Optics and Photonics Journal, 2012, 2, 314-317
http://dx.doi.org/10.4236/opj.2012.24038 Published Online December 2012 (http://www.SciRP.org/journal/opj)
Search for Laser Lines in Sodium-Like Fe Plasmas
Wessameldin S. Abdelaziz1, Mai E. Ahmed2, Tharwat M. El-Sherbini3,
Mohammed Alshaik Ahmed4, Ali S. Khalil5
1National Institute of Laser Enhanced Seiences, Cairo University, Giza, Egypt
2Environmental Affairs Agency, Cairo, Egypt
3Laboratory of Lasers and New Materials, Cairo University, Giza, Egypt
4Al-Azhar University in Palastine, Ghaza, Palastine
5Tebin Institute for Metrological Studies (TIMS), Cairo, Egypt
Email: wessamlaser@yahoo.com
Received August 17, 2012; revised September 23, 2012; accepted October 13, 2012
Energy levels, transition probabilities and effective collision strength for 1s2 2s2 2p6 3l, 4l, 5l (l = 0, 1, 2, 3, 4) states of
sodium-like Fe are used in the determination of the reduced populations for 21 fine structure levels over a wide rang of
electron density values (1018 to 1020 cm–3) and at against electron plasma temperatures. Gain coefficients are evaluated
and plotted against the electron density.
Keywords: XUV; Soft X-Ray; Laser Emission; Gain Coefficient
1. Introduction
Emission lines arising from transitions in ions of the so-
dium iso-electronic sequence can be among the strongest
observed ones in the solar ultraviolet spectrum [1]. The
line ratios involving Na-like iron transitions are poten-
tially very useful for electron temperature (Te) diagnos-
tics of the solar transition region [2], and it is noted that
Na-like ions may be employed as abundance indicators
[3]. Moreover, it has been shown that Na-like ion emis-
sion-line ratios provide electron density (Ne) diagnostics
for high-density laboratory plasmas, such as tokomaks
Over the last decade recombination and resonantly
photopumped X-ray laser schemes on the n = 3 – 6 to n’
= 2 – 4, the lasing transitions in H, He, Li and Na-like
ions were extensively studied both experimentally and
theoretically [5].
The modeling of laboratory X-ray lasers is routinely
based on the calculations of the gain coefficients (G)
involved laser levels [5]. In these calculations of the
populations of the upper and lower laser levels are given
from the kinetics data and the peak values of spectral
functions for the potential lasing transitions.
For recombination and resonantly photopumped X-
ray laser schemes, however, the strongest laser lines are
usually found in the long-wavelength part of the n to n’
emission spectra represented by a number of closely-
spaced or even overlapping lines that may be strongly
affected by the ion Stark broadening. In addition, theo-
retical studies of particular X-ray laser schemes using CR
models have shown that these lines often correspond to
transitions with different values of population inversion
The purpose of this work is to use the atomic data to
calculate reduced populations of sodium-like Fe excited
levels over a wide range of electron densities and at
various electron temperatures. The gain coefficients are
also calculated. In order to search for laser lines in so-
dium-like Fe plasma, these data might help experimen-
talists in developing soft X-ray lasers.
2. Computation of Gain Coefficient
The possibility of laser emission from plasma of ions of
Fe 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 [7-10].
ji ejiji
jijij ij
iij iijiij
eij ijij
 
 
where Nj is the the population of level j,
is the
spontaneous decay rate from level j to level i, e
C is the
electron collisional excitation rate coefficient, and d
is the electron collisional de-excitation rate coefficient,
opyright © 2012 SciRes. OPJ
which is related to electron collisional excitation rate
coefficient by [11,12].
iij jie
where gi and gj are the statistical weights of lower and
upper level, respectively.
The electron impact excitation rates are usually ex-
pressed via the effective collision strengths γij as
8.6287 10expcm sec
ij ij
The actual population density NJ of the jth level is ob-
tained from the following identity,
N is the quantity of ions which reach the ioni-
zation stage I and is given by [13]
Ie avg
NfNZ (5)
where fI is the fractional abundance of the Ni-like ioniza-
tion stages calculated by Goldstein et al. [13], Ne is the
electron density, and Zavg is the average degree of ioniza-
Since the populations calculated from Equation (7) are
normalized such that
where 21 is the number of all the levels of the ion under
Electron collisional pumping has been applied. Colli-
sion in the lasant ion plasma will transfer the pumped
quanta to other levels, and may lead to population inver-
sions between the upper and lower levels.
Once a population inversion has been ensured, a posi-
tive gain whith result F > 0 [14].
uu l
uu l
FNg g
where u
and l
are the reduced populations of the
upper level and lower level respectively. Equation (11)
has been used to calculate the gain coefficient for Dop-
pler broadening of the various transitions in the Na-like
Fe ion, [14].
ul u
 (8)
where M is the ion mass, u
is the transition wave-
length in cm, i is the ion temperature in 0K and u, l
represent the upper and lower transition levels respec-
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 lower state is often the ground
state for a particular ion.
3. Results and Discussions
3.1. Level Populations
The reduced population densities are calculated for 21
levels by solving the coupled rate Equations [14], and
plotted for 21 levels using atomic data from literature
[15]. The gain was calculated using Matlab version 7.3.0
computer program for solving simultaneous coupled rate
Our calculations for the reduced populations as a func-
tion of electron densities are plotted in Figure 1 at one
plasma temperatures (3/4 of the ionization potential) for
Na-like Fe.
We took into account in the calculation spontaneous
radiative 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.
We expect that at high densities (Ne > 1020 cm–3), ra-
diative decay to all levels will be negligible compared to
collisional depopulations and all level population become
independent of electron density and are approximately
the same (see ref. [16-18]). The population inversion is
largest when electron collisional de-excitation rate for the
upper level is comparable to the radiative decay of this
level [10].
From our study, it was found that the gain coefficient
was very low at 1/4 and 1/2 ionization potentials for all
elements, and therefore the obtained gain coefficient and
reduced population have not been included in Figure 2.
6 4s(2s1/2)
7 4p(2p1/2)
13 5s(2s1/2)
14 5p(2p1/2)
15 5p(2p3/2)
16 5d(2d3/2)
reduced population
log Ne(cm-3)
Figure 1. Reduced population of Fe15+ levels after electron
collisional pumping as a function of the electron density at
temperature 367 eV.
Copyright © 2012 SciRes. OPJ
Gain (cm)-1
log Ne (cm)-3
Figure 2. Gain coefficient of possible laser transitio ns against
electron density at temperature 367 eV in Fe15+.
3.2. Inversion Factor
As we mentioned before, laser amplification will occur
only if there is population inversion or in other words for
positive inversion factor F > 0. In order to work in the
XUV and soft X-ray spectral regions, we have chosen
transitions between any two levels producing photons
with wavelength between 30 and 1000 Å. The electron
density at which the population reaches collisional equi-
librium approximately equal to A/D, where A is the ra-
diative decay rate and D is the collisional de-excitation
rate [9]. The population inversion is largest when the
electron collisional de-excitation rate for the upper level
is comparable to the radiative decay rate for this level.
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 de-excitation rate coefficient with Z
3.3. Gain Coefficient
As a result of population inversion there will be a posi-
tive gain in laser medium. Equation (8) has been used to
calculate gain coefficient for the Doppler broadening of
various transitions in the Na-like Fe.
Our results for the maximum gain coefficient in cm–1
of those transitions having a positive inversion factor F >
0 in the case of Fe15+ ion at different temperatures are
calculated and plotted against electron density in Figure
The figure shows that the population inversion occurs
for several transitions in the Fe15+ ion, however the larg-
est gain occurs for the Fe15+ ion at 5d(2D3/2)5p(2P3/2)
For Na-like Fe, the population inversion is due to a
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.
These short wavelength laser transitions were pro-
Table 1. Parameters of the most intense laser transitions in
Fe15+ ion plasma.
Transition Atomic data Fe XVI
4p (2p1/2)4s (2s1/2) Wavelength λ (Å) 904
Maximum gain α (cm–1) 0.105
Electron density Ne (cm–3) 1.50E+19
Electron temperature Te (eV) 367.02
5p (2p1/2)5s (2s1/2) Wavelength λ (Å) 835
Maximum gain α (cm-1) 0.0218
Electron density Ne (cm–3) 2.50E+19
Electron temperature Te (eV) 367.02
5d (2d3/2)5p (2p3/2)Wavelength λ (Å) 493
Maximum gain α (cm–1) 2.00E01
Electron density Ne (cm–3) 6.00E+19
Electron temperature Te (eV) 367.02
duced using plasmas as the lasing medium created by
electron impact excitation.
4. Conclusion
The analysis that has 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 Fe. 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 ion under study
(Fe15+ ion) together with the calculated parameters could
be achieved experimentally, then successful low cost
electron collisional pumping XUV and soft X-ray lasers
could be developed for various applications. The results
suggested that some laser transitions in the Fe15+ plasma
ions, as the most promising laser emission lines in the
XUV and soft X-ray spectral regions (Table 1).
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