Coupled-channel Calculations Research Articles

Effect of Coupling Channel on Elastic Scattering for 6He+208Pb, 7Be+58Ni and 7Li+59Co Systems

Effects of coupling channel on elastic scattering have been studied using the nuclear potential Woods-Saxon (WS) for 6He+208Pb, 7Be+58Ni and 7Li+59Co systems. The Continuum-discretized coupled-channels CDCC method have been used to employed no coupled and coupled-channels (CC) calculations for unstable projectiles. Cross-sections of elastic scattering were extracted from the optical potential fits, to achieve the optimum comparability between the theoretical estimates of dσel/dσR and empirical values for the systems reviewed. The breakup process at near Coulomb barrier energies play essential role in the results of cross section as a function of angle and energy in addition to the distribution of the barrier.

The Ds0(2590)+ as the dressed [formula omitted] meson in a coupled-channels calculation

The recent discovery of the Ds0(2590)+ meson by the LHCb Collaboration has stimulated the analysis of meson-meson channels effects in the two-body quark-antiquark meson spectrum. This resonance, assigned to the radial excitation of the pseudoscalar Ds+ meson, has a mass much lower than the predictions of naive quark models, which could indicate a non-negligible D(⁎)K(⁎) coupling which reduces its mass. Based on the importance of nearby meson-meson thresholds in the dynamics of P-wave Ds mesons such as the Ds0⁎(2317)+ and Ds1(2460)+, in this work we perform a coupled-channels calculation including the D(⁎)K(⁎), Ds(⁎)ω and Ds⁎η channels, and study the impact of incorporating those channels in the mass of the bare cs¯. The coupling between two and four-quark sectors is done by means of the P03 mechanism, with all the parameters constrained from previous studies of the heavy meson spectroscopy. The masses, widths and production line shapes of the resulting state are analyzed.

Improved characterization of Feshbach resonances and interaction potentials between Na23 and Rb87 atoms

The ultracold mixture of \Na and \Rb atoms has become an important system for investigating physics in Bose-Bose atomic mixtures and for forming ultracold ground-state polar molecules. In this work, we provide an improved characterization of the most commonly used Feshbach resonance near 347.64 G between \Na and \Rb in their absolute ground states. We form Feshbach molecules using this resonance and measure their binding energies by dissociating them via magnetic field modulation. We use the binding energies to refine the singlet and triplet potential energy curves, using coupled-channel bound-state calculations. We then use coupled-channel scattering calculations on the resulting potentials to produce a high-precision mapping between magnetic field and scattering length. We also observe 10 additional $s$-wave Feshbach resonances for \Na and \Rb in different combinations of Zeeman sublevels of the $F = 1$ hyperfine states. Some of the resonances show 2-body inelastic decay due to spin exchange. We compare the resonance properties with coupled-channel scattering calculations that full take account of inelastic properties.

Prospects for assembling ultracold radioactive molecules from laser-cooled atoms

Molecules with unstable isotopes often contain heavy and deformed nuclei and thus possess a high sensitivity to parity-violating effects, such as the Schiff moments. Currently the best limits on Schiff moments are set with diamagnetic atoms. Polar molecules with quantum-enhanced sensing capabilities, however, can offer better sensitivity. In this work, we consider the prototypical 223Fr107Ag molecule, as the octupole deformation of the unstable 223Fr francium nucleus amplifies the nuclear Schiff moment of the molecule by two orders of magnitude relative to that of spherical nuclei and as the silver atom has a large electron affinity. To develop a competitive experimental platform based on molecular quantum systems, 223Fr atoms and 107Ag atoms have to be brought together at ultracold temperatures. That is, we explore the prospects of forming 223Fr107Ag from laser-cooled Fr and Ag atoms. We have performed fully relativistic electronic-structure calculations of ground and excited states of FrAg that account for the strong spin-dependent relativistic effects of Fr and the strong ionic bond to Ag. In addition, we predict the nearest-neighbor densities of magnetic-field Feshbach resonances in ultracold 223Fr + 107Ag collisions with coupled-channel calculations. These resonances can be used for magneto-association into ultracold, weakly-bound FrAg. We also determine the conditions for creating 223Fr107Ag molecules in their absolute ground state from these weakly-bound dimers via stimulated Raman adiabatic passage using our calculations of the relativistic transition electric dipole moments.

Role of inelastic couplings in 4He + 208Pb elastic scattering in a wide energy range

The phenomenological strengths of the real part of the optical potential, obtained from elastic scattering data analyses within the optical model (OM) approach, present significant energy-dependence. This behavior has been associated with the intrinsic energy-dependence of the effective nucleon–nucleon interaction. However, in earlier works, we proposed that at least part of this dependence can arise from the effect of couplings to inelastic states of the nuclei. In order to deepen this study, in this paper we present extensive data analyses for the elastic scattering and inelastic excitation of 111 states of 208Pb, for the 4He + 208Pb system in a wide energy range. For the purpose of comparison, the theoretical cross sections are obtained using different approaches for the imaginary part of the potential, and within both contexts: OM (distorted wave Born approximation) and coupled-channel calculations.

Fusion Reaction Study of some Selected Halo Systems

The challenge in studying fusion reaction when the projectile is neutron or proton rich halo nuclei is the coupling mechanism between the elastic and the breakup channel, therefore the motivation from the present calculations is to estimate the best coupling parameter to introduce the effect of coupled-channels for the calculations of the total cross section of the fusion , the barrier distribution of the fusion and the average angular momentum 〈L〉 for the systems 6He+206Pb, 8B+28Si, 11Be+209Bi, 17F+208Pb, 6He+238U, 8He+197Au and 15C+232Th using quantum mechanical approach. A quantum Coupled-Channel Calculations are performed using CC code. The predictions of quantum mechanical approach are comparable with the measured data that is available. Above and below the Coulomb barrier, comparison of theoretical calculations of quantum mechanical with the relevant measured data demonstrates good agreement.

Exploring various features of the reaction mechanism involved in the collision of Li7 on Cu

Nuclear reaction induced by a weakly bound projectile on light-medium mass targets poses open questions on the fusion mechanism. It is seen that fusion suppression in such reactions is negligible, but that stands on the basis of scarcely available experimental data. Thus more experimental data are demanded to bring out a clear understanding. In the course of this, for the first time, a measurement of residual cross sections from $^{7}\mathrm{Li}$ induced reaction on $^{\mathrm{nat}}\mathrm{Cu}$ has been presented in this article within the 2.3--6.0 MeV/nucleon energy range. The residues $^{69,67,66}\mathrm{Ge}$, $^{68,67,66,65}\mathrm{Ga}$, and $^{69m,65,63}\mathrm{Zn}$ produced in the reaction have been identified by $\ensuremath{\gamma}$-ray spectroscopy. The measured excitation function of the residues has been analyzed using equilibrium and preequilibrium reaction models in the framework of empire-3.2.2 to understand the reaction mechanisms involved in the low energy region. The underlying reaction mechanism is shown to be a blend of equilibrium and preequilibrium processes. The intensity of the $\ensuremath{\gamma}$ peak 93.31 keV arising in the decay of $^{67}\mathrm{Ga}$ has been revised experimentally. The experimental intensity turns out to be about half the value reported in different nuclear databases. Fusion cross sections have been estimated using the experimental data and empire-3.2.2 code. The estimated fusion cross sections are in line with coupled channel calculations taking inelastic excitations into account. The large production cross section of medically important $^{67}\mathrm{Ga}$ has been measured.

O18+Se76 elastic and inelastic scattering at 275 MeV

Background: An accurate description of the initial and final state interactions in the O18+Se76 collision is demanded by the NUMEN project. The study of single and double charge exchange nuclear reactions is the main purpose for NUMEN, since these can be used as tools to provide experimentally driven information about nuclear matrix elements of interest in the context of neutrinoless double-β decay. To date, the details of the optical potentials and nuclear response to isospin operators for many of the projectile-target systems proposed for the NUMEN double charge exchange studies are poorly known. The O18+Se76 case, here under study, is particularly relevant due to its connection with the Ge76 neutrinoless double-β decay. Purpose: In this work the authors want to characterize the initial-state interaction for the O18+Se76 reactions at 275 MeV incident energy determining the optical potential and evaluating the effect of couplings with the inelastic scattering on the elastic channel. Methods: The angular distributions of differential cross section were measured in the angular region between 4∘ and 22∘ in the center-of-mass reference frame. The cross sections were compared with theoretical calculations, that adopt different optical potentials. Coupling effects on the elastic channel were determined into the coupled channels formalism. Results: The excitation energy spectrum of the colliding nuclei and the cross section angular distributions were measured with satisfactory energy resolution. The elastic scattering cross section is not well reproduced in the full angular range explored when the optical model approach is adopted. A good agreement is found using coupled channel calculations. The initial state interaction for the O18+Se76 nuclear reactions at 275 MeV is determined. Conclusions: Coupled channels effects are crucial to obtain a good description of the measured elastic and inelastic channels cross sections, even at large transferred momenta where the optical model approach fails in reproducing the experimental data. The role of channel coupling could be relevant also in the analysis of other open reaction channels in the same collision and should be accounted for in double charge exchange analyses as well.Received 19 December 2020Revised 6 October 2021Accepted 10 November 2021DOI:https://doi.org/10.1103/PhysRevC.104.054610©2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDirect reactionsLow & intermediate energy heavy-ion reactionsNuclear reactionsNuclear structure & decaysNucleon-nucleon interactionsTransfer reactionsProperties59 ≤ A ≤ 896 ≤ A ≤ 19TechniquesCollective modelsNuclear data analysis & compilationNuclear reaction analysisOptical, coupled-channel & distorted wave modelsSpectrometers & spectroscopic techniquesNuclear Physics

Resonance-like structure near the ηd threshold in the γd→π0ηd reaction

To investigate the interaction between the nucleon $N$ and nucleon resonance $N(1535)1/2^-$, the $\eta d$ threshold structure connected to the isoscalar $S$-wave $N$-$N(1535)1/2^-$ system has been experimentally studied in the $\gamma{d}${$\to$}$\pi^0\eta{d}$ reaction at incident photon energies ranging from the reaction threshold to 1.15 GeV. A strong enhancement is observed near the $\eta d$ threshold over the three-body phase-space contribution in the $\eta d$ invariant-mass distribution. An analysis incorporating the known isovector resonance $\mathcal{D}_{12}$ with a spin-parity of $2^+$ in the $\pi^0d$ channel reveals the existence of a narrow isoscalar resonance-like structure with $1^-$ in the $\eta d$ system. Using a Flatt\'e parametrization, the mass is found to be $2.427_{-0.006}^{+0.013}$ GeV, close to the $\eta d$ threshold, and the width is $\left(0.029_{-0.029}^{+0.006}{\rm\ GeV}\right)+\left(0.00_{-0.00}^{+0.41}\right) p_\eta c$, where $p_\eta$ denotes the $\eta$ momentum in the rest frame of the $\eta d$ system. The observed structure would be attributed to a predicted isoscalar $1^-$ $\eta NN$ bound state from $\eta NN$ and $\pi NN$ coupled-channel calculation, or an $\eta d$ virtual state owing to strong $\eta d$ attraction.

Effect of Surface Diffuseness Parameter on Quasi-elastic Scattering Calculations for 16O+208Pb,63Cu Systems

A systematic study on the surface characteristic of the nucleus-nucleus potential for some heavy-ion. The nuclear potential has been described by using Woods-Saxon (WS), the single-channel (SC) and the coupled-channels (CC) calculations, which were between the relative motion of the colliding nuclei and their intrinsic motions, were conducted to study its influence on calculation, the ratio of the quasi-elastic to the Rutherford cross sections and probe the surface diffuseness method was used to find the best fitted value of the diffuseness parameters in comparison with the experimental data. We find that the best fitted value of the diffuseness parameter which obtained through a coupled-channel calculation with inert target and excited projectile forIn the current work, the single-channel (SC) and the coupled-channels (CC) calculations, which were between the relative motion of the colliding nuclei and their intrinsic motions, were conducted to study its influence on calculation, the ratio of the quasi-elastic to the Rutherford cross sections and probe the surface diffuseness find that the best fitted value of the diffuseness parameter which obtained through a coupled-channel calculation with excited target and projectile excited also inert projectile-excited target for the 16O+208Pb andexcited projectile and inert target for the 16O+63Cu.

Fusion of O16+Ho165 at deep sub-barrier energies

Fusion cross sections have been measured for the asymmetric system $^{16}\mathrm{O}+^{165}\mathrm{Ho}$ at energies near and deep below the Coulomb barrier with an aim to investigate the occurrence of fusion hindrance for the system. Cross sections down to $\ensuremath{\approx}700$ nb have been measured using the off-beam $\ensuremath{\gamma}$-ray technique. The fusion cross sections have been compared with the coupled channel calculations. The energy onset of fusion hindrance appears to occur at ${E}_{\mathrm{c}.\mathrm{m}.}=57\ifmmode\pm\else\textpm\fi{}0.85$ MeV, where a deviation in the slope of the experimental logarithmic derivative compared to that of coupled channel calculations has been observed. This is in agreement with the value obtained from the touching point configuration of the adiabatic model.

Investigation of neutron transfer in Li7+Sn124 system

The relative importance of neutron transfer and breakup process in a reaction around Coulomb barrier energies has been studied for the $^{7}\mathrm{Li}+^{124}\mathrm{Sn}$ system. Coupled channel calculations have been performed to understand the one neutron stripping and pickup cross sections along with the breakup in the $^{7}\mathrm{Li}+^{124}\mathrm{Sn}$ system. The systematics of one and two neutron stripping and pickup cross sections with a $^{7}\mathrm{Li}$ projectile on several targets show an approximate universal behavior, which has been explained by a simple model based on barrier penetration. Complete reaction mechanism have been studied by comparing the reaction cross sections with the measured total fusion and one neutron transfer cross sections.

Zb structures in a constituent quark model coupled-channels calculation

The $Z_b(10610)^\pm$ and $Z_b(10650)^\pm$ are two bottomonium-like structures discovered in the $\pi h_b(mP)$, $\pi \Upsilon(nS)$ and $B^\ast\bar B^{(\ast)}+h.c.$ invariant mass spectra, where $m=\{1,2\}$ and $n=\{1,2,3\}$. Their nature is puzzling due to their charge, which forces its minimal quark content to be $b\bar b u\bar d$ ($b\bar b d\bar u$). Thus, it is necessary to explore four-quark systems in order to understand their inner structure. Additionally, their strong coupling to channels such as $\pi \Upsilon$ and the closeness of their mass to $B^\ast\bar B^{(\ast)}$-thresholds stimulates a molecular interpretation. Within the framework of a constituent quark model which satisfactorily describes a wide range of properties of (non-)conventional hadrons containing heavy quarks, we perform a coupled-channels calculation of the $I^G(J^{PC})=1^+(1^{+-})$ hidden-bottom sector including $B^{(\ast)}\bar B^{\ast}+h.c.$, $\pi h_b$, $\pi \Upsilon$ and $\rho\eta_b$ channels. We analyze the line shapes in the different channels, describing the $\Upsilon(5S)\to \pi B^{(*)}\bar B^{(*)}$ by means of the $^3P_0$ model. Since our description of the line shapes promising, we perform the same coupled-channels calculation for the $Z_b$'s with $J^{--}$, where $J=\{0,1,2\}$. This allows us to obtain a fair description of the corresponding line shapes. The study of the analytic structure of the $S$-matrix suggests that the experimental $Z_b$ structures arise as a combination of several poles with $J^{PC}=0^{--}$, $1^{\pm-}$ and $2^{--}$ quantum numbers nearby the $B\bar B^\ast$ and $B^\ast\bar B^\ast$ thresholds.

Fusion studies in O16+Nd142,150 reactions at energies near the Coulomb barrier

Background: Enhancement in fusion cross sections over one-dimensional barrier penetration model (1D-BPM) predictions has been observed at sub-barrier energies owing to the coupling of various internal degrees of freedom to the relative motion. On the other hand, hindrance in fusion is noticed at energies well above the barrier in a few cases.Purpose: The purpose is to probe the dynamics of heavy ion fusion at energies below and well above the Coulomb barrier.Methods: Fusion excitation functions for the $^{16}\mathrm{O}+^{142,150}\mathrm{Nd}$ reactions are measured using the Heavy Ion Reaction Analyzer (HIRA) at IUAC, New Delhi. Measurements have been performed in the range $12%$ below to $50%$ above the Coulomb barrier for both systems. The measured fusion cross sections are compared with coupled channels calculations using ccfull code. The cross sections are also compared with reactions using $^{16}\mathrm{O}$ beams with other isotopes of Nd to explore the systematic behavior of fusion.Results: Compared with 1D-BPM predictions, fusion enhancement is observed in both $^{16}\mathrm{O}+^{142}\mathrm{Nd}$ and $^{16}\mathrm{O}+^{150}\mathrm{Nd}$ reactions at below-barrier energies, with the latter reaction showing a higher enhancement. Coupled channels calculations incorporating the collective excitations of the target nuclei reproduce the fusion cross sections in both reactions. The collective excitations of the projectile nucleus do not seem to contribute to the observed fusion enhancement. Calculations using Aky\"uz-Winther potential parameters fail to reproduce the fusion cross sections at energies well above the barrier.Conclusions: Fusion enhancement is observed in both reactions studied. Degree of enhancement of sub-barrier fusion cross section is larger for the reaction using $^{150}\mathrm{Nd}$ target. Fusion hindrance is observed in both reactions at very high energies. The hindrance seems to increase with increasing beam energy. Larger value of diffuseness parameter compared with the value consistent with elastic-scattering measurements is required to reproduce the fusion excitation function at energies well above the barrier. This could be a strong indication of dynamical effects in fusion at very high energies.

Reaction channel contributions to proton scattering at 65 MeV

Background: Well-established coupled channel (CC) and coupled reaction channel (CRC) processes make contributions to elastic scattering that are absent from local density folding models.Purpose: To establish and characterize the contribution to the proton optical model potential (OMP) made by the coupling to neutron pickup channels, in particular the proton OMP for 65 MeV protons on $^{48}\mathrm{Ca}$ and $^{40}\mathrm{Ca}$. Also to relate this contribution to results for ${}^{40}\mathrm{Ca}$ at lower energies; to investigate the dynamical nonlocality of this contribution; to characterize the effect on the OMP of breakup of the deuteron.Methods: CRC calculations of neutron pickup and CC calculations of collective states, provide the elastic channel $S$ matrix ${S}_{lj}$. Inversion of ${S}_{lj}$ produces a local potential that yields, in a single channel calculation, the elastic scattering observables from the CC/CRC calculation. Subtracting the bare potential yields a local and $l$-independent representation of the dynamical polarization potential, DPP. From the DPPs due to a selection of channel couplings the influence of dynamically generated nonlocality can be identified. The effect of coupling to the deuteron breakup continuum is also identified.Results: For $^{40}\mathrm{Ca}$, coupling to pickup channels has an effect on observables somewhat weaker than that at 30 MeV, and much less than for pickup coupling for $^{48}\mathrm{Ca}$. The DPPs have similar general properties in each case, but are much larger in magnitude for $^{48}\mathrm{Ca}$. Subsequent breakup of the deuteron makes a large contribution to the DPP, and hence to the OMP. The formal DPPs due to pickup coupling exhibit dynamical nonlocality.Conclusions: The DPPs challenge local density folding models for elastic scattering. The breakup of the deuteron must henceforth be included in calculations of the DPP due to neutron pickup in proton scattering. Pickup coupling effects are still substantial at 65 MeV. No smoothly varying global OMP could fit proton elastic scattering from both ${}^{40}\mathrm{Ca}$ and ${}^{48}\mathrm{Ca}$.

Investigation of multistep effects for proton inelastic scattering to the 21+ state in He6

Multistep effects among bound, resonant, and nonresonant states have been investigated by the continuum-discretized coupled-channels method (CDCC). In the CDCC, a resonant state is treated as multiple states fragmented in a resonance energy region, although it is described as a single state in usual coupled-channels calculations. For such the fragmented resonant states, one-step, and multistep contributions to the cross sections should be carefully discussed because the cross sections obtained by the one-step calculation depend on the number of those states, which corresponds to the size of the model space. To clarify the role of the multistep effects, we propose the one-step calculation without model-space dependence for the fragmented resonant states. Furthermore, we also discuss the multistep effects among the ground, ${2}_{1}^{+}$ resonant, and nonresonant states in $^{6}\mathrm{He}$ for proton inelastic scattering.

Existence of core excited 8Be* =α+α* cluster structure inα+αscattering

AbstractWe show the existence of the $\alpha+\alpha^*$ cluster structure at the highly excited energy around $E_x=20$ MeV in $^{8}$Be for the first time in the coupled-channel calculations. An extended double-folding model derived using a realistic precise cluster wave function with a well-developed $N+3N$ cluster structure for the first excited state of $^4$He was employed. The calculation reproduces the experimental phase shifts in $\alpha + \alpha$ scattering up to $E_{\rm c.m.}=21$ MeV well. The result shows that the well-developed core-excited $\alpha+\alpha^*$ structure appears as resonances for $L=0$ and $2$ near the $\alpha+\alpha^*$ threshold which correspond to the experimental states at $E_x=20.20$ MeV and $E_x=22.24$ MeV in $^8$Be.

Quasielastic backscattering and barrier distribution for the weakly bound projectile Li6 on Tb159

The excitation function for quasielastic scattering of the weakly bound projectile $^{6}\mathrm{Li}$ on a $^{159}\mathrm{Tb}$ target, at large backward angle, has been measured at energies around the Coulomb barrier. The corresponding quasielastic barrier distribution has been extracted from the experimental cross sections, both including and excluding the $\ensuremath{\alpha}$ particles produced in the reaction. The quasielastic scattering cross sections, excluding the $\ensuremath{\alpha}$ particles, have been analyzed in the framework of coupled channels calculations. The centroid of the quasielastic barrier distribution, including the $\ensuremath{\alpha}$ particles, is found to shift towards higher energy relative to the centroid of the fusion barrier distribution for the system. This has been attributed to the low $\ensuremath{\alpha}$-breakup threshold of the nucleus $^{6}\mathrm{Li}$.

Probing open- and closed-channel p -wave resonances

We study the near-threshold molecular and collisional physics of a strong $^{40}$K p-wave Feshbach resonance through a combination of measurements, numerical calculations, and modeling. Dimer spectroscopy employs both radio-frequency spin-flip association in the MHz band and resonant association in the kHz band. Systematic uncertainty in the measured binding energy is reduced by a model that includes both the Franck-Condon overlap amplitude and inhomogeneous broadening. Coupled-channels calculations based on mass-scaled $^{39}$K potentials compare well to the observed binding energies and also reveal a low-energy p-wave shape resonance in the open channel. Contrary to conventional expectation, we observe a nonlinear variation of the binding energy with magnetic field, and explain how this arises from the interplay of the closed-channel ramping state with the near-threshold shape resonance in the open channel. We develop an analytic two-channel model that includes both resonances as well as the dipole-dipole interactions which, we show, become important at low energy. Using this parameterization of the energy dependence of the scattering phase, we can classify the studied $^{40}$K resonance as broad. Throughout the paper, we compare to the well understood s-wave case, and discuss the significant role played by van der Waals physics. The resulting understanding of the dimer physics of p-wave resonances provides a solid foundation for future exploration of few- and many-body orbital physics.

Quantum Spin State Selectivity and Magnetic Tuning of Ultracold Chemical Reactions of Triplet Alkali-Metal Dimers with Alkali-Metal Atoms.

We demonstrate that it is possible to efficiently control ultracold chemical reactions of alkali-metal atoms colliding with open-shell alkali-metal dimers in their metastable triplet states by choosing the internal hyperfine and rovibrational states of the reactants as well as by inducing magnetic Feshbach resonances with an external magnetic field. We base these conclusions on coupled-channel statistical calculations that include the effects of hyperfine contact and magnetic-field-induced Zeeman interactions on ultracold chemical reactions of hyperfine-resolved ground-state Na and the triplet NaLi(a^{3}Σ^{+}) producing singlet Na_{2}(^{1}Σ_{g}^{+}) and a Li atom. We find that the reaction rates are sensitive to the initial hyperfine states of the reactants. The chemical reaction of fully spin-polarized, high-spin states of rotationless NaLi(a^{3}Σ^{+},v=0,N=0) molecules with fully spin-polarized Na is suppressed by a factor of 10-100 compared to that of unpolarized reactants. We interpret these findings within the adiabatic state model, which treats the reaction as a sequence of nonadiabatic transitions between the initial nonreactive high-spin state and the final low-spin states of the reaction complex. In addition, we show that magnetic Feshbach resonances can similarly change reaction rate coefficients by several orders of magnitude. Some of these resonances are due to resonant trimer bound states dissociating to the N=2 rotational state of NaLi(a^{3}Σ^{+},v=0) and would thus exist in systems without hyperfine interactions.