Multichannel Quantum Defect Theory Models Research Articles

Mqdtfit: A collection of Python functions for empirical multichannel quantum defect calculations

The Python functions distributed with this article can be used for calculating the parameters of multichannel quantum defect theory models describing excited bound states of complex atoms. These parameters are obtained by fitting a model to experimental data provided by the user. The two main formulations of the theory are supported, namely the one in which the parameters of the model are a set of eigen channel quantum defects and a transformation matrix, and the one where these parameters are the elements of a reactance matrix. The distribution includes programs for calculating theoretical energy levels, calculating mixing coefficients and channel fractions and producing Lu-Fano plots. Program summaryProgram Title: mqdtfitCPC Library link to program files:https://doi.org/10.17632/nzgygzd96c.1Developer's repository link:https://github.com/durham-qlm/mqdtfitLicensing provisions: BSD 3-clause.Programming language: Python 3.Nature of problem: Multichannel quantum defect theory aims at giving a unified description of the excited states of multielectron systems whereby the properties of entire series can be predicted from models involving a relatively small number of parameters. These parameters are obtained by fitting theory to experimental data, in the empirical version of the theory. The present programs specifically concern the application of this theory to the bound states of complex atomic systems, such as divalent atoms, for which spectroscopic series are often perturbed by isolated states of a different symmetry, making a multichannel description necessary.Solution method: The functions forming this library are grouped and linked to each other into a single Python module. They can be used to calculate the parameters of a given multichannel quantum defect theory model by fitting the model to experimental data provided by the user. These parameters are either a set of eigen channel quantum defects and a transformation matrix, in the eigenchannel parametrization of the theory, or the elements of an orthogonal matrix, in the reactance matrix parametrization of the theory. Given these parameters, this software can also be used to calculate theoretical energy levels, calculate coefficients describing the mixing between the channels considered and produce Lu-Fano plots.

Laser and microwave spectroscopy of even-parity Rydberg states of neutral ytterbium and multichannel-quantum-defect-theory analysis

New measurements of high-lying even parity $6sns\, {}^1 \! S_0$ and $6snd\,{}^{3,1}\!D_2$ levels of neutral $^{174}$Yb are presented in this paper. Spectroscopy is performed by a two-step laser excitation from the ground state $4f^{14}6s^2 \, {}^1 \! S_0$, and the Rydberg levels are detected by using the field ionization method. Additional two-photon microwave spectroscopy is used to improve the relative energy accuracy where possible. The spectroscopic measurements are complemented by a multichannel quantum defect theory (MQDT) analysis for the J=0 and the two-coupled J=2 even parity series. We compare our results with the previous analysis of Aymar {\it{et al}} \cite{Aymar_1980} and analyze the observed differences. From the new MQDT models, a revised value for the first ionization limit $I_{6s}=50443.07041(25)$ cm$^{-1}$ is proposed.

Multichannel quantum-defect-theory analysis of the observed odd-parity5dndautoionization Rydberg series of neutral europium

The spectra of autoionizing Rydberg series of europium atoms are analyzed by a semiempirical multichannel quantum defect theory (MQDT). Experimental data were obtained using two-step resonance ionization spectroscopy below the 5d {sup 9}D{sub J} ionic multiplet. The MQDT model involving five channels and three limits is used to analyze the observed four odd-parity Rydberg series. Theoretical energies are computed and compared with the experimental values. The Rydberg series are assigned on the basis of calculated admixture coefficients. The series having J=3/2 are interpreted as 5dnd (2,3/2), 5dnd (2,5/2), 5dnd (3,3/2), and 5dnd (3,5/2). Interseries perturbations and perturbations due to levels of configuration 5dnd (4,5/2) are investigated. Some of the unobserved levels are also predicted.

Absolute photoionization cross sectionsof the Ba ground state in the autoionization region:II. 221-209 nm

Absolute photoionization cross sections of Ba in the autoionization regionof 209-221 nm have been measured with a bandwidth of 0.0014 nm. Themeasured cross sections are compared with those obtained fromR-matrix calculations combined with multichannel quantum defecttheory (MQDT) techniques. The variationsof the experimental lineshape parameters for the 5d(2D3/2)np[½] and 5d(2D5/2)np[3/2]series are analysed with empirical three-channel MQDT models. From thecalculation of partial density of states in 6pns channels, the perturberof the 5d(2D3/2)np[½] series is confirmed tobe the 6p7s 3P1 level and that of the 5d(2D5/2)np[3/2] series is identified as the 6p7s 1P1level.

Resonant widths, line intensities and lineshapes for MQDT models with two or more open channels

Approximate analytical formulae describing the energy variation of line intensities, autoionization widths and lineshape asymmetries, are derived for a Phase-Shifted Multichannel Quantum Defect Theory model composed of two closed interacting channels coupled to two effective continua. This is accomplished by putting the two compatibility equation solutions, for the common phase shifts of the two open channels, in such a form so the resonant behavior is attributed to one of them, the other accounting for an energy dependent background. Then, the well-known procedures for the simpler case where only one continuum is considered are applied, using only the resonant solution. The method is quite general and applicable to any MQDT model with two or more open channels. The resulting analytical formulae are tested on experimental spectra of Sr, Ba and Cu and it is shown that they are valid as long as: i) The resonances are non-overlapping, ii) The direct closed channel coupling is much stronger than the indirect one through the continua and (when excitation matrix elements are involved) iii) The open channels excitation strength is smaller or at least comparable to the closed channels one.

Double-resonance spectroscopy of a complex resonance in molecular nitrogen.

Double-resonance spectroscopy via several rotational levels of the ${\mathrm{N}}_{2}$ ${\mathit{a}}^{1}$${\mathrm{\ensuremath{\Pi}}}_{\mathit{g}}$, v'=5 state has been used to probe the rotational structure of a complex autoionizing resonance observed just below the ${\mathit{X}}^{2}$${\mathrm{\ensuremath{\Sigma}}}_{\mathit{g}}^{+}$, ${\mathit{v}}^{+}$=1 threshold. A multichannel-quantum-defect-theory (MQDT) analysis confirms that the complex resonance is produced by the interaction of two interloping states with the np and nf Rydberg series converging to the ${\mathit{X}}^{2}$${\mathrm{\ensuremath{\Sigma}}}_{\mathit{g}}^{+}$, ${\mathit{v}}^{+}$=1 limit. The quantum defects of the interlopers and their interaction strengths with the two Rydberg series are determined from fits of the MQDT model to the energies of the peaks observed in the complex resonance. The experimentally determined interaction parameters and those available from ab initio calculations are in fair agreement. The present results complete the interpretation of the photoionization spectrum of ${\mathrm{N}}_{2}$ in the energy region between the ${\mathit{X}}^{2}$${\mathrm{\ensuremath{\Sigma}}}_{\mathit{g}}^{+}$, ${\mathit{v}}^{+}$=0 and 1 thresholds.

Eigenchannel R-matrix calculation of the photoabsorption spectrum of strontium

Presents the first detailed theoretical calculation of the absorption spectrum of Sr out of the 5s2 1S0 ground state. The autoionisation region below the 5p3/2 threshold is investigated, photoionisation cross sections being calculated by combining the multichannel quantum defect theory (MQDT) with the eigenchannel R-matrix method. Small-scale variational R-matrix calculations enable one to evaluate LS-coupled short-range reaction matrices. A jj-coupled reaction matrix is then deduced using a geometrical frame transformation. Diagonalisation of this matrix gives the energy-dependent MQDT parameters required for the description of the 13 relevant interacting channels. The 13-channel MQDT model built with theoretical R-matrix parameters provides an accurate description of the positions and the shapes of the 4d3/2,5/2 np, nf and 5p1/2,3/2 ns, nd J=1 autoionising lines. Particular attention is paid to high lying resonances observed near the spin-orbit split 4d3/2,5/2 and 5p1/2,3/2 thresholds.

Hyperfine structure and isotope shifts in odd-parity 6snp and 6snf Rydberg series of Ba I

Hyperfine structure and isotope shifts in transitions to odd-parity 6snp and 6snf Rydberg configurations of barium have been investigated using laser-atomic-beam spectroscopy. For the 6snf Rydberg configurations with n=13-42 the complete hyperfine multiplet is observed. The data on isotope shifts have been used to test the total perturber character mixed in the 6snp and 6snf Rydberg states predicted by existing multichannel quantum defect theory (MQDT) treatments of the odd-parity J=0, 1, 2, 3 and 4 spectra. The MQDT predictions are shown to be reliable in this respect. The hyperfine structure of the singlet and triplet states is analysed in terms of singlet-triplet mixing. The extracted mixing coefficients disagree with the predictions of existing MQDT models, even when an interaction between 'singlet' and 'triplet' 6snl channels is introduced. New MQDT models have been developed in which both energies and mixing coefficients are reproduced well. Hyperfine structure data provides information on interactions between mutually perturbing channels. Finally an earlier analysis of the hyperfine structure of the 6snf 3F states in terms of effective parameters is discussed.

Multichannel-quantum-defect theory wave functions of Ba tested or improved by laser measurements

Considerable efforts have been made recently to understand the detailed structure and properties of high Rydberg states of the bound even-parity spectrum of Ba. Interest in Ba is motived by the ability to study interacting Rydberg series. The perturbations of 6snl Rydberg series by 5d7d levels are reflected not only in the level structure but also in various observable quantities recently investigated by high-resolution laser spectroscopy. A wealth of data has been obtained on lifetimes, isotopes shifts, Lande factors, hyperfine structures, photoelectron angular distributions, etc. Previously the multichannel-quantum-defect theory (MQDT) has permitted one successfully to interpret the energies of perturbed series. The new possibilities offered by laser spectroscopy to analyze state mixing have provided an opportunity to probe the MQDT wave functions. This paper shows how the data provided by different measurements complement one another to check or even extend the MQDT models derived from energies. Various examples concerning the perturbed 6sns and 6snd Rydberg series are presented. For the 6snd1,3D2 series, energies are insufficient to get the full MQDT parameters needed to determine the wave functions, and gJ-factor and hyperfine-structure measurements have been essential to improve MQDT wave functions.

Multichannel quantum-defect analysis of the bound even-parity spectrum of neutral barium

Extensive new data on the bound even-parity spectrum of neutral barium have been provided recently by two-step optogalvanic spectroscopy. Highly excited members of the Rydberg series $6sns^{3}S_{1}$, $6snd^{3}D_{1}$, $^{3}D_{3}$, and $6sng$ as well as perturbing levels of these series belonging to $5dnd$ configurations have been observed. By means of multichannel quantumdefect theory (MQDT), we have performed the analysis of all available energy data relevant to the $J=1,3,4,5$ even-parity bound spectra of barium. The behavior of $6sns^{3}S_{1}$, $6snd^{3}D_{1}$, $^{3}D_{3}$ series perturbed by levels pertaining to the $5dns (n=7,8)$ and $5dnd (n=6,7)$ configurations is clearly interpreted. The MQDT analyses of $J=1 \mathrm{and} 3$ spectra complementary to that previously performed on the $J=2$ spectrum offer the possibility of studying the anomalous fine structures of highly excited $6snd$ configurations. The MQDT analyses of $J=3\ensuremath{-}5$ spectra explain the observed energies of $6sng^{3}G_{3,4,5}$ and $^{1}G_{4}$ Rydberg levels; the fine structures of the $6sng$ series are only partially resolved in the vicinity of $5d7d$ $J=3 \mathrm{to} 5$ perturbing levels. Since the $J=0$ and $J=2$ bound even-parity spectra of Ba have already been analyzed by MQDT, we have now a full MQDT description of the bound even-parity spectrum for $J$ values ranging from 0 to 5.

Highly excited levels of neutral ytterbium. II. Multichannel quantum defect analysis of odd- and even-parity spectra

For pt.I see ibid., vol.13, no.6, p.1073 (1980). Data on the bound even-parity spectrum of Yb have been presented in the preceding paper (Carnus et al., 1980). These data provide an excellent opportunity for the application of multichannel quantum defect theory (MQDT) to this spectrum. The bound even-parity J=0, 2 spectra and the bound odd-parity J=1 spectrum are analysed with MQDT models. The behaviour of the even-parity 4f146sns 1S0 (n>or=7) and 4f146snd 1.3D2 (n>or=8) Rydberg series perturbed by levels of the 4f146p2 and 4f135d6s6p configurations is well interpreted; in the same way the known bound levels of the 4f146snp 1.3P1 Rydberg series (n>or=12) perturbed by levels of the 4f135d26s configuration are well interpreted. The mutual interaction of the singlet and triplet levels of J=1 and J=2 Rydberg series is analysed. Some assignments of levels are changed and predicted energy values are given for many levels.