Equivalent Two-body Method Research Articles

Langevin dynamics with energy dissipation on periodic lattice structures modeling as an equivalent two-body method for atom-surface interactions

In this work, we propose a classical equivalent two-body (ETB) model that can capture more detailed dynamic features arising from energy dissipation and atom oscillations, by introducing the Langevin equation of a harmonic oscillator. The trapping probability, scattering angle and the residence time of Ar interacting with Pt (111) and W (110) surfaces predicted by the ETB model agree well with the measured experimental data or molecular dynamics simulations. Moreover, the ETB model is also used to study the influence of oscillating and dissipating properties on the thermal accommodation coefficients and rainbow scattering of gas atoms colliding with the surface. It is found that the dependence of energy accommodation coefficients and rainbow scattering on the oscillating and dissipating parameters shows nonlinear behaviors, and the associated mechanisms are disclosed. ETB model further provides the possibility to explore the physics beyond the existing two-body models for a better description of energy transferring during atom surface interactions.

Efimov effect in a three-body system with attractive r−2 two-body interactions

The equivalent two-body method has been applied to investigate a three-body system with attractive r−2 two-body interactions, which are the division between short- and long-range potentials. An effective potential of the R−2 form is derived. Infinitely many bound states are obtained in the system with two heavy particles and one light particle even if all the two-body subsystems have no bound states. This is possible evidence of the Efimov effect.

Discussion on Weakly Bound States of Three-Body Systems

The equivalent two-body method for a three-body system has been generalized to an arbitrary three-body system with short-range two-body interactions. An analytical expression for the long-range effective potential is obtained for the Gaussian potential, the Yukawa potential and the exponential potential. The asymptotic behavior of the effective potential at very large distance is found to be universal and an explanation on the significance of universality is given. The weakly bound excited state for the system is first obtained although there is no bound state for two-body subsystems.

Spectral geometry and the N-body problem.

Exact N-body solutions for the harmonic-oscillator and lower bounds provided by the equivalent two-body method are used as basis results for an application of spectral geometry. It is proved that the relative energy E of a system of N identical particles interacting by the attractive pair potential Vof(r/a) and obeying non-relativistic quantum mechanics is given approximately by the expression E ≈ min r>0 { K(N) r2 + vf(r) } , where E = mEa (N − 1)h and v = NmVoa 2 2h . A table of K-numbers is provided for bosons and fermions (with arbitrary spin) in one and three dimensions. If the potential shape f(r) is a convex function of − 1r and a concave function of r, then the approximations yield both upper and lower energy bounds. Detailed results are given for the case of gravitating fermion systems.

Long-Range Effective Potential in Halo Nuclei11The project supported by CNRS-K.C. Wong Foundation and National Natural Science Foundation of China.

A way to derive the long-range effective potential and the corresponding eigenequation for halo nuclei and is presented by using the equivalent two-body method for a three-body system. The results agree with both Efimov's conclusions and experimental data for halo nuclei.

New perspective on neutron haloes in exotic light nuclei

The three-body model with central two-body Yukawa interactions is applied to the nuclei 11Li, 14Be and 17B and a long-range effective potential is obtained using the equivalent two-body method for a three-body system. It is first pointed out that the long-range effective potential indicates the effect of the three-body correlation and leads to neutron haloes in these nuclei. Quite good agreement between theoretical results and the experimental data is obtained.

Weakly bound states of a three-body system.

The equivalent two-body method [H. Feshbach and S. I. Rubinow, Phys. Rev. 98, 188 (1955)] has been applied to investigate the weakly bound states of a three-body system. An analytical expression for the long-range effective potential is derived if two-body interactions are short-range potentials with Yukawa forms. The weakly bound excited state for the system with two heavy particles and a light particle is first obtained if there is no bound state for two-body subsystems.

Gravitating boson systems.

We study a system composed of N identical bosons interacting in three dimensions via attractive Yu-kawa pair potentials V(r)=-e -λr /r. Four approaches to the problem are related: the «equivalent two-body method,» translation-invariant Gaussian trial functions, collective field theory, and Hartree trial functions. Upper and lower energy bounds are given and are compared to some recent results which were obtained independently by an optimized Hartree method [M. Membrado, F. Pacheco, and J. Sanudo, Phys. Rev. A 39, 4207 (1989)]

The equivalent two-body and delta energy lower bounds for N-boson systems

Studies the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)= gamma f(x/a) and obey non-relativistic quantum mechanics. A recent energy lower bound based on the known exact solution for the delta potential is compared to an earlier bound provided by the equivalent two-body method. The delta bound is good whenever the potential is very narrow and deep. For any bounded potential, the earlier energy bound is always better for large N. Detailed results and graphs are given for the sech-squared potential and the square-well potential.

Complementary pictures of the N-boson problem.

We consider a system of N identical bosons which interact in one dimension via attractive pair potentials and obey nonrelativistic quantum mechanics. This system is studied from two complementary viewpoints. The equivalent two-body method approximates the system as a collection of (N-1) independent two-particle systems with coupling constants enhanced by the factor N/2, and this yields energy lower bounds. The collective-field method approaches the problem from the standpoint of the limit as N\ensuremath{\rightarrow}\ensuremath{\infty} and leads to energy upper bounds. We find that Gaussian trial functions for the N-particle Hamiltonian and Gaussian trial densities in the collective-field theory lead to precisely the same energy upper bounds. The upper bounds provided by field theory allow for systematic improvement via a variational principle. Details are worked out for two exactly soluble problems, namely, the harmonic oscillator and the attractive \ensuremath{\delta} potential.

Complementary energy bounds for N-boson systems with linear pair potentials.

We study the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)=\ensuremath{\gamma}\ensuremath{\Vert}x\ensuremath{\Vert} and obey nonrelativistic quantum mechanics. It is shown that the energy is given by the formula \ensuremath{\varepsilon}=C(N)(N-1)(${\ensuremath{\Elzxh}}^{2}$/ m${)}^{1/3}$(\ensuremath{\gamma}N${/2)}^{2/3}$, where 1.01879(N)1.02333 for all N\ensuremath{\ge}2. The lower bound is provided by the ``equivalent two-body method'' whereas the upper bound is derived by the use of collective field theory. The general relation between these complementary theories is investigated.

Faddeev theory and the4He trimer problem

The binding energy of (4He)3 has been calculated using the Faddeev theory and a model 4He-4He interaction of Herzfeld form. Both Blatt's SQA3 and Burkhardt's Lennard-Jones phase-equivalent parameters yield bound trimers with ground-state energies of 0·07 and 0·02 K respectively. These results compare very favourably, in speed and accuracy, with those obtained from the equivalent-two-body-method (ETBM) and suggest the applicability of Faddeev methods to these molecular systems.

On the existence of bound helium trimers by the equivalent two-body method

The binding energies of the 3,4He trimers are computed for a range of realistic interatomic potentials using the equivalent two-body method of Bodmer and Ali. We find a 4He trimer bound with a binding energy of about 0.25°K and a 3He trimer which is but loosely bound. We also confirm that the Hall–Post–Stenschke method gives poor lower bounds to the energy and that (4He)3 is large, having a root mean square interparticle spacing of 23 Å.

Calculation of the 3He charge form factor with the Reid potential

The 3He charge form factor has been calculated with the Reid soft-core potential using an altered Feshbach-Rubinow equivalent two-body method. A minimum near the experimental one is found, supporting the conclusion of Yang and Jackson.

Calculation of theHe3Charge Form Factor

The ${\mathrm{He}}^{3}$ charge form factor has been calculated using phenomenological mixed symmetric $S$- and $D$-state wave-function components along with the symmetric $S$-state component developed previously from the Bressel-Kerman-Rouben potential using an altered Feshbach-Rubinow equivalent two-body method. A zero of the form factor near the experimental value is found for reasonable ${S}^{\ensuremath{'}}$ and $D$ admixtures. This appears to weaken a suggestion by Tjon, Gibson, and O'Connell concerning the nucleon-nucleon interaction.

The triton wave function in the feshbach-rubinow approximation

Previous work on the triton wave function using the Feshbach-Rubinow equivalent two-body method has been extended to include all of the symmetric and mixed symmetric S- and D-state components. The modified Feshbach-Rubinow method and the potential deepening procedure used previously are found to yield reasonably accurate results for the triton energy and wave function corresponding to the Feshbach-Pease potentials. A method for handling soft-core potentials is used to find an approximate symmetric S-state component of the triton wave function corresponding to the Bressel-Kerman-Rouben potential. The body form factor calculated with this function has a zero for q 2 = 13.8 fm −2.

Further calculations on a simple triton model

The earlier work by two of the authors on a simple triton is extended in two respects. First of all the equivalent two-body method of Bodmer and Ali is applied to some of the potentials previously considered, and the results are compared. The conclusion is that this method is far superior to previously used methods if applied to systems of three identical spinless particles, and if only product type wave functions are used. In the second place the various methods are applied to the case where the interaction is a Yukawa potential, which has been used by Wong and Zambotti to test their method of solving the Faddeev equation. The conclusion is that our methods give better results than theirs except for very small strength parameters.

On the symmetric S- and D-state components of the triton wave function

Approximate forms for the symmetric S- and D-state components of the triton wave function are found using the Feshbach-Rubinow equivalent two-body method. Two coupled ordinary differential equations for the components are obtained, and are solved numerically with the Feshbach-Pease potentials. A further approximation involving one variational parameter is shown to yield good results. Appendices contain the Feshbach-Pease variational wave function written in Derrick-Blatt and Sachs notations, and the detailed relationships between the Derrick-Blatt and Sachs basis functions.

Two-body methods for three-body problems

The “Equivalent two-body method” first introduced by Wigner for deriving triton wave functions is shown to be obtainable directly from the variation principle for the energy. It is extended to cover the general case of noncentral, hard core interactions, for both the bound state and for scattering states, including the breakup reaction. Methods are given for systematically improving the wave function.

The Equivalent Two-body Method for the Hypertriton

The equivalent two-body method proposed by Feshbach and Rubinow in 1955 has been applied to the calculation of the volume integral of the A-nucleon interaction necessary to explain the observed binding energy of the hypertriton. An appreciably smaller value of the volume integral is required than when a trial function of comparable simplicity is employed in the usual variation method. The effect of introducing a term in the potential representing a three-body A-nucleon-nucleon interaction is investigated.