Optics and Photonics Journal, 2012, 2, 358-366
http://dx.doi.org/10.4236/opj.2012.24045 Published Online December 2012 (http://www.SciRP.org/journal/opj)
The Gain Distribution According to the Theoretical
Structure and Decay Dynamics of Sodium-Like Cu
Wessameldin S. Abdelaziz1, Mai E. Ahmed2, Ali S. Khalil3,
Mohammed Alshaik Ahmed4, Tharwat M. El-Sherbini5
1National Institute of Laser Enhanced Sciences, Cairo University, Giza, Egypt
2Environmental Affairs Agency, Cairo, Egypt
3Tebin Institute for Metrological Studies, Helwan, Egypt
4Al-Azhar University, Gaza, Palastine
5Laboratory of Lasers and New Materials, Cairo University, Giza, Egypt
Received August 31, 2012; revised September 30, 2012; accepted October 10, 2012
ABSTRACT
Level structure, oscillator strengths, transition probabilities and radiative life times are evaluated for 1s2 2s2 2p63l, 4l, 5l
(l = 0, 1, 2, 3, 4) states in sodium-like Cu18+. The calculations are carried out using COWAN code. The calculations
were made are compared with other results in literature where a good agreement is found, we also report on some un-
published energy values and oscillator strengths. Our results are used in the calculation of reduced population of 21 fine
structure levels over a wide rang of electron density values (1018 to 1020) and at various electron plasma temperature.
For those transitions with positive population inversion factor, the gain coefficients are evaluated and plotted against the
electron density.
Keywords: Level Structure; Oscillator Strengths; Sodium-Like Cu
1. Introduction
Successful results on soft X-ray amplification have been
achieved with the electron collisional excitation scheme
including neon-like and nickel-like isoelectronic se-
quences [1-3]. With the neon-like scheme, the required
pumping laser intensity increases rapidly as the wave-
length be comes shorter towards the water window spec-
tral region (4.4 - 2.3 nm) [4].
An important objective in the development of X-ray
lasers is to deliver a coherent, saturated output at wave-
lengths toward the water window [1].
Such saturated X-ray lasers are required for hologra-
phy [2] and microscopy [3], of biological specimens and
for deflectometry [4], interferometry [5], and radiogra-
phy [6] of dense plasma relevant to inertial confinement
fusion and laboratory astrophysics [5]. Sodium-like ions
have prominent emission lines in the UV and XUV spec-
trum of the sun. Highly charged sodium-like ions were
observed in several types of laboratory sources such as
high-voltage vacuum spark tokomak and laser-produced
plasmas [1].
Spectra of these ions have a simple structure (one elec-
tron outside a closed shell); the energy levels are practi-
cally free from effects of configuration mixing and
therefore they are well suited for a theoretical interpreta-
tion of line intensities and for diagnostic purposes. In
recent years, there have been extensive spectroscopic
studies, both experimental and theoretical, of sodium
isoelectronic sequence. [7] have used multi-configuration
Hartree Fock (MCHF) method and Non-Orthogonal
Spline configuration interaction (CI) method to evaluate
energy levels and oscillator strengths for Na-like ions up
to Fe XVI. [8] have used theoretical single—configure-
tion Dirac-Fock method to evaluate oscillator strengths
for E1 transitions in the sodium isoelectronic sequence
(Na I-Ca X). Large-scale calculations were undertaken
within the so-called Opacity Project (OP) [8] and, as a
result of broad international collaboration, a complete set
of oscillator strengths was produced for all optically al-
lowed transitions between states with 1≤ Z ≤ 14 as well
as Z = 16, 18, 20, and 26 in all stages of ionization.
However, relativistic effects were also neglected in OP
calculations and LS coupling was assumed. Therefore,
the OP data are available for multiples only and not for
individual fine-structure components. [9] assigned ex-
perimental wavelengths to 1086 lines which have oscil-
lator strengths calculated in OP and added 1163 lines
which have critically evaluated oscillator strengths. [10]
and from the US National Bureau of Standards (NBS)
publications [11-13]. Experimental lifetime measure-
C
opyright © 2012 SciRes. OPJ

W. S. ABDELAZIZ ET AL. 359
ments for the 4p and 5p levels and calculation of transi-
tion probabilities in Na I have been measured using
time-resolved laser induced fluorescence [14].
The purpose of this work is to present the energies of
21 fine structure levels, the Oscillator strengths, the tran-
sition probabilities between them in sodium-like Cu18+.
The atomic data thus obtained are used to calculate
reduced population of sodium-like Cu18+ excited levels
over a wide range of electron density and at various elec-
trons Temperatures. The gain coefficients are also calcu-
lated.
2. Computation of Atomic Structures
2.1. Model of Central Force Field
In quantum mechanics, various physical processes can be
summed by Schrödinger equation, i.e.
.
iii
HE (1)
In the non-relativistic case (the influence of relativistic
effect will be discussed later), the Hamiltonian of an
atomic system with N electrons is:

22
2
2π.
2
kine nuce e
i
ii
ei
HH HH
h2
ij
ij
Z
ee
mr


 


r

(2)
Here Hkin, He-nuc and He-e refer, respectively, to the ki-
netic energy of electrons, the Coulomb potential and the
energy of electrostatic interaction of electrons, ri is the
distance between the i-th electron and nucleus, and
iji j
rrr.
By substituting the Hamiltonian into Schrödinger
equation and solving the equation in the case of multiple
electrons and multiple energy levels, the wave function is
obtained. Now, due to the appearance of the term of in-
teraction of electrons, an exact solution cannot be ob-
tained. On the other hand, the interaction term is compa-
rable with the Coulomb potential term, so it can by no
means be ignored. An approximate solution is to adopt
the method of central force field. If it is assumed that
every electron moves in the central force field of the nu-
cleus and also in the mean force field produced by other
electrons, then we have the following effective Hamilto-
nian:

22
1
1
2
NN
eff effeff i
ii
ii
ei
pZe
i
H
HV
mr


 


 r


(3)
2.2. Method of Calculation
The key problem in the application of central field is to
find an adequate potential function Veff. For this, in recent
decades many effective method of calculation have been
developed. Among them the more important ones are the
potential model, Hartree-Fock theory, the semi-empirical
methods. In the following we present a brief introduction
of semi-empirical methods.
Semi-empirical methods try to calculate atomic struc-
tures via solving the simplified form of the Hartree-Fock
equation. The most typical is the Hartree-Fock-Slater
method.
Afterwards, Cowan et al. revised this method and de-
veloped the RCN/RCG program used in our work. The
merit of the program is its extreme effectiveness, and the
shortcoming is its inability to estimate the precision.
2.3. Configuration Interaction
In the above-stated model of central force field, every
electron can be described with a simple wave function.
The overall wave function of atoms may be expressed
with the following Slater determinant:

 
 
11 1
1
1
!
N
NN
N
N

  







 



 

(4)
In reality, such a description is not very precise. The
best wave function should be a linear combination of
wave functions with single configurations, and these
wave functions possess the same total angular momen-
tum and spin symmetry. This method is called the inter-
action of configurations. In the computation of atomic
structures, consideration of the configuration interaction
is the basis requirement for a program.
2.4. Relativistic Correction
In a non-relativistic system, the oscillator strengths and
dipole transitions under LS-coupling can be calculated.
In calculating forbidden transitions, jj-coupling must be
used, and for this relativistic effects have to be taken into
account. Generally speaking, the effects may be treated
in two ways. One is inclusion of Breit-Pauli operator in
the non-relativistic equation, and other is direct solution
of the Dirac equation. For the former, a mass velocity
term, the Darwin term caused by the electric moments of
electrons and the spin-orbit term are added to the Hamil-
tonian of the model of central force field [15]. For rela-
tivistic correction, the program RCN/RCG restore to the
Breit-Pauli correction.
2.5. Weighted Oscillator Strengths and Lifetimes
The oscillator strength f(γγʹ) is a physical quantity related
to line intensity I and transition probability W(γγʹ), as
given by Sobelman [16]:


22
\
3
2we
wf
mc



 (5)

W. S. ABDELAZIZ ET AL. 360
with,
 
\\
I
gWg fgf  .
Here m = electron mass, e = electron charge, I = is ini-
tial quantum state,

 
\
WE E   , E(γ) initial
state energy, g = (2J + 1) is the number of degenerate
quantum state with angular momentum J (in the formula
for initial state). Quantities with primes refer to the final
state.
In the above equation, the weighted oscillator strength,
gf, is given by Cowan [17]:
22
0
8π
3
mca
g
f
hS

 (6)
where g is the statistical weight of lower level, f is the
absorption oscillator strength,


 
\
EE h   c,
h is Planck’s constant, c = light velocity, and a0 is Bohr
radius, and the electric dipole line strength is defined by:
2
1\
sJPJ   (7)
This quantity is a measure of the total strength of the
spectral line, including all possible transition between m,
m’ for different Jz Eigen states. The tensor operator P1
(first order) in the reduced matrix element is the classical
dipole moment for the atom in units of ea0.
To obtain gf, we need to calculate S first, (or its square
root):
121 \
J
PJs


 (8)
In a multi-configuration calculation we have to expand
the wave function
J

In terms of single configuration
wave-functions,
J

for both upper and lower levels:
J
J
J




 (9)
therefore, we can have the multi-configurationally ex-
pression for the square root of line strength:
\
\
1\12 \
J
S
J
PJ




   (10)
The probability per unit time of an atom in specific
state
J

to make a spontaneous transition to any state
with lower energy is



\\
,
P
JAJJ  

\\
(11)
where

,
A
J  J is the Einstein spontaneous emis-
sion transition probability rate, for a transition from the
state.
J

to the state \\
J

The sum is over all state \\
J

with


\\ .EJ EJ
 
The Einstein probability rate is related to gf with the
following relation by [17]:
22 2
8πe
g
Agf
mc

 (12)
Since the natural lifetime

J


is the inverse of
transition probability, then:




1
\\
=,TJAJJ 

 (13)
which is applicable to an isolated atom.
Interaction with matter or radiation will reduce the
lifetime of any state.
3. Computation of Gain Coefficient
The possibility of laser emission from plasma of ions of
various members of Na-like Cu via electron collisional
pumping, in the XUV and soft X-ray spectral regions is
investigated at different plasma temperatures and plasma
electron densities.
The reduced population densities are calculated by
solving the coupled rate equations [18-21].
de
jji ejiji
ijij ij
ed
ei iji ijiij
ij ijij
NANC C
NNC NCNA

 



 








 
(14)
here Nj is the population of level j,
j
i
A
is the spontane-
ous decay rate from level j to level i, e
j
i
C is the electron
collisional excitation rate coefficient, and d
j
i
C is the
electron collisional de-excitation rate coefficient, which
is related to electron collisional excitation rate coefficient
by [22,23].
exp
de
j
ie
i
ji ij
CC
g

j
gEKT







(15)
where gi and gj are the statistical weights of lower and
upper levels, respectively
The population of the jth level is obtained from the
identity [19,20,24],
 jit
ie
ite
NNN
N
NNN


N
(16)
where
I
N is the total number density of all levels of the
ion under consideration, and Nt is the total number den-
sity of all ionization stage.
Since the populations calculated from Equation (14)
are normalized such that [19,20,25]
55
1
1
J
JI
N
N





 (17)
where n is the number of all the levels of the ion under
consideration.

W. S. ABDELAZIZ ET AL. 361
Electron collisional pumping has been applied. Colli-
sion in the lasant ion plasma will transfer the pumped
quanta to other levels, and resulted in population inver-
sions then produced between the upper and lower levels.
Once a population inversion has ensured a positive
gain through F > 0 [1].
uu l
uu l
g
NN
FNg g 




(18)
where u
u
N
g
and l
l
N
g
are the reduced populations of the
upper level and lower level respectively. Equation (18)
has been used to calculate the gain coefficient for Dop-
pler broadening of the various transitions in the Na-like
Cu ion.
12
3
8ππ
lu
ul u
i
M
A
NF
KT


 
 (19)
where M is the ion mass, u

 is the transition wave-
length in cm, i is the ion temperature in T
K
 and u, l
represent the upper and lower transition levels respec-
tively.
The gain coefficient is expressed in terms of the upper
state density (Nu). This quantity depends on how the up-
per state is populated, as well as on the density of the
initial source state. The source state is often the ground
state for a particular ion.
4. Results and Discussions
4.1. Energy Levels
Adopting the program COWAN [17], we have computed
the parameters of atomic structures of Cu XIX (Table 1).
The energy levels considered in the calculation have 21
fine structures ranging from ground state 1s2 2s2 2p6 3s to
the excited states1s2 2s2 2p6 3d, 1s2 2s2 2p6 4s, 1s2 2s2 2p6
4d, 1s2 2s2 2p6 5s, 1s2 2s2 2p6 5s, 1s2 2s2 2p6 5d, 1s2 2s2
2p6 5 g even parity levels and 1s2 2s2 2p6 3p, 1s2 2s2 2p6
4p, 1s2 2s2 2p6 4f, 1s2 2s2 2p6 5p, 1s2 2s2 2p6 5f odd parity
levels.
Table 2 presents energy levels and fine-structure split-
ting for Cu18+ also presented the energy levels calcula-
tions of (Younis et al., 2006) the present calculations
differ by less than 0.16% for most of the levels, they
have used Configuration-Interaction Code (CIV3).and
the differ from (Nist, 2009) by less than 0.14% for most
of the levels, except in some cases like the levels 2, 3, 4,
5, 9 and 10 in which the percentage difference between
our value and the last two calculations is (0.66%, 0.66%),
(–1%, 0.28%), (0.35%, 0.27%), (0.3%, 0.35%), (–0.5%,
0.12) and (–0.49%, 0.13) respectively, This mean that
our results are in a good agreement with the theoretical
and experimental value.
Table 1. Parameters (in 1000 cm–1) used in the HFR calcu-
lations for configurations of Cu18+.
Parameter Value
Eav 3s 0
Eav 3d 817.8918
ζ 3d 2.5716
Eav 4s 2538.6141
Eav 4d 2852.2149
ζ 4d 1.1058
Eav 5s 3631.0025
Eav 5d 3784.5436
ζ 5D 0.5686
Eav 5g 3826.6066
ζ 5g 0.0683
Eav 3p 355.1267
ζ 3p 23.4863
Eav 4p 2680.7844
ζ 4p 9.3163
Eav 4f 2926.6767
ζ 4f 0.2885
Eav 5P 3701.4463
ζ 5p 4.6147
Eav 5f 3822.256
ζ 5f 0.1487
4.2. Oscillator Strength and Transition
Probability
In the present work we report results of oscillator
strength and transition probabilities using the Cowan
code taking relativistic corrections into account.
Table 3 presents the wavelength, transition probability
and the values of oscillator strength with a comparison
against another published paper (Younis et al., 2006 [26])
and NIST. There was good agreement with (NIST, 2009
[27) and only small differences about (0.8%) there was
some similarities with the paper and high differences in
some values about (85%).
4.3. Radiative Lifetime
Table 4 contains the present results of radiative lifetime
for the upper and lower laser levels for the sodium-like
Cu. The present calculations predict that the lifetime of
the upper laser level must be longer than the lifetime of
lower one to ensure the fast depletion of the population

W. S. ABDELAZIZ ET AL. 362
Table 2. Calculated HFR energy levels and fine structure spliting (in 1000 cm–1) for Cu18+.
index Level Trem Ecal. E[28] E
[29]
1 1s2 2s2 2p6 3s 2s1/2 0 0 0
2 1s2 2s2 2p6 3p 2p1/2 331.64 329.436 329.436
3 1s2 2s2 2p6 3p 2p3/2 366.87 370.758 365.826
4 1s2 2s2 2p6 3d 2d3/2 814.034 811.14 811.791
5 1s2 2s2 2p6 3d 2d5\2 820.463 817.992 817.56
6 1s2 2s2 2p6 4s 2s1/2 2538.614 2535.44 2535.44
7 1s2 2s2 2p6 4p 2p1/2 2671.468 2667.49 2667.49
8 1s2 2s2 2p6 4p 2p3/2 2685.443 2688.902 2681.6
9 1s2 2s2 2p6 4d 2d3/2 2850.556 2865.349 2847
10 1s2 2s2 2p6 4d 2d5\2 2853.321 2867.269 2849.5
11 1s2 2s2 2p6 4f 2f5/2 2926.1 2924.385 2924.4
12 1s2 2s2 2p6 4f 2f7/2 2927.109 2925.41 2925.4
13 1s2 2s2 2p6 5s 2s1/2 3631.003 3615.981 _ _ _ _ _
14 1s2 2s2 2p6 5p 2p1\2 3696.832 3693.4 3693.4
15 1s2 2s2 2p6 5p 2p3\2 3703.754 3695.943 3699.3
16 1s2 2s2 2p6 5d 2d3\2 3783.691 3779.55 3779.3
17 1s2 2s2 2p6 5d 2d5\2 3785.112 3780.434 3780.6
18 1s2 2s2 2p6 5f 2f5\2 3821.959 3818.201 3818.1
19 1s2 2s2 2p6 5f 2f7\2 3822.479 3818.624 3818.7
20 1s2 2s2 2p6 5g 2g7\2 3826.436 - 3823.1
21 1s2 2s2 2p6 5g 2g9\2 3826.743 - 3823.4
of lower level which helps the laser to sustain continuous
or quasi-continuous wave operation.
4.4. Gain Distributions
4.4.1. Level Populations
The reduced population densities are calculated for 2
levels by solving the coupled rate equations (14) using
the coupled equations CRMO code [28] for solving si-
multaneous coupled rate equations.
Our calculations for the reduced population as a func-
tion of electron densities are plotted in Figures 1 and 2 at
three different plasma temperatures (1/2, 3/4 of the ioni-
zation potential) for Na-like Cu.
We took into account in the calculation spontaneous
radiated decay rate and electron collisional processes
between all levels under study.
The behavior of level populations of the various ions
can be explained as follows: in general, at low electron
densities the reduced population density is proportional
to the electron density, where excitation to an excited
state is followed immediately by radiation decay, and
collisional mixing of excited levels can be ignored.
At high densities (Ne > 1020), radiative decay to all lev-
els will be negligible compared to collisional depopula-
tions and all level population become independent of
electron density and approximately equal (see Figures 1
and 2). The population inversion is largest where electron
collisional deexcitation rate for the upper level is com-
parable to radiative decay for this level [21].
Form our study, it was found that the gain coefficient
was very low at (1/4 ionization potential) for all elements,
so the results of gain coefficient and reduced population
have not been mentioned.
4.4.2. Inversion Factor
As we mentioned before, laser emission will occur only
if there is population inversion or in other words for

W. S. ABDELAZIZ ET AL. 363
Table 3. Wavelength, oscillator strenghth and radiative rate
for allowed transitions in Cu18+.
j λ (Å) Aji (sec)–1 fijcal. Fij[28]
1 2 301.531 8.15E+09 0.1114 0.1083
1 3 272.576 2.21E+10 0.24602 0.2461
1 7 37.433 3.82E+11 0.08016 0.2106
1 8 37.238 7.75E+11 0.16105 0.4209
1 14 27.05 2.21E+11 0.02415 0.0168
1 15 27 4.43E+11 0.04841 0.0336
2 4 207.299 1.99E+10 0.25584 0.2737
2 6 45.311 2.01E+11 0.06194 0.0889
2 9 39.7 7.38E+11 0.34911 0.2178
2 13 30.309 8.95E+10 0.012302 0.0261
2 16 28.968 4.18E+11 0.10494 0.1572
3 4 223.631 3.18E+09 0.02371 0.0251
3 5 220.462 1.98E+10 0.21624 0.2305
3 6 46.046 3.84E+11 0.06094 0.0987
3 9 40.263 1.42E+11 0.03443 0.02407
3 10 40.218 8.52E+11 0.30969 0.2166
3 13 30.636 1.73E+11 0.01216 0.0283
3 16 29.267 8.10E+10 0.01039 0.0171
3 17 29.255 4.87E+11 0.09352 0.1542
4 7 53.838 6.63E+10 0.02877 0.08625
4 8 53.436 1.36E+10 0.005806 0.01736
4 11 47.347 2.80E+12 0.93742 0.7719
4 14 34.689 2.65E+10 0.00477 0.0000131
4 15 34.605 5.33E+09 0.00095 0.0000026
4 18 33.246 1.02E+12 0.16941 0.3043
5 8 53.62 8.05E+10 0.03466 0.1052
5 11 47.492 1.32E+11 0.04455 0.03689
5 12 47.469 2.63E+12 0.89093 0.7379
5 15 34.683 3.18E+10 0.00572 0.000004
5 18 33.317 4.83E+10 0.00805 0.01458
5 19 33.311 9.68E+11 0.161008 0.2918
6 7 752.706 1.89E+09 0.16068 0.146
6 8 681.067 5.10E+09 0.35478 0.3387
6 1486.34 7.65E+10 0.08551 0.03472
6 1585.827 1.56E+11 0.17177 0.06953
7 9 558.3844.38E+09 0.40829 0.4317
7 13104.2176.45E+10 0.104705 0.001194
7 1689.91 1.27E+11 0.30688 0.1081
8 9 605.643 6.85E+08 0.03757 0.03846
8 10595.67 4.32E+09 0.3443 0.3502
8 13105.757 1.23E+11 0.10302 0.002109
8 1691.054 2.44E+10 0.03026 0.009396
8 1790.936 1.47E+11 0.27286 0.08475
9 111323.74 3.78E+08 0.09952 0.09749
9 14118.165 3.15E+10 0.06575 0.05316
9 15117.206 6.45E+09 0.01327 0.01065
9 18102.944 4.68E+11 0.74291 0.04293
10111374.02 1.61E+07 0.00455 0.00449
10121355.22 3.37E+08 0.09243 0.09151
1015117.587 3.83E+10 0.07922 0.06517
1018103.238 2.20E+10 0.03522 0.0021
1019103.182 4.42E+11 0.70607 0.04205
1116116.606 1.22E+10 0.01651 0.17251
1117116.413 5.82E+08 0.00118 0.01235
1120111.07 5.43E+11 1.33613 _____
1217116.55 1.16E+10 0.01773 0.1844
12 20 111.194 2.00E+10 0.03706 _____
1221111.156 5.61E+11 1.29991 _____
13141519.09 6.00E+08 0.20795 0.1972
13151374.55 1.62E+09 0.45916 0.4071
14161151.29 1.37E+09 0.54446 0.4472
15161250.98 2.14E+08 0.05011 0.0433
15171229.13 1.35E+09 0.45913 0.3951
16182613.16 1.74E+08 0.1778 0.1376
17182713.97 7.38E+06 0.00816 0.0064
17192676.17 1.54E+08 0.16551 0.1297
182022335.1 1.10E+05 0.01093 _____
192025272.9 2.80E+03 0.00026 _____
192123451.3 9.83E+04 0.01013 _____
For simplicity, the 1s2 2s2 2p6 core is omitted in the identification.

W. S. ABDELAZIZ ET AL. 364
Table 4. Radiative lifetime for Cu18+ laser levels.
Index Level life time (ns)
6 4s (2s1/2) 1.70E−03
7 4p (2p1/2) 2.22E−03
Table 5. Parameters of the most intense laser transitions in
Cu18+ ion plasma.
Transition Atomic data
Ion Cu XIX
4p (2p1/2)→4s (2s1/2) Wavelength λ (Å) 752.7
Maximum gain α (cm–1) 0.079
Electron density Ne (cm–3) 5.00E+19
Electron temperature Te (eV) 502.94
0.00E+00
2.00E-04
4.00E-04
6.00E-04
8.00E-04
1.00E-03
1.20E-03
1.40E-03
7 4p(2p1/2)
6 4s(2s1/2)
reduced population
Log Ne(cm–3)
Figure 1. Reduced population of Cu18+ levels after electron
collisional pumping as a function of the electron density at
temperature 335.29.
0.00E+00
5.00E-04
1.00E-03
1.50E-03
2.00E-03
2.50E-03
6 4s(2s1/2)
7 4p(2p1/2)
Log Ne(cm–3)
Figure 2. Reduced population of Cu18+ levels after electron
collisional pumping as a function of the electron density at
temperature 502.94.
positive inversion factor F > 0. In order to work in the
XUV and X-ray spectral regions, we have chosen transi-
tions between any two levels producing photons with
wavelength between 30 and 1000 Å.The electron density
at which the population reaches collisional equilibrium
approximately equal to A/D, where A is radiative decay
rate and D is the collisional deexcitation rate [20]. The
population inversion is largest where the electron colli-
sional deexcitation rate for the upper level is comparable
to the radiative decay rate for this level.
19.3010319.3979419.4771213 19.544068 19.60206
Log Ne (cm
-
3
)
(7-6)
4.00E-02
3.50E-02
3.00E-02
2.50E-02
2.00E-02
1.50E-02
1.00E-02
0.50E-02
0.00E+00
Gain (cm
-
1
)
Figure 3. Gain coefficient of possible laser transitions
against electron density at temperature 335.29 eV in Cu18+.
0.00E+00
1.00E-02
2.00E-02
3.00E-02
4.00E-02
5.00E-02
6.00E-02
7.00E-02
8.00E-02
9.00E-02
Gain (cm
-1
)
log Ne (cm
-3
)
7 -6
Figure 4. Gain coefficient of possible laser transitions
against electron density at temperature 502.94 eV in Cu18+.
For increasing atomic number Z, the population inver-
sion occurs at higher electron densities, this is due to the
increase in the radiative decay rate with Z and the de-
crease in collisional deexcitation rate coefficient with Z
[29].
4.4.3. Gain Coefficient
As a result of population inversion there will be positive
gain in laser medium. Equation (19) has been used to
calculate gain coefficient for the Doppler broadening of
various transitions in the Na-like Cu.
Gain (cm-
1
)
Our results for the maximum gain coefficient in cm–1
for those transitions having a positive inversion factor F
> 0 in the case of Cu18+ ion at different temperatures are
calculated and plotted against electron density in (Fig-
ures 3 and 4). The figures show that the population in-
versions occur for several transitions in the Cu18+ ion,
however the largest gain occurs for the Cu18+ ion at
4p(2p1/2)-4s(2s1/2) transition.
For Na-like Cu, the population inversion is due to
strong monopole excitation from the 3s ground state to
the 3s 4d configuration and also the radiative decay of the
3s 4d level to the ground level is forbidden, while the 3s
4p level decays very rapidly to the ground level.
This short wavelength laser transitions was produced
using plasmas created by optical lasers as the lasing me-
dium. It is obvious that the gain increases with the tem-
perature as the maximum gain increases with atomic
number. Moreover, the peak of the gain curves shifts to

W. S. ABDELAZIZ ET AL. 365
higher electron densities with the increase of atomic
number.
5. Conclusion
The analysis that have been presented in this work shows
that electron collisional pumping (ECP) is suitable for
attaining population inversion and offering the potential
for laser emission in the spectral region between 50 and
1000 Å from the Na-like Cu. This class of lasers can be
achieved under the suitable conditions of pumping power
as well as electron density. The positive gains obtained
previously for some transitions in the ions under studies
(Cu18+ ion) together with the calculated parameters could
be achieved experimentally, then successful low cost
electron collisional pumping XUV and soft X-ray lasers
can be developed for various applications. The results
have suggested the following laser transitions in the
Cu18+ plasma ion, as the most promising laser emission
lines in the XUV and soft X-ray spectral regions, Table 5
gives parameters of the most intense laser transitions in
Cu18+ ion plasma.
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