Journal of Applied Mathematics and Physics, 2014, 2, 1123-1129
Published Online November 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.212131
How to cite this paper: Abdelati, M.A., Mahmoud, M.A. and Gamal, Y.E.E. (2014) Numerical Investigation of the Tri-Atomic
Ions Formation during Laser Ionization Based on Resonance Saturation. Journal of Applied Mathematics and Physics, 2,
1123-1129. http://dx.doi.org/10.4236/jamp.2014.212131
Numerical Investigation of the Tri-Atomic
Ions Formation during Laser Ionization
Based on Resonance Saturation
M. A. Abdelati1, M. A. Mahmoud2, Y. E. E. Gamal1
1National Institute of Laser Enhanced Sciences, Cairo University, El Giza, Egypt
2Physics Dep art men t, Faculty of Science, Sohag University, Sohag, Egypt
Email: maa @ni les.ed u.eg
Received August 2014
Abstract
We present a theoretical investigation of plasma generation in sodium vapor induced by laser
radiation tuned to the first resonance line (3S-3P) at λ = 589 ns. A set of rate equations that de-
scribe the rate of change of the ground and excited states population as well as the temporal vari-
ation of the electron energy distribution function (EEDF), beside the formed atomic ion Na+, mo-
lecular ion
+
2
Na
and tri-atomic ions
+
3
Na
are solved numerically. The calculations are carried out
at different laser energy and different sodium atomic vapor densities under the experimental
conditions of Tapalian and Smith (1993) to test the existence of the formed tri-atomic ions. The
numerical calculations of the electron energy distribution function (EEDF) show that a deviation
from the Maxwellian distribution due to the super elastic collisions effect. In addition to the com-
petition between associative ionization (3P-3P), associative ionization (3P-3D) and Molnar-H orn -
beck ionization processes for producing
+
2
Na
, the calculations have also shown that the atomic ions
Na+ are formed through the Penning ionization and photoionization processes. These results are
found to be consistent with the experimental observations.
Keywords
Plasma, Lase r, Collisional Ionization, Association Ionization, Tri-Atomic Ions, Photoionizat ion,
Electron Energy Distribution Func tion
1. Introduction
Resonant laser excitation has played a vital role in coupling energy into vapor where efficient ionization by re-
sonance saturation has been observed in Alkali-metal vapor. This phenomenon has attracted the attention of
many authors due to its importance in different fields such as astrophysics, plasma physics and photochemistry
[1]-[7]. Plasma generation induced by intense laser radiation under such conditions showed high efficiency
when the laser wavelength corresponds to an absorption line of the ionized alkali-vapor. Such interaction in-
volves many collisional ionization and excitation energy as well as radiative transfer processes [16] [17]. Accor-
M. A. Abdelati et al.
1124
dingly in the present work a theoretical study is under taken to investigate the formation of triatomic ions by
tuning a laser source to the transition 3p-4d at a wavelength 569 nm in laser excited sodium vapor at the first
resonance transition 3s1/2-3p1/2. The formation of this process is found to depend mainly on the associative ioni-
zation reaction given by [7]:
( )
23
Na4d +NaNae
+−
→+
in this reaction Na2 is assumed to be present at 0.5%
percent in the sodium atomic beam. This study based on a model which solves numerically a set of rate equa-
tions that describe the temporal variation of the following parameters: the electron energy distribution function,
the electron density, the population of the states 3s, 3p and 4d. The temporal variation of
Na+
,
2
Na
+
and
3
Na
+
are also involved. These calculations are carried out for various sodium atomic densities. The model includes
electron generating processes as well as electron heating processes.
2. Rate Equations
The calculations are based on the system of rate equations similar to those presented in our previous papers [8]
[9]. A set of coupled differential equations” which describe the instantaneous population densities of the consi-
dered energy states”, electron density as a function of electron energy together with the normalization conditions
are developed and solved iteratively. Accordingly the equations which represent the rate of change of instanta-
neous population density of 3S, 3P and N (n) states are given by:
( )()()()( )()( )
( )()()()()
()()()()()( )
21 211221
12 PI
2EP HMI 1
d3 33 3d
d
33
1
333.
2
e
e
ec
NS NPR ANSRnNPK
t
nN SKN PNnK
N PKNnNSKNSnK
ε εε
εε
εε
=+− +
−+
+− −
(1)
( )()()()( )()( )( )()( )
( )( )()( )()( )
( )( )
( )
()()( )()
1221 212112
2
2AI1AI2PI PL
2
22
EP 2226
d3 3 33d3d
d11
33333
22
1
3333.
2
ee
c ec
NP N SRN PRAnN PKnNSK
t
N PKNPNDKNPNnKNPvF
NPKN PFN SnKN PR
εεεεεε
σ
σ εε
= −+−+
−−− −
−−− −
∫∫
(2)
( )( )()( )()( )()( )( )( )
()( )()( )()( )
( )
( )( )( )( )()()()
2
1
2
PIEPHMI 2
2624 Na
Na
d
d
1
333
2
34Na.
e nmemnnm e nc
nmmnnm n
nc
n nn
ee cnRDd
n
Nn n NnKnNnKANnnNnK
t
N PNnKNPKNnNSKNnF
NnnKKN PRNdNK
ε εεεε
σ
ε εεε
+
+
=−−−
− +−−
+++ −


∑∑ ∑∑
∑ ∑∑
 
(3)
While the rate of growth of the molecular ion, atomic ion and the tri-atomic ions is given by
( )
()()()( ) ()
22AIAI2 HMI
d Na13 333
d2
NN PKNPNDKNnNSK
t
+
=++
(4)
( )
( )()( )( )
( )
( )
( )
221
2
PIPL2 2
d Na1
3 33
d2
c nc
n
NNPNnKNPvFNPFNn F
t
σσ σ
+
=+ ++
(5)
(6)
where R21 (sec1) represents the stimulated emission rate coefficient for transition from level 2 to 1.
( )( )()( )
21 212121
d4πRB ILIh
νν ννσυ
≡≅
,
( )
I
ν
is the spectral irradiance of the radiation field at frequency
ν
appropriate to the 2 1 transition, B21
represents Milne coefficient, and L21
( )
ν
represents the corresponding line profile function for the transition. R26
M. A. Abdelati et al.
1125
(sec 1) represents the stimulated emission rate coefficient for transition from level 2 to 6. A21 is the resonance tran-
sition Einstein coefficient for spontaneous emission.
( )
3NS
,
( )
3NP
and
( )
Nn
represents the population den-
sity of levels 3S, 3P and nl respectively. ne
( )
ε
represents the free electron density as a function of electron
energy
ε
. Kmn (cm3.sec1) represents the electron-collision rate coefficient for electron-atom collisional excitation
or de excitation between the states m and n (the former was calculated by Measure’s formula [10] and the latter
employing detailed balance relations). Knc (cm3.sec1) represents the electron collisional ionization rate coefficient
for level n calculated by Vriens and Smeets [11] and Kcn (cm6.sec1) represents the three body recombination rate
coefficient calculated from the detailed balance relations.]. KRD(cm3.sec 1) represents the radiative recombina-
tion rate coefficient to level n determined by Darwin and Felenbok [12]. KAI1, KA I2, KMH I, KPI and KEP are the
rate coefficients of associative ionization (3p-3p), associative ionization (3p-3d) [13], Hornbeck-Molnar ioniza-
tion [14], Penning ionization [2] and energy pooling collisions [15] [16] respectively.
( )
1
nc
σ
is the single-photon
ionization cross section for level n [17],
( )
2
2c
σ
is the two photon [18], resonance state ionization cross section
and
PL
σ
is laser induced Penning ionization cross section [19] and v is average velocity of atoms Units, in cm/sec
3
2.688 10
o
kT
vK
m
= =×
T
is the temperature of saturated vapor. F represents the photon flux density. In
Equation (6)
2
4 Nad
K
+
represents the triatomic associative ionization rate coefficient [7]. The time evolu-
tion of the electron energy distribution function is given by Boltzmann equation including all the collisional
processes in which the plasma electrons are involved
( )()()( )()()()()()()()
( )
( )
( )
( )
( )( )()( )
()()( )( )( )( )()()
PI
21
22
2AI1 AI2
2
HMIRD 2
Na
d3
d
11
33333
22
34Na
eenmemnenc
mnmn nn
c nc
n
ee cn
nn
nnNmKnKnNnKN PNnK
t
NPFNnFN PKNPNDKNPvF
NnN SKNnnKKNdNK
εεεεε εε
σσ
ε εεε
+
⊃⊂
=−++
+++ ++
+ −++


∑∑∑ ∑
∑∑ 2
4 Na
.
d+
(7)
The normalization conditions are
( )
()
0
Na
n
NNn N
+
= +
(8)
( )( )
12
00
d1, d
e ee
n nN
εεεε ε
∞∞
= =
∫∫
(9)
where Ne is the number density of electrons and N0 is the density of Na vapor. Note that the factor 1/2 with KEP
and KAI corrects for possible double counting of each colliding pair of identical particles [20].
The rate coefficients of collisional ionization processes and the cross sections of the photoionization processes
in our model are adopted from our previous paper [9]. The rate constant of the Na3+ formations is taken from the
measurements that carried out in [7].
3. Results and Discussions
The above shown set of Equations (1)-(9) are solved numerically using the Rung-Kutta fourth-order technique
under the experimental conditions of Tapalian and Smith (1993) [7]. In their experiment, they examined a plas-
ma formed by a resonant CW laser at different atomic densities of sodium. A computer program is under taken
to obtain the following relations 1) The population of
3
Na
+
and
2
Na
+
as a function of the sodium atomic den-
sity at different exposed times. 2) The time evolution of the (4d) state in the presence and absence of the tri-
atomic associative ionization process as a function of a sodium atomic density. 2) Electron energy distribution
functions for different values of both sodium atomic density and exposure time in the presence and absence of
triatomic associative ionization process.
3.1. Dependence of
3
Na+
and
2
Na+
on Sodium Vapor Density
There is a clear correlation between the growths of Tri-atomic ions with the Sodium atomic density, since the
growth of Na3 start slowly at the low atomic density up to 1.5 × 1012 cm3 then it followed by a noticeable in-
crease as shown in the Figure 1(a). This behavior is observed over the whole range of the exposure time. It is
M. A. Abdelati et al.
1126
Figure 1.
3
Na
+
ion yield versus atomic density.
also noticed here that the most closer calculated values of the Na3 to the experimentally measured ones are those
obtained at exposure time 500 ns. The agreement shown in this Figure 1(b) assures the validity of the model.
The variation of the formed molecular ions density against the Sodium atomic density is shown in the Figure 2.
For an easy comparison the experimentally measured ones of Tapalian and Smith [7] are also shown in this fig-
ure. It is noticed here that the molecular ions density increases with the increase of the atomic density this is ob-
served for both calculated and measured values.
In addition molecular ions can also be formed through photo ionization of Sodium molecules. The genera-
tion of these ions increases linearly with the atomic density as shown in Figure 3.
3.2. The Effect of Triatomic Formation on the Population Density of the 4d State
The formation of Tri atomic ions depends mainly on the population of 4d state therefore Figure 4 represents the
time evolution of the population of this state in the presence and absence of Tri atomic ions formation is shown
by curves ( a, b and c) and (a , b and c) respectively. The saturation behavior shown up to 250 ns reflects the
less contribution of the Tri atomic ions formation during this period. Moreover the increase of the 4d population
density at the higher atomic density (2 × 1012 cm3) over the time interval 400 - 500 ns gives an evidence for the
formation of Tri atomic ions. On the other hand the omission of this process leads to the decrease of 4d popula-
tion density as shown by curves (a, b and c). As this process produced in laser ionization based on resonance
saturation and this technique depends mainly on the saturation of 3p state of Sodium atoms therefore one expect
associative ionization to take place among the atoms of this saturated level.
3.3. The Time Evolution of Electron Energy Distribution Function
For a deeper understanding of the effect of formation Na3+ on time evolution of plasma formation calculations
are performed first to obtained the EEDF in the presence and absence of this process at different time intervals at
atomic density (2 × 1012 cm3). This is shown in Figure 5. In this figure the number of peaks appeared over the
whole electron energy range can be illustrated as follows. The peak A represent electrons at energy about 0.27
eV generated by associative ionization (AI), the peaks B and C represent electrons of energy about 0.87 eV gen-
erated by penning ionization. While peaks D and E refers to electrons which are generated by associative and
penning ionization and heated up by the first and second super elastic collision processes respectively. The peak
F corresponds to electrons resulted by penning ionization and heated up through first and second super elastic
collision process [2] [8].Comparison between (i) and (ii) in Figure 5 shows that: the inclusions of tri-atomic as-
sociative ionization reaction leads only to the increase of the values electron energy distribution Function.
This appears from the values of the peaks. This means that this reaction is not affecting the processes which
are taking place in its absence. This result can be drawn from the constant location of the peaks in the two parts
of the Figure 5. To give a deeper understanding about the time at which this reaction occurs. Figure 6 show that
5.0x10
11
1.0x10
12
1.5x10
12
2.0x10
12
0.0
2.0x10
3
4.0x10
3
6.0x10
3
8.0x10
3
1.0x10
4
1.2x10
4
1.4x10
4
1.6x10
4
EXP (Tapalian and Smith 1993)
(i) 200ns
(ii) 300 ns
(iii) 400 ns
(iv) 500 ns
Na
3
+
( Na
2
+Na(4d))
Atomic density (cm
-3
)
(i)
(ii)
(iii)
(i v)
(a)
4.0x1011 8.0x1011 1.2x1012 1.6x1012 2.0x1012
0.0
2.0x103
4.0x103
6.0x103
8.0x103
1.0x104
1.2x104
1.4x104
1.6x104
EXP (Tapalian and Smith 1993)
Calculated
Na3
+(Na2+Na(4d))
Atomic density (cm-3)
(b)
M. A. Abdelati et al.
1127
Figure 2.
2
Na
+
ion yield versus atomic density.
Figure 3.
2
Na
+
ion yield (photoionization of Na2) atomic den-
sity.
the omission of this reaction does not affect.
Also the position of the peaks on the energy axis but rather it increases the electron density over the energy
range from 0.87 eV to 1.7 eV. This in turn results in a broadening of peaks B and C i.e. it increases the electrons
density which is generated through Penning ionization process. Meanwhile it is noticed here that this reaction
takes place during the early time interval 100 ns and becomes more effective as the time increases.
4. Conclusions
A modified previously developed a model of dense sodium vapor ionization induced by nanosecond resonant
laser pulses exciting the 3S→3P is applied to study the effect of
3
Na
+
formation on the generated plasma. Ki-
netics of the processes which produce the molecular ion
2
Na
+
and the tri-atomic ion
3
Na
+
in lasers excited so-
dium vapor was investigated theoretically. Formation of
3
Na
+
results in an increase of the electron energy dis-
tribution function and hence the electron density which generates more density plasma in other words leads to
full ionization of the atomic vapor. The results showed non-Maxwell Ian distribution of the electrons for differ-
4.0x10
11
8.0x10
11
1.2x10
12
1.6x10
12
2.0x10
12
0.0
2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5
1.2x10
5
1.4x10
5
Exp ( Taplan and Smith 1993
Calculated
Na2
+(Na(3p) +Na(3p) )
Atomic Density (atoms/cm3)
7x10
11
8x10
11
9x10
11
1x10
12
1x10
12
1x10
12
1x10
12
1x10
12
2x10
12
2x10
12
5x10
4
6x10
4
7x10
4
8x10
4
9x10
4
1x10
5
1x10
5
1x10
5
EXP (Tapalian and Smith 1993)
Calculated
Na
2
+
(phot oioni zat ion) ( ar b. uni t es)
Atomic Density (atoms/cm
3
)
M. A. Abdelati et al.
1128
Figure 4 . The time evolution of population density of 4d level.
Figure 5. Electron energy distribution function calculated at different time intervals for sodium atomic density of 2 × 1012 in
(i) Presence of tri atomic associative ionization process; (ii) Absence tri-atomic associative ionization process.
Figure 6. Electr on energy distribution function calculated for different values of sodium atomic density in (i) Presence of
tri-atomic associative ionization process; (ii) Absence of tri-atomic associative ionization process.
0100200300 400500
10
1
10
3
10
5
10
7
10
9
10
11
(with TRI at 2*10
11
)
(without TRI at 2*10
11
)
(with TRI at 8*10
11
)
(without TRI at 8*10
11
)
(with TRI at 2*10
12
)
(without TRI at 2*10
12
)
(4d) population density
time (ns)
a
a'
b
b'
c
c'
012345
10
-4
10
0
10
4
10
8
10
12
10
16
(i) 100 ns
(ii) 200 ns
(iii) 300 ns
(iv) 400 ns
(v) 500 ns
Electron energy (eV)
( A)
( B)
( C)
( D)
( E)
( F)
(i)
(ii)
(iii)
(iv)
(v)
i
012345
10
-4
10
0
10
4
10
8
10
12
10
16
(i) 100 ns
(ii) 200ns
(iii) 300ns
(iv) 400ns
(v) 500ns
Electron energy (ev)
( A)
( B)
( C)
( D)
( E)
( F)
(i)
(ii)
(iii)
(iv)
(v)
ii
012345
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
10
10
10
12
10
14
10
16
(iii) density 2*1012
(ii) density 8*1011
(i) density 2*1011
Electron energy (eV)
( A)
(B) ( C)
( D)
( E)
( F)
(i)
(ii)
(iii)
i
012345
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
10
10
10
12
10
14
10
16
(iii) density 2*10
12
(ii) density 8*10
11
(i) density 2*10
11
Electron Energy (eV)
( A)
( B)
( C)
( D)
( E)
( F)
(i)
(ii)
(iii)
ii
M. A. Abdelati et al.
1129
ent values of sodium vapor densities. The collisional ionization processes such as the associative ionization and
Hornbeck-Molnar ionization play the important rule in producing the molecular ion density
2
Na
+
. The good
agreement shown in compare the calculated and measurement values of the formed Na3+ and Na2+ reveled the
validity of the model in investigated the plasma generation using LIBORS technique.
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