Odd Partial Waves Research Articles

Control of spin-exchange interaction between alkali-earth-metal atoms via confinement-induced resonances in a quasi-(1+0)-dimensional system

A nuclear-spin exchange interaction exists between two ultracold fermionic alkali-earth (like) atoms in the electronic $^{1}{\rm S}_{0}$ state ($g$-state) and $^{3}{\rm P}_{0}$ state ($e$-state), and is an essential ingredient for the quantum simulation of Kondo effect. We study the control of this spin-exchange interaction for two atoms simultaneously confined in a quasi-one-dimensional (quasi-1D) tube, where the $g$-atom is freely moving in the axial direction while the $e$-atom is further localized by an additional axial trap and behaves as a quasi-zero-dimensional (quasi-0D) impurity. In this system, the two atoms experience effective-1D spin-exchange interactions in both even and odd partial wave channels, whose intensities can be controlled by the characteristic lengths of the confinements via the confinement-induced-resonances (CIRs). In current work, we go beyond that pure-1D approximation. We model the transverse and axial confinements by harmonic traps with finite characteristic lengths $a_\perp$ and $a_z$, respectively, and exactly solve the quasi-1D + quasi-0D scattering problem between these two atoms. Using the solutions we derive the effective 1D spin-exchange interaction and investigate the locations and widths of the even/odd wave CIRs for our system. It is found that when the ratio $a_z/a_\perp$ is larger, the CIRs can be induced by weaker confinements, which are easier to be realized experimentally. The comparison between our results and the recent experiment shows that the two experimentally observed resonance branches of the spin-exchange effect are due to an even-wave CIR and an odd-wave CIR, respectively. Our results are advantageous for the control and description of either the effective spin-exchange interaction or other types of interactions between ultracold atoms in quasi 1+0 dimensional systems.

Josephson oscillation and self-trapping in momentum space

A nuclear-spin exchange interaction exists between two ultracold fermionic alkali-earth (like) atoms in the electronic $^{1}{\rm S}_{0}$ state ($g$-state) and $^{3}{\rm P}_{0}$ state ($e$-state), and is an essential ingredient for the quantum simulation of Kondo effect. We study the control of this spin-exchange interaction for two atoms simultaneously confined in a quasi-one-dimensional (quasi-1D) tube, where the $g$-atom is freely moving in the axial direction while the $e$-atom is further localized by an additional axial trap and behaves as a quasi-zero-dimensional (quasi-0D) impurity. In this system, the two atoms experience effective-1D spin-exchange interactions in both even and odd partial wave channels, whose intensities can be controlled by the characteristic lengths of the confinements via the confinement-induced-resonances (CIRs). In current work, we go beyond that pure-1D approximation. We model the transverse and axial confinements by harmonic traps with finite characteristic lengths $a_\perp$ and $a_z$, respectively, and exactly solve the "quasi-1D + quasi-0D" scattering problem between these two atoms. Using the solutions we derive the effective 1D spin-exchange interaction and investigate the locations and widths of the even/odd wave CIRs for our system. It is found that when the ratio $a_z/a_\perp$ is larger, the CIRs can be induced by weaker confinements, which are easier to be realized experimentally. The comparison between our results and the recent experiment shows that the two experimentally observed resonance branches of the spin-exchange effect are due to an even-wave CIR and an odd-wave CIR, respectively. Our results are advantageous for the control and description of either the effective spin-exchange interaction or other types of interactions between ultracold atoms in quasi 1+0 dimensional systems.

Superfluid pairing in a mixture of a spin-polarized Fermi gas and a dipolar condensate

We consider a mixture of a spin-polarized Fermi gas and a dipolar Bose-Einstein condensate in which s-wave scattering between fermions and the quasiparticles of the dipolar condensate can result in an effective attractive Fermi-Fermi interaction anisotropic in nature and tunable by the dipolar interaction. We show that such an interaction can significantly increase the prospect of realizing a superfluid with a gap parameter characterized with a coherent superposition of all odd partial waves. We formulate, in the spirit of the Hartree-Fock-Bogoliubov mean-field approach, a theory which allows us to estimate the critical temperature when the anisotropic Fock potential is taken into consideration and to determine the system parameters that optimize the critical temperature at which such a superfluid emerges before the system begins to phase separate.

Partial-wave expansions of angular spectra of plane waves

Focused electromagnetic beams are frequently modeled by either an angular spectrum of plane waves or a partial-wave sum of spherical multipole waves. The connection between these two beam models is explored here. The partial-wave expansion of an angular spectrum containing evanescent components is found to possess only odd partial waves. On the other hand, the partial-wave expansion of an alternate angular spectrum constructed so as to be free of evanescent components contains all partial waves but describes a propagating beam with a small amount of standing-wave component mixed in. A procedure is described for minimizing the standing-wave component so as to more accurately model a purely forward propagating experimental beam.

Auger electron spectroscopy of molecules: Theory for spin polarization following photoabsorption in rotating linear molecules

This paper develops theoretical expressions to study angular distribution and spin polarization of those Auger electrons which are emitted in the decay of a vacancy created by the absorption of a photon in a rotating linear molecule. Identical expressions except, of course, for different decay amplitudes, in both the Hund’s coupling schemes (a) and (b), are obtained for the differential Auger current emitted in the transition J→Jf measured by an electron spectrometer sensitive to spin detection. The structure of these angular distributions is exactly the same as that of the spin-resolved photoelectrons from unoriented atoms and molecules. The present paper thus puts the angle- and spin-resolved Auger and photoelectron spectroscopies on the same footing wherein identical geometrical and kinematical analysis is applicable. The four parameters needed to completely characterize such distributions depend, in the present case, on rotational orientation and/or alignment of the photoexcited molecule, in addition to its Auger decay amplitudes. The use of parity-adapted molecular states separates the Auger spectra into even and odd partial wave components of the ejected electron continuum in both of the coupling schemes. Our analysis shows that the integrated Auger current is spin resolved provided it is produced in the decay of oriented vacancies. We further find that Auger electrons which leave the molecular ion in Jf=0 state may have nonzero degree of spin polarization if they follow absorption of only circularly polarized light. In this case, both the angular distribution and spin polarization of emitted electrons become totally independent of Auger dynamics. Thus, angle- and spin-resolved Auger electron spectroscopy can be used to produce polarized electrons, to determine rotational orientation and alignment of linear molecules, to study their structure and dynamics, and to prepare ions of such molecules in selective ro-vibronic states.

Rotational branching ratios and photoelectron angular distributions in resonance enhanced multiphoton ionization of HBr via the F 1Δ2 Rydberg state

Results of theoretical studies of rotational ion distributions in the X 2Π1/2 ground state of HBr+ resulting from (2+1) resonance enhanced multiphoton ionization (REMPI) via the S(2) branch of the F 1Δ2 Rydberg state are reported. These results show a strongly parity-favored ion distribution with about 80% population in the (−) component of the Λ doublet of J+ rotational levels. The 20% population in the other parity component of the Λ doublet can be seen to be due to odd partial wave contributions to the photoelectron matrix elements which arise primarily from non-atomic-like behavior of the electronic continuum. This, in turn, is due to angular momentum coupling in the photoelectron orbital brought about by the torques of the nonspherical molecular ion potential. We demonstrate that the effect of alignment on these ion distributions, although not large, is important. Photoelectron angular distributions and alignment of the J levels of the HBr+ ions are also presented. Rotational branching ratios and photoelectron angular distributions resulting from (2+1′) REMPI of HBr via several S branches of the F 1Δ2 state are also shown for near-threshold photoelectron energies.

Calculation of differential Cross Section for pion absorption on diproton

The asymmetry of the differential cross section for the process of pion absorption onT=1 nucleon pairs in3He and3H is calculated using the experimental partial amplitudes for pion scattering prior to the absorption. The asymmetry is found to be a consequence of the even and odd partial-wave interference.

Angular distributions of 230, 232Th(n, f) and the third-minimum hypothesis

Angular distributions of light (or heavy) fission fragments may exhibit an asymmetry with respect to ϑ c. m. = 90° due to interference effects between even and odd partial waves if only a few transition states contribute to fission. This possibility is explored for neutron induced fission of 230, 232Th in the region of subbarrier resonances. Only in case of 23Th(n, f) a slight but significant asymmetry is observed. It is interpreted in support of the third-minimum hypothesis.

The 20Ne( 16O, 16O) 20Ne excitation function at θc.m. = 90° and parity-dependent interactions

The excitation function for 16O + 20Ne elastic scattering at θ c.m. = 90° has been measured over the energy range 21.5 MeV < E c.m. < 31.2 MeV and analysed within the framework of an extended optical model, in order to place additional constraints on spin assignments to resonant states, and hence constraints on the parity dependent part of the 16O− 20Ne potential. A satisfactory description is obtained of elastic scattering excitation functions at θ c.m. = 90° and θ c.m. = 154° ± 2° with a potential which is deeper for even than for odd partial waves. Such a parity dependence is consistent with the assumption that this term arises from a requirement to simulate the effects of α-transfer amplitudes, and with the expectations of models which argue that parity dependence is to be expected as a consequence of the Pauli principle.

Ionic rotational branching ratios in resonant enhanced multiphoton ionization of NO via the A 2Σ+(3sσ) and D 2Σ+(3pσ) states

We present the results of ab initio calculations of the ionic rotational branching ratios in NO for a (1+1) REMPI (resonant enhanced multiphoton ionization) via the A 2Σ+(3sσ) state and a (2+1) REMPI via the D 2Σ+(3pσ) state. Despite the atomic-like character of the bound 3sσ and 3pσ orbitals in these resonant states, the photoelectron continuum exhibits strong l mixing. The selection rule ΔN+l=odd (ΔN≡N+−Ni) implies that the peaks in the photoelectron spectrum corresponding to ΔN=odd (±1,±3) are sensitive to even partial waves while those corresponding to even ΔN probe the odd partial waves in the photoelectron continuum. Recent experimental high resolution photoelectron studies have shown a strong ΔN=0 peak for ionization via the A 2Σ+ and the D 2Σ+ states, indicating a dominance of odd-l partial waves. While this seems natural for ionization out of the 3sσ orbital, it is quite anomalous for 3pσ ionization. Based on extensive bound calculations, Viswanathan et al. [J. Phys. Chem. 90, 5078 (1986)] attribute this anomaly to a strong l mixing in the electronic continuum caused by the nonspherical molecular potential. We have performed ab initio calculations of the rotational branching ratios and compared them with the experimental results. The electronic continuum shows a significant p-wave component which leads to the large ΔN=0 peak in both cases. Calculations are performed for both rotationally ‘‘clean’’ and ‘‘mixed’’ branches. The relative heights of the peaks are very sensitive to the photoelectron kinetic energy for the D 2Σ+ state and less so for the A 2Σ+ state. This is a direct consequence of the l mixing in the continuum.

Convergence of Faddeev partial-wave series for triton ground state.

An iterative technique was used to solve the Faddeev equations, which determine the bound state of the triton, for the Reid soft core, Argonne ${v}_{14}$, super soft core (C), and de Tourreil-Rouben-Sprung (B) potential models. Separate solutions for each model were generated for 5, 9, 18, 26, and 34 three-body channels, the latter corresponding to all two-nucleon partial waves with J\ensuremath{\le}4. The results indicate that the odd partial waves do not contribute substantially to the energy, and that the J&gt;4 partial waves probably contribute less than 10 keV. Coulomb energies, radii, and probability percentages of various wave function components were also calculated.

A three-body scattering problem in the molecular mass limit

A three-body problem is studied that involves the scattering of a heavy particle from a bound state of a heavy and a light particle. For fairly large mass ratios we find many partial waves involved in the scattering with both elastic and inelastic resonances present. It is also found that virtually all of the breakup inelasticity is in the odd partial waves, if the heavy particles are identical bosons. The same problem is solved in the Born-Oppenheimer approximation and it is found that the effective potential between the heavy particles develops an imaginary part in the odd partial waves and thus gives a quantitative account of the three-body results including the inelasticity. The linear combination of nuclear orbitais method is applied to the same problem and is found to be inadequate.

An l-window formalism for elastic heavy-ion scattering

It is shown that the heavy-ion elastic scattering amplitude may be written as an exact summation over sharp cut-off Coulomb amplitudes with coefficients which are simply the differences of successive nuclear S-matrix elements. Thus in the case of strong absorption the coefficients are non-zero only over a small range of angular momenta, formally making the elastic amplitude similar to those for inelastic scattering and transfer reactions in that it possesses an 'l window'. Some good approximations to the sharp cut-off Coulomb amplitudes are given and these enable us to rederive in a simple way the results obtained by the usual integral techniques for dealing with smooth S matrices. The case of identical spin-zero ions is treated in some detail since here one may have very few terms in the transition region due to the absence of all odd partial waves. In particular, the system 16O+16O is discussed and a qualitative understanding of many of the experimental results and of previous fits to the data is obtained.

90° excitation functions for elastic scattering of 28Si+ 16O and 28Si+ 12C

The gross structures in the excitation functions for elastic scattering of 28Si+ 16O and 28Si+ 12C, measured at 90°, where only even partial waves contribute, are found to be anti-correlated with those at 180°. Excitation functions at both angles, including the enhanced gross structure at 180°, can be reproduced by calculations using a potential which attracts odd partial waves more than even partial waves.

Theoretical studies of H2–H2 collisions. I. Elastic scattering of ground state para- and ortho-H2 in the rigid rotor approximation

Close coupling calculations of integral and differential elastic cross sections of hydrogen ground state molecule collisions have been performed, for c.m. energies below 0.5 eV. It is shown that the isotropic part of the potential, determined by the consistent ab initio potentials of five geometries, provides a very accurate (1%–2%) agreement with measured p-H2/p-H2 integral cross sections in the range of 900–2300 m/sec relative velocity. Detailed analysis of p-H2/p-H2 scattering results and the determination of a series of orbiting resonances provide a set of (virtual) quasi-bound-state energy levels. The fit formula of those levels gives two bound states of the (H2)2 system, for J=0 and 1, at −0.340×10−3 and −0.179×10−3 eV, respectively, which is close to the two binding energies found for the isotropic potential. However, the more attractive plane T configuration of the (H2)2 system gives larger binding energies when the zero point vibrations are neglected. Lifetimes and the periods of orbiting have been evaluated for the resonances. Since the resonaces are due to pure orbiting, we have compared the spacing of the p-H2/p-H2 partial integral cross section peaks with peaks of the opacities for p-H2/o-H2 and o-H2/o-H2 and found quantitative agreement almost everywhere, while the exceptions found for o-H2/o-H2 at very small energies can be explained by slightly different effective moments of inertia. The undulatory structure found for the ground state p-H2/p-H2 integral cross section is clearly due to symmetry restrictions which only allow even partial waves. For o-H2/o-H2, the cross sections of solely the symmetric or the asymmetric problem show undulatory structure as well because the even or odd partial wave contributions, respectively, dominate, while in the averaged curve these structures are completely damped. A few examples of differential cross sections and differential helicity transition cross sections at resonance energies are presented. The jz conservation in the body-fixed frame used in the approximation of McGuire has been found to be not quantitatively valid in the energy range of the strong resonances while for increasing energies jz conserving cross sections become dominant. Generally for o-H2/p-H2 collisions jz is better conserved than for o-H2/o-H2.

O18+C12System: Evidence for Nonstatistical Intermediate Structure with Enhancement of Odd Partial Waves in the Elastic Channel

$^{18}\mathrm{O}$ + $^{12}\mathrm{C}$ elastic-scattering excitation functions have been measured at eight angles over the energy range 12-25 MeV (c.m.). In addition, cross sections for the reaction channels $^{12}\mathrm{C}(^{18}\mathrm{O}, \ensuremath{\alpha})^{26}\mathrm{Mg}$ (${E}_{x}=0.0\ensuremath{-}5.47$ MeV) and $^{12}\mathrm{C}$($^{18}\mathrm{O}$, $^{16}\mathrm{O}$)$^{14}\mathrm{C}$ were measured over nearly the same energy range. The elastic scattering exhibits the novel behavior of being dominated by odd partial waves. Correlated structures were observed at 16.6, 19.0 (${J}^{\ensuremath{\pi}}={15}^{\ensuremath{-}}$), and 21.8 (${J}^{\ensuremath{\pi}}={17}^{\ensuremath{-}}$) MeV (c.m.).

Condition for a vanishing cross section at ?-?/2 in Barshay-Temmer-reactions

The symmetry property of the transition matrix for a reaction producing an isobaric multiplet (or identical particles) is derived from the generalized Pauli principle without any specific assumptions concerning the reaction mechanism. In contrast to previous predictions it is shown that for odd partial waves in the exit channel the cross section will generally not vanish at Θ = π/2 if the channel spin projections of the initial and final channels may be different.

Theory of particle and hole transfer in elastic nucleus-nucleus scattering

The optical model description of elastic nucleus-nucleus scattering is generalized to include the transfer of particles and holes in collisions between heavy nuclei which differ only by one or a few nucleons. An elementary model which consists of two identical cores and a valence particle is solved for small valence-particle masses in the molecular two-state approximation. The molecular states of the valence particle are constructed by a linear combination of nuclear orbitals (LCNO). In order to get around the problem of Coriolis coupling the scattering solution is restricted to cases where nuclear states with j ≦ 1 2 or l = 0 are involved. Because of the possible exchange of the valence particle between the two identical cores, even and odd partial waves are scattered by different potentials. The difference is due to exchange potentials which are associated with the molecular states of definite core-permutation symmetry. The result is a characteristic interference pattern in the angular distribution of the elastic cross section. The elementary model is generalized by describing the cores in the framework of Hartree-Fock theory. This description shows that holes can be exchanged like particles. The derivation of an optical core-core potential is given. The exchange potential which is real in the elementary model becomes complex in the general treatment. For the elastic transfer of two particles, two holes or a particle and a hole, four possible approximations are discussed. Correlations between core particles and valence particles or holes can approximately be taken into account by spectroscopic factors in the exchange potentials.

Survival Probability in Dissociative Attachment

The survival probability in dissociative attachment is investigated with special attention to the ($e$, ${\mathrm{H}}_{2}$) system. It is shown that the simple expression for the dissociative-attachment cross section, as given by the product of a capture cross section and a survival probability, is equivalent to the $s$-wave approximation for the $g\ensuremath{\rightarrow}g$ dissociative attachment. This expression, however, does not constitute an approximation for the $g\ensuremath{\rightarrow}u$ dissociative attachment, since the parity of the initial rotational states of ${\mathrm{H}}_{2}$ is always opposite to that of the relative angular momentum states of H and ${\mathrm{H}}^{\ensuremath{-}}$ and the capture cross section appearing in the simple expression is identically zero. According to the Kronig selection rules and the symmetry requirements, only odd partial waves of the incident electron may contribute to the $g\ensuremath{\rightarrow}u$ dissociative attachment in the ($e$, ${\mathrm{H}}_{2}$) system. Consequently, the lowest contributing partial wave is not the $s$ wave but the $p$ wave of the incident electron. This, then, destroys the simple proportional dependence of the cross section on the survival probability. However, one may still express the cross section as a sum of products of a capture cross section and a survival probability for the various contributing angular momentum states of the constituent nuclei. The dependence of the survival probability on the angular momentum states of the constituent nuclei is also investigated for the ($e$, ${\mathrm{H}}_{2}$) system. It is observed that for the $g\ensuremath{\rightarrow}u$ dissociative attachment the survival probability depends strongly on the angular momentum states. This arises because the $g\ensuremath{\rightarrow}u$ dissociative attachment occurs at such a low energy that variations in the centrifugal barrier become comparable with the breakup energy of the constituent atoms. This then suggests a strong temperature dependence for the $g\ensuremath{\rightarrow}u$ dissociative attachment in the ($e$, ${\mathrm{H}}_{2}$) system. For the $g\ensuremath{\rightarrow}g$ dissociative attachment, such dependence is much weaker since here the process significant at a somewhat higher energy and the variation in centrifugal energy is overshadowed by the larger break-up energy of the constituent atoms. The validity of the commonly adopted approximation for survival probability (involving the auto-ionization width and relative velocity of the nuclei) is also examined.

On the Question of Charge Symmetry inD-DReactions

The question of charge symmetry in D-D reactions is discussed in light of the recently discovered virtual excited states of the alpha particle. We point out their effects on the asymmetries in the angular distributions as well as the provision of a mechanism for P-wave enhancement. Relation with three-nucleon isobars is also discussed. D+D-+p+H 3 , D+D->n+He 3 (la) (lb) are also of interest from the viewpoints of nuclear structure and of general quantum mechanical symmetry principles. Of the latter, the most obvious fact is that the Bose statistics of the initial T = 0 system requires that there be no interference between the even and odd angular momentum partial waves; and that among the initial singlet, triplet and quintet spin states only the triplet state can couple with the odd partial waves. An immediate consequence of this is that the angular distribution of the reaction products must be symmetric about 90° in the center of mass system. 3 ) Another fact which follows from overall angular momentum and parity conservation (neglecting the small ad­ mixture of weak interaction current) is that the ground states of the three-body nuclei have positive parity. Reaction (la) also provides the most accurate test to-date of the reciprocity theorem in strong interactions. 4 ) These observations do not depend on the reaction mechanism or on what goes on in the intermediate state. In addition, recent measurements 1 ) of reactions (la) and (lb) bave aroused some controversy with regard to the question of charge symmetry in the three­ nucleon isobars. These measurements showed that the zero-to-ninety-degree asymmetry for the angular distribution of the final neutrons is strikingly larger than that of. the protons. One might argue that this difference could arise from the difference in the mass radii of He 3 and H 3 • Such a difference in the cor­