Structure and dynamics of few-helium clusters using soft-core potentials

In this work we investigate the structure and dynamics of small clusters of Helium atoms. We consider bound states of clusters having A = 2, 3, 4, 5, 6 atoms and continuum states in the three-atom system. Motivated by the fact that the He-He system has a very large scattering length a compared to the range r ₀ of the He-He potential (r ₀/a < 1/10), we propose the use of a soft-core interparticle potential. We use an attractive gaussian potential that reproduces the values of the dimer binding energy and the atomatom scattering length obtained with one of the widely used He-He interactions, the LM2M2 potential. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and a three-body potential, we show the spectrum of the four, five, and sixparticle systems and phase-shifts and inelasticities in the three-atom system. Comparisons to calculations using realistic He-He potentials are given. In addition some universal relations are explored.     

Solving the differential Yakubovsky equations for p 3He scattering by the cluster-reduction method

The cluster-reduction method has been used to solve numerically the differential equations for the s-wave Yakubovsky components in the nppp system. Proton scattering on a ³He nucleus has been considered for energies below the three-body threshold in the singlet and triplet spin states of the system. Nucleon-nucleon interaction has been simulated by the MT I–III potential, and the Coulomb interaction between the protons has been taken into account. Effective equations that describe the relative motion of clusters have been derived. The low-energy behavior of phase shifts has been analyzed. The values of ¹ A=8.2 fm and ³ A=7.7 fm have been obtained for, respectively, the singlet and triplet scattering lengths. The calculated phase shifts agree well with experimental data.     

Effective-range function for doublet nd scattering from an analysis of modern data

The parameters of the generalized effective-range function K(k ²) having a pole are found by using the results that were obtained by calculating the S-wave phase shift δ(E) for doublet nd scattering and the triton binding energy on the basis of Faddeev equations and within the N/D method and which were presented in the literature. The convergence of the expansion of K(k ²) in powers of momentum is studied. The binding energy of the virtual triton and the residues of the partial-wave scattering amplitudes at the poles corresponding to the bound and virtual states are calculated. Correlations between the binding energies of the bound and virtual states of the triton, on one hand, and the doublet scattering length for nd interaction, on the other hand, are considered. The function K(k ²) is also calculated within a two-body model featuring various potentials.     

Two pion exchange three body potential in three nucleon states

We present the observables for theA=3 bound state with a two-pion exchange three-body force added to the de Tourreil and Sprung realistic nucleon-nucleon interaction. This results mainly in increasing the binding energy by 650 keV. In the framework of the impulse approximation, we find no sensible improvement for the electromagnetic form factors, and no central depression for the He³ charge density with point nucleons. We compute the contribution of the three-body force to then-d doublet scattering length whose agreement with experiment is improved.     

Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order

We use an effective field theory for short-range forces (SREFT) to analyze systems of three identical bosons interacting via a two-body potential that generates a scattering length, a, which is large compared to the range of the interaction, ?. The amplitude for the scattering of one boson off a bound state of the other two is computed to next-to-next-to-leading order (N²LO) in the ?/a expansion. At this order, two pieces of three-body data are required as input in order to renormalize the amplitude (for fixed a). We apply our results to a model system of three Helium-4 atoms, which are assumed to interact via the TTY potential. We generate N²LO predictions for atom-dimer scattering below the dimer breakup threshold using the bound-state energy of the shallow Helium-4 trimer and the atom-dimer scattering length as our two pieces of three-body input. Based on the convergence pattern of the SREFT expansion, as well as differences in the predictions of two renormalization schemes, we conclude that our N²LO phase- shift predictions will receive higher-order corrections of < 0.2 %. In contrast, the prediction of SREFT for the binding energy of the “deep” trimer of Helium-4 atoms displays poor convergence.     

Nuclear compression moduli and density-dependent forces

Density-dependent zero-range forces of the form of the modified delta interaction (MDI) are generalized (MDI3, MDI4) in order to yield reasonable values of the compression modulus in nuclear matter (K_N = 200 MeV). This low value can be fitted by introducing two terms with different density dependence in the force. The four free parameters of MDI3 are adjusted to reproduce the nuclear matter values of the binding energy, density and compression modulus, and to fulfil the condition that the total energy of ¹⁶O in harmonic oscillator wave functions has a minimum at the oscillator length b = 1.75 fm, corresponding to the correct rms radius. MDI4 contains in addition a two-body spin-orbit interaction. The five parameters of MDI4 are fitted to the above three nuclear matter data and by requiring that Hartree-Fock (HF) calculations in ²⁰⁸Pb yield the experimental charge rms radius and reasonable values of certain single-particle spin-orbit splittings. The quality of MDI4 is checked by comparing calculated rms radii, binding energies, and elastic electron scattering cross sections with available experimental data for doubly closed shell nuclei. As a test the energy levels and the nuclear monopole polarization of muonic ²⁰⁸Pb are calculated self-consistently yielding impressive agreement with experiment.     

A simplified approach to three-body correlations in nuclear matter through a non-zero particle spectrum

Self-consistent nuclear-matter calculations are presented which take into account Dahlblom's results for the contribution to the binding energy due to three-body correlations. We propose a justified parametrization of the single-particle potential for particle states, the energy contribution of which cancels approximately the energy from certain three-body correlations. Indications are given of how to fix this particle-state potential for a given two-body interaction. Two nucleon-nucleon potentials are used: the Reid soft-core potential and a fully momentum-dependent one-boson-exchange potential similar to the form proposed by Ingber and Potenza. The mechanism of the increase in the total wound due to three-body correlations is investigated and reasons are given why this does not prevent the saturation densities from moving to higher values. Due to three-body correlations and with self-consistency on the hole spectrum, the increase in nuclear-matter binding energy is 0.60 MeV/A for the Reid soft-core interaction and 0.68 Mev/A for the OBEP. The saturation momentum is shifted from 1.42 fm^?1 to 1.44 fm^?1 for the Reid potential and from 1.58 fm^?1 to 1.62 fm^?1 for the OBEP.     

Solutions of the integral equations for four identical particles

Using the separable representation of the scattering amplitudes for the subsystems 3 + 1 and 2 + 2, the integral equations for four identical particles with a separable two-particle interaction are reduced to a set of single variable integral equations. By solving the equations obtained, the binding energies and wave functions of the low-lying 0⁺ states of the system of four identical bosons, as well as the scattering length of a particle scattered by three bound particles, are calculated. The solution of the set of integral equations, describing the bound state of four nucleons, is performed, approximating the space wave function by a symmetric one, and the binding energy and wave function of the nucleus ⁴He are calculated.     

Momentum distributions in A = 3 and 4 nuclei

We develop a general Monte Carlo method to study momentum distributions of nucleons and nucleon clusters in nuclei. The method is used to calculate the momentum distributions of protons and neutrons in A = 3 and 4, d + p amplitudes in ³He, and t + p and d + d amplitudes in ⁴He nuclei, with improved variational wave functions. The nucleon and d + p momentum distributions in ³He are also calculated from a five-channel Faddeev wave function. The calculations are based on realistic hamiltonians that include the three-nucleon interaction, and give reasonable binding energies and densities for light nuclei and nuclear matter. The calculated proton and d + p momentum distributions in ³He at low k are in good agreement with the results obtained by the Saclay analysis of the electron scattering data in the plane-wave impulse approximation; however, at higher values of k, the calculated momentum distributions are larger. The calculated values of the asymptotic D- to S-wave ratio of the d+n and d+d amplitudes are also in agreement with the values obtained from (d, t) and (d, α) reactions. The number of deuterons is found to be 1.38 and ∼2.4 in A = 3 and 4 nuclei, while the number of tritons in ⁴He is ∼1.6. This large value of the triton number reflects the large contribution (more than 90% at small k) of the t+p state to the total proton momentum distribution in ⁴He.     

Pion-nucleus inelastic scattering and charge exchange

Using a separable fixed scatterer pion-nucleon interaction and the distorted wave impulse approximation we have made predictions for medium energy pion inelastic scattering from ¹⁶O and (π^?, π⁰) charge exchange from ⁴⁸Ca. An optical potential based on the pion-nucleon interaction adopted in this work has been shown previously to provide good fits to pion-nucleus elastic scattering. After a discussion of the basic formulae, we present results of calculations for pion inelastic scattering from ¹⁶O for initial pion lab kinetic energies of 70 and 180 MeV. Because of the strong energy dependence of the pion-nucleon interaction there are qualitative differences between the predictions for the nuclear response in the momentum transfer, energy loss plane for T_π^lab = 70 and 180 MeV. At these energies, states not prominently excited by other probes are predicted to be observable. In particular, J^π = 3^? and 4^?, T = 0 states appear prominently in the excitation spectrum region at large momentum transfer. A comparison of π^? and π⁺ scattering showing the effects of the Coulomb interaction is presented. The predictions for pion single-charge exchange on ⁴⁸Ca indicate that this interaction would be useful for studying the location of the T_> states arising from the splitting of the giant dipole resonance in T ≠ 0 nuclei.     

Nuclear matter approach to the heavy-ion optical potential: (II). Imaginary part

The heavy-ion optical potentials are constructed in a nuclear matter approach, for the ¹⁶O + ¹⁶O, ⁴⁰Ca + ¹⁶O and ⁴⁰Ca + ⁴⁰Ca elastic scattering at the incident energies per nucleon E^lab/A ? 45 MeV. The energy density formalism is employed assuming that the complex energy density of colliding heavy ions is a functional of the nucleon density ?(r), the intrinsic kinetic energy density τ⁽²⁾(r) and the average momentum of relative motion per nucleon K_r(≦ 1.5 fm^?1). The complex energy density is numerically evaluated for the two units of colliding nuclear matter with the same values of ρ, τ⁽²⁾ and K_r. The Bethe-Goldstone equation is solved for the corresponding Fermi distribution in momentum space using the Reid soft-core interaction. The “self-consistent” single-particle potential for unoccupied states which is continuous at the Fermi surface plays a crucial role to produce the imaginary part. It is found that the calculated optical potentials become more attractive and absorptive with increasing incident energy. The elastic scattering and the reaction cross sections are in fair agreement with the experimental data.     

Existence of Mott-Schwinger interaction by means of\vec p - 12C elastic scattering

The aim of this work was the unambiguous proof of the existence of the Mott-Schwinger interaction. The analyzing power of the \(\vec p - \) ¹²C elastic scattering was measured in the energy range from 450 to 600 keV for scattering anglesθ _Lab=90° and 120° with an overall accuracy up toΔA=1·10^?4. The data can be described very well with theR-matrix formalism including Mott-Schwinger interaction. Omitting this interaction large discrepancies occur.     

Large momentum part of a strongly correlated Fermi gas

It is well known that the momentum distribution of the two-component Fermi gas with large scattering length has a tail proportional to 1/k⁴ at large k. We show that the magnitude of this tail is equal to the adiabatic derivative of the energy with respect to the reciprocal of the scattering length, multiplied by a simple constant. This result holds at any temperature (as long as the effective interaction radius is negligible) and any large scattering length; it also applies to few-body cases. We then show some more connections between the 1/k⁴ tail and various physical quantities, including the pressure at thermal equilibrium and the rate of change of energy in a dynamic sweep of the inverse scattering length.     

Spectroscopic properties of SbH

Relativistic configuration interaction calculations which include spin-orbit interaction are carried out for nine low-lying ω-ω states and four λ-s states. Spectroscopic properties of six bound ω-ω states are reported. These calculations not only enable assignment of the experimentally observed X₁, X₂, A₁, A₂, and B states but also predict the properties of other electronic states (0⁺(II), 0⁺(IV), 2, 2(II), 1(II), 0⁻) which are yet to be observed. The dissociation energy of SbH is predicted to be 2.7 ± 0.2 eV.     

Three-nucleon interaction in 3-, 4- and ∞-body systems

We report results of variational calculations of ³H, ³He, ⁴He and nuclear matter with the Urbana v₁₄ two-nucleon interaction and realistic models of the three-nucleon interaction (TNI). These include the Tucson and isobar intermediate-state models of the two-pion exchange TNI. The latter is also studied with an intermediate-range three-nucleon repulsion. In general, realistic TNI helps to bring the theory closer to experiment by giving extra binding energy to the A = 3 and 4 nuclei and providing extra saturation to the nuclear matter binding energy. The Coulomb energy of ³He and the rms radii of A = 3, 4 nuclei are also well described. However, some problems remain unresolved. There is a slight overbinding of ⁴He, an underbinding of nuclear matter, and the charge form factors of ³He and ⁴He, calculated with impulse approximation, deviate from the experimental at q²>5 fm^?2.     

Pion scattering from a bound nucleon at medium energies

In order to examine the validity of the impulse approximation for pion-nucleus scattering in the 33-resonance energy region, we consider pion-scattering from a “nucleus” which consists of a single nucleon bound in a harmonic oscillator potential. A separable πN interaction is assumed. The oscillator parameter is chosen such that the nuclear sizes are fitted for ⁴He ～ ¹⁶O. The binding effect is found to result in a downward shift of the resonance energy (by about 20 MeV), and an increase (by 50 ～ 70%) of the total cross section near the resonance. The angular distribution is also strongly modified. In connection with the binding effect, the importance of a careful treatment of nucleon recoil is emphasized. It is pointed out that the closure approximation which is often used to sum over intermediate nuclear states leads to very misleading results. The effect of the Pauli principle is also examined by excluding some intermediate states.     

Spectrum and energy levels of Cr3+ ions and exchange-coupled Cr3+ pairs in lanthanum aluminate

From the optical spectra the energy levels of the Cr³⁺ ion in LaAlO₃ were determined. In the cubic field approximation we obtained with the parametersDq=1750cm^-1,B=661 cm^-1, andC=2908 cm^-1 a good agreement between the calculated and measured energies. For higher Cr³⁺ concentrations the fluorescence spectrum of the exchange-coupled first nearest (1N) Cr³⁺ pairs was separated from the vibronic fluorescence spectrum of the single ions by selective excitation. From the experimental data the energy level scheme of the 1N pairs is constructed for the states ¦⁴ A ₂·⁴ A ₂〉 and ¦⁴ A ₂·² E〉. In the ground state the pairs are ordered antiferromagnetically due to an isotropic exchange interaction with an exchange integralJ=(?68 ± 1) cm^-1. From the fluorescence decay the intrinsic lifetimes of the fluorescent states of the single ions, the 1N and the 2N pairs were determined to be equal to (86± 4) msec, (1.75± 0.25) msec, and (31 ± 5) msec respectively. The fluorescence decay shows an energy transfer between the single ions and the pairs.     

Modifications in structural and optical properties of Mn-ion implanted CdS thin films

In this paper, we report on modifications in structural and optical properties of CdS thin films due to 190 keV Mn-ion implantation at 573 K. Mn-ion implantation induces disorder in the lattice, but does not lead to the formation of any secondary phase, either in the form of metallic clusters or impurity complexes. The optical band gap was found to decrease with increasing ion fluence. This is explained on the basis of band tailing due to the creation of localized energy states generated by structural disorder. Enhancement in the Raman scattering intensity has been attributed to the enhancement in the surface roughness due to increasing ion fluence. Mn-doped samples exhibit a new band in their photoluminescence spectra at 2.22 eV, which originates from the d-d (⁴T₁ → ⁶A₁) transition of tetrahedrally coordinated Mn²⁺ ions.     

Scaling of the interaction in BECs at large scattering lengths

We have studied the scaling of the interaction in Bose-Einstein condensates of ultracold alkali-metal gases for large scattering lengths and momenta where corrections to the mean field approximation become important. We find that the effective interaction in the metastable, open channel, gaseous phase scales well with the scattering length in the range analyzed. Based on this we show that for increasing scattering lengths, or equivalently increasing densities, the system becomes less correlated, and that at large scattering lengths Bragg scattering experiments can directly measure the effective two-body potential in momentum space. This work is motivated by the recent Bragg-scattering measurements in ⁸⁵Rb by Papp et al. [Phys. Rev. Lett. 101, 135301 (2008)], where the results in the line shifts show clear deviations from the simple contact interaction. We show that those results are well described by a soft spheres potential with parameters chosen to scale in scattering length units. So far the resolution in the experiments does not reveal details on the frequency dependence in the dynamic structure function S(k,ω) and we show that the Feynman spectrum determines the measured line shifts. We also construct the effective atom-atom interaction from two coupled channels, open and closed, assuming that the Feshbach resonance dominates the closed channel. The resonance energy and the scattering length a of the system are tunable by magnetic fields. We derive the T-matrix of such a system and use renormalization to calculate the bound state energy as a function of the magnetic field and make comparison with available experiments. The s-wave phase shifts determine the local, effective open-channel interaction, but if no scaling is used in the cut-off parameters of the renormalization the phase shift resembles more and more the ones obtained from the contact interaction with increasing scattering length. This leads to clear deviations from the measured line shift experiments.     

Connection of Kukulin's nucleon-nucleon deep potential with realistic repulsive core interactions

The on-shell equivalence of the deep quantum-chromodynamically motivated realistic nucleon-nucleon interaction recently proposed by Kukulin et al. with more conventional repulsive-core forces is investigated by eliminating its unphysical deeply bound states, while preserving its scattering properties and the binding energy of the deuteron. The resulting interaction, which is built both in the singlet and triplet channels, displays ar ^?2 singular repulsive core followed by a shallow attraction of intermediate range, in good semiquantitative agreement with existing realistic nucleon-nucleon interactions.