Branching fractions for the Mg-like 3s3p–3s3d and 3s3p–3p2 transition arrays

Semiempirical methods are used to characterize and parametrize the effects of intermediate coupling and configuration interaction on the energy levels of the 3s3p, 3p2 and 3s3d configurations for ions in the Mg isoelectronic sequence. These parametrizations are then used to predict the branching fractions for the 3s3p–3p2 and 3s3p–3s3d transition arrays. The predictions are compared with the MCHF calculations for S V and Fe XV, and good agreement is obtained. The application of this method to deduce transition probabilities and oscillator strengths from lifetime measurements is discussed. A large base of measured lifetime data now exists for excited levels in atomic ions [1]. Many applications of atomic structure data require the use of transition probabilities and oscillator strengths, which can be obtained from lifetimes for branched decays by combining lifetime data with branching fraction values. Unlike lifetime measurements, branching fraction measurements require an intensity versus wavelength calibration of the detection system. For multiply ionized atomic spectra this presents severe challenges, associated both with the characteristics of the spectroscopic light sources, and with the lack of calibration standards in the ultraviolet region where these spectra occur. Correspondingly, branching fraction measurements in multiply charged ions are at present virtually nonexistent [2]. The existing base of atomic lifetime data has provided extensive tests of theoretical calculations, and has demonstrated the importance of the inclusion of intermediate coupling, configuration interaction, electron correlation, and many other theoretical considerations. However, lacking a similar base of branching fraction data, similar tests of theory have not been made for these relative quantities. Thus, the possibility exists that theoretical calculations for the ratios of transition probabilities could be less sensitive to the various perturbations, and a reliable base of transition probability and oscillator strength data might then be developed using precision lifetime measurements and theoretically computed branching fractions. It has been demonstrated that branching fractions can be specified for the transition arrays ns2np2–ns2npn′s in the Si [3], Ge [4], Sn [5] and Pb [6] isoelectronic sequences. This 0953-4075/04/000001+06$30.00 © 2004 IOP Publishing Ltd Printed in the UK 1 2 J Steiner and L J Curtis was achieved through the use of energy level data to semiempirically specify the intermediate coupling amplitudes. In these systems it was shown that the effects of configuration interaction are negligible, and that the branching fractions can be accurately specified by singlet–triplet mixing amplitudes and LS-coupling coefficients. Here we report an extension and testing of these methods for transitions in the Mg isoelectronic sequence. These transitions possess strong configuration mixing in addition to intermediate coupling, and it is shown that reliable branching ratios can be obtained here also by a suitably extended semiempirical formulation. To further develop these semiempirical methods, we consider the magnesium isoelectronic sequence, which is simple enough to be theoretically and semiempirically tractable, yet strongly affected by intermediate coupling, configuration interaction and relativistic corrections. For this study we have chosen the 3s3p–3s3d and 3s3p–3p2 transitions because they display both intermediate coupling and configuration interaction in a strong but analysable way. In addition, since these transition arrays are the only E1 transitions available for decay of the 3s3d and 3p2 configurations, the transition probabilities can be obtained from their lifetimes and these transition array branching fractions alone. This method also tests the non-relativistic Schrödinger approximation, wherein only the radial function depends on the central potential. In this approximation, all members of the transition array involve the same E1 transition moment, so ratios depend only on angular factors. The 3s3p configuration contains the P0, P1, P2, P1 levels, and intermediate coupling mixes the P1 and P1 levels. The 3s3d configuration contains the D1, D2, D3, D2 levels, and intermediate coupling mixes the D2 and D2 levels. The 3p2 configuration contains the P0, P1, P2, D2, S0 levels, and configuration interaction separately mixes the P0 and S0 levels and the P2 and D2 levels. Since both the 3s3d and 3p2 configurations contain D2 levels, there is strong configuration mixing between these two levels. Thus, although there are significant intermediate coupling and configuration interaction effects among these systems, all of the couplings are only pairwise, and the normalized amplitudes of each of the mixings can be characterized by a single quantity. For pure configurations for which no more than two levels possess the same total angular momentum J , intermediate coupling mixing amplitudes can be expressed as a mixing angle θJ . Thus the eigenvectors of the levels of an s configuration can be written as ∣∣3L′ −1 〉 = ∣∣3L −1 〉 ∣∣3L′ 〉 = cos θ ∣∣3L 〉 − sin θ ∣∣1L 〉 ∣∣3L′ +1 〉 = ∣∣3L +1 〉 ∣∣1L′ 〉 = sin θ ∣∣3L 〉 + cos θ ∣∣1L 〉 , (1) where the primes denote that the LS designation is only nominal. Similarly, eigenvectors of levels of a p2 configuration can be written as ∣3P′0 〉 = cos θ0 ∣3P0 〉 − sin θ0 ∣1S0 〉 ∣3P′1 〉 = ∣3P0 〉 ∣3P′2 〉 = cos θ2 ∣3P2 〉 − sin θ2 ∣1D2 〉 ∣1D′2 〉 = sin θ2 ∣3P2 〉 + cos θ2 ∣1D2 〉 ∣1S′0 〉 = sin θ0 ∣3P0 〉 + cos θ0 ∣1S0 〉 . (2) The mixing angles can be formulated in terms of direct and exchange Slater and spin–orbit parameters [7]. Branching fractions for the Mg-like 3s3p–3s3d and 3s3p–3p2 transition arrays 3 For the sp ( = 1) and sd ( = 2) configurations the energy level can be written [7] in terms of the Coulomb energy E , the exchange Slater energy G and the diagonal and off-diagonal spin–orbit energies ζ and ζ ′ , as L −1 = E −G − ( + 1)ζ /2 L = E − ζ /4 − L +1 = E −G + ζ /2 L = E − ζ /4 + , (3)