Scattering Phase Shifts Research Articles

Universal scattering phase shift in the presence of spin-orbit coupling

Scattering phase shift, as a key parameter in scattering theory, plays an important role in characterizing low-energy collisions between ultracold atoms. In this work, we theoretically investigate the universal low-energy behavior of the scattering phase shifts for cold atoms in the presence of spin-orbit coupling. We first construct the asymptotic form of the two-body wave function when two fermions get as close as the interaction range, and consider perturbatively the correction of the spin-orbit coupling up to the second order, in which new scattering parameters are introduced. Then for elastic collisions, the scattering phase shifts are defined according to the unitary scattering $S$ matrix. We show how the low-energy behavior of the scattering phase shifts is modified by these new scattering parameters introduced by spin-orbit coupling. The universality of the scattering phase shifts is manifested as the independence of the specific form of the interatomic potential. The explicit forms of the new scattering parameters are analytically derived within a model of the spherical-square-well potential. Our method provides a unified description of the low-energy properties of scattering phase shifts in the presence of spin-orbit coupling.

Scattering of two heavy Fermi polarons: Resonances and quasibound states

Impurities in a Fermi sea, or Fermi polarons, experience a Casimir interaction induced by quantum fluctuations of the medium. When there is short-range attraction between impurities and fermions, also the induced interaction between two impurities is strongly attractive at short distance and oscillates in space for larger distances. We theoretically investigate the scattering properties and compute the scattering phase shifts and scattering lengths between two heavy impurities in an ideal Fermi gas at zero temperature. While the induced interaction between impurities is weakly attractive for weak impurity-medium interactions, we find that impurities strongly and attractively interacting with the medium exhibit resonances in the induced scattering with a sign change of the induced scattering length and even strong repulsion. These resonances occur whenever a three-body Efimov bound state appears at the continuum threshold. At energies above the continuum threshold, we find that the Efimov state in medium can turn into a quasibound state with a finite decay width.

Accurate bulk properties of nuclei from A=2 to ∞ from potentials with Δ isobars

We optimize $\Delta$-full nuclear interactions from chiral effective field theory. The low-energy constants of the contact potentials are constrained by two-body scattering phase shifts, and by properties of bound-state of $A=2$ to $4$ nucleon systems and nuclear matter. The pion-nucleon couplings are taken from a Roy-Steiner analysis. The resulting interactions yield accurate binding energies and radii for a range of nuclei from $A=16$ to $A=132$, and provide accurate equations of state for nuclear matter and realistic symmetry energies. Selected excited states are also in agreement with data.

$$\boldsymbol{p}\boldsymbol{-}^{\mathbf{12}}$$C and $$\boldsymbol{p}\boldsymbol{-}\boldsymbol{d}$$ Scattering within the Separable Model of Interaction

Differential equation approach is adapted to construct exact analytical expressions for the regular, Jost, and physical states for Hulthen plus separable nonlocal potential. The Jost function and the Fredholm determinant for physical boundary condition are exploited to compute scattering phase shifts for proton–deuteron and proton–carbon systems within this two-body model of interaction. Excellent agreement in phase shift values with other sophisticated calculations is obtained.

The energies and ANCs for 5Li resonances deduced from experimental p-α scattering phase shifts using the effective-range and Δ methods

Recently a new Δ method for deducing the energy and asymptotic normalization coefficient (ANC) from phase-shift data has been formulated and applied to resonance states. This differs from the conventional effective-range function (ERF) method by fitting only the Δl function included in the nuclear part of the effective radius function Kl(k2), which is determined by the Coulomb-nuclear scattering phase shift. Both methods use the same description of the scattering amplitude and the original formulas. However, after inserting the modified function Kl(k2) to the scattering amplitude for charged particles, a term containing the psi function appears in the denominator of the partial amplitude. This leads to the inapplicability of the ERF method when increasing the product of charges of colliding particles. Moreover, even for lower charges it is not clear that the results of the ERF method are accurate enough. The Coulomb part of Kl(k2) forms a background which smooths its energy dependence. Therefore, one needs to find when the ERF method is inaccurate. This requires recalculating some published results using the Δ method. This was done in a recent paper for resonances in the α-4He scattering. Here, the Δ method is applied to 5Li using the experimental p-4He scattering phase-shift data in the ground P3/2 and in the first exited P1/2 resonance states. The results are compared with those obtained by the ERF method. The main changes concern resonance energy and width. The Δ method for resonances differs from the method which was proposed for bound states by Ramírez Suárez and Sparenberg (2017) which was also called the Δ method where a pole condition is defined by the Eq. Δl=0. Here, the standard pole condition, including the Coulomb part in the relate equation, is used for a resonant state.

Quantum well electronic states in spatially decoupled 2D Pb nanoislands on Nb-doped SrTiO3(0 0 1)

Two-dimensional (2D) Pb nanoisland has established an ideal platform for studying the quantum size effects on growth mechanism, electronic structures as well as high-temperature superconductivity. Here, we investigate the growth and quantum well electronic states of the 2D Pb nanoislands on Nb-doped SrTiO3(001) by scanning tunneling microscopy and spectroscopy. In contrast to Pb/Si(111), Pb/Cu(111) and Pb/Ag(111), there is no wetting layer of Pb formed on the Nb-doped SrTiO3(001) surface, resulting in isolated Pb nanoislands with an apparent height of 4 atomic layers as the building blocks for the island growth. According to the thickness-dependent quantum well states resolved in both occupied and unoccupied energy regions, the constant group velocity vg = 1.804 × 106 m/s and the Fermi wavevector kF = 1.575 Å−1, have been extracted from a linear fit of the Pb(111) band dispersion along the Γ-L direction. In addition, the energy-dependent scattering phase shift φ(E) obtained by means of the phase accumulation model shows a metallic-like scattering interface analogous to Pb/Ag(111). These spatially decoupled 2D Pb nanoislands thus realize an opportunity to explore the intrinsic quantum confinement phenomena in nanoscale superconductors on the doped titanium-oxide-type substrate.

Construction of an equivalent energy-dependent potential by a Taylor series expansion

To construct a phase-equivalent energy-dependent local potential corresponding to a sum of local and nonlocal interactions, we use a simple method for expanding the wave function in a Taylor series up to the third order. We apply the constructed potentials to calculate the scattering phase shifts using the phase equation. The results for scattering in nucleon–nucleon, $$\alpha$$ –nucleon, and $$\alpha$$ – $$\alpha$$ systems agree reasonably well with the standard data.

A fully quantum calculation of broadening and shifting coefficients of the D1 and D2 spectral lines of alkali-metal atoms colliding with noble-gas atoms

We use the Baranger model to compute collisional broadening and shift rates for the D1 and D2 spectral lines of M + Ng, where M = K, Rb, Cs and Ng = He, Ne, Ar. Scattering matrix elements are calculated using the channel packet method, and non-adiabatic wavepacket dynamics are determined using the split-operator method together with a unitary transformation between adiabatic and diabatic representations. Scattering phase shift differences are weighted thermally and are integrated over temperatures ranging from 100 K to 800 K. We find that predicted broadening rates compare well with experiment, but shift rates are predicted poorly by this model because they are extremely sensitive to the near-asymptotic behavior of the potential energy surfaces.

Glueball scattering cross section in lattice SU(2) Yang-Mills theory

We calculate the scattering cross section between two $0^{++}$ glueballs in $SU(2)$ Yang-Mills theory on lattice at $\beta = 2.1, 2.2, 2.3, 2.4$, and 2.5 using the indirect (HAL QCD) method. We employ the cluster-decomposition error reduction technique and use all space-time symmetries to improve the signal. In the use of the HAL QCD method, the centrifugal force was subtracted to remove the systematic effect due to nonzero angular momenta of lattice discretization. From the extracted interglueball potential we determine the low energy glueball effective theory by matching with the one-glueball exchange process. We then calculate the scattering phase shift, and derive the relation between the interglueball cross section and the scale parameter $\Lambda$ as $\sigma_{\phi \phi} = (2 - 51) \Lambda^{-2}$ (stat.+sys.). From the observational constraints of galactic collisions, we obtain the lower bound of the scale parameter, as $\Lambda > 60$ MeV. We also discuss the naturalness of the Yang-Mills theory as the theory explaining dark matter.

Two-channel approach to the average retarding force of metals for slow singly ionized projectiles

Based on the fundamental momentum-transfer theorem [Phys. Rev. Lett. 15, 11 (1965)] a novel contribution to the retarding force of metallic systems for slow intruders is derived. This contribution is associated with sudden charge-changing cycles during the path of projectiles. The sum of the novel and the well-known conventional contributions, both expressed in terms of scattering phase shifts, are used to discuss experimental data obtained for different targets. It is found that our two-channel modeling, with two nonlinear channels, improves the agreement between several data and theory and thus, as predictive modeling, can contribute to the desired convergence between experimental and theoretical attempts on the retarding force.

Light element (Z=1,2) production from spontaneous ternary fission of ^{252}Cf

The yields of light elements (Z=1,2) obtained from spontaneous ternary fission of ^{252}Cf are treated within a nonequilibrium approach, and the contribution of unstable nuclei and excited bound states is taken into account. These light cluster yields may be used to probe dense matter, and to infer in-medium corrections such as Pauli blocking which is determined by the nucleon density. Continuum correlations are calculated from scattering phase shifts using the Beth-Uhlenbeck formula, and the effect of medium modification is estimated. The relevant distribution is reconstructed from the measured yields of isotopes. This describes the state of the nucleon system at scission and cluster formation, using only three Lagrange parameters which are the nonequilibrium counterparts of the temperature and chemical potentials, as defined in thermodynamic equilibrium. We concluded that a simple nuclear statistical equilibrium model neglecting continuum correlations and medium effects is not able to describe the measured distribution of H and He isotopes. Moreover, the freeze-out concept may serve as an important ingredient to the nonequilibrium approach using the relevant statistical operator concept.

Lattice QCD and Baryon-Baryon Interactions: HAL QCD Method

In this article, we review the HAL QCD method to investigate baryon-baryon interactions such as nuclear forces in lattice QCD. We first explain our strategy in detail to investigate baryon-baryon interactions by defining potentials in field theories such as QCD. We introduce the Nambu-Bethe-Salpeter (NBS) wave functions in QCD for two baryons below the inelastic threshold. We then define the potential from NBS wave functions in terms of the derivative expansion, which is shown to reproduce the scattering phase shifts correctly below the inelastic threshold. Using this definition, we formulate a method to extract the potential in lattice QCD. Secondly, we discuss pros and cons of the HAL QCD method, by comparing it with the conventional method, where one directly extracts the scattering phase shifts from the finite volume energies through the L\"uscher's formula. We give several theoretical and numerical evidences that the conventional method combined with the naive plateau fitting for the finite volume energies in the literature so far fails to work on baryon-baryon interactions due to contaminations of elastic excited states. On the other hand, we show that such a serious problem can be avoided in the HAL QCD method by defining the potential in an energy-independent way. We also discuss systematics of the HAL QCD method, in particular errors associated with a truncation of the derivative expansion. Thirdly, we present several results obtained from the HAL QCD method, which include (central) nuclear force, tensor force, spin-orbital force, and three nucleon force. We finally show the latest results calculated at the nearly physical pion mass, $m_\pi \simeq 146$ MeV, including hyperon forces which lead to form $\Omega\Omega$ and $N\Omega$ dibaryons.

Reaction Rate of Radiative 3He3H Capture

Within the framework of the modified potential cluster model with forbidden states, radiative 3He3H capture at low and astrophysical energies is considered. It is shown that on the basis of potentials which are matched with scattering phase shifts and the energy of the bound state, overall success is obtained in reproducing the available experimental data. On the basis of the obtained total cross sections, a calculation of the astrophysical S-factor and the capture reaction rate has been performed. All of the obtained results are approximated by curves of a definite type, which simplifies their use in applied studies.

Study of nucleon–nucleon and alpha-nucleon elastic scattering by the Manning–Rosen potential

Although often used in molecular dynamics, in this work the Manning–Rosen potential is parameterized to compute the scattering phase shifts for the nucleon–nucleon and the alpha-nucleon systems by exploiting the standard phase function method. We obtain excellent agreement in phase shifts with the more sophisticated calculations up to partial waves

Partial-wave mixing in Hamiltonian effective field theory

The spectrum of excited states observed in the finite volume of lattice QCD is governed by the discrete symmetries of the cubic group. This finite group permits the mixing of orbital angular momentum quanta in the finite volume. As experimental results refer to specific angular momentum in a partial-wave decomposition, a formalism mapping the partial-wave scattering potentials to the finite volume is required. This formalism is developed herein for Hamiltonian effective field theory, an extension of chiral effective field theory incorporating the L\"uscher relation linking the energy levels observed in finite volume to the scattering phase shift. The formalism provides an optimal set of rest-frame basis states maximally reducing the dimension of the Hamiltonian, and it should work in any Hamiltonian formalism. As a first example of the formalism's implementation, lattice QCD results for the spectrum of an isospin-2 $\pi\pi$ scattering system are analyzed to determine the $s$, $d$, and $g$ partial-wave scattering information.

On- and off-shell Jost functions for the Manning-Rosen potential

The s-wave on-shell Jost solution for the Manning-Rosen potential is constructed via the ordinary differential equation approach to the problem. The corresponding off-shell Jost function is calculated by using the integral representation of the off-shell Jost function in a single channel scattering. The half off-shell T- matrix is obtained from the knowledge of the off-shell Jost function. Our constructed expressions for the Jost functions are applied to compute T-matrix, quasi-elastic phase shifts and the scattering phase shifts for the nuclear systems.

Extracting free-space observables from trapped interacting clusters

The energy spectrum of two short-range interacting particles in a harmonic potential trap has previously been related to free-space scattering phase shifts. But the existing formula for this purpose is exact only in the limit of an infinitely shallow trap. Here we provide a systematically improved formula---describing the low-energy dynamics---that enables the use of finite traps. This paves the way for extracting nuclear scattering phase shifts from {\it ab initio} nuclear many-body structure calculations, a long-sought goal in nuclear physics. The derivation establishes effective field theory as a powerful framework for studying the connection between structure information of a trapped system (with two or more sub-clusters) and continuum physics in the fields of both nuclear and condensed-matter physics.

Interaction potentials and ultracold scattering cross sections for the Li+7–Li7 ion-atom system

We calculate the isotope independent Li$^+$-Li potential energy curves for the electronic ground and first excited states. Scattering phase shifts and total scattering cross section for the $^7$Li$^+$-$^7$Li collision are calculated with emphasis on the ultra-low energy domain down to the $s$-wave regime. The effect of physically motivated alterations on the calculated potential energy curves is used to determine the bound of accuracy of the low-energy scattering parameters for the ion-atom system. It is found that the scattering length for the A$^2\Sigma_u^+$ state, $a_u$ = 1325 a$_0$, is positive and has well-constrained bounds. For the X$^2\Sigma_g^+$ state, the scattering length, $a_g$ = 20465 a$_0$ has a large magnitude as it is sensitive to the restrained change of the potential, due to the presence of a vibrational state in the vicinity of the dissociation limit.

Theory of exciton-electron scattering in atomically thin semiconductors

The realization of mixtures of excitons and charge carriers in van-der-Waals materials presents a new frontier for the study of the many-body physics of strongly interacting Bose-Fermi mixtures. In order to derive an effective low-energy model for such systems, we develop an exact diagonalization approach based on a discrete variable representation that predicts the scattering and bound state properties of three charges in two-dimensional transition metal dichalcogenides. From the solution of the quantum mechanical three-body problem we thus obtain the bound state energies of excitons and trions within an effective mass model which are in excellent agreement with Quantum Monte Carlo predictions. The diagonalization approach also gives access to excited states of the three-body system. This allows us to predict the scattering phase shifts of electrons and excitons that serve as input for a low-energy theory of interacting mixtures of excitons and charge carriers at finite density. To this end we derive an effective exciton-electron scattering potential that is directly applicable for Quantum Monte-Carlo or diagrammatic many-body techniques. As an example, we demonstrate the approach by studying the many-body physics of exciton Fermi polarons in transition-metal dichalcogenides, and we show that finite-range corrections have a substantial impact on the optical absorption spectrum. Our approach can be applied to a plethora of many-body phenomena realizable in atomically thin semiconductors ranging from exciton localization to induced superconductivity.

Swift charged particles in a degenerate electron gas: An estimation for the charge-sign effect in stopping

Motivated by the recently introduced [Phys. Rev. A 94, 042704 (2016)] and already investigated [J. Phys. B: At. Mol. Opt. Phys. 50, 185201 (2017)] new approximate form for the stopping power in terms of scattering phase shifts, here we show analytically that within such a modeling of the observable an exact leading phase shift may contain that information which is completely enough to characterize the asymptotic charge-sign-independent Bethe term (∝Z12) and a charge-sign-dependent (Barkas) term (∝Z13) in the stopping force of a degenerate electron gas for fast projectiles with charge Z1. Our study is based on a spherical potential with velocity-dependent screening as in the above works. The next term in an asymptotic expansion, the Bloch term (∝Z14), measures the difference of the exact and first-order Born treatments with Coulomb potential.