Exponential-trigonometric basis functions in the coulomb four-body problem

Basis functions of a new type—specifically, exponential-trigonometric functions depending on all six interparticle distances—have been proposed for the Coulomb four-body problem. A method has been developed for computing nine-dimensional integrals determining the matrix elements of the Hamiltonian for a four-body system and featuring these functions. The efficiency of the approach that relies on the proposed basis functions has been tested by calculating the e ⁺ e ^? e ⁺ e ^?, p ⁺μ^? p ⁺μ^?, μ⁺ e ^?μ⁺ e ^?, and p ⁺ e ^? p ⁺ e ^? molecules.     

ee Annihilation into two hadrons in the energy interval 1400–2400 MeV

The proof is given for the existence of the reaction e⁺e⁻ → h^±h^? in the energy range 1400–2400 MeV, and its energy dependence is compared with that of e⁺e⁻ → e^±e^?, in the same experimental conditions of observation. The exponent of the s-dependence of the ratio α = (e⁺e⁻ → h^±h^?)/ (e⁺e⁻ → e^±e^?) is measured to be n = 2.08 ± 0.45, in the s-range (1.96 − 5.76) GeV², on the basis of 51 e⁺e⁻ → h^±h^? events and 8918 e⁺e⁻ → e^±e^? events observed.     

Calculations of Coulomb systems with optimization of nonlinear parameters

Precise variational calculations of the energy and diverse physical characteristics are performed for the ground state of a fully nonadiabatic four-particle system—the positronium molecule e⁺e^?e⁺e^?. The nonlinear exponential parameters of correlated Gaussian basis functions dependent on all interparticle separations are systematically optimized. The dependence of the calculated energy values and the expectation values of interparticle separations (as well as their degrees and Dirac delta functions of interparticle separations) on the basis size N is examined. For N=200, the ground-state energy of the positronium molecule is found to be E=?0.5160028 au. For an extended basis set with N=4700 containing 200 basis functions with the optimized parameters and 4500 additional functions with random values of nonlinear parameters, E=?.5160036 au. At present, this result is the most accurate variational energy value of the positronium molecule, and it demonstrates the high efficiency of optimization of the nonlinear parameters in calculations of atomic-molecular systems. The lifetime of the positronium molecule e⁺e^?e⁺e^? with respect to electron-positron annihilation is calculated to be τ=2.26×10^?10 s.     

Optimal Hamiltonian of internal motion of an atomic-molecular system in a magnetic field

The effect of a uniform magnetic field on the state of an atomic-molecular system considered as a united system of N particles (without separation of the motion of acoelightac and particles) is analyzed. It is shown that the Hamiltonian of the internal motion of the system in the magnetic field contains (N-1)(N-2)/2 arbitrary parameters corresponding to the coefficients of the gauge transformation of the vector potential. A method of calculation of the optimal values of these coefficients based on the requirement for the minimum diamagnetic correction to the energy is proposed. New formulas for the diamagnetic susceptibility of atomic-molecular systems are obtained that take into account the simultaneous perturbing effect of the magnetic field on the motion of all particles. The theory is applied for calculation of the diamagnetic susceptibilities of the H, He, Li, and Be atoms; the muonic analogues of helium atoms ^∞He²⁺µ^? e ^?,⁴He²⁺µ^? e ^?, and ³He²⁺µ^? e ^?; and the positron-containing systems t ⁺ e ⁺ e ^? e ^?, d ⁺ e ⁺ e ^? e ^?, p ⁺ e ⁺ e ^? e ^?, µ⁺ e ⁺ e ^? e ^?, and e ⁺ e ⁺ e ^? e ^?, with regard for the motion of all particles.     

Evidence for a resonance in positron-electron scattering at 810 keV centre-of-mass Energy

A monoenergetic positron (e ⁺) beam (ΔT/T<5·10^?4) from the Stuttgart pelletron accelerator and a 4.6 mg/cm² Be target has been employed to investigatee ⁺ e ^?scattering in the MeV region in a large solid angle with high statistics by means of a novel positionsensitive detector system. Superimposed on the Bhabha scattering, a structure has been found at ane ⁺ kinetic energy of 2263 keV (810 keV excitation energy in thee ⁺ e ^?rest frame). From the energy-integrated resonance cross-section of \(\sigma _{res} \cdot \Gamma _{res}^{e^ + e^ - } \) ?30 b·eV (c.m.) and the standard resonance cross-section we estimate partial resonance widths \(\Gamma _{res}^{e^ + e^ - } \) ?72 meV or 24 meV for total angular momenta J=0 or J=1, respectively. The structure, which has not been predicted within the framework of quantum electrodynamics, coincides with one of the sum energies of the correlatede ⁺ e ^?lines observed in heavy-ion collisions.     

Hyperspherical Three-Body Calculation for Exotic Atoms

Ground state energies of atomic three-body systems like negatively charged hydrogen, normal helium, positively charged-lithium, beryllium, carbon, oxygen, neon and negatively charged exotic- muonium and positronium atoms have been calculated adopting hyperspherical harmonics expansion method. Calculation of matrix elements of two body interactions needed in the hyperspherical harmonics expansion method for a three body system is greatly simplified by expanding the bra- and ket-vector states in the hyperspherical harmonics (HH) basis states appropriate for the partition corresponding to the interacting pair. This involves the Raynal–Revai coefficients (RRC), which are the transformation coefficients between the HH bases corresponding to the two partitions. Use of RRC become particularly essential for the numerical solution of three-body Schr?dinger equation where the two-body potentials are other than Coulomb or harmonic. However in the present work the technique is used for two electron atoms ¹H^?(p ⁺ e ^? e ^?), D^?(d ⁺ e ^? e ^?), Mu^?(μ ⁺ e ^? e ^?),⁴He(⁴ He ²⁺ e ^? e ^?),⁶Li(⁶ Li ³⁺ e ^? e ^?),¹⁰Be(¹⁰ Be ⁴⁺ e ^? e ^?),¹²C(¹² C ⁶⁺ e ^? e ^?),¹⁶O(¹⁶ O ⁸⁺ e ^? e ^?) etc. and the exotic positronium negative ion Ps ^?(e ⁺ e ^? e ^?) where the interactions are purely Coulomb. The relative convergence in ground state binding energy with increasing K _max for ²⁰Ne has been demonstrated as a representative case. The calculated energies at K _max ?=?28 using RRC’s have been compared with those obtained by a straight forward manner in some representative cases to demonstrate the appropriateness of the use of RRC. The extrapolated energies have also been compared with those found in the literature. The calculated binding energies agree within the computational error.     

Multipion production below 1.1 GeV by e+e− annihilation

The production of multipion events by e⁺e^? annihilation has been measured at centre of mass energies 915,990 and 1076 MeV. Both channels e⁺e^?→π⁺π^?π^o and e⁺e^?→π⁺π^?π⁺π^? have been analysed. An energy threshold effect analysed. An energy threshold effect around 919 MeV (m_ω + m_π^o) has been evidenced for the π⁺π^?π^oπ^o channel and the cross section is consistent with the quasi two-body process e⁺e^?→ωπ^o. The cross section for π⁺π^?π⁺π^? is lower by an order of magnitude and increases with the energy.     

The energy dependence of σ(e+e− → hadrons) in the total centre-of-mass energy range 1.2 to 3.0 GeV

We have observed 1085 events of the type e⁺e^? → hadrons, in the total centre-of-mass energy range √s = 1.2 to 3.0 GeV. The energy dependence of the total annihilation cross-section, parametrized in the form σ(e⁺e^? → hadrons) = A·sⁿ, is measured to be n = -(1.54_?0.29^+0.17) in the above energy range.     

Heavy ion e+e− pairs to all orders in Zα

The heavy ion cross section for continuum e⁺e^? pair production has been calculated to all orders in Zα. Comparison is made with available CERN SPS and RHIC STAR data. Computed cross sections are found to be reduced from perturbation theory with increasing charge of the colliding heavy ions and for all energy and momentum regions investigated. Au or Pb total cross sections are reduced by 28% (SPS), 17% (RHIC) and 11% (LHC). For very high energy (E _e ⁺, E _e ^?>3 GeV) forward pairs at LHC the reduction from perturbation theory is a bit larger (17%). Use of zero degree calorimeter triggering (and thus small impact parameter weighting) makes impact parameter representation of exact pair production useful. Preliminary exact calculations in the zero impact parameter limit show a much larger reduction from perturbation theory (about 40%) at both RHIC and LHC.     

The muon spectra from decays ofW- andZ-bosons produced ine + e −-collisions

The muon spectra from the decays ofW-andZ-bosons, produced ine ⁺ e ^?→W ⁺ W ^?,e ⁺ e ^?→Zγ,e ⁺ e ^?→ZZ ande ⁺ e ^?→ZH reactions have been obtained (H is the Higgs boson). It has been shown (in terms of Glashow-Weinberg-Salam theory) that the main source of muons are thee ⁺ e ^?→Zγ ande ⁺ e ^?→W ⁺ W ^? processes (over theW-boson production threshold). The contribution of thee ⁺ e ^?→ZZ ande ⁺ e ^?→ZH reactions to the inclusive muon spectra is small. It has been also shown, that it is possible to distinguish contributions of thee ⁺ e ^?→Zγ ande ⁺ e ^?→W ⁺ W ^? reactions if the decay muons move at a small angle to the initial electron momentum.     

Electron-positron pair production by collisions of non-relativistic charged particles

The cross-section for (e ⁺,e ^?)-pair production in collisions of non-relativistic charged particles has been calculated to lowest order in the fine structure constant and leading power of the relative velocity of the colliding particles. Thus the deflection of colliding particles is included in one-photon-exchange approximation, while the Coulomb distortion of the (e ⁺,e ^?)-pair is omitted. We have laid particular emphasis on current conservation. The previously neglected contribution of the spatial part of the current turns out to be dominant.     

Electron-positron bremsstrahlung at ADONE. A new limit for the existence of a heavy electron

A new measurement of the cross section for the reaction e⁺e^? → e⁺e^?γ has been performed at the ADONE storage ring, using a wide angle electromagnetic detector together with tagging counters for detecting small-angle electrons. The results are in agreement with the QED predictons. New limits for the mass and coupling constant (e¹, eγ) of a heavy electron, e¹, have been established.     

Fusion cross section at sub-Coulomb energies

The effective barrier at low energies is approximated by a Coulomb and parabolic barriers. A simple closed formula is obtained for the total reaction cross section which is the natural extension of Wong's results into the low energy domain. Theoretical predictions of averaged energy behaviour agree fairly well with various experimental results. The relation?ω ₀≧π(2kh ²Z₁Z₂ e ²/μ)^1/2 R ₀ ^?3/2 between the curvature and the extremum point of the nucleus-nucleus potential is suggested, based on low energy fusion experiments. By assuming equality sign, it is possible to extract the parameters of the barrier directly from experiment. Incidentally it is found that?ω ₀ remains persistently arround 4.5 MeV for all the reactions which have been investigated.     

Approximate eigensolutions of Dirac equation for the superposition Hellmann potential under spin and pseudospin symmetries

The Hellmann potential is simply a superposition of an attractive Coulomb potential ?a/r plus a Yukawa potential be^{?δ r} /r. The generalized parametric Nikiforov–Uvarov (NU) method is used to examine the approximate analytical energy eigenvalues and two-component wave function of the Dirac equation with the Hellmann potential for arbitrary spin-orbit quantum number κ in the presence of exact spin and pseudospin (p-spin) symmetries. As a particular case, we obtain the energy eigenvalues of the pure Coulomb potential in the non-relativistic limit.     

Study of the ρ, ω, ϕ→ηγ→7γ decays with an SND detector on a VEPP-2M collider

The e ⁺ e ^?→ηγ→7γ process was studied in the energy range 2E=600–1060 MeV with an SND detector on a VEPP-2M e ⁺ e ^? collider. The decay branching ratios B(φ→ηγ)=(1.353±0.011±0.052)×10^?2, B(ω→ηγ)=(4.62±0.71±0.18)×10^?4, and B(ρ→ηγ)=(2.73±0.31±0.15)×10^?4 were measured.     

Effects on spectroscopic properties for several low-lying electronic states of CS molecule by core-valence correlation and relativistic corrections

The potential energy curves (PECs) of six low-lying electronic states (X¹Σ⁺, a³Π, a′³Σ⁺, d³Δ, e³Σ⁻ and A¹Π) of CS molecule have been investigated using the full valence complete active space self-consistent field (CASSCF) method followed by the highly accurate valence internally contracted multireference configuration interaction (MRCI) approach with large correlation-consistent basis sets. Effects on the PECs by the core-valence correlation and relativistic corrections have been taken into account. And the two corrections are performed at the level of cc-pV5Z basis set. The way to consider the relativistic corrections is to use the second-order Douglas-Kroll Hamiltonian approximation. Using the CCSD(T), MRCI and MRCI with the Davidson modification (MRCI + Q), the PECs of electronic states involved are extrapolated to the complete basis set (CBS) limit. With the PECs, the spectroscopic parameters (T_e, R_e, ω_e, ω_ex_e, ω_ey_e, α_e, β_e, γ_e and B_e) of the six low-lying electronic states are determined. These parameters are in excellent agreement with the experimental data. The complete vibrational states are computed for the six low-lying electronic states when the rotational quantum number J equals zero, and the inertial rotation constants of the first 23 vibrational states are reported, which agree favorably with the RKR data. Comparison with the measurements shows that the two-point total-energy extrapolation scheme can obviously improve the quality of spectroscopic parameters and molecular constants.     

Level structure of the 89-neutron nucleus 153Gd: (I). Levels and gamma-transitions in 153Gd excited in the decay of 153Tb

Investigations of the properties of ¹⁵³Gd excited states populated in ¹⁵³Tb decay were continued. The following measurements were performed: coincidence spectra e^?γ with L41, K93 + L52, K110, K129 + L87 + L88 and K195 + L152 keV conversion electron lines, angular correlations of high energy γ-cascades going through the 109.7 keV level, delayed e^?γ and e^?e^? coincidence spectra to determine the half-lives of 41.5, 93.3, 109.7, 129.1,183.5, 212.0 and 216.1 keV states, R(135°, ± B) parameters of IPAC for the 102–110 and 83–129 keV cascades using ¹⁵³Tb sources implanted into Fe foil.A decay scheme of ¹⁵³Tb containing 50 excited levels is proposed. Their spins, parities, $\text{log}\text{?t}$ and, for low-lying levels, also the mean half-lives have been determined. An estimation of the g-factors of the 109.7 and 129.1 keV levels has been given. On the basis of half-lives of investigated states absolute values of reduced γ-transition probabilities for these states have been calculated. The structure of the ground state of ¹⁵³Gd is discussed.     

A study of 75Se by neutron capture and the SU(3)-SU(5) transition in the quadrupole-phonon representation

The γ and e^? spectra following thermal neutron capture in ⁷⁴Se were studied with curved-crystal, β, and pair spectrometers. Precise energies have been obtained for the transitions and levels at low energies. Two primary E2 transitions were found. The neutron separation energy for ⁷⁵Se was determined as 8027.6 keV. Precise γ-energies following the electron capture decay of ⁷⁵Se were also measured, resulting in precise level energies in ⁷⁵As. The calculation of the energy levels in ⁷⁵Se has been performed in the SU(6) particle-vibrational model (PTQM) and 27 theoretical states have been tentatively assigned to the experimental levels. The spectrum of the core nucleus ⁷⁴Se has been calculated in the SU(6) quadrupolephonon model (TQM). The structure of theoretical states, the relation to SU(3) and SU(5) limits, and potential energy surface are discussed. The E2, M1 and E1 transitions have been calculated in PTQM and compared to the experiment. Also, an overview is presented of theoretical explanations of the I = j, j?1, j?2 anomalous triplet emphasizing the rule with shell-model classification corrected for quadrupole phonons.     

Theoretical investigation of the molecular structure and transition dipole moments of the NaK low lying electronic states

The electronic structure and the spectroscopic constants of the low lying electronic states of the NaK⁺ ionic molecule have been determined through using an ab initio approach involving a non-empirical pseudopotential for the Na and K cores and core valence correlation correction. The potential energy of nearly 26 electronic states of ²Σ⁺, ²Π, and ²Δ symmetries has been calculated up to their dissociation limit Na(4d) + K⁺ and Na⁺ + K(6s). Their spectroscopic constants (R_e, D_e, T_e, ω_e, ω_eχ_e, and B_e) are derived and compared with the few available theoretical studies. A good agreement has been found for the ground state and few excited states with previous works. New potential energy curves were presented, for the first time, for the higher excited states. Numerous avoided crossing between electronic states of ²Σ⁺, ²Π symmetries have been localized and analyzed. Their existences are related to the charge transfer between the two ionic molecules Na⁺K and NaK⁺. Furthermore, we have determined the transition dipole moments for several states and analyzed the avoided crossings related to charge transfer between alkaline atoms.