Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties

In this work, the Faddeev equations for three-body scattering at arbitrary angular momentum are exactly solved and the transition matrices for some transition processes, including scattering and rearrangement channels are formulated in terms of free-particle resolvent matrix. A generalized Yamaguchi rank-two nonlocal separable potential has been used to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The partial-wave analysis for some transition processes in a three-particle system is suggested. The partial-wave three-particle transition matrix elements have been constructed via knowledge of the matrix elements of the free motion resolvent. The calculation of a number of scattering properties of interest of the system such as transition matrix and its poles (bound states and resonances) and consequently other related quantities like scattering amplitudes, scattering length, phase shifts and cross sections are feasible in a straightforward manner. Moreover, we obtain a new analytical expression for the third virial coefficient in terms of three-body transition matrix.     

Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties

In this work, the Faddeev equations for three-body scattering at arbitrary angular momentum are exactly solved and the transition matrices for some transition processes, including scattering and rearrangement channels are formulated in terms of free-particle resolvent matrix. A generalized Yamaguchi rank-two nonlocal separable potential has been used to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The partial-wave analysis for some transition processes in a three-particle system is suggested. The partial-wave three-particle transition matrix elements have been constructed via knowledge of the matrix elements of the free motion resolvent.The calculation of a number of scattering properties of interest of the system such as transition matrix and its poles(bound states and resonances) and consequently other related quantities like scattering amplitudes, scattering length,phase shifts and cross sections are feasible in a straightforward manner. Moreover, we obtain a new analytical expression for the third virial coefficient in terms of three-body transition matrix.     

Effects of Composite Pions on the Chiral Condensate within the PNJL Model at Finite Temperature

We investigate the effect of composite pions on the behaviour of the chiral condensate at finite temperature within the Polyakov-loop improved NJL model. To this end we treat quark-antiquark correlations in the pion channel (bound states and scattering continuum) within a Beth–Uhlenbeck approach that uses medium-dependent phase shifts. A striking medium effect is the Mott transition which occurs when the binding energy vanishes and the discrete pion bound state merges the continuum. This transition is triggered by the lowering of the continuum edge due to the chiral restoration transition. This in turn also entails a modification of the Polyakov-loop so that the SU(3) center symmetry gets broken at finite temperature and dynamical quarks (and gluons) appear in the system, taking over the role of the dominant degrees of freedom from the pions. At low temperatures our model reproduces the chiral perturbation theory result for the chiral condensate while at high temperatures the PNJL model result is recovered. The new aspect of the current work is a consistent treatment of the chiral restoration transition region within the Beth–Uhlenbeck approach on the basis of mesonic phase shifts for the treatment of the correlations.     

Quantum dynamics of relativistic bosons through nonminimal vector square potentials

The dynamics of relativistic bosons (scalar and vectorial) through nonminimal vector square (well and barrier) potentials is studied in the Duffin–Kemmer–Petiau (DKP) formalism. We show that the problem can be mapped in effective Schrödinger equations for a component of the DKP spinor. An oscillatory transmission coefficient is found and there is total reflection. Additionally, the energy spectrum of bound states is obtained and reveals the Schiff–Snyder–Weinberg effect, for specific conditions the potential lodges bound states of particles and antiparticles.     

Cluster integrals of convex molecule systems

Cluster integrals assigned to particles interacting via the Kihara non-spherical potential are studied theoretically. An exact formula is derived which allows one to consider the effect of molecular shape separately from the effect of soft interactions. Employing the proposed formalism, the cluster integrals are analysed. The approach is applied to determine the third virial coefficient and an efficient computational method is developed. The third virial coefficient was calculated for a combination of molecules with hard cores of prolate spher-ocylindrical- and spherical shapes interacting via the square-well, triangle-well and 12-6 pair potentials. Comparison with numerical results obtained by Monte Carlo integration is made and fair agreement is found.     

Model three-body problem in the molecular mass limit

The bound states of a three-body molecule composed of two identical heavy nuclei and a light “electron” interacting through short-range s-wave potentials are studied. The spectrum of three-body bound states grows as the mass ratio m between the heavy and light particles increases, and presents a remarkable vibration rotation structure that can be fitted with the usual empirical energy formulas of molecular spectroscopy. The results of the exact three-body calculation for the binding energy and bound-state wavefunction are compared with the predictions of the Born-Oppenheimer method for the same system. We find that for m > 30, the Born-Oppenheimer approximation yields very good results for both the binding energies and wavefunctions. For smaller m (1 <m < 30) the Born-Oppenheimer results are still surprisingly good and this is shown to be related to the range of the two-body interactions.     

Second virial coefficient in one dimension as a function of asymptotic quantities

A result from Dodd and Gibbs (J. Math. Phys., 15, 41 (1974)) for the second virial coefficient of particles in one dimension, subject to delta-function interactions, has been obtained by direct integration of the wave functions. It is shown that this result can be obtained from a phase shift formalism, if one also includes the contribution of oscillating terms. The result is important in work to follow, for the third virial coefficient, for which a similar formalism is being developed. We examine a number of fine points in the quantum mechanical formalisms.     

Coulomb final state interaction in identical Boson interferometry

Recently, the muonic three-body systemdtμ, has aroused considerable attention in relation to the realization of a useful muon-catalyzed fusion [1, 2]. This bound system must be solved up to 6 digits in energy to establish the muon catalyzed fusion process. We applied the ATMS method [3] and the coupled-channel method [4] to the Coulomb three-body system and obtained the detailed information on the energy levels and the wave functions of the bound states. Further we investigated the effect of the nuclear interaction on the fusion rate in the bound states [5] and developed a formalism to calculate the muon sticking to⁴He [6]. In this paper we report the results obtained in our collaborations: Figure 2 shows up our main results.     

Zur formelmäßigen Darstellung des Ionisierungsquerschnitts für den Atom-Atomstoß und über die Ionen-Elektronen-Rekombination im dichten Neutralgas

Thomson's classical theory for the calculation of ionisation cross sections in electron-atom encounters is applied for calculating the ionisation cross section for atom-atom collisions. A very simple formula is obtained permitting a rapid calculation of the total ionisation cross sections as a function of the kinetic energyE _a and the massm _A of the ionizing atom, and of the binding energyE _i and the number ξ_i of equivalent electrons in the electronic shell to be ionized. The formula is in good agreement with quantum mechanical calculations and with experimental results. It can be applied to ionization from ground and from excited states. From this formula one obtains an expression for the rate coefficient for ionization in the center-of-mass system of the colliding particles. Then, the method of detailed balancing is applied to calculate the rate coefficient for ion-electron three-body collisional recombination with a neutral atom acting as the third body. This latter formula is applicable to recombination into the ground and into excited states.     

Precise numerical results for limit cycles in the quantum three-body problem

The study of the three-body problem with short-range attractive two-body forces has a rich history going back to the 1930s. Recent applications of effective field theory methods to atomic and nuclear physics have produced a much improved understanding of this problem, and we elucidate some of the issues using renormalization group ideas applied to precise nonperturbative calculations. These calculations provide 11-12 digits of precision for the binding energies in the infinite cutoff limit. The method starts with this limit as an approximation to an effective theory and allows cutoff dependence to be systematically computed as an expansion in powers of inverse cutoffs and logarithms of the cutoff. Renormalization of three-body bound states requires a short range three-body interaction, with a coupling that is governed by a precisely mapped limit cycle of the renormalization group. Additional three-body irrelevant interactions must be determined to control subleading dependence on the cutoff and this control is essential for an effective field theory since the continuum limit is not likely to match physical systems (e.g., few-nucleon bound and scattering states at low energy). Leading order calculations precise to 11-12 digits allow clear identification of subleading corrections, but these corrections have not been computed.     

First quantized electrodynamics

The parametrized Dirac wave equation represents position and time as operators, and can be formulated for many particles. It thus provides, unlike field-theoretic Quantum Electrodynamics (QED), an elementary and unrestricted representation of electrons entangled in space or time. The parametrized formalism leads directly and without further conjecture to the Bethe–Salpeter equation for bound states. The formalism also yields the Uehling shift of the hydrogenic spectrum, the anomalous magnetic moment of the electron to leading order in the fine structure constant, the Lamb shift and the axial anomaly of QED.     

Three-Potential Formalism for the Atomic Three-Body Problem

Based on a three-potential formalism we propose mathematically well-behaved Faddeev-type integral equations for the atomic three-body problem and describe their solutions in Coulomb-Sturmian space representation. Although the system contains only long-range Coulomb interactions these equations allow us to reach a solution by approximating only some auxiliary short-range potentials. We outline the method for bound states and demonstrate its power in benchmark calculations. We can report a fast convergence in angular-momentum channels. Received September 30, 1997; accepted in final form January 23, 1998     

An essentially exact many body calculation for muonic molecular ions and exotic coulombic systems

The L = 0 bound states and the structural properties of muonic molecular ions and exotic nuclear molecular ions are calculated by a hyperspherical harmonics expansion method, which is an essentially exact method for three-body systems. We examine the convergence rate of this method and also examine its crucial dependence on the mass ratio ? (ratio of the masses of similar to dissimilar particles). An a priori criterion for the rate of convergence and the required size of the expansion basis for a predetermined precision have been suggested. Our calculation reveals the cluster type of structure of these systems.     

A model search for Efimov states

A search for Efimov states is carried out in terms of a one-dimensional model three-body problem. A numerically exact solution is obtained that allows scanning through the resonance condition. No evidence is found for the existence of an infinite number of such bound three-body states. It is deduced from one-dimensional results that such an infinity of bound states should also not arise in a complete three-dimensional calculation.     

基于原子间相互作用的低温硅单晶负热膨胀机制的研究

运用晶格热膨胀的微扰理论,推导了硅晶体的热膨胀系数与原子间两体作用和三体作用的三阶力常数之间的关系公式,在此基础上对热膨胀系数的实验数据进行拟合,计算了硅晶体内的原子间两体和三体的三阶力常数,发现三体作用的三阶力常数为正数,是硅晶体在低温下具有负热膨胀性质的根本原因。计算与分析的结果表明,运用Stillinger-Weber模型得到的三阶力常数为负数,据此不可能计算得到低温下的负热膨胀系数,因此应该对该模型进行修正。     

Dielectric constant of polarizable,nonpolar fluids and suspensions

We study the dielectric constant of a polarizable, nonpolar fluid or suspension of spherical particles by use of a renormalized cluster expansion. The particles may have induced multipole moments of any order. We show that the Clausius-Mossotti formula results from a virtual overlap contribution. The corrections to the Clausius-Mossotti formula are expressed with the aid of a cluster expansion. The integrands of the cluster integrals are expressed in terms of two-body nodal connectors which incorporate all reflections between a pair of particles. We study the two- and three-body cluster integrals in some detail and show how these are related to the dielectric virial expansion and to the first term of the Kirkwood-Yvon expansion.     

基于原子间相互作用的低温硅单晶负热膨胀机制的研究

运用晶格热膨胀的微扰理论,推导了硅晶体的热膨胀系数与原子间两体作用和三体作用的三阶力常数之间的关系公式,在此基础上对热膨胀系数的实验数据进行拟合,计算了硅晶体内的原子间两体和三体的三阶力常数,发现三体作用的三阶力常数为正数,是硅晶体在低温下具有负热膨胀性质的根本原因。计算与分析的结果表明,运用Stillinger-Weber模型得到的三阶力常数为负数,据此不可能计算得到低温下的负热膨胀系数,因此应该对该模型进行修正。     

Calculating the thermodynamic parameters of dense gases and weakly ionized plasmas including three-body interactions

A model is developed for the equation of state of dense gases and nonideal plasmas in the approximation of three-body interactions. The reduced third virial coefficient is calculated for a number of spherically symmetric pairwise additive interaction potentials of neutral and charged particles. Its temperature dependence is constructed for various potentials. The density and composition of plasmas in a number of pure substances are calculated numerically. Zh. Tekh. Fiz. 68, 130–132 (July 1998)     

Statistical mechanics of an ideal gas of non-Abelian anyons

We study the thermodynamical properties of an ideal gas of non-Abelian Chern–Simons particles and we compute the second virial coefficient, considering the effect of general soft-core boundary conditions for the two-body wavefunction at zero distance. The behaviour of the second virial coefficient is studied as a function of the Chern–Simons coupling, the isospin quantum number and the hard-core parameters. Expressions for the main thermodynamical quantities at the lower order of the virial expansion are also obtained: we find that at this order the relation between the internal energy and the pressure is the same found (exactly) for 2D Bose and Fermi ideal gases. A discussion of the comparison of obtained findings with available results in literature for systems of hard-core non-Abelian Chern–Simons particles is also supplied.