Angular Momentum Projection Research Articles

Three-Body Protonium Formation in a Collision Between a Slow Antiproton ( $${\bar{\rm p}}$$ p ¯ ) and Muonic Hydrogen: $${{\rm H}_{\mu}}$$ H μ —Low Energy $${\bar{\rm p} + ({\rm p} \mu^-)_{1s} \rightarrow (\bar{\rm p} {\rm p})_{1s} + \mu^-}$$ p ¯ + ( p μ - ) 1 s → ( p ¯ p ) 1 s + μ - Reaction

A bound state of a proton, p, and its counterpart antiproton, $${\bar{\rm p}}$$ , is a protonium atom $${Pn = (\bar{\rm p} {\rm p})}$$ . The following three-charge-particle reaction: $${\bar{\rm p} +({\rm p} \mu^-)_{1s} \rightarrow (\bar{\rm p} \rm{p})_{1s} + \mu^-}$$ is considered in this work, where $${\mu^-}$$ is a muon. At low-energies muonic reaction $${Pn}$$ can be formed in the short range state with α = 1s or in the first excited state: α = 2s/2p, where $${\bar{\rm p}}$$ and p are placed close enough to each other and the effect of the $${\bar{\rm p}}$$ –p nuclear interaction becomes significantly stronger. The cross sections and rates of the Pn formation reaction are computed in the framework of a few-body approach based on the two-coupled Faddeev-Hahn-type (FH-type) equations. Unlike the original three-body Faddeev method the FH-type equation approach is formulated in terms of only two but relevant components: $${{\it \Psi}_1}$$ and $${\it \Psi_2}$$ , of the system’s three-body wave function $${\it \Psi}$$ , where $${{\it \Psi}={\it \Psi}_1+{\it \Psi}_2}$$ . In order to solve the FH-type equations $${\it \Psi_1}$$ is expanded in terms of the input channel target eigenfunctions, i.e. in this work in terms of the $${(\rm{p} \mu^-)}$$ eigenfunctions. At the same time $${\it \Psi_2}$$ is expanded in terms of the output channel two-body wave function, that is in terms of the protonium $${(\bar{\rm{p}} \rm{p})}$$ eigenfunctions. A total angular momentum projection procedure is performed, which leads to an infinite set of one-dimensional coupled integral–differential equations for unknown expansion coefficients.

Rotational study of the NH3-CO complex: millimeter-wave measurements and ab initio calculations.

The rotational spectrum of the van der Waals complex NH3-CO has been measured with the intracavity OROTRON jet spectrometer in the frequency range of 112-139 GHz. Newly observed and assigned transitions belong to the K = 0-0, K = 1-1, K = 1-0, and K = 2-1 subbands correlating with the rotationless (jk)NH3 = 00 ground state of free ortho-NH3 and the K = 0-1 and K = 2-1 subbands correlating with the (jk)NH3 = 11 ground state of free para-NH3. The (approximate) quantum number K is the projection of the total angular momentum J on the intermolecular axis. Some of these transitions are continuations to higher J values of transition series observed previously [C. Xia et al., Mol. Phys. 99, 643 (2001)], the other transitions constitute newly detected subbands. The new data were analyzed together with the known millimeter-wave and microwave transitions in order to determine the molecular parameters of the ortho-NH3-CO and para-NH3-CO complexes. Accompanying ab initio calculations of the intermolecular potential energy surface (PES) of NH3-CO has been carried out at the explicitly correlated coupled cluster level of theory with single, double, and perturbative triple excitations and an augmented correlation-consistent triple zeta basis set. The global minimum of the five-dimensional PES corresponds to an approximately T-shaped structure with the N atom closest to the CO subunit and binding energy De = 359.21 cm(-1). The bound rovibrational levels of the NH3-CO complex were calculated for total angular momentum J = 0-6 on this intermolecular potential surface and compared with the experimental results. The calculated dissociation energies D0 are 210.43 and 218.66 cm(-1) for ortho-NH3-CO and para-NH3-CO, respectively.

Ground state in the finite Dicke model for interacting qubits

We study the ground state of a finite size ensemble of interacting qubits driven by a quantum field. We find a maximally entangled W-state in the ensemble part of the system for a certain coupling parameters region. The area of this region decreases as the ensemble size increases and, in the classical limit, becomes the line in parameter space that defines the phase transition of the system. In the classical limit, we also study the dynamics of the system and its transition from order to disorder for initial energies close to the ground state energy. We find that a critical energy providing this transition is related to the minimum of the projection of the total angular momentum of the quantum system in the $z$-direction.

Shape-coexisting rotation in neutron-deficient Hg and Pb nuclei

For a shape-soft nucleus, the deformation change with increasing angular momentum of rotation can be significant. Total-Routhian-surface (TRS) calculations include the shape changes, but angular momentum is not conserved (neither is it a good quantum number, nor is it kept unchanged in the whole TRS mesh). In the projected shell model (PSM), the angular momentum appears as a good quantum number, but calculations have usually been performed with fixed deformation. In the present work, by performing angular-momentum projection on the mean-field potential-energy surface (PES), we can obtain an angular-momentum-conserved PES which gives deformation for a rotational state at a given spin. In order to investigate the shape-changing effect, we have chosen neutron-deficient Hg and Pb isotopes in which shape coexistence occurs. We interpret the irregular rotational behavior of the oblate bands at low spin as arising from deformation changes which are induced by collective rotation. At higher spin, the oblate rotational spectrum can also be influenced by the crossing between the $K=0$ ground-state band and a low-$K$ two-quasineutron band. Calculated $g$ factors for the states of oblate bands are given for future experimental testing, and the intrinsic structures of high-$K$ oblate states are investigated.

Spin-asymmetric laser-driven relativistic tunneling frompstates

The tunneling ionization of an electron from a p-state in a highly charged ion in the relativistic regime is investigated in a linearly polarized strong laser field. In contrast to the case of an s-state, the tunneling ionization from the p-state is spin asymmetric. We have singled out two reasons for the spin asymmetry: first, the difference of the electron energy Zeeman splitting in the bound state and during tunneling, and second, the relativistic momentum shift along the laser propagation direction during the under-the barrier motion. Due to the latter, those states are predominantly ionized where the electron rotation is opposite to the electron relativistic shift during the under-the-barrier motion. We have investigated the dependence of the ionization rate on the laser intensity for different projections of the total angular momentum and identified the intensity parameter which governs this behaviour. The significant change of the ionization rate is originated from the different precession dynamics of the total angular momentum in the bound state at high and low intensities.

16O + 16O + valence neutrons in molecular orbitals structures of positive- and negative-parity superdeformed bands in 34S

The structures of superdeformed (SD) states in 34S are investigated using the antisymmetrized molecular dynamics and generator coordinate method (GCM). The GCM basis wave functions are calculated via energy variation with a constraint on the quadrupole deformation parameter β. By applying the GCM after parity and angular momentum projections, the coexistence of two positive- and one negative-parity SD bands are predicted, and low-lying states and other deformed bands are obtained. The SD bands have structures of 16O + 16O + two valence neutrons in molecular orbitals around the two 16O cores in a cluster picture. The configurations of the two valence neutrons are δ2 and π2 for the positive-parity SD bands and π1δ1 for the negative-parity SD band.

Microscopic particle-rotor model for the low-lying spectrum ofΛhypernuclei

We propose a novel method for low-lying states of hypernuclei based on the particle-rotor model, in which hypernuclear states are constructed by coupling the hyperon to low-lying states of the core nucleus. In contrast to the conventional particle-rotor model, we employ a microscopic approach for the core states, that is, the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this microscopic particle-rotor model to $^9_\Lambda $Be employing a point-coupling version of the relativistic mean-field Lagrangian. A reasonable agreement with the experimental data for the low-spin spectrum is achieved using the $\Lambda N$ coupling strengths determined to reproduce the binding energy of the $\Lambda$ particle.

Tetrahedral symmetry in Zr nuclei: calculations of low-energy excitations with Gogny interaction

We report on the results of the calculations of the low energy excitation patterns for three Zirconium isotopes, viz 80Zr40, 96Zr56 and 110Zr70, reported by other authors to be doubly-magic tetrahedral nuclei (with tetrahedral magic numbers Z = 40 and N = 40, 56 and 70). We employ the realistic Gogny effective interactions using three variants of their parametrization and the particle-number, parity and the angular-momentum projection techniques. We confirm quantitatively that the resulting spectra directly follow the pattern expected from the group theory considerations for the tetrahedral symmetric quantum objects. We also find out that, for all the nuclei studied, the correlation energy obtained after the angular momentum projection is very large for the tetrahedral deformation as well as other octupole deformations. The lowering of the energies of the resulting configurations is considerable, i.e. by about 10 MeV or even more, once again confirming the significance of the angular-momentum projections techniques in the mean-field nuclear structure calculations.

Beyond-mean-field calculations of collectivities of neutron-rich Fe and Cr isotopes

The rotational symmetry is broken in mean-field models. The symmetry breaking may lead to significant effects even on the ground states of nuclei. The recent observations of enhanced collectivities in neutron-rich Fe and Cr isotopes are challenging the current calculations of mean-field models. By performing angular-momentum projection on mean-field potential energy surface (PES), we can obtain an angular-momentum-conserved PES at each given spin. With the projected PES, we have investigated the collectivity of neutron-rich Fe and Cr isotopes. The calculations reproduce well the excitation energies and electric quadrupole transitions of the collective excited states of the isotopes, indicating the importance of the restoration of the rotational symmetry.

O16+O16molecular structures of positive- and negative-parity superdeformed bands inS34

The structures of excited states in $^{34}\mathrm{S}$ are investigated using the antisymmetrized molecular dynamics and generator coordinate method (GCM). The GCM basis wave functions are calculated via energy variation with a constraint on the quadrupole deformation parameter $\ensuremath{\beta}$. By applying the GCM after parity and angular momentum projections, the coexistence of two positive- and one negative-parity superdeformed (SD) bands are predicted, and low-lying states and other deformed bands are obtained. The SD bands have structures of $^{16}\mathrm{O}+^{16}\mathrm{O}$ + two valence neutrons in molecular orbitals around the two $^{16}\mathrm{O}$ cores in a cluster picture. The configurations of the two valence neutrons are ${\ensuremath{\delta}}^{2}$ and ${\ensuremath{\pi}}^{2}$ for the positive-parity SD bands and ${\ensuremath{\pi}}^{1}{\ensuremath{\delta}}^{1}$ for the negative-parity SD band. The structural changes of the yrast states are also discussed.

Simple regularization scheme for multireference density functional theories

Background: Extensions of single-reference (SR) energy-density-functionals (EDFs) to multi-reference (MR) applications involve using the generalized Wick theorem (GWT), which leads to singular energy kernels that cannot be properly integrated to restore symmetries, unless the EDFs are generated by true interactions. Purpose: We propose a new method to regularize the MR EDFs, which is based on using auxiliary quantities obtained by multiplying the kernels with appropriate powers of overlaps. Methods: Regularized matrix elements of two-body interactions are obtained by integrating the auxiliary quantities and then solving simple linear equations. Results: We implement the new regularization method within the self-consistent Skyrme-Hartree-Fock approach and we perform a proof-of-principle angular-momentum projection (AMP) of states in odd-odd nucleus 26Al. We show that for EDFs generated by true interactions, our regularization method gives results identical to those obtained within the standard AMP procedure. We also show that for EDFs that do not correspond to true interactions, it gives stable and converging results that are different than unstable and non-converging standard AMP values. Conclusions: The new regularization method proposed in this work may provide us with a relatively inexpensive and efficient tool to generalize SR EDFs to MR applications, thus allowing for symmetry restoration and configuration mixing performed for typical nuclear EDFs, which most often do not correspond to true interactions.

Effects of reagent rotational excitation on the H + CHD₃ → H₂ + CD₃ reaction: a seven dimensional time-dependent wave packet study.

Seven-dimensional time-dependent wave packet calculations have been carried out for the title reaction to obtain reaction probabilities and cross sections for CHD3 in J0 = 1, 2 rotationally excited initial states with k0 = 0 - J0 (the projection of CHD3 rotational angular momentum on its C3 axis). Under the centrifugal sudden (CS) approximation, the initial states with the projection of the total angular momentum on the body fixed axis (K0) equal to k0 are found to be much more reactive, indicating strong dependence of reactivity on the orientation of the reagent CHD3 with respect to the relative velocity between the reagents H and CHD3. However, at the coupled-channel (CC) level this dependence becomes much weak although in general the K0 specified cross sections for the K0 = k0 initial states remain primary to the overall cross sections, implying the Coriolis coupling is important to the dynamics of the reaction. The calculated CS and CC integral cross sections obtained after K0 averaging for the J0 = 1, 2 initial states with all different k0 are essentially identical to the corresponding CS and CC results for the J0 = 0 initial state, meaning that the initial rotational excitation of CHD3 up to J0 = 2, regardless of its initial k0, does not have any effect on the total cross sections for the title reaction, and the errors introduced by the CS approximation on integral cross sections for the rotationally excited J0 = 1, 2 initial states are the same as those for the J0 = 0 initial state.

Role of asymmetry parameter and dissipation coefficient ofKcoordinate in angular distribution of fission fragments

Using multidimensional Langevin equations, we have calculated the angular distribution of fission fragments with nonconstant values for the dissipation coefficient of the $K$ coordinate (the projection of the total angular momentum $I$ onto the symmetry axis of the fissioning nuclear system) for symmetry and asymmetry fission. We investigated the relaxation time of the $K$ coordinate, ${\ensuremath{\tau}}_{K}$, as a function of scission criteria and the asymmetry parameter. Calculations are done for ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}\ensuremath{\rightarrow}{}^{248}\mathrm{Cf}$, ${}^{16}\mathrm{O}+{}^{238}\mathrm{U}\ensuremath{\rightarrow}{}^{254}\mathrm{Fm}$, and ${}^{16}\mathrm{O}+{}^{248}\mathrm{Cm}\ensuremath{\rightarrow}{}^{264}\mathrm{Rf}$ reactions and obtained results are compared with available experimental data. Our calculations with nonconstant values of the $K$ coordinate show that the agreement between theoretical calculations and experimental data for asymmetry fission is better than that for symmetry fission.

Construction of basis vectors for symmetric irreducible representations of O(5) $ \supset$ O(3)

A recursive method for the construction of symmetric irreducible representations of $ O(2l+1)$ in the $ O(2l+1)\supset O(3)$ basis for identical boson systems is proposed. The formalism is realized based on the group chain $ U(2l+1)\supset U(2l-1)\otimes U(2)$ , of which the symmetric irreducible representations are simply reducible. The basis vectors for symmetric irreducible representations of the $ O(2l+1)\supset O(2l-1)\otimes U(1)$ can easily be constructed from those of U(2l + 1) $ \supset$ U(2l - 1) ⊗ U(2) $ \supset$ O(2l - 1) ⊗ U(1) with no l -boson pairs, namely with the total boson number exactly equal to the seniority number in the system, from which one can construct symmetric irreducible representations of $ O(2l+1)$ in the $ O(2l-1)\otimes U(1)$ basis when all symmetric irreducible representations of O(2l - 1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the $ O_{1}(3)\otimes U(1)$ basis, where $ O_{1}(3)\equiv O(2l-1)$ , when l = 2 , which is generated not by the angular momentum operators of the d -boson system, but by the operators constructed from d -boson creation (annihilation) operators $ d^{\dagger}_{\mu}$ ( $ d_{\mu}$ with $ \mu=1$ , 0, -1 . Matrix representations of $ O(5)\supset O_{1}(3)\otimes U(1)$ , together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the $ O(5)\supset O(3)$ basis vectors, where the O(3) group is generated by the angular momentum operators of the d -boson system, in terms of those of $ O_{1}(3)\otimes U(1)$ is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of $ O(5)\supset O(3)$ . Formulae for evaluating the elementary Wigner coefficients of $ O(5)\supset O(3)$ are derived explicitly. Analytical expressions of some elementary Wigner coefficients of $ O(5)\supset O(3)$ for the coupling $ (\tau 0)\otimes(1 0)$ with resultant angular momentum quantum number L = 2 $ \tau$ + 2 - k for $ k=0,2,3,\ldots,6$ with a multiplicity 2 case for k = 6 are presented.

Toward extremes of angular momentum: Application of the Pfaffian algorithm in realistic calculations

In a calculation of rotated matrix elements with angular momentum projection, the generalized Wick's theorem may encounter a practical problem of combinatorial complexity when the configurations have more than four quasi-particles (qps). The problem can be solved by employing the Pfaffian algorithm generally applicable to calculations of matrix elements for Hartree-Fock-Bogoliubov states with any number of qps. This breakthrough in many-body techniques enables studies of high-spin states in a shell-model framework. As the first application of the Pfaffian algorithm, the configuration space of the Projected Shell Model is expanded to include 6-qp states for both positive and negative parities. Taking $^{166}$Hf as an example, we show that 6-qp states become the main configuration of the yrast band beyond spin $I \approx 34\hbar$, which explains the observed third back-bending in moment of inertia. Structures of multi-qp high-$K$ isomers in $^{176}$Hf are analyzed as another example.

Deformation and clustering in excited states of 42Ca

The coexistence of various low-lying deformed states in |$^{42}$|Ca and the |$\alpha $|–|$^{38}$|Ar correlations in those deformed states have been investigated using deformed-basis antisymmetrized molecular dynamics. Wave functions of the low-lying states are obtained via parity and angular momentum projections and the generator coordinate method (GCM). Basis wave functions of the GCM calculation are obtained via energy variations with constraints on the quadrupole deformation parameter |$\beta $| and the distance between |$\alpha $| and the |$^{38}$|Ar clusters. The rotational bands built on the |$J^\pi = 0_2^+$| (1.84 MeV) state as well as the |$J^\pi = 0_3^+$| (3.30 MeV) state are both reproduced. The coexistence of two additional |$K^\pi = 0^+$| rotational bands is predicted; one band is shown to be built on the |$J^\pi = 0_3^+$| state. Members of the ground-state band and the rotational band built on the |$J^\pi = 0_3^+$| state contain |$\alpha $|–|$^{38}$|Ar cluster structure components.

Numerical algorithm for the standard pairing problem based on the Heine–Stieltjes correspondence and the polynomial approach

We present a detailed study of the computational complexity of a numerical algorithm based on the Heine–Stieltjes correspondence following the new approach we proposed recently for solving the Bethe ansatz (Gaudin–Richardson) equations of the standard pairing problem. For k pairs of valence nucleons in n non-degenerate single-particle energy levels, the approach utilizes that solutions of the Bethe ansatz equations can be obtained from two matrices of dimensions (k+1)×(k+1) and (n−1)×(k+1), which are associated with the extended Heine–Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem. The method reduces to solving a system of k polynomial equations, which can be efficiently implemented by the fast Newton–Raphson algorithm with a Monte Carlo sampling procedure for the initial guesses. By extension, the present algorithm can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multi-particle systems. Program summaryProgram title: exactPairingHSCatalogue identifier: AETD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5411No. of bytes in distributed program, including test data, etc.: 39758Distribution format: tar.gzProgramming language: Mathematica.Computer: Laptop, workstation.Operating system: Tested with MATHEMATICA version 9.0 on Mac OS X and Windows 7.RAM: Less than 10 MBClassification: 17.15.Nature of problem:The program calculates exact pairing energies based on the Heine–Stieltjes polynomial approach. Existing conventional exact-pairing approaches require solving systems of highly nonlinear equations, which are difficult and often impossible to solve beyond the simplest of the quantum-mechanical many-particle systems. In this study, the Heine–Stieltjes polynomial approach is employed to provide solutions for more than one or two pairs of particles residing in many energy levels.Solution method:The new Heine–Stieltjes polynomial approach transforms the pairing problem to one that involves the handling of only two matrix equations. This, combined with an efficient numerical algorithm implemented by the fast Newton–Raphson method with a Monte Carlo sampling procedure for the initial guesses, makes exact pairing solutions feasible even when more energy levels or heavy nuclei (many pairs) are considered.Running time:Less than a hundred seconds using a 2.80 GHz processor. The notebook takes approximately 23 min to complete.

Low-velocity transient-field technique with radioactive ion beams:gfactor of the first excited 2+state in72Zn

The $g$ factor of the first excited ${2}^{+}$ state in ${}^{72}\mathrm{Zn}$ has been measured using the transient-field (TF) technique in combination with Coulomb excitation in inverse kinematics. This experiment presents only the third successful application of the TF method to a short-lived radioactive beam in 10 y, highlighting the intricacies of applying this technique to present and future isotope separator on-line facilities. The significance of the experimental result, $g({2}_{1}^{+})=+0.47(14)$, for establishing the structure of the Zn isotopes near $N=40$ is discussed on the basis of shell-model and beyond-mean-field calculations, the latter accounting for the triaxial degree of freedom, configuration mixing, and particle number and angular momentum projections.

Effect of spin polarization of excitons on the energy spectra of GaAs/AlGaAs heterostructures

The evolution of the energy position of the maximum of the free exciton photoluminescence line in high-quality GaAs/AlGaAs heterostructures under optical excitation by short laser pulses of high density 5 · 1014–2 · 1018 cm−3 is considered. The influence of screening of the Coulomb interaction and interexciton exchange interaction are discussed. These effects give corrections to the exciton peak energy position of opposite signs. The second effect is sensitive to the exciton spin orientation and is manifested as splitting of the energy peaks of excitons with angular momentum projections +1 and −1. The value of splitting is proportional to the density of excitons and to the degree of their spin polarization and reaches 1.5 meV.

Relativistic effects in time delay of atomic photoionization

We study relativistic effects in the time delay of atomic photoionization using an example of the Xe atom subjected to a circularly polarized electromagnetic pulse. We find that while relativistic effects play only a fairly small role for both the ionization probabilities and the time delay of the corotating electron with the same sign of the angular momentum projection as the circularly polarized light, these effects produce substantial change in the time delay for a counter-rotating electron with the opposite sign of the angular momentum projection.