Gaussian approximations for the nuclear Coulomb interaction

The kernel 1/¦r-r′¦=1/y in the direct term of the average Coulomb potential of the nuclear Hartree-Fock model is approximated by a sum of gaussians iny. For 0.5≦y≦30 Fm, a sixteen term expression is found such the direct Coulomb energy is obtained to one part in 10³. The exchange Coulomb potential is estimated in the statistical model. Applications of these accurate and practical approximations to fission calculations are discussed.     

Long range absorption and other optical-model effects from strong inelastic coupling

An optical potential component is constructed to represent the effect of a strongly coupled inelastic excitation upon elastic scattering. In the particular case of quadrupole Coulomb excitation a long range imaginary potential component is derived in closed form. The effects of long range absorption upon the elastic scattering are considered in a general way by inserting this potential into a weak absorption model and deriving an elastic scattering cross section in closed form. Below the Coulomb barrier the formula takes a simple form which may be related to the semiclassical theory of Coulomb excitation. The potential component arising from nuclear excitation of an inelastic state may be evaluated numerically on a computer. Two examples computed (50 MeV α-scattering on ¹⁵⁴Sm and 60 MeV ¹⁶O scattering on ⁴⁰Ca) exhibit strong l-dependence in the potential component.     

Relativistic mean-field study on proton skins and proton halos in exotic nuclei

We investigate the ground state properties of proton-rich nuclei in the framework of the relativistic mean-field model. Calculations show that the experimental proton halo in the nuclei ^26,27,28P can be reproduced by the model. The proton halos can appear in proton-rich nuclei because the total nuclear potential is attractive up to the radial distance r ≈ 5.5 fm. But the size of proton halos is finite due to the limitation of the Coulomb potential barrier. The mean-square radius of a halo proton is not very sensitive to the separation energy of the last proton in some very proton-rich nuclei due to the effect of the Coulomb barrier. This behavior is different from the case of a neutron halo where the mean-square radius of a halo neutron is inversely proportional to the separation energy of the last halo neutron. We have also analysed the differences of the relativistic mean-field potentials of ²⁵Al and ²⁶P and found that the isovector potential from the p meson has an important effect on the differences.     

Quantum-statistical mechanics of a system of charged particles at high temperatures

If a quantum-mechanical potential is introduced the calculation of the quantummechanical binary distribution function for a system with Coulomb interaction is reduced to the well-known mathematical formalism of classical statistical mechanics in the case ofnλ³?1 (λ being the thermal wavelength). The two-particle quantummechanical potential is determined by the two-particle Slater sum. In this paper we calculated the two-particle Slater sum using an expansion according toe ² and the resolvent formalism. From the binary distribution function the correlation energy and the free energy as well were determined up to ordere ⁶. Symmetry effects were taken into account.     

Two-particle density matrix for Coulomb interaction

An integral representation is derived for the density matrix of two particles interacting via a Coulomb potential. In particular this representation yields a simple expression for the exchange term of the Slater sum of two identical particles which is suitable for numerical calculations. Results are given for several values of the parameter ξ=e ²/kT      

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.     

A strangeonium potential model with completeO(v 2/c 2) hyperfine splittings

We model the strangeonium system by numerically solving the Schrödinger equation with a Coulomb plus linear scalar confining potential. All hyperfine terms toO(v ²/c ²) are treated in detail; we show that neglect of spin-independent terms is unjustified. The model is fit to the ?(1019.6) andf′ (1516), and masses for excitedss states are predicted: the strangeonium parameters are contrasted with the values found in charmonium.     

Approximate scattering solutions of the dirac equation for electron-nucleus processes in light nuclei

Scattering solutions of the second-order Dirac equation for the case of the Coulomb potential and which are correct to first order in the coupling constantZe ²/hc are investigated and found to describe pure Coulomb scattering equally well as the Sommerfeld-Maue wave functions. Errors introduced by the use of these solutions are studied in a numerical calculation of cross sections for nuclear electric-quadrupole excitation by high-energy electrons. The use of these wave functions is suggested for simplified calculations of lowest-order Coulomb corrections to Born approximation results for various electron-nucleus processes.     

Time-dependent Hartree-Fock calculation of 12C + 12C with a realistic potential

We have used a realistic single-panicle K-matrix model to compute the head-on scattering of ¹²C + ¹²C at incident projectile lab energies of 3.2, 6.4, 12.8, 19.2, 25.6, 32, 51.2 and 64 $\text{MeV}\text{nucleon}$, above the Coulomb barrier, in the time-dependent Hartree-Fock approximation. Direct and exchange Coulomb forces as well as spin-orbit forces are included. A large deformed harmonic oscillator basis is used. Spatial density and current distributions at various times are shown. The outgoing energy is found to be E₀ = 0.8E_in?28 (MeV), in the c.m. system. Fusion and fully relaxed scattering are observed at low energy. Some compression is seen at higher energies but no shock waves can be detected. Consequences for heavy-ion reactions are discussed.     

Second Coulomb energy differences of isospin triplets in the 2s-1d shell

Second Coulomb energy differences, which in the present case are proportional to the tensor Coulomb energy, are calculated for 0⁺, T = 1 ground states in the region 18 ≦ A ≦ 42 using a shell model that includes a pairing interaction. The calculation is done with a mathematical formalism that includes p-n pairs as well as p-p and n-n pairs. Besides an enhancement of proton-pair Coulomb energies, the pairing interaction is responsible for lowering the Coulomb energy of N = Z members of isospin triplets and also gives rise to an important term in the second energy difference. Using pairing strengths derived from fitting energy levels for mass-18 and mass-42 nuclei, results of the calculation reproduce experimental second energy differences extremely well.     

Spatial correlations in the electron gas: Path integral Monte Carlo simulation

Thermally excited states of the three-dimensional electron gas in a neutralizing background are computed by path integral Monte Carlo simulation for values of the Wigner-Seitz radius within the interval 5 < r _s < 15. Coulomb and exchange interactions, permutation symmetry, and spin state are treated explicitly. Variation of electron correlation functions with density and temperature is analyzed. Quantum effects suppress and enhance spatial correlation at low and high densities, respectively. Transition between the electron-gas states characterized by these opposite trends corresponds to a density of approximately 2.5 × 10²¹ cm^?3. A transition line between liquid-like and gaslike phases is determined in the temperature-density diagram. Weak anisotropy of many-body correlations in the liquid-like state stimulates excitation of spherically symmetric collective rotational modes. The effective short-range pseudopotential exhibits strong temperature dependence due to exchange effects. For strongly correlated systems, the characteristic screening length deviates from that predicted by the Thomas-Fermi screening model ( $ \sim \sqrt {r_s } $ ), approaching a linear function of r _s. The effective short-range interaction substantially differs from the Yukawa potential in mean field theory. Coulomb interaction shifts the Fermi level up by an order of magnitude or higher, and this effect becomes stronger with decreasing density.     

Asymptotic Normalization Coefficients in a Potential Model Involving Forbidden States

It is shown that values obtained for asymptotic normalization coefficients by means of a potential fitted to experimental data on elastic scattering depend substantially on the presence and the number n of possible forbidden states in the fitted potential. The present analysis was performed within exactly solvable potential models for various nuclear systems and various potentials without and with allowance for Coulomb interaction. Various methods for changing the number n that are based on the use of various versions of the change in the parameters of the potential model were studied. A compact analytic expression for the asymptotic normalization coefficients was derived for the case of the Hulthén potential. Specifically, the d + α and α + ¹²C systems, which are of importance for astrophysics, were examined. It was concluded that an incorrect choice of n could lead to a substantial errors in determining the asymptotic normalization coefficients. From the results of our calculations, it also follows that, for systems with a low binding energy and, as a consequence, with a large value of the Coulomb parameter, the inclusion of the Coulomb interaction may radically change the asymptotic normalization coefficients, increasing them sharply.     

Continuum resonances with shielded Coulomb-like potential and Efimov effect

Motivated by the possibility of the second energy level (0 ₂ ⁺ ) of ¹²C (in a three-alpha model) to turn into an Efimov state, we study a simple non-realistic toy model formed by three bosons interacting by the phenomenological s-wave Ali-Bodmer potential plus a Coulomb interaction. An artificial three-body potential was used to create a resonance with energy close to the energy of the 0 ₂ ⁺ of ¹²C, 0.38 MeV. The strength of the Coulomb potential is decreased until the energies of the two alpha pairs are zero. The system was placed inside a harmonic trap and a stabilization method has been used to calculate the energies of the resonances. We found that the shielded-Coulomb potential, which keeps the long tail, is not able to produce the Efimov effect. The energy of the three alphas decreases only to 0.19 MeV when the two-body energy crosses the threshold to become bound.     

Hyperfine splitting in positronium and muonium

We calculate corrections to the ground state hyperfine structure of relative order α² ln α^?1 in positronium and relative order $\text{(}\text{m}{}_{\mathrm{<mtext>e</mtext>}}\text{m}{}_{\mathrm{\mu }}\text{)α}{}^2\text{lnα}{}^{\mathrm{?1}}$ in muonium due to the exchange of virtual photons. Our results are in agreement with those of Lepage. Contributions arising from the Coulomb potential and from the exchange of one transverse photon along with any number of ladder Coulomb photons are discussed in detail. In treating the single transverse photon-multiple Coulomb photon exchanges, we sum the contributions involving different numbers of Coulomb photons and reexpand the resulting expression in terms of a quantity that is inherently smaller than the Coulomb potential in the non-relativistic region. The procedure enables us to take into account from the beginning important cancellations that occur between the various terms in an expansion in powers of the Coulomb potential. The techniques developed here may be useful in calculating higher order corrections.     

The stopping of slow75As recoil atoms in He,Ne and H2

Recent measurements of the 2p—1s X-ray line in theK ^? p atom indicate that the 1s level shift is very different from the theoretical prediction based on the extrapolation from theK ^? p scattering data. Deloff and Law have suggested that an anomalously large Coulomb effect could be responsible for this discrepancy. We examine this possibility by carrying out an explicit model calculation, using the method by which Sauer demonstrated the possible existence of an anomalously large Coulomb effect in the proton-proton system. Our conclusion is that if there exists an anomalous Coulomb effect which explains theK ^? p atomic level shift result, then the same effect would cause a serious conflict with theK ^? p scattering data above threshold.     

Double folding model analysis of elastic scattering of halo nucleus 11Be from 64Zn

Calculations of elastic scattering cross-sections for ^9,10,11Be + ⁶⁴Zn at near-Coulomb barrier energy have been performed using a potential obtained from the double folding model and are compared with the experiment. In the framework of the double folding model, the nuclear matter densities of ^9,10,11Be projectiles and a ⁶⁴Zn target are folded with the complex energy-dependent effective M3Y interaction. The angular distributions of the differential cross-section for ^9,10Be scattering from ⁶⁴Zn at E _c.m≈ 24.5 MeV agree remarkably well with the data, while in case of ¹¹Be, calculations show a Coulomb–nuclear interference peak which is not observed in the data.     

Folding model analysis of the nucleus–nucleus scattering based on Jacobi coordinates

This paper presents the results of scattering of ¹⁶O+²⁰⁹Bi interaction near the Coulomb barrier. The interaction potential between two nuclei is calculated using the double folding model with the effective nucleon–nucleon (NN) interaction. The calculations of the exchange part of the interaction were assumed to be of finite-range and the density dependence of the NN interaction is accounted for. Also the results are compared with the zero-range approximation. All these calculations are done using the wave functions of the two colliding nuclei in place of their density distributions. The wave functions are obtained by the D-dimensional wave equation using the hyperspherical calculations on the basis of Jacobi coordinates. The numerical results for the interaction potential and the differential scattering are in good agreement with the previous works.     

The dependence of scattering length on van der Waals interaction and reduced mass of the system in two-atomic collision at cold energies

The static exchange model (SEM) and the modified static exchange model (MSEM) recently introduced by Ray in Pramana – J. Phys.83, 907 (2014) are used to study the elastic collision between two hydrogen-like atoms when both are in ground states by considering the system as a four-body Coulomb system in the centre of mass frame, in which all the Coulomb interaction terms in direct and exchange channels are treated exactly. The SEM includes the non-adiabatic short-range effect due to electron exchange. The MSEM added in it, the long-range effect due to induced dynamic dipole polarizabilities between the atoms e.g., the van der Waals interaction. Applying the SEM code in different H-like two-atomic systems, a reduced mass (μ) dependence on the scattering length is observed. Again, applying the MSEM code on H(1s)–H(1s) elastic scattering and varying the minimum values of interatomic distance R₀, the dependence of scattering length on the effective interatomic potential consistent with the existing physics is observed. Both these basic findings in low and cold energy atomic collision physics are quite useful and are being reported for the first time.