Model Study of Three-Body Forces in the Three-Body Bound State

The Faddeev equation for the three-body bound state with two- and three-body forces is solved directly as three-dimensional integral equation. The numerical feasibility and stability of the algorithm, which does not employ partial wave decomposition is demonstrated. The three-body binding energy and the full wave function are calculated with Malfliet-Tjon-type two-body potentials and scalar two-meson exchange three-body forces. For two- and three- body forces of ranges and strengths typical of nuclear forces the single-particle momentum distribution and the two-body correlation function are similar to the ones found for realistic nuclear forces.     

Three-Body Scattering Below Breakup Threshold: An Approach Without Using Partial Waves

 The Faddeev equation for three-body scattering below the three-body breakup threshold is directly solved without employing a partial-wave decomposition. In the simplest form it is a three-dimensional integral equation in four variables. From its solution the scattering amplitude is obtained as function of vector Jacobi momenta. Based on Malfliet-Tjon-type potentials differential and total cross sections are calculated. The numerical stability of the algorithm is demonstrated and the properties of the scattering amplitude discussed. Received March 9, 1999; revised July 29, 1999; accepted for publication September 6, 1999     

Poincaré Invariant Three-Body Scattering

Relativistic Faddeev equations for three-body scattering are solved at arbitrary energies in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated withing the framework of Poincaré invariant quantum mechanics. Based on a Malfliet–Tjon interaction, observables for elastic and breakup scattering are calculated and compared to non-relativistic ones.     

Three-body elastic and inelastic scattering at intermediate energies

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. For identical bosons this results in a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. The cross sections for both elastic and breakup processes in the intermediate energy range up to about 1 GeV are calculated based on a Malfliet-Tjon type potential, and the convergence of the multiple scattering series is investigated.     

On the validity of dwba for deuteron stripping: (I). The Mitra three-body model

Previous work showing that there exists an exact formulation of the DWBA for stripping in the S-wave, separable potential, three-body model of Mitra is discussed and extended. The one-body equation obeyed by the c.m. wave function used in the reformulated DWBA is derived and compared with the equation obeyed by the wave function used in the standard formulation of DWBA, viz., the deuteron elastic scattering wave function. Results obtained by other workers on application of three-body methods to direct reactions are discussed in light of the fact that an exact DWBA exists for the separable potential model.     

Three-body model for stripping nuclear reactions

A three-body model for the deuteron stripping nuclear reactions is presented. A set of three integral equations is obtained for the wave functions of the three-body problem by introducing a decomposition into angular momentum states into the Lippmann-Schwinger equation. Simple two-particle interactions with separable potentials are used. These separable potentials reduce the three-body problem to the solution of coupled sets of one-dimensional Fredholm integral equations. The angular distributions for²⁸Si(d,p)²⁹Si and⁴⁰Ca(d, p)⁴¹Ca stripping reactions are calculated. From the extracted spectroscopic factors, good agreement with the experimental measurements is obtained.     

Coupled Rearrangement Channel Calculation of the Hyperfine Structure of the dtμ Molecule

Hyperfine structure splittings are calculated for the J = v = 1 state of the dtμ molecule. The splittings are determined by the accurate three-body wave function obtained by coupled rearrangement channel method using the updated physical constants. The result obtained is in good agreement with the previous calculation within ∼0.07 meV. The discrepancy is due to the accuracy of the three-body wave function. This revised version was published online in August 2006 with corrections to the Cover Date.     

A New Approach to the 3D Faddeev Equation for Three-body Scattering

A novel approach to solve the Faddeev equation for three-body scattering at arbitrary energies is proposed. This approach disentangles the complicated singularity structure of the free three-nucleon propagator leading to the moving and logarithmic singularities in standard treatments. The Faddeev equation is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using, is an integral equation in five variables, magnitudes of relative momenta and angles. The singularities of the free propagator and the deuteron propagator are now both simple poles in two different momentum variables, and thus can both be integrated with standard techniques.     

Dispersion theory of nucleon transfer reactions induced by heavy ions

A new approach, the distorted wave pole approximation (DWPA) with the three-body Coulomb effects, is developed by combining the dispersion method and DWBA to analyse the heavy ion-induced neutron transfer reactions. The influence of the three-body Coulomb dynamics on the peripheral partial wave amplitudes is investigated. Differential cross sections of the neutron transfer reactions are calculated to compare the proposed model with the conventional DWBA. The values of nuclear vertex constants for virtual separation of neutron from various nuclei are obtained. The results of the calculations show that DWPA can be applied to analyse the heavy ion-induced neutron transfer reactions and that the three-body Coulomb effects are taken into account with acceptable accuracy in DWBA.     

α8Be cluster model for the 0 2 + resonance in the 12C nucleus

The 0 ₂ ⁺ resonance in the ¹²C nucleus is treated on the basis of the α+⁸Be two-cluster model. An equation for the function describing the relative motion of the clusters is derived by using the s-wave differential Faddeev equations for the 3α system and by relying on the simplest version of the resonating-group method. A phenomenological potential is taken to simulate the pair αα interaction. A three-body potential binding three alpha particles together gives rise to a resonance in two-cluster α+⁸Be scattering. The calculated resonance features and the calculated parameters of the wave function of the system are compared with the results obtained by other authors.     

基于三维Fokker-Planck方程的电子回旋波加热与电流驱动模拟

利用反弹平均的三维Fokker-Planck方程,对电子回旋波加热和电流驱动进行数值模拟.考虑超热电子径向扩散对电流驱动的影响,在方程中加入径向扩散输运项,采用九点格式的中心差分对方程进行数值离散得到系数矩阵,采用不完全LU分解对系数矩阵进行预处理,利用双共轭梯度稳定法求解得到分布函数.在不考虑电子径向扩散输运条件下,得到电子回旋波驱动电流密度与功率沉积密度的分布;考虑径向扩散输运的计算结果与BANDIT3D进行比较,驱动电流分布的趋势基本一致.     

Critical Temperature of Lequid-gas Phase Transition for Hot Nuclear Matter and Three-body Force Effect

The finite temperature Brueckner-Hartree-Fock (FTBHF) approach is extended by introducing a microscopic three-body force. Within the extended approach, the three-body force effects on the equation of state of hot nuclear matter and its temperature dependence have been investigated. The critical properties of the liquid-gas phase transition of hot nuclear matter have been calculated. It is shown that the three-body force provides a repulsive contribution to the equation of state of hot nuclear matter. The repulsive effect of the three-body force becomes more pronounced as the density and temperature increase and consequently inclusion of the three-body force contribution in the calculation reduces the predicted critical temperature from about 16MeV to about 13MeV. By separating the contribution originated from the 2σ-exchange process coupled to the virtual excitation of a nucleon-antinucleon pair from the full three-body force, the connection between the three-body force effect and the relativistic correction from the Dirac-Brueckner-Hartree-Fock has been explored. It turns out that the contribution of the 2σ-NN part is more repulsive than that of the full three-body force and the calculated critical temperature is about 11MeV if only the 2σ-NN component of the three-body force is included which is lower than the value obtained in the case of including the full three-body force and is close to the value predicted by the Dirac-Brueckner-Hartree-Fock (DBHF) approach. Our result provides a reasonable explanation for the discrepancy between the values of critical temperature predicted from the FTBHF approach including the three-body force and the DBHF approach.     

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.     

A solution of the Coulomb three-body problem

In this work, a final state wave function is constructed which represents a solution of the three-body Schr?dinger equation. The formulated wave function is superimposed of one basic analytical function with various parameters. The coefficients of these basic functions involved in final state wave function can be easily calculated from a set of linear equations. The coefficients depend only on incident energy of the system. The process can also be prolonged for application to the problems more than three bodies.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

New Multiple Soliton-like Solutions to (3+1)-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1 )-dimensional Burgers equation with variable coefficients.     

New Multiple Soliton-like Solutions to （3＋1）-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1)-dimensional Burgers equation with variable coefficients.     

A three-body model of the <Superscript>11</Superscript>B nucleus

The binding energy and the rms charge and mass radii have been calculated in terms of the single-channel three-body ⁴He⁴He³H model of the ¹¹B nucleus with an expansion of the three-body wave function in a nonorthogonal Gaussian basis. Parameters of the wave function are presented and convergence of the three-body energy depending on the number of expansion terms is demonstrated.     

Potential splitting approach to the three-body coulomb scattering problem

The potential splitting approach is extended to a three-body Coulomb scattering problem. The distorted incident wave is constructed and the driven Schrödinger equation is derived. The full angular momentum representation is used to reduce the dimensionality of the problem. The phase shifts for e⁺?H and e⁺?He⁺ collisions are calculated to illustrate the efficiency of the presented method.     

Solving Faddeev-Merkuriev equations within the J-matrix approach: Application to coulomb problems

A version of the J-matrix method for solving three-body differential Faddeev-Merkuriev equations is proposed. This version permits taking into account the full spectrum of a two-body Coulomb subsystem. Laguerre functions are used as a basis in expanding the total wave function of the problem being considered. In order to test the efficiency of the proposed method, the differential cross section for single ionization of a helium atom is calculated where the emerging He⁺ ion remains in an excited state. The result agrees satisfactorily with experimental data both in the shape of the respective curve and in magnitude.