Search for Laser Lines in Sodium like Fe Plasmas

doi:10.4236/opj.2012.24038

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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: [email protected]

Received August 17, 2012; revised September 23, 2012; accepted October 13, 2012

ABSTRACT

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

[4].

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

[5,6].

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

NANCC

NNCNC NA



 





















 





(1)

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,

W. S. ABDELAZIZ ET AL. 315

which is related to electron collisional excitation rate

coefficient by [11,12].

exp

iij jie

CC EKT

 



 









(2)

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

CKT







1



(3)

The actual population density NJ of the jth level is ob-

tained from the following identity,

NNN

(4)

where

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-

tion.

Since the populations calculated from Equation (7) are

normalized such that











 (6)

where 21 is the number of all the levels of the ion under

consideration.

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

FNg g













(7)

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].

8π2π

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-

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 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

equations.

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.

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

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.

W. S. ABDELAZIZ ET AL.

316

0.000

0.050

0.100

0.150

0.200

0.250

Gain (cm)-1

log Ne (cm)-3

7-6

14-13

16-15

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

[19].

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)

transition.

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) 2.00E−01

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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