Multichannel Quantum Defect Research Articles

Absorption spectrum of H2 between the third and the fourth dissociation thresholds (132 500 – 139 000 cm−1)

Over a thousand spectral lines in the absorption spectrum of molecular hydrogen (H2) to the npσ 1Σu+ and npπ 1Πu± Rydberg levels (n ≥ 4) were measured and assigned for rotational levels with N’=0 – 4 in the spectral range between 132 500 and 139 000 cm−1. The experimental energies, the absorption cross sections, and Einstein A coefficients were compared with ab initio multi-channel quantum defect (MQDT) calculations. The assigned lines include 31P(1), 225 R(0) or P(2), 291 R(1) or P(3), 109 R(2) lines, 71 R(3) lines, and over 300 Q(1–3) lines. Transition energies were determined up to excitation energies of 139 000 cm−1 above the ground state, thereby extending earlier works and leading to several reassignments. The overall agreement between experiment and first principles calculations, without adjustable parameters, is very good in view of the multi-state interferences treated within the MQDT-framework. The line intensities in terms of Einstein A coefficients are well represented in the MQDT-framework. These line intensities follow, in general, the 1/n3 scaling behavior as characteristic in Rydberg series, but many deviations occur which are explained by MQDT. Despite the mostly very good agreement between the observations and the outcome of the MQDT calculations for both level energies and line intensities, some pronounced deviations were found which are discussed. The shortcomings of the MQDT calculations are ascribed to the treatment of the excited states in terms of a 1snp single electron configuration, therewith neglecting possible interferences with 1snf or 2s core excited states. So far, some 108 lines remained unassigned.

Multichannel photoelectron phase lag across atomic barium autoionizing resonances

Phase lag associated with coherent control where an excited system decays into more than one product channel has been subjected to numerous investigations. Although previous theoretical studies have treated the phase lag across resonances in model calculations, quantitative agreement has never been achieved between the theoretical model and an experimental measurement of the phase lag from the $\ensuremath{\omega}\ensuremath{-}2\ensuremath{\omega}$ ionization of atomic barium. Yamazaki and Elliot, Phys. Rev. Lett. 98, 053001 (2007); Phys. Rev. A 76, 053401 (2007), suggesting that a toy model with phenomenological parameters is inadequate to describe the observed phase lag behavior. Here the phase lag is treated quantitatively in a multichannel coupling formulation, and our calculation based on a multichannel quantum defect and $R$-matrix treatment achieves good agreement with the experimental observations. Our treatment also develops formulas to describe the effects of hyperfine depolarization on multiphoton ionization processes. Moreover, we identify resonances between ${\text{Ba}}^{+}\phantom{\rule{4pt}{0ex}}5{d}_{3/2}$ and $5{d}_{5/2}$ thresholds that have apparently never been experimentally observed and classified.

Resonant control of photoelectron directionality by interfering one- and two-photon pathways

Coherent control of interfering one- and two-photon processes has for decades been the subject of research to achieve the redirection of photocurrent. The present study develops two-pathway coherent control of ground state helium atom above-threshold photoionization for energies up to the $N=2$ threshold, based on a multichannel quantum defect and R-matrix calculation. Three parameters are controlled in our treatment: the optical interference phase $\Delta\Phi$, the reduced electric field strength $\chi=\mathcal{E}_{\omega}^2/{\mathcal{E}_{2\omega}}$, and the final state energy $\epsilon$. A small energy change near a resonance is shown to flip the emission direction of photoelectrons with high efficiency, through an example where $90\%$ of photoelectrons whose energy is near the $2p^2\ ^1S^e$ resonance flip their emission direction. However, the large fraction of photoelectrons ionized at the intermediate state energy, which are not influenced by the optical control, make this control scheme challenging to realize experimentally.

Potential Energies of the Orbitally Degenerate Atmospheric Rydberg Complexes

A method was developed for calculating the vibronic potential energy surfaces (PESs) of atmospheric complexes consisting of orbitally degenerate Rydberg nitrogen and oxygen molecules and the molecules of a neutral medium in the ground electronic state. The degenerate states are formed as a result of l-mixing in the D and E layers of the atmosphere during the periods of increased solar activity. The complexes are populated in the nonequilibrium two-temperature plasma and are responsible for the incoherent additional background radiation in the decimeter (microwave) and terahertz (IR) bands at an altitude of 80–110 km from the Earth’s surface. To describe the interaction of a weakly bound electron with a singly charged molecular ion and a neutral molecule of a gas medium, the formalism of the multichannel quantum defect (MCD) theory was used. Quantum-chemical calculations of the dependences of the scattering lengths, polarizabilities, and quadrupole moments of the main atmospheric molecules N2 and O2 on the interatomic distance were performed. The specific features of the behavior of vibronic PESs of Rydberg complexes for large values of the principal quantum number (n ≫ 1) were analyzed. The vibronic PESs of orbitally degenerate states were constructed. They are necessary for determining the positions and shape of the vibronic minima of the l-mixing cross sections of the N2 and O2 Rydberg molecules in the D and E layers of the Earth’s atmosphere, where the delay times of satellite positioning signals should be minimum. The possibility of “quantum chaos” appearing in the Rydberg complexes at sufficiently large n values and angular momenta of the weakly bound electron was noted.

Experimental and theoretical studies of the [formula omitted] and [formula omitted] ([formula omitted]–6) states of D2: Energies, natural widths, absorption line intensities, and dynamics

Over a thousand spectral lines in the photoexcitation spectrum of molecular deuterium (D2) to np1Σu+ and 1Πu+ Rydberg levels (n⩾4) were measured for rotational levels N′=1–6 in the 117000–137000cm−1 spectral range by two different types of experiments at two synchrotron radiation sources: a vacuum ultraviolet (VUV) Fourier-transform (FT) spectrometer at SOLEIL, Paris and a 10m-normal-incidence monochromator (NIM) at BESSY II, Berlin. The experimental energies, the absorption cross sections, Einstein A-coefficients, and line widths are compared with ab initio multi-channel quantum defect (MQDT) calculations for these levels. More than 350 R(0) or P(2) lines were assigned, some 280 R(1) or P(3) lines, some 270 R(2) or P(4) lines, over 100 R(3) or P(5) lines, over 90 R(4) lines, and 24 R(5) lines to extract information on the N′=1–6 excited levels. Transition energies were determined up to excitation energies of 137000cm−1 above the ground state, thereby extending earlier work by various authors and considerably improving the spectral accuracy (<0.1cm−1), leading to several reassignments. The absorption and the dissociation, ionization and fluorescence excitation cross sections from the NIM experiment are measured on absolute scale and are used to calibrate intensities in the VUV-FT spectra. The overall agreement between experiment and first principles calculations, without adjustable parameters, is excellent in view of the multi-state interferences treated within the MQDT-framework: For the low N′ values the averaged deviations between those observed in the FT-SOLEIL spectra and those calculated with MQDT are ∼0.1cm−1 with a spread of ∼0.5cm−1. The line intensities in terms of Einstein coefficients are well represented in the MQDT-framework, as are the level widths representing the lifetimes associated with the sum of the three decay channels. These line intensities follow, in general, the 1/n3 scaling behavior as characteristic in Rydberg series, but deviations occur and those are explained by MQDT. The decay dynamics of the excited N Rydberg levels is analyzed on the basis of the measured quantum yields for ionization, dissociation and fluorescence observed in the NIM experiment in terms of absolute cross sections for the distinctive channels. In particular in the n=4 manifolds dissociation is found to play a major role, where in the n=5 manifolds the behavior is most erratic due to strong competition between decay channels. At n=6, ionization takes over as the dominant channel. Despite the excellent agreement between observations and the outcome of the MQDT calculations for both level energies and dynamics, some pronounced deviations are found as in the splitting of the 5pπ,v=4–6, N′=1 levels. The shortcomings of the MQDT calculations are ascribed to the treatment of the excited states in terms of a 1snp single electron configuration, therewith neglecting possible interferences with 1snf or 2s core excited states. Some 27 lines remained unassigned; in view of their observation in fluorescence it is stipulated that these lines probe levels in the nf manifold.

Experimental and theoretical study of the [formula omitted] ([formula omitted]) [formula omitted] excited states of D2: Absolute absorption cross sections and branching ratios for ionization, dissociation and fluorescence

The absolute absorption cross sections for photoexcitation of D2 to N=0 Rydberg levels have been measured at a resolution of 0.001nm in the 123,500–135,000cm−1 spectral range. The experimental energies and line intensities are compared with ab initio multichannel quantum defect (MQDT) calculations of the n⩾4npσ1Σu+N=0 levels of D2. The calculations provide theoretical values of level positions, line intensities and autoionization widths. They are based on quantum–mechanical clamped-nuclei potential energy curves and dipole transition moments available for the lowest Rydberg members of the npσ series. The overall agreement between experiment and theory is good. The decay dynamics of the excited N=0 Rydberg levels is discussed based on the observed quantum yields for ionization, dissociation and fluorescence, and is compared with the yields previously observed in the analogous states of the H2 isotopomer.

Near-Threshold Photoionization of Molecular Nitrogen

SynopsisPhotoabsorption of molecular nitrogen (N2) is investigated near the first ionization threshold using an R-matrix, multi-channel quantum defect (MQDT) approach.

Quantum defect theory of dipole and vibronic mixing in Rydberg states of CaF

The Rydberg spectra of CaF combine the simplicity of a single electron outside a doubly closed-shell Ca2+F- ion core with the exceptional polarity of the ion core. A global multichannel quantum defect (MQDT) fit to 612 previously assigned levels, 507 from n approximately = 12-18, N=0-14, v+=1, 97 from n approximately = 9-10, N=0-14, v+=2, and 8 from n approximately = 7, N=3-10, v+=3, produces the complete L=0-3 quantum defect matrix mu (with the exception of one element) and 19 of 20 elements of the partial differentialmu/differentialR matrix, as well as the molecular constants of the CaFX 1sigma+ state [omega(e)+=694.58(14), omega(e)x(e+)=2.559(40), B(e+)=0.373 07(16) cm(-1), and the v=0, N=0 to v(+)=0, N(+)=0 ionization energy, 46,996.40(8) cm(-1)]. This experimentally determined mu(R) matrix is unusual in the completeness of its representation of the spectrum of both core-penetrating and nonpenetrating Rydberg series, including both local perturbations and vibrational autoionization rates, as well as all dynamical processes encoded in the spectrum that result from the scattering (at negative energy) of the Rydberg electron off the Ca2+F- ion core. The MQDT theory is presented in a form that clarifies the relationships of the reaction (K) and phase (P) matrices of MQDT to effective Hamiltonian models for local interactions between accidentally near degenerate levels. In particular, a Hund's case (b) like representation of the Hamiltonian is described in which the rovibronic K matrix is diagonalized and the P matrix, which contains information about the v+, N+ eigenstates of the ion, becomes nondiagonal.

The rare K-decays in the Multiscale Walking Technicolor Model

We report a theoretical approach based on a combination of jj-coupled eigenchannel R-matrix and multichannel quantum-defect (MQDT) methods suitable to determine Raman couplings, dynamical Stark shifts and threshold-resolved ionization rates in heavy alkaline-earth atoms. These atomic parameters are needed to analyze the partial photoelectron spectra and the ionization dynamics of multiphoton processes in a perturbative treatment of the field-atom interaction to the lowest non-vanishing order of perturbation theory. Numerical results are presented in barium, pertaining to the experimental level scheme used by Wang, Chen and Elliott [Phys. Rev. Lett. 77, 2416 (1996)] to study coherent control through two-color interfering paths. Particular emphasis is given on the comparison of length and velocity formulations for the electric dipole operator.

4pnp J =0 e -2 e autoionizing series of calcium: experimental and theoretical analysis

The even parity 4pnp J=0, 1, 2 doubly excited autoionizing states of neutral calcium in an atomic beam are investigated by a two-step isolated core excitation (ICE) method using two different combinations of polarization of the laser beams. The different excited energy levels are assigned to nine autoionizing Rydberg series 4p1/2, 3/2np J=0, 1, 2 for ≤ n ≤ 22. The theoretical interpretation is achieved by a combination of the eigenchannel R-matrix theory and the multichannel quantum defect (MQDT) method. Two, five and six closed interacting channels are introduced for the J=0, J=1 and J=2 series respectively. Theoretical energy level positions, autoionization widths and excitation profiles are compared with the experimental data, confirming the identification of the observed structures and providing evidence of extended mixing between the 4p1/2np and 4p3/2np series.

Dissociative recombination of electrons and molecular ions

The existing techniques for the calculation of the dissociative recombination (DR) of electrons and molecular ions were compared. The advantages of the method of multichannel quantum defect (MQD), in which equations are formulated directly for the T-matrix of collisions and the unitarity of the scattering S-matrix is thus ensured, were demonstrated. The effect of molecular rotation and of the nonadiabatic electron-rotation coupling on the e− + H2+ →, H* + H reaction was investigated. A procedure was suggested based on the use of the adiabatic approximation (with respect to the nuclear rotation) in the near-threshold area while taking into account the contributions of the excited vibronic states of the Rydberg complex formed in an intermediate stage of the reaction. It is notable that the partial rate constants (and the corresponding cross-sections) arc very sensitive to the initial rotation excitation. However, the temperature-averaged rate constants under equilibrium conditions are only slightly affected by rotation.

3dnl J=0e-2e autoionizing levels of calcium: observation by laser optogalvanic spectroscopy and theoretical analysis

The even parity J=0-2 autoionizing spectra of calcium are investigated below the 3d threshold by a two-step laser excitation from the 3d4s metastables through the 3d4p 3P0.1.2 intermediate states. The 3d4s are populated by electronic collisions in a DC glow discharge through a Ca heat-pipe. Around 300 resonance transitions are measured with an accuracy of approximately 0.2 cm-1 for the narrow ones using standard laser calibration techniques. Their upper levels are assigned to 17 autoionizing 3dns, 3dnd and 3dng Rydberg series and several perturbers belonging to the 4p2 and 4p5p configurations. The theoretical interpretation is achieved by a combination of the eigenchannel R-matrix and multichannel quantum defect (MQDT) methods. Five, twelve and fourteen interacting channels are introduced for J=0, J=1 and J=2 series respectively. Theoretical energy level positions and excitation profiles are compared with the experimental data, confirming the identification of the observed structures and pointing out the behaviour of the different perturbed Rydberg series.

Eigenchannel R-matrix study of the J=0 and J=2 even parity spectra of calcium below the Ca+ 3d3/2 threshold

The J=0 and J=2 even parity spectra of calcium are studied through a combination of the eigenchannel R-matrix and multichannel quantum defect (MQDT) methods. The bound and autoionizing spectra below the 3d3/2 threshold are investigated. The calculation reproduces well the perturbations of the 4sns 1S0 , 4snd 1.3D2 series by low-lying doubly excited states observed experimentally and provides new insight on the identification of the perturbers. Theoretical predictions of the positions of the 3dnl and low-lying 4pnp J=0e and J=2e resonances are given. Results of photoionization spectra from low-lying states are presented. Particular attention is paid to analyse the effects of the strong interaction between the 3dnd and 4pnp singlet channels which dominates the bound and autoionizing spectra above the first limit. Comparisons are made with the results of previous experimental and theoretical studies.

Eigenchannel R-matrix calculation of the J=1 odd-parity spectrum of barium

The J=1 odd-parity spectrum of barium is studied through a combination of the eigenchannel R-matrix and multichannel quantum defect (MQDT) methods. Short-range parameters used by MQDT to describe electronic channel interactions are determined by performing R-matrix calculations in LS coupling and the (LS/jj) frame transformation is used to include fine-structure effects. This approach has proven to be an efficient tool for describing lighter alkaline earths but its ability to handle Ba, where spin-orbit interaction is stronger, is analysed. The calculations reproduce well the perturbations of the principal Rydberg series 6snp 1.3P1 by eight low-lying doubly excited levels as well as most of the features observed in the photoabsorption spectrum 6s2-J=1' up to the 6p3/2 threshold. Although some discrepancies were found, the considerable overall success obtained by describing spin-orbit effects in Ba with the (LS/jj) frame transformation instead of including them explicitly in the short-range calculation is a great surprise.

R-matrix calculation of photoionisation from Rydberg states in strontium

This paper presents the first detailed theoretical calculation of numerous observables investigated in Sr using stepwise laser excitation techniques. Measurements concern the 5p12/ns and 5p32/ns J=1 autoionising levels excited through the 5sns 1S0 Rydberg states; isolated resonances as well as interacting autoionising 5pns levels have been investigated. Results on the excitation spectra, the positions and widths of the levels as well as on the branching ratios and the angular distributions of the photoelectrons have been reported. All the above-mentioned observables are calculated through a combination of the eigenchannel R matrix and the multichannel quantum defect (MQDT) methods. Short-range parameters used by MQDT to describe final-state channel interactions are determined by performing R-matrix calculations in LS coupling and including spin-orbit effects through a geometric (jj/LS) frame transformation. The calculation of the required dipole matrix elements is based on the isolated core excitation approximation. All the measured quantities are accurately reproduced except the asymmetry parameters characterising the angular distributions of electrons ejected to the Sr+ 5s level, these latter quantities being found to be the most sensitive to the interseries interaction. In addition, the calculated MQDT parameters are compared with those previously obtained empirically from experiment.

R-matrix calculation of eigenchannel multichannel quantum defect parameters for strontium

Within the framework of the eigenchannel R-matrix method, theoretical calculations of the multichannel quantum defect (MQDT) parameters have been performed for 1.3Po, 1.3De and 1.3Se spectra of Sr. For the singlet spectra, both discrete and autoionisation regions below the Sr+ 4d threshold are investigated, while the analysis of the triplets is restricted to the bound spectra below the Sr+ 5s threshold. Near the Sr+5s threshold, the calculated parameters are in good agreement with those previously fitted to spectroscopic data. The main improvements with respect to the empirical MQDT studies concern evidence of the large energy dependence of the parameters and identification of isolated perturbers of the 1D2 and 1S0 spectra. In addition, the energy levels and the spectral intensities are accurately reproduced and channel mixing and correlation effects in doubly excited states are analysed in detail. A close similarity is found between the 5snp-4d np 1Po mixing in Sr and the msnp-mpns 1Po mixing in the light alkaline earths Be and Mg. Such near invariance does not hold for the 1De and 1Se spectra.

The photoionisation of the Si+(2P0) ground state: a combined application of the R-matrix and quantum defect theories

Accurate results for the photoionisation of the Si+ (3s23p 2P0) ground state are reported for the photon energy region between the first two ionisation thresholds, Si2+ (3s2 1Se, 3s3p 3P0), at 750 and 540 AA. The cross section is dominated by the autoionised 3P0 np, nf (n>or=4) Rydberg series, whose shapes and oscillator strengths are drastically perturbed by the 1P0 4p 2Se, 2De interlopers. These results are the first outcome of an elaborate calculation, based on a consistent LS multichannel description of the initial bound state and the final continua. The ejection of one valence electron is allowed for, together with flexible relaxation within the remaining valence shell, and possible monoexcitation into a near-spectroscopic orbital (3d, 4s, 4p). This extensive calculation (12 ionic states, up to 22 channels) is the first application of a new numerical approach combining the advantages of the R-matrix and the multichannel quantum defect (MQDT) theories.

Vibrational and rotational excitation of molecular ions by slow electrons

The exact asymptotic solution of the problem for scattering of slow electrons by positively charged ions M + is obtained using the multichannel quantum defect (MQD) method and results of the adiabatic approximation.