Received 3 May 2016; accepted 17 July 2016; published 20 July 2016

ABSTRACT

The P XIII spectrum has been analyzed by several authors using different light sources. The semi- empirical oscillator strengths (gf) and the lifetimes presented in this work for all known P XIII spectral lines and energy levels were carried out in a multi-configuration Hartree-Fock relativistic (HFR) approach. In this calculation, the electrostatic parameters were optimized by a least- squares procedure in order to improve the adjustment of theoretical to experimental energy levels. The method produces gf-values that are in agreement with intensity observations and lifetime values closer to the experimental ones.

Keywords:

P XIII Spectrum, Atomic Transitions, Energy Levels, Oscillator Strengths, Lifetimes

1. Introduction

The ground state configuration of twelve times ionized phosphorus, P XIII is 1s²2s with the term ²S_1/2 being this spectrum a member of the Li-like isoelectronic sequence with a complete core plus a single-electron valence shell. The ionization limit for this ion is estimated as being 4,934,000 ± 600 (611.74± 0.07 eV). This spectrum was originally analyzed in 1948 by H. A. Robinson to find the energy levels. These studies have being used in the compilation of Atomic Energy Levels published by C. E. Moore [1] , although a wavelength list has never been published. R. L. Kelly in Ref. [2] has included some of these lines as given by Moore in his first compilation. Later in 1970, Fawcett [3] , using a theta-pinch as light source, identified wavelengths of transitions occurring in the interval 22 - 41 Å and 278 - 594 Å. Fawcett et al. [4] using laser-produced plasma experiments confirmed the identity of many of their previous obtained lines classification. Goldsmith et al. [5] added, revised and extended the analyses performed by Robinson and Fawcett, classifying 13 lines in the range 23 - 39 Å with the transitions 2s-np, 2p-3s, and 2p-nd. Kasyanov et al. [6] and Dere [7] realized measurements of the transition 1s²2p²P_3/2 to the ground level 1s²2s ²S_1/2. Deschepper et al. [8] using a Doppler-tuned X-ray absorption technique realized measurements to the transition 1s²2s ²S_1/2-1s2s2p ⁴P_1/2 and ⁴P_3/2. Edlén [9] , using series formulae, has given results for the 1s²nl systems that are in agreement with the expected experimental errors. Wavelengths observed by Fawcett and Ridgeley [10] and Goldsmith et al. [5] in the region 102 - 111 Å, were used in the spectral analysis to evaluate the 4p and 4f levels, being that the 4s levels were evaluated from isoelectronic sequence values for 2p - 4s separations. Aglitskii et al. [11] and Boiko et al. [12] , using laser-produced plasma, have contributed to the spectral features that have been observed in transitions of doublet term of the configurations 1s2s², 1s2p², 1s2s2p, 1s2s3p, 1s2s4p, 1s2p3p, 1s2p4p, where the levels were derived from energy separations calculated by Vainshtein and Safronova [13] [14] . The wavelength tables given by these publications [15] [16] , are lines arranged by spectra, being some lines of P XIII spectra which are not in other references. Martin et al. [17] compiles all the energy levels of all the phosphorus ions, revising the earlier version of Moore of all the published data so far, as well as the unpublished data of Robinson. Hayes and Fawcett [18] give a new original contribution by using the same spectrum obtained in theta-pinch experiment earlier. Kelly [19] reviews all P XIII line identifications in his compilation. Recently, Wang et al. [20] - [22] has published data of oscillator strengths for 1s²2s to 1s²np, and 1s²2p to 1s²nd transitions of the Lithium isoelectronic sequence.

2. Methodology

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 [23] 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 [24] . 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:

, (1)

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

, (2)

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:

(3)

This quantity is a measure of the total strength of the spectral line, including all possible transitions between and J_z 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

. (4)

In a multiconfiguration calculation we have to expand the wavefunction in terms of single configuration wavefunction, , for both upper and lower levels:

(5)

Therefore, we can have the multiconfigurational expression for

(6)

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

(7)

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:

(8)

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 [25] , , 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.

4. Conclusion

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

(^*) 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 [26] .

Cite this paper

A. J. Mania,F. R. T. Luna, (2016) Oscillator Strengths and Lifetimes for the P XIII Spectrum. Spectral Analysis Review,04,1-10. doi: 10.4236/sar.2016.41001

References