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DETERMINING THE THEORETICAL CONDITIONS UNDER WHICH TWO, THREE,AND FOUR NEUTRONS BIND IN A HARMONIC OSCILLATOR TRAP

Neeraj Vij

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

We consider the general two, three, and four neutron systems in a harmonic oscillator (HO) trap. By constructing a model for the 2, 3, and 4-neutron interactions, we make predictions about the theoretical conditions that produce a 2, 3, and 4 body bound state. We do so using an Effective Field Theory (EFT) in a No Core Shell Model (NCSM) basis space. The HO dependence is added to discretize the problem into a finite number of energy eigenstates in which the NCSM basis states are written. We find that as the width parameter (b) of the well tends to infinity, a bound 2-body state reproduces a bound 3 and 4-body state, that 2- body bound states do not reproduce 3 and 4 body bound states in the regime of 0.6760 < b/a < 1.7467, and we also show below scattering lengths of a = 4.438 X 10-15 m, 1.700 X 10- 15 m, and 1.866 X 10-15 m the 2, 3, and 4 body states are bound, respectively, for a width parameter of 3 Fermi’s.

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We consider the general two, three, and four neutron systems in a harmonic oscillator (HO) trap. By constructing a model for the 2, 3, and 4-neutron interactions, we make predictions about the theoretical conditions that produce a 2, 3, and 4 body bound state. We do so using an Effective Field Theory (EFT) in a No Core Shell Model (NCSM) basis space. The HO dependence is added to discretize the problem into a finite number of energy eigenstates in which the NCSM basis states are written. We find that as the width parameter (b) of the well tends to infinity, a bound 2-body state reproduces a bound 3 and 4-body state, that 2- body bound states do not reproduce 3 and 4 body bound states in the regime of 0.6760 < b/a < 1.7467, and we also show below scattering lengths of a = 4.438 X 10-15 m, 1.700 X 10- 15 m, and 1.866 X 10-15 m the 2, 3, and 4 body states are bound, respectively, for a width parameter of 3 Fermi’s.
We study systems of few two-component fermions interacting via short-range interactions within a harmonic-oscillator trap. The dominant interactions, which are two-body interactions, are organized according to the number of derivatives and defined in a two-body truncated model space made from a bound-state basis. Leading-order (LO) interactions are solved for exactly using the formalism of the no-core shell model, whereas corrections are treated as many-body perturbations. We show explicitly that next-to-LO and next-to-next-to-LO interactions improve convergence as the model space increases. We present results at unitarity for three- and four-fermion systems, which show excellent agreement with the exact solution (for the three-body problem) and results obtained by other methods (in the four-body case). We also present results for finite scattering lengths and nonzero range of the interaction, including (at positive scattering length) observation of a change in the structure of the three-body ground state and extraction of the atom-dimer scattering length.
We apply the general principles of effective field theories to the construction of effective interactions suitable for few- and many-body calculations in a no-core shell model framework. We calculate the spectrum of systems with three and four two-component fermions in a harmonic trap. In the unitary limit, we find that three-particle results are within 10% of known semianalytical values even in small model spaces. The method is very general, and can be readily extended to other regimes, more particles, different species (e.g., protons and neutrons in nuclear physics), or more-component fermions (as well as bosons). As an illustration, we present calculations of the lowest-energy three-fermion states away from the unitary limit and find a possible inversion of parity in the ground state in the limit of trap size large compared to the scattering length. Furthermore, we investigate the lowest positive-parity states for four fermions, although we are limited by the dimensions we can currently handle in this case.
We consider an analytic way to make the interacting N -body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two-body terms of the N -body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, centre position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N -body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N -boson problem in two and three dimensions where we fit the two-body interactions to agree with the well-known zero-range model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as a function of dimension, boson number N , and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose–Fermi mixtures, and to more exotic geometries.
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In this work we study unitary neutron systems confined in a harmonic-oscillator trap, based on the BCS theory and the quasiparticle ring-diagram method. The low-momentum interaction Vlow-k{V}_{\text{low-}k}Vlow-k​ renormalized from a fine-tuned CD-Bonn potential with the 1S0^{1}S_{0}1S0​ scattering length approaching the infinity is adopted. The energy ratios of trapped unitary neutron systems with various particle numbers, both those with magic numbers and those with nonmagic ones, to corresponding noninteracting systems, are shown to all flow to a constant value. The BCS wave functions for these trapped unitary systems, with different trap parameters and different particle numbers, manifest themselves with universal patterns: the effect of harmonic-oscillator shells is significant, and moreover, the orbits of the lowest valence shell are occupied with a uniform probability. We also discuss the self-energy given by the BCS calculation, and present a weighted sum rule satisfied by the monopole components of the unitary neutron-neutron interaction. In terms of these, the constant energy ratio independent of the trap parameter and the particle number can be interpreted.
We calculate the renormalized effective two-, three- and four-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming two-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a t (0) defined at zero collision energy, which is necessary to obtain both the leading-order effective four-body interaction and consistently include corrections for realistic two-body interactions. The leading-order, effective three- and four-body interaction energies are and , where ω and σ ( ω ) are the harmonic oscillator frequency and length, respectively, and energies are in units of ℏ ω . The one-standard deviation error ±0.0001 for the third-order coefficient in U 3 ( ω ) is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective three- and four-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range (FR) boson–boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.