Three-body Parameters Research Articles

Universality of Efimov states in highly mass-imbalanced cold-atom mixtures with van der Waals and dipole interactions

We study three-body systems in a mass-imbalanced, two-component, cold-atom mixture, and we investigate the three-body parameter of their Efimov states for both bosonic and fermionic systems, with a major focus on the Er-Er-Li Efimov states. For a system interacting solely via van der Waals interactions, the van der Waals universality of the three-body parameter is analytically derived using the quantum defect theory. With the addition of a perturbative dipole interaction between the heavy atoms, the three-body parameters of the bosonic and fermionic Efimov states are found to behave differently. When the dipole interaction is as strong as the van der Waals interaction, corresponding to realistic Er-Er-Li Efimov states, we show that the van der Waals universality persists once the effects of the nonperturbative dipole interaction are into the s-wave and p-wave scattering parameters between the heavy atoms. For a dipole interaction much stronger than the van der Waals interaction, we find that the universality of the Efimov states can be alternatively characterized by a quasi-one-dimensional scattering parameter due to a strong anisotropic deformation of the Efimov wave functions. Our work thus clarifies the interplay of isotropic and anisotropic forces in the universality of the Efimov states. Based on the renormalized van der Waals universality, the three-body parameter is estimated for specific isotopes of Er-Li cold-atom mixtures. Published by the American Physical Society 2024

Development and application of ReaxFF methodology for understanding the chemical dynamics of metal carbonates in aqueous solutions.

A new ReaxFF reactive force field has been developed for metal carbonate systems including Na+, Ca2+, and Mg2+ cations and the CO32- anion. This force field is fully transferable with previous ReaxFF water and water/electrolyte descriptions. The Me-O-C (Me = metal) three-body valence angle parameters and Me-C non-reactive parameters of the force field have been optimized against quantum mechanical calculations including equations of state, heats of formation, heats of reaction, angle distortions and vibrational frequencies. The new metal carbonate force field has been validated using molecular dynamics simulations to study the solvation and reactivity of metal and carbonate ions in water at 300 K and 700 K. The coordination radius and self-diffusion coefficient show good consistency with existing experimental and simulation results. The angular distribution analysis explains the structural preference of carbonate ions to form carbonates and bicarbonates, where Na+ predominantly forms carbonates due to weaker angular strain, while Ca2+ and Mg2+ prefer to form bicarbonate monodentate in nature. Residence time distribution analyses on different systems reveal the role of ions in accelerating and decelerating the dynamics of water and carbonate ions under different thermodynamic conditions. The formation and dissolution of bicarbonates and carbonates in solution were explored on the basis of the protonation capability in different systems. The nucleation phenomenon of metal carbonates at ambient and supercritical conditions is explained from the perspective of cluster formation over time: Ca2+ ions can form prenucleation clusters at ambient temperature but show saturation with increasing temperature, whereas Na+ and Mg2+ ions show a rapid increase in cluster size and amount upon increasing time and temperature.

Impact of three-body interactions in a ReaxFF force field for Ni and Cr transition metals and their alloys on the prediction of thermal and mechanical properties

In this work, we discuss the temperature-dependent elastic constants, thermal expansion and melting temperature of Ni and Cr transition metals using a ReaxFF reactive force field. While the ReaxFF force field has been successfully applied to various chemistries on these transition metals, it cannot replicate the experimentally observed relationship between C12 and C44. In addition, ReaxFF predicts the negative stacking-fault energy for fcc Ni, accelerating the fcc → hcp phase transformation. We show that by introducing three-body interaction parameters to the metal force field, which have not been included previously for transition metals, ReaxFF can successfully predict experimental elastic constants of fcc Ni and bcc Cr at finite temperatures.We then evaluate the thermal lattice expansion at T = 0 – 1700 K and compare to experimental data available in the literature – the higher thermal expansion coefficient corresponds to Ni, which is known to have a lower melting temperature than Cr. To estimate the melting temperature of Ni and Cr using the ReaxFF force field, we performed molecular dynamics (MD) simulations by adopting the hysteresis method. With a sufficiently large system size and void concentration, the melting temperature is predicted to be 1698 K for Ni (agreeing within 1.7%) and 2410 K for Cr (within 10%). Furthermore, we develop a Ni/Cr alloy force field by combining Ni and Cr force fields and optimizing their cross-interaction parameters. It is shown in both ReaxFF and DFT that fcc-based Ni3Cr is the most stable phase with a negative heat of formation while all bcc alloys have positive heats of formation, indicating a possible phase separation for Ni and Cr. The elastic constants at 0 K calculated by ReaxFF agree with DFT values within 5–10%, except for the large deviation in C33.

Charged multihadron systems in lattice QCD+QED

Systems with the quantum numbers of up to twelve charged and neutral pseudoscalar mesons, as well as one-, two-, and three-nucleon systems, are studied using dynamical lattice quantum chromodynamics and quantum electrodynamics (QCD+QED) calculations and effective field theory. QED effects on hadronic interactions are determined by comparing systems of charged and neutral hadrons after tuning the quark masses to remove strong isospin breaking effects. A non-relativistic effective field theory, which perturbatively includes finite-volume Coulomb effects, is analyzed for systems of multiple charged hadrons and found to accurately reproduce the lattice QCD+QED results. QED effects on charged multi-hadron systems beyond Coulomb photon exchange are determined by comparing the two- and three-body interaction parameters extracted from the lattice QCD+QED results for charged and neutral multi-hadron systems.

Two- and three-body problem with Floquet-driven zero-range interactions

We study the two-body scattering problem in the zero-range approximation with a sinusoidally driven scattering length and calculate the relation between the mean value and amplitude of the drive for which the effective scattering amplitude is resonantly enhanced. In this manner we arrive at a family of curves along which the effective scattering length diverges but the nature of the corresponding Floquet-induced resonance changes from narrow to wide. Remarkably, on these curves the driving does not induce heating. In order to study the effect of these resonances on the three-body problem we consider one light and two heavy particles with driven heavy-light interaction in the Born-Oppenheimer approximation and find that the Floquet driving can be used to tune the three-body and inelasticity parameters.

Efimov–van der Waals universality for ultracold atoms with positive scattering lengths

We study the universality of the three-body parameters for systems relevant for ultracold quantum gases with positive $s$-wave two-body scattering lengths. Our results account for finite-range effects and their universality is tested by changing the number of deeply bound diatomic states supported by our interaction model. We find that the physics controlling the values of the three-body parameters associated with the ground and excited Efimov states is constrained by a variational principle and can be strongly affected by $d$-wave interactions that prevent both trimer states from merging into the atom-dimer continuum. Our results enable comparisons to current experimental data and they suggest tests of universality for atomic systems with{positive scattering lengths.

Unitary boson-boson and boson-fermion mixtures: third virial coefficient and three-body parameter on a narrow Feshbach resonance

We give exact integral expressions of the third cluster or virial coefficients of binary mixtures of ideal Bose or Fermi gases, with interspecies interactions of zero range and infinite s-wave scattering length. In general the result depends on three-body parameters R t appearing in three-body contact conditions, because an Efimov effect is present or because the mixture is in a preefimovian regime with a mass ratio close to an Efimov-effect threshold. We give a new, exact integral expression of R t for the microscopic narrow Feshbach resonance model. A divergence of R t in the preefimovian regime at a scaling exponent s = 1 / 2 is predicted and physically discussed. The analytical results are applied to typical species used in cold atom experiments.

Description of light nuclei in pionless effective field theory using the stochastic variational method

We construct a coordinate-space potential based on pionless effective field theory with a Gaussian regulator. Charge-symmetry breaking is included through the Coulomb potential and through two- and three-body contact interactions. Starting with the effective field theory potential, we apply the stochastic variational method to determine the ground states of nuclei with mass number $A\leq 4$. At next-to-next-to-leading order, two out of three independent three-body parameters can be fitted to the three-body binding energies. To fix the remaining one, we look for a simultaneous description of the binding energy of $^4$He and the charge radii of $^3$He and $^4$He. We show that at the order considered we can find an acceptable solution, within the uncertainty of the expansion. We find that the EFT expansion shows good agreement with empirical data within the estimated uncertainty, even for a system as dense as $^4$He.

Universal few-body physics in resonantly interacting spinor condensates

Optical trapping techniques allow for the formation of bosonic condensates with internal degrees of freedom, so-called spinor condensates. Mean-field models of spinor condensates highlight the sensitivity of the quantum phases of the system to the relative strength of the two-body interaction in the different spin-channels. Such a description captures the regime where these interactions are weak. In the opposite and largely unexplored regime of strongly correlated spinor condensates, three-body interactions can play an important role through the Efimov effect, producing possible novel phases. Here, we study the three-body spinor problem using the hyperspherical adiabatic representation for spin-1, -2 and -3 condensates in the strongly-correlated regime. We characterize the Efimov physics for such systems and the relevant three-body mean-field parameters. We find that the Efimov effect can strongly affect the spin dynamics and three-body mean-field contributions to the possible quantum phases of the condensate through universal contributions to scattering observables.

Revealing the origin of super Efimov states in the hyperspherical formalism

Super-Efimov states are a new kind of universal three-body bound states predicted for three identical fermions with $p$-wave resonant interactions in two dimensions by a recent field-theoretic calculation [Phys.~Rev.~Lett.~\textbf{110}, 235301 (2013)]. The binding energies of these states obey a dramatic double exponential scaling $E_n=E_*\exp(-2 e^{\pi n/s_0+\theta})$ with universal scaling $s_0=4/3$ and three-body parameters $E_*$ and $\theta$. We use the hyperspherical formalism and show that the super-Efimov states originate from an emergent effective potential $-1/4\rho^2-(s_0^2+1/4)/\rho^2\ln^2\left(\rho\right)$ at large hyperradius $\rho$. Moreover, for pairwise interparticle potentials with van der Waals tails, our numerical calculation indicates that the three-body parameters $E_*$ and $\theta$ are also universal and the ground super-Efimov state shall cross the threshold when the $2$D $p$-wave scattering area is about $-42.0\, l_\text{vdW}^2$ with $l_\text{vdW}$ the van der Waals length.

Three-body recombination in heteronuclear mixtures at finite temperature

Within the universal zero-range theory, we compute the three-body recombination rate to deep molecular states for two identical bosons resonantly interacting with each other and with a third atom of another species, in the absence of weakly bound dimers. The results allow for a quantitative understanding of loss resonances at finite temperature and, combined with experimental data, can be used for testing the Efimov universality and extracting the corresponding three-body parameters in a given system. Curiously, we find that the loss rate can be dramatically enhanced by the resonant heavy-heavy interaction, even for large mass ratios where this interaction is practically irrelevant for the Efimov scaling factor. This effect is important for analysing the recent loss measurements in the Cs-Li mixture.

Contact parameters in two dimensions for general three-body systems

We study the two-dimensional three-body problem in the general case of three distinguishable particles interacting through zero-range potentials. The Faddeev decomposition is used to write the momentum-space wave function. We show that the large-momentum asymptotic spectator function has the same functional form as derived previously for three identical particles. We derive the analytic relations between the three different Faddeev components for the three distinguishable particles. We investigate the one-body momentum distributions both analytically and numerically and analyze the tail of the distributions to obtain two- and three-body contact parameters. We specialize from the general cases to the examples of two identical, interacting or non-interacting, particles. We find that the two-body contact parameter is not a universal constant in the general case and show that the universality is recovered when a subsystem is composed of two identical non-interacting particles. We also show that the three-body contact parameter is negligible in the case of one non-interacting subsystem compared to the situation where all the subsystems are bound. As an example, we present the results for mixtures of lithium with two cesium or two potassium atoms, which are systems of current experimental interest.

Dynamics of spin-1 bosons in an optical lattice: Spin mixing, quantum-phase-revival spectroscopy, and effective three-body interactions

We study the dynamics of spin-1 atoms in a periodic optical-lattice potential and an external magnetic field in a quantum quench scenario where we start from a superfluid ground state in a shallow lattice potential and suddenly raise the lattice depth. The time evolution of the non-equilibrium state, thus created, shows collective collapse-and-revival oscillations of matter-wave coherence as well as oscillations in the spin populations. We show that the complex pattern of these two types of oscillations reveals details about the superfluid and magnetic properties of the initial many-body ground state. Furthermore, we show that the strengths of the spin-dependent and spin-independent atom-atom interactions can be deduced from the observations. The Hamiltonian that describes the physics of the final deep lattice not only contains two-body interactions but also effective multi-body interactions, which arise due to virtual excitations to higher bands. We derive these effective spin-dependent three-body interaction parameters for spin-1 atoms and describe how spin-mixing is affected. Spinor atoms are unique in the sense that multi-body interactions are directly evident in the in-situ number densities in addition to the momentum distributions. We treat both antiferromagnetic (e.g. $^{23}$Na atoms) and ferromagnetic (e.g. $^{87}$Rb and $^{41}$K) condensates.

Universality constraints on three-body recombination for cold atoms: FromHe4toCs133

For atoms with large scattering length, the dependence of the three-body recombination rate on the collision energy is determined by the scattering length and the Efimov three-body parameters and can be expressed in terms of universal functions of a single scaling variable. We use published results on the three-body recombination rate for $^{4}\mathrm{He}$ atoms to constrain the universal functions. We then use those universal functions to calculate the three-body recombination rate for other atoms with large scattering length at nonzero temperature. The constraints from the $^{4}\mathrm{He}$ results are strong if the scattering length is near the minimum of the three-body recombination rate at threshold. We apply our results to $^{133}\mathrm{Cs}$ atoms with a large positive scattering length, and compare them with experimental results from the Innsbruck group.

Effective three-body potentials for Li+(aq) and Mg2+(aq)

A method for the extraction of effective three-body potential parameters from high-level ab initio cluster calculations is presented and compared to effective pair potentials extracted at the same level. Dilute Li+(aq) and Mg2+(aq) solutions are used as test cases and long molecular-dynamics simulations using these newly developed potentials were performed. Resulting thermodynamical, structural, and dynamical properties are compared to experiment as well as to the empirical effective pair potentials of Åqvist. Moreover, a new time-saving method for the correction of cluster energies computed with a relatively cheap ab initio method, to yield expensive, high-level ab initio energies, is presented. The effective pair approach is shown to give inconsistent results when compared to the effective three-body potentials. The performance of three different charge compensation methods (uniform charge plasm, Bogusz net charge correction, and counter ions) is compared for a large number of different system sizes. For most properties studied here, the system-size dependence is found to be small for system sizes with 256 water molecules or more. However, for the self-diffusion coefficients, a 1/L dependence is found, i.e., a very large system-size dependence. A very simple method for correcting for this deficiency is proposed. The results for most properties are found to compare reasonably well to experiment when using the effective three-body potentials.

DYNAMIC SCREENING OF THE (e,2e) PROCESSES FOR LOW-ENERGY ELECTRON IMPACT IONIZATION OF HELIUM

Based on the modified three-body Sommerfeld parameters,the effective shield of the residual electron in the final state of He+ is further modified for (e,2e) processes- Such a modification represents a dynamic screening of the four bodies in the final state- The triple differential cross sections for electron impact ionization of atomic helium at an incident energy of 50eV in the asymmetric geometry are calculated by use of the modified Sommerfeld parameters-The results were compared with those of the absolute measurements and the only existing theoretical results of the convergent colse-coupling method-It was found that the present results give a better description for the experimental data at fixed ejected energy of 10eV-

Comparative analysis of free-ion energy levels of Er 3+ (4f 11) in various crystal hosts

The best energy level data available in the literature for the trivalent erbium-doped systems of about 30 crystal hosts are analysed in terms of a uniform model free-ion Hamiltonian H fi. H fi consists of 20 free-ion parameters which include the central field E avg, electrostatic repulsion F k ( k = 2, 4, 6), spin-orbit interaction parameter ξ, two-body configuration interaction parameters α, β and γ, three-body configuration interaction parameters T i ( i = 2, 3, 4, 6, 7, 8), spin—other-orbit interaction parameters M j ( j = 0, 2, 4) and electrostatically correlated spin-orbit interaction parameters P k ( k = 2, 4, 6). Our systematic theoretical analysis of crystal free-ion levels results in good correlation between calculated and observed energies. The reanalysis of some systems reported here yields improved fits of calculated to experimental energy levels compared with fits reported previously in the literature. The effects of the crystalline hosts on the empirical parameters are examined.

Potential energy functions for atomic solids

A potential energy function has been proposed for atomic solids consisting of two-body and three-body terms which over its range of parameters encompasses all simple crystal structures. Phase diagrams have been constructed as a function of a single parameter in the two-body term and one, of several, parameters in the three-body term, and these include as the most stable structures hcp, fcc, bcc, sc and diamond. The hcp and fcc lattices have been shown to distort to lower symmetries with appropriate three-body parameters and these distortions are in accord with known structures. A brief study has been made of the appropriate parameters for silicon.

Energies of N equivalent electrons expressed in terms of two-electron energies and independent three-electron parameters: a new complete set of orthogonal operators. III. Ab initio calculations

For pt.I see ibid., vol.17, p.23-75 (1984). As an illustration of the theory developed in the first paper in this series, a set of five orthogonal parameters is used to describe all electrostatic two-particle excitations in the 3dN configurations completely to second order. These parameters represent the term energies of d2 and, consequently, the 3dN configurations are perceived as a collection of 1/2N(N-1)d2 pairs, the total energy of the configuration being the sum of the corresponding pair energies. As regards their effects, the parameters are partitioned into a structural and an additive part; the structural part is found to behave as a linear function of N, whereas the additive part is proportional to the average energy of the configuration. In order to take into account all electrostatic one-particle effects as well, two orthogonalised three-body parameters T1 and T2 are used, the first corresponding to the well known Trees parameter and the second covering the remaining class of one-electron excitations. With a least-squares fitting procedure, both the d2 energies and the three-electron parameters are determined for the dN ground configuration in the III, IV, V and VI spectra of the iron group elements. Surprisingly, the effects of T2 turn out to be important, especially for the lower ionisation stages. Amongst other things, old problems concerning the far-off position with respect to calculations of the 12D term in the 3d6, and parameter irregularities in the 3d7 configurations, seem to be solved with the introduction of T2. As a result, the mean error of the fit in some of these cases is reduced by factors of three to four. Using the above set of parameters, a prediction of the 3d4 ground configuration of Mn IV is made.

A simple relation for the excess functions of nonrandom mixtures

Semi-empirical three-parameter relations for the average intermolecular energy ū m and the excess functions of nonrandom mixtures have been derived from a hypothesis that the asymmetry of the functions result from the formation of molecular clusters. It is postulated that changes in cluster concentration upon dilution in an inert solvent can be interpreted as changes in the apparent three-body interaction parameters. Comparison with data for binary systems where there is unusual composition dependence of the excess chemical potentials demonstrates the accuracy of the relations. Conclusions regarding the sources of deviations from random mixing come from computer simulations and other data.