By using the fully relativistic flexible atomic code (FAC) [14] we obtained energy level values for the 1s² 2s² 2pnl (n = 3, l = s, p & d) and ml (m = 4, l = s, p, d & f) configurations in C-like Ar¹⁸⁺, Ti²²⁺ and Fe²⁶⁺ ions. This data presented in Tables 1-3. The first column of each table provides an index for the levels, the second column presented the main components of the computed eigenvectors and the third column presented our calculations of energy levels. Tables 4-6 presented the comparing data between our calculations and with the experimental values compiled by NIST. We obtained the agreement between FAC, Bhatia, Seely and Feldman [15] for Ar¹⁸⁺, Ti²²⁺ and Fe²⁶⁺ energy levels and with the other experimental energies [16] [17] with the values available at the National Institute of Standards and Technology (NIST) [18] and is within values less than 0.5% for a majority of levels.

The reduced population densities are calculated for 69 fine structure levels arising from 1s² 2s² 2pnl (n = 3, l = s, p & d) and ml (m = 4, l = s, p, d & f) configurations that emit radiation in the XUV and soft X-ray spectral regions. The calculations were performed by solving the coupled rate Equation (1) simultaneously using MAT- LAB version 7.10.0 (R2010a) computer program. The reduced populations density are calculated as a function of electron densities and plotted at different plasma temperatures for Ar¹⁸⁺, Ti²²⁺ and Fe²⁶⁺ ions.

The behavior of level populations density of the various ions (Ar XIII, Ti XVII & Fe XXI) 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. This result is in agreement with that of Feldman et al. [9] [10] [19] . At high population densities (10⁺²³), radiative decay to all levels will be negligible compared to collisional depopulations and all level populations become independent of electron density and are approximately equal. The 10⁺¹⁷ electron density shows a peak at before the other levels then decreases to the saturation faster than the other levels, which mean that then on radiative transitions dominant the de-excitation because of its higher energy and fast decay time (see Figures 1-9). The population inversion is largest where electron collisional de-excitation rate for the upper level is comparable to radiative decay for this level [9] [19] .

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 X-ray spectral regions, we have selecting transitions between any two levels producing photons with wavelengths between 5˚ and 100˚A. The electron density at which the population reaches corona equilibrium approximately equals to A/D, where A is the radiative decay rate and D is the collisional de-excitation rate [13] . The population inversion is largest where the electron collisional de-excita- tion rate for the upper level is comparable to the radiative decay rate for this level.

As population inversion will be positive in laser medium. Equation (8) has been used to calculate gain coefficient for the Doppler broadening of various transitions in the Ar XIII, Ti XVII & Fe XXI ions. For F > 0 transition having positive inversion with the maximum gain coefficient in cm⁻¹. The maximum gain was calculated and plotted against electron density Figures 10-18 these short wavelength laser transitions can be produced using plasmas created by optical lasers as the lasing medium.

For Ar XIII, Ti XVII & Fe XXI ions the rates for electron collisional excitation from the 1s² 2s² 2p2 ground

Note: (a) NIST [18] data; (b) CIV3 program by K. M. Aggarwal [5] (c) SS. The data from the work done by Bhatia et al. [15] & (d) Our calculations calculated by the fully relativistic flexible atomic code (FAC).

state to the 1s² 2s² 2p 3l (l = s, p, d) configuration are greater than the rates for excitation from the ground state to the 1s² 2s² 2p 4l state. For electron densities and electron temperatures that are typical of laboratory high- density plasma sources this agreement with Feldman et al. [9] , such as laser-produced plasmas, it is possible to create a quasi-stationary population inversion in this ion.

Under favorable conditions large laser gains for this transition in the XUV and soft X-ray regions of the spectrum can be achieved in the carbon-like Ar XIII, Ti XVII & Fe XXI ions from our calculation. The gain calculations were performed at various electron temperatures and at various electron densities. It is obvious that the gain increases with the temperature.

Note: (a) NIST [18] data, (b) GRASP code by K. M. Aggarwal [4] (c) SS. The data from the work done by Bhatia et al. [15] & (d) Our calculations calculated by the fully relativistic flexible atomic code (FAC).

The results have suggested the following laser transitions in the Ar XIII, Ti XVII & Fe XXI plasma ions, wavelength, radiative life time of the upper and lower laser levels in the possible laser transitions and maximum gain coefficient at various temperatures presented in Tables 7-9, as the most promising laser emission lines in the XUV and soft X-ray spectral regions.

The lifetimes are determined almost entirely from the allowed and the strong inter combination transitions. The radiative lifetime of an excited atomic state j, is related to the atomic transition probability by:

where the sum is extended over all the lower states which can be reached from the upper state by radiative decay.

Note: (a) NIST [18] data, (b) GRASP code by K. M. Aggarwal [20] (c) SS. The data from the work done by Bhatia et al. [15] . (d) Our calculations calculated by the fully relativistic flexible atomic code (FAC).

Tables 7-9 contains the present results of radiative lifetime for the upper and lower laser levels for the (Ar XIII), (Ti XVII) and (Fe XXI).