Four-body Equations Research Articles

α-cluster states and 4α-particle bose condensate in 16O

In order to explore the 4α-particle condensate state in 16O, we solve a full four-body equation of motion based on the 4α OCM (Orthogonality Condition Model) in a large 4α model space spanned by Gaussian basis functions. A full spectrum up to the 06+ state is reproduced consistently with the lowest six 0+ states of experimental spectrum. The 06+ state is obtained at 2 MeV above the 4α breakup threshold and has a dilute density structure, where the rms radius is more than 5 fm. The state has an appreciably large α condensate fraction 61%, and a highly amount of α+12C(02+) components, both of which are strong pieces of evidence of the state being the 4α condensate state.

Ab initiofour-body calculation ofn-He3,p-H3, andd-dscattering

Four-body equations in momentum space are solved for neutron-$^{3}\mathrm{He}$, proton-$^{3}\mathrm{H}$, and deuteron-deuteron scattering; all three reactions are coupled. The Coulomb interaction between the protons is included using the screening and renormalization approach as was recently done for proton-deuteron and proton-$^{3}\mathrm{He}$ scattering. Realistic potentials are used between nucleon pairs. For the first time fully converged results for the observables pertaining to the six different elastic and transfer reactions are obtained and compared with experimental data.

Four-Body Calculation of Proton-He3Scattering

The four-body equations of Alt, Grassberger, and Sandhas are solved, for the first time, for proton-3He scattering including the Coulomb interaction between the three protons using the method of screening and renormalization as was done recently for proton-deuteron scattering. Various realistic two-nucleon potentials are used. Large Coulomb effects are seen on all observables. Comparison with data at different energies shows large deviations in the proton analyzing power but quite reasonable agreement in other observables. The effect of the nucleon-nucleon magnetic moment interaction and correlations between p-d and p-3He analyzing powers are studied.

Four-nucleon scattering:Ab initiocalculations in momentum space

The four-body equations of Alt, Grassberger, and Sandhas are solved for $n\text{\ensuremath{-}}$$^{3}\mathrm{H}$ scattering at energies below three-body breakup threshold using various realistic interactions including one derived from chiral perturbation theory. After partial wave decomposition the equations are three-variable integral equations that are solved numerically without any approximations beyond the usual discretization of continuum variables on a finite momentum mesh. Large number of two-, three-, and four-nucleon partial waves are considered until the convergence of the results is obtained. The total $n\text{\ensuremath{-}}$$^{3}\mathrm{H}$ cross section data in the resonance region is not described by the calculations which confirms previous findings by other groups. Nevertheless the numbers we get are slightly higher and closer to the data than previously found and depend on the choice of the two-nucleon potential. Correlations between the ${A}_{y}$ deficiency in $n\text{\ensuremath{-}}d$ elastic scattering and the total $n\text{\ensuremath{-}}$$^{3}\mathrm{H}$ cross section are studied.

Relativistic two-, three- and four-body wave equations in scalar QFT

We use the variational method within the Hamiltonian formalism of QFT to derive relativistic two-, three- and four-body wave equations for scalar particles interacting via a massive or massless mediating scalar field (the scalar Yukawa model). The Lagrangian of the theory is reformulated by using Green's functions to express the mediating field in terms of the particle fields. The QFT is then constructed from the resulting reformulated Hamiltonian. Simple Fock-space variational trial states are used to derive relativistic two-, three- and four-body equations. The equations are shown to have the Schrödinger non-relativistic limit, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Ground-state solutions of the relativistic equations are obtained approximately for various strengths of coupling, for both massive and massless mediating fields, and a comparison of the two-, three- and four-particle binding energies is presented.

The α-particle in nuclear matter

Among the light nuclear clusters the α-particle is by far the strongest bound system and therefore expected to play a significant role in the dynamics of nuclei and the phases of nuclear matter. To systematically study the properties of the α-particle we have derived an effective four-body equation of the Alt–Grassberger–Sandhas (AGS) type that includes the dominant medium effects, i.e. self energy corrections and Pauli-blocking in a consistent way. The equation is solved utilizing the energy dependent pole expansion for the subsystem amplitudes. We find that the Mott transition of an α-particle at rest differs from that expected from perturbation theory and occurs at approximately 1/10 of nuclear matter densities.

Contribution of Nucleon-NucleonPWaves tont−nt,dd−pt, anddd−ddScattering Observables

The four-body equations of Alt, Grassberger, and Sandhas [Phys. Rev. C 1, 85 (1970)] are solved for a system of four nucleons, using realistic NN interactions in channels $j\ensuremath{\le}2$. The results of the calculation are compared with data for the reactions $\mathrm{dd}\ensuremath{\rightarrow}\mathrm{dd}$, $\mathrm{dd}\ensuremath{\rightarrow}{p}^{3}\mathrm{H}$, and ${n}^{3}\mathrm{H}\ensuremath{\rightarrow}{n}^{3}\mathrm{H}$. The calculations indicate that the NN P waves have a strong effect on 4N observables including a $10%$ contribution to the total ${n}^{3}\mathrm{H}\ensuremath{\rightarrow}{n}^{3}\mathrm{H}$ cross section at the peak of the low energy resonance.

Microscopic calculation of four-nucleon scattering observables in dd→dd and dd→p 3H

The four-body equations of Alt, Grassberger and Sandhas are solved for a system of four nucleons, using realistic NN interactions. The results of the calculations are compared with data for the reactions and dd→dd and dd→p 3H. Preliminary calculations indicate that the nucleon-nucleon p-waves have a strong effect on 4N observables.

Exact solution of the four-body Faddeev-Yakubovsky equations for the harmonic oscillator.

The Faddeev-Yakubovsky equations for four identical spinless particles in three-dimensional space interacting via harmonic-oscillator two-body potentials are solved analytically for any excitation and for an arbitrary value of the total angular momentum.

Analysis of 2H(\vecd, p)3H Reaction at 30-90 KeV by Four-Body Faddeev-Yakubovsky Equation

Low-energy 2 H(d, p) 3 H reactions are investigated by the four-body Faddeev-Yakubovsky equations. Cross sections and tensor analyzing powers are calculated at 30-90 keV energies. The PEST-1 potentials are used for nucleon-nucleon interactions. The [2+2] and [3+1] subamplitudes are treated by the Hilbert-Schmidt expansions. Numerical results give qualitative explanation of experimental data

Integrodifferential equation approach. I. Triton and alpha -particle binding energies.

A single integrodifferential equation in two variables, valid for {ital A} nucleons interacting by pure Wigner forces, which has previously only been solved in the extreme and uncoupled adiabatic approximations, is now solved exactly for three- and four-nucleon systems. The results are in good agreement with the values obtained for the binding energies by means of an empirical interpolation formula. This validates all our previous conclusions, in particular that the omission of higher (than two) order correlations in our four-body equation only produces a rather small underbinding. The integrodifferential equation approach is here also extended to spin-dependent forces of the Malfliet-Tjon type, resulting in two coupled integrodifferential equations in two variables. The exact solution and the interpolated adiabatic approximation are again in good agreement. The inclusion of the hypercentral part of the two-body interaction in the definition of the Faddeev-type components again leads to substantial improvement for fully local potentials, acting in all partial waves.

4He binding energy calculation including full tensor-force effects.

The four-body equations of Alt, Grassberger, and Sandhas are solved in the version where the (2)+(2) subamplitudes are treated exactly by convolution, using one-term separable Yamaguchy nucleon-nucleon potentials in the $^{1}\mathrm{S}_{0}$ and $^{3}\mathrm{S}_{1}$${\mathrm{\ensuremath{-}}}^{3}$${\mathrm{D}}_{1}$ channels. The resulting ${j}^{p}$${=1/2}^{+}$ and ${(3/2}^{+}$ three-body subamplitudes are represented in a separable form using the energy-dependent pole expansion. Converged bound-state results are calculated for the first time using the full interaction, and are compared with those obtained from a simplified treatment of the tensor force. The Tjon line that correlates three-nucleon and four-nucleon binding energies is shown using different nucleon-nucleon potentials. In all calculations the Coulomb force has been neglected.

Nonmesonic photonuclear reactions in the three-nucleon system.

We derive sets of equations which describe \ensuremath{\gamma}${+}^{3}$${\mathrm{H}}_{\mathrm{e}}$${\ensuremath{\rightarrow}}^{3}$${\mathrm{H}}_{\mathrm{e}}$, p+d, and 2p+n, where the photon is either real or virtual, by requiring four-body unitarity. The initial and final wave functions and the final-state interactions (in the case of p+d and 2p+n) are the solutions of the three-body (Faddeev) equation in the bound and continuum states, respectively. The driving terms in the sets of equations for the reactions consist of two parts: the impulse one-baryon current [the nucleon, \ensuremath{\Delta}(1232), and ${N}^{\mathrm{*}}$(1440),. . . ] and an amplitude which satisfies a four-body equation and which includes the meson-exchange currents in lowest order.

Contribution ofp-wave (3)+1 subamplitudes to4He binding energy and scattering observables below four-body breakup threshold

The four-body equations of Alt, Grassberger and Sandhas, in the form that the (2)+(2) subsystem amplitudes are treated by the convolution method, are solved for separable two-body interactions between nucleon pairs in the channels3S1-3D1 (t00 alone) and1S0. AllL=0 andL=1 (3)+1 subamplitudes are represented in a separable form using the energy dependent pole expansion. The contribution ofp-wave (3)+1 subamplitudes to the4He binding energy, threshold scattering observables and low energy cross sections forn3H→n3H,p3He→p3He,dd→dd anddd→p3H reactions is studied for different two-body potentials.

Four-body calculation of the four-nucleon system: Binding energies and scattering results

Using a two-body separable $t$ matrix between pairs, we solve the resulting four-body equations for four spinless bosons in the form that the 2+2 subsystem contribution is treated exactly by the convolution method. The 3+1 subamplitudes are represented as finite rank operators. We compare the utility of these methods both in four-body bound state and scattering calculations. We also develop an approximation that neglects four-body intermediate states in the 2+2 subsystem contribution and compare it with exact results. Finally, we present the results for a four-nucleon calculation with spin dependent $s$-wave separable interactions between pairs. These formally exact four-body equations are subsequently used to develop one parameter models to describe low energy phase shifts and cross sections for the reactions n $^{3}\mathrm{H}$\ensuremath{\rightarrow}n $^{3}\mathrm{H}$, dd\ensuremath{\rightarrow}dd, and dd\ensuremath{\rightarrow}p $^{3}\mathrm{H}$.

Eigenvalues of the Yakubovskii equation kernel for a four-nucleon system

Low-lying states of $^{4}\mathrm{He}$ have been studied by calculating the trajectories of the first two eigen-values of the kernels of the Yakubovskii four-body equation as the total energy $E$ increases from $\ensuremath{-}\ensuremath{\infty} \mathrm{to} \ensuremath{\infty}+i\ensuremath{\epsilon}$. The two-particle interactions used are of the separable Yamaguchi type and include spin-dependent forces. The integral equations are derived for each state with values of spin $S$, isospin $T$, and total angular momentum $L$. To obtain a set of single variable integral equations, the Schmidt expansion is applied. The deformation of the integration contour is performed for the complex eigenvalues, and the eigenvalue problems for these equations are solved to determine the binding energy or the resonance energy including $l=1$ subamplitudes for 3 + 1 subsystems. The binding energies for the ground and the first excited state are -45.009 MeV and -11.529 MeV, respectively. A resonance state is found to be about -4.889 MeV in the state with $ST=10$, $L=1$ corresponding to the degenerate state of the second, third, and ninth excited states of the $^{4}\mathrm{He}$ nucleus.[NUCLEAR STRUCTURE $^{4}\mathrm{He}$; calculated levels. Four-body, separable potential model.]

Elastic and rearrangement scattering between two interacting deuterons as a four-body problem

A four-body model is used in studying the four-nucleon system. This model is solvable and can be applied to nuclear reactions involving four nucleons. Intermediate quasiparticle states are used for the two- and three-body scattering leading to $T$ matrices which are separable. The model leads to four-body equations which, by using partial wave decomposition, are reduced to single variable integral equations. Numerical calculations of the differential cross sections of the nuclear reactions $^{2}\mathrm{H}(d, p)^{3}\mathrm{H}$, $^{2}\mathrm{H}(d, n)^{3}\mathrm{He}$, and $^{2}\mathrm{H}(d, d)^{2}\mathrm{H}$ are carried out for different deuteron laboratory energies between 6.1 and 51.5 MeV. Inclusion of the $p$-wave three-body amplitudes is tested. Also, the simultaneous exchange of two nucleons between the incoming deuterons is investigated. The theoretically calculated angular distributions are in good agreement with the experimental measurements. The extracted values of the spectroscopic and normalization factors are reasonable.NUCLEAR REACTIONS $^{2}\mathrm{H}(d, p)^{3}\mathrm{H}$, $^{2}\mathrm{H}(d, n)^{3}\mathrm{He}$, $^{2}\mathrm{H}(d, d)^{2}\mathrm{H}$; ${E}_{d}=6.1\ensuremath{-}51.5$ MeV; four-body model. Calculated $\ensuremath{\sigma}(\ensuremath{\theta})$. Extracted spectroscopic factors.

Binding energy estimates for charmed few-body systems

By solving exact three-and four-body equations, we obtain estimates of the binding energies of light nuclei containing one charmed ${C}_{0}^{+}$ baryon.NUCLEAR STRUCTURE Few-body hypernuclei, charmed analogs of hypernuclei, binding energy estimates.

Derivation of factorization properties for highly connectedN-body equations

Resorting to cluster expansions for the fullN-body resolvent, we show that highly-connected-kernel equations, can be obtained by multiplying two-body connected-kernel equations, having only physical solutions, by an appropriate multiplier, which can introduce spurious homogeneous solutions. It is also explicitly shown that the factorization property for the four-body Rosenberg-like equations is the four-body analogue (matrix version) of the factorization property for the three-body Weinberg-Van Winter equation.

$${}_{\Lambda \Lambda }^{10} Be$$ in Faddeev-Yakubovsky formalismin Faddeev-Yakubovsky formalism

The four-body dynamical equations for two distinct pairs of identical particles derived earlier are applied to investigate the system\({}_{\Lambda \Lambda }^{10} Be\). The two-body potentials have been taken to be of the Yamaguchi form, and the Bateman approximation has been used for the other amplitudes. From the set of coupled integral equations, the separation energy, BΛΛ, for the two Λ particles in\({}_{\Lambda \Lambda }^{10} Be\) is obtained as 43·97 MeV.