Single-channel Quantum Defect Theory Research Articles

Millimeter-wave spectroscopy of Zn I in triplet F Rydberg states

We continue our experiments on millimeter-wave spectroscopy of higly-excited Rydberg atoms with two valence electrons. In this work we have measured frequencies of 16 transitions between Rydberg triplet 4snf3FJ and singlet 4snd1D2, 4sng1G4 levels of neutral zinc (Zn I) in the frequency range 87–141 GHz and in the principal quantum number range n=30–37. The observed transitions were driven predominantly from triplet 4snf3F3 levels that were excited by a three-step laser excitation scheme 4s21S0 → 4s4p3P1 → 4s4d3D2 → 4snf3F3. The measured frequencies as well as those obtained in our previous work (Kutsenko et al., 2015, J. Phys. B: At. Mol. Opt. Phys. 48, 245,005) were used to derive the Ritz expansion coefficients for triplet F states in the framework of the single-channel quantum defect theory.

Millimeter-wave spectroscopy of Zn I in 1D2, 1F3 and 1G4 Rydberg states

We have measured the frequencies of 24 transitions between Rydberg singlet L = 2–4 levels of zinc in the frequency range 79–285 GHz. The transitions were driven from 4snf 1F3 levels that were excited by two-step laser excitation through 4s4d 1D2. Single-photon 4snf 1F3 4s(n + 2)d 1D2, 4snf 1F3 4s(n + 3)d 1D2, 4snf 1F3 4s(n + 1)g 1G4, and two-photon 4snf 1F3 4s(n + 1)f 1F3 series for principal quantum number n = 28–40 were observed. Ritz expansion coefficients in the single-channel quantum defect theory were obtained for D, F and G states by least square fitting analysis of the spectra. An estimate of the dipole polarizability of Zn+ has been obtained.

Energy levels of 1s2n d (n ≤ 9) states for lithium-like ions

The full-core plus correlation method with multi-configuration interaction wave functions is extended to the calculation of the non-relativistic energies of 1s2n d (n ≤ 9) states for the lithium isoelectronic sequence from Z = 11 to 20. Relativistic and mass-polarization effects on the energy are calculated as the first-order perturbation correction. The quantum-electrodynamics correction is also included. The fine structure splittings are determined from the expectation values of spin—orbit and spin—other-orbit interaction operators in the Pauli-Breit approximation. Combining the term energies of lowly excited states obtained with the quantum defects calculated by the single channel quantum defect theory, each of which is a smooth function of energy and approximated by a weakly varying function of energy, the ion potentials of highly excited states (n ≥ 6) are obtained with the semi-empirical iteration method. The results are compared with experimental data in the literature and found to be closely consistent with the regularity.

Transition energy and dipole oscillator strength of 1s23d-1s2 nf transitions for Sc18+ ion

The transition energies of the 1s23d-1s2nf (4⩽n⩽9) transitions and fine structure splittings of 1s2nf (n⩽9) states for Sc18+ ion are calculated with the full-core plus correlation method. The quantum defect of 1s2nf series is determined by the single-channel quantum defect theory. The energies of any highly excited states with n⩾10 for this series can be reliably predicted using the quantum defect as function of energy. Three alternative forms of the dipole oscillator strengths for the 1s23d-1s2nf (n⩽9) transitions of Sc18+ ion are calculated with the transition energies and wave functions obtained above. Combining the quantum defect theory with the discrete oscillator strengths, the discrete oscillator strengths for 1s23d-1s2nf (n > 9) transitions and the oscillator strengths densities corresponding to the bound-free transitions are obtained.

Energy and Oscillator Strength of V20+ Ion

The ionization potentials and fine structure splittings of 1s2nl (l = s, p, and d; n ⩽ 9) states for lithium-like V20+ ion are calculated by using the full-core plus correlation (FCPC) method. The quantum defects of these three Rydberg series are determined according to the single-channel quantum defect theory (QDT). The energies of any highly excited states with n ⩾ 10 for these series can be reliably predicted using the quantum defects that are function of energy. The dipole oscillator strengths for the 1s22s–1s2np and 1s22p–1s2nd (n ⩽ 9) transitions of V20+ ion are calculated with the energies and FCPC wave functions obtained above. Combining the QDT with the discrete oscillator strengths, the discrete oscillator strengths for the transitions from the given initial state to highly excited states (n ⩾ 10) and the oscillator strength density corresponding to the bound-free transitions are obtained.

Transition energy and oscillator strength of Sc 18+ ion

The transition energies and the dipole oscillator strengths for the 1s22s–1s2np (2 ≤n≤ 9) and 1s22p–1s2nd (3 ≤n≤ 9) of lithium-like Sc18+ ion are calculated by using the full core plus correlation method. The fine structure splittings of 1s2np and 1s2nd (n≤ 9) states are determined from the expectation values of spin-orbit and spin-other-orbit interaction operators. The quantum defects of these series, as function of principal quantum number n, are obtained. The agreement between the f-values obtained from three alternative formulae is excellent. Comparisons of our results with experimental data available in the literature are carried out. Combining the single-channel quantum defect theory with the discrete oscillator strengths obtained in this work, this paper obtains the behaviour of discrete oscillator strengths and oscillator strength densities corresponding to the bound-free transitions adjacent to ionization threshold.

Oscillator strengths for 2 2 P- n 2 D transitions of lithium-like systems with Z =11 to 20

The dipole-length, velocity and acceleration absorption oscillator strengths for the 1s22p–1s2nd (3 ⩽n⩽9) transitions of the lithium-like systems with Z=11 to 20 are calculated using the energies and the multiconfiguration interaction wavefunctions obtained from a full core plus correlation method. The agreement between the f-values calculated from the length, velocity and acceleration formulae is excellent. Our results agree closely with the experimental data available in the literature. Combining these discrete oscillator strengths with the single-channel quantum defect theory, the discrete oscillator strengths for the transitions from 1s22p state to highly excited states (n⩾10) and the oscillator strength densities corresponding to the bound-free transitions are obtained.

Exact analytic relation between quantum defects and scattering phases with applications to Green's functions in quantum defect theory

The relation between the quantum defects, µ λ , and scattering phases, δ λ , in the single-channel Quantum Defect Theory (QDT) is discussed with an emphasis on their analyticity properties for both integer and noninteger values of the orbital angular momentum parameter λ. To derive an accurate relation between µ λ and δ λ for asymptotically-Coulomb potentials, the QDT is formally developed for the Whittaker equation in its general form “perturbed” by an additional short-range potential. The derived relations demonstrate that µ λ is a complex function for above-threshold energies, which is analogous to the fact that δ λ is complex for below-threshold energies. The QDT Green’s function, G λ , of the “perturbed” Whittaker equation is parameterized by the functions δ λ and µ λ for the continuous and discrete spectrum domains respectively, and a number of representations for G λ are presented for the general case of noninteger λ. Our derivations and analyses provide a more general justification of known results for nonrelativistic and relativistic cases involving Coulomb potentials and for a Coulomb plus point dipole potential.

Oscillator strengths for 3 [formula omitted]D– [formula omitted]F transitions of the lithium isoelectronic sequence

The nonrelativistic dipole-length, -velocity and -acceleration absorption oscillator strengths for the 1s 23d–1s 2 nf (4≤ n≤9) transitions of the lithium isoelectronic sequence up to Z=10 are calculated by using the energies and the multiconfiguration interaction wave functions obtained from a full core plus correlation (FCPC) method. In most cases, the agreement between the f-values from the length and velocity formulae is up to forth or fifth digits. Combining these discrete oscillator strengths with the single channel quantum defect theory (QDT), the discrete oscillator strengths for the transitions from 1s 23d state to highly excited levels ( n≥10) and the oscillator strength densities corresponding to the bound-free transitions are obtained.

Oscillator strengths for 2 transitions of the lithium isoelectronic sequence

The nonrelativistic dipole-length, -velocity and -acceleration absorption oscillator strengths for the 1s22s-1s2np\(\) transitions of the lithium isoelectronic sequence up to Z=10 are calculated by using the energies and the multiconfiguration interaction wave functions obtained from a full core plus correlation (FCPC) method. In most cases, the agreement between the f-values from the length and velocity formulae is up to forth or fifth digit. Combining these discrete oscillator strengths with the single channel quantum defect theory (QDT), the discrete oscillator strengths for the transitions from the 1s22s state to highly excited levels (\(\)) and the oscillator strength densities corresponding to the bound-free transitions are obtained.

Oscillator strengths for 2

The nonrelativistic dipole-length, -velocity and -acceleration absorption oscillator strengths for the transitions of the lithium isoelectronic sequence up to Z=10 are calculated by using the energies and the multiconfiguration interaction wave functions obtained from a full core plus correlation (FCPC) method. In most cases, the agreement between the f -values from the length and velocity formulae is up to forth or fifth digit. Combining these discrete oscillator strengths with the single channel quantum defect theory (QDT), the discrete oscillator strengths for the transitions from the state to highly excited levels () and the oscillator strength densities corresponding to the bound-free transitions are obtained.

Radiative properties of diatomic Rydberg states in quantum defect theory. Application to the hydrogen molecule

Single-channel quantum defect theory (QDT) is used to approximate the Born-Oppenheimer electronic wavefunction of Rydberg diatomic states. The development is based on an extension of quantum-mechanical and semiclassical approaches for the estimation of molecular transition dipole matrix elements as a function of internuclear distance R. The theory is tested by a calculation of the singlet and triplet transition moments of the molecule involving Rydberg states with n = 2-4; l = 0-2. For small and moderate internuclear distance the agreement with ab initio data is found to be generally good, and in some cases identical results are obtained. The disagreement is attributed to a breakdown of the one-channel QDT representation of the interacting electronic states for large R values. The predicted moments for - and - transitions remove existing discrepancies between theoretical and accurate experimental lifetimes for and states.

Theory of oscillator strength of the lithium isoelectronic sequence

Using the energies and the multiconfiguration interaction wavefunctions obtained from a full core plus correlation method, the nonrelativistic dipole-length, -velocity and -acceleration absorption oscillator strengths for the transitions are calculated. In most cases, the agreement between the f-values from the length and velocity formulae is up to the fourth or fifth digit. Our acceleration f-values also agree closely with the length and velocity f-values. Combining these discrete oscillator strengths with the single-channel quantum defect theory, the discrete oscillator strengths for the transitions from the state to highly excited levels and the oscillator strength densities corresponding to the bound-free transitions are obtained.

Analytical approximations for adiabatic and non-adiabatic matrix elements of homonuclear diatomic Rydberg states. Application to the singlet p-complex of the hydrogen molecule

Closed expressions for the adiabatic and non-adiabatic matrix elements for coupling between the Rydberg states of diatomic molecules are derived by a combination of single-channel quantum defect theory and semiclassical mechanics. Validity conditions on the derived results and limiting behaviours as are discussed. Applications of the theory to low-lying and states of the singlet p-complex of hydrogen are in good agreement with ab initio results.

Stark mapping of H2 Rydberg states in the strong-field regime with dynamical resolution.

We have acquired spectra of high Rydberg states of molecular hydrogen in a static external field, in the energy region from below the energy at which field ionization becomes classically possible (${\mathit{E}}_{\mathit{c}}$) to well above this energy. Simultaneous spectra of ionization and dissociation were acquired, thereby allowing direct information on the excited-state decay dynamics to be obtained. We have found that states with energies below ${\mathit{E}}_{\mathit{c}}$ undergo field-induced predissociation, while states with energies well above ${\mathit{E}}_{\mathit{c}}$ decay predominantly by field ionization. Field ionization and dissociation compete effectively as decay channels for states with energies in a restricted region just above ${\mathit{E}}_{\mathit{c}}$. Comparison of our ionization spectra to the results of a single-channel quantum-defect theory Stark calculation shows quantitative agreement except near curve crossings, indicating that inclusion of different core rotational state channels will be required to properly account for coupling between the Stark states. Several states in the spectra undergo pronounced changes in their dynamical properties over a narrow range of field values, which we interpret as being due to interference cancellation of the ionization rates for these states.

Single-channel quantum-defect theory for the description of doubly excited states of helium.

A single-channel quantum-defect theory for the doubly excited states of helium is formulated in hyperspherical coordinates. For doubly excited states associated with each excited ${\mathrm{He}}^{+}$(N) threshold, the outer electron experiences asymptotically a combined Coulomb and dipole potential, -1/R+\ensuremath{\alpha}/${\mathit{R}}^{2}$. The two independent solutions in the asymptotic region for energies below and above the threshold are explicitly given for both the attractive and the repulsive dipole potentials. Quantum defects for $^{1}$P\ifmmode^\circ\else\textdegree\fi{} doubly excited states of five channels in the N=3 manifold and of the dominant channels below the N=2, 3, and 4 thresholds of ${\mathrm{He}}^{+}$ are calculated in a single-channel hyperspherical potential, and the results are compared with quantum defects extracted from other theoretical calculations and recent experimental results. Procedures for obtaining diabatic curves from the adiabatic potentials are also discussed.

Schrödinger-like equation for relativistic particles

The Dirac equation in a spherically symmetric screened Coulomb potential is transformed to a modified Schrödinger equation of the form d2u/dr2+k2(r)u=0. This transformation is induced by expressing the Dirac function as a linear combination of the function u and its derivative du/dr. Various properties of the transformation and of the resulting equations are studied. The close similarity between the modified Schrödinger equation and the Schrödinger equation suggests that methods applied to the Schrödinger equation to derive nonrelativistic relations can be applied to the modified Schrödinger equation to derive the analogous relativistic relations. As an example, this approach is applied to the single channel quantum defect theory to give a new derivation of its relativistic form.

Quantum defect theory in a double-valley potential

Single-channel quantum defect theory is extended to handle rapid variations of the quantum defect in a case where the functional form can be inferred, namely when a shape resonance occurs among the bound states, and an experimental example is analysed.