ARTICLE IN PRESS
Journal of Quantitative Spectroscopy &
Radiative Transfer 90 (2005) 29–41
www.elsevier.com/locate/jqsrt
Short communication
Modeling of laser-induced excitation and ionization in
cesium atoms
Y.E.E. Gamala, M.A. Mahmoudb,, H.A. Abd El-Rahmanb
a
National Institute of Laser Enhanced Science, Cairo University, Cairo, Egypt
Physics Department, Faculty of Science, South Valley University, Sohag, Egypt
b
Received 1 July 2003; accepted 21 May 2004
Abstract
A theoretical model is presented to describe kinetics of the plasma formation in cesium undergone to
resonant laser excitation (D1;2 line). The model is based on a rate equations approach where the following
populations are considered: ground state (6s level), laser excited level (6p), a series of high excited levels
close to the ionization limit, and the electron density. We show temporal evolution of these populations and
provide an explanation of the kinetic governing the ionization path-ways. Moreover, we compare the
behavior of the electron density as a function of the laser power with the experimental data by Hunnenkens
et al. This comparison for the electron density with irradiation time 40 ns is proved a good agreement with
the experimental results.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Laser-induced; Ionization; Cesium
1. Introduction
Laser-induced ionization is of considerable interest both from the stand point of understanding
the mechanisms involved and from the attempts to exploit these interactions in a diverse range of
applications. In the plasma field, the most well known examples are, laser fusion, X-ray laser
Corresponding author. Fax: +20-093-601-950.
E-mail address: [email protected] (M.A. Mahmoud).
0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jqsrt.2004.05.044
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development, and laser heating of magnetically confined plasmas [1,2]. In addition to these
applications, the laser saturation of resonance transition has long been recognized as a very
effective means of coupling laser energy into a gaseous medium. In essence, the dense population
of resonance-state atoms resulting from laser saturation represents both a source of energy for
rapidly heating the medium and a large pool of atoms having their ionization energy reduced by
laser photon energy. In the case of an un-ionized gas, this interaction suited for the creation of
long plasma channels that will be needed for electron (or ion) beam transportation in inertial
fusion [3].
Within the past 20 years there has been a great deal of interest in the production of plasmas and
electrical discharges by laser irradiation of vapors, especially Na [4–7], Li [8], Ca, Sr [9], Ba [10,11]
and Cs [12–15]. These vapors irradiated by laser light tuned to an atomic resonance, under similar
conditions of atomic density and laser intensity suggests that the basic mechanism of ionization is
likely to be the same for the different alkali species. The mechanism proposed by Measures [16]
seems to apply very convincingly in these different cases. It is based on electron impact ionization
by super-elastically heated electrons. The saturation of the transition by laser provides a dense
population of excited atoms, which serves to couple the energy of the field to the free electrons
through superelastic collisions with the excited atoms. The super-elastically heated electrons can
then collisionally excite the resonance states to high-lying states which are further ionized by the
laser field or can directly collisionally ionize the resonance states, producing secondary electrons
which participate in the same sequential processes as a chain reaction.
On the other hand, Mahmoud and Gamal [17] have presented a modeling of the phenomenon
of resonance ionization in laser-excited metallic vapor to describe the transient kinetics of
ionization mechanisms, focusing on the time-dependent electron energy distribution function. The
computational model indicated that, the major processes in the different stages of the plasma
creation are purely collisional for both excitation and ionization.
The purpose of the present work is to reveal a relatively simple physical model of laser-induced
excitation and ionization in cesium vapor. This model allows us to study in detail the ionization
and population of high-lying atomic states which occur in cesium vapor excited at the first
resonance transition, 6S–6P. Also we have studied the laser-power dependencies of level
populations and electron densities which created during the interactions.
2. Description of the model
A modification of Mahmoud and Gamal model [17] will be developed to study the case of
resonance ionization of cesium atomic vapor irradiated with a tuned cw laser radiation to the
6s–6p transition. The cesium atom is presented as an 11 atomic-level system namely a ground
state, eight excited states as well as molecular and atomic ion states. Here the fine structure of the
energy levels is neglected. A diagram illustrating the energy level system, which is used in this
model, is shown in Fig. 1. These levels are numbered from 1 to 11 in ascending order of energy.
Accordingly, the previous model is modified to include more atomic excited states to suit the case
of Cs atom. Therefore the various physical processes which take place in the interaction region
and its rate coefficients will be given as follows.
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31
Fig. 1. Energy-level diagram of cesium atom involved in the model.
2.1. The physical processes
As a result of illumination of the atomic Cs vapor with a tuned cw laser source, the following
processes may take place.
1. Laser saturation
Csð6sÞ þ hv ! Csð6pÞ
2. Associative ionization
2
Csð6pÞ þ Csð6pÞ ! Csþ
2 ð Sg;v Þ þ e
3. Penning ionization
Csð6pÞ þ CsðnlÞ ! Csþ þ Csð6sÞ þ e
4. Superelastic collision
CsðnlÞ þ e ! Csð6sÞ þ e
where is the electron energy.
5. Electron impact excitation
CsðnlÞ þ e ! Csðn‘l 0 Þ þ e
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6. Electron impact ionization
Csð6sÞ þ e ! Csþ þ 2e
7. Three body recombination
Csþ þ 2e ! CsðnlÞ þ e
8. Radiative recombination
Csþ þ e ! CsðnlÞ þ hv
where ðnl ¼ 6p, 7d, 9s, 8d, 10s, 9d, 10d, 11d)
The power of the cw laser was low enough, so that the multiphoton or laser-assisted processes
are negligible.
2.2. Cesium data
The data required for the computation are obtained from various sources. In this work we used
the experimental measured value of the associative ionization rate coefficient obtained by
Dobrolezh et al. [18]. The value of rate coefficient is given by:
K AI ¼ ð2 0:2Þ 1013 cm3 s1 :
For the Penning ionization process rate coefficient as described by Hunnenkens et al. [19]. Its
value is given by
K PI 109 cm3 s1 :
By the use of suitable approximations for the energy dependence of the cross-sections, the
corresponding rate coefficients can be expressed by simple analytical formula.
The cross-sections for excitation by electron impact are linearly approximated in the range of
interest [20]:
Qnm ðÞ ¼ C nm ð E s Þ;
4E s ¼ E n E m :
Here, C nm is the excitation cross-section constant and E s the energy threshold. The excitation rate
coefficient is obtained by integration with respect to Maxwellian energy distribution:
kB T e 1=2
C nm ðE s þ 2kB T e Þ exp½E s =kB T e ðcm3 s1 Þ:
ð1Þ
K nm ¼ 4
2pme
The rate coefficient K mn for the reverse reaction, superelastic collision is obtained by the principle
of detailed balance (e.g. [21]):
g
K mn ¼ K nm n exp½ðE m E n Þ=ðkB TÞ ðcm3 s1 Þ;
ð2Þ
gm
where T is the heavy-particle temperature and, gn and gm are the statistical weights of the lower
and upper levels of the transition, respectively.
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For the ionization of the excited states by electron impact we use the empirical formula given by
Drawin [22] as:
10
K nc ¼ 1:46 10
zn
EH
n
En
2
3 1
T 1=2
e cn ðU n ; bn Þ ðcm s Þ;
ð3Þ
where zn denotes the number of equivalent electrons in the n level, E ¼ 13:59 eV, U ¼ =E ion ; is
the electron energy and E ion is the ionization energy of the atom and T e is the electron
temperature in K.
The analytical function cn was approximated by Drawin [23] to be,
un e
1
1
jn ðun ; bn Þ ffi
þ ln 1:25bn 1 þ
;
ð4Þ
un
1 þ un 20 þ un
where bn ffi 1 generally.
The reverse reaction, recombination by three-particle collision, is obtained by the detailed
balancing of ionization:
K cn ¼ 2:07 1016
gn
K nc eun
gþ
ðcm6 s1 Þ;
ð5Þ
where gn and gþ are the statistical weights of the neutral atom and atomic, respectively.
For the radiative recombination process we use the empirical formula given by Drawin [22] for
the rate coefficient:
K ¼ 2:07 1011 Z 2 T e fðux Þ ðcm3 s1 Þ;
where fðux Þ ¼ X ux
2
eðux =n Þ E b ðux =n2 Þ
3
n
n¼1
ð6Þ
ð7Þ
2
and ux ¼ 1:58 105 E H
n Z =T;
where Z is the atomic number of neutral atom and T the absolute temperature of the heavy
particle.
3. Method of calculations
The above-mentioned processes are cooperated together into four sets of equations. The first
represents the rate of change of the population density of the 6s level, the second represents the
temporal variation of the 6p state, the third describes the rate of change of the population density
of the nl states of the Cs atom while the fourth describes the time-dependent Boltzmann’s
equation which give the actual energy distribution function for the free electrons (EEDF) created
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due to these processes. These are given by
dNð6sÞ
¼ Nð6pÞðR31 þ A21 Þ Nð6sÞR12 þ N e ðÞNð6pÞK 21 ðÞ
dt
N e ðÞNð6sÞK 12 ðÞ þ Nð6pÞNðnÞK PI ;
(8)
dNð6pÞ
¼ Nð6sÞR12 Nð6pÞðR21 þ A21 Þ N e ðeÞNð6pÞK 21 ðeÞ
dt
þ N e ðeÞNð6sÞK 12 ðeÞ N 2 ð6pÞK AI
Nð6pÞNðnÞK PI Nð6sÞN e ðeÞK nc ðeÞ;
(9)
X
X
X
dNðnÞ
N e ðeÞNðnÞK nm ðeÞ N e ðeÞNðmÞK mn ðeÞ NðnÞAnm
¼
dt
nm
mn
nm
X
X
N e ðeÞNðnÞK nc ðeÞ Nð6pÞNðnÞK PI
þ
n
N 2e ðeÞ½N e ðeÞK cn ðeÞ
n
þ K rd ðeÞ;
(10)
X
X
dN e ðeÞ
¼
N e ðeÞNðmÞK nm ðeÞ N e ðeÞNðnÞK mn ðeÞ
dt
mn
nm
X
X
þ
N e ðeÞNðnÞK nc ðeÞ þ
Nð6pÞNðnÞK PI þ N 2 ð6pÞK AI
n
N 2e ðeÞ½N e ðeÞK cn ðeÞ
n
þ K rd ðeÞ:
(11)
Where Nðm cm3 Þ represents the population density of level m, N e ðcm3 Þ represents the free
electron density, Amn ðs1 Þ represents the spontaneous emission probability for m to n transition,
K mn ðcm3 s1 Þ represents the m ! n transition electron collisional rate coefficient, K nc ðcm3 s1 Þ
the collisional ionization rate coefficient, K cn ðcm6 s1 Þ the three-body recombination rate
coefficient, and K rd ðcm3 s1 Þ represents the radiative recombination rate coefficients.
K AI ðcm3 s1 Þ and K PI ðcm3 s1 Þ are the rate coefficients of associative Rionization and Penning
ionization, respectively. According to Measures et al. [2] R21 B21 I l ðvÞL21 ðvÞ dv=4p ðs1 Þ
represents the stimulated emission rate coefficient for the (2–1) resonance transition, IðvÞ is the
spectral irradiance of the radiation field at frequency v appropriate to the resonance transition,
B21 represents the Milne coefficient, and L21 ðvÞ represents the resonance line profile function.
4. Results and discussions
The computations are carried out under the experimental conditions of Hunnenkens et al. [19].
In their experiment the cesium density was taken as 1 1016 cm3 and the cw dye laser with
output 200 mW tuned to excited the Cs atoms from the ground state (6s) to the first resonance
excited state (6p) at l ¼ 894:35 nm to study the various collisional processes which responsible for
the creation and growth of the free electrons leading to plasma formation.
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In these calculations the electron energy ðeÞ lies between 0 and 4 eVð4the ionization energy of
cesium atom). Considering the energy step to of order 0:1 eV, therefore about 40 rate equations
representing the growth rate of the electron energy distribution function have to be calculated
each time step. This in addition to the set of the population density as well as the ionic species
formed during the interaction. These equations are solved numerically using a Runge Kutta
technique, taking into account the rate coefficients for the various processes as mentioned before.
4.1. The electron energy distribution function
The electron energy distribution function at different intervals of irradiation is plotted and
shown in Fig. 2. This figure enables us to study the time evolution of the electrons seeding and
growing processes. This can be seen from the observed peaks labeled (a, b, c, d, e, f, g ) which are
changing in height as the irradiation time increases. Therefore, it was important to study the
behavior of each peak as function of time. A correlation between the position of the peak and
electron energy specifies exactly the physical process responsible for each peak. So that peak
labeled (a) at energy 0:25 eV is referred to electrons created by an associative ionization process.
Since in this process electrons created with energies in the range 0–0:5 eV. While the peaks labeled
(b,c) at energies 0.7 and 1:25 eV, respectively are attributed to electrons produced by Penning
ionization process with energies around 0.5–1:7 eV. The peak (d) is appears at energy 1:73 eV
referred to the first superelastic collision between free electrons created by associative ionization
and atoms in the Cs(6p) State. Peak (e) appeared at energy 3:1 eV represents the electrons
gained their energies from the second superelastic collision. In this case, the free electrons, which
are created by associative ionization, can be heated through double collisions with Cs(6p) atoms.
Fig. 2. Time development of the electron energy distribution function in cesium vapor excited with cw laser at laser power 200 mW.
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The peak labeled (f) is referred to electrons undergo the superelastic collisions gaining an amount
of energy 4:3 eV (i.e.) free electrons may collide three times with Cs(6p) to obtain such energy.
Peak (g) may be referred to accumulation of electron having 5:5 eV which are produced by
ionization of ground- and excited-state atoms. Since, this peak appears only after 40–50 ns
irradiation times were enough to collisional ionization processes to take place to produce such
electrons density with energy 5:5 eV.
The overall feature of the distribution function indicates that electrons are mainly created by
both associative and Penning ionization heated up through several superelastic collisions leading
to almost full-ionization attainment of electrons density 1013 cm3 (which is about 103 of
the atomic density) at about 40–50 ns.
4.2. The electron density as a function of laser power
The electron density at different time intervals from 1 to 50 ns is calculated as a function of the
laser power and it is shown in Fig. 3. From this figure we notice that as the laser power increases
the electron density increases and it seems to have the same trend over all the range of time
duration considered. The only observable effect of the time duration was that the low values of the
Fig. 3. The total number of electrons as a function of laser power at various time.
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electron density associated with t ¼ 1 ns where a value of 2 105 cm3 is obtained at laser power
as 20 mW to 2 107 cm3 at laser power 200 mW. As the irradiation time increases the initial and
final values of the electron density increase with almost the same rate over the duration time from
1 to 10 ns. At 40 and 50 ns the growth rate of the electron density becomes faster following
approximately an exponential increase when the laser power exceed 150 mW. The behavior of this
curve can be explained by the fact that as the interaction time increases the possibility of collision
increases and hence more electrons are created faster during this time intervals. This means that
increasing the irradiation time with the increase of laser power lead to an immediate plasma
formation.
To confirm these results we compared our relation with that which obtained by Hunnenkens et
al. [19] when they applied their simple model to calculate the dependence of the electron density
on the laser power. This comparison for the electron density with irradiation time 40 ns is
represented in Fig. 4 where our calculations (dot line) proves a good agreement with Hunnenkens
et al. [19] (circle line). At this point we may conclude that, to produce high density plasma by laser
excitation an irradiation time of about 40 ns is sufficient for such process.
Fig. 4. Comparison between the calculated electron density of our model and Hunnenkens et al. model [19] at time 40 ns.
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4.3. The population density of excited states as function of laser power
The population density for the excited states (6p, 7d and 9s) at energies 1.37, 3.23, 3:33 eV,
respectively) at different time intervals are calculated as a function of laser power and shown in
Figs. 5, 6 and 7. Apart from the different behavior of the population density of the state 6p at the
different time intervals the overwhole behavior of the population density as a function of laser
power for the various excited states in Figs. 6 and 7 is almost the same. That to say as the laser
power increases the population density increases. Beyond this time, moreover, this population
density grows fastly as the interaction time increases from 1 to 5 ns, after that the growth shows a
slow increase as the time varies from 5 up to 50 ns. In these figures we notice that the population
density at 200 mW (the maximum power used experimentally) shows an abrupt increase at
interaction time 40 and 50 ns. Also we notice that the higher the energy state the less population
obtained. This may be explained due to the effect of inelastic collisional excitation of the free
electrons having energies greater than 3:23 eV created through the superelastic collision. The
higher energy states may also be populated through recombination processes in particular at the
period of interaction. Also these states are very close to each other (from 3.23 to 3:7 eV) their
populations represent a considerable amount of irradiation source. This inconsistent with the
Fig. 5. The population density of 6p state as a function of laser power at various time.
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Fig. 6. The population density of 7d state as a function of laser power at various time.
experimental measure (for transitions 7D3=2 ! 6P1=2 at l ¼ 6800 mm, 9S1=2 ! 6P1=2 at l ¼
6350 mm and 6P1=2 ! 6S1=2 at l ¼ 8943 mm) are observed. These results are pronouncely
observed at laser power 200 mW and interaction time in the order of 40 and 50 ns. On the other
hand, regarding Fig. 5 which describes the population density of state (6p) as a function of laser
power at different time intervals. We observe that the growth of the population densities
noticeable from 1 to 5 ns above this time interval the population density shows a very slow
increase over laser power ranged from 20 to 100 mW for the whole range of time intervals.
Beyond this power a very fast increase of the population density is observed at interaction time 40
and 50 ns reaching a value of about 2 1014 cm3 . This result clarifies the fact that, as the laser
power increases the population density increases but to a certain value ð150 mWÞ above which the
increase of the population density becomes nonlinear starting from 1:1 1013 cm3 reaching to a
value 1:57 1014 cm3 in about 50 ns which is very high compared to this population density
obtained at 50 mW where the difference is only from 2:74 1012 to 4:29 1012 cm3 . In general
we notice from this figure the saturation of the state (6p) 1012 cm3 by resonance irradiation
may be proceed by a very fast pumping during the early stages (less than 1 ns) of the radiation
time. We can also see that, this state populated mainly by resonance excitation and depopulated
by associative ionization.
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Fig. 7. The population density of 9s state as a function of laser power at various time.
5. Conclusions
We have studied kinetics of the collisional ionization processes that occur in laser-irradiated
cesium vapor. To investigate of each physical process during the plasma generation, the temporal
evolution of each excited states as well as the electron density are calculated. From these
computations we concluded that, the electrons are mainly generated by associated and
contributed to grow through Penning ionization. Superelastic collisions are the mechanisms
which are responsible for the electron energy gain through which hot electrons can be created in
the plasma with energies up to 5 eV. Electron impact collisional processes are responsible for
depopulating of the highly excited states by electrons gaining their energies through superelastic
collisional processes. As a general conclusion, we can notify that laser-induced ionization by
resonant excitation of alkali vapor atoms is a very useful technique for producing high-density
plasma with a considerably low laser power during time duration of the order of ten’s of
nanosecond.
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