Gain Coefficient Calculation for Short Wave Laser Emission from Sodium like Co ()
1. Introduction
An important objective in the development of X-ray lasers is to deliver a coherent, saturated output at wavelengths toward the water window [1].
Such saturated X-ray lasers are required for holography [2] and microscopy [3], of biological specimens and for deflectometry [4], interferometry [5], and radiography [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 spectrum of the sun. Highly charged sodium-like ions are observed in several types of laboratory sources such as high-voltage vacuum spark tokamak and laser-produced plasmas [1].
Spectra of these ions have a simple structure (one electron outside a closed shell); the energy levels are practically free from effects of configuration mixing and therefore they are well suited for a theoretical interpretation 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] has 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-configuration 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 allowed 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 multiplets only and not for individual fine-structure components. [9] assigned experimental wavelengths to 1086 lines which have oscillator 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 measurements for the 4p and 5p levels and calculation of transition 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 transition probabilities between them in sodium like Co16+.
The atomic data thus obtained are used to calculate reduced population of sodium like Co16+ excited levels over a wide range of electron density at various electron temperatures. The gain coefficients are also calculated.
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.
(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 [1]:
(2)
Here Hkin, He-nuc and He-e refer, respectively, to the kinetic 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 ri,j = |ri – rj|.
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 interaction of electrons, an exact solution cannot be obtained. On the other hand, the interaction term is comparable 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 nucleus and also in the mean force field produced by other electrons, then we have the following effective Hamiltonian [1]:
(3)
where Z is the atomic number
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 structures 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 developed 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 [1]:
(4)
Ф: is a total function with respect to exchange of electrons.
ϕ(χ): is a single particle function or spin orbitals.
N: is a number of non-interacting electrons with no spin-orbit interaction.
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 momentum and spin symmetry. This method is called the interaction 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 Hamiltonian of the model of central force field [15]. For relativistic correction, the program RCN/RCG restore to the Breit-Pauli correction.
2.5. Weighted Oscillator Strengths and Lifetimes
The oscillator strength f(γγ\) is a physical quantity related to line intensity I and transition probability W(γγ\), as given by Sobelman [16]:
(5)
with, I α gW(γγ\) α g|f(γγ\)| = gf.
Here m = electron mass, e = electron charge, = is initial quantum state, 
, 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]:
(6)
where g is the statistical weight of lower level, f is the absorption oscillator strength, 
, h is planck’s constant, c = light velocity, and a0 is Bohr radius, and the electric dipole line strength is defined by:
(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):
(8)
In a multi-configuration calculation we have to expand the wave-function
.
In terms of single configuration wave-functions, 
for both upper and lower levels:
(9)
therefore, we can have the multiconfigurational expression for the square root of line strength:
(10)
The probability per unit time of an atom in specific state 
to make a spontaneous transition to any state with lower energy is
(11)
where 
is the Einstein spontaneous emission transition probability rate, for a transition from the state. 
to the state
.
The sum is over all state 
with 
The Einstein probability rate is related to gf with the following relation by [17]:
(12)
Since the natural lifetime 
is the inverse of transition probability, then:
(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 Co 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].
(14)
where Nj is the population of level j, 
is the spontaneous decay rate from level j to level i, 
is the electron collisional excitation rate coefficient, and 
is the electron collisional de-excitation rate coefficient, which is related to electron collisional excitation rate coefficient by [22-23].
(15)
where gi and gj are the statistical weights of lower and upper levele, respectively.
The population of the j th level is obtained from the identity [19,20,24],
(16)
where 
is the total number density of all levels of the ion under consideration, and Nt is the total number density of all ionization stage.
Since the populations calculated from Equation (14) are normalized such that [19,20,25]
(17)
where n is the number of all the levels of the ion under consideration.
Electron collisional pumping has been applied. Collision in the lasant ion plasma will transfer the pumped quanta to other levels, and resulted in population inversions then produced between the upper and lower levels.
Once a population inversion has ensured a positive gain through F > 0 [1].
(18)
where 
and 
are the reduced populations of the upper level and lower level respectively. Equation (18) has been used to calculate the gain coefficient for Doppler broadening of the various transitions in the Na like Co ion.
(19)
where M is the ion mass, 
is the transition wavelength in cm, 
is the ion temperature in 
and u,l represent the upper and lower transition levels respectively.
The gain coefficient is expressed in terms of the upper state density (Nu). This quantity 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 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 Co XVII. 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 5g 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 1).
Table 2 present energy levels and fine-structure splitting for Co16+ also presented the energy levels calculations of (Younis et al., 2006) the present calculations differ by less than 0.15% for most of the levels, they
Table 1. Calculated HFR energy levels and fine structure splitting.
have used Configuration–Interaction Code (CIV3).and the differ from (Nist, 2009) by less than 0.15% for most of the levels, except in some cases like the levels 2, 3, 4 and 5 in which the percentage difference between our value and the last two calculations is (0.63%, 0.64%), (−46%, 29%), (0.35%, 0.26%) and (0.27%, 0.33%) respectively, This mean that our results are in a good agreement with the theoretical and experimental value.
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 present the wavelength, transition probability and the values of oscillator strength with a comparison against another published paper (Younis et al., 2006 [Reff.1]) and NIST. There was good agreement with (NIST, 2009 [Reff.2]) and only small differences about (0.8%) there was some similarities with the paper and high differences in some values about (85%).