Calculation of Atomic Data and Gain Coefficient for XUV & Soft X-Ray Laser Emission from Ge XXIII

Energy levels, transition probabilities, oscillator strengths, and collision strengths have been calculated for transitions in Ne-like Ge. The data refer to a 241 fine-structure levels belonging to the configurations 1s2 2s2 2p5nl, 1s2 2s1 2p6nl (n = 3, 4, 5, 6; l = s, p, d, f, g and h), which have been calculated by the fully relativistic flexible atomic code (FAC). These data are used in the determination of the reduced population for the 241 fine structure levels and gain coefficients over a wide range of electron densities (from 2 × 10+20 to 4 × 10+22 cm−3) and at various electron plasma temperatures (650, 850, 1050, 1250, 1450, 1650, 1850) eV by using the MATLAB R2013a Computer program for solving simultaneous coupled rate equations. The reduced population for the 241 fine structure levels the gain coefficients for those transitions with positive population inversion factor are determined and plotted against the electron density.


Introduction
The concept for x-ray lasers went back to the 1965s, when it was based on the laser-produced plasmas by electron de-excitation having a promising interest science, which was first proposed by the Soviet scientists Gudzenko and Shelepin [1].They thought that the short wavelength laser in the x-ray region of the electromagnetic spectrum needed a large energy gap which was sustained in the highly ionized ions.After ionization, in the equilibrium plasma, ions having specific number of electrons such as 2 (helium-like), 10 (neon-like), 28 (nickel-like) and 46 (palladium-like) were relatively stable and survived in a wide range of temperature and density [2] [3].X-ray lasers pumping methods are electron collisional excitation, photo excitation, charge transfer, electron collisional recombination and de-electronic recombination pumping using a picosecond chirped pulse amplification (CPA) pulse [4], a capillary discharge [5] [6] and a free electron laser.However, the electron collisional excitation pumping of the inner shell and outer shell of highly ionization states have shown a more stable and a higher output [7] [8].Thus, the electron collisional pumping proved to be the most efficient method in producing the x-ray lasers.
Experimentally there exist in the literature some studies trying to develop high-efficiency X-ray laser with significant gain, for example, [3] [9] proposed the original mechanism for demonstrating X-ray lasing by resonant photo pumping.Several authors during the past three decades [10] [11] have studied this lasing mechanism experimentally and theoretically, in the hope of developing high-efficiency X-ray laser.In another study by Qi, N. and Krishnan, M. [12], the shortest wavelength at which the significant gain had been measured using the resonant photo pumping was in the beryllium-like carbon at 2163 Å, which was far from the X-ray spectral region.
In this paper, we calculate energy levels for 241 fine-structure states using a fully relativistic approach based on Dirac equation (see Table 1).Weighted Oscillator strengths, spontaneous radiative decay rates are calculated in the single multipole approximation (see Table 2), and Collision strengths by electron impact using the factorization-interpolation method are calculated in the distorted wave approximation.Effective collision strengths are calculated by interpolating the data from the collision strengths and integrating over Max wellian distribution at different temperatures.Rate coefficients are calculated from effective collision strengths using a formula that will be described later in this paper.Then, we predict the reduced population and gain coefficient for Ge XXIII by a steady state equation in the collisional radiative model after achieving a population inversion between the allowed transition states.

Gain Coefficient Computations
The possibility of laser emission from plasma of Ge XXIII ion 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 [13] [14].
where j N and i N is the fractional population of level j and i respectively, e N is the electron density, ji A is the Einstein coefficient for spontaneous radiative decay from j to i; and e ij C and d ji C represent the rate coefficient for collisional excitation and de-excitation respectively.The actual population density J N of the th j level can be calculated from the equation of identity [15] [16].
where i g and j g are the statistical weights of the lower and upper levels, respectively.The electron impact excitation rates usually are expressed via the effective collision strengths ji γ as 6 1 2 8.6287 10 where the values of ji γ and ji A are obtained by [17].The actual population density J N of the th j level is obtained from the following identity [17], where I N is the quantity of ions which reached to the ionization stage I [17], where e N is the electron density, avg Z is the average degree of ionization and I f is the fractional abundance of the ionization states which can be calculated from the relation [17].Since the populations calculated from Equation ( 1) are normalized such that, 241 1 1 where 241 is the number of all the levels of the ion under consideration, the quantity actually obtained from Equation (1) is the fractional population J I N N .After the calculation of levels population, the quantities j j N g and i i N g can be calculated.Application of electron collisional pumping, the collision in the laser ion plasma will transfer the pumped quanta to other levels, and will result in population inversions between the upper and lower levels.Once a population inversion has been ensured a positive gain through 0 F > [18] is obtained.
where j j N g and i i N g are the reduced populations of the upper level and lower level respectively.Equation (7) has been used to calculate the gain coefficient (α) for Doppler broadening of the various transitions in the Ge XXIII ion.
where M is the ion mass ij λ is the transition wavelength in cm (see Table 3), i T is the ion temperature in K and j , I represents the upper and lower transition levels respectively.
As seen from Equation ( 8), the gain coefficient is expressed in terms of the upper state density ( ) j N .This quantity j N depends on how the upper state is populated, as well as on the density of the initial source state.The source state is often the ground state for the particular ion.
2s 1 2p 6 nl (n = 3, 4, 5, 6; l = s, p, d, f, g and h) configurations in Ne-like Ge ions are presented in Table 1.The main components of the computed Eigen vectors are also given in these tables in the jj-coupling schemes.For Ge XXIII, the agreement between FAC, MCDF, and other theoretical and experimental energies [19]- [23] with the values available at the National Institute of Standards and Technology (NIST) [24] and is within values less than 0.5% for a majority of levels.

Radiative Decay Rates
The oscillator strengths ( ) 1.49 10 8π where m and e are the electron mass and charge, respectively, c is the velocity of light, ij λ is the transition wa- velength in Å and i ω and j ω are the statistical weights of the lower i and upper j levels, respectively.Simi- larly, the oscillator strength ( ) ij f … (Dimensionless) and the line strength S (in atomic units, 2.0261 10 303.75 and The wavelengths, transition probabilities, and weighted oscillator strengths, for E1transitions calculated using the (FAC) are reported in Table 2 and Table 3 for Ge XXIII.We present a comparison between our results and other theoretical transition probabilities for allowed E1 transitions for Ge XXIII in tables respectively.It shows that our results are in a good agreement with the other theoretical and experimental results [22] [23].Table 2 shows a comparison between the FAC, MCDF, RCI, and other theoretical weighted oscillator strength [22] [23] values for some transitions among the levels of the 1s 2 2s 2 2p 5 nl, 1s 2 2s 1 2p 6 nl (n = 3, 4, 5, 6; l = s, p, d, f, g and h) configurations for Ge XXIII.Generally, there are no discrepancies between our FAC, COWAN, CIV3, MCDF, RCI, and experimental calculations and the agreement is within ≤20% for strong transitions.

Level Population
The reduced population densities are calculated for 241 finestructure levels arising from 1s 2 2s 2 2p 6 nl, 1s 2 2s 2 2p 5 nl, 1s 2 2s 1 2p 6 nl (n = 3, 4, 5, 6; l = s, p, d, f, g and h) 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 MATLAB R2013acomputer program.The present calculations for the reduced populations as a function of electron densities are plotted in Figure 1 to Figure 4 for levels (5,15,33,37)  1650, and 1850) eV for Ge XXIII ion.In the calculation we took into account spontaneous radiative decay rate and electron collisional processes between all levels under the 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 radiative decay, and collisional mixing of excited levels can be ignored.This result is in agreement with that of Feldman et al. [19]- [21].At high electron densities (Ne > 5 × 10 22 ), the radiative decay to all the levels will be negligible compared to collisional depopulations and all the level populations become independent of the electron density and are approximately equal (see Figures 1 to 4).

Radiative Lifetime
The lifetimes are determined almost entirely from the allowed and the strong inter combination transitions.The radiative lifetime j τ of an excited atomic state j, is related to the atomic transition probability ji A by: where the sum is extended over all the lower states which can be reached from the upper state by radiative decay.Table 4 contains the present results of radiative lifetime for the upper and lower laser levels for the Ge XXIII.

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. However, large amplification, the gain exceeds all losses and ensures laser emission.In order to work in the XUV and X-ray spectral regions, we have selected transitions between any two levels producing photons with wavelengths between 12 and 52 nm.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 [19] [21].The population in version is largest where the electron collisions de-excitation rate for the upper level is comparable to the radiative decay rate for this level.

Gain Coefficient
As a result of population inversion there will be positive gain in laser medium.Equation (8) has been used to calculate gain coefficient for the Doppler broadening of various transitions in the Ge XXIII ion.Our results for the maximum gain coefficient in cm −1 for those transitions having appositive inversion factor F > 0 in the case of Ge XXIII ion at different temperatures are calculated (see Table 4) and plotted against electron density in Figure 5 and  XXIII ion, however, the largest gain occurs for the Ge XXIII ion at (2s 1\2 3d 5\2 ) 2 -----(2s 1\2 3p 3\2 ) 1 transition.These short wavelength laser transitions can be produced using plasmas created by optical lasers as the lasing medium.For Ge XXIII ion the rates for electron collisional excitation from the 1s 2 2s 2 2p 6 ground state to the 1s 2 2s 1 2p 6 3d configuration are greater than the rates for excitation from the ground state to the 1s 2 2s 1 2p 6 3p state.For electron densities and electron temperatures that are typical of laboratory high-density plasma sources, such as laser produced plasmas, it is possible to create a quasi-stationary population inversion between the 2s 1 2p 6 3d and 2s 1 2p 6 3p states in Ge XXIII ion.Our calculations have shown that 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 Ge XXIII.The gain calculations were performed at electron temperatures equal to (650, 850, 1050, 1250, 1450, 1650, and 1850) eV at different electron densities.It is obvious that the gain increases with the temperature.We also find that some wavelengths for the lasing transitions in Table 4 are much closed to the values of it in the ref.[23] which give us the accurate wavelength calculated experimentally, although we Ignores some physical processes in the our rate equation.

Conclusions
The analysis of how the electron collisional pumping (ECP) is suitable for attaining population inversion and offering the potential for laser emission in the spectral region between 12 and 53 nm from the Ge XXIII ion.This class of lasers can be achieved under the suitable conditions of pumping power as well as electron density.If the positive gains obtained previously for some transitions in the ions under studies (Ge XXIII ion) together with the calculated parameters can 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 laser transitions in the Ge XXIII plasma ion (see Table 4), as the most promising laser emission lines in the XUV and soft X-ray spectral regions.
for a transition i→ j are related by the follow- by the following standard equations.For the electric dipole (E1) transitions,

Table 1 .
Energy levels of Ge XXIII, and their threshold energies (in eV).

Table 2 .
Comparison between some weighted oscillator strengths, and transition probabilities for E1 transitions of Ge XXIII.