Quantum Defect Method Research Articles

An energy-modified quantum defect method for the analysis of Rydberg spectra: Application to 2-butyne.

The high resolution Rydberg absorption spectrum of 2-butyne C4H6 recorded previously at the SOLEIL synchrotron facility has been interpreted using multichannel quantum defect theory (MQDT). The calculations are based on the continuum scattering calculations of Xu et al., J. Chem. Phys. 136, 154303 (2012) and of Jacovella et al., J. Phys. Chem. A 119, 12339 (2015) pertaining to the dipole-allowed excited state symmetries in absorption from the ground state. In contrast to the traditional approach of calculating low-lying electronic states first and then attempting to extend the calculations to ever higher energy, here the analysis proceeds through the extension of these detailed calculations of the electronic continuum scattering down into the discrete region of the spectrum. The continuum reaction matrices and dipole transition moments are adapted to the discrete Rydberg region via the use of an energy-modified formulation of MQDT theory and associated energy dependences of the quantum defects. The analysis reproduces more than 40 Rydberg states from n ≈ 10 down to the 3d and 4s levels with an rms error of better than 20 cm-1. These belong to five Rydberg series with three different molecular symmetries. While the approach predicts many additional series, most of these are calculated and observed to carry only little oscillator strength. The analysis shows that the Rydberg spectrum is dominated by the excitation of an e″ symmetry electron of fδ and gπ type, in line with what previous studies of the above-threshold shape resonance of 2-butyne have shown. The present study is intended to serve as an example showing how first principles continuum calculations may be useful for the interpretation of highly bound discrete states in a range that poses problems for the standard abinitio techniques. The quantitative treatment of the dipole absorption cross sections is deferred to a future paper.

Precision Spectroscopy of Radiation Transitions between Singlet Rydberg States of the Group IIb and Yb Atoms

The measurements of microwave (μw) and radio-frequency (RF) radiation quantitative parameters may be based on the quantum–optical approach to determine the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions between opposite-parity Rydberg states nL 1L and n′L±1 1L±1 (where n′= n,n±1,n±2) of the singlet series in the alkaline–earth–metal-like atoms of group IIb (Zn, Cd, Hg) and Yb. The matrix elements determine the shifts of Rydberg-state energy levels in the field of resonance μw or RF radiation, splitting the resonance of electromagnetically induced transparency (EIT) for intensely absorbed probe radiation. Numerical computations based on the single-electron quantum defect method (QDM) and the Fues’ model potential (FMP) approach with the use of the most reliable data from the current literature on quantum defect values are performed for frequencies and matrix elements of transitions between singlet Rydberg states of 1S0-, 1P1-, 1D2-, and 1F3-series in Zn, Cd, Hg, and Yb atoms. The calculated data are approximated by polynomials in the powers of the principal quantum numbers. The polynomial coefficients are determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically calculated functions. These approximation expressions provide new possibilities for accurately evaluating the frequencies and matrix elements of dipole transitions between Rydberg states over a wide range of quantum numbers n >> 1, accompanied by the emission and absorption of μw and RF photons.

Microwave Field Metrology Based on Rydberg States of Alkali-Metal Atoms

The high-precision determination of microwave radiation parameters may be based on measurements of the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions from even-parity nS1/2 and nD5/2 to odd-parity n′P3/2 and n′F7/2 (where n′ = n, n ± 1, n ± 2) for the Rydberg states of alkali-metal atoms. The matrix elements determine the splitting of Rydberg-state energy levels in the field of a resonance microwave (μw) radiation, which results in the splitting of the resonance in electromagnetic induced transparency (EIT). Numerical computations based on the single-electron quantum defect method (QDM) and the Fues’ model potential (FMP) approach with the use of the most reliable data of the current literature on quantum defect values were performed for the 2S, 2P, 2D and 2F series of the Rydberg states of Li, Na, K, Rb and Cs atoms. The calculated data were approximated by quadratic polynomials of the principal quantum number. The polynomial coefficients were determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically presented functions. The approximation equations may be used for the accurate evaluation of the frequencies and matrix elements of μw transitions in wide ranges of the Rydberg-state quantum numbers n >> 1.

The use of photoionization cross section for evaluating contribution of continuum to the blackbody radiation induced shift and broadening of Rydberg-state energy levels of group IIb ions

General properties of n-dependence at the threshold and regularities of above-threshold ionization cross sections for states with high principal quantum numbers n and arbitrary angular momenta l are determined and used for evaluating the blackbody-radiation(BBR)-induced ionization broadening of nS, nP, nD and nF Rydberg-state energy levels of the group IIb ions Zn+, Cd+, Hg+ at temperatures from T=100 K to T=3000 K. The contributions to the energy level shifts from transitions into continuum are also determined with the use of the ionization cross sections. The single-electron quantum defect method (QDM) was used, which appears rather precise and suitable for calculating the amplitudes of radiation transitions from Rydberg states, and specifically for summation of matrix-element-dependent terms over infinite number of discrete states. The dependencies on temperature and binding energy are determined and approximate analytic equations are proposed for simplified evaluations of the BBR-induced ionization broadening and shifts. Numerical parameters of approximate equations are tabulated for S, P, D and F Rydberg states of the group IIb ions.

Broadening of energy levels of Rydberg states with small orbital angular momenta in ions of group IIb by ambient thermal radiation

We present the derivation of simple analytical expressions for calculating the widths of Rydberg states with small orbital angular momenta in Zn+, Cd+, Hg+ ions of group IIb of the periodic system of elements induced by blackbody radiation (BBR) at ambient temperatures from 100 to 3000 K. The probabilities of radiative transitions from excited nS, nP, nD and nF states to all dipole-accessible states of ions are calculated. The wave functions of the quantum defect (QD) method for the initial and final states of the Rydberg electron are used to calculate the amplitudes of transitions between bound states. The dependences of the probabilities of induced decays and excitations on the BBR temperature, the principal and orbital quantum numbers of the Rydberg ion are determined. Analytical expressions are derived for numerical estimates of the contributions of the probabilities of thermally induced decays and excitations to the width of the Rydberg energy level. The numerical values of the coefficients of interpolation polynomials representing asymptotic expansions in powers of the principal quantum number n for the relative probabilities of decays and excitations of Rydberg states with large values of n and small orbital angular momenta l = 0, 1, 2, 3 are calculated.

On the Two-Phonon Relaxation of Excited States of Boron Acceptors in Diamond

The relaxation of holes from excited states of boron acceptors in diamond with the emission of two optical phonons is studied theoretically. To describe the wave function of acceptor states, an electron-like Hamiltonian with an isotropic effective mass is used. The wave function of the ground state is determined by the quantum-defect method. The probability of the transition is calculated in the adiabatic approximation. It is assumed that the phonon dispersion law is isotropic and the phonon frequency is quadratically dependent on the wave-vector modulus, with the maximum and minimum frequencies ωmax and ωmin at the center and boundary of the Brillouin zone, respectively. A high sensitivity of the probability of the transition to the characteristic of phonon dispersion ωmax – ωmin is revealed, especially for the transition with the energy ET in the range 2ℏωmin ≤ ET 1012 s–1) in the “resonance” range ℏωmin + ℏωmax ≤ ET ≤ 2ℏωmax.

Двухфононная релаксация возбужденных состояний акцепторов бора в алмазе

The relaxation of holes in the excited states of boron acceptors in diamond is theoretically investigated when two optical phonons are emitted. An electron-like Hamiltonian with an isotropic effective mass was used to describe the wave function of the acceptor states. The wave function of the ground state was found by the quantum defect method. The transition probability was calculated in adiabatic approximation. The dispersion law of phonons was considered isotropic, and the frequency of phonons is quadratically dependent on the wave vector module, with the maximum value, omegamax, achieved in the center of the Brillouin zone, and the minimum, omegamin, at the edge of the Brillouin zone. The high sensitivity of the rates to the characteristic of phonon dispersion, omegamax-omegamin, especially for the transition energy, ET, in the 2homegamin≤ ET<homegamin +homegamax interval was revealed. Depending on the transition energy and dispersion of phonons, the rate of two-phonon relaxation in the low-temperature limit varies from ultra-low values (less than 108 s-1) near the threshold, ET=2homegamin, to ultra-high values (more than 1012 s-1) in the "resonance" region homegamin+homegamax≤ ET≤2homegamax.

Multiphonon Intracenter Relaxation of Boron Acceptor States in Diamond

The relaxation rates are calculated in the adiabatic approximation, in which the steady-state impurity states are taken to be electronic-vibrational (vibronic) states. The probabilities of transitions between these states with the emission (or absorption) of one or several phonons are calculated in first-order perturbation theory on the assumption that the transitions are a result of the violation of adiabaticity. The electron part of the wave function of the vibronic state is described by a simple Hamiltonian with an isotropic effective mass. The wave function of the ground state is determined by the quantum defect method. According to the calculations, a hole relaxes from the excited boron acceptor state, whose energy is 304 meV higher than the energy of the ground state, to the ground state with the emission of two optical phonons with a rate of ~1011 s–1. This value is an estimate from above, since the model of nondispersive optical phonons used in the study overestimates the number of phonon modes, whose participation in relaxation is allowed by the energy conservation law. However, despite the rough approximation, it can be concluded that the multiphonon relaxation of boron acceptor states in diamond is a fast process.

Многофононная внутрицентровая релаксация состояний акцепторов бора в алмазе

AbstractThe relaxation rates are calculated in the adiabatic approximation, in which the steady-state impurity states are taken to be electronic-vibrational (vibronic) states. The probabilities of transitions between these states with the emission (or absorption) of one or several phonons are calculated in first-order perturbation theory on the assumption that the transitions are a result of the violation of adiabaticity. The electron part of the wave function of the vibronic state is described by a simple Hamiltonian with an isotropic effective mass. The wave function of the ground state is determined by the quantum defect method. According to the calculations, a hole relaxes from the excited boron acceptor state, whose energy is 304 meV higher than the energy of the ground state, to the ground state with the emission of two optical phonons with a rate of ~10^11 s^–1. This value is an estimate from above, since the model of nondispersive optical phonons used in the study overestimates the number of phonon modes, whose participation in relaxation is allowed by the energy conservation law. However, despite the rough approximation, it can be concluded that the multiphonon relaxation of boron acceptor states in diamond is a fast process.

The argon spectrum in the range of 1200–2000 cm–1

Atomic argon spectrum in the range of 1200–2000 cm–1 has been recorded by time-resolved Fourier transform spectroscopy. The lines have been identified using the oscillator strengths calculated by the quantum-defect method. Previously unknown energies of the 6h levels of Ar I have been found from the spectrum.

CHARGE EXCHANGE-INDUCED X-RAY EMISSION OF Fe xxv AND Fe xxvi VIA A STREAMLINED MODEL

ABSTRACT Charge exchange (CX) is an important process for the modeling of X-ray spectra obtained by the Chandra, XMM-Newton, and Suzaku X-ray observatories, as well as the anticipated Astro-H mission. The understanding of the observed X-ray spectra produced by many astrophysical environments is hindered by the current incompleteness of available atomic and molecular data—especially for CX. Here, we implement a streamlined program set that applies quantum defect methods and the Landau–Zener theory to generate total, n-resolved, and -resolved cross sections for any given projectile ion/target CX collision. By using these data in a cascade model for X-ray emission, theoretical spectra for such systems can be predicted. With these techniques, Fe25+ and Fe26+ CX collisions with H, He, H2, N2, H2O, and CO are studied for single-electron capture (SEC). These systems have been selected because they illustrate computational difficulties for high projectile charges. Furthermore, Fe xxv and Fe xxvi emission lines have been detected in the Galactic center and Galactic ridge. Theoretical X-ray spectra for these collision systems are compared to experimental data generated by an electron-beam ion trap study. Several ℓ-distribution models have been tested for Fe25+ and Fe26+ SEC. Such analyses suggests that commonly used ℓ-distribution models struggle to accurately reflect the true distribution of electron capture as understood by more advanced theoretical methods.

Blackbody-radiation-induced shifts and the broadening of Rydberg states in the ions of group IIa elements

The rates of blackbody-radiation(BBR)-induced transitions from excited nS, nP, nD and nF states into bound states and into continua of the group IIa ions Ca+, Sr+, Ba+ at temperatures from T = 100 K to T = 3000 K are calculated in consecutive comparison of the Fues model potential (FMP) with quantum defect method (QDM). The temperature and binding-energy dependencies are determined and analytical asymptotic approximations are proposed for evaluating numerically the contributions of the bound states and continua to the Rydberg-level widths and positions. The general properties of n-dependence at the threshold and regularities of above-threshold ionization cross sections for states with high principal quantum numbers n are used for determining the rates of BBR-induced ionization. Relations are determined between deviations from general regularities for the sum rules of oscillator- strengths and specific non-local dependence of the FMP on optical-electron angular momentum. The QDM appears preferable in comparison with the FMP approach for calculating the amplitudes of radiation transitions from Rydberg states, and specifically sums of matrix-element-dependent terms over very large (in particular, infinite) numbers of discrete states. The fractional (related to natural decay) rates of BBR-induced decay into lower bound states, excitation into upper bound states and ionization transitions into the continuum are presented asymptotically with parameters tabulated for the S, P, D and F Rydberg states.

FTIR laboratory measurement of Ne i Rydberg states in 1.43−14.3μm spectral range

Context. The Ne i spectrum in the 800−1800 cm −1 (5.55−14.3 μm) domain has not been previously investigated. Aims. To address this spectral gap, we performed high-resolution laboratory measurements of the Ne i emission spectrum over the 800−7000 cm −1 (1.43−14.3 μm) range. Methods. The Ne i laboratory spectra were obtained from a pulsed discharge plasma and recorded using time-resolved Fourier transform spectroscopy. The identification of the spectral lines was performed according to the transition probability values calculated using the quantum defect method. Results. Based on these measurements, we report the wavenumbers of 151 Ne i lines and energy values for 14 Ne i levels that are not listed in the available databases.

The calculation method of interaction between metal atoms under influence of the radiation

A method of calculation of interatomic interaction potentials in the presence of ionized states has been developed. They have been obtained for the atoms with different ionization degree on example of aluminum. The Heine-Abarenkov-Animalu model potential form factors was employed. The form factor parameters of ionized atoms was determined on the base of the quantum defect method using the atomic-spectroscopy data. The potential of interatomic interaction for different charged states with different degree of ionization were determined.

Valence-shell single photoionization of chlorine-likeK2+ions: Experiment and theory

The absolute single photoionization cross-section was measured for Cl-like K$^{2+}$ over the photon energy range from 44.2 - 69.7 eV at a constant energy resolution of 0.045 eV. The experiments were performed by merging an ion beam with a beam of synchrotron radiation from an undulator. The ground-state ionization threshold was measured at 0.004 eV energy resolution to be 45.717 $\pm$ 0.030 eV. The measurements are rich in resonance structure due to multiple Rydberg series of transitions to autoionizing states. These series are assigned spectroscopically using the quantum defect method, guided by pseudo-relativistic Hartree-Fock calculations for the energies and oscillator strengths of transitions to autoionizing states. The experimental results, which include significant contributions from K$^{2+}$ ions initially in metastable states, are in satisfactory agreement with a linear superposition of semi-relativistic R-matrix calculations of photoionization cross sections from these initial states.

HYPERFINE-CHANGING TRANSITIONS IN3He II AND OTHER ONE-ELECTRON IONS BY ELECTRON SCATTERING

We consider the spin-exchange (SE) cross section in electron scattering from $^3$He\,{\scriptsize II}, which drives the hyperfine-changing \hbox{3.46 cm} (8.665 GHz) line transition. Both the analytical quantum defect method --- applicable at very low energies --- and accurate R-matrix techniques for electron-He$^+$ scattering are employed to obtain SE cross sections. The quantum defect theory is also applied to electron collisions with other one-electron ions in order to demonstrate the utility of the method and derive scaling relations. At very low energies, the hyperfine-changing cross sections due to e$-$He$^+$ scattering are much larger in magnitude than for electron collisions with neutral hydrogen, hinting at large rate constants for equilibration. Specifically, we obtain rate coefficients of $K(10\,{\rm K}) = 1.10 \times 10^{-6}\,\rm cm^3/s$ and $K(100\,{\rm K}) = 3.49\times 10^{-7}\,\rm cm^3/s$.

Theoretical study on intensities in the discrete and continuum electronic spectrum of acetone

Intensities, expressed as oscillator strengths for the discrete and, as cross-sections for the continuum electronic spectrum of acetone are reported. The calculations have been performed with the molecular quantum defect method (MQDO), which has proved to be a very adequate tool for this type of studies in other organic molecules. The present work covers transitions to more highly excited Rydberg states than had previously been reported. The quality of the oscillator strengths has been assessed by comparison with scarce data available in the literature, and by its compliance with the expected trends along the Rydberg series. Electronic partial photoionization cross-sections for acetone over the 10–50eV photon energy are also calculated. Cross-section profiles for Rydberg series that constitute the ionization channels of acetone from the outermost orbital in its ground state are reported for the first time.

Energy Levels and Atomic Lifetimes of Rydberg States in Neutral Indium

(Received December 22, 2011; revised version July 24, 2012; in nal form September 25, 2012) Atomic lifetimes and energy levels were calculated using weakest bound potential model theory for 5sns S1/2 (n ≥ 6), 5snp P 0 1/2 (n ≥ 5), 5snp P 0 3/2 (n ≥ 5), 5snd D 3/2 (n ≥ 5), 5snd D 5/2 (n ≥ 5) series of the Rydberg states in neutral indium. The use of the quantum defect method and Martin's expression allowed us to supply lifetime values along by means of the series above. Some lifetimes and energy values not existing in the literature for high Rydberg states in neutral indium atom were obtained using this method. Our results nicely agree with the available experimental results and theoretical results.

Precision spectroscopy of high rotational states in H2investigated by Doppler-free two-photon laser spectroscopy in theEF 1Σg+–X 1Σg+system

Recently a high precision spectroscopic investigation of the EF^1\Sigma_g^+ - X^1\Sigma_g^+ system of molecular hydrogen was reported yielding information on QED and relativistic effects in a sequence of rotational quantum states in the X^1\Sigma^+_g$ ground state of the H_2 molecule [E.J. Salumbides et al., Phys. Rev. Lett. 107, 043005 (2011)]. The present paper presents a more detailed description of the methods and results. Furthermore, the paper serves as a stepping stone towards a continuation of the previous study by extending the known level structure of the EF^1\Sigma^+_g state to highly excited rovibrational levels through Doppler-free two photon spectroscopy. Based on combination differences between vibrational levels in the ground state, and between three rotational branches (O, Q and S branches) assignments of excited EF^1\Sigma^+_g levels, involving high vibrational and rotational quantum numbers, can be unambiguously made. For the higher EF^1\Sigma^+_g levels, where no combination differences are available, calculations were performed using the multi-channel quantum defect method, for a broad class of vibrational and rotational levels up to J=19. These predictions were used for assigning high-J EF-levels and are found to be accurate within 5 cm^-1.

Photoionization absorption and zero-field spin splitting of acceptor-bound magnetic polaron in p-type Hg1-xMnxTe single crystals

Temperature-dependent magnetic (2–300 K), DC Hall (10–300 K), and infrared transmission (11.5–300 K) measurements are performed on a series of p-type Hg1-xMnxTe (0.12 ≤ x ≤ 0.26) single crystals in the spin-glass regime. Photoionization absorption (PIA) of acceptor-bound magnetic polarons (acceptor-BMPs) is observed to evolve with temperature, which is better accounted for by the classical oscillator model than by the quantum defect method. At low temperatures, p-type Hg1−xMnxTe manifests distinct phenomena of paramagnetic enhancement, negative magnetoresistance, and decrease of the effective binding energy and blueshift of the PIA of the acceptor-BMPs with nearly the same degree as temperature declines. A spin-splitting model is proposed, which can well reproduce the experimentally observed zero-field spin splitting of the acceptor-BMP level at low temperatures and the increase of the spin splitting as temperature drops. The results suggest that the acceptor-BMPs in Hg1−xMnxTe may have potential applications in light-driven polaronic memories, tunable far-infrared lasers, and detectors.