Deformed shell model study of heavy N=Z nuclei and dark matter detection
R. Sahu, V. K. B. Kota
备注：
12 pages, 3 figures and one table; Article for the invited talk (by R Sahu) given in the 35th International Workshop on "Nuclear Theory" held at Rila Mountains (Sofia, Bulgaria) during June 26 to July 2nd, 2016
Deformed shell model (DSM) based on Hartree-Fock intrinsic states is applied to address two current problems of interest. Firstly, in the $f_{5/2}pg_{9/2}$ model space with jj44b effective interaction along with isospin projection, DSM is used to describe the structure of the recently observed low-lying $T=0$ and $T=1$ bands in the heavy odd-odd N=Z nucleus $^{66}$As. DSM results are close to the data and also to the shell model results. For the $T=1$ band, DSM predicts structural change at $8^+$ just as in the shell model. In addition, the lowest two $T=0$ bands are found to have quasi-deuteron structure above a $^{64}$Ge core and the $5^+$ and $9^+$ levels of the third $T=0$ band are found to be isomeric states. Secondly, in a first application of DSM to dark matter, detection rates for the lightest supersymmetric particle (a dark matter candidate) are calculated with $^{73}$Ge as the detector.
The structure of Ne, Na, Mg, Al, Si and S nuclei near the neutron drip-line region is investigated in the frame-work of relativistic meannfield (RMF) and non-relativistic Skyrme Hartree-Fock formalisms. The drip-line of these nuclei are pointed out. We analysed the large deformation structures and many of these neutron rich nuclei are quite deformed. New magic number are seen for these deformed nuclei.
Shell structure of the single-particle spectrum for reflection-asymmetric deformed cavity is investigated. Remarkable shell structure emerges for certain combinations of quadrupole and octupole deformations. Semiclassical periodic-orbit analysis indicates that bifurcation of equatorial orbits plays an important role in the formation of this new shell structure.
An analysis of the construction of a q-deformed version of the 3-dimensional harmonic oscillator, which is based on the application of q-deformed algebras, is presented. The results together with their applicability to the shell model are compared with the predictions of the modified harmonic oscillator.
Covariant density functional theory is used to analyze the evolution of low-lying M1 strength in superfluid deformed nuclei in the framework of the self-consistent Relativistic Quasiparticle Random Phase Approximation (RQRPA). In nuclei with a pronounced neutron excess two scissor modes are found. Besides the conventional scissor mode, where the deformed proton and neutron distributions oscillate against each other, a new soft M1 mode is found, where the deformed neutron skin oscillates in a scissor like motion against a deformed proton-neutron core.
Using deformed shell model based on Hartree-Fock intrinsic states with $^{70}$Zn as an example and employing two realistic effective interactions, namely jj44b that produces deformed shape and JUN45 that generates spherical shape, it is demonstrated that the $0ν$ and $2ν$ nuclear transition matrix elements for double beta decay will reduce considerably as we go from spherical shapes to deformed shapes. This result is further substantiated by using $^{150}$Nd as another example.
An extension of the previously proposed Variable Moment of Inertia with Softness model VMIS has been applied to describe successfully the effect of backbending in some deformed e-e nuclei. The model shows good results in the rare earth and actinide regions.
A highly specialized two-center shell model has been developed accounting for the splitting of a deformed parent nucleus into two ellipsoidaly deformed fragments. The potential is based on deformed oscillator wells in direct correspondance with the shape change of the nuclear system. For the first time a potential responsible for the necking part between the fragments is introduced on potential theory basis. As a direct consequence, spin-orbit {\bf ls} and {\bf l$^2$} operators are calculated as shape dependent. Level scheme evolution along the fission path for pairs of ellipsoidaly deformed f…
The 62Sm, 64Gd, 64Dy, 70Y b, 72Hf and 74W nuclei are classified as deformed nuclei. Low-lying bands are one of the most fundamental excitation modes in the energy spectra of deformed nuclei. In this paper a theoretical analysis of the experimental data within the phenomenological model is presented. The energy spectra of ground states are calculated. It is found the low-lying spectra of ground band states are in good agreement with the experimental data.
A study of the shape transition from spherical to axially deformed nuclei in the even Ce isotopes using the nucleon-pair approximation of the shell model is reported. As long as the structure of the dominant collective pairs is determined using a microscopic framework appropriate to deformed nuclei, the model is able to produce a shape transition. However, the resulting transition is too rapid, with nuclei that should be transitional being fairly well deformed, perhaps reflecting the need to maintain several pairs with each angular momentum.
We calculate the binding energy and deformation parameter for light nuclei with and without pairing using a deformed relativistic mean field model. The role of BCS-type pairing effect is analyzed for Ne and F isotopes. The calculated odd-even staggering and the deformation parameters argue strongly against the role of pairing in the light nuclei.
We discuss the notion of partial dynamical symmetry in relation to nuclear spectroscopy. Explicit forms of Hamiltonians with partial $SU(3)$ symmetry are presented in the framework of the interacting boson model of nuclei. An analysis of the resulting spectrum and electromagnetic transitions demonstrates the relevance of such partial symmetry to the spectroscopy of axially deformed nuclei.
We present analytic expressions of the Coulomb interaction between a spherical and a deformed nuclei which are valid for all separation distance. We demonstrate their significant deviations from commonly used formulae in the region inside the Coulomb radius, and show that they remove various shortcomings of the conventional formulae.
An article by Covello, Gargano and Itaco [Phys.Rev. C 56, 3092 (1997)] tries to find evidence for the important role of the residual tensor force between the valence proton and neutron in doubly odd deformed nuclei. It is shown that observable effects discussed by these authors do not fully justify their rather strong conclusions.
Weakly-bound deformed nuclei have been studied by the Skyrme Hartree-Fock-Bogoliubov (HFB) approach in large coordinate-space boxes. In particular, the box-size dependence of the HFB calculations of weakly-bound deformed nuclei are investigated, including the particle density and pairing density distributions at nuclear surfaces, the near-threshold resonant and continuum quasiparticle spectra, and energetic properties. The box size may have larger influences in pairing properties than in other bulk properties. We demonstrate that large-box calculations of weakly-bound nuclei are important to p…
Octupole deformed shapes in neutron-rich actinides and superheavy nuclei as well as extremely deformed shapes of the N~Z light nuclei have been investigated within the framework of covariant density functional theory. We confirmed the presence of new region of octupole deformation in neutron-rich actinides with the center around Z~96, N~196 but our calculations do not predict octupole deformation in the ground states of superheavy Z~108 nuclei. As exemplified by the study of 36Ar, the nodal structure of the wavefunction of occupied single-particle orbitals in extremely deformed structures allo…
Magnetic form factors corresponding to elastic electron scattering from odd-A nuclei are presented. The calculations are carried out in plane-wave Born approximation. The one-body properties are obtained in a deformed self-consistent mean-field calculation based on a Skyrme HF+BCS method. Collective effects are also included in the cranking approximation. Results on several stable nuclei are compared with the available experimental information. It is shown that a deformed formalism improves the agreement with experiment in deformed nuclei, while reproducing equally well spherical nuclei by tak…
The consequences of the spontaneous breaking of rotational symmetry are investigated in a field theory model for deformed nuclei, based on simple separable interactions. The crucial role of the Ward-Takahashi identities to describe the rotational states is emphasized. We show explicitly how the rotor picture emerges from the isoscalar Goldstone modes, and how the two-rotor model emerges from the isovector scissors modes. As an application of the formalism, we discuss the M1 sum rules in deformed nuclei, and make connection to empirical information.
Potential energies, moments of inertia, quadrupole and octupole moments of dinuclear systems are compared with the corresponding quantities of strongly deformed nuclei. As dinuclear system we denote two touching nuclei (clusters). It is found that the hyperdeformed states of nuclei are close to those of nearly symmetric dinuclear systems, whereas the superdeformed states are considered as states of asymmetric dinuclear systems. The superdeformed and hyperdeformed states constructed from two touching clusters have large octupole deformations. The experimental measurement of octupole deformation…
We present a study of deformed nuclei in the framework of the sdg interacting boson model utilizing both numerical diagonalization and analytical $1/N$ expansion techniques. The focus is on description of high-spin states which have recently become computationally accessible through the use of computer algebra in the $1/N$ expansion formalism. A systematic study is made of high-spin states in rare-earth and actinide nuclei.
The single-particle spectrum of deformed shell-model states in nuclei, is shown to exhibit a supersymmetric pattern. The latter involves deformed pseudospin doublets and intruder levels. The underlying supersymmetry is associated with the relativistic pseudospin symmetry of the nuclear mean-field Dirac Hamiltonian with scalar and vector potentials.
The thermal evolution of the shell correction energy is investigated for deformed nuclei using Strutinsky prescription in a self-consistent relativistic mean-field framework. For temperature independent single-particle states corresponding to either spherical or deformed nuclear shapes, the shell correction energy $Δ_{sc}$ steadily washes out with temperature. However, for states pertaining to the self-consistent thermally evolving shapes of deformed nuclei, the dual role played by the single-particle occupancies in diluting the fluctuation effects from the single-particle spectra and in drivi…
Non-trivial geometrical effects in relativistic central collisions of deformed nuclei are studied using a simple version of optical Glauber model. For very small impact parameters large centrality and eccentricity fluctuations are observed. In very high multiplicity collisions of oblate nuclei (e.g. Au and Cu) significant fraction of events with elliptic flow strength v_2 dependent on oblateness beta_2 is predicted.
Shell structure of low-lying neutron resonant levels in axially-symmetric quadrupole-deformed potentials is discussed, which seems analogous to that of weakly-bound neutrons. As numerical examples, nuclei slightly outside the neutron-drip-line, $^{39}_{12}$Mg$_{27}$ and $^{21}_{6}$C$_{15}$, are studied. For the lowest resonance I obtain $I^π$ = $Ω^π$ = 5/2$^{-}$ for $^{39}$Mg which is likely to be prolately deformed, while $I^π$ = $Ω^π$ = 1/2$^{+}$ may be assigned to the nucleus $^{21}$C which may be oblately deformed. Consequently, $^{21}$C will not be observed as a recognizable resonant stat…
Background: Collective excitations of nuclei and their theoretical descriptions provide an insight into the structure of nuclei. Replacing traditional phenomenological interactions with unitarily transformed realistic nucleon-nucleon interactions increases the predictive power of the theoretical calculations for exotic or deformed nuclei. Purpose: Extend the application of realistic interactions to deformed nuclei and compare the performance of different interactions, including phenomenological interactions, for collective excitations in the sd-shell. Method: Ground-state energies and charge r…
From the semi-empirical formalisms of Bohr-Mottelson, a new model, based on the effect of beta- and gamma- head energies and the variable moment of inertia, was developed to calculate the ground state rotational band of almost all deformed e-e nuclei. The model can be tuned for nuclei with experimentally available beta- and gamma-vibrational energies as well as for those with unmeasured beta- and gamma-vibrations.
Reaction cross-sections are calculated using the Coulomb modified Glauber model for deformed target nuclei. The deformed nuclear matter density of the target is expanded into multipoles of order k = 0,2,4.The reaction cross-sections between some spherical and deformed nuclei are studied at energy range (10-1000 MeV/nucleon). The most significant effects in the intermediate energy range are the Coulomb field and in-medium effect that modified the trajectory of the incident beams. Introducing the deformation effect beside the Coulomb field and in-medium effect improves the agreement with the exp…
We extend the shell model Monte Carlo approach to heavy deformed nuclei using a new proton-neutron formalism. The low excitation energies of such nuclei necessitate calculations at low temperatures for which a stabilization method is implemented in the canonical ensemble. We apply the method to study a well deformed rare-earth nucleus, 162Dy. The single-particle model space includes the 50-82 shell plus 1f_{7/2} orbital for protons and the 82-126 shell plus 0h_{11/2}, 1g_{9/2} orbitals for neutrons. We show that the spherical shell model reproduces well the rotational character of 162Dy within…
The quon algebra, which interpolates between the Bose and Fermi algebras and depends on a free paramenter $q$, is used to generate a deformed Dyson boson expansion of the quadrupole operator. Then we obtain a quadrupole-quadrupole hamiltonian, for a single j-shell, in terms of this deformed bosonic operator. The hamiltonian is diagonalized and its eigenvalues are compared with the ones obtained from the fermionic quadrupole-quadrupole hamiltonian. The deformation parameter helps in achieving the correct energy levels, what cannot be encountered in practice with the usual non-deformed Dyson exp…
We investigate the deformation properties of atomic nuclei in a hadronic chiral SU_f(3) model approach. The parameters are fitted to hadron mass properties and adjustments for spherical finite nuclei have been performed. Using these parameters the deformation of a series of light and heavy nuclei are obtained in a two-dimensional self-consistent calculation. In addition we discuss a case of superdeformation in a heavy nucleus.
A general new technique to solve the two-center problem with arbitrarily-orientated deformed realistic potentials is demonstrated, which is based on the powerful potential separable expansion method. As an example, molecular single-particle spectra for $^{12}$C + $^{12}$C $\to$ $^{24}$Mg are calculated using deformed Woods-Saxon potentials. These clearly show that non-axial symmetric configurations play a crucial role in molecular resonances observed in reaction processes for this system at low energy.
This work presents an older, but still interesting description of a superfluid condensate with two components (protons and neutrons), confined by time-dependent deformed harmonic oscillator potentials either in collective uniform rotation, or opposite angular oscillation.
Relationship between quantum shell structure and classical periodic orbits is briefly reviewed on the basis of semi-classical trace formula. Using the spheroidal cavity model, it is shown that three-dimensional periodic orbits, which are born out of bifurcation of planar orbits at large prolate deformations, generate the superdeformed shell structure.
Shell corrections of the finite deformed Woods-Saxon potential are calculated using the Green's function method and the generalized Strutinsky smoothing procedure. They are compared with the results of the standard prescription which are affected by the spurious contribution from the unphysical particle gas. In the new method, the shell correction approaches the exact limit provided that the dimension of the single-particle (harmonic oscillator) basis is sufficiently large. For spherical potentials, the present method is faster than the exact one in which the contribution from the particle con…
We formulate a new scheme of the Hartree-Fock-Bogoliubov mean-field theory applicable to weakly bound and pair correlated deformed nuclei using the coordinate-space Green's function technique. On the basis of a coupled-channel representation of the quasiparticle wave function expanded in terms of the partial waves, we impose the correct boundary condition of the asymptotically out-going waves on the continuum quasiparticle states. We perform numerical analysis for $^{38}$Mg to illustrate properties of the continuum quasiparticle states and the pair correlation in deformed nuclei near the neutr…
We consider deformed nuclei within an effective theory that exploits the small ratio between rotational and vibrational excitations. For even-even nuclei, the effective theory predicts small changes in the rotational constants of bands built on multi-phonon excitations that are linear in the number of excited phonons. In 166Er and 168Er, this explains the main variations of the rotational constants of the two-phonon gamma vibrational bands. In 232Th, the effective theory correctly explains the trend that the rotational constants decrease with increasing spin of the band head. We also study the…
We discuss characteristic features of SU(3) partial dynamical symmetry in relation to nuclear spectroscopy and compare with previous broken-SU(3) calculations for ^{168}Er.
The effect of deformation on the excitation of multiple giant dipole resonances is studied. Analytical expressions are derived in the framework of the interacting boson model for the energies and E1 properties of giant dipole resonances in spherical and deformed nuclei, and a numerical treatment of transitional nuclei is proposed. Coulomb-excitation cross sections are calculated in $^{238}$U and in the samarium isotopes.
In the frame work of the hydrodynamical model, a new model of the ground state rotational bands of deformed e-e nuclei is developed by introducing the variable moment of inertia, and the effect of beta- and gamma- vibrational bands. The model is applied to calculate the energies of the ground state band of 158Dy. The results of our calculations are in close agreement with data compared with other existing models.
The properties of kaonic nuclei are studied using a two-dimensional Skyrme-Hartree-Fock model with a $KN$ Skyrme force. We focus in particular on the instability of the solutions for a too strong $KN$ interaction, which determines a maximum value of the kaon binding in this approach. We then analyze the change of the deformation properties of several core-deformed nuclei caused by the added kaon, and find a shrinking of the core and in some cases a complete loss of deformation.
We explore the rotational feature of deformed halos by performing the angular momentum projection (AMP) on the ground state wave functions obtained from the deformed relativistic Hartree-Bogoliubov theory (DRHBc) in continuum. The DRHBc+AMP approach self-consistently describes the coupling between single particle bound states and the continuum not only in the ground state but also in rotational states. The rotational modes of deformed halos in $^{42}$Mg and $^{44}$Mg are investigated by studying properties of rotational states such as excitation energy, configuration, and density distribution.…
Pseudospin symmetry is a relativistic symmetry of the Dirac Hamiltonian with scalar and vector mean fields equal and opposite in sign. This symmetry imposes constraints on the Dirac eigenfunctions. We examine extensively the Dirac eigenfunctions of realistic relativistic mean field calculations of deformed nuclei to determine if these eigenfunctions satisfy these pseudospin symmetry constraints.
Initial eccentricity of the interacting volume of QCD matter created in collisions of prolate and oblate nuclei is studied using optical Glauber simulation. Collisions of prolate 165-Ho + 165-Ho nuclei and Au+Au nuclei are investigated (assuming oblate ground-state deformation for 197-Au nucleus). Qualitative predictions are obtained for RHIC energies.
A formula to evaluate the effects of a general deformation on the Coulomb direct contribution to the energy of the Isobaric Analog State (IAS) is presented and studied via a simple yet physical model. The toy model gives a reasonable account of microscopic deformed Hartree-Fock-Bogolyubov (HFB) calculations in a test case, and provides a guidance when predicting unknown IAS energies. Thus, deformed HFB calculations, to predict the IAS energies, are performed for several neutron-deficient medium-mass and heavy nuclei which are now planned to be studied experimentally.
The relationship between the deformed harmonic oscillator and the formation of molecular cluster structures, whereby valence neutrons are exchanged between cluster cores, is examined. It is found that there is a strong connection between the properties of the valence orbitals associated with deformed structures in the deformed harmonic oscillator and the molecular orbitals created by linear combinations of single-centre orbitals around nuclear clusters. The conclusion is that in addition to the appearance of clustering in the deformed harmonic oscillator that \emph{every} prolate deformed clus…
Examples of the change of neutron shell-structure in both weakly-bound and resonant neutron one-particle levels in nuclei towards the neutron drip line are exhibited. It is shown that the shell-structure change due to the weak binding may lead to the deformation of those nuclei with the neutron numbers $N \approx$ 8, 20, 28 and 40, which are known to be magic numbers in stable nuclei. Nuclei in the "island of inversion" are most easily and in a simple manner understood in terms of deformation. As an example of spectroscopic properties other than single-particle energies, magnetic moments of so…
A collective vector-boson model with broken SU(3) symmetry is applied to several deformed even-even nuclei. The model description of ground and $γ$ bands together with the corresponding B(E2) transition probabilities is investigated within a broad range of SU(3) irreducible representations (irreps) $(λ,μ)$. The SU(3)-symmetry characteristics of rotational nuclei are analyzed in terms of the bandmixing interactions.
The giant dipole resonance (GDR) in deformed nuclei is analyzed using the self-consistent separable random-phase-approximation (SRPA) with Skyrme forces SkT6, SkM$^*$, SLy6 and SkI3. The deformed nuclei $^{150}$Nd and $^{238}$U are used as representative rare-earth and actinide samples. Dependencies of the dipole strength distributions on some basic characteristics of the Skyrme functional and nuclear matter properties (isoscalar and isovector effective masses, time-odd contributions) are discussed. Particular attention is paid to the fragmentation structure of the GDR strength which are shown…
Using an angular momentum projected single particle basis, a pnQRPA approach is used to study the $2νββ$ properties of ten isotopes, exhibiting various quadrupole deformations. The mother and daughter nuclei exhibit different quadrupole deformations. Since the projected basis enables a unified description of deformed and spherical nuclei, situations where the nuclei involved in the double beta decay process are both spherical, both deformed or one spherical and another deformed, can be treated through a sole formalism. Dependence of single $β^-$ and $β^+$ strength distribution on atomic mass n…
We have investigated the effects of the neutron-proton interaction in several doubly odd deformed nuclei within the framework of the particle-rotor model. In this paper, we show some selected results of our study which evidence the importance of the tensor-force effects.
Spontaneous symmetry breaking in non-relativistic quantum systems has previously been addressed in the framework of effective field theory. Low-lying excitations are constructed from Nambu-Goldstone modes using symmetry arguments only. We extend that approach to finite systems. The approach is very general. To be specific, however, we consider atomic nuclei with intrinsically deformed ground states. The emergent symmetry breaking in such systems requires the introduction of additional degrees of freedom on top of the Nambu-Goldstone modes. Symmetry arguments suffice to construct the low-lying …
Techniques from effective field theory are applied to nuclear rotation. This approach exploits the spontaneous breaking of rotational symmetry and the separation of scale between low-energy Nambu-Goldstone rotational modes and high-energy vibrational and nucleonic degrees of freedom. A power counting is established and the Hamiltonian is constructed at next-to-leading order.
We describe and apply a version of the finite amplitude method for obtaining the charge-changing nuclear response in the quasiparticle random phase approximation. The method is suitable for calculating strength functions and beta-decay rates, both allowed and forbidden, in axially-deformed open-shell nuclei. We demonstrate the speed and versatility of the code through a preliminary examination of the effects of tensor terms in Skyrme functionals on beta decay in a set of spherical and deformed open-shell nuclei. Like the isoscalar pairing interaction, the tensor terms systematically increase a…
The Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei (RHFBz) is introduced. The model is based on an effective Lagrangian with density-dependent meson-nucleon couplings in the particle-hole channel, and the pairing part of the Gogny force is used in the pairing channel. The RHFBz quasiparticle equations are solved by expansion in the basis of a deformed harmonic oscillator. Illustrative RHFBz calculations are performed for Carbon, Neon and Magnesium isotopes. The effect of the explicitly including the pion field is investigated for binding energies, deformation parameters…
Deformed Hartree-Fock-Bogoliubov calculations for finite nuclei are carried out. As residual interaction, a Brueckner G-matrix derived from a meson-exchange potential is taken. Phenomenological medium modifications of the meson masses are introduced. The binding energies, radii, and deformation parameters of the Carbon, Oxygen, Neon, and Magnesium isotope chains are found to be in good agreement with the experimental data.
A semi-microscopic model to study the neutron and proton induced backbending phenomena in some deformed even-even nuclei from the rare earth region, is proposed. The space of particle-core states is defined by the angular momentum projection of a quadrupole deformed product state. The backbending phenomena are described by mixing four rotational bands, defined by a set of angular momentum projected states, and a model Hamiltonian describing a set of paired particles moving in a deformed mean field and interacting with a phenomenological deformed core. The ground band corresponds to the configu…
We discuss three practical methods for performing Eulerian rotations of Slater determinants in a three-dimensional Cartesian geometry. In addition to the straightforward application of the active form of the quantum mechanical rotation operator, we introduce two methods using a passive position-space rotation followed by an active spin-space rotation, one after variation and the other before variation. These methods can be used to initialize reactions involving deformed nuclei where a particular alignment of the deformed nuclei with respect to the collision axis is desired. We show that doing …
Large scale shell model calculations in two major oscillator shells (sd and pf) describe simultaneously the super-deformed excited band of 36Ar and its low-lying states of dominant sd character. In addition, several two particle two hole states and a side band of negative parity are also well reproduced. We explain the appearance of the super-deformed band at such low excitation energy as a consequence of the very large correlation energy of the configurations with many particles and many holes (np-nh) relative to the normal filling of the spherical mean field orbits (0p-0h). We study the mech…
Half-lives $T_{1/2}^{2ν}$ for two neutrino positron double beta decay modes $β^{+}$EC/ECEC are calculated for $^{74}$Se, a nucleus of current experimental interest, using deformed shell model based on Hartree-Fock states employing a modified Kuo interaction in ($^{2}p_{3/2}$, $^{1}f_{5/2}$, $^{2}p_{1/2}$, $^{1}g_{9/2}$) space. The calculated half-life for the ECEC mode is $\sim 10^{26}$yr and it may be possible to observe this in the near future with improved sensitivity of experiments.
We formulate a new Hartree-Fock-Bogoliubov method applicable to weakly bound deformed nuclei using the coordinate-space Green's function technique. An emphasis is put on treatment of quasiparticle states in the continuum, on which we impose the correct boundary condition of the asymptotic out-going wave. We illustrate this method with numerical examples.
