Quasiclassical analysis of the spectra of two groups of central potentials

A method is suggested for analyzing the spectra of central attractive potentials either with Coulomb singularity (intra-atomic potentials) or finite at zero point (potentials in spherical clusters and nuclei). It is shown that, if the orbital degeneracy is removed, then $\varepsilon _{nl} - \varepsilon _{n0} \cong a_{\varepsilon _{n0} } (l + 1/2)^2 $ for small l in the shell n. In atoms and ions, the coefficient a _ε is nonnegative, so that the energy in the n shell increases with l. The validity of this formula for the inner electrons is illustrated by calculating the spectrum of the mercury atom. In cluster potentials, the opposite situation, as a rule, occurs: the larger l, the lower the corresponding level (a _ε<0). However, in the soft potentials of small clusters, spectral regions with different signs of a _ε coexist and the orbitally degenerate level exists in the spectral region where a _ε=0. Aluminum clusters Al_N are taken as an example to find out how the position of the region with the degenerate level varies with varying cluster size N, and it is found that this region is “pushed out” to higher energies with an increase in N. In this connection, the presence of multiply ionized Al_N clusters of the corresponding size in a low-temperature aluminum plasma is discussed.     

Optical properties of nano-multi-layered quantum dot: oscillator strength, absorption coefficient and refractive index

In this paper, the exact solution of Schrödinger equation for multi-layered quantum dot (MLQD) within the effective mass approximation and dielectric continuum model is obtained with finite and infinite confining potential (CP). The MLQD is a nano-structured semiconductor system that consists of a spherical core (GaAs) and a coated spherical shell (Ga $_{1-x}$ Al $_{x}$ As) as the whole dot is embedded inside a bulk material (Ga $_{1-y}$ Al $_{y}$ As). Using the obtained energies, wave functions and taking advantage of numeric calculations, the oscillator strength, refractive index and absorbtion coefficient change associated with intersubband electronic transition from the ground state to the first allowed excited state are investigated for different CPs (both finite and infinite) and shell thicknesses. The results show that all values of ground state energy for large core dot radius approach the same value (the energy of bulk material) independent of CPs and shell thicknesses. Also it is shown that the optical properties are strongly affected by the changes in CPs and shell thicknesses.     

The effect of density dependent Av18 effective interaction on the ground state properties of heavy closed shell nuclei

The channel-dependent Argonne Av₁₈ effective two-body interactions (CDEI) which are generated through the lowest order constrained variational (LOCV) calculation for asymmetric nuclear matter with the charge-dependent Av₁₈ bare nucleon–nucleon potential are used to calculate the ground state properties of heavy closed shell nuclei such as ⁴⁸Ca, ⁹⁰Zr, ¹²⁰Sn and ²⁰⁸Pb. The harmonic oscillator basis, and the local density approximation (LDA) are applied to create the relative and the center of mass dependent effective two-body potential. We get more binding with respect to the similar calculation with the Reid types potentials. It is tried to omit the LDA and perform full calculation with the Av₁₈CDEI for light nuclei. The results indicate that the LDA works quite well. It is also shown that in case of heavy closed shell nuclei and unlike our previous report with Reid68Day   interaction, the contributions of higher partial waves (J>2$J>2$) are very important for the calculations with Av₁₈ potential and we get reasonable agreement between our calculated binding energies and RMS radii, with those predicated by the others methods, and the experimental data. Finally, the various aspects of channel and density dependent two-body effective interactions are discussed.     

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.     

An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Three-alpha force and the 3-α model of12C

We postulate a Gaussian three-body potential amongα particles and adjust its parameters so that, when it is added to the Ali-Bodmerα-α potential, a good fit to experimental energies of low-lying 0⁺ and 2⁺ states of¹²C is achieved. With these potentials we made a linear variational calculation in a basis of harmonic oscillator functions which are translationally invariant, completely symmetric, and have a definite orbital angular momentum. We study the influence of this three-body potential on elastic and inelastic form factors, transition widths, Coulomb energy and charge radius of the 3-α system. The 3-α potential improved results found with the Ali-Bodmer potential alone. We find the 0 ₂ ⁺ state to be a (non-rigid) linear chain and the ground state to be a triangle ofα particles.     

On the elliptic flow for nearly symmetric collisions and nuclear equation of state

We present the results of elliptic flow for the collision of nearly symmetric nuclei (₁₀Ne²⁰+ ₁₃Al²⁷_{10}{\rm Ne}^{20}+\,_{13}{\rm Al}^{27}, ₁₈Ar⁴⁰+ ₂₁Sc⁴⁵_{18}{\rm Ar}^{40}+\,_{21}{\rm Sc}^{45}, ₃₀Zn⁶⁴+ ₂₈Ni⁵⁸_{30}{\rm Zn}^{64}+\,_{28}{\rm Ni}^{58}, ₃₆Kr⁸⁶+ ₄₁Nb⁹³_{36}{\rm Kr}^{86}+\,_{41}{\rm Nb}^{93}) using the quantum molecular dynamics (QMD) model. General features of elliptic flow are investigated with the help of theoretical simulations. The simulations are performed at beam energies between 45 and 105 MeV /nucleon. A significant change can be seen from in-plane to out-of-plane elliptic flow of different fragments with incident energy. A comparison with experimental data is also made. Further, we show that elliptic flow for different fragments follows power-law dependence as given by the function C(A_tot)^tC{(A_{\rm tot})^\tau}.     

Relativistic corrections to Coulomb energy calculations

Coulomb energies of nuclei have been calculated using a recently introduced relativistic nuclear shell model¹). The results are very close to those of the usual non-relativistic, isotropic harmonic oscillator shell model, showing the most deviation for heavy elements such as lead.     

Thomas-Fermi theory for atomic nuclei revisited

The recently developed semiclassical variational Wigner-Kirkwood (VWK) approach is applied to finite nuclei using external potentials and self-consistent mean fields derived from Skyrme interactions and from relativistic mean field theory. VWK consists of the Thomas-Fermi part plus a pure, perturbative ?² correction. In external potentials, VWK passes through the average of the quantal values of the accumulated level density and total energy as a function of the Fermi energy. However, there is a problem of overbinding when the energy per particle is displayed as a function of the particle number. The situation is analyzed comparing spherical and deformed harmonic oscillator potentials. In the self-consistent case, we show for Skyrme forces that VWK binding energies are very close to those obtained from extended Thomas-Fermi functionals of ?⁴ order, pointing to the rapid convergence of the VWK theory. This satisfying result, however, does not cure the overbinding problem, i.e., the semiclassical energies show more binding than they should. This feature is more pronounced in the case of Skyrme forces than with the relativistic mean field approach. However, even in the latter case the shell correction energy for e.g., ²⁰⁸Pb turns out to be only ∼−6 MeV what is about a factor two or three off the generally accepted value. As an ad hoc remedy, increasing the kinetic energy by 2.5%, leads to shell correction energies well acceptable throughout the periodic table. The general importance of the present studies for other finite Fermi systems, self-bound or in external potentials, is pointed out.     

Supermultiplicity and the relativistic Coulomb problem with arbitrary spin

The Hamiltonian for n relativistic electrons without interaction but in a Coulomb potential is well known. If in this Hamiltonian we take r _′ ^u =r′, P _′ ^u =P′ with u=1,2,..., n, we obtain a one-body problem in a Coulomb field, but the appearance of n of the α _u , u=1,..., n, each of which corresponds to spin $\tfrac{1}{2}$ , indicates that we may have spins up to (n/2). We analyze this last problem first by denoting the 4×4 matrices α, β as direct products of 2×2 matrices which correspond to the ordinary spin, and a new concept, also related to the SU(2) group, which we call sign spin. In this new notation our problem depends on the sixteen generators of a U(4) group reduced along the chain Û(2)??(2) sub-groups associated with the ordinary and sign spins. We now make a change of variables in our Hamiltonian so a term ε related to the frequency ω of an oscillator, which will be our variational parameter, appears in it, and later construct the full states of the problem with a harmonic oscillator of frequency 1 and ordinary and sign spin parts. Finally we obtain the matrix representation of our Hamiltonian with respect to the states mentioned and discuss the energy spectra of the problem where the partition {h} representing the irrep of U(4) and j the total angular momentum, take the values {h}=[1], j= $\tfrac{1}{2}$ ; {h}=[11], j=0; {h}=[2], j=0.     

Variational Calculation of 4He Tetramer Ground and Excited States Using Realistic 4He–4He Potentials

We calculated binding energies and wave functions of the ⁴He tetramer ground and excited states employing various realistic ⁴He?⁴He potentials which includes the currently most accurate one with the adiabatic, relativistic, QED and residual retardation corrections. We used our Gaussian expansion method (GEM) for ab initio variational calculations of few-body systems. We found that precisely the same shape of the short-range correlation (r _ij < 4Å) in the dimer appear in the ground and excited states of trimer and tetramer. The four kinds of the binding energies of the trimer and tetramer ground and excited states, ${B_3^{(0)}, B_3^{(1)}, B_4^{(0)}}$ and ${B_4^{(1)}}$ , for the different potentials exhibit perfect linear correlations over the range of binding energies relevant for ⁴He atoms; namely, six types of the generalized atomic Tjon lines were observed.     

ΛΛ 6 He and Λ 9 Be systems in the three-body cluster model treated on the basis of differential Faddeev equations

The _ΛΛ ⁶ He and _Λ ⁹ Be hypernuclei are treated as the S=0, T=0 (for the former) and S=1/2, T=0 (for the latter) bound states of the three-cluster systems ΛΛα and Λαα, respectively. The cluster-reduction method is used to solve the s-wave differential Faddeev equations for these systems. On the basis of the MT I–III model, the ΛΛ interaction potential is specified in the form $V_{\Lambda \Lambda } = \frac{2}{3}V_{NN} $ . Phenomenological potentials are used to describe Λα and αα interactions. The binding energies of the _ΛΛ ⁶ He and _Λ ⁹ Be hypernuclei and the parameters of low-energy Λ-hyperon and α-particle scattering on a _Λ ⁵ He hypernucleus are calculated. It is shown that the proposed ΛΛ interaction potential makes it possible to reproduce faithfully the binding energy of the _ΛΛ ⁶ He hypernucleus and that scattering in the Λ _Λ ⁵ He system is similar to neutron scattering on a deuteron.     

Effect of isospin-dependent cross-section on fragment production in the collision of charge asymmetric nuclei

To understand the role of isospin effects on fragmentation due to the collisions of charge asymmetric nuclei, we have performed a complete systematical study using isospin-dependent quantum molecular dynamics model. Here simulations have been carried out for $^{124}{\rm X}_{n}+^{124}{\rm X}_{n}$ , where n varies from 47 to 59 and for $^{40}{\rm Y}_{m}+^{40}{\rm Y}_{m}$ , where m varies from 14 to 23. Our study shows that isospin-dependent cross-section shows its influence on fragmentation in the collision of neutron-rich nuclei.     

Half-life measurements of 137, 139Cs excited nuclear states

A fast-timing setup has been developed to measure the nuclear-state half-lives of neutron-rich nuclei. The first on-line measurements were performed on ^{137, 139}Cs obtained from the β decay of ^{137, 139}Xe. These neutron-rich Xe nuclei were produced by the ²³⁸U photofission induced by the 50MeV electron beam delivered by the ALTO facility. The half-lives of the first excited state in ¹³⁷Cs and of six excited states in ¹³⁹Cs were measured for the first time. Taking into account new information given by the deduced transition probabilities, we discuss the structure of the first excited states in ^{137, 139}Cs : it is shown that, in both nuclei, the first excited $\tfrac{5} {2}^ +$ level corresponds mainly to the one-proton state issued from the π2d _5/2 orbital and that the most probable spin values for the states located at 289.8, 393.5, and 393.8keV in ¹³⁹Cs are $\tfrac{5} {2}^ +$ , $\tfrac{3} {2}^ +$ , and $\tfrac{1} {2}^ +$ , respectively.     

Energy dependence of {\bar{K}N} interaction in nuclear medium

When the $\bar{K}N$ system is submerged in nuclear medium the $\bar{K}N$ scattering amplitude and the final state branching ratios exhibit a strong energy dependence when going to energies below the $\bar{K}N$ threshold. A sharp increase of $\bar{K}N$ attraction below the $\bar{K}N$ threshold provides a link between shallow $\bar{K}$ -nuclear potentials based on the chiral $\bar{K}N$ amplitude evaluated at threshold and the deep phenomenological optical potentials obtained in fits to kaonic atoms data. We show the energy dependence of the in-medium K ^??? p amplitude and demonstrate the impact of energy dependent branching ratios on the Λ-hypernuclear production rates.     

Effect of shell structure in the fusion reactions

The fusion reactions are studied in the central collisions ⁸²Se+ + ¹³⁴Ba and ⁸²Se+ + ¹³⁸Ba by the improved isospin-dependent quantum molecular-dynamics model, where the nucleus ¹³⁸Ba has a closed neutron shell N = 82 . Comparing the shell correction energies and fusion probabilities of these two reactions with the ones of other asymmetric or more symmetric reaction systems that form the same compound nuclei, we find the dependence of the fusion reaction on the nuclear shell structure of the colliding nuclei. The experimental data of the fusion probabilities are described well by the present model. The result suggests that the neutron shell closure N = 82 promotes fusion.     

Quark mixing angles in the left-ringt symmetric model with lightW R fromK 0-K 0 mixing

K _L?K _S mass difference and the CP violation parameter, ?, of theK ⁰ ? \(\overline {K^0 } \) system are used to set bounds on the right-handed Cabibbo-like angle and the CP violating phase angle in the left-right symmetric electroweak model of four quarks. The corresponding mixing and phase angles in typical left-right asymmetric models (g _L≠g _R) are also determined.     

Self-Similar Strcture Shell Model

In Order to extend the conventional shell model(SM) calculation with harmonic oscillator bases to halo nuclei, a self-similar, structure shell model(SSM) is proposed. The SSM is achieved by a rescaling of both the Kinetic and potential energy term of the harmonic oscillator and a mean field imitation, so that the single particle orbit in SSM has state (orbit) -dependent frequency. The large r. m. s. radius and the thick neutron skin for halo nuclei as well as the bound state properties of Borromean nuclei such as ⁶He, ¹¹Li and ¹⁴Be can be reproduced.     

Nuclear structure of the heavyN=82 isotones:144Sm and146Gd

For single-closed shell nuclei with a large number of extra-protons, outside theZ=50,N=82 double-closed shell nucleus, a standard shell-model calculation yields dimensions of the Hamiltonian matrices beyond the scope of present computer technology. A quasi-particle (QP) calculation however, is able to describe the most important proton-excitations in a much restricted configuration space. Extended BCS+TDA calculations are performed on the single-closed shell¹⁴⁴Sm and¹⁴⁶Gd nuclei with a Gaussian force as effective two-body interaction. The influence of the nonconservation of the exact number of extra-protons in the BCS-approach is studied by performing a projection onto the states with an exact number of extra-protons (BCS+P). The first excited 2⁺, 4⁺ and 6⁺ states in¹⁴⁶Gd are described mainly by the (1g7/2) ₀ ⁸ (2d5/2) ₀ ⁴ \((1h11/2)_{J^\pi }^2 \) configuration.