Wigner Threshold Law Research Articles

Quantum dynamics of ultracold Na+ Na2 collisions.

Ultracold collisions between spin-polarized Na atoms and vibrationally excited Na2 molecules are investigated theoretically, using a reactive scattering formalism (including atom exchange). Calculations are carried out on both pairwise additive and nonadditive potential energy surfaces for the quartet electronic state. The Wigner threshold laws are followed for energies below 10(-5) K. Vibrational relaxation processes dominate elastic processes for temperatures below 10(-3)-10(-4) K. For temperatures below 10(-5) K, the rate coefficients for vibrational relaxation (v=1-->0) are 4.8x10(-11) and 5.2x10(-10) cm(3) s(-1) for the additive and nonadditive potentials, respectively. The large difference emphasizes the importance of using accurate potential energy surfaces for such calculations.

Magnetic-field effects in ultracold molecular collisions

We investigate the collisional stability of magnetically trapped ultracold molecules, taking into account the influence of magnetic fields. We compute elastic and spin-state-changing inelastic rate constants for collisions of the prototype molecule ${}^{17}{\mathrm{O}}_{2}$ with a ${}^{3}\mathrm{He}$ buffer gas as a function of the magnetic field and the translational collision energy. We find that spin-state-changing collisions are suppressed by Wigner's threshold laws as long as the asymptotic Zeeman splitting between incident and final states does not exceed the height of the centrifugal barrier in the exit channel. In addition, we propose a useful one-parameter fitting formula that describes the threshold behavior of the inelastic rates as a function of the field and collision energy. Results show a semiquantitative agreement of this formula with the full quantum calculations, and suggest useful applications also to different systems. As an example, we predict the low-energy rate constants relevant to evaporative cooling of molecular oxygen.

Control of near-threshold detachment cross sections via laser polarization.

The behavior of near-threshold cross sections for dissociation of a target into a pair of particles, as described by Wigner's threshold law, can depend sensitively on the angular momentum of the particles. In this Letter, we investigate the near-threshold nonresonant two-photon detachment process in the negative ion of gold. The expected s-wave threshold behavior is observed with linearly polarized light. Closure of the s-wave channel is realized by using circular polarization, allowing the first observation of a d-wave threshold. Practical applications are discussed, including extensions which could prove valuable for investigations of negative ions with near-threshold structure.

Ultracold inelastic atomic collisions: Threshold relaxation of O(^{3}P_{0}) by He

Relaxation of $\mathrm{O}{(}^{3}{P}_{j=0})$ by collisions with He at ultracold temperatures is investigated, and it is illustrated that cross sections and rate constants for the $j=\stackrel{\ensuremath{\rightarrow}}{0}j=2$ and $j=\stackrel{\ensuremath{\rightarrow}}{0}j=1$ transitions exhibit qualitatively different threshold behaviors stemming from the different mechanisms of these nonadiabatic transitions. The results allow us to make the conclusion that, in general, rate constants for the nonadiabatic transitions governed by electrostatic couplings should obey Wigner's threshold law for s-wave scattering approaching a finite value at $\stackrel{\ensuremath{\rightarrow}}{T}0\mathrm{K},$ while cross sections and rate constants for the transitions determined by Coriolis couplings should vanish in the zero-temperature limit.

Dynamic hyperpolarizability and two-photon detachment in the presence of a strong static electric field: Application to H^{-}

Our recent analysis [J. Phys. B 33, R141 (2000)] of the problem of ${\mathrm{H}}^{\ensuremath{-}}$ in both laser and strong static electric fields, which treated effects that are linear in the laser intensity I (e.g., the dynamic polarizability and the single-photon detachment cross section), is extended here to treat effects of higher order $(\ensuremath{\sim}{I}^{2})$ in the laser intensity (e.g., the dynamic hyperpolarizability, the two-photon detachment cross section, the linear in I corrections to the single-photon detachment rate, etc). We introduce the concept of the dynamic hyperpolarizability of an atom in the presence of a strong electric field based on the complex quasienergy approach with properly normalized, quasistationary quasienergy wave functions. Our analysis of the general structure of the dynamic hyperpolarizability tensor of an atom in a nondegenerate $(S$-) state is performed for arbitrary field geometries and laser polarizations. The connection of the hyperpolarizability $\ensuremath{\gamma}(\mathcal{F};\ensuremath{\omega})$ to the complex quasienergy and to atomic ionization rates is established. Analytic results (in terms of Airy functions) are obtained for five irreducible components of a hyperpolarizability tensor (that are independent of the laser polarization and the field geometry) for the case of a weakly bound electron in a three-dimensional, zero-range potential. These results are used for the analysis of the frequency, field geometry, and laser polarization dependence of the two-photon detachment rate as well as of the linear in laser intensity corrections to the single-photon detachment rate. It is shown that the oscillatory behavior of the frequency dependence of both the real and the imaginary parts of $\ensuremath{\gamma}(\mathcal{F};\ensuremath{\omega})$ exists for both the coplanar and orthogonal field geometries, and that this behavior is qualitatively different for frequencies below and above the single-photon detachment threshold. The threshold behavior of the two-photon detachment rate is analyzed in detail and the static electric-field-induced modification of Wigner's threshold law for a short-range potential is discussed.

Effective range analysis of positron-hydrogen collisions

Humberston [Can. J. Phys. 60, 591 (1982)], using the Kohn variational method, showed that the range of validity of Wigner's threshold law for positronium formation in positron-hydrogen collisions is extremely narrow for the S wave. To examine the near-threshold behavior, we analyzed ab initio calculations using the multichannel effective range theory of Watanabe and Greene [Phys. Rev. A 22, 158 (1980)] and the single-channel effective range theory of Fabrikant [Opt. Spectrose 53, 131 (1982)] for the positronium-proton channel. We confirmed the presence of a Ramsauer minimum in the $L=0$ elastic cross section for positronium-proton scattering, and found a similar minimum for $L=1.$ The near-threshold structure is interpreted in terms of a virtual state, resonances, tunneling through and transmission over a barrier, and quantum suppression.

Theoretical studies of threshold features in the cross-sections for low-energy e +–H and e +–He scattering

The Kohn variational method is used to investigate structure in the elastic scattering and positronium formation cross-sections close to the positronium formation threshold for low energy positron scattering by hydrogen and helium. The results agree with Wigner's threshold law, but show that its validity only extends over a very restricted energy range. A great similarity is found in the s-wave contribution to the positronium formation cross-section for the two systems in both energy dependence and magnitude, and a significant feature is found in the total cross-section for positron–helium scattering at the positronium formation threshold.

Near-threshold laser spectroscopy of iridium and platinum negative ions: Electron affinities and the threshold law

The electron affinity of Ir is measured to be 12617.4(12) ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$$\phantom{\rule{0ex}{0ex}}[1.56436(15)\mathrm{eV}],$ from photodetachment studies on ${\mathrm{Ir}}^{\mathrm{\ensuremath{-}}}.$ Previous measurements of the electron affinity of platinum reported results which were inconsistent within quoted error bars. A photodetachment study with a very improved energy resolution, signal-to-noise ratio, and signal-to-background ratio, was conducted on ${\mathrm{Pt}}^{\mathrm{\ensuremath{-}}},$ and yields a much more accurate electron affinity for Pt of 17140.1(4) ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ $[2.12510(5)\mathrm{}\mathrm{eV}],$ in good agreement with the most recent measurement. Possible explanations for the poor agreement between the earlier results are discussed. In both the ${\mathrm{Ir}}^{\mathrm{\ensuremath{-}}}$ and ${\mathrm{Pt}}^{\mathrm{\ensuremath{-}}}$ spectra, the data indicate that the detachment cross section deviates from the expected Wigner threshold law, even near the detachment threshold. This behavior cannot be explained by the correction terms to the Wigner law proposed by the currently available threshold detachment models.

Fitting line shapes in photoassociation spectroscopy of ultracold atoms: A useful approximation

Line shapes observed in photoassociation spectroscopy of ultracold $(T<1\mathrm{mK})$ atoms are asymmetric and the peak of the line is shifted by a temperature-dependent amount from the zero-temperature threshold. In principle, calculating the line shape requires a full knowledge of both the ground- and excited-state potentials. We demonstrate here that an approximation based on the Wigner threshold law for scattering yields a relatively simple expression which allows one to extract line positions and linewidths directly from the data. The formula contains a few parameters (threshold frequency, linewidth, temperature, height) and does not require a detailed knowledge of the molecular potentials involved. We illustrate the utility of this approach with examples for Na photoassociation at 500 \ensuremath{\mu}K and also discuss the limitations of the approach.

New empirical rate expression for reactions without a barrier: Analysis of the reaction of CN with O2

The rate coefficients of reactions that occur on potential energy surfaces without a barrier often exhibit a negative temperature dependence at low temperatures. Generally, this behavior is modeled with either the Harcourt–Essen equation, k(T)=AT−m, or a “negative” activation energy, k(T)=ATm exp{ΔE/kBT}. Neither of these expressions is consistent with the Wigner threshold law. The general expression k(T)=(1+T/TW)−m∑l=0∞Al(1+T/TW)−l(T/TW)l is proposed where the relative angular momentum of the reacting species is l, TW and m are independent parameters to be extracted from the data, and the amplitude of each partial wave is Al. This expression may be approximated by k(T)=A0(1+T/TW)−m exp[(T/TW)/(1+T/TW)]. For CN+O2→ NCO+O and CO+NO the above expression reproduces the rate data, the branching ratio to the CO+NO channel, and the reactive cross section for the NCO+O channel. The rate coefficient for the NCO+O channel is given by k(cm3 s−1)=1.79×10−10(+T/21.7)−1.38{exp[(T/21.7)/(1+T/21.7)]−1}+4.62×10−12 exp[(T/21.7)/(1+T/21.7)] while for CO+NO we obtain k(cm3 s−1)=1.79×10−10(1+T/21.7)−1.38. An analytic form of the C–O bonding potential and the electric dipole–quadrupole interaction is used to show that the quantum threshold region extends up to 7 K. These results demonstrate the need of a complete quantum treatment for reactions that proceed on potential surfaces without a barrier.

Rovibrational relaxation in collisions between H2 molecules: II. Influence of the rotational state of the perturber

We have computed cross sections and rate coefficients for rovibrational transitions in ortho- and para-H2, induced by ground state (v = 0, J = 1) ortho-H2. These results complement our previous calculations relating to excitation by para-H2 (v = 0, J = 0). The cross sections are found to be insensitive to the rotational state of the perturber. Discrepancies with measurements of the rate coefficient for v = 10 vibrational relaxation at low temperatures persist. The value of this rate coefficient at low temperatures is shown to be determined by a combination of the Wigner threshold law and the presence of a quantal interference minimum. Further experimental and theoretical work on the H2-H2 system is called for in order to resolve the remaining discrepancies.

A simple analytical estimate of the bound-free Franck–Condon factors for a transition to a repulsive exponential potential in a diatomic molecule

An analytical study of the bound-free (b-f) Franck–Condon (FC) factors is proposed for an electronic transition in a diatomic molecule. The case where the continuous state is characterized by an exponential repulsive potential curve is investigated. The paper focuses on the qualitative characterization of the b-f FC factors in terms of physical parameters. A simple analytical estimate is proposed which matches exactly the b-f FC factors for small kinetic energy. This analysis allows us to define the short- and long-range domains of the exponential repulsive potential curve. Depending on the equilibrium position of the bonding curve (which may fall into the short- or long-range domain), the behaviour of the b-f FC factors is qualitatively different. The relation with the Wigner threshold laws is established.

Quenching ofH2Vibrations in Ultracold3Heand4HeCollisions

We present results of quantum mechanical scattering calculations including full rovibrational coupling on the relaxation of vibrationally excited ${\mathrm{H}}_{2}$ molecules in collision with ultracold ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}$ atoms. The exothermic vibrational deactivation cross sections exhibit minima at translational temperatures near 10 K and show an inverse velocity dependence in accordance with Wigner's threshold law in the limit of zero kinetic energy. The corresponding rate coefficients exhibit minima before attaining constant values below ${10}^{\ensuremath{-}3}$ K. The rate coefficients increase by 3 orders of magnitude when the initial vibrational level of the molecule is raised from $v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ to $v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}10$. The results suggest that quasibound resonance states of ${}^{3}{\mathrm{HeH}}_{2}$ and ${}^{4}{\mathrm{HeH}}_{2}$ exist, associated with each excited vibrational state of ${\mathrm{H}}_{2}$, and that both the triatomic species ${}^{3}{\mathrm{HeH}}_{2}$ and ${}^{4}{\mathrm{HeH}}_{2}$ have one bound vibrational state.

Multichannel effective-range theory with long-range interactions

Effective-range theory, for both single-channel and multichannel scattering by systems interacting with short-range forces at low energies, provides a simple representation of the energy dependence of scattering matrix elements near reaction thresholds in conformity with the Wigner threshold law. A modification of effective-range theory for single-channel electron-atom scattering, required to account for long-range polarization forces, was developed some years ago. A multichannel extension of that theory is presented here, allowing for a superposition of asymptotic power-law potentials, with asymptotic wave functions expressed as sums of Bessel functions. Threshold branch-point singularities are extracted explicitly, leaving modified reaction-matrix elements that vary smoothly with energy. For target states that are not spherically symmetric the asymptotic wave functions account for inverse-cube law interactions that couple the various channels. The theory provides an unambiguous prescription for analytic continuation of scattering parameters from just above to just below a reaction threshold, a feature that may be useful, for example, in studying resonant cross sections based on above-threshold calculations. The present work represents an extension, to a wider class of scattering systems and long-range interactions, of an earlier analysis of threshold behavior [M. Gailitis and R. Damburg, Proc. Phys. Soc. London 82, 192 (1963)] applicable to electron-hydrogen scattering in the field of the inverse-square potential arising from the linear Stark effect.

Mechanism for the production of vibrationally excited ultracold molecules of 7Li 2

A scheme is described by which translationally cold molecules can be created in specific rotational and vibrational levels and it is suggested they be employed to test Wigner's threshold laws for inelastic collisions. Calculations are presented of the spontaneous emission probabilities and the absorption oscillator strengths of transitions between the vibrational energy levels of the ground and excited triplet states of the molecules 7Li 2. The fractions of transitions into the continuum and into the discrete levels are estimated and it is shown that the translationally cold molecules can be produced efficiently in photoassociation experiments. The photoabsorption rate coefficients are computed.

Threshold phenomena in ultracold atom–molecule collisions

Vibrational relaxation cross sections for an atom–molecule collision are calculated by explicit numerical solution of the coupled equations of scattering theory and shown to follow an inverse velocity dependence at very low initial kinetic energies of the incident atom, in accordance with Wigner's threshold law. Rate constants for H+H 2( v, j=0)→H+H 2( v′< v, j=0) are calculated for all the vibrational levels of H 2 and found to vary by seven orders of magnitude between the deepest level v=0 and the vibrational level v=12.

A comment on the validity of an approximation method for two-colour photodetachment

We show that by using the approximation method of Bivona et al, two-colour photodetachment cross sections are obtained that behave according to the Wigner threshold law even if the dressing effect of the low-frequency radiation field on the bound state is ignored. Within the range of the low frequency and intensity of the radiation field considered, it is found that the values of the cross sections calculated by using the field-free bound state are not significantly modified when the corrections to the wavefunction of the bound state due to the driving action of the low-frequency field are accounted for.

Near-threshold infrared photodetachment of Al-: A determination of the electron affinity of aluminum and the range of validity of the Wigner law.

The relative photodetachment cross section of ${\mathrm{Al}}^{\mathrm{\ensuremath{-}}}$ has been measured in the wavelength range 2420--2820 nm (0.440--0.512 eV), using a coaxial ion-laser beams apparatus, in which a 2.98-keV ${\mathrm{Al}}^{\mathrm{\ensuremath{-}}}$ beam is merged with a beam from an F-center laser. The cross-section data near the $^{3}$${\mathit{P}}_{0,1,2}$${\ensuremath{\rightarrow}}^{2}$${\mathit{P}}_{1/2,3/2}$ photodetachment threshold have been fitted to the Wigner threshold law and to the zero-core-contribution theory of photodetachment. The electron affinity of aluminum was determined to be 0.44094(+0.00066/-0.00048) eV, after correcting the experimental threshold for unresolved fine structure in the ground states of ${\mathrm{Al}}^{\mathrm{\ensuremath{-}}}$ and Al. The new measurement is in agreement with the best previous measurement (0.441\ifmmode\pm\else\textpm\fi{}0.010 eV) and is 20 times more precise. The Wigner law agrees with experiment within a few percent for photon energies within 3% of threshold. A proposed leading correction to the Wigner law is discussed. \textcopyright{} 1996 The American Physical Society.

Quantum suppression of cold atom collisions.

Very-low-energy collisions between two atoms are usually suppressed, in that the probability of close approach of the atoms becomes greatly reduced as the collision energy vanishes, even if the potential is completely attractive (with the exception of the Coulomb interaction). The suppression is a quantum effect, related to the Wigner threshold law. It is gauged by comparing the ratio of the probability of being inside the well to the probability of being outside for both the classical and quantum regimes. As the asymptotic kinetic energy vanishes, the approaching atoms reach a minimum distance of typically 20 or 30 a.u. Here we study attractive interaction potentials of the form -\ensuremath{\alpha}/${\mathit{r}}^{\mathit{n}}$, and give some numerical results for accurate X${\mathrm{}}^{1}$${\mathrm{\ensuremath{\Sigma}}}_{\mathit{g}}^{+}$ and a${\mathrm{}}^{3}$${\mathrm{\ensuremath{\Sigma}}}_{\mathit{u}}^{+}$ states of ${\mathrm{Li}}_{2}$ and ${\mathrm{Na}}_{2}$ molecules. We show that in some circumstances it is possible to use Wentzel-Kramers-Brillouin theory in the suppression regime (where it fails) and to correct for its failure with a simple factor. \textcopyright{} 1996 The American Physical Society.

Line shapes of high resolution photoassociation spectra of optically cooled atoms.

High resolution photoassociation spectra of colliding ultracold trapped atoms contain detailed information about ground and excited state interactions. We calculate the asymmetric line shapes for ultracold ${\mathrm{Na}}_{2}$ absorption using a resonance scattering expression, proportional to a free-bound Franck-Condon factor, that is justified by full quantum scattering calculations of a collision in a radiation field. The line shapes illustrate Wigner threshold law behavior, which is characteristic of the quantum limit as $T\ensuremath{\rightarrow}0$. Using an adiabatic hyperfine analysis, we calculate and compare a model spectra for $J=1,2,3, \mathrm{and} 4$ features of the ${\mathrm{Na}}_{2}$ $^{1}\ensuremath{\Pi}_{g}(\ensuremath{\nu}=48)$ state to recent high resolution experimental data.