega$-manifolds
in the a${}^{3}\Pi,v^{\prime}=0$ state with a wavemeter (HighFinesse WS6-600).
P and R lines reach $e$-levels and Q-lines reach $f$-levels (see Fig. 10). In
particular, we choose the P${}_{1}(1)$, the R${}_{2}(8)$ and the Q${}_{3}(9)$
lines because they reach levels in the metastable state for which the total
span of the hyperfine structure is smaller than 40 MHz. Therefore, they appear
as single lines and the resolution is mainly limited by the bandwidth of the
narrow-band PDA. The P${}_{1}(1)$ transition appears 1,762,797 $\pm$ 100 MHz
lower and the Q${}_{3}(9)$ transition appears 1,098,984 $\pm$ 120 MHz higher
in frequency than the R${}_{2}(8)$ transition. The uncertainty in the relative
frequencies is determined by tracking the wavelength of a temperature
stabilized helium neon laser coupled to the same wavemeter. Using the known
rotational energies in the electronic ground state, this gives a value of
$A=47.395\pm 0.007$ cm-1 and $\lambda=0.0887\pm 0.0011$ cm-1. We also fit the
rotationally resolved absorption spectra for the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band measured by Kopp, Lindgren and Malmberg [71], to the eigenvalues of our
Hamiltonian for the a${}^{3}\Pi$ state. However, we keep the values for the
rotational constant and the $\Lambda$-doubling parameters fixed to the values
determined from our fit to the microwave data. This way we obtain a value of
$A=47.393\pm 0.031$ cm-1 and $\lambda=0.086\pm 0.020$ cm-1, in agreement with
the values derived from our own measurements. The term-value extracted from
these Doppler-limited spectra is $E_{0}=27253.04\pm 0.01$ cm-1.

Using the second order gradient technique, we determine the hyperfine
structure parameters, the parameters describing the $\Lambda$-doubling, the
spin-rotation interaction parameter $\gamma$ and the rotational constant with
high accuracy, and the results are summarized in Table 1. For each of the
parameters, the standard deviation (SD) as well as the product of the standard
deviation with the square-root of the quality-factor Q is given; the latter is
the better measure for the accuracy with which each parameter is determined by
a fit whose parameters are correlated [93]. The parameter for the spin-spin
interaction between the nuclei is calculated from the known equilibrium
distance between the nuclei and their magnetic moments to be $D_{1}$ = 0.0066
MHz. The centrifugal distortion parameter $D_{0}$ = 0.0314 MHz in the $v=0$
level is obtained by refitting the data presented by Kopp et al. [71] using
the Hamiltonian presented above. For convenience, the vibronic energy $E_{0}$
is set to zero. In the final fit of the radio-frequency and microwave data,
both $A$ and $\lambda$ are fixed at $A_{0}=47.395$ cm-1 and $\lambda$ = 0.0887
cm-1. The centrifugal distortion term to the spin-orbit splitting is
calculated from known spectroscopic constants to $A_{D}$ = 0.1 MHz [81, 94].

Table 1:  Spectroscopic parameters of the a${}^{3}\Pi,v=0$ state. Rotational constant $B_{0}$, fine-structure constant $\gamma$, $\Lambda$-doubling parameters and hyperfine structure parameters for the a${}^{3}\Pi,v=0$ state obtained from the best fit to the experimental data together with their standard deviation (SD) and the product of SD and $\sqrt{Q}$ (all values in MHz). For the $\Lambda$-doubling parameter $o$ and for four hyperfine structure parameters, the centrifugal distortion term has been included. These terms are labelled with a superscript $(R)$ and are given directly underneath the parameter. The term named $eq_{0}Q_{LS}(\textrm{Al})$ describes the spin-orbit correction to $eq_{0}Q\textrm{(Al)}$. In the fit, the parameters $A_{0}=47.395$ cm-1, $\lambda=0.0887$ cm-1 and $D_{0}=0.0314$ MHz are fixed to the best known experimental values. The centrifugal distortion term of the spin-orbit splitting, $A_{D}$, as well as the parameter for the spin-spin interaction between the nuclei, $D_{1}$, are fixed to the calculated values of $A_{D}$ = 0.1 MHz and $D_{1}$ = 0.0066 MHz. Parameter | Value (MHz) | SD | $\textrm{SD}\cdot\sqrt{\textrm{Q}}$
---|---|---|---
$B_{0}$ | $16634.7458$ | $0.0010$ | $0.0024$
$\gamma$ | $-7.6089$ | $0.0526$ | $0.1369$
$o$ | $4968.3175$ | $0.0510$ | $2.9243$
$o^{(R)}$ | $-0.0061$ | $0.0012$ | $0.0097$
$p$ | $-24.3462$ | $0.0510$ | $4.7224$
$q$ | $-1.8176$ | $0.0023$ | $0.1798$
$a\textrm{(Al)}$ | $199.1620$ | $0.0353$ | $0.4525$
$b_{F}\textrm{(Al)}$ | $1247.9697$ | $0.2783$ | $9.2715$
$b_{F}^{(R)}\textrm{(Al)}$ | $0.0222$ | $0.0045$ | $0.1638$
$c\textrm{(Al)}$ | $-21.0093$ | $0.4124$ | $13.7410$
$c^{(R)}\textrm{(Al)}$ | $-0.0568$ | $0.0114$ | $0.5826$
$d\textrm{(Al)}$ | $121.9077$ | $0.0155$ | $0.0935$
$eq_{0}Q\textrm{(Al)}$ | $-12.9921$ | $0.0159$ | $0.0286$
$eq_{0}Q_{LS}\textrm{(Al)}$ | $-0.0392$ | $0.0042$ | $0.0097$
$eq_{2}Q\textrm{(Al)}$ | $51.1137$ | $0.0054$ | $0.0062$
$C_{I}\textrm{(Al)}$ | $-0.0569$ | $0.0169$ | $1.3288$
$C^{\prime}_{I}\textrm{(Al)})$ | $-0.0115$ | $0.0007$ | $0.0043$
$a\textrm{(F)}$ | $207.1350$ | $0.0109$ | $0.0156$
$a^{(R)}\textrm{(F)}$ | $-0.0688$ | $0.0038$ | $0.0338$
$b_{F}\textrm{(F)}$ | $169.6289$ | $0.1605$ | $0.5853$
$b_{F}^{(R)}\textrm{(F)}$ | $-0.0346$ | $0.0019$ | $0.0152$
$c\textrm{(F)}$ | $122.8043$ | $0.2596$ | $1.3787$
$d\textrm{(F)}$ | $119.2769$ | $0.0868$ | $0.4241$
$C^{\prime}_{I}\textrm{(F)}$ | $0.0346$ | $0.0041$ | $0.0202$

To reach a standard deviation of the fit comparable with the accuracy of the
radio-frequency and microwave measurements of a few kHz, we first include
centrifugal distortion terms to each $\Lambda$-doubling and hyperfine
parameter. We then analyze the correlations between the parameters and exclude
them one by one, testing each time whether the standard deviation of the fit
increases significantly. The five remaining centrifugal distortion terms,
labelled with a superscript $(R)$, are listed in Table 1, directly underneath
the corresponding parameters; if any of these is set to zero, the standard
deviation of the fit increases significantly. We find that it is relatively
straightforward to get a good fit to all the measured hyperfine levels within
a certain $\Omega$-manifold, but that some hyperfine parameters need a
correction depending on the product of $\mathbf{L}\cdot\mathbf{S}$, i.e.
depending on the $\Omega$-manifold that they are in. We first include spin-
orbit correction terms to several hyperfine parameters, but find that only the
term $(L_{0}\cdot S)\cdot D^{(1)}\cdot I^{2}\textrm{(Al)}$ is significant. It
describes the spin-orbit correction to $eq_{0}Q\textrm{(Al)}$. We name this
parameter $eq_{0}Q_{LS}\textrm{(Al)}$ and list it underneath
$eq_{0}Q\textrm{(Al)}$ in Table 1. This term has, to the best of our
knowledge, not been described in the literature before. It means that the
quadrupole component of the electrostatic potential ($q_{0}$) is slightly
different in the different $\Omega$-manifolds due to different admixing of
other electronic wave functions via the $\mathbf{L}\cdot\mathbf{S}$
interaction.

Figure 10 summarizes the results of this section. It shows the calculated
hyperfine structure for each $\Omega$-level and for rotational levels up to
$J=10$ using the parameters given in Table 1. The energies of the hyperfine
levels for a given $J$ are shown relative to the energy of the gravity center,
i.e. relative to the energy of the $J$-level in absence of any hyperfine
interaction. The accuracy of the predicted energy levels is better than 10
kHz, much less than the thickness of the lines that indicate the energy levels
in the figure. The size of the hyperfine splitting varies significantly across
the different $\Omega$-manifolds. In particular, the different behaviour of
the $e$\- and the $f$-levels in the $\Omega=1$ manifold is remarkable.

Figure 10: Hyperfine structure for rotational levels up to $J=10$ in the
a${}^{3}\Pi,v=0$ state, calculated using the parameters given in Table 1. The
energies of the levels, $E$, are given relative to the energy of the gravity
center, $E_{gc}$. Note that each $\Omega$-manifold has its own energy scale,
which is the same for the $e$\- and the $f$-levels.

Brown and co-workers observed a sub-structure in Doppler-limited rotational
lines of optical spectra that involved the a${}^{3}\Pi,v=0$ state [59]. They
neglected the interaction of the fluorine nucleus with the electron, and
determined a value for the Fermi contact parameter of the Al nucleus in the
a${}^{3}\Pi,v=0$ state of $b_{F}(\textrm{Al})=0.046\pm 0.005$ cm-1. This value
agrees with the Fermi contact parameter of $b_{F}(\textrm{Al})=0.041674\pm
0.000004$ cm-1 found here. As can be seen in Fig. 10, the energy levels in the
$\Omega=0$ manifold follow a regular pattern with relatively large hyperfine
splittings, which are mainly determined by $b_{F}(\textrm{Al})$. The
"unblended lines" that Brown and co-workers analyzed originated exclusively
from high-$J$ ($\geq 15$) rotational levels in either the $\Omega=0$ or the
$\Omega=2$ manifold. This allowed them to extract a first value for
$b_{F}(\textrm{Al})$ from the optical spectra, despite their limited
resolution [59].

##  VII Hyperfine measurements in the $X^{1}\Sigma^{+}$ state

To measure the hyperfine structure in the
X${}^{1}\Sigma^{+},v^{\prime\prime}=0$ state we drive and detect the
$J^{\prime\prime}=2\rightarrow 1$ transition near 66 GHz using the setup
described in Sec. III and illustrated in Fig. 3 (b). To maximize the signal-
to-noise ratio, we first deplete the population in the $J^{\prime\prime}=1$
level by optical pumping on the R(1) line of the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band. We
intersect the molecular beam with a laser beam (30 mW, 3 mm diameter, CW) in
the preparation chamber to empty the population in the $J^{\prime\prime}=1$
level. Subsequently, the molecules interact with microwave radiation for a
duration of 60 $\mathrm{\SIUnitSymbolMicro s}$, to transfer population from
the $J^{\prime\prime}=2$ to the $J^{\prime\prime}=1$ level. The molecules in
the $J^{\prime\prime}=1$ level are detected in the first detection chamber by
driving a specific hyperfine component of the Q${}_{2}(1)$ line to the
a${}^{3}\Pi,v^{\prime}=0$ state using a CW laser near 367 nm. From the
a${}^{3}\Pi,v^{\prime}=0$ state the molecules are ionized via the
c${}^{3}\Sigma^{+},v=0$ state and mass-selectively detected. This detection
scheme enables the selective recording of the hyperfine components of the
$J^{\prime\prime}=2\rightarrow 1$ transition even when these are spectrally
overlapping. This is demonstrated by the measurements shown in Fig. 11.

Figure 11: Observed hyperfine resolved components of the $J^{\prime\prime}=\rightarrow 1$ rotational transition in the X${}^{1}\Sigma^{+},v^{\prime\prime}=0$ state. Shown are the $F^{\prime\prime}=4,F^{\prime\prime}_{1}=9/2\rightarrow F^{\prime\prime}=3,F^{\prime\prime}_{1}=7/2$ component, detected via the $F^{\prime}=3,F^{\prime}_{1}=7/2$ hyperfine component of the a${}^{3}\Pi_{1},v^{\prime}=0,J^{\prime}=1$ level (a) and the $F^{\prime\prime}=5,F^{\prime\prime}_{1}=9/2\rightarrow F^{\prime\prime}=4,F^{\prime\prime}_{1}=7/2$ component, detected via the $F^{\prime}=4,F^{\prime}_{1}=7/2$ hyperfine component of the a${}^{3}\Pi_{1},v^{\prime}=0,J^{\prime}=1$ level (b). Table 2:  Observed frequencies f${}_{\textrm{o}}$ of the hyperfine resolved components of the $J^{\prime\prime}=2\rightarrow 1$ transition in the X${}^{1}\Sigma^{+},v^{\prime\prime}=0$ state together with the observed-minus-calculated values f${}_{\textrm{o}}$-f${}_{\textrm{c}}$, their calculated intensity (in Debye2) and the $F$, $F_{1}$ assignment of the levels involved. f${}_{\textrm{o}}$ (MHz) | f${}_{\textrm{o}}$-f${}_{\textrm{c}}$ | Int. (D2) | $F$ | $F_{1}$ | $\rightarrow$ | $F$ | $F_{1}$
---|---|---|---|---|---|---|---
$65,944.464\pm 0.002$ | -0.002 | $1.30$ | 2 | 5/2 | $\rightarrow$ | 1 | 3/2
$65,944.510\pm 0.002$ | -0.002 | $2.04$ | 3 | 5/2 | $\rightarrow$ | 2 | 3/2
$65,945.968\pm 0.004$ | 0.003 | $2.24$ | 3 | 7/2 | $\rightarrow$ | 3 | 7/2
$65,945.972\pm 0.002$ | 0.002 | $2.93$ | 4 | 7/2 | $\rightarrow$ | 4 | 7/2
$65,949.835\pm 0.004$ | -0.002 | $2.94$ | 2 | 3/2 | $\rightarrow$ | 2 | 3/2
$65,953.229\pm 0.001$ | -0.001 | $8.20$ | 4 | 9/2 | $\rightarrow$ | 3 | 7/2
$65,953.271\pm 0.001$ | -0.001 | $10.30$ | 5 | 9/2 | $\rightarrow$ | 4 | 7/2
$65,953.854\pm 0.002$ | 0.002 | $3.98$ | 3 | 7/2 | $\rightarrow$ | 2 | 5/2
$65,953.912\pm 0.002$ | 0.002 | $5.41$ | 4 | 7/2 | $\rightarrow$ | 3 | 5/2
$65,961.061\pm 0.004$ | -0.001 | $1.30$ | 2 | 3/2 | $\rightarrow$ | 3 | 5/2
Table 3:  Spectroscopic parameters determined from the best fit to the data, together with their standard deviation (SD) and the product of SD with $\sqrt{Q}$ (all values in MHz). The rotational centrifugal distortion parameter and the nuclear-nuclear spin-spin interaction term are fixed at $D_{0}=0.0312$ MHz and $D_{1}=0.0066$ MHz, respectively. Parameter | Value (MHz) | SD | $\textrm{SD}\cdot\sqrt{Q}$
---|---|---|---
$B_{0}$ | $16488.3548$ | $0.0003$ | $0.0003$
$eq_{0}Q\textrm{(Al)}$ | $-37.5260$ | $0.0069$ | $0.0074$
$C_{I}(\textrm{Al})$ | $0.0104$ | $0.0004$ | $0.0004$
$C_{I}(\textrm{F})$ | $0.0360$ | $0.0016$ | $0.0016$
$D_{1}$ | $0.0066$ |  |

Only the terms $eq_{0}Q$, $C_{I}$(Al), $C_{I}$(F) and the nuclear-nuclear
spin-spin interaction term $D_{1}$ contribute to the hyperfine structure in
the X${}^{1}\Sigma^{+},v^{\prime\prime}=0$ state. The quadrupole interaction
splits the $J^{\prime\prime}=1$ or $J^{\prime\prime}=2$ level into three or
five $F^{\prime\prime}_{1}$ components, respectively. The parameter
$C_{I}$(Al) describes a small perturbation to the main splitting. The
$C_{I}$(F) term gives rise to a splitting of the $F^{\prime\prime}_{1}$ levels
into the final $F^{\prime\prime}$ levels. Fig. 11 (a) shows a resonance
shifted by 42 kHz from the resonance shown in Fig. 11 (b). This shift is the
frequency difference of the splittings due to the interaction of the fluorine
nuclear spin in the $J^{\prime\prime}=1$, $F^{\prime\prime}_{1}=7/2$ and
$J^{\prime\prime}=2$, $F^{\prime\prime}_{1}=9/2$ levels. Table 2 lists the ten
observed hyperfine resolved transitions together with the difference between
the observed and calculated transition frequencies,
f${}_{\textrm{o}}$-f${}_{\textrm{c}}$, their calculated intensity (in Debye2)
and the $F$, $F_{1}$ assignment of the levels involved. We fit the data to the
eigenvalues of the Hamiltonian to determine the spectroscopic parameters. In
the fit we fix the rotational centrifugal distortion parameter at its best
known value of $D_{0}$ = 0.0312 MHz [71], and calculate the nuclear-nuclear
spin-spin interaction parameter $D_{1}$ from the known distance between the
nuclei and their magnetic moments as $D_{1}$ = 0.0066 MHz. The results of the
fit are presented in Table 3. The values obtained for $eq_{0}Q(\textrm{Al})$
and $C_{I}(\textrm{Al})$ agree with the values reported in the literature
[64]. There has been no previous value for $C_{I}(\textrm{F})$, which is found
to be considerably larger than $C_{I}(\textrm{Al})$.

Figure 12: Hyperfine energy level structure for the $J=0$ and $J=1$ rotational
levels in the X${}^{1}\Sigma^{+},v=0$ state.

The hyperfine energy level structure for the lowest two rotational levels in
the X${}^{1}\Sigma^{+},v=0$ state is shown in Fig. 12. The $F=2$ and $F=3$
hyperfine components of the rotational ground-state level are nearly
degenerate. There is a small splitting of less than 1 kHz due to second order
interactions between levels with different $J$ but equal $F$ quantum numbers.

##  VIII Hyperfine measurements in the $A^{1}\Pi$ state

The accuracy with which we can measure the hyperfine structure in the
A${}^{1}\Pi$ state is intrinsically limited by its short radiative lifetime.
The hyperfine structure can only be partly resolved for the lowest rotational
levels in the A${}^{1}\Pi$ state. This can be seen in the Q-branch spectrum
shown in Fig. 13. The Q(1) transition shows substructure, whereas all other
Q-lines appear broadened.

To record the LIF spectra with negligible Doppler broadening, we decrease the
transverse velocity spread in the molecular beam. For this we reduce the speed
of the molecular beam to about 600 m/s by using Ar as a carrier gas, we reduce
the skimmer opening to 2 mm diameter, and we mount a 3 mm wide vertical slit
in the molecular beam directly in front of the LIF detection zone. We measure
the residual Doppler broadening in this setup by recording several hyperfine
resolved rotational lines of the
A${}^{2}\Sigma^{+},v^{\prime}=0\leftarrow\textrm{X}^{2}\Pi_{1/2},v^{\prime\prime}=0$
band of NO. We choose NO because its optical transition occurs at 226 nm, very
close to that of AlF, whereas its excited state lifetime is about a hundred
times longer [95]. We use a mixture of 5% NO and 2% SF6 in Ar, and we also
fire the ablation laser, to reproduce the experimental conditions of the AlF
measurements. The NO spectral lines have a Gaussian lineshape with a full
width at half maximum (FWHM) of 12 MHz. This is an upper limit for the Doppler
contribution to the lineshape of the
A${}^{1}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ band of the heavier AlF
molecule.

The lowest four lines in the Q-branch as well as the isolated R(0) line of the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band of
AlF are shown in Fig. 13. Figure 13 (a) appears as a low-resolution version of
the spectrum shown in Fig. 6 (a); the spectrum of the R(0) line shown in Fig.
13 (b) appears as a low-resolution version of the spectrum of the R2(0) line
shown in Fig. 6 (b). Based on this resemblance we can assign the $F_{1}$ and
$F$ quantum numbers to the hyperfine components of the $J^{\prime}=1$ level in
the A${}^{1}\Pi,v=0$ state.

Figure 13: Observed laser-induced fluorescence excitation spectra of the Q(1),
Q(2), Q(3) and Q(4) lines (a) and of the R(0) line (b) of the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band of
AlF, together with the simulated spectra (solid curves). Note that the
frequency scale in panel (b) is a factor two smaller than in panel (a). The
sticks underneath the spectrum show the position of the individual hyperfine
components of the rotational lines and their intensities.

To determine both $B_{0}$ and $q$ in the A${}^{1}\Pi,v=0$ state, we measure
the relative frequency of the six lowest rotational lines in the Q-branch and
the frequency separation between the closely spaced Q(15), R(0) and Q(16)
lines. From the fit to the data we determine the rotational constant to
$B_{0}=0.55378\pm 0.00001$ cm-1 and the $\Lambda$-doubling parameter to
$q=-2.94\pm 0.06$ MHz. The latter value is consistent with the value of
$q=-3.0\pm 0.3$ MHz, found by refitting the original data of Naudé and Hugo
[68].

The spectra presented in Fig. 13 show the partly resolved hyperfine structure
in the A${}^{1}\Pi,v=0$ state. The observed lineshape is well approximated by
a Voigt profile. The Gaussian contribution to the Voigt profile is determined
from the NO spectra and fixed to 10 MHz (FWHM). We use four parameters to fit
to the data: the two hyperfine parameters $a(\textrm{Al})$ and
$a(\textrm{F})$, an overall frequency offset and the Lorentzian contribution
to the Voigt profile. $B_{0}$ and $q$ are fixed to the values mentioned above.
The simulated spectrum reproduces the data well with $a(\textrm{Al})=113\pm 5$
MHz and $a(\textrm{F})=181\pm 5$ MHz. The hyperfine parameters, together with
the rotational and $\Lambda$-doubling parameters for the
X${}^{1}\Sigma^{+},v=0$ state, are presented in Table 4. The $eq_{0}Q$ and
$eq_{2}Q$ terms only play a minor role in the energy level structure in the
A${}^{1}\Pi$ state and are fixed at 0 MHz and 40 MHz, respectively. The
standard deviation of the fit does not change significantly when these
parameters are changed by about 10 MHz. The simulated spectra, assuming a
rotational temperature of 6 K, are shown as solid blue curves in Fig. 13 and
fit the experimental data well. The Lorentzian contribution to the Voigt
profile has a full width at half maximum of 84 $\pm$ 1 MHz, equivalent to a
radiative lifetime of the A${}^{1}\Pi,v=0$ state of $1.90\pm 0.03$ ns. This
value is in good agreement with the calculated value of 1.89 ns [56].

Table 4:  Spectroscopic constants obtained from the best fit to the experimental data, together with their standard deviation (SD). Parameter | Value (MHz) | SD (MHz)
---|---|---
$B_{0}$ | $16601.9$ | $0.3$
$q$ | $-2.94$ | $0.06$
$a(\textrm{Al})$ | $113$ | $5$
$a(\textrm{F})$ | $181$ | $5$

The hyperfine energy level structure for the lowest rotational level in the
A${}^{1}\Pi,v=0$ state is shown in Fig. 14. The energies of the hyperfine
levels are only known to within 10 MHz.

Figure 14: Hyperfine energy level structure for the J=1 rotational level in
the A${}^{1}\Pi,v=0$ state together with the calculated $g_{F}$ factors

##  IX  Optical cycling

The number of photons an AlF molecule scatters on the Q-lines of the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band is
limited by branching to states that are not coupled by the excitation laser.
In this section we discuss these loss channels in more detail.

###  IX.1 Rotational branching

All Q-lines are rotationally closed, due to angular momentum selection rules.
However, the magnetic hyperfine interaction of the Al nuclear spin mixes the
lowest rotational levels in the A${}^{1}\Pi,v=0$ state, i.e. it mixes states
with the same total angular momentum $F$. This mixing causes a small leak from
the main optical cycling transition to higher rotational levels in the ground
electronic state. The calculated rotational branching ratios for the Q(1) line
are listed in table 5.

Table 5:  Calculated branching ratio $r_{JJ^{\prime\prime}}$ from a given $(F_{1},F)$ component of the $J=1$ level in the A${}^{1}\Pi,v=0$ state to the $J^{\prime\prime}=3$ ($r_{13}$) or to the $J^{\prime\prime}=1$ ($r_{11}$) level in the ground state. $(F^{\prime}_{1},F^{\prime})$ | $(r_{13}/r_{11})\times 10^{6}$
---|---
$(3/2,2)$ | $0.15$
$(3/2,1)$ | $0.43$
$(5/2,3)$ | $3.0$
$(5/2,2)$ | $0.99$
$(7/2,3)$ | $1.31$
$(7/2,4)$ | $3.42$

###  IX.2  Vibrational branching

There are no selection rules that limit the branching to vibrational states in
the ground electronic state. To determine the number of vibrational repump
lasers that are necessary for laser slowing and cooling, we measure the
Einstein coefficient $A_{01}$ of the
A${}^{1}\Pi,v=0\rightarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=1$ (0-1)
band relative to the Einstein $A_{00}$ coefficient of the 0-0 band. We do this
by dispersing the laser-induced fluorescence of the molecules using a compact
spectrometer. As an alternative method, we measure the ratio of Einstein
$B^{ab}_{01}/B^{ab}_{11}$ coefficients by comparing the absorption strengths
of rotational lines of the 0-1 band to the absorption strength of the same
rotational lines of the 1-1 band. In this case we measure the ratio
$A_{01}/A_{11}=\nu^{3}_{01}B^{ab}_{01}/(\nu^{3}_{11}B^{ab}_{11})$, with
$\nu_{01}$ and $\nu_{11}$ the transition frequencies of the rotational lines
of the 0-1 and 1-1 band, respectively. In a molecular beam, the population
distribution across vibrational levels in the electronic ground state might
change on the timescale of an experiment. Therefore, it is more precise to
measure the ratio $A_{01}/A_{11}$ instead of $A_{01}/A_{00}$. The experimental
results of the two methods are compared to numerical calculations.

For both methods, it is useful to increase the molecular flux and the
interaction time with the laser beams. Therefore, we use a cryogenic helium
buffer gas molecular beam [96, 97] instead of the supersonic molecular beam
described in Sec. 3. The cryogenic beam typically delivers more than two
orders of magnitude more molecules per pulse to the detection region with a
velocity that is four times lower. The design of the cryogenic beam source is
similar to the one presented in the literature [98]: a pulsed Nd:YAG laser
(Minilite II, 1064 nm, 10-20 mJ, 0.5 mm) ablates a solid aluminum target in
the presence of a constant flow of 0.01 sccm room temperature SF6 gas which is
mixed with 1 sccm of cryogenic helium gas inside a buffer gas cell. By
changing the source conditions we can increase the population in the
$\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=1$ state from about 4% to up to 15%
of the overall population. This, however, also increases the mean velocity of
the AlF molecules to over 300 m/s.

For the first method, we tune a CW laser to the Q(1) line of the 0-0 band near
227.5 nm (1 mm diameter beam, 90 mW) and cross it with the molecular beam,
about 320 mm from the source. The laser-induced fluorescence is collected,
coupled into an optical fiber and dispersed using a compact Czerny-Turner
spectrometer (Avantes AvaSpec-ULS2048-EVO, 600 $\mathrm{\SIUnitSymbolMicro m}$
fiber, 25 $\mathrm{\SIUnitSymbolMicro m}$ entrance slit) with a resolution of
1.5 nm. A typical dispersed fluorescence spectrum is shown in Fig. 15 (a). We
average 4000 shots and subtract the background, measured without the molecular
beam but with 90 mW of CW laser light present (Fig 15 (b)). To fit the data,
we first determine the line shape of the spectrometer at 227.5 nm and 231.7 nm
by coupling narrow-band laser light directly into the spectrometer. We then
use the two spectral response functions to fit the molecular spectrum. The
only fit parameters are the two emission amplitudes. The fit to the data is
shown as the red curve in Fig. 15. Figure 15 (c) shows the emission of the
molecules on the 0-1 band around 231.7 nm more clearly. The fit gives a ratio
of emission amplitudes of $\nu_{01}A_{01}/(\nu_{00}A_{00})=\left(7\pm
3\right)\times 10^{-3}$. We account for a slightly different detection
efficiency of the spectrometer at 231.7 nm and 227.5 nm. Light at 231.7 nm is
detected with a $10\pm 3\%$ higher efficiency than light at 227.5nm. To reduce
the uncertainty, we repeat this measurement 14 times which gives a final
result for the R(2) line of $A_{01}/A_{00}=\left(7.3\pm 1\right)\times
10^{-3}$.

Figure 15: Dispersed fluorescence spectrum after excitation on the Q(1) line
of the 0-0 band. The LIF is coupled into an optical fiber and dispersed using
a grating spectrometer (a). The red curve shows a fit to the data (circles)
using the measured spectral response of the spectrometer. The background
spectrum (b) is recorded without the molecular beam but with the excitation
laser present. The inset (c) shows the 0-1 emission band on an enlarged scale.

For the second method, we overlap two lasers and cross them with the molecular
beam orthogonally. The frequency of one laser is locked to the center of the
R$(2)$ line of the 0-1 band near 231.7 nm and the frequency of the second
laser is locked to the R$(2)$ line of the 1-1 band near 227.5 nm. We choose
these lines specifically because they appear as isolated single lines. The
spatial mode of each laser is cleaned up using pinholes resulting in a
Gaussian intensity distribution in the interaction region with a $1/e^{2}$
diameter of 0.46 mm. The LIF from each laser is imaged on a PMT, recorded on
alternate shots to minimize the effect of source fluctuations and averaged
over 50 shots. The two ToF profiles are shown in Fig. 16. The circles show the
ToF profile of the molecules excited on the 0-1 band with a laser power of 7.3
mW. The solid black curve results from excitation on the 1-1 band with a laser
power of 0.045 mW. The power in each laser beam is adjusted such that the
amplitudes of the two ToF profiles are nearly identical. The weak excitation
is necessary to prevent optical pumping into dark rotational states. The
difference of the two ToF profiles is shown in Fig. 16 (b). The ratio of laser
powers directly reflects the ratio of Einstein coefficients
$B^{abs}_{01}/B^{abs}_{11}=6.2\times 10^{-3}$, because
$\nu_{00}A_{00}/(\nu_{11}A_{11})\approx 1$. We repeat this measurement for 16
different pairs of laser powers. The mean and standard error of these
measurements gives the final result of $A_{01}/A_{11}=(5.72\pm 0.08)\times
10^{-3}$. We repeat the measurement using the R(1) rotational lines, giving
$A_{01}/A_{11}=(5.59\pm 0.02)\times 10^{-3}$. The results are summarized in
table 6 and compared to theoretical calculations.

First, we derive RKR potentials by using Le Roy’s program [99] to fit both
Morse and expanded Morse oscillator (EMO) functions to precise spectroscopic
data [58, 100]. These potentials are then used to calculate the Franck-Condon
factors (fAX) which are listed in Table 7 and compared to the Franck-Condon
factors derived from ab initio calculations [55]. To predict the transition
probability of the weak off-diagonal bands more precisely, we include the
variation of the transition dipole moment with internuclear distance [101]. We
use the multi-reference-configuration-interaction (MRCI) method available in
MOLPRO 2019 [102] to perform ab initio calculations of the relevant transition
dipole moment 111The MRCI calculations are fed with the natural orbitals from
a Multi-Configuration Self-Consistent Field (MCSCF) calculation with a
complete active space (CAS) consisting of nine orbitals with $A_{1}$ symmetry,
three with $B_{1}$ symmetry and three with $B_{2}$ symmetry associated with
the point group $C_{2v}$. The calculations employ the AV5Z [112] basis set for
each atom.. The results of this calculation are listed in Table 6.

Figure 16: (a) Two ToF profiles. One laser is locked to the R$(2)$ line of the 1-1 band (white circles) and a second laser is locked to the R$(2)$ line of the 0-1 band (solid black curve) of the A${}^{1}\Pi\leftarrow\textrm{X}^{1}\Sigma^{+}$ transition. The two laser beams are overlapped and have the same Gaussian intensity profile. The powers are set to 0.045 mW for the 1-1 band and 7.3 mW for the 0-1 band. The difference between the two ToFs is shown in (b) Table 6:  Calculated and measured ratios of transition probabilities. | Measured | Calculated
---|---|---
$\left(A_{01}/A_{00}\right)\times 10^{3}$ | $7.3\pm 1$ | $4.7$
$\left(A_{01}/A_{11}\right)\times 10^{3}$ | $5.59\pm 0.02$ | $4.8$
$\left(A_{02}/A_{00}\right)\times 10^{3}$ | - | $0.1$
Table 7:  Calculated Franck-Condon factors. $f_{AX}$ | Ab initio [55] | EMO | Morse
---|---|---|---
$f_{00}$ | 0.99992 | 0.996 | 0.9948
$f_{01}$ | $9\times 10^{-6}$ | 0.0040 | 0.0005
$f_{02}$ | $7\times 10^{-5}$ | $2.6\times 10^{-5}$ | $2.7\times 10^{-5}$

###  IX.3  Electronic branching - the $A^{1}\Pi,v=0\leftarrow
a^{3}\Pi,v^{\prime}=0$ band

Another loss channel from the main optical cycle is branching to the
metastable a${}^{3}\Pi$ state. To quantify this loss channel we measure the
strength of the spin-forbidden
A${}^{1}\Pi,v=0\leftarrow\textrm{a}^{3}\Pi,v^{\prime}=0$ band relative to the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band.
Figure 17 (a) shows the spectrum of the
A${}^{1}\Pi,v=0,J=1,+\leftarrow\textrm{a}^{3}\Pi_{0},v^{\prime}=0,J^{\prime}=1,-$
transition near 597 nm. This is the first time that this very weak band in AlF
is observed. To obtain this spectrum we first excite the molecules on the
R1(0) line to the $\textrm{a}^{3}\Pi_{0},v^{\prime}=0,J^{\prime}=1,-$ level
using the PDA. A few mm further downstream, at the center of the LIF detector,
the triplet molecules are excited to the A${}^{1}\Pi,v=0,J=1,+$ level by
driving the Q1(1) line. We use a CW dye laser with about 300 mW in a 1.6 mm
diameter ($1/e^{2}$) beam to drive this transition. The resulting laser-
induced fluorescence on the Q(1) line of the
A${}^{1}\Pi,v=0\rightarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band
near 227.5 nm is recorded as a function of the dye laser frequency. The
hyperfine structure in both electronic states is known from the measurements
presented above, and the simulated spectrum is shown as the solid curve in
Fig. 17 (a). We leave the Doppler width and the population distribution across
the hyperfine components in the
$\textrm{a}^{3}\Pi_{0},v^{\prime}=0,J^{\prime}=1,-$ level as fitting
parameters. The best fit is obtained by assuming a Doppler broadening of 82
MHz and an equal population distribution across hyperfine levels. Fig. 17 (c)
shows the ToF profile when the laser frequency is locked to the highest
frequency peak of the Q1(1) line (410 mW in a 1.6 mm diameter ($1/e^{2}$)
beam). There is a large peak due to stray light from the PDA, followed by the
fluorescence signal of the molecules. We can now compare the strength of this
weak, spin-forbidden transition to the strong R(0) line of the
A${}^{1}\Pi,v=0,\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band to
quantify this loss channel. We lock the CW UV laser frequency to the center of
the R(0) line and set the laser intensity such that each molecule scatters two
photons on average (about 2.7 mW in a 2.0 mm ($1/e^{2}$) diameter beam). For
molecules with diagonal Franck-Condon factors, the maximum number of photons
emitted per molecule $n$ before being optically pumped into a dark rotational
state is determined by the Hönl-London factor $p$ to $n=1/(1-p)$. For the R(0)
line, $p=2/3$ and the maximum number of emitted photons per molecule is $n=3$.
The resulting ToF profile is shown in Fig. 17 (b). The dip in the center of
the ToF profile is caused by optical pumping with the PDA on the R${}_{1}(0)$
line of the
$\textrm{a}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. It measures the fraction of ground-state molecules that are excited to
the triplet state. By comparing the amplitude of the dip in Fig. 17 (b) to the
Q${}_{1}(1)$ ToF profile shown in Fig. 17 (c) we can determine the relative
transition strength. The ratio of the two ToF amplitudes is $5\pm 0.5\times
10^{-4}$. By accounting for the different laser intensities and wavelengths we
measure a decay rate from the A${}^{1}\Pi,v=0,J=1,+/-$ level to the
$\textrm{a}^{3}\Pi_{0}$ state of $29\pm 3$ s-1 or a relative loss from the
cycling transition of $(0.56\pm 0.1)\times 10^{-7}$.

As a consistency check, we repeat the experiment for the
A${}^{1}\Pi,v=0,J=1,-$ level. Here, we drive the P1(1) transition with the PDA
to excite the molecules to the a${}^{3}\Pi_{0},v=0,J=0$ level. This is
followed by the CW dye laser excitation on the R${}_{1}(0)$ line of the
A${}^{1}\Pi,v=0\leftarrow\textrm{a}^{3}\Pi,v^{\prime}=0$ band. The resulting
decay rate of the A${}^{1}\Pi,v=0,J=1,+/-$ level to the
$\textrm{a}^{3}\Pi_{0}$ state is $38\pm 4$ s-1. In total there are six
possible decay channels to six different rotational states in the
$\textrm{a}^{3}\Pi,v^{\prime}=0$ state. We can neglect decays to higher
vibrational levels. By measuring the transition strengths of the Q${}_{1}(1)$
and R${}_{1}(0)$ lines, we can calculate the other four decay channels and
give a total decay rate from the A${}^{1}\Pi,v=0,J=1,+/-$ level to the
$\textrm{a}^{3}\Pi$ state of $53\pm 5$ s-1, or a relative loss from the
cycling transition of $(1.0\pm 0.1)\times 10^{-7}$.

Figure 17: (a) Spectrum of the
A${}^{1}\Pi,v=0,J=1,+\leftarrow\textrm{a}^{3}\Pi_{0},v^{\prime}=0,J^{\prime}=1,-$
transition near 597 nm. (b) LIF signal as a function of time when the
frequency of the 227.5 nm laser is locked to the center of the R(0) line of
the A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. The laser power is set such that each molecule scatters on average two
UV photons. Optical pumping by the PDA on the R${}_{1}(0)$ line of the
$\textrm{a}^{3}\Pi_{0},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band creates a dip in the center of the ToF profile. (c) LIF signal as a
function of time when the CW dye laser is locked to the highest frequency
component of the Q${}_{1}(1)$ line of the
A${}^{1}\Pi,v=0\leftarrow\textrm{a}^{3}\Pi_{0},v^{\prime}=0$ band. Each
excited molecule emits only a single UV photon. The signal is about 2000 times
weaker and the stray light from the PDA excitation can be seen as a sharp peak
detected 10 $\mathrm{\SIUnitSymbolMicro s}$ before the signal.

###  IX.4 Ionization

Similar to MOTs of Cd and Mg there is a loss channel due to photoionization.
The rate is given by

$\Gamma_{\textrm{ion}}=\frac{\sigma P(I,\delta)I}{\hbar\omega},$ (7)

where $P(I,\delta)$ is the fraction of AlF molecules in the excited state. We
estimate the fraction of excited state molecules using a rate model to be
about 0.02 for a peak laser intensity of $I=1$ W/cm2. Assuming that the
photoionization cross-section from the A${}^{1}\Pi$ state is similar to the
one we measure for the a${}^{3}\Pi$ state ($\sigma=3.8\times 10^{-17}$ cm-2)
the expected photo-ionization rate is $\Gamma_{\textrm{ion}}=4$ s-1. This is
equivalent to a relative loss from the optical cycling transition of
$\Gamma_{\textrm{ion}}/\Gamma=2\times 10^{-9}$.

##  X Electric dipole moment measurements

We determine the electric dipole moments in the X${}^{1}\Sigma^{+},v=0$
($\mu(\textrm{X})$), the a${}^{3}\Pi,v=0$ ($\mu(\textrm{a})$) and the
A${}^{1}\Pi,v=0$ ($\mu(\textrm{A})$) states by measuring how selected
rotational lines of optical transitions split and shift in external electric
fields. First, we excite the AlF molecules with the CW laser on the
R${}_{2}(0)$ line of the
a${}^{3}\Pi_{1},v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band in relatively low electric fields of up to 2 kV/cm. The laser
polarization is perpendicular to the direction of the external electric field.
In this setup we can only drive transitions with $\Delta M_{F}=\pm 1$. In Fig.
18, the measured R${}_{2}(0)$ transition is shown for three different electric
field strengths. The observed splitting of the $J=1$ level in the
a${}^{3}\Pi,v=0$ state directly determines $\mu(\textrm{a})$. The entire
spectrum shifts to higher frequency with increasing electric field due to the
negative Stark shift of the $J^{\prime\prime}=0$ level in the electronic
ground state. This overall frequency shift is subtracted in the spectra shown
in Fig. 18. The frequency axis in the spectra is linearized and scaled by
recording the transmission of a temperature stabilized confocal Fabry-Pérot
cavity with a free spectral range of 149.9 MHz. The observed linewidth in the
spectra is limited by the residual Doppler broadening to about 30 MHz. The
simulated spectra, assuming $\mu(\textrm{a})=1.770$ Debye, are shown in blue
(pointing downwards) underneath the experimental spectra. The observed
structure is reproduced well by the simulations. The $\Delta M_{F}=\pm 1$
components with positive and negative Stark shift mix through the hyperfine
interaction which gives intensity to a small central component. It is not a
residual $\Delta M_{F}=0$ component due to a small misalignment of the laser
polarization and the direction of the electric field. The amplitude of the
central peak decreases with increasing Stark splitting. We repeat this
measurement for different spacings between the high voltage electrodes. All
measurements agree within their experimental uncertainty with a mean value of
$\mu(\textrm{a})$ = 1.770 $\pm$ 0.010 Debye.

Figure 18: Measurement of the R${}_{2}(0)$ line of the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band under field-free conditions (a), in an electric field of 499 V/cm (b) and
in an electric field of 1987 V/cm (c), together with the simulated spectra
(blue, pointing downwards), assuming $\mu(\textrm{a})=1.770$ Debye. The
overall frequency shift of the spectra due to the negative Stark shift of the
$J^{\prime\prime}=0$ level in the electronic ground state is removed.

Dipole moments can be measured more accurately when the Stark shifts are
large, i.e. in high electric fields. However, it is challenging to measure
large frequency shifts in optical spectra with high accuracy. In high electric
fields, the Stark shifts of AlF are comparable to the spacings between
rotational energy levels. These spacings are known to kHz accuracy. Instead of
measuring the frequency shift as a function of the electric field strength, we
measure the electric field strength at which rotational lines overlap. We can
determine the field at which the component of a specific rotational line with
positive Stark shift overlaps with the component of a second rotational line
with negative Stark shift. Alternatively, we can determine the field at which
the Stark shifted components of a specific rotational line overlap with the
zero-field center frequency of a second rotational line. For the latter
method, we record the ion signal as a function of the excitation frequency in
a high electric field and under field-free conditions for each molecular pulse
simultaneously. The Stark shift of a rotational line depends linearly on both
dipole moments of the two electronic states involved. The slope of this line
changes for different rotational lines. Therefore, by comparing the Stark
shifts of different rotational lines we can determine the values for both
dipole moments. First we use this method to determine $\mu(\textrm{a})$ and
$\mu(\textrm{X})$, i.e. by driving rotational lines in the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. With the laser polarization perpendicular to the electric field, the up-
shifted component of the R${}_{2}(0)$ line overlaps with the zero-field center
frequency of the R${}_{2}(1)$ line at $58.1\pm 0.3$ kV/cm. With the laser
polarization parallel to the electric field, the $\Delta M_{F}=0$ component of
the Q${}_{2}(1)$ line with the largest Stark shift overlaps with the central
hyperfine component of the zero-field R${}_{2}(0)$ line at $86.84\pm 0.20$
kV/cm. The component of the Q${}_{2}(2)$ line with the largest Stark shift
overlaps with the central hyperfine component of the zero-field R2(0) line in
an electric field of $128.4\pm 0.6$ kV/cm. We also determine the electric
field strength at which different components of low-$J$ Q2 and R2 lines
overlap. This gives additional linear relationships between $\mu(\textrm{a})$
and $\mu(\textrm{X})$. Figure 19 shows simulated Stark spectra of the
Q${}_{2}(1)$ and Q${}_{2}(2)$ line with respect to the R${}_{2}(0)$ line.

Figure 19: Stark shift of the Q${}_{2}(1)$ and Q${}_{2}(2)$ lines with the
laser polarization orthogonal to the electric field. The $\Delta M_{F}=0$
components of the Q${}_{2}(1)$ and Q${}_{2}(2)$ lines with the largest Stark
shift overlap with the R${}_{2}(0)$ line (field-free) at approximately 87
kV/cm and 128 kV/cm, respectively. Figure 20: Density plot of
$\mu(\textrm{X})$ versus $\mu(\textrm{a})$, depicting the joint probability of
the various dipole moment measurements. The cross shows the most probable
values for the dipole moments in Debye. The 1$\sigma$ standard deviation is
shown as a dashed contour.

The resulting density plot of $\mu(\textrm{X})$ versus $\mu(\textrm{a})$ is
shown in Fig. 20. The dashed contour indicates the $1\sigma$ standard
deviation. The most probable values for the dipole moments are indicated by
the cross as $\mu(\textrm{X})=1.515\pm 0.004$ Debye and
$\mu(\textrm{a})=1.780\pm 0.003$ Debye. The value of $\mu(\textrm{X})$
reported here is in agreement with the only experimental value reported to
date of $1.53\pm 0.10$ Debye [61]. Quantum chemistry calculations give a value
of $\mu(\textrm{X})=1.54$ Debye [104]. Knowing the value of $\mu(\textrm{X})$,
a single measurement determines the value for $\mu(\textrm{A})$. We measure
the electric field at which a Stark shifted component of the R(0) line of the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band
overlaps with the field-free R(1) line. We lock the frequency of the 227.5 nm
CW laser to the zero-field center frequency of the R(1) line and monitor the
LIF signal in the second detection chamber. The CW laser beam interacts with
the molecular beam for a second time in the high electric field region in the
preparation chamber. At an electric field of $61.85\pm 0.20$ kV/cm the CW
laser is resonant with the Stark shifted component of the R(0) line, thereby
depleting the population in the $J^{\prime\prime}=0$ level in the ground
state. We monitor this depletion via two-color ionization in the first
detection chamber. This way we determine the dipole moment in the
A${}^{1}\Pi,v=0$ state to $\mu(\textrm{A})=1.45\pm 0.02$ Debye.

Figure 21: Spectrum of the R${}_{2}(0)$ line of the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band in an electric field of 152.5 kV/cm with the laser polarization parallel
to the electric field. The simulated spectrum (pointing downwards) is shown
for $\alpha_{2}=0$ ${\mathrm{\SIUnitSymbolAngstrom}}^{3}$ (dashed) and for
$\alpha_{2}=-1.25$ ${\mathrm{\SIUnitSymbolAngstrom}}^{3}$ (solid).

So far we have not included effects due to the polarizability of the molecule,
which are expected to be small. Figure 21 shows a spectrum of a Stark shifted
component of the R${}_{2}(0)$ line of the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band in an electric field of 152.5 kV/cm, with the laser polarization parallel
to the electric field. The simulated spectrum agrees better with the data if a
small polarizability anisotropy of $\alpha_{2}=-1.25$
${\mathrm{\SIUnitSymbolAngstrom}}^{3}$ is included.

##  XI Summary and Outlook

In summary, we have characterized the AlF molecule spectroscopically to
determine its suitability for laser cooling and trapping.

We have measured the spin-forbidden
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band. The spectra show that the band becomes predominantly allowed due to
spin-orbit coupling with the A${}^{1}\Pi$ state. The spectra also show that
the interaction with ${}^{1}\Sigma$ states is about one order of magnitude
weaker. We have ionized the molecules from the a${}^{3}\Pi,v=0$ state with ArF
excimer light and measured the 193 nm ionization cross-section to be $36\pm 5$
Mbarn. We have used a frequency-doubled pulsed dye laser for state-selective
resonance-enhanced multi-photon ionization from the a${}^{3}\Pi,v=0$ state.
This enabled us to record laser-radio-frequency-laser triple resonance spectra
of AlF molecules in the a${}^{3}\Pi,v=0$ state. We determine the hyperfine
energy levels with kHz accuracy and deduce precise hyperfine and rotational
constants and the magnetic g-factors.

We have also measured laser-microwave-laser-laser quadruple resonance spectra
of the X${}^{1}\Sigma^{+},v=0$ state to determine its hyperfine structure and
rotational constants. The detailed knowledge of the a${}^{3}\Pi,v=0$ state has
enabled us to resolve hyperfine splittings in the X${}^{1}\Sigma^{+},v=0$
state that are comparable to the widths of the microwave transitions. The
hyperfine splitting in the X${}^{1}\Sigma^{+},v=0$ state is smaller than the
linewidth of the main laser cooling Q(1) line of the strong
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band.

We have measured laser-induced fluorescence excitation spectra of the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band to
determine the hyperfine interaction, rotational constants and lifetime of the
A${}^{1}\Pi,v=0$ state. The spontaneous decay rate from the A${}^{1}\Pi,v=0$
state is measured to be $\Gamma=2\pi\times(83.8\pm 1.3)$ MHz. The hyperfine
level structure in the $J=1$ level resembles the structure in the $J=1$ level
of the a${}^{3}\Pi_{1},v=0$ state. All Q-lines of the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band are
rotationally closed and can be used for laser cooling. There is a small
residual rotational branching due to rotational mixing via the hyperfine
interaction in both electronic states. However, the relative loss from the
optical cycling transition is calculated to be below $10^{-5}$. We have
measured the vibrational branching ratio to
X${}^{1}\Sigma^{+},v^{\prime\prime}=1$ to be $(5.6\pm 0.02)\times 10^{-3}$ in
good agreement with the theoretical prediction. By addressing this vibrational
loss channel with a repump laser it is possible to scatter about $10^{4}$
photons before the molecules are pumped into
X${}^{1}\Sigma^{+},v^{\prime\prime}=2$. The weak R1(0) and Q1(1) lines of the
spin-forbidden A${}^{1}\Pi,v=0\leftarrow\textrm{a}^{3}\Pi,v^{\prime}=0$ band
of AlF have been observed for the first time. From their intensity, the
spontaneous decay rate from the A${}^{1}\Pi,v=0,J=1$ level to the
a${}^{3}\Pi,v=0$ state is determined as $53\pm 5$ s-1. This is equivalent to a
relative loss from the optical cycling transition of about $10^{-7}$.

We have measured laser excitation spectra on the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band and on the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band in
electric fields to determine the electric dipole moments. In the
X${}^{1}\Sigma^{+},v=0$ state, in the a${}^{3}\Pi,v=0$ state and in the
A${}^{1}\Pi,v=0$ state we determined the electric dipole moments as
$\mu(\textrm{X})=1.515\pm 0.004$ Debye, $\mu(\textrm{a})=1.780\pm 0.003$ Debye
and $\mu(\textrm{A})=1.45\pm 0.02$ Debye, respectively. Knowing the energy
level structure and dipole moments in each of the electronic states enabled us
to simulate the Stark spectra for fields up to 150 kV/cm. The accurate
spectroscopic parameters and electric dipole moments for the lowest singlets
and the lowest triplet state of AlF are an excellent benchmark for quantum
chemistry calculations.

We plan to slow AlF molecules from a molecular beam and trap and cool them in
a magneto-optical trap to produce a dipolar gas with a high phase space
density. The Q-lines on both the
A${}^{1}\Pi,v=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$ band and
on the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}=0$
band are rotationally closed and can be used for laser cooling, slowing and
magneto-optical trapping. With one vibrational repump laser it is possible to
scatter about 104 photons on the A${}^{1}\Pi-\textrm{X}^{1}\Sigma^{+}$ band
and about 4000 photons on the Q${}_{2}(1)$ line of the
a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+},v^{\prime\prime}$
band. Scattering 104 photons on the A${}^{1}\Pi-\textrm{X}^{1}\Sigma^{+}$ band
corresponds to a velocity change of 382 m/s, sufficient to slow a cryogenic
buffer gas beam or even a supersonic molecular beam to rest. The achievable
photon scattering rate is only limited by the currently available laser power
in the UV. However, CW lasers with an output power of up to 0.56 W at 229 nm
[105] and 1.4 W at 243 nm [106] have recently been demonstrated. We expect an
effective photon scattering rate on the Q(1) line of
$\Gamma_{\textrm{eff}}=\Gamma/35=2\pi\times 2.4$ MHz [107, 108]. At this rate
the stopping distance is 2 cm for a 150 m/s cryogenic buffer gas beam and 8 cm
for a 300 m/s supersonic molecular beam. To compensate for the Doppler shift
as the molecules slow down, a Zeeman slower, chirped laser slowing or white
light slowing can be used. A static, single frequency type-II MOT can be used
to trap and cool a large number of AlF molecules to a temperature of a few mK
[109]. Using a rate model [107] we estimate the capture velocity to be about
40 m/s, limited by the currently available laser power. Sub-Doppler cooling
can be used to lower the temperature to the $\mathrm{\SIUnitSymbolMicro K}$
range. Narrow-line cooling to the recoil limit is feasible on the Q2(1) line
of the a${}^{3}\Pi,v^{\prime}=0\leftarrow\textrm{X}^{1}\Sigma^{+}$ band.
However, it is challenging due to the small restoring force resulting from the
long radiative lifetime of the $J=1$ level in the a${}^{3}\Pi_{1},v=0$ state
of approximately 1 ms.

Besides optical manipulation, we can also use electric fields to slow and trap
AlF molecules. AlF compares well to metastable CO, which has been used
extensively to test and demonstrate various electric field manipulation tools
[15, 110, 90, 111]. The Stark shift to mass ratio, an important performance
metric for Stark manipulation, in the a${}^{3}\Pi_{1},v=0,J=1$ state is only
$20\%$ smaller than for the corresponding level in CO. A decisive advantage of
AlF compared to CO is that it can be controlled with electric fields not only
in its triplet but also in its electronic ground state.

##  Appendix A Observed and calculated transition frequencies in
$a^{3}\Pi,v=0$

Table LABEL:frequencies lists the 138 measured radio-frequency and microwave
transitions in the a${}^{3}\Pi,v=0$ state of AlF. The measured transition
frequency f${}_{\textrm{o}}$ is listed in the first column, followed by the
experimental uncertainty $\delta\textrm{f}_{\textrm{o}}$ (in MHz). The latter
includes the statistical uncertainty from the fit and a systematic uncertainty
determined by the Zeeman broadening of the line due to the small residual
magnetic field in the interaction region. The third column lists
$\delta\textrm{f}_{\textrm{loop}}$, which is used as an experimental
consistency check. A combination of three transitions probe an interval that
can also be measured directly, in a single transition. Their difference should
be zero within the total combined experimental uncertainty. We list this
deviation from zero for each line that is part of such a loop as
$\delta\textrm{f}_{\textrm{loop}}$. This deviation from zero is never larger
than the combined experimental uncertainty of the four lines that make up a
loop. The next twelve columns list the $\Omega$, $J$, parity ($p$) and $F$
labels of the levels involved in the transition, together with their
calculated $g_{F}$-factors. The labels with (without) a prime refer to the
upper (lower) level of the transition. However, this labelling scheme is not
unique because there can be multiple levels with the same parity and $F$
quantum number within a given $J$ and $\Omega$. To be able to distinguish
levels with the same quantum numbers and parity, we introduce an index $n$.
The lowest energy level for a given value of $F$ gets the index $n=1$ and is
in the a${}^{3}\Pi_{0}$ manifold. The column with the header $d^{2}$ (D2)
lists the calculated square of the reduced dipole moment $d$ of the transition
in Debye2. These transition strengths vary over four orders of magnitude. The
difference between the observed and calculated transition frequency,
f${}_{\textrm{o}}-\textrm{f}_{\textrm{c}}$ is given (in MHz) in the second to
last column. The last column lists the difference between the calculated
$g_{F}$-factors of the upper and the lower level of the transition. This value
correlates with the experimental uncertainty to determine the line center,
because transitions with larger $\delta g=g^{\prime}_{F}-g_{F}$ show more
Zeeman broadening.
We use all 138 transitions in the fit to the eigenvalues of the Hamiltonian to
determine the 24 spectroscopic parameters given in Table 1.

Table 8: Observed and calculated rf and microwave transition frequencies in the a^3Π,v=0 state of AlF, transition dipole moments and magnetic g_F factors. |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
f${}_{\textrm{o}}$ (MHz) |  $\delta$f${}_{\textrm{o}}$ |  $\delta$f${}_{\textrm{loop}}$ | $\Omega^{\prime}$ | $J^{\prime}$ | $p^{\prime}$ | $F^{\prime}$ | $n^{\prime}$ | $g_{F}^{\prime}$ | $\Omega$ | $J$ | $p$ | $F$ | $n$ | $g_{F}$ | d2 (D2) |  $\textrm{f}_{\textrm{o}}$-f${}_{\textrm{c}}$ |  $g_{F}^{\prime}$-$g_{F}$
$~{}9815.516$ | 0.003 | 0.003 | 0 | 0 | - |  3 |  1 | $-0.028$ | 0 | 0 | + |  2 |  1 | $-0.022$ | $~{}0.0050$ | $-0.004$ | $-0.006$
$~{}9818.512$ | 0.003 | 0.003 | 0 | 0 | - |  3 |  1 | $-0.028$ | 0 | 0 | + |  3 |  1 | $-0.016$ | $~{}0.1100$ | $-0.001$ | $-0.012$
$~{}9822.671$ | 0.003 | 0.003 | 0 | 0 | - |  2 |  1 | $-0.039$ | 0 | 0 | + |  3 |  1 | $-0.016$ | $~{}0.0050$ | $~{}0.002$ | $-0.023$
$~{}9819.673$ | 0.003 | 0.003 | 0 | 0 | - |  2 |  1 | $-0.039$ | 0 | 0 | + |  2 |  1 | $-0.022$ | $~{}0.0750$ | $-0.003$ | $-0.017$
$~{}9686.817$ | 0.005 | 0.004 | 0 | 1 | + |  2 |  2 | $-0.024$ | 0 | 1 | - |  3 |  3 | $~{}0.025$ | $~{}0.0350$ | $-0.006$ | $-0.049$
$~{}9692.796$ | 0.004 | 0.007 | 0 | 1 | + |  1 |  1 | $-0.040$ | 0 | 1 | - |  2 |  3 | $~{}0.035$ | $~{}0.0222$ | $-0.002$ | $-0.075$
$~{}9807.485$ | 0.003 | 0.003 | 0 | 1 | + |  2 |  3 | $~{}0.015$ | 0 | 1 | - |  3 |  3 | $~{}0.025$ | $~{}0.0028$ | $-0.004$ | $-0.010$
$~{}9808.913$ | 0.003 | 0.004 | 0 | 1 | + |  3 |  2 | $~{}0.011$ | 0 | 1 | - |  3 |  3 | $~{}0.025$ | $~{}0.0459$ | $-0.007$ | $-0.014$
$~{}9812.306$ | 0.002 | 0.007 | 0 | 1 | + |  2 |  3 | $~{}0.015$ | 0 | 1 | - |  2 |  3 | $~{}0.035$ | $~{}0.0327$ | $-0.006$ | $-0.020$
$~{}9821.992$ | 0.004 | 0.001 | 0 | 1 | + |  4 |  1 | $~{}0.006$ | 0 | 1 | - |  3 |  3 | $~{}0.025$ | $~{}0.0355$ | $~{}0.002$ | $-0.019$
$~{}9822.583$ | 0.004 | 0.003 | 0 | 1 | + |  3 |  3 | $~{}0.008$ | 0 | 1 | - |  3 |  3 | $~{}0.025$ | $~{}0.0025$ | $~{}0.004$ | $-0.017$
$~{}9827.406$ | 0.003 | 0.003 | 0 | 1 | + |  3 |  3 | $~{}0.008$ | 0 | 1 | - |  2 |  3 | $~{}0.035$ | $~{}0.0253$ | $~{}0.005$ | $-0.027$
$~{}9841.880$ | 0.003 | 0.001 | 0 | 1 | + |  3 |  2 | $~{}0.011$ | 0 | 1 | - |  4 |  1 | $~{}0.007$ | $~{}0.0101$ | $-0.004$ | $~{}0.004$
$~{}9845.927$ | 0.004 | 0.003 | 0 | 1 | + |  2 |  3 | $~{}0.015$ | 0 | 1 | - |  3 |  2 | $~{}0.009$ | $~{}0.0066$ | $-0.007$ | $~{}0.006$
$~{}9854.960$ | 0.005 | 0.001 | 0 | 1 | + |  4 |  1 | $~{}0.006$ | 0 | 1 | - |  4 |  1 | $~{}0.007$ | $~{}0.0107$ | $~{}0.006$ | $-0.001$
$~{}9861.027$ | 0.005 | 0.003 | 0 | 1 | + |  3 |  3 | $~{}0.008$ | 0 | 1 | - |  3 |  2 | $~{}0.009$ | $~{}0.0073$ | $~{}0.004$ | $-0.001$
$~{}9894.181$ | 0.001 | 0.007 | 0 | 1 | + |  1 |  1 | $-0.040$ | 0 | 1 | - |  1 |  1 | $-0.036$ | $~{}0.0026$ | $~{}0.001$ | $-0.004$
$~{}9899.272$ | 0.002 | 0.004 | 0 | 1 | + |  2 |  2 | $-0.024$ | 0 | 1 | - |  2 |  2 | $-0.023$ | $~{}0.0050$ | $~{}0.004$ | $-0.001$
$10013.696$ | 0.002 | 0.007 | 0 | 1 | + |  2 |  3 | $~{}0.015$ | 0 | 1 | - |  1 |  1 | $-0.036$ | $~{}0.0096$ | $~{}0.003$ | $~{}0.051$
$10021.372$ | 0.003 | 0.004 | 0 | 1 | + |  3 |  2 | $~{}0.011$ | 0 | 1 | - |  2 |  2 | $-0.023$ | $~{}0.0155$ | $~{}0.008$ | $~{}0.034$
$~{}9749.561$ | 0.003 | 0.001 | 0 | 2 | - |  4 |  2 | $~{}0.017$ | 0 | 2 | + |  5 |  1 | $~{}0.016$ | $~{}0.0018$ | $~{}0.002$ | $~{}0.001$
$~{}9760.598$ | 0.004 |  | 0 | 2 | - |  3 |  5 | $~{}0.022$ | 0 | 2 | + |  4 |  3 | $~{}0.020$ | $~{}0.0014$ | $-0.002$ | $~{}0.002$
$~{}9840.139$ | 0.004 | 0.001 | 0 | 2 | - |  5 |  1 | $~{}0.016$ | 0 | 2 | + |  5 |  1 | $~{}0.016$ | $~{}0.0031$ | $~{}0.002$ | $~{}0.000$
$~{}9851.702$ | 0.003 |  | 0 | 2 | - |  4 |  3 | $~{}0.020$ | 0 | 2 | + |  4 |  3 | $~{}0.020$ | $~{}0.0024$ | $~{}0.003$ | $~{}0.000$
$~{}9853.710$ | 0.002 |  | 0 | 2 | - |  1 |  2 | $-0.038$ | 0 | 2 | + |  2 |  4 | $~{}0.007$ | $~{}0.0018$ | $~{}0.003$ | $-0.045$
$~{}9854.234$ | 0.003 | 0.001 | 0 | 2 | - |  4 |  2 | $~{}0.017$ | 0 | 2 | + |  4 |  2 | $~{}0.019$ | $~{}0.0010$ | $-0.003$ | $-0.002$
$~{}9877.260$ | 0.005 |  | 0 | 2 | - |  3 |  4 | $~{}0.015$ | 0 | 2 | + |  3 |  4 | $~{}0.017$ | $~{}0.0026$ | $-0.005$ | $-0.002$
$~{}9881.786$ | 0.004 |  | 0 | 2 | - |  2 |  5 | $~{}0.021$ | 0 | 2 | + |  2 |  5 | $~{}0.027$ | $~{}0.0018$ | $-0.005$ | $-0.006$
$~{}9944.813$ | 0.004 | 0.001 | 0 | 2 | - |  5 |  1 | $~{}0.016$ | 0 | 2 | + |  4 |  2 | $~{}0.019$ | $~{}0.0035$ | $-0.002$ | $-0.003$
$~{}9953.603$ | 0.003 |  | 0 | 2 | - |  4 |  3 | $~{}0.020$ | 0 | 2 | + |  3 |  5 | $~{}0.025$ | $~{}0.0027$ | $-0.001$ | $-0.005$
$~{}9966.078$ | 0.002 |  | 0 | 2 | - |  2 |  4 | $~{}0.006$ | 0 | 2 | + |  1 |  2 | $-0.045$ | $~{}0.0015$ | $-0.004$ | $~{}0.051$
$~{}~{}~{}~{}1.079$ | 0.002 | 0.001 | 1 | 1 | - |  1 |  7 | $-0.227$ | 1 | 1 | + |  1 |  7 | $-0.227$ | $~{}1.2300$ | $~{}0.002$ | $~{}0.000$
$~{}~{}~{}~{}6.484$ | 0.002 | 0.007 | 1 | 1 | - |  3 | 13 | $~{}0.039$ | 1 | 1 | + |  2 | 12 | $~{}0.073$ | $~{}0.0014$ | $~{}0.001$ | $-0.034$
$~{}~{}~{}10.902$ | 0.002 | 0.005 | 1 | 1 | - |  2 | 11 | $-0.151$ | 1 | 1 | + |  2 | 11 | $-0.149$ | $~{}2.6900$ | $~{}0.002$ | $-0.002$
$~{}~{}~{}13.428$ | 0.003 | 0.005 | 1 | 1 | + |  3 | 14 | $~{}0.150$ | 1 | 1 | - |  3 | 14 | $~{}0.148$ | $~{}7.4000$ | $-0.007$ | $~{}0.002$
$~{}~{}~{}21.379$ | 0.001 | 0.007 | 1 | 1 | - |  2 | 12 | $~{}0.074$ | 1 | 1 | + |  2 | 12 | $~{}0.073$ | $~{}0.6420$ | $-0.003$ | $~{}0.001$
$~{}~{}~{}21.744$ | 0.001 | 0.006 | 1 | 1 | - |  3 | 13 | $~{}0.039$ | 1 | 1 | + |  3 | 13 | $~{}0.038$ | $~{}0.4910$ | $-0.003$ | $~{}0.001$
$~{}~{}~{}21.826$ | 0.003 | 0.002 | 1 | 1 | + |  4 | 13 | $~{}0.113$ | 1 | 1 | - |  4 | 13 | $~{}0.113$ | $~{}9.0800$ | $-0.006$ | $~{}0.000$
$~{}~{}~{}36.644$ | 0.002 | 0.005 | 1 | 1 | - |  2 | 12 | $~{}0.074$ | 1 | 1 | + |  3 | 13 | $~{}0.038$ | $~{}0.0018$ | $-0.001$ | $~{}0.036$
$~{}~{}~{}74.353$ | 0.004 | 0.004 | 1 | 1 | - |  4 | 13 | $~{}0.113$ | 1 | 1 | + |  3 | 14 | $~{}0.150$ | $~{}0.0458$ | $~{}0.006$ | $-0.037$
$~{}~{}101.298$ | 0.004 | 0.002 | 1 | 1 | + |  1 |  7 | $-0.227$ | 1 | 1 | - |  2 | 11 | $-0.151$ | $~{}0.0128$ | $~{}0.003$ | $-0.076$
$~{}~{}109.610$ | 0.005 | 0.004 | 1 | 1 | + |  4 | 13 | $~{}0.113$ | 1 | 1 | - |  3 | 14 | $~{}0.148$ | $~{}0.1050$ | $-0.004$ | $-0.035$
$~{}~{}113.279$ | 0.004 | 0.005 | 1 | 1 | - |  1 |  7 | $-0.227$ | 1 | 1 | + |  2 | 11 | $-0.149$ | $~{}0.0373$ | $~{}0.007$ | $-0.078$
$~{}~{}120.377$ | 0.008 | 0.005 | 1 | 1 | + |  2 | 12 | $~{}0.073$ | 1 | 1 | - |  1 |  7 | $-0.227$ | $~{}3.6300$ | $~{}0.001$ | $~{}0.300$
$~{}~{}142.835$ | 0.007 | 0.002 | 1 | 1 | - |  2 | 12 | $~{}0.074$ | 1 | 1 | + |  1 |  7 | $-0.227$ | $~{}3.6500$ | $~{}0.001$ | $~{}0.301$
$~{}~{}188.903$ | 0.005 | 0.007 | 1 | 1 | + |  3 | 14 | $~{}0.150$ | 1 | 1 | - |  2 | 12 | $~{}0.074$ | $~{}3.8100$ | $~{}0.001$ | $~{}0.076$
$~{}~{}196.856$ | 0.006 | 0.005 | 1 | 1 | - |  3 | 14 | $~{}0.148$ | 1 | 1 | + |  2 | 12 | $~{}0.073$ | $~{}3.8300$ | $~{}0.007$ | $~{}0.075$
$~{}~{}203.805$ | 0.006 | 0.007 | 1 | 1 | + |  3 | 14 | $~{}0.150$ | 1 | 1 | - |  3 | 13 | $~{}0.039$ | $~{}0.1010$ | $~{}0.005$ | $~{}0.111$
$~{}~{}207.488$ | 0.007 | 0.006 | 1 | 1 | + |  3 | 13 | $~{}0.038$ | 1 | 1 | - |  2 | 11 | $-0.151$ | $~{}5.4200$ | $~{}0.004$ | $~{}0.189$
$~{}~{}212.117$ | 0.006 | 0.005 | 1 | 1 | - |  3 | 14 | $~{}0.148$ | 1 | 1 | + |  3 | 13 | $~{}0.038$ | $~{}0.0245$ | $~{}0.005$ | $~{}0.110$
$~{}~{}222.754$ | 0.005 | 0.006 | 1 | 1 | + |  2 | 12 | $~{}0.073$ | 1 | 1 | - |  2 | 11 | $-0.151$ | $~{}0.0459$ | $~{}0.007$ | $~{}0.224$
$~{}~{}240.135$ | 0.007 | 0.005 | 1 | 1 | - |  3 | 13 | $~{}0.039$ | 1 | 1 | + |  2 | 11 | $-0.149$ | $~{}5.4000$ | $~{}0.004$ | $~{}0.188$
$~{}~{}255.034$ | 0.006 | 0.005 | 1 | 1 | - |  2 | 12 | $~{}0.074$ | 1 | 1 | + |  2 | 11 | $-0.149$ | $~{}0.0321$ | $~{}0.004$ | $~{}0.223$
$~{}~{}299.902$ | 0.006 | 0.004 | 1 | 1 | - |  4 | 13 | $~{}0.113$ | 1 | 1 | + |  3 | 13 | $~{}0.038$ | $~{}5.4700$ | $~{}0.009$ | $~{}0.075$
$~{}~{}299.985$ | 0.006 | 0.004 | 1 | 1 | + |  4 | 13 | $~{}0.113$ | 1 | 1 | - |  3 | 13 | $~{}0.039$ | $~{}5.4100$ | $~{}0.006$ | $~{}0.074$
$~{}~{}430.512$ | 0.010 | 0.005 | 1 | 1 | - |  3 | 14 | $~{}0.148$ | 1 | 1 | + |  2 | 11 | $-0.149$ | $~{}0.0160$ | $~{}0.015$ | $~{}0.297$
$~{}~{}433.038$ | 0.010 | 0.005 | 1 | 1 | + |  3 | 14 | $~{}0.150$ | 1 | 1 | - |  2 | 11 | $-0.151$ | $~{}0.0099$ | $~{}0.007$ | $~{}0.301$
$~{}~{}~{}~{}5.823$ | 0.002 | 0.002 | 1 | 2 | + |  4 | 14 | $~{}0.055$ | 1 | 2 | - |  4 | 14 | $~{}0.054$ | $~{}1.9800$ | $~{}0.006$ | $~{}0.001$
$~{}~{}~{}20.560$ | 0.005 |  | 1 | 2 | - |  2 | 13 | $~{}0.002$ | 1 | 2 | + |  1 |  8 | $-0.154$ | $~{}1.0800$ | $~{}0.000$ | $~{}0.156$
$~{}~{}~{}23.798$ | 0.003 |  | 1 | 2 | - |  4 | 15 | $~{}0.076$ | 1 | 2 | + |  4 | 15 | $~{}0.075$ | $~{}3.8900$ | $-0.009$ | $~{}0.001$
$~{}~{}~{}25.431$ | 0.001 |  | 1 | 2 | - |  2 | 14 | $~{}0.075$ | 1 | 2 | + |  1 |  9 | $~{}0.077$ | $~{}0.9420$ | $-0.005$ | $-0.002$
$~{}~{}~{}28.224$ | 0.001 |  | 1 | 2 | + |  3 | 15 | $~{}0.039$ | 1 | 2 | - |  3 | 15 | $~{}0.039$ | $~{}0.5000$ | $~{}0.003$ | $~{}0.000$
$~{}~{}~{}46.617$ | 0.003 |  | 1 | 2 | - |  3 | 15 | $~{}0.039$ | 1 | 2 | + |  2 | 13 | $~{}0.001$ | $~{}1.6200$ | $-0.006$ | $~{}0.038$
$~{}~{}~{}49.659$ | 0.002 |  | 1 | 2 | - |  3 | 16 | $~{}0.076$ | 1 | 2 | + |  2 | 14 | $~{}0.076$ | $~{}1.0800$ | $-0.003$ | $~{}0.000$
$~{}~{}~{}61.370$ | 0.003 | 0.002 | 1 | 2 | + |  4 | 15 | $~{}0.075$ | 1 | 2 | - |  4 | 14 | $~{}0.054$ | $~{}0.0343$ | $~{}0.009$ | $~{}0.021$
$~{}~{}~{}93.480$ | 0.003 | 0.002 | 1 | 2 | - |  5 | 13 | $~{}0.061$ | 1 | 2 | + |  4 | 15 | $~{}0.075$ | $~{}0.0760$ | $-0.012$ | $-0.014$
$~{}~{}115.544$ | 0.003 |  | 1 | 2 | + |  5 | 13 | $~{}0.061$ | 1 | 2 | - |  4 | 14 | $~{}0.054$ | $~{}1.1700$ | $~{}0.009$ | $~{}0.007$
$~{}~{}116.955$ | 0.003 |  | 1 | 2 | + |  1 |  9 | $~{}0.077$ | 1 | 2 | - |  1 |  8 | $-0.152$ | $~{}0.0034$ | $-0.004$ | $~{}0.229$
$~{}~{}120.115$ | 0.002 | 0.002 | 1 | 2 | + |  4 | 14 | $~{}0.055$ | 1 | 2 | - |  3 | 15 | $~{}0.039$ | $~{}1.6400$ | $~{}0.002$ | $~{}0.016$
$~{}~{}133.810$ | 0.004 |  | 1 | 2 | - |  3 | 16 | $~{}0.076$ | 1 | 2 | + |  2 | 13 | $~{}0.001$ | $~{}0.0116$ | $-0.008$ | $~{}0.075$
$~{}~{}149.029$ | 0.003 | 0.002 | 1 | 2 | - |  5 | 13 | $~{}0.061$ | 1 | 2 | + |  4 | 14 | $~{}0.055$ | $~{}1.1100$ | $-0.007$ | $~{}0.006$
$~{}~{}175.662$ | 0.004 | 0.002 | 1 | 2 | + |  4 | 15 | $~{}0.075$ | 1 | 2 | - |  3 | 15 | $~{}0.039$ | $~{}0.0403$ | $~{}0.005$ | $~{}0.036$
$~{}~{}181.424$ | 0.004 |  | 1 | 2 | + |  3 | 16 | $~{}0.076$ | 1 | 2 | - |  2 | 13 | $~{}0.002$ | $~{}0.0011$ | $-0.005$ | $~{}0.074$
$~{}~{}~{}26.397$ | 0.002 |  | 1 | 3 | - |  4 | 16 | $~{}0.040$ | 1 | 3 | + |  4 | 16 | $~{}0.039$ | $~{}1.0400$ | $~{}0.007$ | $~{}0.001$
$~{}~{}~{}58.145$ | 0.003 |  | 1 | 3 | + |  5 | 15 | $~{}0.047$ | 1 | 3 | - |  5 | 14 | $~{}0.040$ | $~{}0.3110$ | $-0.011$ | $~{}0.007$
$~{}~{}~{}58.206$ | 0.002 |  | 1 | 3 | + |  5 | 14 | $~{}0.039$ | 1 | 3 | - |  4 | 16 | $~{}0.040$ | $~{}0.6560$ | $-0.011$ | $-0.001$
$~{}~{}~{}89.015$ | 0.002 |  | 1 | 3 | - |  4 | 16 | $~{}0.040$ | 1 | 3 | + |  3 | 17 | $~{}0.040$ | $~{}0.7340$ | $-0.002$ | $~{}0.000$
$~{}~{}110.090$ | 0.003 |  | 1 | 3 | + |  5 | 15 | $~{}0.047$ | 1 | 3 | - |  4 | 16 | $~{}0.040$ | $~{}0.0051$ | $-0.007$ | $~{}0.007$
$~{}~{}~{}~{}0.294$ | 0.003 |  | 1 | 7 | + |  8 | 17 | $~{}0.016$ | 1 | 7 | - |  8 | 17 | $~{}0.016$ | $~{}0.9120$ | $~{}0.000$ | $~{}0.000$
$~{}~{}~{}~{}5.226$ | 0.003 |  | 1 | 7 | - |  9 | 13 | $~{}0.015$ | 1 | 7 | + |  8 | 17 | $~{}0.016$ | $~{}0.0178$ | $-0.004$ | $-0.001$
$~{}~{}~{}~{}5.362$ | 0.002 |  | 1 | 7 | - |  7 | 18 | $~{}0.017$ | 1 | 7 | + |  8 | 16 | $~{}0.015$ | $~{}0.0801$ | $~{}0.000$ | $~{}0.002$
$~{}~{}~{}~{}8.043$ | 0.003 |  | 1 | 7 | - |  9 | 14 | $~{}0.015$ | 1 | 7 | + |  8 | 17 | $~{}0.016$ | $~{}0.0134$ | $~{}0.001$ | $-0.001$
$~{}~{}~{}11.373$ | 0.003 |  | 1 | 7 | + |  8 | 17 | $~{}0.016$ | 1 | 7 | - |  7 | 19 | $~{}0.018$ | $~{}0.0440$ | $~{}0.002$ | $-0.002$
$~{}~{}~{}18.589$ | 0.002 |  | 1 | 7 | - |  8 | 16 | $~{}0.016$ | 1 | 7 | + |  8 | 16 | $~{}0.015$ | $~{}0.8280$ | $~{}0.000$ | $~{}0.001$
$~{}~{}~{}27.045$ | 0.003 | 0.002 | 1 | 7 | + |  9 | 13 | $~{}0.014$ | 1 | 7 | - | 10 | 10 | $~{}0.013$ | $~{}0.0444$ | $~{}0.000$ | $~{}0.001$
$~{}~{}~{}33.727$ | 0.003 | 0.002 | 1 | 7 | + |  9 | 13 | $~{}0.014$ | 1 | 7 | - |  9 | 14 | $~{}0.015$ | $~{}0.5340$ | $-0.002$ | $-0.001$
$~{}~{}~{}45.060$ | 0.002 |  | 1 | 7 | - |  6 | 20 | $~{}0.020$ | 1 | 7 | + |  7 | 18 | $~{}0.017$ | $~{}0.0606$ | $~{}0.000$ | $~{}0.003$
$~{}~{}~{}46.428$ | 0.003 |  | 1 | 7 | + |  9 | 13 | $~{}0.014$ | 1 | 7 | - |  8 | 16 | $~{}0.016$ | $~{}0.0736$ | $~{}0.002$ | $-0.002$
$~{}~{}~{}55.504$ | 0.002 | 0.002 | 1 | 7 | - |  5 | 22 | $~{}0.025$ | 1 | 7 | + |  6 | 21 | $~{}0.021$ | $~{}0.0212$ | $-0.001$ | $~{}0.004$
$~{}~{}~{}62.319$ | 0.002 | 0.002 | 1 | 7 | - |  5 | 23 | $~{}0.025$ | 1 | 7 | + |  6 | 21 | $~{}0.021$ | $~{}0.0097$ | $-0.003$ | $~{}0.004$
$~{}~{}~{}72.488$ | 0.003 |  | 1 | 7 | - |  8 | 16 | $~{}0.016$ | 1 | 7 | + |  7 | 18 | $~{}0.017$ | $~{}0.0877$ | $~{}0.002$ | $-0.001$
$~{}~{}~{}75.855$ | 0.002 |  | 1 | 7 | - |  5 | 22 | $~{}0.025$ | 1 | 7 | + |  6 | 20 | $~{}0.020$ | $~{}0.0291$ | $~{}0.000$ | $~{}0.005$
$~{}~{}~{}78.724$ | 0.003 |  | 1 | 7 | + |  9 | 14 | $~{}0.015$ | 1 | 7 | - |  8 | 16 | $~{}0.016$ | $~{}0.0006$ | $~{}0.008$ | $-0.001$
$~{}~{}~{}80.565$ | 0.003 | 0.002 | 1 | 7 | - |  4 | 24 | $~{}0.031$ | 1 | 7 | + |  5 | 23 | $~{}0.025$ | $~{}0.0006$ | $-0.004$ | $~{}0.006$
$~{}~{}~{}88.363$ | 0.002 | 0.002 | 1 | 7 | - |  5 | 22 | $~{}0.025$ | 1 | 7 | + |  5 | 23 | $~{}0.025$ | $~{}0.3950$ | $~{}0.000$ | $~{}0.000$
$~{}~{}~{}95.180$ | 0.003 | 0.002 | 1 | 7 | - |  5 | 23 | $~{}0.025$ | 1 | 7 | + |  5 | 23 | $~{}0.025$ | $~{}0.2150$ | $~{}0.000$ | $~{}0.000$
$~{}~{}102.205$ | 0.002 |  | 1 | 7 | - |  6 | 20 | $~{}0.020$ | 1 | 7 | + |  5 | 23 | $~{}0.025$ | $~{}0.0154$ | $~{}0.001$ | $-0.005$
$~{}~{}106.043$ | 0.003 | 0.002 | 1 | 7 | + | 10 | 10 | $~{}0.013$ | 1 | 7 | - | 10 | 10 | $~{}0.013$ | $~{}1.1700$ | $~{}0.004$ | $~{}0.000$
$~{}~{}109.587$ | 0.003 |  | 1 | 7 | - |  6 | 21 | $~{}0.021$ | 1 | 7 | + |  5 | 23 | $~{}0.025$ | $~{}0.0157$ | $~{}0.002$ | $-0.004$
$~{}~{}112.727$ | 0.003 | 0.002 | 1 | 7 | + | 10 | 10 | $~{}0.013$ | 1 | 7 | - |  9 | 14 | $~{}0.015$ | $~{}0.0172$ | $~{}0.005$ | $-0.002$
$~{}~{}131.992$ | 0.003 | 0.002 | 1 | 7 | - |  4 | 24 | $~{}0.031$ | 1 | 7 | + |  4 | 24 | $~{}0.031$ | $~{}0.4810$ | $-0.005$ | $~{}0.000$
$~{}~{}139.792$ | 0.001 | 0.002 | 1 | 7 | - |  5 | 22 | $~{}0.025$ | 1 | 7 | + |  4 | 24 | $~{}0.031$ | $~{}0.0206$ | $~{}0.001$ | $-0.006$
$~{}~{}146.608$ | 0.002 | 0.001 | 1 | 7 | - |  5 | 23 | $~{}0.025$ | 1 | 7 | + |  4 | 24 | $~{}0.031$ | $~{}0.0244$ | $~{}0.000$ | $-0.006$
$~{}~{}~{}~{}0.524$ | 0.006 |  | 2 | 4 | + |  5 | 28 | $~{}0.166$ | 2 | 4 | - |  5 | 28 | $~{}0.166$ | $~{}4.5000$ | $~{}0.001$ | $~{}0.000$
$~{}~{}~{}~{}0.569$ | 0.006 |  | 2 | 4 | + |  4 | 30 | $~{}0.210$ | 2 | 4 | - |  4 | 30 | $~{}0.210$ | $~{}3.9700$ | $-0.002$ | $~{}0.000$
$~{}~{}~{}~{}1.721$ | 0.005 |  | 2 | 7 | - | 10 | 22 | $~{}0.041$ | 2 | 7 | + | 10 | 22 | $~{}0.041$ | $~{}4.5700$ | $~{}0.002$ | $~{}0.000$
$~{}~{}~{}~{}2.034$ | 0.006 |  | 2 | 7 | - |  9 | 26 | $~{}0.046$ | 2 | 7 | + |  9 | 26 | $~{}0.046$ | $~{}4.1700$ | $~{}0.001$ | $~{}0.000$
$~{}~{}~{}~{}2.779$ | 0.004 | 0.001 | 2 | 7 | - |  9 | 25 | $~{}0.044$ | 2 | 7 | + |  9 | 25 | $~{}0.044$ | $~{}3.8800$ | $-0.002$ | $~{}0.000$
$~{}~{}~{}~{}3.086$ | 0.004 | 0.004 | 2 | 7 | - |  8 | 29 | $~{}0.050$ | 2 | 7 | + |  8 | 29 | $~{}0.050$ | $~{}3.5500$ | $-0.002$ | $~{}0.000$
$~{}~{}~{}~{}3.397$ | 0.003 | 0.001 | 2 | 7 | - |  8 | 28 | $~{}0.048$ | 2 | 7 | + |  8 | 28 | $~{}0.048$ | $~{}3.3000$ | $-0.004$ | $~{}0.000$
$~{}~{}~{}~{}3.556$ | 0.004 |  | 2 | 7 | - |  5 | 34 | $~{}0.075$ | 2 | 7 | + |  5 | 34 | $~{}0.075$ | $~{}2.1800$ | $-0.006$ | $~{}0.000$
$~{}~{}~{}~{}3.666$ | 0.100 | 0.004 | 2 | 7 | - |  7 | 30 | $~{}0.054$ | 2 | 7 | + |  7 | 30 | $~{}0.054$ | $~{}2.8200$ | $-0.006$ | $~{}0.000$
$~{}~{}~{}~{}3.666$ | 0.100 | 0.004 | 2 | 7 | - |  7 | 31 | $~{}0.056$ | 2 | 7 | + |  7 | 31 | $~{}0.056$ | $~{}3.0300$ | $-0.005$ | $~{}0.000$
$~{}~{}~{}~{}3.684$ | 0.100 |  | 2 | 7 | - |  6 | 32 | $~{}0.062$ | 2 | 7 | + |  6 | 32 | $~{}0.062$ | $~{}2.4400$ | $-0.007$ | $~{}0.000$
$~{}~{}~{}~{}3.720$ | 0.005 |  | 2 | 7 | - |  4 | 35 | $~{}0.095$ | 2 | 7 | + |  4 | 35 | $~{}0.095$ | $~{}1.9100$ | $-0.007$ | $~{}0.000$
$~{}~{}~{}~{}3.754$ | 0.004 |  | 2 | 7 | - |  5 | 35 | $~{}0.078$ | 2 | 7 | + |  5 | 35 | $~{}0.078$ | $~{}2.3300$ | $-0.004$ | $~{}0.000$
$~{}~{}~{}~{}3.861$ | 0.004 |  | 2 | 7 | - |  6 | 33 | $~{}0.064$ | 2 | 7 | + |  6 | 33 | $~{}0.064$ | $~{}2.6200$ | $-0.008$ | $~{}0.000$
$~{}~{}~{}85.166$ | 0.004 |  | 2 | 7 | + |  9 | 25 | $~{}0.044$ | 2 | 7 | - |  8 | 29 | $~{}0.050$ | $~{}0.0096$ | $-0.001$ | $-0.006$
$~{}~{}118.002$ | 0.004 |  | 2 | 7 | - |  6 | 32 | $~{}0.062$ | 2 | 7 | + |  5 | 34 | $~{}0.075$ | $~{}0.2400$ | $-0.002$ | $-0.013$
$~{}~{}129.340$ | 0.004 |  | 2 | 7 | - |  5 | 35 | $~{}0.078$ | 2 | 7 | + |  4 | 35 | $~{}0.095$ | $~{}0.0494$ | $~{}0.000$ | $-0.017$
$~{}~{}137.062$ | 0.004 |  | 2 | 7 | - |  6 | 33 | $~{}0.064$ | 2 | 7 | + |  5 | 35 | $~{}0.078$ | $~{}0.1540$ | $~{}0.000$ | $-0.014$
$~{}~{}139.858$ | 0.004 |  | 2 | 7 | - |  7 | 30 | $~{}0.054$ | 2 | 7 | + |  6 | 32 | $~{}0.062$ | $~{}0.3070$ | $-0.002$ | $-0.008$
$~{}~{}148.712$ | 0.005 |  | 2 | 7 | + |  7 | 31 | $~{}0.056$ | 2 | 7 | - |  6 | 33 | $~{}0.064$ | $~{}0.2030$ | $~{}0.011$ | $-0.008$
$~{}~{}153.672$ | 0.004 |  | 2 | 7 | + |  8 | 28 | $~{}0.048$ | 2 | 7 | - |  7 | 30 | $~{}0.054$ | $~{}0.3170$ | $~{}0.008$ | $-0.006$
$~{}~{}168.989$ | 0.004 | 0.004 | 2 | 7 | + |  8 | 29 | $~{}0.050$ | 2 | 7 | - |  7 | 31 | $~{}0.056$ | $~{}0.1940$ | $~{}0.004$ | $-0.006$
$~{}~{}174.809$ | 0.004 | 0.001 | 2 | 7 | + |  9 | 25 | $~{}0.044$ | 2 | 7 | - |  8 | 28 | $~{}0.048$ | $~{}0.2700$ | $~{}0.004$ | $-0.004$
$~{}~{}175.737$ | 0.004 | 0.004 | 2 | 7 | - |  8 | 29 | $~{}0.050$ | 2 | 7 | + |  7 | 31 | $~{}0.056$ | $~{}0.1940$ | $-0.007$ | $-0.006$
$~{}~{}180.986$ | 0.004 | 0.001 | 2 | 7 | - |  9 | 25 | $~{}0.044$ | 2 | 7 | + |  8 | 28 | $~{}0.048$ | $~{}0.2700$ | $~{}0.000$ | $-0.004$
$~{}~{}195.015$ | 0.004 |  | 2 | 7 | - |  9 | 26 | $~{}0.046$ | 2 | 7 | + |  8 | 29 | $~{}0.050$ | $~{}0.1260$ | $-0.009$ | $-0.004$
$~{}~{}200.643$ | 0.005 |  | 2 | 7 | - | 10 | 22 | $~{}0.041$ | 2 | 7 | + |  9 | 25 | $~{}0.044$ | $~{}0.1650$ | $-0.003$ | $-0.003$
$64825.398$ | 0.001 |  | 0 | 2 | + |  2 |  5 | $~{}0.027$ | 0 | 1 | - |  2 |  3 | $~{}0.035$ | $~{}3.7500$ | $-0.001$ | $-0.008$
$64825.964$ | 0.001 |  | 0 | 2 | + |  3 |  4 | $~{}0.017$ | 0 | 1 | - |  3 |  3 | $~{}0.025$ | $~{}5.4100$ | $~{}0.003$ | $-0.008$
$64830.784$ | 0.002 |  | 0 | 2 | + |  3 |  4 | $~{}0.017$ | 0 | 1 | - |  2 |  3 | $~{}0.035$ | $~{}0.1730$ | $~{}0.001$ | $-0.018$
$65067.937$ | 0.002 |  | 0 | 2 | + |  4 |  3 | $~{}0.020$ | 0 | 1 | - |  3 |  2 | $~{}0.009$ | $11.4000$ | $-0.004$ | $~{}0.011$
$65074.328$ | 0.002 |  | 0 | 2 | + |  5 |  1 | $~{}0.016$ | 0 | 1 | - |  4 |  1 | $~{}0.007$ | $14.3000$ | $-0.001$ | $~{}0.009$
$66466.114$ | 0.001 |  | 1 | 2 | + |  2 | 14 | $~{}0.076$ | 1 | 1 | - |  2 | 12 | $~{}0.074$ | $~{}2.8500$ | $~{}0.000$ | $~{}0.002$
$66471.700$ | 0.001 |  | 1 | 2 | + |  3 | 15 | $~{}0.039$ | 1 | 1 | - |  3 | 13 | $~{}0.039$ | $~{}4.2300$ | $~{}0.000$ | $~{}0.000$
$66563.594$ | 0.002 |  | 1 | 2 | + |  4 | 14 | $~{}0.055$ | 1 | 1 | - |  3 | 13 | $~{}0.039$ | $~{}5.5000$ | $-0.001$ | $~{}0.016$

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