Dalitz plot analysis of three-body fragmentation of Na3+ excited by He impact

Article abstract of DOI:10.1063/1.1455623

Theoretical and experimental Dalitz plots for three-body fragmentation of Na⁺ excited by He impact were studied. The fragmentation involves three adiabatic electronic states of Na₃⁺ . Each of the three sectors of the Dalitz plots correspond to a different product adiabatic electronic state.

Full text of DOI:10.1063/1.1455623

Dalitz plot analysis of three-body fragmentation of Na 3 + excited by He impact
D. Babikov, E. A. Gislason, M. Sizun, F. Aguillon, V. Sidis, M. Barat, J. C. Brenot, J. A. Fayeton, and Y. J. Picard
Citation: The Journal of Chemical Physics 116, 4871 (2002); doi: 10.1063/1.1455623
View online: http://dx.doi.org/10.1063/1.1455623
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 116, NUMBER 12
22 MARCH 2002
Dalitz plot analysis of three-body fragmentation of Na¿3 excited
by He impact
D. Babikova)
Theoretical Chemistry and Molecular Physics Group, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545
E. A. Gislason
Department of Chemistry, University of Illinois at Chicago, SES (M/C 111), Chicago, Illinois 60607-7061
M. Sizun, F. Aguillon, V. Sidis, M. Barat, J. C. Brenot, J. A. Fayeton, and Y. J. Picard
´
ˆ
´
Laboratoire des Collisions Atomiques et Moleculaires, Bat. 351, Unite Mixte de Recherche C8625, Universite´
Paris-Sud XI, 91405 Orsay Cedex, France
͑Received 10 December 2001; accepted 8 January 2002͒
Three-body fragmentation of Naϩ3 ions to NaϩϩNa(3s)ϩNa(3s) following excitation by He is
studied experimentally and theoretically. The three reduced kinetic energies of the products in the
center-of-mass are determined for each fragmentation event, and the results are displayed in a Dalitz
plot. The fragmentation involves three adiabatic 1A electronic states of Na3ϩ that become
Ј
degenerate at the detector. It is possible to determine the final electronic state for each event, and
here we show that each of the three product states appears in a particular sector of the Dalitz plot.
Theoretical and experimental Dalitz plots for the three-body fragmentation of Naϩ3 are presented,
and the results are related to various mechanisms for three-body fragmentation of this
system. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1455623͔
I. INTRODUCTION
ity vectors, a study of, for example, 10 000 dissociations will
produce an enormous amount of data. The best way to dis-
play such data is by using Dalitz plots.11 These plots have
been used in the past to interpret several of the three-body
fragmentation processes described above.1–4,7
The dissociation of an excited polyatomic molecule into
three fragments is a process of great interest in molecular
physics.1–3 Recently, a few experiments have been done
where all three products are detected in coincidence. One
such experiment is the polar fragmentation of Hϩ3 following
a collision with He studied by Jaecks and co-workers,4
In processes ͑1͒ and ͑4͒ it is clear that at the end there
are three degenerate singlet electronic product states. ͓For
example, in ͑1͒ the negative charge can be on any of three H
nuclei; each possibility corresponds to a different electronic
state.͔ It is also clear that early in the fragmentation process
the three adiabatic electronic states are not degenerate and,
presumably, the populations of the three states are not the
same. As the fragmentation process evolves there is no com-
pelling reason why the populations of the three adiabatic
states should become equal as they reach the region where
they are equal in energy. In a recent paper10 we have shown
for process ͑4͒ that it is straightforward both experimentally
and theoretically to determine the populations of the three
adiabatic states at the detector and, more importantly, that the
populations are not equal.
Hϩ3 ϩHe͒→HϩϩHϩϩHϪ ϩHe͒.
͑1͒
͑
͑
A related study of the same system was carried out by Hino-
josa et al.5 Continetti and co-workers6 have used dissociative
photodetachment to induce three-body fragmentation in the
systems,
OϪ6 ϩh␯→O2ϩO2ϩO2ϩeϪ,
͑2͒
͑3͒
OϪ3 D O͒ϩh␯→OϩO ϩD OϩeϪ.
͑
2
2
2
A third study involves the collisionless fragmentation of a
photoexcited H3 molecule to yield three ground state H
atoms.7 In our laboratory we have studied the collision-
induced fragmentation of Naϩ3 ,
In this paper we determine experimental and theoretical
Dalitz plots for process ͑4͒. We present a more useful con-
vention for labeling the energy axes in the plots. Finally, we
show that a particular area of the Dalitz plot can be associ-
ated with each product adiabatic electronic state.
Naϩ3 ϩHe͒→NaϩϩNa 3s͒ϩNa 3s͒ ϩHe͒,
͑4͒
͑
͑
͑
͑
both experimentally8,10 and theoretically.9,10
In the coincidence experiments described above the
laboratory velocities of all three product particles are deter-
mined, and from those values the translational energies of the
particles relative to their common center-of-mass can be de-
termined. Since each fragmentation event yields three veloc-
II. THEORY
A. Dalitz plots for Na¿3 three-body fragmentation
The construction of the Dalitz plot is straightforward for
three Na particles of equal mass m.11 We begin by calculat-
ing the kinetic energy of the three fragments in the cluster
a͒
Author to whom correspondence should be addressed. Electronic mail:
babikov@lanl.gov
0021-9606/2002/116(12)/4871/6/$19.00
4871
© 2002 American Institute of Physics
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4872
J. Chem. Phys., Vol. 116, No. 12, 22 March 2002
Babikov et al.
c.m. frame after the collision with He. Suppose the final lab
velocities of the three fragments are Va , Vb , and Vc . Then
the velocity of the c.m. of the product cluster is
V
CMϭ 13 V ϩV ϩV ͒,
͑5͒
͑
a
b
c
and the kinetic energy of particle i in the cluster c.m. is
2
Eiϭm/2
͉
ViϪVCM
͉
.
͑6͒
The reduced c.m. kinetic energies of the particles are then
given by
␧iϭEi /Etot
EtotϭEaϩEbϩEc .
,
͑7͒
͑8͒
Clearly the three values of ␧ sum to one. The Dalitz plot is
i
a plot that shows as a single point the three values of ␧ for
i
each fragmentation event. The points lie inside the tangent
circle of radius 1/3 inscribed in an equilateral triangle of unit
height ͓see Fig. 1͑a͔͒. Each Dalitz axis i is shown as a vector
of unit length drawn from the center of one side of the tri-
angle ͑where ␧iϭ0͒ to the opposite vertex ͑where ␧iϭ1͒.
For clarity in Fig. 1͑a͒ each axis is shown a second time
outside the triangle. Conservation of energy restricts all
points to lie within the triangle; conservation of energy and
momentum restricts all points to lie within the circle in-
scribed in the triangle.11 Thus, the maximum possible value
for ␧ is 2/3. For any point in the circle the value of ␧ is
i
i
given by the perpendicular projection of the point along Dal-
itz axis i ͓see Fig. 1͑a͔͒. We choose the reduced energy of the
Naϩ product ͓␧(Naϩ)͔ to lie along the vertical Dalitz axis
͑axis Naϩ͒, and we choose the reduced energy of the faster
Na neutral ͓␧(Nafast)͔ in the c.m. system to lie along the
Dalitz axis that goes from the right side of the triangle to the
vertex at the upper left ͑axis Nafast͒. The energy of the slower
Na neutral is denoted ␧(Naslow) and lies along axis Naslow
.
This new convention restricts all product points to the left
side of the Dalitz plot. In general, due to symmetry, any
three-body fragmentation process that produces two identical
products as in processes ͑1͒ and ͑4͒ requires only half the
Dalitz circle to display the data. Similarly, if three identical
products are produced, as in the fragmentation of H3 to three
H atoms or in process ͑2͒, only one sixth of the Dalitz circle
is required.7 Restricting the data to one-half ͑or one-sixth͒ of
the circle has the advantage of doubling ͑or multiplying by
six͒ the experimental signal to noise. As discussed by Braun
et al.,7͑b͒ a Dalitz plot conserves momentum phase space
density. Thus, if a three-body fragmentation process pro-
duces products uniformly distributed in phase space, the cor-
responding Dalitz plot has a uniform density.
FIG. 1. ͑a͒ General Dalitz plot for process ͑4͒. The tangent circle of radius
1/3 is inscribed in an equilateral triangle of unit height. Each point is deter-
mined from the values of the reduced c.m. kinetic energies ␧ (0р␧iр1) of
i
the three particles as defined in Eq. ͑7͒. Each Dalitz axis i is shown as a
vector of unit length drawn from the center of one side of the triangle
͑where ␧iϭ0͒ to the opposite vertex ͑where ␧iϭ1͒. For clarity each axis is
shown a second time outside the triangle. For any point in the circle the
value of ␧ is given by the perpendicular projection of the point along axis
i
We note that knowledge of the three ␧ values combined
i. The sum of the three ␧ equals one. The vertical axis gives ␧(Naϩ); the
i
i
axis running from lower right to upper left gives ␧(Nafast), the reduced
energy of the faster Na neutral in the center-of-mass; and the axis running
from lower left to upper right gives ␧(Naslow). Because of this convention
for the Na energies all of the data points are found in the left half of the
circle. An example point is shown, where the values are ␧(Naϩ)ϭ0.33,
␧(Nafast)ϭ0.54, and ␧(Naslow)ϭ0.13. ͑b͒ A number of examples of the
Naϩ –Na–Na triangle shape are shown below the Dalitz plot for particular
points on the left hand side of the plot, which is shaded. As discussed in the
text all fragmentation events that give products in the top segment of the
Dalitz plot correspond to product in the ground adiabatic electronic state
͑state 1͒. Similarly, points in the middle ͑lowest͒ segment correspond to
products in state 2 ͑state 3͒.
with conservation of energy and momentum allows one to
reconstruct the three velocity vectors relative to the c.m. ve-
locity, ViϪVCM , subject to a scaling factor that is deter-
mined by Etot . These three vectors also determine the shape
of the NaϩϪNaϪNa triangle at the detector. ͑This point is
discussed further in Sec. II B.͒ It is instructive to show the
triangle shape at certain points on the Dalitz plot to indicate
what type of fragmentation is taking place at those points.
Several possibilities are shown in Fig. 1͑b͒. If a point lies on
an axis, then two values of ␧ are equal. For example, if a
i
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Fragmentation of Na3ϩ
4873
J. Chem. Phys., Vol. 116, No. 12, 22 March 2002
point lies on the vertical Naϩ axis, this means that
␧(Nafast)ϭ␧(Naslow). The three axes meet at the center of
the circle, and there ␧(Naϩ)ϭ␧(Nafast)ϭ␧(Naslow)ϭ1/3. A
point at the bottom of the circle along the Naϩ axis with
␧(Naϩ)ϭ2/3 represents the case where a fast Naϩ ion is
produced, and the two Na neutrals separate with a very small
velocity relative to each other. Similarly, a point along the
Nafast axis with ␧(Nafast)ϭ2/3 at the edge of the circle cor-
responds to a fast Na neutral with the remaining Na and Naϩ
separating with a very small velocity relative to each other. A
third interesting limit occurs when ␧(Naϩ)ϭ0 at the top of
the circle. In that case the Naϩ is produced at rest in the c.m.
system, and the two Na neutrals fly apart with equal speeds
but opposite directions. A comparable situation occurs when
␧(Naslow)ϭ0. Any point along the circumference of the
circle gives three products that dissociate along a line. The
Dalitz plot naturally divides into three sectors. The
Naϩ –Na–Na triangle shapes for typical points near the cen-
ter of each sector are also shown in Fig. 1͑b͒. The triangles
show the relative placement of the Naϩ ion in the triangle.
͑For example, in point #1 the ion is opposite the longest side
of the triangle.͒ In general the shape of the triangle is not
mentally. The adiabatic states ͑1, 2, or 3͒, on the other hand,
are determined by the energy ordering with E1ϽE2ϽE3 . If
the diabatic Hamiltonian matrix is set up and diagonalized,
one obtains the adiabatic energies.12 In general the positive
charge is delocalized over all three Na nuclei for each adia-
batic state. Once all three atom–atom distances are suffi-
ciently large ͑greater than 12 Å for Naϩ3 ͒, however, the posi-
tive charge is fixed on one of the three nuclei, and the
diabatic Hamiltonian matrix is diagonal.
As the Naϩ3 system evolves to larger separations Rk , the
system may hop from one adiabatic surface to another at
avoided crossings. We define the asymptotic region as the
region where all three Rk values are sufficiently large that
two conditions are simultaneously satisfied: first, the positive
charge is fixed on one nucleus ͑i.e., the diabatic state is fixed͒
and, second, the Rk values are ordered in the same way as
the k . We have shown10 that hops between adiabatic states
v
are forbidden in the asymptotic region, because there are no
additional avoided crossings in this region and because there
is no electronic coupling between the adiabatic states. It is
possible to determine the one–one mapping between diabatic
and adiabatic states in this region. The only important con-
tributions to the interaction energy are the attractive ion-
induced dipole potential terms between Naϩ and each Na,
which are proportional to 1/R4. The total energy is the sum
of these two terms. It can be proven10 that if RaϽRbϽRc ,
then EcϽEbϽEa . This case is shown in Fig. 2. Since the
asymptotic region persists to the detector, this is the mapping
at the detector as well. Thus, we see that the adiabatic state
can be identified by the shape of the triangle at the detector
and by which vertex of the triangle corresponds to Naϩ.
One new and very useful feature of the Dalitz plot for
process ͑4͒ is that the position of a point on the plot also
indicates the product adiabatic electronic state for the event.
To see this consider the point labeled ‘‘1’’ in the upper sector
of the Dalitz plot in Fig. 1͑b͒ outlined by the points HABC.
An examination of the triangle formed by the three Na nuclei
for point ‘‘1’’ shows that the side opposite the Naϩ ion is the
largest of the three sides. Consequently, point 1 corresponds
to a product in the ground adiabatic electronic state ͑see Fig.
2͒. It is straightforward to show in fact that any point in the
upper HABC sector corresponds to a product in the ground
state ͑state 1͒. Similar considerations for point ‘‘2’’ in Fig.
1͑b͒ shows that the Naϩ ion is opposite the second largest
side of the triangle. Consequently, point ‘‘2’’ and all other
points in the HCDE sector of the Dalitz plot correspond to
products in the first excited adiabatic electronic state ͑state
2͒. Finally, all points in the sector HEFG including point 3 in
Fig. 1͑b͒ indicate products in the second excited adiabatic
electronic state ͑state 3͒.
intuitively obvious from the three ␧ values, and, conse-
i
quently, Fig. 1͑b͒ proves to be very useful as a reference
guide for interpreting Dalitz plots for the Naϩ3 system.
It is common to show all experimental data in a single
Dalitz plot, regardless of the Etot value. For experiments that
look at fragmentation from a known excited state,7 Etot has a
single value. However, for experiments that use collisional
activation to induce fragmentation, there will be a range of
E
tot values, and it may be instructive to create Dalitz plots for
different values of Etot . Alternatively, we shall see that it is
useful to create various Dalitz plots based upon particular
values of other experimental variables such as the deflection
angle of the Naϩ3 in the collision with He.
B. Determination of the product adiabatic electronic
state
We can determine the product adiabatic electronic state
for each event of reaction ͑4͒ from the experimental coinci-
dence data. The analysis has been described earlier;10 only a
1
brief summary is given here. There are three A electronic
Ј
states involved. The three Na nuclei form a triangle. We label
the nuclei a, b and c and denote the length of the triangle
side opposite nucleus k as Rk . Once the particles are far
apart they fly in straight lines, and the relative distances Rk
are given by
R ϭ ktϩR0.
͑9͒
v
k
k
Here R0k is on the order of a few angstroms and is negligible
compared to Rk at the detector. At sufficiently large t the Rk
values are ordered in the same way as the atom–atom rela-
C. Theoretical calculations of Dalitz plots for Na3¿
fragmentation
tive speeds k . For example, if
ϽRc at large t, and this ordering is maintained until the
particles reach the detector.
The diabatic state of the system ͑a, b, or c with energy
Ea , Eb , or Ec͒ is identified by which nucleus ͑a, b, or c͒
has the positive charge, and this state is determined experi-
Ͻ Ͻ c , then RaϽRb
va vb v
We have carried out a series of calculations for reaction
͑4͒. They have been described in detail elsewhere9 and only
a brief summary is given here. The potential energies and
couplings have been calculated using the DIM procedure of
Kuntz,13 but extended as described in Refs. 9, 10, and 14.
The theoretical calculation is carried out in three steps. First,
v
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4874
J. Chem. Phys., Vol. 116, No. 12, 22 March 2002
Babikov et al.
position of the point on the Dalitz plot ͓see Fig. 1͑b͔͒.
III. EXPERIMENTAL PROCEDURE
A detailed description of the experiment has been pre-
sented elsewhere.8 Only the main features will be noted here
with special attention paid to the modifications required for
the multifragmentation studies. The Naϩ3 clusters are pro-
duced in an expansion at the exit of an oven heated to
820 °C. They are ionized by 60 eV electrons, accelerated to
2.4 keV and mass selected by a Wien filter. The beam is
chopped at 3.3 MHz before the collision zone. The cluster
ion beam crosses at 90° a ‘‘cold’’ He target beam produced
by a supersonic expansion. The fragmented clusters then en-
ter a parallel plate electrostatic analyzer. The ionic fragment
Naϩ is deflected in the electric field and detected on the first
position sensitive detector ͑PSD͒. Since only one ionic frag-
ment per event is expected, we use a ‘‘slow’’ PSD ͑3 ␮s dead
time for position encoding͒ based on a Quantar resistive an-
ode localization device as in the previous experiments.8 The
two neutral fragments fly on a straight line through the ana-
lyzer and are detected in coincidence with the ionic fragment
on a second PSD. For this experiment the earlier detector
was replaced by a ‘‘faster’’ PSD based on a delay line local-
ization technique. This allows detection and localization of
two neutral fragments coming from the same physical event
provided that the second fragment reaches the PSD at least
20 ns after the first fragment. The loss of events due to the
detection dead time has been estimated for the Naϩ3 case8͑c͒
to be about 5%. The present combination of a heavy projec-
tile with a light target provides a nearly total collection of
fragments on the two PSDs ͑4␲ collection in the center-of-
mass reference frame͒. The chopper clock also triggers a
multistop time-to-digital converter ͑TDC͒ that records the
time of flight ͑TOF͒ of both neutral fragments as well as of
the ionic fragment for each event. The velocity vectors of the
three fragments are determined, in an event-by-event analy-
sis, from their locations on the two PSDs and from their
FIG. 2. Examples of the three possible triangular configurations of the par-
ticles Na, Na, and Naϩ at the detector. The nuclei are labeled a, b, and c, and
the sides opposite the nuclei are Ra , Rb , and Rc , respectively. Here Ra
ϽRbϽRc . The location of the Naϩ ion is indicated by a ‘‘ϩ.’’ The triangles
are identical except for the location of the charge. ͑a͒ Naϩ ion at c. As
discussed in the text this configuration has the lowest energy and corre-
sponds to the ground electronic state ͑state 1͒; ͑b͒ Naϩ ion at b. This con-
figuration has the second lowest energy and corresponds to the first excited
electronic state ͑state 2͒; ͑c͒ Naϩ ion at a. This configuration has the highest
energy and corresponds to the second excited state ͑state 3͒.
the Naϩ3 cluster ion classical reactant state is prepared for a
fixed vibrational energy. Second, the Naϩ3 ϪHe collision is
treated using the classical path approximation for the He
atom and the vibrationally sudden approximation for Naϩ3 .
The Naϩ3 –He interaction introduces couplings between the
three adiabatic states of Naϩ3 , and, in addition, gives an im-
pulse to each Na nucleus. Solution of the time-dependent
TOFs. This data allows us to construct the three values of ␧
i
needed for the Dalitz plot. The collision volume and the
analyzing device are biased at 2.4 kV in order to get rid of
the fragments produced by collision with the background gas
or by evaporation of hot clusters along the incident trajec-
tory. Because of their different energies, fragments produced
¨
Schrodinger equation gives the populations of the three elec-
tronic states after the collision as well as the new momentum
of each Na nucleus. Third, the fragmentation of the Na3ϩ
after the collision with He is treated using the trajectory sur-
face hopping ͑TSH͒ procedure. The trajectory may pass
through one or more avoided crossings, and it may hop from
one adiabatic surface to another. Earlier work9 has shown
that the vibrational energy of the Naϩ3 in the experiment is
about 1 eV, so this amount was used in the present work.
The TSH calculations are carried out until all three val-
ues of Rk exceed 23 a.u. Our calculations show that at this
point the positive charge is fixed on one nucleus, which de-
fines the diabatic state of the system. The diabatic state then
remains the same until the detector. Each trajectory also
TABLE I. Experimental and theoretical distributions of product adiabatic
electronic states.
Theoretical results, contributions ͑%͒
Experimental results,a
contributions ͑%͒
State
At detectorb After dead-time adjustmentc
1
2
3
42.4
33.1
24.5
39.9
34.0
26.1
44.1
35.8
20.1
aA total of 10 512 experimental three-body fragmentation events were ob-
served.
bThe calculated percentage of product adiabatic electronic states at the de-
tector.
cFor comparison with experiment the theoretical results have been adjusted
to account for the dead time of 20 ns of the experimental detector ͑see text
for further details͒.
gives the three ␧ values for the Dalitz plot. The adiabatic
i
electronic state at the detector is then determined from the
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Fragmentation of Na3ϩ
4875
J. Chem. Phys., Vol. 116, No. 12, 22 March 2002
FIG. 4. A comparison of the theoretical Dalitz plot with the experimental
plot for process ͑4͒. The middle circle is a theoretical plot that has been
corrected for the dead time of the experimental detector.
great majority of three-body events originate in state 1. It is
also apparent that there is much hopping between adiabatic
states as the trajectories evolve. Finally we note that all three
product states have appreciable populations, consistent with
column three of Table I.
Figure 4 compares the experimental and theoretical Dal-
itz plots for process ͑4͒. The theoretical results on the left
were obtained by adding together the nine Dalitz plots in Fig.
3. The Dalitz plot in the center is the theoretical result but
corrected for the dead time of the experimental detector. This
plot can be directly compared with the experimental Dalitz
plot on the right. The experimental plot shows considerable
intensity throughout the entire plot, but state 1 is the most
populated and state 3 the least populated. One reason the
entire plot is populated is that the reactant Naϩ3 ions have 1.0
eV internal energy. Thus, even before the collision with He
the Naϩ3 has a wide variety of shapes different from the
equilateral ground state, and also each nucleus has consider-
able vibrational momentum. The collision with He, in addi-
tion to possibly producing an excited electronic state of
Naϩ3 , will give each nucleus an additional impulse, further
broadening the distribution of Na momenta in the three-body
phase space. There is a very strong feature peaked along the
Nafast axis near the edge of the circle. This corresponds to a
Na atom being ejected rapidly from the Naϩ3 followed by a
slower dissociation of the remaining Naϩ2 species to Na and
Naϩ. Assuming that the Naϩ2 is randomly oriented in space
as it dissociates, we would expect a peak symmetric about
the Nafast axis, with equal amounts of products in states 1 and
2. Such a symmetric peak is seen in both the experimental
and theoretical Dalitz plots ͓near point C in Fig. 1͑b͔͒. The
mechanism for producing this peak has been discussed in our
earlier paper.9͑d͒ One of the most efficient ways for process
͑4͒ to occur is for the He to hit one of the Na nuclei in Na3ϩ
in a hard, small impact parameter collision. Sometimes this
‘‘direct binary’’ collision leaves behind a stable diatomic
product ͑Naϩ2 or Na2͒, but often the diatomic dissociates giv-
ing the three-body fragmentation seen here. The ‘‘direct bi-
nary’’ mechanism should also produce the analogous process
whereby a fast Naϩ ion is ejected from Naϩ3 followed by a
slower dissociation of the remaining Na2 species to Na and
Na. This process should appear as a peak along the Naϩ axis
near the edge of the circle ͓near point G in Fig. 1͑b͔͒. This
peak is clearly seen in the theoretical Dalitz plot. Most of the
peak is lost, however, when the theoretical results are cor-
rected for the dead time of the detector. Such a large correc-
tion is to be expected for the case that the two Na atoms have
FIG. 3. Theoretical Dalitz plots for process ͑4͒. The initial state is the
adiabatic electronic state of the trajectory immediately after the Naϩ3 colli-
sion with He, and the final state is the adiabatic electronic state of the
trajectory at the detector. The plots have not been corrected for the dead time
of the experimental detector. The right-hand side of each plot gives the
percentage of all three-body fragmentation events that appear in the particu-
lar Dalitz plot.
in the collision volume are easily separated from background
fragments either by the electrostatic selector or by the TOF
analysis. Consequently the lab collision energy with He is
4.8 keV; the relative collision energy is 263 eV. For each set
of experiments with Naϩ3 clusters, the electrostatic selector is
tuned on the mass of the Naϩ fragment. Ion detection in
coincidence with two neutral fragments insures an unam-
biguous selection of the Naϩ3 →NaϩϩNaϩNa fragmentation
pathway.
IV. RESULTS AND DISCUSSION
A summary of the product electronic state distribution is
given in Table I. A total of 10 512 three-body fragmentation
events were obtained in the experiments. It is seen that state
1, which corresponds to the ground adiabatic electronic state,
is favored, but states 2 and 3 also have appreciable popula-
tions. Two theoretical results are given. The populations ex-
pected at the detector are shown in the third column. As
discussed earlier, the neutral detector has a 20 ns dead time,
so some three-body events are not detected. We have ad-
justed our theoretical calculations to remove all such events,
and the adjusted values are shown in column four. Clearly,
the detector dead time has the greatest effect on products
produced in state 3. The values in column four can be di-
rectly compared with the experimental results in column two.
The agreement is quite good. Further discussion of the re-
sults in this table can be found in Ref. 10.
Figure 3 gives nine state-to-state Dalitz plots obtained
from the theoretical calculations. The initial state is the elec-
tronic state of the trajectory immediately after the Naϩ3 col-
lides with He, and the final state is the adiabatic electronic
state of the trajectory at the detector. The plots have not been
corrected for the dead time of the detector. We see that the
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4876
J. Chem. Phys., Vol. 116, No. 12, 22 March 2002
Babikov et al.
body fragmentation process. We have demonstrated that each
of the three sectors of the Dalitz plot corresponds to a differ-
ent product adiabatic electronic state. This analysis can be
applied to certain other three-body fragmentation experi-
ments such as process ͑1͒. The new convention for labeling
the axes of the plot has the advantage of doubling the signal
to noise in the plots. All areas of the Dalitz plots have ap-
preciable population, due in part to the fact that the Na3ϩ
reactant ion is quite hot. In addition, two intensity peaks
corresponding to sequential decay are observed. One corre-
sponds to a fast ejection of Na from Naϩ3 followed by a
slower dissociation of the Naϩ2 species. The other is the
analogous process where a fast Naϩ is ejected followed by
the slower dissociation of the Na2 . Both processes are
caused by the ‘‘direct binary’’ mechanism whereby the fast
He atom hits one Na nucleus hard and ejects it rapidly from
Naϩ3 , either as Na or Naϩ. Two additional Dalitz plots have
been presented, one for products that arise from Naϩ3 scat-
tered at small angles and the other for Naϩ3 scattered at large
angles. The first plot shows primarily fragmentation that
arises from electronic excitation of the Naϩ3 , whereas the
second is dominated by the ‘‘direct binary’’ processes de-
scribed above. The theoretical results agree well with the
experiments.
FIG. 5. Theoretical and experimental Dalitz plots for restricted values of the
scattering angle ␹. Here ␹ is the deflection angle of the Naϩ3 species in the
center of mass after scattering off a He atom. ͑Top plots͒ Dalitz plots for
fragmentation events with ␹Ͻ6°. ͑Bottom plots͒ Dalitz plots for fragmen-
tation events with ␹Ͼ6°.
little relative separation energy. The experimental results in
this region are very sensitive to the detector response8 when
two neutrals arrive at approximately the same time. In spite
of this problem both the corrected theoretical and the experi-
mental Dalitz plots show this ‘‘direct binary’’ peak near the
bottom of the circle. This two-step fragmentation process
whereby one ion or atom is ejected rapidly followed by a
slower dissociation of the remaining diatomic is seen often in
three-body fragmentation processes and is referred to as ‘‘se-
quential decay.’’ 2
The importance of the ‘‘direct binary’’ mechanism in
producing the peaks seen in Fig. 4 is further supported by the
theoretical and experimental Dalitz plots shown in Fig. 5.
The experiments can easily determine the c.m. deflection
angle ␹ of the Naϩ3 after scattering with the He atom. Both
experiments and theory ͓see, for example, Fig. 7 in Ref.
9͑d͔͒ show that the ␹ values for three-body fragmentation
separate nicely into two peaks, one sharp peak with ␹Ͻ6°
and the other more diffuse peak with ␹Ͼ6°. The large angle
scattering occurs when the He atom hits one of the Na nuclei
in a hard collision that then leads to complete fragmentation.
The small angle peak, on the other hand, is seen when the He
does not hit any Na nucleus hard but instead gives electronic
excitation to ͑primarily͒ state 3 ͓Ref. 9͑a͔͒ which then yields
three-body fragmentation. Figure 5 shows that the Dalitz
plots for ␹Ͻ6° have a very diffuse population over the en-
tire allowed plot, as would be expected for electronic exci-
tation of the reactant Naϩ3 . By comparison, the plots for ␹
Ͼ6° show clearly the two peaks associated with ‘‘direct bi-
nary’’ scattering, one in the upper left part of the plot ͓near
point C in Fig. 1͑b͔͒ and one at the bottom ͑near point G͒.
In summary, we have presented the first experimental
and theoretical Dalitz plots for process ͑4͒. In fact, the cal-
culated result is the first theoretical Dalitz plot for any three-
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