Use of intermediate coupling relationships to test measured branching fraction data

The extensive arc emission measurements of transition probabilities by Corliss and Bozman are known to contain errors due to flaws in the determination of level populations in the source. However, if no similar errors were present in the photometric calibration, then branching fractions from the same upper level deduced from these measurements should be valid. It is shown that the branching fractions for the ns2np2–ns2np(n+ 1)s transitions in Si I, Ge I, Sn I and Pb I can be accurately estimated using intermediate coupling amplitudes obtained from spectroscopic data, and thus provide a test of the validity of these measurements. The results of Corliss and Bozman are examined in the context of comparisons with these estimated values and with other measurements, and it is demonstrated that branching fractions from the same upper level obtained from these data can be quite reliable. Atomic oscillator strengths can be determined experimentally either by absolute emission, absorption or dispersion measurements, or through the combined measurement of relative branching fractions and level lifetimes. The absolute measurements require sample equilibrium, a knowledge of the absolute number density and an absolute photometric calibration, and the lifetime measurements yield oscillator strengths only in cases where a single decay channel exists. Thus high-precision measurements have often involved the combined measurement of lifetimes and branching fractions [1]. While many methods have been brought to bear on the precision measurement of lifetimes [2] and much progress has been made in the measurement of branching ratios [3], branching fraction data remain sparse and urgently needed. A very extensive tabulation of transition probability data derived from arc spectra line intensities exists in the monograph of Corliss and Bozman [4]. However, it is well known [5] that these transition probabilities provide neither an absolute nor a self-consistent set of values, so care must be exercised in their use. In general, the normalization must be corrected for two types of errors. The first is in the determination of the concentration of radiating atoms in the arc source (which can be corrected by summing transition probabilities over final states and renormalizing to match measured lifetime data). The second is in the determination of the level populations in the arc source due to either a lack of thermodynamic equilibrium or an inaccurate temperature determination, which causes the renormalization factor to depend on the excitation energy of the upper level. Since branching ratios involve the comparison of relative transition probabilities from the same upper level, neither the overall normalization nor the arc temperature should affect their validity. However, it does require an accurate photometric calibration of the detection equipment over the very wide range of wavelengths spanning the decay channels. 0953-4075/98/190769+06$19.50 c © 1998 IOP Publishing Ltd L769 L770 Letter to the Editor In general, the comparison of experimental branching fractions with theoretical calculations requires the specification of radial transition moments, which are sensitive to the details of the potential. However, in the non-relativistic approximation, the relative intensities of the lines within a supermultiplet all involve the same radial transition moment, which cancels when branching ratios are formed. If there is no significant branching to other configurations the branching fractions can be specified from angular factors and the intermediate coupling (IC) mixing amplitudes, which can be determined from spectroscopic energy level data. The ns2np2–ns2np(n + 1)s manifold of transitions in Si I, Ge I, Sn I and Pb I provide such a case, and comprehensive measurements for this supermultiplet are contained both in the tabulation of Corliss and Bozman [4], and in more recent studies [6–9]. While the lack of configuration interaction (CI) in these systems simplifies the calculational specification of the branching fractions, the transitions cover a wide range of both wavelength regions and intensity ratios, and are no less challenging to experimental measurement than any other system studied in [4]. We have therefore made data-based empirical IC calculations of these branching fractions and compared them with the measurements of [4] and others, in order to evaluate the reliability of the tabulation of Corliss and Bozman as a source of branching fraction data. For a pure configuration, the intermediate coupling amplitudes are manifested both by the energy levels and by the transition probabilities of the levels. Thus, if the singleconfiguration picture is valid, the measured energy level splittings within the upper and the lower configuration can be used to determine the mixing amplitudes, and these can then be used to specify (to within factors of the radial transition matrix) the relative transition probabilities. In the case of the sp and p2 configurations, there are at most two normalized mixing amplitudes for a given value of J , which can be characterized by a singlet–triplet mixing angle θJ . For sp the mixing between P1 and P1 can be characterized by θ1 (primes denote that the LS notation is only nominal for the physical states) |P0〉 = |P0〉 (1) |P1〉 = cos θ1|P1〉 − sin θ1|P1〉 (2) |P2〉 = |P2〉 (3) |P1〉 = sin θ1|P1〉 + cos θ1|P1〉 (4) whereas for p2 the mixing can be characterized between P0 and S0 by θ0 and between P2 and D2 by θ2: |P0〉 = cos θ0|P0〉 − sin θ0|S0〉 (5) |P1〉 = |P1〉 (6) |P2〉 = cos θ2|P2〉 − sin θ2|D2〉 (7) |D2〉 = sin θ2|P2〉 + cos θ2|D2〉 (8) |S0〉 = sin θ0|P0〉 + cos θ0|S0〉. (9) A formalism has been developed previously [10, 11] by which these mixing angles are first extracted from measured energy level data and then used to predict transition probabilities. For a p2–sp manifold, the transitions from the upper level sp to the levels of the ground configuration p2 can be deduced from this formalism using the LS-coupling angular transition matrices [12, 13]. The nonvanishing values are 〈P0|r|P1〉 = −〈P1|r|S0〉 = 〈P1|r|P0〉 = − √ 20 (10) − 2〈P2|r|P1〉 = 2〈P1|r|P2〉 = 〈P1|r|D2〉 = 10 (11) √ 5〈P1|r|P1〉 = 〈P2|r|P2〉 = √ 75. (12) Letter to the Editor L771 These equations yield, for the upper level P1 ′ 〈P0|r|P1〉 = − √ 20 cos(θ1 + θ0) 〈p2|r|sp〉 (13) 〈P1|r|P1〉 = √ 15 cos θ1 〈p2|r|sp〉 (14) 〈P2|r|P1〉 = 5(2 sin θ1 sin θ2 + cos θ1 cos θ2) 〈p2|r|sp〉 (15) 〈D2|r|P1〉 = −5(2 sin θ1 cos θ2 − cos θ1 sin θ2) 〈p2|r|sp〉 (16) 〈S0|r|P1〉 = − √ 20 sin(θ1 + θ0) 〈p2|r|sp〉 (17) for the upper level P2 ′ 〈P1|r|P2〉 = −5 〈p2|r|sp〉 (18) 〈P2|r|P2〉 = 5 √ 3 cos θ2 〈p2|r|sp〉 (19) 〈D2|r|P2〉 = 5 √ 3 sin θ2 〈p2|r|sp〉 (20) and for the upper level P1 ′ 〈P0|r|P1〉 = − √ 20 sin(θ1 + θ0) 〈p2|r|sp〉 (21) 〈P1|r|P1〉 = √ 15 sin θ1 〈p2|r|sp〉 (22) 〈P2|r|P1〉 = −5(2 cos θ1 sin θ2 − sin θ1 cos θ2) 〈p2|r|sp〉 (23) 〈D2|r|P1〉 = 5(2 cos θ1 cos θ2 + sin θ1 sin θ2) 〈p2|r|sp〉 (24) 〈S0|r|P1〉 = √ 20 cos(θ1 + θ0) 〈p2|r|sp〉. (25) It should be noted that in a fully relativistic Dirac treatment the corresponding expressions will involve two separate jj -coupled radial transition matrices, and reduce to equations (13)– (25) only if these two radial matrices are equal. Theoretical studies of these relativistic corrections have been presented elsewhere [14]. For pure sp and p2 configurations the energy levels (and thereby the mixing angles) are specified [10] by three parameters (F0, G1, ζp for sp and F0, F2, ζpp for p2, in the notation of [13]). Since the sp and p2 configurations contain four and five levels, respectively, the specification of these three parameters is overdetermined. Here this was treated by using the average energies εJ of the J = 0, 1, 2 levels to make an exactly determined parametrization, computing the singlet–triplet splittings from this parametrization, and then using the deviations as a measure of the validity of the single-configuration picture. Within this framework, the mixing angles θJ can be determined from the relationships [10] cot(2θJ ) = WJ (26) where the sp mixing J = 1 level is given by W1 = [ε2 − 3ε1 + 2ε0]/[ √ 2(ε2 − ε0)] (27) and the p2 mixing of the J = 0 and J = 2 levels is given by W0 = −[10ε2 − 21ε1 + 11ε0]/[4 √ 2(5ε2 − 3ε1 − 2ε0)] (28) W2 = −[5ε2 + 3ε1 − 8ε0]/[2 √ 2(5ε2 − 3ε1 − 2ε0)]. (29) In terms of the transition elements 〈k|r|i〉 given by equations (13)–(25), the transition probabilities are obtained from Aik(ns −1) = 3 [1265.38/λ(Å)]3 |〈k|r|i〉|2 (30) L772 Letter to the Editor Table 1. Spectroscopic database and intermediate coupling parametrization of energy levels (in cm−1). Sources of spectroscopic data: Si I [15]; Ge I [16]; Sn I [17, 18]; Pb I [19]. Level Eobs EIC 1E Eobs EIC 1E