Double Ionization Symmetry Considerations
M. A. Coplan*, R. W. van Boeyen¶, J. H. Moore† , and J. P. Doering¶
*Institute for Physical Science and Technology, University of Maryland,
College Park, MD 20742, USA
†
Department of Chemistry and Biochemistry, University of Maryland,
College Park, MD 20742, USA
¶
Department of Chemistry, The Johns Hopkins University, Baltimore, MD 21218, USA
Abstract. In order to understand the mechanisms of electron impact double ionization it is necessary to establish
experimental conditions that correspond to a well defined scattering regime. We have investigated high
momentum transfer double ionization collision for which a relatively high-energy incident electron ejects two
electrons from an atomic target with energies substantially larger than their binding energies. The target under
investigation is magnesium, a quasi two-electron atom, with two 3s electrons outside of a closed electron shell.
The incident electron energy is 1000 eV with the 867 eV scattered electron detected at 20° with respect to the
incident electron direction. Ejected electrons have 55 eV energy. In this regime, two probable mechanisms, TS1
and TS2, corresponding to one and two collisions of the incident electron with the target, produce different
ejected electron distributions about the momentum transfer direction. A new spectrometer has been constructed
that uses elements of previous designs in a configuration where the momentum transfer vector lies along the axis
of symmetry of a spherical electrostatic energy analyzer that can accommodates up 8 discrete ejected electron
detectors. From the symmetry of the double ionization cross-section about the analyzer axis the relative
contributions of the different ionization mechanisms can be assessed.
electrons with binding energies ε 1 and ε 2, and momenta
q 1 and q 2. At the instant of collision the momentum of
the ion core is q i . After the double ionization collision
there are three free electrons, the incident electron that
has been scattered through angle θs and leaves the
collision with energy Es and momentum k s, the two
ejected electrons that leave the collision with energies
and momenta E1, k 1 and E2, k 2, and the residual
doubly charged ion that is left with momentum q i’ .
The momentum transferred by the incident electron
during the course of the collision is K = k 0 – k s. The
energy deposited by the incident electron is E0 – E s =
∆E.
The application of conservation laws and
reasonable assumptions about the collision parameters
are applied to the following discussion; atomic units
are used throughout.
At the instant of the collision the total momentum of
the system is k 0 + q 1 + q 2 + q i . After the collision
has taken place the momentum is k s + k 1 + k 2 + q i’ .
Assuming that the target system (ion core and two
electrons) is initially at rest (q 1 + q 2 + q i = 0), that no
momentum is transferred to the ion core in the course
of the collision (q i = q i’ ) gives K – k 1 - k 2 = q 1 + q 2.
For each collision, measurement of the momentum
transferred by the incident electron and the momenta of
the two ejected electrons are sufficient to determine the
net momentum of the electrons in the target at the
instant of each collision.
INTRODUCTION
There are a large number of ways an electron can be
scattered from a target atom or molecule with the
accompanying ejection of two of the electrons from the
target. The object of our experiments is to investigate
through a suitable choice of experimental conditions
those double ionization events that can be separated
into a collision term, that describes the way the
incident electron interacts with the electrons in the
target, and a structure term that describes the state of
the electrons in the target at the instant of the
interaction with the incident electron. If the collision
mechanism can be established then the structure portion
can be investigated in detail. For the experiments
described here the scattered and two ejected electrons
are detected in triple coincidence. These experiments
are called (e,3e). By combining triple coincidence
detection with energy and momentum measurements,
the kinematics of a given double ionization collision
can be specified. A variation of the (e,3e) experiment is
the (e, (3-1)e) experiment [1] where two of the three
product electrons are detected in coincidence. These
experiments are useful for identification of double
ionization mechanisms.
In electron impact double ionization an electron with
energy E0 and momentum k 0 is incident on a target that
consists of a doubly charged core and two orbital
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
68
Initially the energy of the system is the kinetic
energy of the incident electron E0. This assumes that
the ground state of the target is the zero of energy.
After the collision, energy is transferred from the
incident electron to the two electrons in the target.
This energy is sufficient to remove them from the
target and leave them with kinetic energies E1 and E 2.
The application of conservation of energy (assuming no
energy transferred from the incident electron to the ion
core) yields E0 = E s + ε 1 + ε 2 + E 1 + E 2 so that ∆E =
ε 1 + ε 2 + E 1 + E 2.
For shakeoff, an incident electron strikes a target
electron and ejects it in the direction of the momentum
transfer vector [2]. The residual singly-charged ion is
left in an excited state which, upon decay, produces a
doubly charged ion in the ground state and a second
ejected electron. For this mechanism one expects one
ejected electron along the momentum transfer direction
with the second electron uniformly distributed over the
unit sphere. The excited ion decay must leave the
second electron with considerable kinetic energy in
order to fulfill the ∆E requirement. . Ion excitation to
high lying states in the double ionization continuum is
unlikely. This mechanism can be discriminated against
experimentally by having no ejected electron energy
analyzer/detector in the momentum transfer direction.
The composite particle model treats the two ejected
electrons as a single particle [3] that is ejected by the
impact of the incident electron. The relative motion of
the two electrons that make up the composite particle
are reflected in the energies and angular distributions of
the two electrons. If both ejected electrons are at rest at
the instant of ejection, this model predicts that they
will be found along the momentum transfer direction.
The extent to which their directions of travel deviate
from the momentum transfer direction is a measure of
their internal motion. This internal motion can be
measured by the observation of the angular
distributions of the two electrons about the momentum
transfer direction, however the cross sections for this
mechanism place it below the sensitivity threshold of
the present instrument.
In the TS1 mechanism the incident electron strikes
and ejects one target electron transmitting to it
momentum equivalent to the energy lost in the
collision. This struck electron on its way out of the
target atom or molecule strikes a second electron
transferring momentum to it and the two electrons
leave the target [4]. The angular distribution of the two
ejected electrons in this case must be consistent with
the total momentum delivered from the incident
electron. If the second collision is also one in which
all of the momentum transferred from the first electron
to the second is transformed into kinetic energy then
the two ejected electrons will leave the collision at
nearly 90° to each other. If the ejected electrons have
equal energies, they should be observed at positions
180° from each other around the base of a 90° cone
whose axis lies along the momentum transfer direction.
This is shown in Fig. 1a.
TS2 is similar to the TS1 mechanism because two
collisions occur inside the target [4]. In this case the
incident electron participates in both, first ejecting one
target electron and then the second target electron. The
requirement that each collision result in the transfer of
momentum into kinetic energy and that the energies of
the ejected electrons are identical limit the kinematics
of the collision as shown in Fig. 1b. In this figure k s’
is the momentum vector of the incident electron after
EXPERIMENTAL CONSIDERATIONS
If the angle through which the incident electron is
scattered is θs, and the angle between the two ejected
electrons is ω 12, K 2 = k 02 + k s2 – 2 k 0 ks cosθs. If the
sum of the momenta of the ejected electrons is P where
P = k 1 + k 2, and P2 = |P|2, then P 2 = k 12 + k 22 – 2 k1 k 2
cos ω 12,. In electron impact double ionization there is
independent control over the momentum transfer, K,
and the energy lost by the incident electron, ∆E. The
energy and momentum of the incident electron can be
fixed making it possible to select only those scattered
electrons that have deposited a predetermined quantity
of momentum and energy into the target by the
positioning an energy analyzer/detector at a fixed angle
(θs) with respect to the incident direction. If all the
momentum transferred from the incident electron in the
course of the collision goes into the momentum of the
ejected electrons (K = P) and the two ejected electrons
have no momentum in the target at the instant they are
ejected (q 1 = q 2 = 0), then ks2 + ε 1 + ε 2 + k 1 k 2 cos ω 12
= k 0 ks cosθs. Thus, a knowledge of θs and the amount
of momentum transferred from the incident electron to
the ejected electrons is not sufficient establish the angle
between the momentum vectors of the ejected electrons.
To decide on an experimental geometry for the study of
double ionization, some knowledge of the mechanism
is needed. Also there are the general requirements that
the double ionization occur on a time scale that is short
compared to an electron orbit and that the ejected
electrons not have their trajectories distorted by the
presence of the residual ion or the other electrons in the
target or each other. These requirements can be
satisfied if ∆E is large compared to the binding energy
of the ejected electrons.
The geometry of our experiments restricts
observations to those collisions with high momentum
transfer from the incident electron to the target followed
by conversion of the transferred momentum to kinetic
energy of the ejected electrons. To simplify the
situation further we have required equal energies for the
two ejected electrons. There are four, large-momentumtransfer, double-ionization mechanisms that are
consistent with our experimental geometry. They are
shakeoff, composite particle ejection, TS1 and TS2.
69
the first scattering event. Only the ejected electron
from this collision is observed and its momentum
vector k 1 lies along the surface of the shallow cone
with opening in the direction of k 0. If the second
collision ejects an electron at 45° to the overall
momentum transfer direction, the only (e,3e) events
that can be detected in this arrangement are those where
the two equal energy ejected electrons are at 90° to each
other in a plane approximately perpendicular to the
plane defined by k 0 and k s.
valence electrons.
Double ionization of the 3s
electrons results in a doubly charged magnesium atom
with a closed shell structure.
The electron
configuration of the ground state atom and relevant
singly and doubly charged ions are given in Table 1.
TABLE 1. Magnesium Ion States and Scattered Electron
Energies for E0 = 1000 eV and E 1 = E2 = 55 eV.
Final State
E (eV) Es (eV)
Configuration
2
2
6
Single
7.6
938.0
1s 2s 2p 3s
2
2
5
Ionization 1s 2s 2p 3s 2
57
833
2
6
2
99
846
1s 2s2p 3s
2
2
6
22.7
867.4
1s 2s 2p
2
2
5
75.4
814.6
1s 2s 2p 3s
2
2
4
2
Double
123
767
1s 2s 2p 3s
2
6
Ionization
110
780
1s 2s2p 3s
2
5
2
150
740
1s 2s2p 3s
2
0
6
2
225
665
1s 2s 2p 3s
ks
(a)
k0
K
k 1, k 2
ks'
k1
(b)
For the experiment under consideration the incident
electron energy is 1000 eV and the ejected electron
analyzer is set to transmit 55 eV electrons. This
ejected electron energy was chosen in order to avoid
interference with the intense auger electron signal at 35
eV. Table I also lists scattered electron energies, E s, for
both single and double ionization and different final
ion states. The energy resolution of the scattered and
ejected analyzers exclude all double ionization events
other than ejection of the two 3s electrons. The single
ejection of a 2p electron with a kinetic energy of 55 eV
is within the energy bandwidth of the scattered electron
analyzer, but does not result in a triple coincidence
signal.
Those electrons from the incident beam that are
scattered through 20° are detected by a scattered
analyzer/detector. Electrons ejected from the target
magnesium atoms by the incident electron beam that
have trajectories at 45° to the momentum transfer
direction at 70° enter a spherical electrostatic analyzer
with those electrons with 55 eV of kinetic energy
transmitted by the analyzer to the imaging plane where
eight detectors are located. The detectors are spaced
uniformly around the periphery of the detector plane
with their angular positions around the symmetry axis
of the analyzer specified by the angle η. The operating
parameters for the experiment are given below:
k0
FIGURE 1. Schematic representation of TS1 (a) and TS2
(b) double ionization geometries
The
experimental
arrangement
is
shown
schematically in Fig. 2. An electron gun produces an
electron beam of well-defined energy and direction that
is incident on a beam of target atoms that emitted from
the target source positioned perpendicular to the
electron beam axis.
A coaxial Faraday collector
measures the incident electron current and aids in the
focusing of the electron beam. A trap located directly
above the interaction of the electron beam and stream of
target atoms collects atoms from the oven, preventing
them from escaping into the vacuum chamber and
depositing on critical surfaces.
ks
k0
s
k1
k2
K
Incident electron beam
Energy
k0
Current
Diameter
FIGURE 2. Schematic diagram of the double ionization
experimental geometry.
The target used for these experiments is magnesium,
a quasi-two-electron atom. The electronic structure
consists of a closed shell outside of which are two 3s
70
1000 eV
8.58 a.u.
0.3 to 3 µA
0.5 mm
Scattered electron analyzer/detector
Energy
ks
Position (θs)
Angular Acceptance
Transmitted energy
Energy pass-band
INITIAL RESULTS
867 eV
7.98 a.u.
20°
0.024 sr
867 eV
12 eV
Tests of the energy resolution, time resolution, and
background level were consistent with the design goals
of the experiment. Estimates of the double ionization
cross section depend critically on the mechanism used
to model the collision and for this reason vary over
several orders of magnitude. In order to optimize the
operating parameters of the instrument the (e,(3-1)e)
process was used. For this experiment the incident
electron energy was varied from 922 to 1032 eV in 10
eV increments while maintaining the pass energies of
the scattered and ejected electron analyzers at 867 and
55 eV respectively. Only double coincidences between
the scattered electron and 55 eV ejected electrons were
recorded. In addition to peaks at 922, 982, and 1022
eV corresponding to single ionization of the 3s, 2p,
and 2s electrons there was additional ejected electron
signal beginning at 942 eV corresponding to the
threshold for the ejection of two 3s electrons, one with
55 eV (the detected electron) and the other with 0 eV.
(the undetected electron). From the (e,(3-1)e) signal
rate we estimate the (e,3e) rate for 3s double ionization.
to be of order 10-3 Hz, substantially less than the
measured background rate.
Ejected electron analyzers
Angular acceptance
Transmitted energy
k1, k 2
Energy pass-band
Detector locations (η)
14° x 7°
55 eV
2.01 a.u.
0.5 eV (FWHM)
0°, 45°, 90°, 135°, 180°,
225°, 270°, 315°
The electronics used for the experiment consist of
channel electron multipliers (Sjuts, Optotechnik) at the
image planes of the scattered and ejected electrostatic
energy analyzers. Their output pulses are amplified by
video amplifiers. Pulses from the scattered electron
detector are sent to a single level discriminator where
they are converted to ECL compatible signals provided
they exceed the lower level threshold. The amplified
pulses from the ejected electron detectors are routed to a
separate level discriminator that converts them to
standard ECL levels and stretches them to 200 ns in
duration. The ejected electron discriminator has, in
addition to the direct ECL outputs, an output (Σ) that
is proportional in amplitude to the number of
simultaneous events at the discriminator inputs. A
single event produces a 50 mV pulse of 200 ns
duration and two events produce a 100 mV pulse of
duration equal to the time overlap of the pulses. When
the Σ signal amplitude indicates the presence of two
ejected electron events within 200 ns of each other it is
combined via an AND gate with the scattered electron
signal to activate the time-to-digital converter (TDC).
The TDC will not be activated if there is only a single
ejected electron event; it will not be activated if there
are two ejected electron events within 200 ns of each
other but no scattered electron event within 200 ns of
the two ejected electron events. By this means the
TDC only processes triple coincidences, significantly
reducing dead time.
CONCLUSIONS
Electron impact double ionization has the potential
for providing information about the detailed motion of
the electrons in atoms and molecules if the mechanism
of the ionization can be well established. Collisions
by high-energy electrons that transfer momentum and
energy to the target electrons in an atom or molecule in
an impulsive manner can produce ejected electrons with
sufficient energies to simplify the description of the
mechanism. The application of conservation of energy
and momentum to models of these collisions reveals
basic symmetries that can be used to extract structure
information from the experimental cross section data.
Of the four different large-momentum-transfer,
collision mechanisms that have been considered, only
TS1 and TS2 are likely to occur with high probability
under typical experimental conditions. Investigation of
the kinematics of the TS1 and TS2 mechanisms shows
that an experimental geometry with ejected electron
detectors located around the momentum transfer
direction provides information about both mechanisms
while allowing the TS1 mechanism to be separated
from TS2.
A spectrometer for investigating high momentum
transfer double ionization collisions has been
constructed and placed into operation. Initial tests
have met design goals and have set a lower limit on
the size of the double ionization cross sections that can
be observed. Estimates of fully determined double
ionization cross sections have been obtained from
System performance
Momentum transfer
2.94 a.u.
Collision volume
.5 mm x 1.0 mm x 1.0 mm
Overall energy resolution
12 eV
Triple coincidence time resolution
4.5 ns
Singles rate (0.3 µA incident beam current)
scattered detector
3 kHz
ejected detectors (8)
35 kHz
Background rate (coincidence window)
10-5 Hz
71
(e,(3-1)) experiments. These estimates give an (e,3e)
count rate well within the capabilities of the
spectrometer and confirm the possibility of obtaining
data about the detailed motion of electrons in atoms.
ACKNOWLEDGEMENTS
Theoretical and computational contributions by N.
Watanabe and J. Cooper have provided very useful
guidance for this work. Support from National Science
Foundation Grant PHY-99-3787 has made this research
possible.
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