Semi-Empirical Oscillator Strengths and Lifetimes for the P IV Spectrum ()

Antônio Jamil Mania^{}

Universidade Estadual de Santa Cruz, DCET, Ilhéus, Brazil.

**DOI: **10.4236/sar.2017.54005
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Universidade Estadual de Santa Cruz, DCET, Ilhéus, Brazil.

In this work numerical codes carried out in a multiconfiguration Har-tree-Fock relativistic (HFR) approach for the P IV ion are used to obtain the oscillator strengths of each transition as well as the lifetimes of each energy level. With the existing data from several authors that contributed to the spectrum using different light sources, and optimizing the electrostatic parameters by a least-squares procedure when replacing the theoretical values by the experimental ones in the energy matrices, one obtains closer values and according to the observations for the intensities, and also of the lifetimes closer to those that would be obtained experimentally.

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Mania, A. (2017) Semi-Empirical Oscillator Strengths and Lifetimes for the P IV Spectrum. *Spectral Analysis Review*, **5**, 49-80. doi: 10.4236/sar.2017.54005.

1. Introduction

P IV spectrum is a member of the Mg-like isoelectronic sequence with a complete core plus a double-electron valence shell giving singlets and triplets terms. This sequence has several terms of perturbations, a numerous spin-forbidden lines, and also configuration interactions. Some transitions can be explained by a mixing of one or both involved terms with a perturbing term, indicating that each of them possesses a large portion of the wave function of the other ones.

The first analysis of P IV was made by Bowen and Millikan [1] (1925), who identified 23 lines connecting 14 levels belonging to 8 terms. Wavelengths above 2000 Å, were taken from Geuter [2] (1907). The observations in the vacuum ultraviolet region were made by using a vacuum spark between electrodes of magnesium or silicon containing compounds of phosphorus. Another recording of phosphorus in the vacuum ultraviolet was made by Queney [3] in 1929, classifying one line (1902). In 1932 Bowen [4] published observations of 32 additional lines in the same region and enlarged the term system to 29 levels of 15 terms which have been verified in more recent work, although the label of one term has been changed. A new contribution to the analysis of P IV was made by Robinson [5] (1937) using a vacuum spark with beryllium electrodes filled with red phosphorus and extending the observations towards shorter wavelengths. Toresson [6] suggested the change of the terms 3s3d ^{l}D_{2} and 3p^{2 l}D_{2} in earlier analyses. Zare [7] [8] indicated that there is a strong mutual interaction between these two ^{l}D terms along the all isoelectronic sequence, and there is no way to distinguish the terms experimentally. Spectral analysis given by Fawcett [9] for the term 3p3d ^{l}P has been rejected and replaced by a level now established by 12 combinations. The new value has confirmed the suggestion given by Victor et al. [10] to interchange the names 3p3d ^{l}P and 3p4s ^{1}P in the íon Si III, since 3p3d ^{l}P will then run more nearly parallel to the other 3p3d terms in the isoelectronic sequence along. Spectral analysis given by Fawcett for higher members of the isoelectronic sequence, from C1 VI to Cr XII, became to be incompatible with the most recent data. Zetterberg and Magnusson [11] studied this ion with complete spectral analysis with new measurements in the whole region from 256 to 9600 Å. The used light source and the spectrographs had a sliding spark in vacuum. Red phosphorus was packed in drilled holes in beryllium electrodes. Between 9800 and 2000 Å the spectrum was recorded by a Jarrell-Ash spectrograph with a plate factor of 5 Å/mm in the first order. In the vacuum region from 2550 to 200 Å, the spectrograms have been taken with a 3 m normal incidence spectrograph using two different gratings and the plate factor of 2.77 Å/mm in the first order. Lines of different ionization stages were separated by varying the inductance in the discharge circuit. P IV lines appear to be best with one or two turns of the coil and a voltage of 6 kV. Fisher and Godefroid [12] [13] have done extensive MCHF calculations of transitions probabilities for lines that involved singlets reporting an appreciable mixing of the configuration when the energies are very close. The plunging configurations and Rydberg series of the 3snf ^{1}F lower members are strongly mixed. The relativistic and non-relativistic analysis of percentage composition of levels have shown that the upper and lower eigenvectors of the interaction are respectively dominated for 3s4f and 3p3d with the terms ^{1}F. Martin et al. [14] compiled all the known energy levels of all the phosphorus ions.

The ionization energy limit for this ion derived by means of a polarization formula applied to terms of the 3snh, 3sni and 3snk series is estimated as being 414,922.8 ± 1.0 cm^{−1} (51.4443 ± 0.0002 eV) [11] .

So then, the purpose of this study is to present the oscillator strengths and lifetimes for experimentally known electric dipole transitions and energy levels for this spectrum. In order to obtain these values, the reduced matrix elements are calculated by using an optimization of the energy parameters which were adjusted from a least-squares procedure. In this adjustment, the code fits experimental levels by varying the electrostatic parameters. This procedure improves the values of the wavelengths σ in,

$gf=\frac{8\text{\pi}mc{a}_{0}\sigma}{3h}$ (1)

with S being the electric dipole line strength. Also the quantities, ${y}_{\beta J}^{\gamma}$ and ${y}_{{\beta}^{\prime}{J}^{\prime}}^{{\gamma}^{\prime}}$ that measure the total strength of the spectral line in

${S}_{\gamma {\gamma}^{\prime}}^{{1}^{\prime}/2}={\displaystyle \underset{\beta}{\sum}{\displaystyle \underset{{\beta}^{\prime}}{\sum}{y}_{\beta J}^{\gamma}}\langle \beta J\Vert {{\rm P}}^{1}\Vert {\beta}^{\prime}{J}^{\prime}\rangle {y}_{{\beta}^{\prime}{J}^{\prime}}^{{\gamma}^{\prime}}}$ (2)

were improved in this new fitting. Considering that,

$P\left(\gamma J\right)={\displaystyle \sum A\left(\gamma J,{\gamma}^{\prime}{J}^{\prime}\right)}$ (3)

as the probability per unit time of an atom in a specific state γJ to make a spontaneous transition to any state with lower energy being $A\left(\gamma J,{\gamma}^{\prime}{J}^{\prime}\right)$ the Einstein spontaneous emission transition probability rate related to the natural lifetime $\tau \left(\gamma J\right)$ of a state by,

$\tau \left(\gamma J\right)={\left({\displaystyle \sum A\left(\gamma J,{\gamma}^{\prime}{J}^{\prime}\right)}\right)}^{-1}$ (4)

These equations must to be applied to an isolated atom. Matter or radiation interaction will tend to reduce their values. In this way, the values for gf and lifetime given in this work were calculated according to previous equations.

For to convert the wavelengths ${\lambda}_{vac}=n{\lambda}_{air}$ given by the code, were used the relation [15] ,

$n=1.0+8342.13\times {10}^{-8}+\frac{2406030.0}{130.0\times {10}^{8}-{\sigma}^{2}}+\frac{15997.0}{38.9\times {10}^{8}-{\sigma}^{2}}$ (5)

for the index of refraction (dry air containing 0.03 CO_{2} by volume at normal pressure and T = 15˚C).

2. Methodology of Calculation

The theoretical predictions for the energy level values were obtained diagonalizing the energy matrices with appropriate Hartree-Fock relativistic (HFR) values for the electrostatic parameters. In these computations all strong configuration interactions were included and HFR method is used to give a better accuracy in many cases. For this purpose, the computer code developed by Cowan [16] was employed. The main purpose is to reach a fitting to the experimental energy levels, minimizing the uncertainties as much as possible, using the least-squares method for each parity. The standard deviation is less than one percent of the energy range covered by the energy levels. The accuracy is related in the computation of the gf and lifetimes values. The propagated experimental uncertainties of the input data, in the optimization of the energy levels, does not influence the process, being small values when are compared to the uncertanties coming from the fitting. The radial integrals E_{AV}, F^{k}, G^{k} and R^{k} are considered simply as adjustable parameters, whose values are to be determined empirically so as to give the best possible fitting between the calculated eigenvalues and the observed energy levels. The values for the optimized electrostatic parameters substitute their corresponding theoretical values, and the are used again to calculate energy matrices.

3. Results and Discussion

The P IV spectrum is characterized by strong interactions among their configurations yelding a mixture of levels that difficult a severe analysis. These effects of perturbations are stronger on the singlet levels, namely in 3p^{2}, 3s3d, 3s4d, 3s5g, 3s6s, 3p4p and 3d^{2}. He same effect appears in the 3p3d, 3s4f, 3p4s, 3s5p, 3s5f, 3p4d and 3s8f configurations. There are many examples of spin-forbidden lines caused by near coincidence of singlet and triplet levels with the same J-value.

Also, configuration interactions are numerous. Apparent two-electron transitions, such as 3s5p ^{l}P - 3p4p ^{1}P, 3s4d ^{3}D - 3p4s ^{3}P or 3p^{2 1}D - 3snf ^{1}F, 3s6s ^{3}S - 3p4p ^{3}S, can be explained by a mixing of one or both of the involved terms with a perturbing term. The term 3p^{2l}D combines just like 3s3d ^{1}D, indicating that each of them possesses a large portion of the wavefunction of the other one. The perturbation of 3s7p ^{3}P by the inverted 3p4d ^{3}P term is also evident. The energy levels used in our fitting method are from spectral analyses [11] , even though the code has exchanged values making it difficult to adjust the electrostatic parameters. The fittings were achieved by considering some simplifications such as keeping fixed those singlet energy levels that exchanged their values due to mutual interactions, ―3p3d ^{1}F ( fitted 290 3268; experimental 314 4237), 3s4f ^{1}F (fitted 314 4251 - experimental 290 3277), 3p4s ^{1}P (fitted 316 8828 - experimental 327 8735), 3s5p ^{1}P ( fitted 321 0106 - experimental 316 8888), 3s8f ^{1}F ( fitted 385 9827 - experimental 388 1242), 3p4d ^{1}F ( fitted 388 1209 - experimental 385 9800), ―and where there was no success in reproducing the observed structure these levels were not included in the calculations of gf and lifetimes, however, we retain the percentage compositions with the values indicated by the adjustment.

Standard deviation reached for each parity as 10 cm^{−1} and 17 cm^{−1}, for even and odd configurations, respectively, were satisfactory for the aims of this work [16] . The leading eigenvector percentages are in accordance with those provided by the numerical code [16] .

In Table 1 (for the vacuum region) and Table 2 (in the air), are shown the results for the oscillator strengths. A comparison with the observed wavelegenths values derived from previous works, as well as their nomenclatures of intensities adopted are also shown. Table 3 (even configurations) and Table 4 (odd configurations) for the lifetimes values, are shown as well as the eigenvectors percentages composition.

4. Conclusion

Semi-empirical values of oscillator strengths and lifetimes for the known spec-

Table 1. P IV wavelengths and semi-empirical oscillator strengths calculated in the region vacuum-ultraviolet.

Abbreviations used for the intensities [11] : b = broad, bl = blended, h = hazy, m = masked by, d = diffuse. References: R = Robinson; B = Bowen; R* Robinson’s identification revised; Q = Queney.

Table 2. PIV semi-empirical oscillator strengths calculated in the region above 2000 Å.

Abbreviations used for the intensities [11] : b = broad, bl = blended, h = hazy, m = masked by, d = diffuse. Ref. [11] : B* - Line taken from Geuter’s measurements and classified by Bowen and Millikan or Bowen.

Table 3. P IV energy levels and lifetimes for the even configurations.

^{1}Percentage composition lower than 3% were omitted. N.I.: Level no identified. *As suggested by the fitting.

Table 4. P IV energy levels and lifetimes for the odd configurations.

^{1}Percentage composition lower than 3% were omitted. N.I.: Level no identified. *As suggested by the fitting.

trum of the ion P IV are presented by the method of attempting to reproduce as much as possible the observed values and extracting information about the values of gf and lifetimes that are closer to the experimental ones. Phosphorus is an astrophysically important element. The present work is part of an ongoing program, whose goal is to obtain oscillator strength and lifetimes for elements of astrophysical importance. Phosphorus occupies the fifteenth place with respect to cosmic distribution [17] .

Conflicts of Interest

The authors declare no conflicts of interest.

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