Energy Thresholds of Stability of Three-Particle Systems (original) (raw)
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Critical Stability of Few-body Systems
Series on Knots and Everything, 2013
When a two-body system is bound by a zero-range interaction, the corresponding three-body system-considered in a non-relativistic framework-collapses, that is its binding energy is unbounded from below. In a paper by J.V. Lindesay and H.P. Noyes [1] it was shown that the relativistic effects result in an effective repulsion in such a way that three-body binding energy remains also finite, thus preventing the three-body system from collapse. Later, this property was confirmed in other works based on different versions of relativistic approaches. However, the three-body system exists only for a limited range of two-body binding energy values. For stronger two-body interaction, the relativistic three-body system still collapses. A similar phenomenon was found in a two-body systems themselves: a two-fermion system with one-boson exchange interaction in a state with zero angular momentum J = 0 exists if the coupling constant does not exceed some critical value but it also collapses for larger coupling constant. For a J = 1 state, it collapses for any coupling constant value. These properties are called "critical stability". This contribution aims to be a brief review of this field pioneered by H.P. Noyes.
Journal of Physics B: Atomic, Molecular and Optical Physics, 1990
Near-threshold collision-induced dissociation is simulated numerically for the process Xe + Xe, + 3Xe by carrying out detailed classical trajectory calculations. A static model of the target is chosen and the system is assumed to have zero total angular momentum. Then by varying a single parameter, the threshold law v -E'.6 for the cross section is derived, and this is consistent with analytical results obtained previously. We also consider the distributions of energy, mutual angles and individual angular momenta in the final channel.
Small-energy three-body systems. V. threshold laws when Wannier theory fails
Journal of Physics B-atomic Molecular and Optical Physics - J PHYS-B-AT MOL OPT PHYS, 1994
We investigate cases of Coulombic systems near the break-up threshold for which the Wannier model holds, but not Wannier theory. Making use of the classical trajectory method, we derive threshold laws for a model system of fractional charge (Z= 1/4 au) nucleus and electrons, and a real (though perhaps impractical) system of two beryllium nuclei and an antiproton. For the first system we find the threshold law of the form exp(- lambda / square root E), where E is the total energy, and for the second one a number of characteristic features above the classical threshold have been obtained. Finally we investigate numerically a realistic case of an electron and two beryllium nuclei and discuss some general features of the ionization probability above the classical threshold.
Critical Stability of Three-Body Relativistic Bound States with Zero-Range Interaction
Few-Body-Systems, 2004
For zero-range interaction providing a given mass M 2 of the twobody bound state, the mass M 3 of the relativistic three-body bound state is calculated. We have found that the three-body system exists only when M 2 is greater than a critical value M c (≈ 1.43 m for bosons and ≈ 1.35 m for fermions, m is the constituent mass). For M 2 = M c the mass M 3 turns into zero and for M 2 < M c there is no solution with real value of M 3 .
Universal low-energy behavior in three-body systems
Journal of Mathematical Physics, 2015
We consider a pairwise interacting quantum 3-body system in 3-dimensional space with finite masses and the interaction term V 12 + λ(V 13 + V 23), where all pair potentials are assumed to be nonpositive. The pair interaction of the particles {1, 2} is tuned to make them have a zero energy resonance and no negative energy bound states. The coupling constant λ > 0 is allowed to take the values for which the particle pairs {1, 3} and {2, 3} have no bound states with negative energy. Let λ cr denote the critical value of the coupling constant such that E(λ) → −0 for λ → λ cr , where E(λ) is the ground state energy of the 3-body system. We prove the theorem, which states that near λ cr , one has E(λ) = C(λ − λ cr)[ln(λ − λ cr)] −1 + h.t., where C is a constant and h.t. stands for "higher terms." This behavior of the ground state energy is universal (up to the value of the constant C), meaning that it is independent of the form of pair interactions.
Stability for Lagrangian relative equilibria of three-point-mass systems
Journal of Physics A: Mathematical and General, 2006
In the present paper we apply geometric methods, and in particular the reduced energy-momentum (REM) method, to the analysis of stability of planar rotationally invariant relative equilibria of three-point-mass systems. We analyse two examples in detail: equilateral relative equilibria for the three-body problem, and isosceles triatomic molecules. We discuss some open problems to which the method is applicable, including roto-translational motion in the full three-body problem.
Scaling and universality in two dimensions: three-body bound states with short-ranged interactions
Journal of Physics B: Atomic, Molecular and Optical Physics, 2011
The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body systems are characterized by energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the threebody system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22 we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio and we map out an energy-mass phase-diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.
Formation and relaxation of quasistationary states in particle systems with power-law interactions
Physical review, 2017
We explore the formation and relaxation of so-called quasi-stationary states (QSS) for particle distributions in three dimensions interacting via an attractive radial pair potential V (r → ∞) ∼ 1/r γ with γ > 0, and either a soft-core or hard-core regularization at small r. In the first part of the paper we generalize, for any spatial dimension d ≥ 2, Chandrasekhar's approach for the case of gravity to obtain analytic estimates of the rate of collisional relaxation due to two body collisions. The resultant relaxation rates indicate an essential qualitative difference depending on the integrability of the pair force at large distances: for γ > d − 1 the rate diverges in the large particle number N (mean field) limit, unless a sufficiently large soft core is present; for γ < d − 1, on the other hand, the rate vanishes in the same limit even in the absence of any regularization. In the second part of the paper we compare our analytical predictions with the results of extensive parallel numerical simulations in d = 3 performed with an appropriate modification of the GADGET code, for a range of different exponents γ and soft cores leading to the formation of QSS. We find, just as for the previously well studied case of gravity (which we also revisit), excellent agreement between the parametric dependence of the observed relaxation times and our analytic predictions. Further, as in the case of gravity, we find that the results indicate that, when large impact factors dominate, the appropriate cutoff is the size of the system (rather than, for example, the mean inter-particle distance). Our results provide strong evidence that the existence of QSS is robust only for longrange interactions with a large distance behavior γ < d − 1; for γ ≥ d − 1 the existence of such states will be conditioned strongly on the short range properties of the interaction.