Malfliet-Tjon Potential Research Articles

Elastic scattering study of $$\alpha$$- $$^3$$H and $$\alpha$$- $$^3$$He modeled by Malfliet–Tjon potential using phase function method

Elastic scattering study of $$\alpha$$- $$^3$$H and $$\alpha$$- $$^3$$He modeled by Malfliet–Tjon potential using phase function method

A three-dimensional momentum-space calculation of three-body bound state in a relativistic Faddeev scheme

In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body t–matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.

Four-body bound state calculations using three-dimensional low-momentum effective interaction Vlow-k

In this paper, we solve the coupled Yakubovsky integral equations for four-body (4B) bound state using the low-momentum effective two-body interaction [Formula: see text] in a three-dimensional (3D) approach, without using a partial wave (PW) decomposition. The renormalization group (RG) evolved interaction is constructed from spin-independent Malfliet–Tjon potential using the Lee–Suzuki method. The cutoff dependence of the 4B binding energy and wave function is investigated for a wide range of the momentum cutoff [Formula: see text] of [Formula: see text] interaction from 1.0 to 8.0[Formula: see text][Formula: see text].

Calculation of relativistic nucleon-nucleon potentials in three dimensions

In this paper, we have applied a three-dimensional approach for calculation of the relativistic nucleon-nucleon potential. The quadratic operator relation between the non-relativistic and the relativistic nucleon-nucleon interactions is formulated as a function of relative two-nucleon momentum vectors, which leads to a three-dimensional integral equation. The integral equation is solved by the iteration method, and the matrix elements of the relativistic potential are calculated from non-relativistic ones. Spin-independent Malfliet-Tjon potential is employed in the numerical calculations, and the numerical tests indicate that the two-nucleon observables calculated by the relativistic potential are preserved with high accuracy.

Effect of final state interaction in 3He (π−, π+) 3n reaction

The calculation of the doubly differential cross section was performed using sequential single charge exchange mechanism for the double charge exchange reaction 3He (π−, π+) 3n. The effect of the final state interaction between two active neutrons was investigated using Malfliet Tjon potential. Our results were compared with experimental results at intermediate energy range for the values 120, 180, 210 and 240 MeV. In this study, pion production was used to interpret the low energy peak that was observed on the forward angles at the incident energies of 210 and 240 MeV.

Coulombic Transformation in Momentum Space

We studied the Coulombic transformation of potential in momentum space. The Coulombic transformation is defined as a unitary transformation in momentum space, which is equivalent of the Coulomb–Fourier transformation in coordinate space. The analytic continuation scheme avoids the difficulty which is occurred from the singularity of the Coulomb wave function in momentum space. We adopted the point method to perform the analytic continuation. The validity of the new scheme is checked by comparing with the analytic solution for the Malfliet-Tjon potential. Numerical calculation of the integration was done by separating into four intervals. We demonstrate the high accuracy of our calculation.

Three-Dimensional Low-Momentum Interaction in Two-Body Bound State Calculations

The deuteron binding energy and wave function are calculated by using the recently developed three-dimensional form of low-momentum nucleon-nucleon ( NN ) interaction. The homogeneous Lippmann- Schwinger equation is solved in momentum space by using the low-momentum two-body interaction, which is constructed from Malfliet-Tjon potential. The results for both, deuteron binding energy and wave function, obtained with low-momentum interaction, are compared with the corresponding results obtained with bare potential.

Translationally Invariant Single Particle Picture of the Three-Nucleon System

An ansatz of a single particle picture, known for example from the shell model, has been used to construct a model wave function which is as close as possible to an exact three-body wave function. The exact wave function is obtained by solving the Faddeev equation with the Malfliet–Tjon potential. In order to judge the quality of the model wave function, we compare correlation functions of the model wave function and the exact solution. The correlation functions differ significantly at small distances but are close to each other for larger values of their arguments.

Solving Faddeev equations for a bound state and a continuous spectrum of a three-nucleon system by the method of K-harmonic expansions

A method for solving Faddeev equations in configuration space for a bound state and a continuous spectrum of the system of three nucleons was developed on the basis of expansions in K harmonics. Coulomb interaction and particle spins were not taken into account in this study. The method in question was used to describe the triton bound state and differential cross sections for neutron-deuteron scattering at subthreshold incident-neutron energies. The Volkov, Malfliet-Tjon, and Eikemeier-Hackenbroich local nucleon-nucleon potentials were employed in the present calculations.

Low-momentum effective interaction in the three-dimensional approach

The formulation of the low-momentum effective interaction in the model space Lee–Suzuki and the renormalization group methods is implemented in the three-dimensional approach. In this approach the low-momentum effective interaction V low k has been formulated as a function of the magnitude of momentum vectors and the angle between them. As an application the spin–isospin independent Malfliet–Tjon potential has been used into the model space Lee–Suzuki method and it has been shown that the low-momentum effective interaction V low k reproduces the same two-body observables obtained by the bare potential V N N .

Inclusive electron scattering offHe4

Inclusive electron scattering off $^{4}\mathrm{He}$ is investigated for low and medium energy and momentum transfers. The final state interaction, given by the simple semirealistic Malfliet-Tjon potential, is treated rigorously applying the Lorentz integral transform (LIT) method. In addition to the nonrelativistic one-body current a consistent meson exchange current is constructed and implemented. Results are presented for both longitudinal and transverse response functions at various momentum transfers. Good agreement with experimental data is found for the longitudinal response function, whereas some strength is missing in the transverse response function on the low energy side of the quasi-elastic peak.

Integral equation calculations for the photodisintegration process 4He( gamma,n)3He.

Results obtained by solving Alt-Grassberger-Sandhas (AGS)-type integral equations for the photodisintegration of {sup 4}He, employing the Malfliet-Tjon potential, are compared with the latest experimental data. Good agreement between theory and experiment is found in electric dipole approximation for the total cross section, but the differential cross sections differ at higher energies. This discrepancy is reduced, but not fully removed by taking into account the electric quadrupole contributions. In order to get some feeling for the sensitivity to the underlying potential, we also show calculations based on the Yamaguchi potential. They differ from the Malfliet-Tjon results in a way which resembles the trends known from triton photodisintegration. {copyright} {ital 1996 The American Physical Society.}

Collective excited states of a model 16O nucleus

As a first step toward developing a coordinate-space based, microscopic theory of nuclear giant resonances, we apply Feynman's theory of collective excitation to the study of excited states of a model 16O nucleus. This is a system of 16 nucleons interacting via a purely central Malfliet-Tjon potential. For this Fermi system, our modified, fixed-node diffusion Monte Carlo algorithm yields excellent results for the ground-state energy and one- and two-body densities. By solving the Feynman eigenvalue equation using these densities as inputs, we can further determine the energy and the transition densities of the collective monopole, dipole and quadrupole excitations.

Correlated hyperspherical-harmonic expansion for three-nucleon systems

The hyperspherical-harmonic-expansion method is applied to solve the Schrodinger equation for a three-particle system interacting via central spin-dependent potentials. The convergence of the expansion has been improved by multiplying the hyperspherical basis by an appropriate correlation factor, chosen as a product of one-dimensional functions fixed by means of a two-body Schrodinger equation. The results obtained for three nucleons interacting via the Malfliet-Tjon potential are in close agreement with those given by the most accurate methods.

Euler and correlated harmonic-oscillator wave functions for three-nucleon systems

A variational method is applied for the solution of the Schrödinger equation for a three-particle system interacting via central spin-dependent potentials. The trial wave function is calculated by means of a Euler procedure and it is improved by adding to it a set of correlated basis wave functions. The results obtained for three nucleons interacting via the Malfliet-Tjon potential are in close agreement with those given by the Faddeev theory.

Neutron-deuteron breakup calculations with W-matrix representation of the two-body input.

This paper provides a further test of employing the W-matrix representation of the two-body T matrix in three-nucleon calculations. Calculating differential neutron-deuteron breakup cross sections with the Malfliet-Tjon potentials I and III, we find that the single-term separable approximation for the two-body T matrix which follows from the W-matrix approach, when used as input in the three-nucleon Alt-Grassberger-Sandhas equations, leads to results which agree extremely well with benchmark results provided by Wital-dash-bara, Gloeckle, and Cornelius. The present investigations, thus, corroborate and extend the findings obtained previously for elastic neutron-deuteron scattering. Our theoretical results are also compared with recent experimental data obtained at Erlangen, and it is found that the Malfliet-Tjon potential supports a very good reproduction of the overall features of these data. We conclude that the W-matrix approach provides a remarkably simple and efficient tool for high-quality three-nucleon calculations.

Neutron-deuteron scattering calculations with W-matrix representation of the two-body input.

This paper provides a further test of employing the W-matrix representation of the two-body T matrix in three-nucleon calculations. Calculating differential neutron-deuteron breakup cross sections with the Malfliet-Tjon potentials I and III, we find that the single-term separable approximation for the two-body T matrix which follows from the W-matrix approach, when used as input in the three-nucleon Alt-Grassberger-Sandhas equations, leads to results which agree extremely well with benchmark results provided by Wital-dash-bara, Gloeckle, and Cornelius. The present investigations, thus, corroborate and extend the findings obtained previously for elastic neutron-deuteron scattering. Our theoretical results are also compared with recent experimental data obtained at Erlangen, and it is found that the Malfliet-Tjon potential supports a very good reproduction of the overall features of these data. We conclude that the W-matrix approach provides a remarkably simple and efficient tool for high-quality three-nucleon calculations.

Analytic Sturmian functions and convergence of separable expansions.

A method for solving integral equations is developed and applied to the homogeneous Lippmann-Schwinger equation in momentum space. It has been used with Yukawa-type potentials, V(r)=\ensuremath{\Sigma}, and yields solutions that are analytic expressions rational in the variable ${k}^{2}$. More specifically, the principal-value form of the homogeneous Lippmann-Schwinger equation is solved by making an analytic series expansion of the integral, which is then summed using Pad\'e approximants. An application to the Malfliet-Tjon potential V in s wave is given. In a finite subspace of rational functions with fixed denominators, the solutions, referred to as Sturmian functions, are obtained corresponding to the energy -0.35 MeV, which is the physical bound state energy for this potential. With these analytic eigenfunctions as form factors and with the associated eigenvalues, a separable expansion, namely, the unitary pole expansion, is constructed for the local potential. The unitary pole expansion is then used for analytic k-matrix calculations. At intermediate energies through ${E}_{\mathrm{c}.\mathrm{m}.}$=666 MeV, and at ultrahigh momenta, as the rank of the unitary pole expansion approaches 13, analytic wave functions (or, equivalently, half-shell k matrices) and phase shifts are found that are in good agreement with exact results.This is in accord with our observation that separable expansions should not be regarded as low energy approximations, but instead are finite l approximations, for the following reason. For the class of potentials we consider, the partial wave two-body operators ${V}_{l}$(p,q), ${t}_{l}$(p,q,E), and ${k}_{l}$(p,q,E) are compact even though V(r), T(p,q,E), and K(p,q,E) are not compact. Therefore, in principle, these partial wave operators can be approximated by sequences of separable expansions which converge to them in norm. We show that the unitary pole expansion may be such a sequence. There is a practical need for analytic separable expansions that converge more rapidly. In this light we illustrate how our solutions can be used with the Ernst-Shakin-Thaler formulation of separable expansions, and we also discuss the application of our method to the calculation of Gamow states.

Microscopic calculation of a model 16O nucleus

We present variational calculations for a hypothetical system of 16 nucleons interacting via the Malfliet-Tjon potential and compare our results with the Green-function Monte Carlo (GFMC) data of Helmbrecht et al. For a system of 16 bosons, we find that the (approximate) ground-state energy of our optimized hypernetted-chain calculation agrees with the GFMC data within 1%. For fermions, we find that the optimized Fermi hypernetted chain calculation leads to a ground state energy between the GFMC data and the variational Monte Carlo result with non-optimized correlations; the one-body density predicted by the FHNC theory agrees well with the one predicted by the GFMC calculation.

Path integral Monte Carlo calculations of4He and6Li

We have performed Monte Carlo calculations to estimate the exact energies of model problems for4He and theL=0,1, and 2 states of6Li. Using a Feynman path-integral expression for the imaginary-time evolution operator, we recast the ground state energy as a sum over histories, which are then sampled stochastically. Use of a trial wave function dramatically improves the efficiency of the Monte Carlo method. For a state-independent Malfliet-Tjon potential, together with the Coulomb interaction, we find a ground state energy of −28.00+0.20 MeV for4He, and a degeneracy of theL=0,1, and 2 states in6Li at about −59.65+-0.50 MeV. Density distributions for these nuclei are also calculated.