JOURNAL OF CHEMICAL PHYSICS
VOLUME 113, NUMBER 21
1 DECEMBER 2000
Alkali–helium exciplex formation on the surface of helium nanodroplets.
II. A time-resolved study
J. Reho, J. Higgins, K. K. Lehmann, and G. Scoles
Department of Chemistry, Princeton University, Princeton, New Jersey 08544
共Received 9 December 1999; accepted 7 September 2000兲
We have monitored the time evolution of the fluorescence of K* He exciplexes formed on the
surface of helium nanodroplets using reversed time-correlated single photon counting. In modeling
the present data and that from our previous work on Na* He, we find that partial spin–orbit coupling
as well as the extraction energy of helium atoms from the droplet contribute to the observed
dynamics of both K* He and Na* He formation, which differ considerably after either D 1 (n 2 P 1/2
←n 2 S 1/2) or D 2 (n 2 P 3/2←n 2 S 1/2) excitation for both K(n⫽4) and Na(n⫽3). Our quantitative
prediction of the Na* He formation dynamics coupled with preliminary data on and modeling of the
formation dynamics of K* He allow for extrapolation to the case of Rb* He. Spin–orbit
considerations combined with a simple model of helium atom extraction from the matrix reveal the
following predicted trend: as the choice of the alkali guest atom is moved down the periodic table,
alkali atom–He exciplex formation along the 1 2 ⌸ 3/2 surface occurs faster while formation along the
1 2 ⌸ 1/2 surface occurs more slowly, ceasing to occur at all in the case of Rb. © 2000 American
Institute of Physics. 关S0021-9606共00兲01245-9兴
I. INTRODUCTION
tive to extend our model to the case of the Rb* He exciplex.
The work described below, along with the Na* He results
presented in an earlier publication4 共where we have reported
the rise times of the Na* He fluorescence in Table I兲, represent the first time-resolved measurements of exciplex formation on a liquid helium surface. In addition to improving our
understanding of alkali atom–liquid helium interactions, this
work also provides a well-characterized example of a desorption induced by an electronic transition 共DIET兲, a common
surface science process5 which has been characterized for the
related system of H atoms adsorbed on He surfaces.6,7 Since
the time of these experiments, studies of K on helium droplets have been done in the femtosecond regime and likewise
offer promising results for furthering the understanding of
atom–liquid helium interactions.8
The work presented here has allowed us to investigate
the effect of long-range potential barriers induced by spin–
orbit coupling. In our previous study of Na* He
fluorescence,4 we had concluded that while spin–orbit effects could be shown to be important, they alone were not
able to account for the observed dynamics. While in the
present work the theoretical treatment is extended to include
other contributions to the dynamics not related to spin–orbit
effects, the latter are still found to play a crucial role in the
dynamic behavior of NaHe, and have an even greater effect
on KHe. The cold fluid environment 共0.38 K兲9 of He nanodroplets is ideally suited to the study of such small barriers to
recombination. As the existence of such barriers would allow
for increasing concentrations of light metal dopants in ultracold hydrogen-based fuels, precise knowledge of such lowlying barriers is relevant to improving the properties and
increasing the efficiency of solid rocket propellants.10
As shown in the preceding article, the dispersed emission which follows the excitation of the D lines of K or Na
atoms located on the surface of helium nanodroplets provides good evidence for alkali atom–He exciplex formation.
Indeed, along with sharp emissions corresponding to the
D 2 (n 2 P 3/2→n 2 S 1/2) and D 1 (n 2 P 1/2→n 2 S 1/2) lines of gas
phase K(n⫽4) or Na(n⫽3), in each case we detect a broad
structured feature which extends from the atomic D lines
several thousand cm⫺1 into the red. All but the red tail of this
emission can be assigned to the bound–free emission from
K* He and Na* He exciplexes on the basis of spectral simulations calculated using existing interaction potentials.
While alkali atom–He1,2 exciplex formation is exclusively a surface phenomenon, the formation of the Ag*He2
exciplex has been observed upon excitation of Ag atoms in
bulk liquid helium.1 However, for Na* (3 P) atoms solvated
in bulk liquid helium, no fluorescence has been detected,2
leading to the conclusion that a ring of several He atoms
forms around the node of the p orbital in a time much shorter
than the fluorescence lifetime of the 3 P state of Na.3 Since in
the equilibrated excited state Na* Hen structure the distance
between the He atoms of the ring and the Na* core is very
short, the ground state NaHen repulsive surface crosses the
excited state well, leading to the observed total quenching of
fluorescence. On the droplet surface, however, the same excited state potential causes the formation of an alkali atom–
He1,2 exciplex, as desorption 共impossible in the bulk兲 occurs
after the binding of the alkali atom to one or two He atoms,
as previously reported.4
We have used the emission of K* He and Na* He in order to follow the formation dynamics of these species with a
temporal resolution of tens of picoseconds. The successful
modeling of the data from these species is taken as an incen0021-9606/2000/113(21)/9694/8/$17.00
9694
© 2000 American Institute of Physics
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J. Chem. Phys., Vol. 113, No. 21, 1 December 2000
II. EXPERIMENT
The experiments reported here involve the formation of
helium nanodroplets doped by gas phase pick up of alkali
atoms and their spectroscopic probing by laser-induced fluorescence. Reversed time-correlated photon counting is used
here to temporally resolve the fluorescence generated upon
D-line excitation of these surface-bound alkali atoms by
mode-locked quasi-cw lasers. The temporal resolution
achievable in these experiments depends not only on the
width and stability of the laser pulses but also on the quality
of the fast electronics employed when using the timecorrelated photon counting detection system. As the nanodroplet formation and doping with alkali atoms has been
described in the preceding article, here below we will limit
ourselves to the description of the laser systems employed to
excite the alkali atoms, and the reversed time-correlated
single photon counting apparatus used for detection.
A. Laser systems
A mode-locked, frequency-doubled YAG laser
共Quantronix 416兲 synchronously pumps a folded-cavity dye
laser 共Spectra Physics 35兲 lasing on Rhodamine 6G dye, producing pulses of 10–12 ps full width at half maximum
共FWHM兲 as determined by autocorrelation. The line width of
the laser is 2 cm⫺1, which is twice the Fourier transform
limit. A home-built pulse picker employing a Crystal Technology 3500 acoustooptic modulator 共AOM兲 is used to decrease the pulse repetition rate of the dye output from 76 to
3.8 MHz, which is the rate used in the NaHe exciplex experiments.
Acquisition of the time evolution of K* He fluorescence
is accomplished through the use of a Coherent Antares
mode-locked, doubled YAG laser which pumps a cavitydumped Coherent 700 dye laser equipped with a model 7220
cavity dumper running LDS 751 dye. The repetition rate of
the dye laser is altered from its nominal 76 MHz rate to 3.8
MHz by use of the cavity dumper, while maintaining output
powers on the order of nanojoules per pulse. The profile of
the output pulse again shows a spectral FWHM of 20 cm⫺1
and a temporal width of 10–12 ps.
B. Detection and signal processing
Laser-induced fluorescence of the Na-doped He droplets
is collected by a single-mirror optic and transported to a
microchannel plate detector 共Hammamatsu R2807U-07兲
through a multimode, incoherent fiber bundle. The fluorescence signal arising from the plate detector is amplified and
passed through a constant fraction discriminator 共Tennelec
TC454兲, providing the ‘‘start’’ signal for an EG&G Ortec
457 time-to-amplitude converter 共TAC兲. The laser pulse that
triggers the fluorescent event is detected by a fast photodiode, amplified and appropriately delayed, and processed by
another channel of the constant fraction discriminator. This
signal is then used as a ‘‘stop’’ signal in the TAC. This
reversed method insures that each ‘‘start’’ pulse is followed
by a ‘‘stop’’ pulse inside the time window of the TAC. The
output of the TAC is processed by a multichannel analyzer
Alkali–helium exciplex formation. II
9695
共Oxford Nucleus PCA II兲 and binned, thus compiling the
experimental histograms that give the emission intensity as a
function of time after each excitation pulse.
The multichannel analyzer 共MCA兲 contains 8192 channels which we have set to cover an 80 ns time interval for the
Na*He experiments and a 320 ns time period for the K* He
work. The instrument was calibrated by a pulse generator
共Stanford Research Systems兲 and was found to be linear over
the 80 and 320 ns time windows with standard deviations of
the fit less than 20 ps. Count rates encountered in our experiments using this photon counting assembly varied from 200
to 3000 photons per second. At these count rates, the pileup
error in the TAC is negligible.11
We have measured the instrument response function by
passing H2 through a 34.7 K nozzle with a 20 ␮m diameter at
backing pressures in excess of 300 psi. This produces a beam
made of large frozen hydrogen particles that scatter our laser
light probe. The instrument response function is then the
histogram of this cold (H2 ) n beam scatter signal. In all cases,
the central peak of the instrument response function is found
to have a width of approximately 200 ps FWHM. When
using the external AOM, side peaks occur at the natural frequency of the laser 共75.7 MHz兲, but undergo a suppression
of 20:1 in comparison with the main pulse. There is no evidence of such side peaks for the cavity-dumped laser system.
A factor of one-tenth the FWHM of the instrument response
function is often claimed as the shortest time resolvable by a
time-correlated single photon counting instrument.12 Thus
we claim the ability to resolve times of 20 ps or higher.
C. Data analysis
Modeling of our data is carried out through an iterative
convolution method written using the MATHCAD suite of
programs13 in which the numerical instrument function is
convoluted with a kinetic model by use of a pair of fast
fourier transforms. The following kinetic model is used:
G 共 t 兲 ⫽A 共 e ⫺t/␶ f ⫺e ⫺t/ ␶ r 兲 ⫹D,
in which ␶ f and ␶ r represent the fall and rise times, respectively, of the collected fluorescence. D is a background factor
that accounts for the dark counts in the MCD, calculated
from the measured total dark count rate and the total signal
averaging time for each bin. Weights of each bin are calculated based upon Poisson statistics; a weighting factor of
(N i ) ⫺1 is used to weight each bin where N i is the observed
number of counts in the ith bin. The global ␹ 2 minimum is
sought through the Marquardt algorithm. A model is deemed
successful if the reduced ␹ 2 ( ␹ r2 ) is close to one. A typical
result is ␹ r2 ⫽1.2. When standard ␹ 2 analysis of the results
obtained with the above model is unsatisfactory, a model that
is biexponential in rise and fall is used. The standard deviations as well as correlations between model parameters 共less
than 0.95 in all cases兲 are found through standard equations
based upon a linear approximation for the variation of the
model with parameters.14 This fitting method avoids the
problems of noise amplification prevalent in the deconvolution of an instrument response function from the experimental histogram by Fourier transform methods.
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9696
J. Chem. Phys., Vol. 113, No. 21, 1 December 2000
Reho et al.
Use of this iterative convolution method for measurements of gas-phase K 4 2 P 3/2,1/2→4 2 S 1/2 and Na 3 2 P 3/2,1/2
→3 2 S 1/2 atomic fluorescence generated in the absence of a
helium cluster beam yields instrument-limited rise times
which, as expected, are well fit by a single exponential with
a time constant of less than 20 ps in all cases, as expected.
The measured lifetime of 4 2 P 3/2,1/2 K (27.0⫾0.1 ns) is in
good agreement with the literature value 共27.0 ns兲,15 as is the
case for the 16.3⫾0.1 ns measured for the 3 2 P Na lifetime
共16.4 and 16.3 ns for the 3 2 P 1/2 and 3 2 P 3/2 states,
respectively兲.15
III. EXPERIMENTAL RESULTS: K* He AND Na* He
EXCIPLEXES
Time-correlated single photon counting has been employed to study the emission of K atoms attached to helium
nanodroplets using a similar method as in our previous NaHe
studies.4 Here a bandpass filter centered at 13 046.3 cm⫺1
with a FWHM of 170 cm⫺1 was employed to select free K*
atomic emission. As discussed in the previous article, the K
atoms emitting at the D-line wavelengths have desorbed
from the droplet surface prior to emission and thus correspond to bound–free transitions in the framework of a
K–Hen pseudodiatomic potential energy surface. As expected, the fall times agree with the literature values for gas
phase K(4 2 P) lifetime as expected, and the rise times were
all 60⫾20 ps.
A bandpass filter with a FWHM of 554 cm⫺1 centered at
11 765 cm⫺1 was introduced to select the K* He exciplex
emission. Using the K transition dipole moment, a radiative
lifetime of 36.9 ns can be predicted for emission at 11 765
cm⫺1 共the center of the filter transmission兲. However, the
measured lifetime centers around 28 ns 共near the free atom
value兲, deviating significantly from the prediction. We note
here that in the case of Na* He exciplex emission, in which
the experimental values ranged from 19 to 21 ns, the fall
times are quantitatively predicted by assuming the same transition dipole as for the free atom and a mean emission wavelength at the center of the emission feature 共15 798 cm⫺1兲.4
The excellent fit obtained with the expected radiative rate in
the case of Na* He argues that the fraction of exciplexes
remaining bound to the helium droplet surface is small in the
region investigated. Further, it would seem probable that any
exciplexes remaining on the cluster would go on to bind
further helium atoms in times on the order of several picoseconds, causing quenching of the observed red fluorescence.
This quenching would, in turn, appear in the measured decay
as a faster fall time component.
The disagreement with the predicted fall time in the case
of K, however, is due to an instrumental artifact, i.e., to the
excessive width of the bandpass filter, which also allows the
free atom fluorescence to reach the detector. Calculations
taking into account the quantum efficiency of the microchannel plates, the relative transmission of the filter at atomic and
exciplex values, and the average count rates recorded in the
experiments, predict that the majority 共93% for J⫽1/2 and
89% for J⫽3/2兲 of the fluorescence collected in the above
manner is actually due to K* 4 2 P 3/2,1/2→4 2 S 1/2 atomic
emission. While this is inconvenient, it is still possible to
FIG. 1. 共a兲 Time evolution of red-shifted K* /K* He emission collected upon
excitation corresponding to J⫽1/2 line along with the best fit to the data
using a biexponential model 共see text兲. 共b兲 Residuals of best fit to the data
using a single-rise exponential model. 共c兲 Residuals of the best fit to the data
using a model with a biexponential rise 关as in 共a兲兴. The abscissa scales of
共a兲, 共b兲, and 共c兲 are the same.
model the data using a biexponential form in the iterative
convolution procedure to yield the synthetic data function
I(t) to be fit to experiment.
The results of these calculations 共in which the K* /K* He
ratio is a free parameter but the rise and fall time of the K*
component are held constant兲 come close to the K* /K* He
fluorescence ratio calculated above. In the case of D 1 excitation, the best fit to the data attributes 8% of the emission to
the exciplex 共while 92% is attributed to K* emission兲 and
yields an exciplex rise time of 7.9 ns. The data and the best
fit using the biexponential model are shown in Fig. 1共a兲. In
Fig. 1共b兲 the residuals of the best fit to the data using a model
with a single exponential for both rise and fall is shown for
comparison with Fig. 1共c兲 in which the residuals of the best
fit to the data using a biexponential rise/biexponential fall
model are shown.
The sensitivity of the ␹ r2 as a function of the relative
K* /K* He emission amplitudes has also been investigated. It
is found that ␹ r2 increases by 10% if the exciplex population
is reduced to 5.8% or increased to 9.8%. Having established
the necessity of the biexponential model, we next investigated the sensitivity of the fit to the value of the excimer
emission rise time. To do this, ␹ 2 values were obtained by
varying the exciplex rise time and constraining all other parameters with the exception of the amplitude coefficient. It
was found that minimization of ␹ 2 for exciplex emission
after D 1 excitation occurs with a rise time value of ␶ rise
⫽7.9 ns for the exciplex component and that ␹ 2 increases by
two percent when this ␶ rise is lowered to 4.7 ns or is increased to 17.6 ns. The best fit to the data corresponding to
excitation of the D 2 -line of K shows instead a fluorescence
rise time (k 3 ) ⫺1 of 50 ps with a ␹ r2 value close to that reported for the J⫽1/2(D 1 ) data. The addition of the second
rise time component in the case of the excitation at the
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J. Chem. Phys., Vol. 113, No. 21, 1 December 2000
Alkali–helium exciplex formation. II
9697
D 2 -line does not alter the ␹ 2 value, and setting it as a free
parameter again yields a result near 50 ps.
The measured onsets of exciplex fluorescence upon excitations corresponding to the D 1 - and D 2 -lines of the alkali
atom correspond to different formation rates of the exciplex
along the 1 2 ⌸ 1/2 and 1 2 ⌸ 3/2 surfaces, respectively.4 There
is a sizable delay for exciplex formation along the 1 2 ⌸ 1/2
molecular surface of K* He in contrast to relatively quick
formation along the 1 2 ⌸ 3/2 molecular surface, continuing a
trend first observed for the Na* He emission dynamics.4 The
delay of K* He exciplex formation along the 1 2 ⌸ 1/2 surface
appears to be about a factor of 10 greater than that previously
measured for Na* He, in which a delay of ⬃700 ps was
found for exciplex formation upon excitation at and to the
red of the Na D 1 -line, with a delay of only 60 ps found for
exciplex formation at energies equal to and greater than the
Na D 2 -line.4
IV. DISCUSSION
A. The influence of spin–orbit coupling
We have shown in previous work4 that an explanation of
the differences in exciplex formation times described above
requires consideration of the relevant potential energy
curves. In the preceding article, it was shown that the potential energy surfaces calculated by Pascale16 succeed in predicting the experimental exciplex emission. These surfaces
do not take into account spin–orbit effects, and thus produce
a single 2 ⌸ surface corresponding to the K, Na (4 P,3P)
⫹He asymptote. Spin–orbit effects can be accounted for
through the introduction of a constant 共i.e., independent of
bond length兲 perturbation equal to the alkali atomic 2 P spin–
orbit splitting ⌬ SO⫽57.7 cm⫺1 共for K兲 and 17.19 cm⫺1 共for
Na兲 as has been done for the AgHe system by Jakubek and
Takami.17 In terms of the two potentials V ⌸ (R) and V ⌺ (R)
of Pascale, the effective Hamiltonian in terms of the basis
states 关 1 2 ⌸ 3/2,1 2 ⌸ 1/2,1 2 ⌺ 1/2兴 is18
冋
V ⌺共 R 兲
&
•⌬ SO
3
&
⌬ SO
•⌬ SO V ⌸ 共 R 兲 ⫺
3
3
0
0
0
0
V ⌸共 R 兲 ⫹
⌬ SO
3
册
,
in which R is the NaHe internuclear distance. Diagonalization of this Hamiltonian for each R leads to the three curves
given in Fig. 2 as solid lines labeled by their dominant character at short R. Two of these curves, labeled 1 2 ⌸ 1/2 and
1 2 ⌸ 3/2 共separated at long range by an energy equal to the
atomic spin–orbit splitting兲, are responsible for formation of
the NaHe exciplex.
Unlike the 1 2 ⌸ 3/2 surface which is purely attractive at
long range, the 1 2 ⌸ 1/2 surface has a small ‘‘outer’’ well of
0.5 cm⫺1 at R⫽9.0 Å and, centered near 7.4 Å, a small barrier of height 0.19 cm⫺1 relative to the 1 2 ⌸ 1/2 asymptote
共although the barrier height relative to the minimum of the
outer well is 0.7 cm⫺1兲. The existence of this barrier can be
understood from the fact that the Pauli repulsion on the
FIG. 2. The spin–orbit averaged 共dotted curves兲 共Ref. 16兲 and spin–orbit
decoupled 共solid curves兲 potential energy surfaces of NaHe which correspond to the Na(3 2 P)⫹He(1 1 S) asymptote. The energy is normalized to
the spin–orbit averaged asymptote.
2 2 ⌺ 1/2 curve becomes significant at larger distances than
does the attraction on the 1 2 ⌸ 1/2 surface. A similar barrier in
the AgHe exciplex curves was noted by Persson et al.1 and
used to explain the lack of exciplex formation when the 2 P 1/2
level of Ag was optically excited. In that case, however, the
barrier was much larger 共66 cm⫺1兲, as could be expected
given the much larger atomic spin–orbit splitting 共921
cm⫺1兲15 of Ag.
It is, therefore, natural to try to explain the slower formation rate for the Na* He exciplex on the 1 2 ⌸ 1/2 surface by
the presence of this 0.7 cm⫺1 barrier. However, we have
previously determined that this barrier is insufficient for
quantitative prediction of the NaHe exciplex dynamics
within the Wentzel–Kramers–Brillouin 共WKB兲 approximation.4 An expanded model that takes into account the
presence of the He atoms that remain in the droplet will be
presented in the next section.
In order to consider the dynamics of K* He formation,
the potential energy surfaces of Pascale16 have again been
employed in creating spin–orbit coupled 1 2 ⌸ 1/2 and 1 2 ⌸ 3/2
molecular surfaces for K* He in the same manner as has been
described in detail for Na* He. The Pascale K(4 2 P)
⫹He(1 2 S) potential energy surfaces are shown as dotted
lines in Fig. 3 where the spin–orbit coupled surfaces, generated through diagonalization of the effective Hamiltonian in
terms of the basis states 关 1 2 ⌸ 3/2,1 2 ⌸ 1/2,2 2 ⌺ 1/2兴 , are shown
as solid curves.
The spin–orbit coupling procedure generates a barrier of
5.53 cm⫺1 along the 1 2 ⌸ K* He potential energy
surface.17,18 Using the WKB approximation and a zero-point
energy of 3.9 cm⫺1 共with the 1 2 ⌸ 1/2 surface normalized to
the 1 2 ⌸ 1/2 asymptote兲 a small delay of ⬃2 ps is generated
along the 1 2 ⌸ 1/2 molecular surface. While the experimental
formation time is affected by a rather large error, it is three
orders-of-magnitude larger than 2 ps. We are, therefore, confronted with the need to consider the effect of the presence of
the droplet on the exciplex formation rates.
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9698
J. Chem. Phys., Vol. 113, No. 21, 1 December 2000
FIG. 3. The spin–orbit averaged 共dotted curves兲 共Ref. 16兲 and spin–orbit
decoupled 共solid curves兲 potential energy surfaces of KHe which correspond
to the K(4 2 P)⫹He(1 1 S) asymptote. The energy is normalized to the spin–
orbit averaged asymptote.
B. Cluster effects in exciplex formation
An important shortcoming of the model for exciplex formation described above is that it treats the He atom that will
participate in it as initially ‘‘free,’’ ignoring the interaction
with the droplet. Formation of the exciplex, however, includes extracting the He atom from the droplet and this costs
energy in an amount nearly equal to the heat of evaporation
of He from the bulk liquid, which is ⬃5 cm⫺1. In order to
include this extraction energy, we need to model the shape of
the He atom–droplet interaction potential.
We adopt a very simple model, which was previously
used by Eichenauer and LeRoy.19 We treat the droplet as
having an abrupt interface and bulk density,20 ␳
⫽.0218 Å ⫺3 , below this interface. The leaving He atom is
treated as an impenetrable sphere of radius a with a 1/r 6
attraction with the He fluid with the well-known dispersion
coefficient21 C 6 ⫽6892 cm⫺1 Å 6 (1.42 a.u.). The total interaction of the departing He can then be calculated by integration over all He outside the excluded radius, a. This yields
the following expression for the potential, V(h), where h is
the height above the liquid helium 共droplet兲 surface 共h⬍0
being below the surface and h⫽0 being on the surface兲:
V共 h 兲⫽
冦
⫺
⫺
⫺
冉
冋
冉
␲␳ C 6 1
• 3
6
h
冊
a⬍h⬍⬁,
冉 冊册
冊冉
冊
␲␳ C 6 1 a⫺h
• ⫹
3a 2
3
a
⫺a⭐h⭐a,
4 ␲␳ C 6
␲␳ C 6 1
• 3
⫹
2
3a
6
兩h兩
h⬍⫺a.
The parameter a is set equal to 3.35 Å to give the correct
attractive potential in the bulk22 共⫺16.72 cm⫺1兲 as h→
⫺⬁. Figure 4 shows the resulting potential where the potential energy of a He atom inside the droplet is taken as the
Reho et al.
FIG. 4. Relative position of the He atom–helium droplet (He–Hen ) potential energy curve 共dotted line兲 along the Na–He internuclear distance coordinate. This corresponds to the removal of a helium atom from the droplet
and is plotted as a function of the distance of the helium atom from the
‘‘flat’’ surface of the droplet. Note that the distance parameter R He–Hen corresponds to ⫺h used in the text for modeling the potential. The spin–orbit
decoupled potential energy curve for NaHe 共based on the Pascale surfaces,
Ref. 16兲 is given by the solid line. The 1 2 ⌸ 1/2 potential energy surface of
NaHe at long range calculated from the spin–orbit averaged surfaces of
Pascale 共Ref. 16兲 with the addition of the He–Hen potential interaction is
given by the dashed line. Energy is normalized to the Na(3 2 P 1/2)
⫹He(1 1 S) asymptote for the interaction of NaHe while the zero of energy
is referenced to a He atom inside the droplet for the He–Hen interaction.
zero of energy. It is seen that in this simple model, most of
the attraction of the He to the droplet develops over the range
⫺4 Å⬍h⬍⫹4 Å.
While it will be shown that this simple model can function quite well in predicting exciplex formation there are
several major approximations implicit in it that should be
pointed out for the sake of completeness. First, we take the
density of the surface of the helium droplet to be a step
function. Density functional calculations predict that the
sloping density region actually extends over several Å.23
Such a gradient would serve to ‘‘smooth’’ out the He
atom–He droplet potential curve of Fig. 4 and would not
alter the mean position or height of the barrier produced
共although the barrier width would be affected兲.
Another approximation which we make is to preserve
isotropy, i.e., to assume that the kinetic energy of the exiting
helium atom is one-third of the difference (E k ) between the
potential energy of a helium atom in the bulk liquid 共16.72
cm⫺1兲 and the binding energy 共4.94 cm⫺1兲 of the helium
atom to the droplet. While this approximation allows us to
model our time-resolved results, this value of the kinetic energy serves as a lower limit for the kinetic energy of the He
atom leaving the cluster surface. This is because it is known
that approach to the surface of the dimple, where the density
can be assumed to slightly increase locally, causes more energy to become available as kinetic energy than would be
expected if the case of the bulk truly applied in the region of
the dimple.
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J. Chem. Phys., Vol. 113, No. 21, 1 December 2000
C. Modeling Na* He and K* He exciplex formation
1. Modeling the Na * He exciplex dynamics
As the He–Hen potential curve discussed above was created without recourse to the position or existence of a Na
atom, the relative placement of the He atom–Hen surface
with respect to the NaHe surfaces depends upon the choice
of the distance of the alkali atom from the cluster. As stated
above, the depth and width of the dimple where the Na atom
resides have been calculated previously through use of a discontinuous deformable surface model24 that predicts the
depth to be 3.42 Å and the radius to be 4.1 Å. The model
also positions a ground state Na atom 1.64 Å above the surface of the undeformed droplet, so that a total distance of
5.06 Å is calculated to be between a Na(3 2 S) atom and the
bottom of the well of the helium dimple.
It is also necessary to know the change in the Na–
nanodroplet distance as the Na atom is excited to its 3 2 P
atomic state. This information is available through calculations performed by Kanorsky25 who concluded that, upon
excitation, the Na atom moves 3.5 Å away from the surface.
Using the dimple profile calculations of Ref. 24 in conjunction with the calculated displacement of the Na atom caused
by 3 2 P 3/2,1/2←3 2 S 1/2 excitation, a final Na(3 2 P) atom–
helium droplet distance of 8.56 Å is established. Using this
distance, the abscissa of the He atom–helium cluster potential can be related to that of the He atom–Na atom potential
as shown in Fig. 4, which illustrates the case for the
1 2 ⌸ 1/2Na–He surface.
When added together 共the solid and dotted lines in Fig.
4兲, the two curves provide a potential function that includes
both the Na–He spin–orbit coupled surface and the He atom
extraction from the droplet 共dashed line, Fig. 4兲. This new
potential energy surface will be labeled as the X–He–Hen
potential surface 共X being the alkali atom兲 maintaining the
symmetry designations used for the corresponding X–He
surfaces.
It can be seen from Fig. 4 that addition of the He atom
extraction increases the height of the barrier along the
1 2 ⌸ 1/2 surface from 0.7 to 11.2 cm⫺1 and causes the disappearance of the outer well at R⫽9.0 Å. The location of the
barrier changes from a distance of 7.42 Å to a distance of
6.96 Å. Addition of the He atom extraction potential curve to
the 1 2 ⌸ 3/2Na–He surface results in the formation of a previously nonexistent barrier along this surface which has a
height of 7.77 cm⫺1 and is located at a distance of 7.67 Å.
Tunneling times have been calculated for the new barriers along both spin–orbit coupled ⌸ surfaces for
Na–He–Hen . The WKB approximation is used in expressing the semiclassical tunneling probability. The tunneling
probability T(E) calculated for the 1 2 ⌸ 1/2 surface using this
model is
T 共 E 兲 ⫽1.542⫻10⫺3 ,
while for the 1 2 ⌸ 3/2 surface, the calculated tunneling probability is found to be
T 共 E 兲 ⫽1.012⫻10⫺2
at E⫽E k /3⫽3.96 cm⫺1, the assumed zero-point energy in
the direction of the reaction coordinate.
Alkali–helium exciplex formation. II
9699
As the tunneling probability represents the probability of
crossing the barrier at each attempt made by the He atom,
assuming that the Debye frequency of liquid helium is a
good estimate of the frequency of attempts, the inverse of the
product of the tunneling probability and the Debye frequency
(⬃1012 Hz) 26 represents the fluorescent onset time. Table I
lists the fluorescent onset times for both spin–orbit coupled
Na–He–Hen ⌸ surfaces, and includes some important parameters of the barriers involved in each case. In the case of
the 1 2 ⌸ 1/2 surface, a calculated fluorescent onset time of
650 ps deviates by only 7% from the experimental value of
700 ps, and is within the statistical error of the fit. The calculated onset time of 100 ps in the case of excimer formation
along the 1 2 ⌸ 3/2 surface deviates by only 30% from the
experimental time constant of 70 ps. While this close agreement with experiment must be viewed as fortuitous due to
the approximate nature of the He atom–helium cluster potential surface and neglect of the manybody effects in the He
droplet, the agreement reached for the ratio of the two time
constants 共which is about a factor of 10 in both cases兲 is
possibly more robust with respect to whatever adjustments to
the potential may be needed. While the slower rate of formation 共700 ps兲 has been measured with a relatively high level
of confidence, the fast rise time 共70 ps兲 is more uncertain as
it is closer to our experimental resolution.
2. Modeling of the K * He exciplex dynamics
Having established the validity of our method of calculating exciplex fluorescence onset times in the case of
Na* He, the case of K* He exciplex emission may be taken
up as an independent test of the validity of our model. For
the case of a K(4 2 S 1/2) atom dopant, the discontinuous deformable surface model24 predicts the depth of the dimple to
be 3.50 Å deep with a radius of 4.3 Å. The model also
positions a ground state K atom 2.12 Å above the surface of
the cluster, so that a total distance of 5.62 Å is derived between a K(4 2 S 1/2) atom and the well of the helium dimple.
Because calculations such as those performed by Kanorsky for Na*Hen are unavailable in the case of potassium, the
displacement of the K atom upon 4 2 P 3/2,1/2←4 2 S 1/2 excitation is taken as a free parameter in modeling the fluorescence
onset for K* He emission. We find that fits to the fluorescent
onset data are best predicted by assuming that the K(4 2 P)
atom sits the same distance above the helium dimple bottom
as does the Na(3 2 P), i.e., 8.56 Å. This distance leads to a
barrier along the 1 2 ⌸ 1/2K–He–Hen surface with a height of
17.5 cm⫺1 and a FWHM of 3.23 Å. This new barrier is
compared to the original 1 2 ⌸ 1/2K* –He barrier in Fig. 5. The
tunneling probability through the barrier arising along the
1 2 ⌸ 1/2K* –He–Hen surface is found to be
T 共 E 兲 ⫽1.23⫻10⫺4 ,
in which E⫽E k /3 and E k is the zero-point energy. From this,
the fluorescence onset time is calculated to be 8.1 ns, in good
agreement with the fit to the experimental rise time 共7.9 ns兲.
An increase in K* -dimple well distance of 0.25 Å from the
value of 8.56 Å increases the predicted fluorescence onset by
almost a factor of two 共to 15.3 ns兲.
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9700
Reho et al.
J. Chem. Phys., Vol. 113, No. 21, 1 December 2000
FIG. 5. The 1 2 ⌸ 1/2 potential energy surface of KHe at long range calculated from the spin–orbit averaged surfaces of Pascale 共Ref. 16兲. The figure
shows this surface without 共solid line兲 and with 共dashed line兲 the addition of
the He–Hen potential interaction 共dotted line兲. Energy is normalized to the
K(4 2 P 1/2)⫹He(1 1 S) asymptote for the interaction of KHe while the zero
of energy is referenced to a He atom inside the droplet for the He–Hen
interaction.
Exciplex formation along the K* – He–Hen 1 2 ⌸ 3/2 surface can also be predicted using the spin–orbit coupled
K*He1 2 ⌸ 3/2 surface. Addition of this potential to the
He–Hen potential surface creates a barrier of 7.14 cm⫺1 with
a FWHM of 4.58 Å when 8.56 Å is used for the distance of
K* from the dimple well. In this case the tunneling times for
exciplex formation along this 1 2 ⌸ 3/2 surface are roughly
two orders-of-magnitude faster 共56 ps兲 than was the case for
formation along the 1 2 ⌸ 1/2K* –He–Hen surface, in agreement with the experimental results 共⬃50 ps兲. The parameters
of both 1 2 ⌸ barriers are presented in Table I. The calculation of the tunneling times through the barriers on the 1 2 ⌸ 3/2
and 1 2 ⌸ 1/2 surfaces are based on the K* –Hen pseudodiatom
in the v ⬘ ⫽0 level of the excited state potential surface. Vibrationally excited levels of the K* –Hen system will lead to
higher tunneling rates and faster formation of the K* – He
exciplex. This is due to the smaller K* –Hen distance in vibrationally excited levels of the excited electronic state
TABLE I. Barrier parameters generated by addition of the He atom–He
cluster potential energy to the spin–orbit coupled potential energy surfaces
of NaHe and KHe. The barrier center is referenced to the Na–He and K–He
distance coordinates.
Na–He–Hen
Barrier parameter
Height 共cm⫺1兲
Center 共Å兲
FWHM 共Å兲
Tun. prob. (⫻10⫺3 )
Rise time 共calc.兲
Rise time 共expt.兲
2
⌸ 1/2
11.2
6.96
4.09
1.542
650 ps
700 ps
2
⌸ 3/2
7.8
7.64
4.79
10.12
100 ps
75 ps
K–He–Hen
2
⌸ 1/2
17.5
6.55
3.23
0.123
8.1 ps
7.9 ns
2
⌸ 3/2
7.14
8.00
4.58
18
56 ps
50 ps
共compared to the v ⬘ ⫽0 level兲 after vertical excitation from
the ground state.
In comparing the fit exciplex rise times 共7.9 ns and 50
ps兲 with the respective predicted values 共8.1 ns and 56 ps兲,
we see that the model reproduces the relative orders-ofmagnitude for exciplex formation along the two 1 2 ⌸ surfaces found from fits to the experiment. In the case of K* He,
it is necessary to keep in mind that the experimental rise
times have a larger error bar than was the case in Na* He, as
the rise is biexponential and the fraction ascribed to exciplex
formation 共0.07兲 is small. Further, it has been shown in the
preceding article that the spin–orbit coupled potential energy
surfaces for K–He of Pascale16 did a poorer job in predicting
the emission spectrum of K* He than in the case of Na* He.
Thus, while the excellent agreement between the calculated
and measured values 共which depends on the choice of the
value of the K* -nanodroplet distance兲 may be somewhat fortuitous, again we feel that the rise time for exciplex formation along the 1 2 ⌸ 1/2K* –He–Hen potential 共7.9 ns兲 was
measured with a relatively high level of confidence since the
magnitude of the measurement is much greater than our experimental resolution.
3. The Rb * He exciplex: An extension of the model
As the model used above seems to be fairly quantitative
for Na* He and K* He, the Rb* He exciplex will also be
treated. The spin–orbit averaged potential energy surfaces
for the 1 2 ⌸ state of RbHe have also been calculated by
Pascale.16 Using the fine structure splitting between the
5 2 P 1/2 and 5 2 P 3/2 lines of Rb* 共138 cm⫺1兲 as the mixing
constant ⌬ SO in the spin–orbit coupling procedure as illustrated above, the 1 2 ⌸ 1/2 and 1 2 ⌸ 3/2 RbHe potential energy
surfaces can be generated. As expected, the resultant 1 2 ⌸ 1/2
surface exhibits a larger 共17.77 cm⫺1, 2.19 Å FWHM兲 barrier than in the case of both Na* He and K* He.
The He–Hen potential surface may again be combined
with both spin–orbit coupled Rb–He 1 2 ⌸ molecular potentials, using the same 8.56 Å distance taken for Na(3 2 P) and
K(4 2 P) for the distance between the excited alkali atom and
the bottom of the helium dimple. The resultant barrier to
exciplex formation along the 1 2 ⌸ 1/2 surface, located at an
internuclear distance of 6.0 Å, has increased in amplitude to
31 cm⫺1 with a FWHM of 2.9 Å. Exciplex formation along
the 1 2 ⌸ 3/2 Rb* –He–Hen surface is predicted to encounter a
barrier of 6.6 cm⫺1 at a Rb* He distance of 8.2 Å with a
FWHM of 4.5 Å. Tunneling times calculated through the
Rb* –He–Hen 1 2 ⌸ 3/2 barrier predict a fluorescence onset
time of 31 ps, while the much larger barrier present along the
Rb* –He–Hen 1 2 ⌸ 1/2 surface yields a predicted fluorescence
onset time of 500 ns.
The fluorescence lifetime of Rb(5 2 P 1/2) is known to be
29.4 ns and that of Rb(5 2 P 3/2) to be 27.0 ns.15 Thus, in
extending the model of exciplex formation to Rb* –He–Hen ,
it has been possible to predict that while exciplex emission
should be seen upon excitation of the D 2 (5 2 P 3/2←5 2 S 1/2)
atomic transition of Rb* attached to helium nanodroplets,
not much exciplex emission should be visible for excitations
of Rb–Hen corresponding to the D 1 (5 2 P 1/2←5 2 S 1/2)
atomic Rb transition, as the predicted onset time of exciplex
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J. Chem. Phys., Vol. 113, No. 21, 1 December 2000
fluorescence is more than an order-of-magnitude slower than
the fluorescent lifetime. As it is likely that the actual
Rb* –Hen distance would be larger than in the case of K* ,
the quoted value of 500 ns should be considered to be a
lower limit for the fluorescence onset time of exciplex emission along the 1 2 ⌸ 1/2Rb–He–Hen surface, thus pointing
even more strongly towards the absence of exciplex emission
along this surface.
The predictions presented here concerning Rb–He exciplex dynamics have been confirmed by recent work of
Trasca and Ernst,27 in which Rb* – He exciplex formation
has been observed upon optical excitation at the J⫽3/2Rb
line 共i.e., corresponding to exciplex formation along a
1 2 ⌸ 3/2Rb–He–Hen surface兲, while excitations along the J
⫽1/2 Rb line 共which would correspond to formation along
the 1 2 ⌸ 1/2 Rb–He–Hen surface兲 yield no exciplex emission.
As the work of Trasca and Ernst was carried out after our
calculation, our exciplex formation model has been shown to
be predictive in the case of the Rb* – He exciplex dynamics.
V. CONCLUSIONS
As evidenced from the three systems studied above, exciplex formation upon optical n 2 P 3/2,1/2←n 2 S 1/2 excitation
of an alkali atom residing on the surface of a helium droplet
can occur by one of two paths. If the n 2 P 1/2←n 2 S 1/2 (J
⫽1/2) transition of the alkali atom is excited, formation occurs along the 1 2 ⌸ 1/2 X–He–Hen potential energy surface,
which takes into account the He extraction energy from the
droplet but has a profile heavily influenced by the spin–orbit
mixing of the original 1 2 ⌸ 1/2 X–He surface with the
2 2 ⌺ 1/2 X–He surface. Thus the formation rates for the various alkali atom–He exciplexes along this surface scale with
the spin–orbit operator, which in our case is taken to be the
spin–orbit splitting for each atomic J⫽3/2,1/2 pair.
Exciplex formation can also occur due to excitation of
the n 2 P 3/2←n 2 S 1/2 (J⫽3/2) transition of the alkali atom
along the 1 2 ⌸ 3/2 X–He–Hen potential energy surface. The
1 2 ⌸ 3/2 X–He molecular surface from which the corresponding X–He–Hen surface is derived does not spin–orbit mix
with the 2 2 ⌺ 1/2 state. Therefore, relative exciplex formation
rates along these surfaces scale as the dispersion 共i.e., longrange or van der Waals attraction兲. Whereas the spin–orbit
mixing term increases as the alkali atom becomes larger, and
thus creates a larger barrier along the alkali atom–He 1 2 ⌸ 1/2
pair potential, the change in dispersion has the opposite effect. As the alkali atoms become larger, dispersion along the
spin–orbit averaged 1 2 ⌸ alkali atom–He pair potential 共and
thus along the 1 2 ⌸ 3/2 surface兲 ‘‘turns on’’ at larger and
larger internuclear distances. The effect is that the He–Hen
extraction potential becomes increasingly less significant in
barrier formation, as it adds to a more and more negative
region of the alkali atom–He pair potential as the alkali becomes larger.
Therefore, while exciplex formation along the
X–He–Hen 1 2 ⌸ 3/2 surfaces occurs more and more quickly
as the alkali atom becomes larger, the concomitant increase
in the spin–orbit mixing for the larger alkali atoms creates
Alkali–helium exciplex formation. II
9701
longer delays in exciplex formation along the
1 2 ⌸ 1/2 X–He–Hen surfaces. Soon after formation the exciplexes desorb from the nanodroplet surface. This is demonstrated by the fact that the emission decays are well fit by the
single exponential gas phase lifetime 共corrected for frequency shifts兲 and the fact that were the exciplexes to remain
on the surface, they would be expected to bind further He
atoms and undergo fluorescence quenching,3 as discussed
above for alkali atoms in bulk liquid helium.
ACKNOWLEDGMENTS
We would like to acknowledge the assistance of Carlo
Callegari, John Eng, Wolfgang Ernst, and Donna Strickland.
Y. T. Lee and Arthur Suits are gratefully acknowledged for
the loan of laser equipment, as are Wolfgang Ernst and Warren S. Warren. J.R. wishes to thank Arthur Suits for his help
and hospitality. Useful discussions with F. Stienkemeier are
gratefully acknowledged. This work was made possible
through the Air Force HEDM 共High Energy Density Materials兲 program, Grant F49620-98-1-0045.
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