Faddeev Equations In Momentum Space Research Articles

Trion clustering structure and binding energy in two-dimensional semiconductor materials: Faddeev equations approach

In this work, we develop the basic formalism to study trions in semiconductor layered materials using the Faddeev equations in momentum space for three different particles lying in two dimensions. We solve the trion Faddeev coupled integral equations for both short-range one-term separable Yamaguchi potential and Rytova-Keldysh (RK) interaction applied to the MoS$_2$ layer. We devise two distinct regularization methods to overcome the challenge posed by the repulsive electron-electron RK potential in the numerical solution of the Faddeev equations in momentum space. The first method regulates the repulsive interaction in the infrared region, while the second regulates it in the ultraviolet region. By extrapolating the trion energy to the situation without screening, the two methods gave consistent results for the MoS$_2$ layer with a trion binding energy of $-49.5(1)$~meV for the exciton energy of $-753.3$~meV. We analyzed the trion structure for the RK and Yamaguchi potentials in detail, showing their overall similarities and the dominant cluster structure, where the strongly bound exciton is weakly bound to an electron. We found that this property is manifested in the dominance of two of the Faddeev components over the one where the hole is a spectator of the interacting electron pair.

Li6 in a three-body model with realistic Forces: Separable versus nonseparable approach

{\bf Background:} Deuteron induced reactions are widely used to probe nuclear structure and astrophysical information. Those (d,p) reactions may be viewed as three-body reactions and described with Faddeev techniques. {\bf Purpose:} Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. However, it needs to be demonstrated that their solution based on separable interactions agrees exactly with solutions based on non-separable forces. {\bf Results:} The ground state of $^6$Li is calculated via momentum space Faddeev equations using the CD-Bonn neutron-proton force and a Woods-Saxon type neutron(proton)-$^4$He force. For the latter the Pauli-forbidden $S$-wave bound state is projected out. This result is compared to a calculation in which the interactions in the two-body subsystems are represented by separable interactions derived in the Ernst-Shakin-Thaler framework. {\bf Conclusions:} We find that calculations based on the separable representation of the interactions and the original interactions give results that agree to four significant figures for the binding energy, provided an off-shell extension of the EST representation is employed in both subsystems. The momentum distributions computed in both approaches also fully agree with each other.

Separable representation of multichannel nucleon-nucleus optical potentials

One important ingredient for many applications of nuclear physics to astrophysics, nuclear energy, and stockpile stewardship are cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not feasible, indirect methods, e.g. (d,p) reactions, should be used. Those (d,p) reactions may be viewed as three-body reactions and described with Faddeev techniques. Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. Optical potentials representing the effective interactions in the neutron (proton) nucleus subsystem are usually non-Hermitian as well as energy-dependent. Including excitations of the nucleus in the calculation requires a multichannel optical potential. The purpose of this paper is to introduce a separable, energy-dependent multichannel representation of complex, energy-dependent optical potentials that contain excitations of the nucleus and that fulfill reciprocity exactly. Momentum space Lippmann-Schwinger integral equations are solved with standard techniques to obtain the form factors for the separable representation. Starting from energy-dependent multichannel optical potentials for neutron and proton scattering from $^{12}$C, separable representations based on a generalization of the Ernst-Shakin-Thaler (EST) scheme are constructed which fulfill reciprocity exactly. Applications to n$+^{12}$C and p$+^{12}$C scattering are investigated for energies from 0 to 50~MeV.

Mean-square radii of two-component three-body systems in two spatial dimensions

We calculate root-mean-square radii for a three-body system confined to two spatial dimensions and consisting of two identical bosons ($A$) and one distinguishable particle ($B$). We use zero-range two-body interactions between each of the pairs, and focus thereby directly on universal properties. We solve the Faddeev equations in momentum space and express the mean-square radii in terms of first-order derivatives of the Fourier transforms of densities. The strengths of the interactions are adjusted for each set of masses to produce equal two-body bound-state energies between different pairs. The mass ratio, ${\cal A}=m_B/m_A$, between particles $B$ and $A$ are varied from $0.01$ to $100$ providing a number of bound states decreasing from $8$ to $2$. Energies and mean-square radii of these states are analyzed for small ${\cal A}$ by use of the Born-Oppenheimer potential between the two heavy $A$-particles. For large ${\cal A}$ the radii of the two bound states are consistent with a slightly asymmetric three-body structure. When ${\cal A}$ approaches thresholds for binding of the three-body excited states, the corresponding mean-square radii diverge inversely proportional to the deviation of the three-body energy from the two-body thresholds. The structures at these three-body thresholds correspond to bound $AB$-dimers and one loosely bound $A$-particle.

Testing semilocal chiral two-nucleon interaction in selected electroweak processes

The recently developed semilocal improved chiral nucleon-nucleon interaction is used for the first time to study several electromagnetic and weak processes at energies below the pion production threshold. Cross sections and selected polarization observables for deuteron photodisintegration, nucleon-deuteron radiative capture, three-body $^{3}\mathrm{He}$ photodisintegration, as well as capture rates for decays of the muonic $^{2}\mathrm{H}$ and $^{3}\mathrm{He}$ atoms are calculated. The Lippmann-Schwinger and Faddeev equations in momentum space are solved to obtain nuclear states. The electromagnetic current operator is taken as a single nucleon current supplemented by many-body contributions induced via the Siegert theorem. For muon capture processes the nonrelativistic weak current together with the dominant relativistic corrections is used. Our results compare well with experimental data, demonstrating the same quality as is observed for the semiphenomenological Argonne V18 potential. Compared to the older version of the chiral potential with a nonlocal regularization, a much smaller cut-off dependence is found for the state-of-art chiral local interaction employed in this paper. Finally, estimates of errors due to the truncation of the chiral expansion are given.

Separable representation of energy-dependent optical potentials

Background. One important ingredient for many applications of nuclear physics to astrophysics, nuclear energy, and stockpile stewardship are cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not feasible, indirect methods, e.g. (d,p) reactions, should be used. Those (d,p) reactions may be viewed as three-body reactions and described with Faddeev techniques. Purpose. Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. Optical potentials representing the effective interactions in the neutron (proton) nucleus subsystem are usually non-Hermitian as well as energy-dependent. Potential matrix elements as well as transition matrix elements calculated with them must fulfill the reciprocity theorem. The purpose of this paper is to introduce a separable, energy-dependent representation of complex, energy-dependent optical potentials that fulfill reciprocity exactly. Results. Starting from a separable, energy-independent representation of global optical potentials based on a generalization of the Ernst-Shakin-Thaler (EST) scheme, a further generalization is needed to take into account the energy dependence. Applications to n$+^{48}$Ca, n$+^{208}$Pb, and p$+^{208}$Pb are investigated for energies from 0 to 50~MeV with special emphasis on fulfilling reciprocity. Conclusions. We find that the energy-dependent separable representation of complex, energy-dependent phenomenological optical potentials fulfills reciprocity exactly. In addition, taking into account the explicit energy dependence slightly improves the description of the $S$ matrix elements.

Separable representation of phenomenological optical potentials of Woods-Saxon type

Background: One important ingredient for many applications of nuclear physics to astrophysics, nuclear energy, and stockpile stewardship are cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not feasible, indirect methods, e.g. (d,p) reactions, should be used.} Those (d,p) reactions may be viewed as three-body reactions and described with Faddeev techniques. Purpose: Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. While there exist several separable representations for the nucleon-nucleon interaction, the optical potential between a neutron (proton) and a nucleus is not readily available in separable form. The purpose of this paper is to introduce a separable representation for complex phenomenological optical potentials of Woods-Saxon type. Results: Starting from a global optical potential, a separable representation thereof is introduced based on the Ernst-Shakin-Thaler (EST) scheme. This scheme is generalized to non-hermitian potentials. Applications to n$+^{48}$Ca, n$+^{132}$Sn and n$+^{208}$Pb are investigated for energies from 0 to 50 MeV and the quality of the representation is examined. Conclusions: We find a good description of the on-shell t-matrix for all systems with rank up to 5. The required rank depends inversely on the angular momentum. The resulting separable interaction exhibits a different off-shell behavior compared to the original potential, reducing the high momentum contributions.

Non-Strange Baryon Spectra and Confinement in the Constituent Quark Model

In the frame of the constituent quark model with screened color confinement, non-strange baryon spectra are studied by using non-relativistic and semi-relativistic Faddeev equations in momentum space. The results show that a qualitative description of the high-energy baryon spectra can be obtained, especially for states at excitation energies from 1 to 1.5 GeV. Compared with the linear confinement potential, the discrepancy between non-relativistic and semi-relativistic approaches becomes smaller in the present model with screened color confinement.

Three-body description of direct nuclear reactions: Comparison with the continuum discretized coupled channels method

The continuum discretized coupled channels (CDCC) method is compared with the exact solution of the three-body Faddeev equations in momentum space. We present results for (i) elastic and breakup observables of $d+$$^{12}\mathrm{C}$ at ${E}_{d}=56$ MeV, (ii) elastic scattering of $d+$$^{58}\mathrm{Ni}$ at ${E}_{d}=80$ MeV, and (iii) elastic, breakup, and transfer observables for $^{11}\mathrm{Be}$$+p$ at ${E}_{{}^{11}\mathrm{Be}}/A=38.4$ MeV. Our comparative studies show that in the first two cases, the CDCC method is a good approximation of the full three-body Faddeev solution, but for the $^{11}\mathrm{Be}$ exotic nucleus, depending on the observable or the kinematic regime, it may miss some of the dynamic three-body effects that appear through the explicit coupling to the transfer channel.

Testing nuclear forces by polarization transfer coefficients ind(p→,p→)dandd(p→,d→)preactions atEplab=22.7MeV

The proton to proton polarization transfer coefficients K_x^{x'}, K_y^{y'}, K_z^{x'} and the proton to deuteron polarization transfer coefficients K_x^{x'}, K_y^{y'}, K_z^{x'}, K_x^{y'z'}, K_y^{z'z'}, K_z^{y'z'}, K_y^{x'z'} and K_y^{x'x'-y'y'} have been measured in d(\vec p, \vec p)d and d(\vec p, \vec d)p reactions at E^{lab}_p = 22.7 MeV, respectively. The data have been compared to predictions of modern nuclear forces obtained by solving the three-nucleon Faddeev equations in momentum space. Realistic (semi) phenomenological nucleon-nucleon potentials combined with model three-nucleon forces and modern chiral nuclear forces have been used. The AV18, CD Bonn, Nijm I and II nucleon-nucleon interactions have been applied alone or combined with the Tucson-Melbourne 99 three-nucleon force, adjusted separately for each potential to reproduce the triton binding energy. For the AV18 potential also the Urbana IX three-nucleon force have been used. In addition chiral NN potentials in the next-to-leading-order and chiral two- and three-nucleon forces in the next-to-next-to-leading-order have been applied. Only when three-nucleon forces are included a satisfactory description of all data results. For the chiral approach the restriction to the forces in the next-to-leading order is insufficient. Only when going over to the next-to-next-to-leading order one gets a satisfactory description of the data, similar to the one obtained with the (semi) phenomenological forces.

Three bosons in one dimension with short-range interactions: Zero-range potentials

We consider the three-boson problem with $\delta$-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle a off of a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired S-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the 1+2 scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pair-wise interactions are short-range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.

Three-nucleon bound states using realistic potential models

The bound states of $^3$H and $^3$He have been calculated using the Argonne $v_{18}$ plus the Urbana three-nucleon potential. The isospin $T=3/2$ state have been included in the calculations as well as the $n$-$p$ mass difference. The $^3$H-$^3$He mass difference has been evaluated through the charge dependent terms explicitly included in the two-body potential. The calculations have been performed using two different methods: the solution of the Faddeev equations in momentum space and the expansion on the correlated hyperspherical harmonic basis. The results are in agreement within 0.1% and can be used as benchmark tests. Results for the CD-Bonn interaction are also presented. It is shown that the $^3$H and $^3$He binding energy difference can be predicted model independently.

Benchmark calculations for polarization observables in three-nucleon scattering

High precision benchmark calculations for phase shifts and mixing parameters as well as observables in elastic neutron-deuteron scattering below the deuteron breakup threshold are presented using a realistic nucleon-nucleon potential. Two totally different methods, one using a variational principle in configuration space and the other solving the Faddeev equations in momentum space, are used and compared to each other. The agreement achieved in phase shifts and mixing parameters as well as in the polarization observables is excellent. The extreme sensitivity of the vector analyzing power A{sub y} to small changes of the phase shifts and mixing parameters is pointed out. thinsp {copyright} {ital 1998} {ital The American Physical Society}

Charge form factors and root mean square radii of 3He and 3H with the new Bonn potential.

We calculate the charge form factors and the rms charge radii of /sup 3/H and /sup 3/He using the wave functions generated from the solutions of the Faddeev equations in momentum space with the new Bonn one-boson-exchange momentum space potential. For the charge radii, we obtain, r/sub c/(/sup 3/H) = 1.72 fm and r/sub c/(/sup 3/He) = 1.89 fm, which are in excellent agreement with the experimental data. However, our results for the trinucleon charge form factors in the impulse approximation indicate no improvement with regard to the experimental situation.

Trinucleon asymptotic normalization constants with the new Bonn potential.

The trinucleon asymptotic normalization constants are calculated from solutions to the momentum space Faddeev equations using the new static Bonn two-nucleon potential. We find chemically bondC/sub 0/chemically bond/sup 2/ = 3.39, C/sub 2//C/sub 0/ = 0.043 (/sup 3/H), and D/sub 2/ = -0.22 fm/sup 2/. These values are in agreement with most of the experimental data. Furthermore, our results support the capability of the static Bonn potential to predict low-energy three-nucleon observables reasonably well without the introduction of a three-body force.

Momentum space Faddeev calculation for N-d scattering below the breakup threshold.

Momentum space Faddeev equations for different models that also include the Coulomb force between two protons and a spin triplet nucleon-nucleon tensor force within an angular momentum approximation are solved at ${E}_{\mathrm{lab}=0.2}$ to 2.0 MeV. From a low energy expansion of the N-d S-wave phase shifts, effective range parameters are extracted. The Coulomb contribution to the relation between the trinucleon binding energy and doublet scattering length remains qualitatively unchanged when either the separable nucleon-nucleon tensor potential or the singlet potential is varied. All our doublet p-d S-wave effective range functions show a nonlinear behavior that can be ascribed to a pole connected with a virtual trinucleon bound state, whereas phase shift analysis rejects such a pole. The calculated p-d differential cross sections are roughly compatible with experiment, suggesting that phase shift analysis might not be accurate enough to detect the virtual bound state.

Influence of Coulomb polarization on pd low energy parameters

Modified momentum space Faddeev equations are solved for pd scattering using variable tensor force in the 3S 1- 3D 1 NN state. Utilizing an angular momentum truncation pd phase shifts between E p = 0.2 and 2.0 MeV are calculated and low energy parameters are extracted. Contributions of polarization were subtracted to allow for an application of the Coulomb modified effective range functions. The calculated scattering lengths and the pd Phillips line are compatible with results from a configuration space method.

Nucleon-deuteron low energy parameters

Momentum space Faddeev equations are solved for nucleon-deutron scattering and effective range parameters are calculated. A reverse trend is found in the two spin states by 4a nd < 4a pd and 2a pd < 2a nd which is in agreement with a configuration space calculation, but in conflict with all existing experiments. The Coulomb contributions to the effective range are small in quartet but sizeable in doublet scattering.

Deuteron breakup by α-particles at 13,15 and 18 MeV

Kinematically complete measurements on the D(α, αp)n reaction have been made at three low energies for the incident α-particle and for a number of coplanar angular combinations at each energy. The experimental data, which thus cover a substantial phase space volume, provide material for a severe test of low-energy reaction theories. The data have been analysed in a three-body model, which relies on the numerical solution of the Faddeev equations in momentum space. Rank-one separable two-body potentials for the alpha-nucleon and nucleon-nucleon systems have served as the input information. Agreement between calculated and measured cross sections ranges from excellent to rather poor. Due to the strong effect of the Coulomb correction, which at present is quite uncertain, the appearance of cases with excellent agreement may be fortuitous. The variation of on-shell properties of the two-body potentials produce small changes in the cross sections, implying that more refined models must be employed to achieve good agreement, for instance, the explicit inclusion of effects due to the Pauli principle. However, before firm conclusions in this respect can be drawn, much greater confidence has to be gained in how to treat the Coulomb interaction in few-body breakup reactions.

Reliability of Faddeev calculations in momentum space for the bound three-nucleon system

The accuracy of solving the Faddeev equations in momentum space for the three-nucleon bound state is studied. The source of numerical instabilities in previous calculations is discussed and removed. 3H and 3He results obtained from the Reid soft-core and Paris potentials including all two-body partial waves up to total angular momentum I = 2 are given.