Wigner Threshold Law Research Articles

Universal trimers with p-wave interactions and the faux-Efimov effect

Abstract An unusual class of equal mass p-wave universal trimers with symmetry L Π = 1 ± is identified, for both a two-component fermionic trimer with s- and p-wave scattering length close to unitarity and for a one-component fermionic trimer at p-wave unitarity. Moreover, fermionic trimers made of atoms with two internal spin components are found for L Π = 1 ± , when the p-wave interaction between spin-up and spin-down fermions is close to unitarity and/or when the interaction between two spin-up fermions is close to the p-wave unitary limit. The universality of these p-wave universal trimers is tested here by considering van der Waals interactions in a Lennard–Jones potential with different numbers of two-body bound states; our calculations also determine the value of the scattering volume or length where the trimer state hits zero energy and can be observed as a recombination resonance. The faux-Efimov effect appears with trimer symmetry L Π = 1 − when the two fermion interactions are close to p-wave unitarity and the lowest 1 / R 2 coefficient gets modified, thereby altering the usual Wigner threshold law for inelastic processes involving three-body continuum channels.

Bond dissociation energy of N2 measured by state-to-state resolved threshold fragment yield spectra.

The precise determination of the bond dissociation energy of N2 is crucial for thermochemistry database and theoretical calculations. However, there has been ongoing debate regarding its exact value. In this study, we used the velocity map imaging method combined with an extreme ultraviolet laser to measure the threshold fragment yield (TFY) spectra of N2 in the N(2D) + N(2D) photodissociation channels. By integrating the signals within a small circular area on the fragment velocity map images, we were able to obtain TFY spectra at nine different dissociation thresholds. These spectra are rotational state-resolved for the N2(J″) molecules and spin-orbit state-resolved for the dissociation channels involving N(2D) fragments. By employing the Wigner threshold law to simulate the TFY spectra and conducting statistical analysis on the comprehensive dataset, we determined the N2 bond dissociation energy to be 78 691.09 ± 0.15cm-1. This work now places N2 among the few diatomic molecules with bond dissociation energies measured at sub-wavenumber precision.

Resonances in the Hulthén potential: benchmark calculations, critical behaviors, and interference effects

We perform benchmark calculations of resonance states in the Hulthén potential by employing the uniform complex-scaling generalized pseudospectral method. Complex resonance energies for states with the lowest four orbital angular momenta are reported for a wide range of screening parameters where their positions lie above the threshold. Our results are in good agreement with previous J-matrix predictions, but differ significantly from the complex-scaling calculations based on oscillator basis set. By tracing the resonance poles via bound-resonance transition as the screening parameter increases, we successfully identify the electronic configurations of the numerically obtained resonances. The asymptotic laws for resonance position and width near the critical transition region are extracted, and their connections with the bound-state asymptotic law and Wigner threshold law, respectively, are disclosed. We further find that the birth of a new resonance will distort the trajectories of adjacent higher-lying resonances, while even if two resonances are exactly degenerate in real energy position, they can still be treated as near-isolated resonances provided their widths are significantly different in magnitude.

Low-energy scattering parameters: A theoretical derivation of the effective range and scattering length for arbitrary angular momentum

The most important parameters in the study of low-energy scattering are the s-wave and p-wave scattering lengths and the s-wave effective range. We solve the scattering problem and find two useful formulas for the scattering length and the effective range for any angular momentum, as long as the Wigner threshold law holds. Using that formalism, we obtain a set of useful formulas for the angular-momentum scattering parameters of four different model potentials: hard-sphere, soft-sphere, spherical well, and well-barrier potentials. The behavior of the scattering parameters close to Feshbach resonances is also analyzed. Our derivations can be useful as hands-on activities for learning scattering theory.

Influence of a Supercritical Electric Dipole Moment on the Photodetachment of C_{3}N^{-}.

Threshold photodetachment spectroscopy of the molecular ion C_{3}N^{-} has been performed at both 16(1) and 295(2) K in a 22-pole ion trap. The 295(2) K spectrum shows a large increase in the cross section with an onset about 200 cm^{-1} below threshold, which is explained by significant vibrational excitation of the trapped ions at room temperature. This excitation disappears at cryogenic temperatures leading to an almost steplike onset of the cross section at threshold, which cannot be adequately described with a Wigner threshold law. Instead, we show that the model developed by O'Malley for photodetachment from neutrals with large permanent dipoles [Phys. Rev. 137, A1668 (1965)PHRVAO0031-899X10.1103/PhysRev.137.A1668] fits very well to the data. A high-resolution scan of the threshold region yields additional features, which we assign to the rotational P and R branches of an electronic transition to a dipole-bound state with ^{1}Σ^{+} symmetry. This state is found 2(1) cm^{-1} below threshold in very good agreement with a recent computational prediction. We furthermore refine the value of the electron affinity of C_{3}N to be 34 727(1) cm^{-1}.

Measurement of electron affinity of iridium atom and photoelectron angular distributions of iridium anion

The latest electron affinity value of an iridium atom is 1.564 36(15) eV, determined via a method based on the Wigner threshold law by Bilodeau and co-workers. However, they observed a significant deviation from the Wigner threshold law in the threshold photodetachment experiment. To address this dilemma, we conducted high-resolution photoelectron spectroscopy of Ir- via the slow-electron velocity-map imaging method in combination with an ion trap. The electron affinity of Ir was measured to be 12 614.97(9) cm-1 or 1.564 057(11) eV. We find that the Wigner threshold law is still valid for the threshold photodetachment of Ir- through a p-wave fitting of the photodetachment channel Ir-5d86s23F4→Ir5d86sb4F9/2. The photoelectron angular distributions of photodetachment channels Ir-5d86s23F4→Ir5d76s2a4F9/2 and Ir-5d86s23F4→Ir5d86sb4F9/2 were also investigated. The behavior of anisotropy parameter β indicates a strong interaction between the two channels. Moreover, the energy level 3P2 of Ir-, which was not observed in the previous works, was experimentally determined to be 4163.24(16) cm-1 above the ground state.

A new accurate potential energy surface for HeTiO system and rotational quenching of TiO due to He collisions

A new accurate ab initio potential energy surface (PES) of the HeTiO system is constructed. The multi-reference configuration interaction with the Davidson correction is used to obtain the energy points, and the neural network method is employed to fit the PES. State-to-state rotational quenching cross sections and rate coefficients are obtained over a wide range based on the PES. Wigner threshold law is proved to be valid in the ultralow energy regime. In comparison with 3He, 4He is more efficient for the TiO molecule cooling process in the dominant transition j = 1 → 0 due to the isotope effect.

Channel-closing effects in strong-field ionization by a bicircular field

Channel-closing effects, such as threshold anomalies and resonantlike intensity-dependent enhancements in strong-field ionization by a bicircular laser field are analyzed. A bicircular field consists of two coplanar corotating or counter-rotating circularly polarized fields having different frequencies. For the total detachment rate of a negative ion by a bicircular field we observe threshold anomalies and explain them using the Wigner threshold law and energy and angular momentum conservation. For the corotating bicircular case, these effects are negligible, while for the counter-rotating case they are pronounced and their position depends on the magnetic quantum number of the initial state. For high-order above-threshold ionization of rare-gas atoms by a counter-rotating bicircular laser field we observe very pronounced intensity-dependent enhancements. We find all four types of threshold anomalies known from collision theory. Contrary to the case of linear polarization, channel-closing effects for a bicircular field are visible also in the cutoff region of the electron energy spectrum, which is explained using quantum-orbit theory.

Two-body loss rates for reactive collisions of cold atoms

We present an effective two-channel model for reactive collisions of cold atoms. It augments elastic molecular channels with an irreversible, inelastic loss channel. Scattering is studied with the distorted-wave Born approximation and yields general expressions for angular momentum resolved cross sections as well as two-body loss rates. Explicit expressions are obtained for piece-wise constant potentials. A pole expansion reveals simple universal shape functions for cross sections and two-body loss rates in agreement with the Wigner threshold laws. This is applied to collisions of metastable ${^{20}}$Ne -- ${^{21}}$Ne atoms, which decay primarily through exothermic Penning-, or Associative- ionization processes. From a numerical solution of the multichannel Schr\"odinger equation using the best currently available molecular potentials, we have obtained synthetic scattering data. Using the two-body loss shape functions derived in this paper, we can match these scattering data very well.

Rotational quenching of CS in ultracold 3He collisions

Quantum mechanical scattering calculations of rotational quenching of CS (v=0) collision with 3He are performed at ultracold temperatures and results are compared with isotopic 4He collision. Rotational quenching cross sections and rate coefficients have been calculated in the ultracold region for rotational levels up to j=10 using the He–CS potential energy surface computed at the CCSD(T)/aug-cc-pVQZ level of theory. The quenching cross sections are found to be two orders of magnitude larger for the 3He than the 4He isotope under ultracold conditions. Wigner threshold law is found to be valid below 10−3K temperature.

Quantum Mechanical Rate Coefficient of Formation of HD Molecule at Ultracold Temperatures: Its Importance in Interstellar Cooling

Molecular hydrogen and its isotope HD acted as one of the most important interstellar coolants in the primordial gas medium. In this paper, we present accurate time-independent quantum mechanical (TIQM) rate coefficients of formation of ultracold HD molecules by \({\rm D} +{\rm H}_2(v,j)\to {\rm HD}(v', j')+{\rm H}\) reaction at very low collision energy. State resolved integral cross sections between different rotational \((j)\) and vibrational \((v)\) levels and corresponding Boltzmann-averaged thermal rate coefficients are computed between temperature \(\rm T = 10^{-8}K\)-\(\rm 10K\). We found the exponential decrease of the rate coefficients with reducing temperature following Arrhenius' empirical equation is not valid at ultracold temperature limit. At lower temperatures, the rate coefficients become independent of temperature (constant) and Wigner's threshold laws are obeyed. Since cooling of the primordial gases lead to the formation of the first structures of the universe, inclusion of the accurate low-temperature rate coefficients will lead to improved modeling for the evolution of the early universe.

Reactive scattering calculations for (87)Rb+(87)RbHe→Rb2((3)Σ(u)(+),v)+He from ultralow to intermediate energies.

We investigate atom-diatom reactive collisions, as a preliminary step,in order to assess the possibility of forming Rb(2) molecules in their lowest triplet electronic state by cold collisions of rubidium atoms on the surface of helium nanodroplets [corrected]. A simple model related to the well-known Rosen treatment of linear triatomic molecules [N. Rosen, J. Chem. Phys. 1, 319 (1933)] in relative coordinates is used, allowing to estimate reactive probabilities for different values of the total angular momentum. The best available full dimensional potential energy surface [Guillon et al., J. Chem. Phys. 136, 174307 (2012)] is employed through the calculations. Noticeable values of the probabilities in the ultracold regime, which numerically fulfill the Wigner threshold law, support the feasibility of the process. The rubidium dimer is mainly produced at high vibrational states, and the reactivity is more efficient for a bosonic helium partner than when the fermion species is considered.

Reactive collisions in confined geometries

We consider low energy threshold reactive collisions of particles interacting via a van der Waals potential at long range in the presence of external confinement and give analytic formulas for the confinement modified scattering in such circumstances. The reaction process is described in terms of the short range reaction probability. Quantum defect theory is used to express elastic and inelastic or reaction collision rates analytically in terms of two dimensionless parameters representing phase and reactivity. We discuss the modifications to Wigner threshold laws for quasi-one-dimensional and quasi-two-dimensional geometries. Confinement-induced resonances are suppressed due to reactions and are completely absent in the universal limit where the short-range loss probability approaches unity.

H + LiH+ collision dynamics at ultracold temperature conditions

Dynamics of H+LiH+ (v=0, j) collisions is investigated at cold and ultracold temperature conditions by a time-independent quantum mechanical method. The ab initio potential energy surface of the electronic ground state of the collisional system developed by Martinazzo et al. (2003) is used in the dynamics study. The collision dynamics at energy as low as 10−9eV has been examined. The Wigner threshold law holds in the limit of vanishing collision energy. Extremely large rotational and vibrational excitation of the product H2, formed via the LiH+ depletion channel, is found from the theoretical results. Integral cross sections and temperature dependent rate constants are reported for the H2 formation path. While elastic collision remains the dominant process, the depletion and inelastic processes also contribute to the dynamics with rotational excitation of the reagent LiH+ at ultracold temperature conditions and for s-wave scattering.

Cold magnetically trapped2Dgscandium atoms. II. Scattering dynamics

The binary collision dynamics of ${}^{2}{D}_{g,3/2}$ ground state scandium atoms is studied from first principles. We employ 30 coupled diabatic ab initio potentials in a coupled-channels study of the scattering dynamics of cold and ultracold scandium atoms in external magnetic fields. Due to the long-ranged magnetic dipolar interaction, the field dependence of the cross section does not follow the threshold laws derived by Volpi and Bohn [Phys. Rev. A 65, 052712 (2002)]. In the field-free case, the near-threshold cross section is independent of the collision energy, and hence the cross section does not follow the well-established Wigner threshold laws. The observed threshold behavior is explained in the Born approximation. For energies above $1\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, inelastic collisions are driven by the anisotropic nonrelativistic electronic interaction. For energies below $100\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, the ratio of elastic-to-inelastic collisions is likely to be favorable for evaporative cooling. Both anisotropy in the long-range interaction and in the short-range potential contribute to large cross sections for inelastic collisions at higher energies and lead to a small ratio of elastic-to-inelastic collisions. This is in agreement with the large rates for Zeeman relaxation of submerged-shell atoms observed experimentally. The effect of the uncertainty in the ab initio potential is sampled by scaling the reduced mass and is found to have little influence on the conclusions drawn from this work.

Photoelectron angular distributions for states of any mixed character: an experiment-friendly model for atomic, molecular, and cluster anions.

We present a model for laboratory-frame photoelectron angular distributions in direct photodetachment from (in principle) any molecular orbital using linearly polarized light. A transparent mathematical approach is used to generalize the Cooper-Zare central-potential model to anionic states of any mixed character. In the limit of atomic-anion photodetachment, the model reproduces the Cooper-Zare formula. In the case of an initial orbital described as a superposition of s and p-type functions, the model yields the previously obtained s-p mixing formula. The formalism is further advanced using the Hanstorp approximation, whereas the relative scaling of the partial-wave cross-sections is assumed to follow the Wigner threshold law. The resulting model describes the energy dependence of photoelectron anisotropy for any atomic, molecular, or cluster anions, usually without requiring a direct calculation of the transition dipole matrix elements. As a benchmark case, we apply the p-d variant of the model to the experimental results for NO(-) photodetachment and show that the observed anisotropy trend is described well using physically meaningful values of the model parameters. Overall, the presented formalism delivers insight into the photodetachment process and affords a new quantitative strategy for analyzing the photoelectron angular distributions and characterizing mixed-character molecular orbitals using photoelectron imaging spectroscopy of negative ions.

Dynamic processes and polarizability of sodium atom in Debye plasmas

Dynamic processes including excitation and ionization, and spectrum parameters including the oscillator strengths, dipole polarizabilities from the orbital 3s,3p of sodium atom embedded in weakly coupled plasma are investigated in the entire energy range of a non-relativistic regime. The interaction between the valence electron and the atomic core is simulated by a model potential, and the plasma screening of the Coulomb interaction between charged particles is described by the Debye-Hückel model. The screening of Coulomb interactions reduces the number of bound states, decreases their binding energies, broadens their radial distribution of electron wave functions, and significantly changes the continuum wave functions including the amplitudes and phase-shift. These changes strongly affect the dipole matrix elements between the bound-bound and bound-continuum states, and even the oscillator strengths, the photo-ionization cross sections and the dipole polarizabilities. The plasma screening effect changes the interaction between the valence electron and the atomic core into a short-range potential. The energy behaviors of photo-ionization cross sections are unfolded, for instance, its low-energy behavior (obeying Wigner threshold law), and the appearance of multiple shape and virtual-state resonances when the upper bound states emerge into the continuum. The Combet-Farnoux and Cooper minima in the photo-ionization cross sections are also investigated, and here, the Cooper minima appear not only for the l→l+1 channel but also for l→l−1 one, different from that of hydrogen-like ions in a Debye plasma, which appear only in the l→l+1 channel. The total static electric dipole polarizabilities monotonously and dramatically increase with the plasma screening effect increasing, which are similar to those of hydrogen-like ions and lithium atom. Comparison of calculated results for the oscillator strength, the photo-ionization cross section and polarizability with the results of other authors, when available, is made.

Femtosecond Photodetachment of Silver Anions

The two- and three-photon detachment of negative silver ions in a femtosecond infrared laser field is studied using photoelectron velocity map imaging methods. Photoelectron angular distributions (PADs) are obtained for these detachment channels; these PADs change dramatically when the laser wavelength and intensity are changed. Theoretical predictions, which are based on the adiabatic Keldysh-Faisal-Reiss saddle point method, are in good agreement with our experiment. The dependence of the PAD on the laser wavelengths and intensities is due to the interference between the different partial wave functions. The relative contributions of the different partial waves to the detachment amplitude are altered by changing the laser parameters and, as a result, the shape of the PAD. Close to the detachment threshold, the two-photon detachment process also follows the Wigner threshold law. Near the detachment threshold, the large differences between the calculated results and our experimental results indicate that the ponderomotive energy shifts caused by the femtosecond laser fields must be taken into account in the theoretical model. The three-photon detachment of Ag(-) is also observed and compared with theoretical calculations.

Experimental observations ofs−pcoupling in copper dimer anions

The photoelectron velocity-map image technique was employed to examine the one-photon detachment of Cu monomer and dimer anions using a linearly polarized infrared femtosecond laser. While the photoelectron angular distribution (PAD) in a single-photon detachment experiment of Cu${}^{\ensuremath{-}}$ shows no dependence on photon energy, the dimer anion PAD does show a strong dependence on photon energy. As the laser wavelength increases from 532 to 817 nm, the photoelectron angular distributions of the main one-photon detachment channel of the prominent detachment channel of Cu${}_{2}$${}^{\ensuremath{-}}$ changes its direction from parallel to the direction of polarization for the 532-nm laser to perpendicular to the direction of polarization for the 817-nm laser. Based upon a numerical analysis of the anisotropy parameters, the relative contributions of the $s$, $p$, and $d$ orbitals are obtained. The fully occupied $d$ orbitals contribute less to the detachment amplitude. The contribution of the $p$ waves to the detachment amplitude increases with the photon energy, while that of $s$ waves decreases, which is in agreement with the prediction of the Wigner threshold law. The relative contributions of the different partial waves, $s$ and $p$, to the detachment amplitude are altered by the excess energy of the detached electron, resulting in the determined shape of the photoelectron angular distributions. Our one-photon photodetachment experiments on the energy-dependent angular distributions of the copper anionic dimers indicate that $s$-$p$ coupling is important to the photodetachment of copper dimer anions, especially at increased photon energies.

Inverse scattering for inelastic atomic processes at thresholds

We have observed resonance structures in detached electron spectra arising from fast collisions of Li− with gas targets and from photo-detachment of Li−. Using a generalization of the Wigner threshold law, we were able to fit the experimental data to a high degree of accuracy. Moreover, we were able to extract numerical values of different electron-atom parameters as the scattering length, the effective range of the potential and the number of supported bound states. Results for the scattering length obtained from this two differentiated experiments agree with each other within a maximum error of 2%.