Electron collisions with cesium atoms--

Plasma Sources Science and Technology
Plasma Sources Sci. Technol. 23 (2014) 035011 (7pp)
Electron collisions with cesium
atoms—benchmark calculations and
application to modeling an excimer-pumped
alkali laser
Oleg Zatsarinny1 , Klaus Bartschat1,2 , Natalia Yu Babaeva3 and
Mark J Kushner3
Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor,
MI 48109, USA
E-mail: [email protected], [email protected], [email protected] and
[email protected]
Received 26 February 2014, revised 1 April 2014
Accepted for publication 2 April 2014
Published 19 May 2014
The B-spline R-matrix (BSR) with pseudostates method is employed to describe electron
collisions with cesium atoms. Over 300 states are kept in the close-coupling expansion,
including a large number of pseudostates to model the effect of the Rydberg spectrum and,
most importantly, the ionization continuum on the results for transitions between the discrete
physical states of interest. Predictions for elastic scattering, momentum transfer, excitation
and ionization are presented for incident energies up to 200 eV and compared with results from
previous calculations and available experimental data. In a second step, the results are used to
model plasma formation in an excimer-pumped alkali laser operating on the
Cs (62 P3/2,1/2 → 62 S1/2 ) (852 nm and 894 nm) transitions. At sufficiently high operating
temperature of a Cs–Ar containing quartz cell, pump power, and repetition rate, plasma
formation in excess of 1014 –1015 cm−3 occurs. This may reduce laser output power by
electron collisional mixing of the upper and lower laser levels.
Keywords: electron–cesium collisions, cross section, elastic scattering, excitation, ionization,
momentum transfer, excimer-pumped alkali laser
(Some figures may appear in colour only in the online journal)
devoted to developing suitable model potentials to account
for the interaction of the projectile with the valence electron
and of both with the core. To mention just one example,
adding a small dielectronic term [12] to the core potential
obtained by optimizing the Rydberg spectrum proved to be the
critical ingredient for moving the (6s6p)3 P0,1,2 negative-ion
configuration from previously predicted bound states to very
low-energy (a few meV) resonances in the elastic channel, a
prediction that was ultimately confirmed experimentally [21].
Apart from its role as a suitable target to study fundamental
collision physics, in particular as a testing ground for state-of-
1. Introduction
As noted in our previous paper on the subject [1], electron
scattering from Cs has received significant attention, both
experimentally [2–9] and theoretically [10–20]. Being a
heavy target with nuclear charge Z = 55, relativistic
effects are expected to be important. On the other hand,
being an alkali atom, it is likely that many aspects of the
collision can be modeled well as a quasi-two-electron problem,
especially for energies below the threshold for excitation and
ionization of inner shells. Hence, much effort has been
© 2014 IOP Publishing Ltd Printed in the UK
O Zatsarinny et al
Plasma Sources Sci. Technol. 23 (2014) 035011
the-art scattering theory, Cs is also of practical importance in
many applied areas. One example is an excimer-pumped alkali
laser (XPAL), where lasing on the Cs (6p)2 P1/2 → (6s)2 S1/2
transition at 852 nm occurs when pumping on the blue satellite
of the Cs (6s)2 S1/2 → (6p)2 P3/2 transition, peaking at 837 nm
for mixtures of Cs vapor and Ar [22–24]. The intermediate
state is the alkali-rare gas excimer CsAr (B 2 1/2
A potentially important issue in high power pumping
of XPAL is the formation of plasma by electrons that are
superelastically heated by collisions with the resonant states of
Cs. This is a process analogous to LIBORS (laser ionization
by optical resonance saturation), whereby alkali atoms may
be nearly fully ionized by laser excitation of the resonance
states [25]. The analysis and modeling of XPAL systems based
on Cs have been challenged by the lack of a comprehensive
set of electron impact cross sections for Cs [26]. The utility
of the Cs cross sections reported here will be demonstrated
by analysis of the Cs/Ar XPAL system and the prediction of
plasma formation under high power pumping.
This paper is organized as follows. In section 2 we briefly
describe our numerical method to treat e–Cs collisions. This is
followed in section 3 by the presentation and discussion of our
predicted elastic, momentum-transfer, excitation, ionization
and total cross sections, where the initial state can be either
the (6s)2 S1/2 ground state or one of the excited (6p)2 P1/2,3/2
states. The modeling is discussed in section 4, and we finish
with a brief summary (section 5).
The second approach is a fully relativistic, all-electron
B-spline R-matrix with pseudostates (DBSR) ansatz. This
model is a significant extension of the work described by
Zatsarinny and Bartschat [1]. In the latter, most results
were obtained in a 12-state model, including the (6s)2 S1/2
ground state and the physical excited states (6p)2 P1/2,3/2 ,
(5d)2 D3/2,5/2 , and (7s)2 S1/2 states, respectively. Another
six pseudostates were then selected, since they had a fairly
significant dipole connection to the ground state. Some larger
models, including one with 20 physical bound states up to
(5f)2 F5/2,7/2 and one with another 10 pseudostates, were also
set up. However, the effect of these additional states was only
tested for a few partial waves, due to the large computational
effort required and the limited resources available at the time.
In this work, the 30-state model (labeled DBSR-30 below)
was run to convergence with respect to the number of partial
waves (up to total electronic angular momentum J = 45
for the projectile–target system), and a much larger model,
coupling a total of 311 states, was set up and also pushed to
convergence. This DBSR-311 approach is not only expected
to handle ionization via reinterpretation of the results for
excitation of pseudostates above the ionization threshold, but
it should also provide a very good indication about the likely
convergence of the results for transitions between the lowlying physical states with respect to the number of states
retained in the close-coupling expansion. In contrast to
electron collisions with noble gases, where this convergence
can be very slow [27, 28], one generally expects a much faster
convergence for alkali targets, especially for transitions from
the ground state. Nevertheless, it is important to check this
expectation, especially for transitions from already excited
initial states. In high density plasmas (see below) such
transitions can be important, since there is a good chance of
further electron-induced excitation, de-excitation or ionization
before the excited atom relaxes back to the ground state by
optical decay.
As mentioned above, relativistic effects in the DBSR
calculations were accounted for at the level of the Dirac–
Coulomb Hamiltonian, exchange effects between all electrons
were treated with properly antisymmetrized wavefunctions,
and the most important core polarization effects were
accounted for by opening up the 5s2 and 5p6 subshells.
For low-energy elastic scattering below the first excitation
threshold, where polarization effects are most critical, a
special polarized pseudostate model was developed in order
to reproduce the static dipole polarizability of the Cs ground
state with an accuracy matching the best currently available
experimental data. Many more details about the BSR approach
can be found in [29] and a recent Topical Review [30].
2. Theoretical approach
The calculations performed for this work are based on two
independent ab initio R-matrix (close-coupling) models. The
first one is a semi-relativistic approach, in which relativistic
effects for the valence and the projectile electrons were
accounted for through the one-electron terms of the Breit–
Pauli Hamiltonian. This BPRM model is an extension of the
work described by Bartschat and Fang [17] to incident energies
up to 50 eV. Depending on the incident energy, between 5 and
40 states (including pseudostates to account for the possibility
of ionization) were retained in the close-coupling expansion
of the quasi-two-electron scattering system. By carefully
analyzing the results from the various models, we chose what
we believe is the ‘best set’ of our BPRM predictions.
The principal purpose of the Breit–Pauli calculations,
however, was to present an independent check for the fullrelativistic model described below. One would generally
expect that accounting for relativistic effects at the Breit–Pauli
level is sufficiently accurate for low-energy e–Cs scattering
involving only the valence electron. This expectation was
confirmed in [14], where more details can be found. The effect
of the core electrons in the BP model was described by a local
model potential, which was optimized to generate a highly
accurate valence electron spectrum for neutral Cs, as well as
the electron affinity of Cs− by adding the dielectronic term
suggested in [12]. As a result, it is likely that the very lowenergy regime (below about 0.1 eV) is described very well by
this model.
3. Results for angle-integrated cross sections
Figure 1 shows the angle-integrated elastic cross section for
electron collisions with Cs atoms in their (6s)2 S1/2 ground
state as a function of the collision energy. There is good
agreement between the DBSR-311 and the BPRM results,
except for energies below 0.1 eV. Here the DBSR-311 model
predicts a resonance maximum in the p-wave, but in contrast
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Plasma Sources Sci. Technol. 23 (2014) 035011
Momentum-Transfer Cross Section (10-16 cm2)
Elastic Cross Section (10-16 cm2)
Cs (6s)
expt., Visconti et al. (1971)
Incident Electron Energy (eV)
Incident Electron Energy (eV)
Figure 1. Angle-integrated elastic cross section for electron
collisions with Cs atoms in their (6s)2 S1/2 ground state. The
full-relativistic DBSR-311 results are compared with the predictions
from the semi-relativistic BPRM model, the optical potential (OP)
approach [20], and the experimental data of Visconti et al [3]. For
elastic scattering, the DBSR-30 results are very close to those from
the DBSR-311 model and hence not shown for clarity of the figure.
Figure 2. Same as figure 1, except that there are no experimental
data available.
Cross Section (10-16 cm2)
to the BPRM model [14] (and also the 5-state full-relativistic
calculation of Thumm and Norcross [12]), the (6s6p)3 P0,1,2
fine-structure components of this feature overlap and hence the
fine-structure is not resolved. The position of this maximum
is also somewhat high compared with experiment [21]. This
demonstrates the difficulty to obtain such a fine detail in a
fully ab initio model, without any assistance from experiment
(in this case structure information), which is included in the
phenomenological core potential. As found in many other
calculations as well, the experimental data of Visconti et al
[3] (available in the 0.3–2 eV range), lie below the theoretical
predictions. Finally, the ‘optical potential’ (OP) results of
Gangwar et al [20] are significantly (up to a factor 2) larger
than both our BPRM and DBSR-311 results.
The corresponding predictions for the momentum-transfer
cross section are shown in figure 2. The general features are
the same as for the elastic cross section shown in figure 1,
except that the OP results are significantly closer to those
obtained with BPRM and DBSR-311. No experimental data
are available for this case.
Figure 3 depicts results for excitation from the (6s)2 S1/2
ground state.
In particular, for the strong optically
allowed (6s)2 S1/2 → (6p)2 P1/2,3/2 resonance transitions, the
convergence is excellent with the number of coupled states
included in the close-coupling expansion. This is a wellknown feature of electron-alkali scattering, where the large
oscillator strength of these transitions not only leads to large
cross sections, but also accounts for about 99% of the dipole
polarizability of the ground state. For the other transitions,
the convergence is still encouraging, with the smaller models
generally overestimating what we believe is the actual cross
section. Not surprisingly, the reduction effect due to additional
Incident Electron Energy (eV)
Figure 3. Angle-integrated cross section for electron-impact
excitation of the (6p)2 P1/2,3/2 , (5d)2 D3/2,5/2 , (7s)2 S1/2 and
(7p)2 P1/2,3/2 states in Cs atoms from the (6s)2 S1/2 ground state. The
full-relativistic DBSR-30 and DBSR-311 results are compared with
the predictions from the semi-relativistic BPRM model.
channel-coupling is most visible for the smallest cross sections,
in this case those for excitation of the (7p)2 P1/2,3/2 states.
Predictions for excitation from the (6p)2 P1/2 and (6p)2 P3/2
excited states are presented in figure 4. Again, the agreement
between the DBSR-30 and DBSR-311 predictions is generally
good, while larger deviations are noticeable compared with
some of the BPRM results. We have no explanation for the
latter observation, but we do believe that the DBSR-311 data
represent the most reliable set and should be used in modeling
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Plasma Sources Sci. Technol. 23 (2014) 035011
6p1/2 - 6p3/2
Ionization Cross Section (10-16 cm2)
6p1/2 - 5d3/2
6p1/2 - 5d5/2
6p1/2 - 7s1/2
6p1/2 - 7p1/2
6p3/2 - 7s1/2
6p3/2 - 7p1/2
6p1/2 - 7p3/2
Incident Electron Energy (eV)
Figure 5. Angle-integrated cross section for electron-impact
ionization of Cs atoms in their (6s)2 S1/2 ground state. The
full-relativistic DBSR-311 results are compared with the predictions
from the semi-relativistic BPRM model and non-relativistic
convergent close-coupling (CCC) approach. Also shown are
experimental data from Tate and Smith [2], which were visually
normalized to the DBSR-311 predictions and thus are about 10%
lower than the normalization to the CCC results presented by
Lukomski et al [8]. The ragged structures in the DBSR-311 results
at the few-per cent level are due to the finite density of states in the
pseudostate spectrum.
6p3/2 - 7p3/2
Tate & Smith (1934)
6p3/2 - 5d5/2
Ionization Cross Section (10-16 cm2)
Cross Section (10-16 cm2)
6p3/2 - 5d3/2
Cs (6s1/2)
Incident Electron Energy (eV)
Figure 4. Same as figure 3 for transitions from the (6p)2 P1/2 and
(6p)2 P3/2 excited states.
Results for electron-impact ionization of ground-state Cs
atoms are exhibited in figure 5. In the energy region where
only the 6s electron can be ionized, there is reasonably good
agreement between the BPRM and DBSR-311 predictions, as
well as the non-relativistic convergent close-coupling (CCC)
results [8]. The DBSR-311 model even reproduces, at least
qualitatively, the structure seen experimentally [2] around
15 eV incident energy, which is due to significant excitation
of a wealth of autoionizing 5p5 n1 1 n2 2 states [31]. Since
obtaining the branching ratios for autoionization versus optical
decay of these autoionizing states is very complex, we have
assumed a 100% contribution to the ionization signal from their
excitation. It is, therefore, not surprising that the calculation is
slightly overestimating the ionization cross section for incident
energies in the 12–20 eV range.
The energy dependence of the cross section data was
experimentally determined by Tate and Smith [2]. We
normalized their data via a visual fit to the DBSR-311
predictions in the energy region below 20 eV. The absolute
values are about 10% lower than the normalization to the CCC
results chosen in [8].
Results for ionization from the (6p)2 P3/2 excited state are
presented in figure 6. Not surprisingly, the agreement between
Cs (6p3/2)
Lukomski et al. (2006)
Incident Electron Energy (eV)
Figure 6. Angle-integrated cross section for electron-impact
ionization of Cs atoms in the (6p)2 P3/2 excited state. The
full-relativistic DBSR-311 results are compared with the
predictions from the semi-relativistic BPRM model and the
non-relativistic CCC approach. Also shown are the experimental
data of Lukomski et al [8].
DBSR-311, BPRM, and CCC is very good, and hence the wellknown discrepancy of nearly an order of magnitude with the
experimental data of Lukomski et al [8] persists. We refrain
from speculating about possible reasons for this discrepancy,
but we are pleased by the agreement among the three theoretical
datasets, which were obtained in entirely independent ways.
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Plasma Sources Sci. Technol. 23 (2014) 035011
The model used in this investigation is Global Kin, a
global plasma kinetics model, which is essentially the same
as described in [32]. In Global Kin, rate equations for a
species densities, temperatures, pump intensities and laser
intensities are integrated as a function of time over successive
pulsed pump periods. Electron-impact processes are included
for elastic and inelastic collisions, accounting for electronic
and vibrational excitation, superelastic collisions, ionization
and recombination. All rate coefficients for electron-impact
processes are functions of electron temperature, Te , which
is determined by solving an electron energy conservation
equation. The rate coefficients for electron processes are
obtained from solutions of Boltzmann’s equation for the
electron energy distribution function (EED) using the just
described cross sections.
The EEDs and electron impact rate coefficients for an
Ar/Cs = 1/10−4 mixture for the values of Te expected
during XPAL operation are shown in figure 8. (For these
non-Maxwellian distributions, Te = (2/3)ε, where ε is
the average electron energy.) The momentum transfer cross
section is in excess of 100 Å2 at energies below 1.5 eV, and
the inelastic threshold for excitation of the Cs (6p)2 P1/2 state
is 1.38 eV. The end result is that the EED is cut off for electron
temperatures Te < 0.5–0.7 eV. At the same time, the low
threshold energies and large inelastic cross sections produce
significant rate coefficients for electron-impact excitation at
moderate Te . The superelastic rate coefficients, which quench
the Cs (6p) states and produce electron heating, have rate
coefficients in excess of 10−7 cm3 s−1 .
In the demonstration XPAL system, pump radiation
propagates through a quartz cell heated to 450 K containing
500 Torr of Ar and Cs vapor (mole fraction 6.85 × 10−5 ). The
pump pulses at 837 nm are 4 ns (full width half maximum)
with a 3 µs inter-pulse period. A sufficient number of pump
cycles were calculated to achieve a pulse-periodic steady state
in excited state and plasma densities. The electron densities
over successive pulses for different maximum pump intensities
are shown in figure 9. Plasma is formed by heating of
background electrons via superelastic relaxation of dominantly
the Cs (6p)2 P3/2,1/2 states. A pulse-periodic steady state
is reached after 10–60 pump pulses with electron and ion
densities of (2–5) × 1014 cm−3 .
The densities of states responsible for lasing are also
shown in figure 9 for the 1st and 81st pulses for an 8 MW cm−2
pump pulse. Te peaks at 0.4 eV during early pulses of the
pumping and 0.7 eV in later pulses, and it decreases during
the inter-pulse period to about 0.2 eV. The nearly horizontal
dashed line in figure 9 shows the sum of the densities of
Cs+ and Cs2+ . The density of CsAr (B 2 1/2
) follows the
intensity of the pump pulse, as its dissociative lifetime is <1 ns.
The laser transition is saturated by a high intra-cavity laser
intensity, indicated by the densities of the Cs (6p)2 P3/2 and
the Cs (6s)2 S1/2 states being in the ratio of their degeneracies.
The plasma density of 2 × 1014 cm−3 on the 81st pulse is
predominantly Cs+ with a small contribution from Cs2+ . With
an initial Cs density of 7 × 1014 cm−3 , there is significant
ionization of the Cs, which contributes to the depletion of the
ground state. The laser pulse length decreases from 15 ns on
Total Cross Section (10-16 cm2)
DBSR-311, elastic
elastic + excitation
Visconti et al. (1971)
Jaduszliwer and Chan (1992)
Kauppila and Stein (unpublished)
MacAskill et al. (2002)
Electron Energy (eV)
Figure 7. Total electron scattering cross section from the ground
(6s)2 S1/2 ground state of Cs. The present DBSR-311 results are
compared with various sets of experimental data [3, 4, 6, 7]. Also
shown are the contributions from elastic scattering as well as elastic
scattering plus excitation of discrete states and, finally, the
momentum-transfer cross section.
We finish this section with a presentation of the total
cross section for electron collisions with Cs atoms in their
ground state. The DBSR-311 results are shown in figure 7,
where we also include the individual contributions from elastic
scattering alone and together with an estimate for excitation of
all discrete physical states below the ionization threshold. Note
the dominance of excitation over elastic scattering and also
ionization, essentially as soon as excitation to the (6p)2 P1/2,3/2
is energetically allowed. As mentioned above, this is due to the
large oscillator strengths of the resonance transitions and yields
a completely different picture when compared with electron
collisions with noble gases [27, 28]. For completeness, we also
include the momentum-transfer cross section in the figure, in
order to compare it directly with the elastic result. Finally, all
existing experimental data lie systematically below theory. We
have no explanation for this, but we re-emphasize that other
calculations [13, 14] have found the same qualitative result.
4. Application to modeling an excimer-pumped laser
In this section, we demonstrate the utility of the Cs electron
impact cross sections by their use in a first-principles global
model for the Ar/Cs XPAL system. This effort investigated
the possible formation of plasma during high repetition rate,
high power optical pumping and the consequences on laser
performance. In the Cs/Ar XPAL system the CsAr (B 2 1/2
state is optically pumped by 837 nm pulses. The excimer
rapidly dissociates, which produces atoms in the Cs (6p)2 P3/2
state. Lasing can then occur on the Cs (6p)2 P3/2 → (6s)2 S1/2
(D2 ) transition at 852.1 nm. Plasma can be formed by
superelastic heating of sparse seed electrons and by associative
ionization of high-lying states of Cs. Once the plasma is
formed, stepwise ionization of all excited states of Cs is also
an important source of plasma.
O Zatsarinny et al
Plasma Sources Sci. Technol. 23 (2014) 035011
Figure 8. Transport properties of an Ar/Cs = 1/10−4 plasma. Top:
electron energy distributions (EEDs) for different average electron
temperatures. The inset shows detail of the EEDs at low electron
energy. Bottom: rate coefficients for electron-impact processes as a
function of electron temperature (1 eV ≈ 11 600 K).
Figure 9. Properties of the XPAL operating on Cs vapor.
(a) Electron density for high repetition rate optical pumping for
different pump intensities. Density of species responsible for lasing
(852 nm) during the (b) 1st and (c) 81st pump pulse.
work, both regarding the number of coupled states, the
processes considered and the range of energies covered.
The results were then used to model an excimer-pumped
alkali laser with Cs as one of the species. The utility
of the electron impact cross section set was demonstrated
through global modeling of the Cs/Ar excimer-pumpedalkali laser (XPAL), which oscillates on the Cs (6p)2 P3/2 →
(6s)2 S1/2 (D2 ) transition at 852.1 nm. Heating of electrons by
superelastic relaxation of the excimer excited Cs (6p)2 P3/2,1/2
resonant states lead to significant plasma formation, which
ultimately reduces laser power by depletion of the ground state
the first pump pulse to 9 ns on the 81st pump pulse due to
the depletion of the ground state by ionization and electron
collision mixing of the laser levels.
5. Summary and conclusion
We have presented datasets for electron collision cross sections
with Cs atoms initially in either the (6s)2 S1/2 ground state
or the excited (6p)2 P1/2,3/2 states. The present DBSR-311
results, obtained in a full-relativistic close-coupling with
pseudostates approach, are a significant extension of previous
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Plasma Sources Sci. Technol. 23 (2014) 035011
through ionization and by electron collision mixing of the laser
This work was supported by the NSF under grants No PHY1068140, PHY-1212450, and the XSEDE allocation PHY090031 (OZ, KB), and by the DoD High Energy Laser
Multidisciplinary Research Initiative (NYB, MJK).
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