sr88_vaillant2024_k_matrix_model_data
Full module path: rydstate.species.strontium.sr88_vaillant2024_k_matrix_model_data
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Combined 6-channel model for the strongly mixed 5snd 1D2 and 5snd 3D2 series. |
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5snf 3F2 series, described by its Rydberg-Ritz quantum defects, since Vaillant 2024 has no triplet F models. |
5snf 3F3 series, described by its Rydberg-Ritz quantum defects, since Vaillant 2024 has no triplet F models. |
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5snf 3F4 series, described by its Rydberg-Ritz quantum defects, since Vaillant 2024 has no triplet F models. |
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5sns 3S1 model fitted to the data set (b) of the Addendum (high precision data of Couturier 2019 for n >= 13). |
Module description
MQDT models for Sr88 in the reactance matrix (K-matrix) formulation of Vaillant, Jones and Potvliege.
The models were originally published in C. L. Vaillant, M. P. A. Jones and R. M. Potvliege, J. Phys. B: At. Mol. Opt. Phys. 47, 155001 (2014) and updated in the Addendum J. Phys. B: At. Mol. Opt. Phys. 57, 199401 (2024), https://doi.org/10.1088/1361-6455/ad76f0 (arXiv:2403.08742) all K-matrix parameters below are taken from the Addendum. Table III of the Addendum quotes the parameters with 7 significant digits, we use the values with 9 significant digits from the mqdtfit scripts (https://github.com/durham-qlm/mqdtfit).
The K-matrix elements are given in the frame of the dissociation channels of the paper (outer channels here): jj-coupled channels where the fine structure of the ionic core is resolved (5s_1/2 nl_j, 4d_5/2 nl_j, 4d_3/2 nl_j) and LS-coupled channels where an averaged ionization threshold is used (4dnl, 5pnp), see Table I of the Addendum.
Sign convention of the off-diagonal elements
The channel kets of the paper and of rydstate differ by channel dependent signs, because the two valence electrons are coupled in the opposite order. The paper couples the core electron (1) first and the Rydberg electron (2)second, see the jj->LS recoupling formula in Sec. 2 of the 2014 paper (matrix U_{i alphabar} of mqdtfit):
<(l_c s_c) j_c, (l_r s_r) j_r; J | (l_c l_r) L, (s_c s_r) S; J>
= sqrt((2 j_c + 1) (2 j_r + 1) (2 L + 1) (2 S + 1)) * 9j{l_c s_c j_c; l_r s_r j_r; L S J}
whereas the rydstate kets couple the Rydberg electron first, see AngularKetBase.calc_reduced_overlap()
(note the order of the arguments of the 9j symbol):
<(l_r s_r) j_r, (l_c s_c) j_c; J | (l_r l_c) L, (s_r s_c) S; J>
= sqrt((2 j_r + 1) (2 j_c + 1) (2 L + 1) (2 S + 1)) * 9j{l_r s_r j_r; l_c s_c j_c; L S J}
Exchanging the coupling order of two angular momenta j_1, j_2 -> j_12 multiplies the coupled ket by (-1)^(j_1 + j_2 - j_12), so the kets of the paper are related to the rydstate kets by:
|i>_paper = d_i |i>_rydstate, with d_i = (-1)^(j_c + j_r - J) for jj-coupled channels,
d_i = (-1)^(l_c + l_r - L) * (-1)^(1 - S) for LS-coupled channels.
Consequently the K-matrix in the rydstate frame is K_rydstate = D K_paper D with D = diag(d_i), i.e. the off-diagonal elements between two channels with different d_i are multiplied by -1 compared to Table III of the Addendum (marked with (*) in the comments below). This reproduces the recoupling matrices U_{i alphabar} of mqdtfit exactly (see the test test_vaillant2024_recoupling_phase_convention), and is necessary to get the correct singlet/triplet character and matrix elements. The energies of all states are independent of these signs, since det(D M D) = det(M).
Validity ranges
The lower bound of nu_range of each model is chosen such that all states used in the fit are included, but no lower lying states, which the models do not reproduce (e.g. 5s6s 1S0 or 5s5d 1D2).
Triplet F series
The triplet F series are not covered by the Addendum, which revisited only the singlet and triplet S, P and D series and the singlet F series. The three-channel models of the original 2014 paper (table IX, with the 5snf 3FJ, 4dnp 3FJ and 4dnf 3FJ channels) are not usable instead: they are among the parameters affected by the sign error of the fitting program (they give quantum defects of ~0.19 instead of the measured ~0.12 for n = 10-25) and were not refitted. We therefore describe the triplet F series by single channel models with the Rydberg-Ritz quantum defects of Robertson 2021 (ARC 3.0, table B.1), fitted to the 5snf 3FJ levels of Rubbmark 1978 and Sansonetti 2010 (NIST) with the same ionization threshold I_s = 45932.2002 1/cm as used here. We use the unrounded values of the ARC source code. These quantum defects reproduce the NIST levels with n = 10-20 to < 0.15 1/cm (rms 0.04-0.06 1/cm, i.e. the NIST uncertainty), see test_vaillant2024_triplet_f_energies_match_nist.