Received 3 May 2016; accepted 17 July 2016; published 20 July 2016
Computations of wavelengths made with the aid of a Hartree-Fock Relativistic (HFR) computer program package and a program of least-square procedure as given by Cowan  to adjust the values of the energetic parameters, comparing the data and calculating its consistency with the identification of known energy levels. The adjustable parameters are to be determined empirically to give the best possible fitting between the calculated eigenvalues and the observed energy levels. The fitting process is carried out by a self-consistent procedure until the parameter values no longer change from one iteration cycle to the next. The main purpose is to reach a fitting to the experimental energy levels, which minimizes the uncertainties as much as possible, using the least-squares method for each parity in which the standard deviation is less than one percent of the energy range covered by the energy levels. The optimized electrostatic parameters substitute their corresponding theoretical values and they are used again to calculate energy matrices, the determination of the oscillator strengths and lifetimes values. All strong configuration interactions are to be included and HFR method is used to given a better accuracy  . It should also be noticed, that at higher levels, the j-j notation is better and it should be used to estimate the percentages of compositions.
The oscillator strengths is a physical quantity related to line intensity I and transition probability, by:
With, being m the electron mass, e their charge, γ the initial quantum state, , E(γ) the initial state energy and g = (2J + 1) is the number of degenerate quantum states with angular momentum J. Quantities with primes refer to the final state. In the equation above, the weighted oscillator strength, gf, is
where, h is Planck’s constant; c is the light velocity; and a0 is the Bohr radius. The electric dipole line strength is defined by:
This quantity is a measure of the total strength of the spectral line, including all possible transitions between and Jz eigenstates. 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
In a multiconfiguration calculation we have to expand the wavefunction in terms of single configuration wavefunction, , for both upper and lower levels:
Therefore, we can have the multiconfigurational expression for
The probability per unit time of an atom in a specific state γJ to make a spontaneous transition to any state with lower energy is
where is the Einstein spontaneous emission transition probability rate for a transition from the to the state. The sum is over all states with. The Einstein probability rate is related to gf through the following relation by:
Since the natural lifetime. The natural lifetime is applicable to an isolated atom.
The interaction with matter or radiation will reduce the lifetime of a state. The values for gf and lifetime given in Table 1 and Table 2, respectively, were calculated according to these equations. In order to obtain better values for oscillator strengths, we calculated the reduced matrix elements P1 by using optimized values of energy parameters which were adjusted from a least-squares calculation. In this adjustment, the code tries to fit experimental energy values by varying the electrostatic parameters. This procedure improves σ values used in Equation (2) and and values used in Equation (6).
Wavelength values in vacuum were converted to air by the relation  , , where the index of refraction of standard air (dry air containing 0.03CO2 by volume at normal pressure and) is
3. Results and Discussion
In our fitting process, the standard deviation reached for each parity as 12 cm−1 and 5 cm−1, for even and odd configurations, respectively, is satisfactory for the aims of this work. Values for gf and lifetime given in Table 1 and Table 2, respectively, were calculated by the previously described method. Table 1 shows the results of the comparison between wavelength values as calculated by the method and the observed. In Table 2, we present lifetimes, energy levels and an estimation of their percentage composition. For the even-parity configurations we have the following picture: 1s25g, 1s26g, 1s26s, 1s27s, 1s28s, 1s27d, 1s28d, 1s2s2, and the series 1s2pnp (3 ≤ n ≤ 4). For the odd-parity case we study the configuration 1s26h, and the series 1s2nf (4 ≤ n ≤ 6), 1s2np (5 ≤ n ≤ 8), 1s2snp (2 ≤ n ≤ 4). The interpretation of the configuration levels structure was made by least-squares fit of the observed levels and we propose the new values possible of the energy levels marked with asterisk (*) in the table. The oscillator strengths and lifetimes for the lithium-like ions are of astrophysical interest for photo-ionization modelling of elemental abundances in cosmic objects since an extensive data source is not currently available. Transitions in this ion have been of particular importance in extrapolation analysis especially for the dense spectra from N-like sequence in which the phosphorus is one element in isoelectronic sequence linking lighter elements where the analysis is more extensive. Is also an important testing ground for the development of theoretical methods which attempt to calculate atomic structure of many-electron systems.
We have presented oscillator strengths and lifetimes for all known transitions in P XIII. The gf-values are better agreement with line intensity observations and lifetime values that are closer to the experimental ones. We have been stimulated by the need to determine both important parameters in the study of plasma laboratory and solar
Table 1. Oscillator strengths and spectral lines for P XIII in the vacuum.
Table 2. (a) Even configurations. (b) Odd configurations.
(*) Indicates an attempt to identify.
(*) Indicates an attempt to identify.
spectra, as also phosphorus is an astrophysically important element. The present work is part of an ongoing program, whose goal is to obtain weighted oscillator strength, gf, and lifetimes for elements of astrophysical importance. Phosphorus occupies the fifteenth place with respect to cosmic distribution  .