Theory of Particle and Cluster Emission
Abstract
Phenomenological Description of Emission Processes.- Binary Emission Processes.- Core-Angular Harmonics.- Coupled Channels Methods.- Semiclassical Approach.- Fine Structure of Emission Processes.- Ternary Emission Processes.- Microscopic Description of Emission Processes.- Microscopic Emission Theories.- Preformation Amplitude.- Selfconsistent Emission Theory.- QRPA Description of the ?-Decay to Excited States.- Heavy Cluster Decays.- Conclusions.- Appendices.
Chapters (12)
We introduce the main tool to describe emission processes, namely Gamow resonances. The reduced width and penetrabilitry are
defined. Then we analyze the experimental material in terms of the Geiger–Nuttall law for different emission processes. We
introduce the double folding procedure, as the most general method to compute the inter-fragment phenomenological potential.
The problem how the binary system is born from the initial nucleus is described in terms of the phenomenological spectroscopic
factor, defined as the ratio between computed and experimental half lives. It turns out, that half lives of α-decays or heavy
cluster emission processes, predicted by phenomenological potentials describing scattering data, are too short. This feature
is a signature that such clusters do not exist as free components on the nuclear surface.
We introduce the core-angular wave functions, involved in the description of various binary emission processes. These functions
are also called surface functions. In the last section we discuss the angular distribution of emitted fragments
The most general procedure to describe the emission of deformed fragments within a phenomenological approach is the coupled
channels method. We analyze various methods to integrate the coupled channels system of differential equations describing
emission processes, namely (a) numerical integration, (b) diagonalisation method, (c) analytical continuation method, (d)
distorted wave approach and (e) two potential method. These methods are general, not depending upon the concrete structure
of the emitted fragments. We then discuss the intrinsic system of coordinate, adiabatic approach, emission from triaxial nuclei,
the coupling with rotation and vibration of the heavy fragment.
For most of decay processes the relative large difference between the height of the Coulomb barrier and Q-value allows the use of the semiclassical method to solve the system of equations describing emission processes. We introduce
the penetration formula for spherical and for deformed emitters as well. Based on this method we then analyze various models
like Cluster Model (CM), Super Asymmetric Fission Model (SAFM), Effective Liquid Drop Model (ELDM) and Fragmentation Theory
(FT).
We describe the fine structure of the α-decay and proton emission by considering rotational and vibrational degrees of freedom
of emitted fragments. Axial and triaxial symmetry is considered. Then we analyze the most general case, namely the double
fine structure in the binary cold fission process
In this chapter we analyze the radioactive processes with three simultaneously emitted fragments. The coupled systems of equations,
describing two proton emission and ternary fission, are separately derived. We discuss adiabatic approach and angular distribution
of the light cluster in the ternary fission.
In this chapter, devoted to microscopic approaches, we derive the general expression of the decay width, known as the Fermi
golden rule, by using the time-dependent Schrödinger equation. We also introduce the equivalent surface formula containing
the preformation amplitude. Then, we show that these relations can be recovered within the reaction Feshbach theory and R-matrix
approach. We describe the Resonating Group Method, as the most general microscopic approach to analyze the emission of composite
objects.
In this Chapter, we present the α-like microscopic theory based on the concept of the preformation probability of the light
partner. This approach can be applied to the emission of light clusters like α-particle, 8Be, or 12,14C and 16O. We extensively describe the Multi-step Shell Model (MSM) technique to built preformation amplitude, by applying this procedure
to 208Pb + α and 40Ca + α systems. Then, we analyze superfluid emitters by using the BCS approach. We also present α-decay processes from superheavy
nuclei and analyze the preformation factor for two-proton emission within the BCS approach.
In a selfconsistent emission theory, the product between the reduced width and penetrability in the decay width should not
depend upon the matching radius. This condition is not a trivial one in the microscopic theory and this point is extensively
discussed in this chapter. We show that the standard shell model approach is not able to satisfy this property along neutron
chains. Only the inclusion of an α-cluster part, depending exponentially upon the Somerfeld parameter, is able to cure this
deficiency
The decay width is not very sensitive to the nuclear mean field parameters, because the preformation amplitude is a coherent
superposition of many single-particle configurations, including states in continuum. The situation changes for transitions
to excited states. In this chapter, we show that the decay width is very sensitive to the structure of the wave function in
the daughter nucleus. We analyze the fine structure to 2+ vibrational states in the α-decay in terms of the Quasiparticle Random Phase Approximation. It is shown that this method
is able to describe the main experimental features concerning both electromagnetic and α transitions.
Microscopic description of heavier decays than α-particle decay is a much more difficult task within the α-like theory. Still
the preformation amplitude can be estimated in an approximate way for heavy cluster decays, until the 14C emission. This procedure is extensively presented here. We give numerical results for the emission of 14C. Then we describe the fission-like theory of emission processes, based on the Two Center Shell Model.
... Microscopic theories have been developed (Harada, 1961;Fliessbach & Okabe, 1985;Fröman, 1957;Delion, 2010;Delion, Insolia, & Liotta, 1992a, 1992cDodig-Crnkovic, Janouch, Liotta, & Sibanda, 1985). ...
... In 1991-1992 Delion et al. (Delion, Insolia, & Liotta, 1992a;Delion, Insolia, & Liotta, 1992b;Insolia, Curutchet, Liotta, & Delion, 1991;Delion, Insolia, & Liotta, 1992c;Delion, 2010) considered deformed parent nuclei (Ra, Rn and Th isotopes). They took a large configuration space, Nilsson + BCS wave functions in a Woods-Saxon potential with parameters introduced by Dudek and Nazarewicz. ...
... 16), proton radioactivity was discovered experimentally at GSI Darmstadt (Hofmann & Münzenberg, 2000;Hofmann, Reisdorf, Münzenberg, Heßberger, Schneider, & Armbruster, 1982;Hofmann, 1996). Among other many publications we also quote (Delion, 2010). ...
... Even though this potential allows shallow bound states (for v 0 ≤ −30 MeV and r 0 = 2.0 fm), we found no influence of this potential on our calculations. The system (16) can be solved by the usual Numerov method. However, we found that much better stability can be achieved by employing the renormalized Numerov algorithm [14]. ...
... In order to avoid using the complicated 3-body framework presented above in the very complex internal region, we first compute the energy levels and wave functions of bound and resonant protons inside the parent nucleus in the 2-body framework. Details of the procedure can be found in [16] together with the Woods-Saxon(WS) parametrization including spin-orbit interaction. Here we briefly outline the procedure. ...
... We first diagonalized the WS mean field for protons by adjusting its real part in order to obtain at the Fermi level the positive experimental proton energy = Q 2p /2 (the paired nucleons have equal energies). Then we solved BCS equations by using the inter-proton force given by the nuclear gaussian interaction in Eq. (16). By changing the nuclear strength v 0 , we obtained the pairing gap at ...
We propose a semi-microscopic model for the simultaneous emission of two protons. This model has the advantage of avoiding certain technical aspects of a fully microscopic 3-body framework, while also allowing the investigation of the influence of proton pairing on the total lifetime of the decaying nucleus. Thus, we use the standard singlet two-proton wave function on the nuclear surface, provided by the Bardeen-Cooper-Schrieffer (BCS) approach, as a boundary condition for the propagator operator. Our model allows for the estimation of all quantities related to the 2p emission process, since it provides the 3-body wave function over most of the domain. We show that reasonable agreement with experimental values can be reached by varying the pp pairing strength outside the nucleus in an interval close to the "bare" singlet value.
... The general theory of resonant states is expanded upon in Refs. [12,13]. Here, we will give a concise overview of the coupled-channels equation and its application to α-decay in the case of even-even nuclei. ...
... The wave function that is under consideration has the form of a clustered α-daughter ansatz [13] with the total spin of the initial state ...
... contains the kinetic operator depending on the reduced mass, a term describing the dynamics of the daughter nucleus H D (b † 2 ) and the α-core interaction V (b † 2 , R), which is generally split into spherical and deformed parts. The spherical component follows from the matching of a harmonic oscillator with the double folding potential integrated from the M3Y nucleon-nucleon plus Coulomb force (see Ref. [13] for computational details, along with references contained therein), while for the deformed component it is enough to retain the QQ coupling. ...
We present an overview of a unified description of the structure and α-emission properties of even-even nuclei. The low-energy spectrum relevant for α-emission is described within the framework of the Coherent State Model (CSM). The treatment of the α-emission process is based on an α-daughter interaction containing a monopole component, calculated through a double folding procedure with a M3Y interaction plus a repulsive core simulating the Pauli principle, and a quadrupole-quadrupole (QQ) interaction. The decaying states are identified with the lowest narrow outgoing resonances obtained through the coupled channels method. The α-branching ratio to the first excited state is reproduced by means of the QQ strength. Simultaneously, a reasonable agreement is obtained for the α-branching ratios to the second and third excited states.
... As mentioned in Ref. [7], this picture involves a problem of non-orthogonality, being difficult to show in a convincing manner how a microscopic description generated by a shell model evolves to finally give a combination of an α particle and a residual nucleus in a collective potential. The decaying states manifest as Gamow resonances in the potential of the binary system [8][9][10]. ...
... The last quantity describes only the smooth behavior of the total energy as function of the particles number and of the deformation and it is considered to be already included in the liquid drop energy. The shell correction is then 8 where the sum of single particle energies ǫ i ...
The decay half-lives and the fine structure phenomenon are investigated with fission-like models. A superasymmetric fission path in a configuration space spanned by five degrees of freedom is determined in accordance with the least action principle. The deformation energy is evaluated within the macroscopic–microscopic approach while the inertia is obtained in the framework of the cranking model. The single particle levels schemes are calculated connecting the ground state of the parent nucleus and the asymptotic configuration of two separated nuclei. The probabilities to find different seniority configurations are obtained by solving time-dependent pairing equations generalized by including the Landau–Zener effect and the Coriolis coupling. The microscopic equations of motion for even numbers of particles are deduced, those for odd-nuclear systems being obtained in previous works. The models used in the calculations are reviewed within a detailed description. The microscopic equations of motion are solved by starting from the ground state configuration and arriving at the scission point. A description of all the possible configurations at scission together with their realization probabilities is given. By fitting the inter-nuclear velocity, the best agreement between experimental and theoretical hindrance factors is retained. The theoretical results for the decay half-lives for Po and Bi are compared with experimental data showing discrepancies ranging over three orders of magnitude. The accuracy of the model concerning the calculations of the half-lives for different channels is discussed. The connections between the classical theories concerning the preformation of the particle and the fission-like descriptions are highlighted.
... This process is fundamental to the study of nuclear stability and structural properties, and may have profound implications across various scientific and potentially technological domains. Theoretical models aim to refine the prediction of alpha decay half-lives, in order to facilitate a more nuanced comprehension of nuclear forces and the inter-nucleon interactions [1,2,3,4,5]. ...
Three isotone groups (N=128,130,132) with alpha-decaying, short-lifetime even-even nuclei of large mass number are studied in order to show that the tunneling phase of the alpha decay process is appropriately describable by a specific non-hermitian mean-field model. In this model the mean-field lifetime of the alpha cluster is extracted directly by the complex diagonalization of the non-hermitian Hamiltonian of the system. In this paper, we demonstrate that by calculating the mean-field width of the alpha clusters, in each isotone groups, by non-perturbative complex spectral calculations, then, due to the distinct patterns the isotones are naturally organized into with respect to the half-life and proton number, the total width of the alpha-decay can be approximated by matching the calculated results to experimental data.
... Among the most fundamental properties of an unstable system at the edge of nuclear stability, the first to be established is the decaying mode and corresponding half-life [1,55]. In order to capture the decaying properties, the continuum effect needs to be taken into account. ...
One of characteristic phenomena for nuclei beyond the proton dripline is the simultaneous emission of two protons (2\emph{p}). The current status of our knowledge of this most recently observed and the least known decay mode is presented. First, different approaches to theoretical description of this process, ranging from effective approximations to advanced three-body models are overviewed. Then, after a brief survey of main experimental methods to produce 2\emph{p}-emitting nuclei and techniques to study their decays, experimental findings in this research field are presented and discussed. This review covers decays of short-lived resonances and excited states of unbound nuclei as well as longer-lived, ground-state radioactive decays. In addition, more exotic decays like three- and four-proton emission are addressed. Finally, related few-body topics, like two-neutron and four-neutron radioactivity, and the problem of the tetraneutron are shortly discussed.
... De nombreux modèles permettant l'étude du processus de désintégration α ont été développés au cours du siècle dernier, allant des approches semi-classiques aux approches microscopiques. L'un des premiers modèles concluant a été la loi de Geiger-Nuttall [GN11] et son interprétation impliquant l'effet tunnel par Gamow [Gam28], ainsi que la classe des modèles WKB (Wentzel, Kramers and Brillouin) [Del10]. D'un point de vue expérimental, même l'identification de tous les noyaux émetteurs α n'a peut-être pas (sûrement pas) encore été réalisée, comme l'a montré la découverte d'une émission α du 209 Bi en 2003 [MER03], ou la question restante d'une éventuelle radioactivité du 208 Pb [Bee+13]. ...
Les systèmes nucléaires exhibent de nombreuses propriétés individuelles et collectives héritées de deux caractéristiques principales : les noyaux sont des systèmes auto-liés par l'interaction nucléaire forte et ils sont composés d'un nombre important de particules. La première de ces spécificités empêche la dérivation simple d'un potentiel nucléon-nucléon. La seconde fait des systèmes nucléaires des problèmes à N-corps quantiques, problème intrinsèquement complexe à résoudre. De cette complexité émerge aussi toute la richesse des systèmes nucléaires. La distribution des nucléons au sein du noyau en est un parfait exemple. D'une structure parfaitement homogène décrite par un liquide de Fermi à la présence d'inhomogénéités importantes appelées clusters, les nucléons s'arrangent dans le noyau de façon complexe. Ces différentes propriétés peuvent être étudiées à l'aide du formalisme de l'énergie fonctionnelle de la densité covariant (cEDF) au niveau champ moyen (RMF) ou bien en incluant les corrélations particule-particule (RHB). Cette approche permet d'obtenir une description satisfaisante des propriétés générales du noyau (énergie, rayon, spectre, ...), et en particulier de rendre compte de l'apparition de structures en clusters au sein du noyau. Les EDF sont construites à partir de Lagrangiens phénoménologiques dont les différents paramètres doivent être ajustés pour reproduire les résultats expérimentaux. Cet ajustement peut s'avérer particulièrement complexe et couteux numériquement si l'on souhaite obtenir une interaction capable de décrire l'entièreté de la carte des noyaux avec une bonne précision. Une méthode combinant approche Monte-Carlo et minimisation simplex est présentée ici afin d'ajuster différentes fonctionnelles relativistes. Ces nouvelles fonctionnelles sont ensuite testées sur environ 1000 noyaux dont l'énergie, le rayon, les gaps, la déformation et d'autres propriétés sont confrontées aux données expérimentales disponibles. L'approximation du champ moyen permet de reproduire un grand nombre de propriétés avec un bon accord théorie-expérience dans un cadre aujourd'hui peu couteux numériquement. Cependant, cette approximation néglige un grand nombre de corrélations dont la prise en compte peut s'avérer nécessaire pour obtenir une description de certains phénomènes. Plusieurs méthodes existent pour aller au-delà du champ moyen statique et nous nous intéressons particulièrement ici au formalisme QRPA (quasiparticle random phase approximation) dont une reformulation récente, la QFAM (quasiparticle finite amplitud method), permet des calculs à grande échelle. Les fonctions de réponses calculées dans ce formalisme montrent que l'étude des structures en clusters mentionnées précédemment s'avère particulièrement importante dans la compréhension des excitations de basse énergie dans les systèmes nucléaires. D'autres mécanismes d'apparition de résonances sont étudiés comme l'excès de neutrons ou l'appariement. Ces structures en clusters possèdent aussi une grande importance dans la description du phénomène de radioactivité. En effet, une compréhension profonde de la préformation et de l'émission de ces clusters permet d'approcher le phénomène de radioactivité d'un point de vue microscopique. En particulier, le phénomène d'émission spontanée de particules alpha ne disposait pas, jusqu'à présent, d'une description entièrement microscopique en termes de nucléons seulement. La puissance du formalisme utilisé ici a de plus permis d'identifier un nouveau mode de désintégration théorique où deux particules alpha sont simultanément émises dos à dos.
... The subsequent tunneling through the Coulomb barrier, surrounding the potential well, ensures the leaking of -particle from the nucleus. Such a situation is described by the Schrödinger equation having the same form as the equations for stationary bound levels [3,5]. The necessity to provide the decay of the nucleus forced Gamov and his followers to introduce the complex energy 0 − Γ/2 instead of the real energy of the level 0 , where Γ stands for the width of the level. ...
In the framework of the Goldberger-Watson decay theory, we consider the-decay of nuclei as a transition between the initial bound state of the nucleus and scattering states of the continuum spectrum with-particle. The scattering wave functions for the-decay with arbitrary orbital angular momentum are derived in the quasiclassical approximation. The-particle is described by the square-integrable wave packet formed by these functions, whose amplitude exponentially grows outside the nucleus up to the wave front. The Moshinsky's distortions of the-wave front are analyzed. The derived general expression for the decay rate is approximated by the quasiclassical formula. K e y w o r d s: decay theory, alpha-decay, tunneling, scattering theory.
... The following types of radioactivities have been experimentally confirmed worldwide (Oxford, Moscow, Orsay, Berkeley, Geneva, Dubna, Argonne, Milano, Viena, Lanzhou, Beijing and Livermore) [26]: 14 [27], P. Buford Price [28]; Eid Hourany and Michel Hussonnois [4,29]; Roberto Bonetti [26], A. Ogloblin [30] and others. Among the books there are [31][32][33][34][35]; chapters in books: [36][37][38][39][40][41]. Many theoretical works have been performed by other colleagues, e.g. ...
Cluster radioactivity (spontaneous emission of heavy particles from nuclei) is presented from theoretical point of view in good agreement with experimental results. After a brief hystorical account we give details about the analytical super asymmetric fission (ASAF) model extensively used for predicting the half-lives of heavy and superheavy (Z ≥ 104) elements. For the already measured 26 cluster decays (from 14C to 32,34Si of parent nuclides with Z = 87-96) it is clear that cluster radioactivity is a rare phenomenon in the best case about 9 orders of magnitude weaker than the competing alpha decay. Then we show the theoretical possibility of a strong cluster decay compared to alpha decay for some superheavy nuclei with Z ≥ 122, e.g. 306122; 310-314122; 306-324124, and 311-323124.
... The α-like structures were experimentally evidenced in light nuclei [8] and therefore they were extensively analysed in the low-lying energy region [9][10][11][12][13], as well as in dipole resonance area [14,15]. In medium and heavy nuclei α-clustering can experimentally be correlated with the α-decay phenomenon [16]. It was understood that an α-clustering component is necessary in addition to the single-particle basis in order to describe the absolute value of the α-decay width [17,18]. ...
Background: Within the quartet condensation model (QCM), the isovector pairing correlations for N=Z nuclei are described with a very high accuracy by a "condensate" of α-like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes. Purpose: We present the final analytical expressions for the above-mentioned quantities, for all cases up to four quartets in the valence shell. Method: The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system cadabra2. Results: The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as a matter of copying and pasting the presented formulas. Conclusions: We introduce in this work a method of computer-aided analytical calculus for a many-body setting. In particular, we provide precise and easy-to-use tools for the description of isovector pairing correlations.
... At the same time, a simplified version considering proton and neutron pairs within the boson approximation was successful in explaining the even-odd pair staggering of binding energies [12]. In medium and heavy nuclei the α-clustering can experimentally be related to the α-decay phenomenon [13]. It became clear that an α-clustering component was necessary in addition to the standard single-particle basis in order to describe the absolute value of the α-decay width [14,15]. ...
We present a new bosonic approximation for the projected-BCS (PBCS) and Quartet Condensation Model (QCM). In each case, the starting point is the reformulation of the pair/quartet condensate state in terms of particle-hole excitations with respect to the completely occupied Fermi sea. The main simplification of our approach is the assumption that the pair operators corresponding to both particle and hole states obey bosonic commutation relations. This simplifies tremendously the computations and allows for an analytic derivation of the averaged Hamiltonian on the condensate state as a function of the mixing amplitudes. We study both the pure bosonic approach and the renormalized version. In the case of a picket fence model of doubly degenerate states, we find a very good agreement between the fermionic and the renormalized bosonic case, regarding both the ground state energy and the pair mixing amplitudes.
... The α-like structures were experimentally evidenced in light nuclei [8] and therefore they were extensively analysed in the low-lying energy region [9][10][11][12][13], as well as in dipole resonance area [14,15]. In medium and heavy nuclei α-clustering can experimentally be correlated with the α-decay phenomenon [16]. It was understood that an α-clustering component is necessary in addition to the single-particle basis in order to describe the absolute value of the α-decay width [17,18]. ...
Within the Quartet Condensation Model (QCM), the isovector pairing correlations for nuclei are described with a very high accuracy by a condensate of -like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes. We present the final analytical expressions for the above mentioned quantities, for all cases up to four quartets in the valence shell. The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system Cadabra2. The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as matter of copying and pasting the presented formulas. We introduce in this work the method of computer aided analytical calculus for a many body setting. In particular, we provide precise and easy to use tools for the description of isovector pairing correlations.
The influence of the daughter nuclei deformations on the α decay's half-life is modeled by adding the term depending on the quadrupole deformation parameter value to the empirical relations for the α -decay half-lives. It is shown that the addition of the deformation-dependent term led to a better description of the experimental data by the empirical expressions for the half-lives of the α decay than the ones without it. A new empirical expression for a description of the α -decay half-lives is proposed. The empirical relation for the α decay half-lives for the transitions from the ground state to the ground state and the lowest 2 + state of the even-even nuclei is discussed.
Published by the American Physical Society 2024
It is shown that the proton formation probabilities, extracted from experimental decay half-lives, can be well reproduced by a simple multilinear formula with only three parameters. The parameters obtained by considering the standard root mean square deviation and the mini-max criteria are very similar to each other. In addition, we applied Bayesian analysis to study the uncertainties of the parameters and the model predictions. In this way we explain the systematics of proton decay half-lives. The multilinearity of the model also provides a way to classify the relative hindrance of different proton decays. All the recent experimental data agree very well with the model prediction. Our Bayesian analysis suggests that those new data do help constrain the uncertainty of the model parameters.
We review the phenomenological versus microscopic description of -clustering in nuclei above A=100. The even–odd staggering in pair binding energies along -lines is explained in terms of the so-called isospin pairing. We compare the pocket-like potential on the nuclear surface, where -clusters are formed with significant probability, to the local equivalent potential provided by the microscopic cluster mean field approach. The systematics of the -decay width shows significantly larger cluster probabilities above magic numbers. The relation of -clustering to electromagnetic and weak processes is analyzed.
Nuclear symmetry energy is a fundamental quantity currently under intense investigation in both nuclear physics and astrophysics. The softness or stiffness of symmetry energy is still under debate and the extraction of symmetry energy from neutron skin thickness Rskin remains a challenge. Parity-violating measurements PREX and CREX provide important opportunities for constraining Rskin in Pb208 and Ca48. We investigate the occurrence of an α cluster at the surface of nuclei and its impact on the extraction of symmetry energy from Rskin. Our result indicates that the α-clustering probability in Pb208 is small and the extracted density slope of symmetry energy L is almost unaffected. In contrast, the α-clustering probability in Ca48 is sizable and the corresponding correction to L should be taken into account. This correction progressively increases with the α-clustering probability, leading to a modification of the L−Rskin correlation, a fact that may have important implications in constraining nuclear symmetry energy.
α and cluster decays are analyzed for heavy nuclei located above Pb208 on the chart of nuclides: Rn216–220 and Ra220–224, which are also candidates for observing the 2α decay mode. A microscopic theoretical approach based on relativistic energy density functionals (EDF), is used to compute axially symmetric deformation-energy surfaces as functions of quadrupole, octupole, and hexadecupole collective coordinates. Dynamical least-action paths for specific decay modes are calculated on the corresponding potential-energy surfaces. The effective collective inertia is determined using the perturbative cranking approximation, and zero-point and rotational energy corrections are included in the model. The predicted half-lives for α decay are within one order of magnitude of the experimental values. In the case of single-α emission, the nuclei considered in the present study exhibit least-action paths that differ significantly up to the scission point. The differences in α-decay lifetimes are not only driven by Q values, but also by variances of the least-action paths prior to scission. In contrast, the 2α decay mode presents very similar paths from equilibrium to scission, and the differences in lifetimes are mainly driven by the corresponding Q values. The predicted C14 cluster decay half-lives are within three orders of magnitudes of the empirical values, and point to a much more complex pattern compared with the α-decay mode.
We show that the Hartree-Fock-Bogoliubov (HFB) method is able to describe experimental values of α decay widths by including a residual nucleon-nucleon surface Gaussian interaction (SGI) within the standard procedure used to calculate the nuclear mean field. We call this method the cluster HFB (CHFB) approach. In this way we correct the deficient asymptotic behavior of the corresponding single-particle wave functions generated by the standard mean field. The corrected mean field becomes a sum between the standard mean Woods-Saxon–like field and a cluster Gaussian component centered at the same radius as the SGI. Thus, we give a confirmation of the mean field plus cluster potential structure, which was assumed in our previous work on α-decay widths. Systematic calculations evidence the linear correlation between the SGI strength and fragmentation potential, allowing for reliable predictions concerning the half-lives of superheavy emitters.
We study the description of single-species and isovector pairing correlations in the framework of the projected-BCS (PBCS) and the quartet condensation model (QCM) from a particle-hole perspective, and we introduce the representation of the QCM quartet condensate state in terms of particle-hole excitations with respect to the Hartree-Fock state. We also present a new bosonic approximation for both PBCS and QCM. In each case, the starting point is the reformulation of the pair and quartet condensate states in terms of particle-hole excitations with respect to the Hartree-Fock state. The main simplification of our approach is the assumption that the pair operators corresponding to both particle and hole states obey bosonic commutation relations. This simplifies tremendously the computations and allows for an analytic derivation of the averaged Hamiltonian on the condensate state as a function of the mixing amplitudes. We study both the pure bosonic approach and the renormalized version and compare the particle-hole bosonic version to the naive prescription of applying the boson approximation directly to the original condensate state. We compare the fermionic and the renormalized particle-hole bosonic approaches in the case of a picket-fence model of doubly degenerate states and in a realistic shell-model space with an effective interaction for the N=Z nuclei above Sn100.
We report on a study of the α-decay fine structure and the associated Eα−Eγ correlations in the decays of Os171,172 and Ir171,172,174. In total, 13 new α-decay energy lines have been resolved, and three new γ-ray transitions have been observed following the new decay branches to Re168 and W167. The weak α-decay branch from the bandhead of the νi13/2 band in Os171 observed in this work highlights an unusual competition between α, β, and electromagnetic decays from this isomeric state. The nucleus Os171 is therefore one of few nuclei observed to exhibit three different decay modes from the same excited state. The nuclei of interest were produced in Mo92(Kr83,xpyn) fusion-evaporation reactions at the Accelerator Laboratory of the University of Jyväskylä, Finland. The fusion products were selected using the gas-filled ion separator RITU and their decays were characterized using an array of detectors for charged particles and electromagnetic radiation known as GREAT. Prompt γ-ray transitions were detected and correlated with the decays using the JUROGAM II germanium detector array surrounding the target position. Results obtained from total Routhian surface (TRS) calculations suggest that α-decay fine structure and the associated hindrance factors may be a sensitive probe of even relatively small shape changes between the final states in the daughter nucleus.
We have studied the proton decay in almost all superheavy nuclei with atomic number Z = 104–126. We have calculated the energy released during the proton decay (QP), penetration factor (P), normalization factor (F), and the proton decay half-lives. The latter are also longer than that of other decay modes such as the alpha decay and spontaneous fission. The competition of the proton decay with different decay modes reveals that the proton decay is not the dominant decay mode in the superheavy nuclei region. This means that superheavy nuclei are stable against the proton decay.
The model for the description of the simultaneous emission of two α-particles from the opposite sides of the nucleus (the double α-decay) is discussed in detail. The 32 smallest values of the half-lives of the double α-decay of nuclei are calculated. The daughter nuclei formed after the double α-decay are even-even and spherical in the ground state in the model. It is shown that the half-lives of the double α-decay are much smaller than the emission of ⁸Be cluster with sequential decay of ⁸Be into two α-particles. The calculated values of the double α-decay half-lives show that measuring this decay is possible in the accelerator experiments by forming the proton-rich nuclei. A detailed comparison of the characteristics of the single and double α-decays as well as the emission of ⁸Be cluster is presented.
The quantum tunneling and emission of a single constituent nucleon provide a beautifully simple and unique window into the complex properties of atomic nuclei at the extreme edge of nuclear existence. In particular, for odd-odd proton emitting nuclides, the associated decay energy and partial half-life can be used to probe the correlations between the valence neutrons and protons which have been theoretically predicted to favour a new type of nuclear superfluidity, isoscalar neutron-proton pairing, for which the experimental “smoking gun" remains elusive. In the present work, proton emission from the lanthanum isotope 57116La59, 23 neutrons away from the only stable isotope 57139La82, is reported. 116La nuclei were synthesised in the fusion-evaporation reaction 58Ni(64Zn, p5n)116La and identified via their proton radioactivity using the mass spectrometer MARA (Mass Analysing Recoil Apparatus) and the silicon detectors placed at its focal plane. Comparisons of the measured proton energy (Ep = 718 ± 9 keV) and half-life (T1/2 = 50 ± 22 ms) with values calculated using the Universal Decay Law approach indicate that the proton is emitted with an orbital angular momentum l = 2 and that its emission probability is enhanced relative to its closest, less exotic, odd-even lanthanum isotope (57117La60) while the proton-emission Q-value is lower. We propose this to be a possible signature for the presence of strong neutron-proton pair correlations in this exotic, neutron deficient system. The observations of γ decays from isomeric states in 116La and 117La are also reported. Neutron-proton pairing is a topic of continuous interest in nuclear physics and open questions remain. The authors experimentally observe direct proton decay from the ground state of odd-odd 116La, providing support for the presence of strong neutron-proton pair correlations in this exotic, neutron deficient system.
We review microscopic approaches describing α-like resonances in nuclei with Z ≥ 50. We give a microscopic explanation of α-like rotational bands in the 40Ca+α system in terms of single particle Gamow resonances. On the other hand, we show that for α-decaying nuclei the decay width can be described only in terms of a preformed α-cluster, existing on the nuclear surface in addition to the standard mean field cluster. Thus, we use a semi-microscopic hybrid model combining the mean field formation with a preformed α-cluster in order to explain the order of magnitude of the experimental decay width. Finally, we analyze proton-neutron versus α-like correlations.
"The low-lying rotational bands in light nuclei were explained a long time ago by α-clusters orbiting around the remaining core. These quasimolecular states are related to an enhancement of the elastic α-particle cross section at large angles, the so-called Anomalous Large Angle Scattering (ALAS) phenomenon. On the other hand, α-particles were evidenced by the discovery of the α-decay for nuclei with Z > 50. We used a phenomenological model of an α-particle moving in a pocket-like potential, able to explain the ALAS phenomenon, in order to understand the main features of the decay data between ground and excited states. Then we extended the Brink model, explaining giant resonances, and presented a possibility to detect α-like collective resonances."
We propose a semimicroscopic model for the simultaneous emission of two protons. This model has the advantage of avoiding certain technical aspects of a fully microscopic three-body framework, while also allowing the investigation of the influence of proton pairing on the total lifetime of the decaying nucleus. Thus, we use the standard singlet two-proton wave function on the nuclear surface, provided by the Bardeen-Cooper-Schrieffer (BCS) approach, as a boundary condition for the propagator operator. Our model allows for the estimation of all quantities related to the 2p emission process, since it provides the three-body wave function over most of the domain. We show that reasonable agreement with experimental values can be reached by varying the pp pairing strength outside the nucleus in an interval close to the “bare” singlet value.
Background: While many phenomenological models for nuclear fission have been developed, a microscopic understanding of fission has remained one of the most challenging problems in nuclear physics.
The alpha decay half-lives of some even-even nuclei have been calculated using WKB method and including Bohr-Sommerfeld quantization for nuclear potential. Three forms of the nucleon-nucleon, alpha-nucleon, and alpha-alpha folding potentials have been adopted for calculation of the nuclear potential. Moreover, the phenomenological two parameter Fermi (2pF) and the microscopic Hartree-Fock-Bogoliubov (HFB) density distributions have been employed for nuclear matter. The effect of the folding potentials and matter densities have been investigated. The obtained results showed good agreement between theory and experiment and confirmed the applicability of the alpha-folding potentials in alpha decay calculations.
We investigate proton systematics in terms of the barrier penetration and formation probability. To this purpose we approximate the realistic proton-core interaction by a parabolic dependence describing the nuclear part matched to a pure Coulomb potential. In this way we can use analytic semiclassical solutions to describe the proton dynamics inside the barrier. We then carefully investigate effects induced by the centrifugal barrier and quadrupole deformation. It turns out that the nuclear centrifugal and deformation effects are much smaller than their Coulomb counterparts. On the other hand, the Coulomb action defining the semiclassical penetrability is larger by one order of magnitude than its nuclear counterpart. We evidence a dependence of the Coulomb action upon the Coulomb parameter in terms of two parallel lines separated by A=145. Such a dependence was previously discovered for the logarithm of the experimental decay width corrected by the Coulomb centrifugal barrier. We also evidence an additional but smaller residual linear decrease of the proton formation probability versus A1/3 along two parallel lines separated by A=145, induced by the internal proton wave function. Therefore, the dependence between the experimental decay width corrected by the centrifugal barrier and the Coulomb penetrability multiplied by the proton formation probability in the doubly logarithmic scale can be fitted within a factor of two by a universal straight line.
Relativistic energy density functional approaches are known to well describe nuclear states which involve alpha clusters. Here, alpha emitting nuclei are analyzed through the behavior of the spatial localisation of nucleonic states, calculated with an axially deformed relativistic Hartree–Bogoliubov approach over the nuclear chart. The systematic occurrence of more localised valence states, having an n = 1 radial quantum number, allows to pinpoint nuclei in agreement with experimentally known alpha-emitters. The cases of ²¹² Po and ¹⁰⁴ Te are investigated, showing the concomitant contributions of the pseudospin symmetry and the presence of n = 1 states, on the alpha preformation probability. The impact of the localisation of valence states, on alpha preformation probability, is then analyzed. It allows to study shell effects on this probability, over isotopic and isotonic chains. Finally, a phenomenological law is also provided, relating this probability to the radial quantum number of the valence states.
α decay and β-delayed fission are two important decay modes of heavy exotic nuclei. Experimental α decay and β-delayed fission studies deliver significant nuclear-structure information in regions of the nuclear chart with limited accessibility. This information is important to improve the predictability of contemporary nuclear models used for e.g. nuclear astrophysics calculations. The basic principles and the current understanding of α and β-delayed fission decay are introduced. Examples of recent experiments and their impact on the understanding of heavy nuclei are presented.
We approximate the realistic α-core nuclear interaction between the inner turning point and Coulomb barrier, derived within the double folding procedure, by a parabolic dependence. It turns out that the corresponding harmonic oscillator frequency is concentrated in a narrow interval around ℏω1≈ 9 MeV for all analyzed transitions from even-even and odd-mass α emitters. The penetrability through the nuclear barrier has an exponential dependence on the ratio between the fragmentation potential and this harmonic oscillator frequency. On the other hand, the Coulomb penetrability has the standard dependence on the Coulomb parameter. Our analysis revealed that the reduced width, extracted by using the nuclear plus Coulomb penetrability factors, exponentially decreases versus the number of α clusters along all analyzed α lines, except one region where it increases by approaching the neutron magic number Nmag=126. In that case, the number of valence proton pairs is much smaller than the number of valence neutron pairs. The reduced width exponentially depends on the magic neutron-proton asymmetry above doubly magic nuclei. These dependencies allow us to propose a systematics for reduced widths in terms of the quartet number and magic asymmetry evaluated above the closest doubly magic nuclei.
The Wildermuth-Tang (WT) prescription is used to verify the Bohr-Sommerfeld (BS) quantization condition in the
α
-decay problem. It gives the global quantum number that relates the number of nodes of the quasibound radial wave function of the
α
-daughter system to the shell model and Pauli exclusion principle. Here we examine the applicability of the WT rule in the
α
-decay microscopic calculations that start with solving stationary Schr"{o}dinger wave equation, for different types of the interaction potentials. We found that applying the BS quantization condition along with the WT prescription for the potentials that have no internal pocket yields a large number of nodes in the radial wave function compared to the potentials characterized with an automatic physical internal pocket, which likely produce nodeless or as maximum as two-node interior wave function. This gives confidence in the latter type of the potentials that efficaciously simulates the Pauli principle by considering the change in the intrinsic kinetic energy. However, it is possible to reproduce the observed half-life data using the potentials that have no automatic internal pocket with applying the BS quantization condition with quantum numbers which are significantly less than that obtained from the WT rule, upon properly normalizing the potential.
We investigate the influence of a monochromatic strong laser electromagnetic field on α clustering and emission, by analyzing the Fourier components of the time-dependent α-core realistic potential in the Henneberger representation. It turns out that the resulting potential becomes deformed and the static component is by far dominant beyond the nuclear surface where the α cluster is formed, while higher Fourier terms are important in the internal region, where the α-particle probability is hindered by the Pauli principle. This fact, combined with the observation that an α cluster lives much longer before its emission than the laser period, allows us the use of the stationary coupled channels approach in the system of coordinates given by the laser beam direction. We predict that the angular distribution of emitted α particles becomes anisotropic due to the deformation of the α-core potential induced by the laser field, even for the spherical emitter Po212.
As for the dominant decay channel of superheavy nuclei, we have reshaped the crucial nuclear potential in the α-decay process through the systematical analysis of available experimental decay data. Specifically, a Fermi-type renormalization factor brings us an effective α-daughter potential so that we can obtain the decay energy and half-life on the same footing, which exactly corresponds to the stability of nuclei. During this procedure, the α preformation factor is related with the recalculated pairing and shell corrections. Besides the excellent agreement between theory and experiment for both the decay energy and the half-life, we extend the study to available short-lived nuclei in the superheavy-mass region as a predictive set. It is found that the present α-core potential can figure out the decay scheme of these superheavy α emitters very well. Encouraged by this, it is expected that the following predictions on the heaviest nuclei can be valuable for discussing their robustness and shed some light on the “island of stability” plus the fundamental mass of unknown nuclides from a novel perspective.
We use the available experimental Gamow-Teller β− and β+/EC (electron-capture) decay rates between 0+ and 1+ ground states in neighboring even-even and odd-odd nuclei, combined with 2νββ half-lives, to analyze the influence of the nuclear environment on the weak axial-vector strength gA. For this purpose, the proton-neutron deformed quasiparticle random-phase approximation (pn-dQRPA), with schematic dipole residual interaction is employed. The Hamiltonian contains particle-hole (ph) and particle-particle (pp) channels with mass-dependent strengths. In deriving the equations of motion we use a self-consistent procedure in terms of a single-particle basis with projected angular momentum provided by the diagonalization of a spherical mean field plus the quadrupole-quadrupole interaction. Our analysis evidenced a quenched average effective value 〈gA〉≈0.7 with a root-mean-square deviation of σ≈0.3 for transitions from even-even emitters and σ≈0.6 for transitions from odd-odd emitters.
α clustering in nuclei is considered with the quartet model (QM) where four valence nucleons (the quartet) move on the top of the core (daughter) nucleus. In the QM approach, it is assumed that the intrinsic wave function of the quartet is changed from the pure cluster configuration to the shell-model configuration when it crosses some critical radius and enters into the core nucleus. The QM approach could give not only the level scheme, the electromagnetic transition, and the nuclear radius but also the α-cluster formation probability. Numerical results are calculated for Ne20, Ti44, and Po212, where a quartet moves on top of a double magic nucleus. Good agreement with experimental data and previous theoretical studies is obtained. The QM approach is a useful complement to the present phenomenological and microscopic models and could help deepen our understanding of α clustering across the nuclide chart.
We investigate the influence of a strong laser electromagnetic field on the decay rate of the α-emission processes. We define the parameter D = S0/R0, depending on the geometrical nuclear radius R0 and S0∼I/ω2, where I is the beam intensity and ω the frequency. The barrier penetrability within the static approach has an important increase for D > Dcrit = 1, due to the fact that the resulting Coulomb potential becomes strongly anisotropic. The influence of the laser field can be expressed in terms of a shifted Geiger-Nuttal law by a term depending on S0 and deformation. On the other hand, the adiabatic approach also predicts an important change of the decay width under the influence of the laser beam.
Superstrong static electric fields could deform Coulomb barriers between α clusters and daughter nuclei, and bring up the possibility of speeding up α decays. We adopt a simplified model for the spherical α emitter ²¹²Po and study its responses to superstrong static electric fields. We find that superstrong electric fields with field strengths | E | ∼ 0.1 MV/fm could turn the angular distribution of α emissions from isotropic to strongly anisotropic, and speed up α decays by more than one order of magnitude. We also study the influences of superstrong electric fields along the Po isotope chains, and discuss the implications of our studies on α decays in superstrong monochromatic laser fields. The study here might be helpful for future theoretical studies of α decay in realistic superstrong laser fields.
The Woods-Saxon-Gaussian (WSG) potential is proposed as a new phenomenological potential to systematically describe the level scheme, electromagnetic transitions, and alpha-decay half-lives of the alpha-cluster structures in various alpha + closed shell nuclei. It modifies the original Woods-Saxon (WS) potential with a shifted Gaussian factor centered at the nuclear surface. The free parameters in the WSG potential are determined by reproducing the correct level scheme of ²¹²Po=²⁰⁸Pb+α. It is found that the resulting WSG potential matches the M3Y double-folding potential at the surface region and makes corrections to the inner part of the cluster-core potential. It was also determined that the WSG potential, with nearly identical parameters to that of ²¹²Po (except for a rescaled radius), could also be used to describe alpha-cluster structures in ²⁰Ne=¹⁶O+α and ⁴⁴Ti=⁴⁰Ca+α. In all three cases, the calculated values of the level schemes, electromagnetic transitions, and alpha-decay half-lives agree with the experimental data, which indicates that the WSG potential could indeed capture many important features of the alpha-cluster structures in alpha + closed shell nuclei. This study is a useful complement to the existing cluster-core potentials in literature. The Gaussian form factor centered at the nuclear surface might also help to improve our understanding of the alpha-cluster formation, which occurs in the same general region. © 2018 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.
The α-decay is considered as a superasymmetric fission process. A fragmentation path is obtained by mean of the least action principle in a configuration space spanned by five degrees of freedom. The potential barrier is obtained within the macroscopic-microscopic model while the effective mass and the momentum of inertia within the cranking approach. The single-particle wave functions and the single-particle energies are supplied by the Woods-Saxon two-center shell model. The fine structure of ²¹¹Bi α-decay is treated within a set of generalized time-dependent pairing equations that takes into account the Landau-Zener effect and the Coriolis coupling. A low value of the internuclear collective velocity was used. The theoretical results showed a good agreement with the experimental data. Essentially, in the framework of the formalism, the fine structure for the ²¹¹Bi is due to the occurrence of the Coriolis coupling.
Quasimolecular α-like ground rotational bands were evidenced a long time ago in light nuclei, but they cannot be detected in heavy nuclei due to large Coulomb barriers. In order to search for rotational bands built on excited states in these nuclei, we investigate the shape of an α-nucleus quasimolecular potential matched to a realistic external α-daughter interaction by using as input data α-decay widths. It turns out that its Gaussian length parameter lies in a narrow interval, b0∈[0.6,0.8] fm, and the equilibrium radius is slightly larger than the predicted Mott transition point from nucleonic to the α-cluster phase in finite nuclei, confirming that α clusters are born on the nuclear surface at low densities. We point out that the α emitters above magic nuclei have the largest spectroscopic factors Sα∼10%. In addition, we predict that for nuclei with b0>0.75 fm, the first excited vibrational resonant state in the quasimolecular potential is close to the Coulomb barrier and therefore the rotational band built on it can be evidenced by the structure of the α-scattering cross section versus energy. Moreover, its detection by a highly sensitive γ-ray beam produced by laser facilities would provide an additional proof for the existence of α molecules in heavy nuclei.
We review the coupled channels approach of α transitions to excited states. The α-decaying states are identified as narrow outgoing Gamow resonances in an α-daughter potential. The real part of the eigenvalue corresponds to the Q-value, while the imaginary part determines the half of the total α-decay width. We first review the calculations describing transitions to rotational states treated by the rigid rotator model, in even-even, odd-mass and odd-odd nuclei. It is found that the semiclassical method overestimates the branching ratios to excited 4⁺ for some even-even α-emitters and fails in explaining the unexpected inversion of branching ratios of some odd-mass nuclei, while the coupled-channels results show good agreement with the experimental data. Then, we review the coupled channels method for α-transitions to 2⁺ vibrational and transitional states. We present the results of the Coherent State Model that describes in a unified way the spectra of vibrational, transitional and rotational nuclei. We evidence general features of the α-decay fine structure, namely the linear dependence between α-intensities and excitation energy, the linear correlation between the strength of the α-core interaction and spectroscopic factor, and the inverse correlation between the nuclear collectivity, given by electromagnetic transitions, and α-clustering.
Starting from the variational principle, the time-dependent pairing equations are generalized by including the Landau-Zener effect and the Coriolis coupling. A system of microscopic equations of motion for configuration mixing is deduced, allowing the determination of quantities that have the same meaning as the preformation factors of the α particle. These equations are solved in order to reproduce the hindrance factors of the α decay of an odd-A mass nucleus. The α decay of Po211 is treated as a superasymmetric fission process, by following the rearrangement of the nuclear orbitals from the parent ground state up to the scission configuration. The probabilities of finding the excited states of the daughter at scission are obtained from the microscopic equations of motion. The intensities of the transitions to the excited states of the daughter were evaluated theoretically. The experimental data were compared with the theoretical findings. A very good agreement was obtained. A mean value of the tunneling velocity of about 2×104 fm/fs was extracted.
We investigate the influence of a strong laser electromagnetic field on the α-decay rate by using the Hennenberger frame of reference. We introduce an adimensional parameter D=S0/R0, where R0 is the geometrical nuclear radius and S0∼I/ω2 is a length parameter depending on the laser intensity I and frequency ω. We show that the barrier penetrability has a strong increase for intensities corresponding to D>Dcrit=1, due to the fact that the resulting Coulomb potential becomes strongly anisotropic even for spherical nuclei. As a consequence, the contribution of the monopole term increases the barrier penetrability by 2 orders of magnitude, while the total contribution has an effect of 6 orders of magnitude at D∼3Dcrit. In the case of deformed nuclei, the electromagnetic field increases the penetrability by an additional order of magnitude for a quadrupole deformation β2∼0.3. The influence of the electromagnetic field can be expressed in terms of a shifted Geiger-Nuttal law by a term depending on S0 and deformation.
The process of the synthesis of superheavy elements (SHEs) is not yet understood completely. In the presented work we make an attempt to describe the cold fusion reactions of the type X+(Pb,Bi)-->SHE Sin at subbarrier energies. The process of the formation of SHEs is subdivided into three steps. (1) The capture of two spherical nuclei and the formation of a common shape of the two touching nuclei. Low-energy surface vibrations and transfer of few nucleons are taken into account in the first step of the reaction. (2) The formation of a spherical or near spherical compound nucleus. (3) The survival of the excited compound nucleus due to evaporation of neutrons and gamma-ray emission in competition with fission. A lowering of the fission barrier was taken into account, which arises from a reduction of shell effects at increasing excitation energy of the compound nucleus. The following reactions were analyzed in detail: (Fe-58, Ni-64, Zn-70, Ge-78) + Pb-207, (Ti-50, Cr-54, Fe-58, Co-59, Ni-62,Ni-64, Cu-65, Zn-66,Zn-68,Zn-70, Ga-71, Ge-74,Ge-76,Ge-78, As-75, Se-80,Se-82) + Pb-208, (Fe-58, Ni-64, Zn-70, Ge-78) + Pb-210, and (Ti-50, Cr-54, Fe-58, Ni-64, Zn-70, Ge-78) + Bi-209. The presented model describes well the available experimental cross-section data and allows for predicting cross-section values for the synthesis of so-far unknown heavier elements.
A microscopic description of the α decay to intruder 0{sup +}â states of Pb isotopes in terms of proton-neutron pairing vibrations is performed. The calculated hindrance factors with respect to α transitions into the ground state are in good agreement with the available experimental data. Predictions for further measurements are given. Hindrance factors corresponding to the decay towards Po, Hg, and Pt isotopes are calculated within a simple model. The calculated quantities agree well with the experimental data, indicating that α decay is a powerful tool to precisely determine the transition from spherical to deformed shapes in the region around {ital Z}=82.
Calculations based on the relativistic mean field framework have been carried out for the ground state properties of the relevant nuclei appearing in the alpha -emission chain of superheavy element Z=112 . The calculations compare well with the experiment. The calculated densities along with the energy and the density dependent M3Y effective nucleon-nucleon interaction are used in the double folding model to compute the respective total interaction energies. These, in turn are used to calculate the half-lives of the parent nuclei against the split into an alpha and the daughter in the WKB approximation. The calculations are repeated for the nuclei appearing in the 4n+2 alpha chain for a comparative study. The sensitivity of the calculated half-lives on the input information, especially the Q values are quantitatively investigated.
Half-lives for proton emission from proton-rich nuclei have been calculated by using the effective liquid drop model of heavy-particle decay of nuclei. It is shown that this model is able to offer results for spontaneous proton-emission half-life values in excellent agreement with the existing experimental data. Predictions of half-life values for other possible proton-emission cases are presented for null orbital angular momentum.
The widths for one- and two-proton decay of the 12- state in 18Ne are calculated. Shell-model wave functions are used to obtain the spectroscopic factors. The R-matrix theory of Barker which incorporates the final-state interaction between the two protons is used for the di-proton decay model. The calculated widths for both one- and two-proton decay are in qualitative agreement with experiment. We find that the decay width for sequential two-proton decay through the ghost of the 1/2+ bound state in 17F is comparable to the width of the direct di-proton decay.
The width of the 12O ground state is calculated, using R-matrix formulas for one-proton sequential decay through the ground state of 11N to the 10C ground state. From simple models for the 12O and 11N states, upper limits are placed on the reduced widths for the two decays, giving an upper limit on the 12O width that is less than about 100 keV. It is suggested that the available experimental data may not be inconsistent with this limit.
Cold fission and cluster decay processes are studied for [iopmath latex=""] 234,236,238U [/iopmath] nuclei in the super-asymmetric mass region of exotic cluster radioactivity. The idea is to see whether the clusters already observed as cluster decays (here for [iopmath latex=""] 234,236U [/iopmath] parents) could also be observed via the cold fission process. Our calculations are made on the well accepted quantum mechanical fragmentation theory and the saddle-point fission model and the preformed-cluster model that are also based on this theory.
Proton decay from is analyzed in the perspective of decay from a deformed nucleus. The experimental values of the half-life for decay from the ground state and the recently found isomeric state are very well described as decay from a {K=5/2}^{-{}} ground state and a isomeric state with an oblate deformation -0.18-0.14. With this deformation one is able to describe consistently both decays, using the correct spectroscopic factors, which is not possible in a spherical interpretation.
The half-life of 45Fe calculated using an R-matrix formula for the contribution due to di-proton decay agrees with experimental values, provided the available decay energy is near the upper limit of the experimental range.
The spontaneous fission of 252Cf has been studied via γ γ γ coincidences and γ γ-light charged particle coincidences with Gammasphere. The yields of correlated Mo-Ba pairs in binary fission with 0–10 neutron emission have been remeasured with an uncompressed γγγ coincidence data. The previous hot fission mode with 8–10 neutron emission seen in the Mo-Ba split is found to be smaller than earlier results but still present. New 0n binary SF yields are reported. By gating on the light charged particles detected in ΔE-E Si detectors and γγ coincidences with Gammasphere, the relative yields of correlated pairs in alpha ternary fission with zero to 6n emission are observed for the first time. A number of correlated pairs are identified in ternary fission with 10Be as the LCP. We observed only cold, 0n 10Be and little, if any, hot, xn 10Be channel.
The equivalence of the generator coordinate method and the resonating group method is used in the derivation of two new methods to describe the scattering of spin-zero fragments. Both these methods use generator coordinate kernels, but avoid the problem of calculating the generator coordinate weight function in the asymptotic region. The scattering of two a-particles is studied as an illustration.
The hindrance factors and the reduced wave amplitudes for alpha transitions to β-states are computed. It is shown that the alpha intensities are strongly enhanced by the quadrupole forces and that only the superfluid model with pairing and quadrupole-quadrupole residual interactions can explain the experimental data, i.e. β-states are not pure pairing vibrational ones.
Plots of 4-nucleon (α) separation energies along lines of constant isospin are presented. They show strong systematic features. For a dozen such lines there is a gap in the known masses. A polynomial interpolation is proposed to predict some 100 new binding energies.
In the alpha decay theories of Thomas and Mang, the alpha reduced width is expressed in terms of the integral of a product of the wave functions of the initial nucleus, the final nucleus, the alpha particle and the relative orbital angular momentum. In the present paper a method is indicated which permits the calculation of the reduced widths for favoured alpha transitions. For the initial and final nuclei we use the wave functions given by the collective model in Nilsson's scheme and for the spatial internal wave function of the alpha particle an exponential-type functiom.
A method is discussed which makes it possible to see that in nuclei many different cluster structures are present. Which cluster structures are preferred depends on the form and strength of the interaction forces, the type of nucleus and the excitation levels of this nucleus. Some examples are discussed. The influence on the asymmetrical fission is mentioned briefly.
A theory is developed, within the context of the DavydovFillppov model
of the nucleus, to describe the even-parity alpha decay of nonaxlal nuclel. The
wave equation of the system consisting of a daughter nucleus and an alpha
particle is solved under the boundary conditions that on the nuclear surface the
wave function shall be equal to a given function and that the function shall
represent only an outgoing wave for large values of r. It is found that if the
given functlon is assumed to be equal to a constant, the observed variations of
the alpha transition probabilities with respect to Z in even-even nuclei cannot
be explained. But if one assumes that this glven function can be represented as
an expansion in spherical harmonics, keeping only quadrupole terms, the expansion
coefficients being determined by the empirical data (that is, setting the
boundary conditions by the empirical data), the observed variations can be
theoretically reproduced. Even-parity alpha transitions in odd-A nuclei are also
considered. The comparison between theory and the empirical data is quite
satisfactory. (auth)
The experimental evidence supporting the double-humped character of the fission barrier in actinide nuclei is reviewed and compared to theoretical predictions. The discussion covers the existence and half-life systematics of spontaneously fissioning isomers, shape-isomeric gamma decay, rotational transitions and the moment of inertia of isomers, fragment angular distributions in isomeric fission, intermediate structure in fission cross sections, and finally the systematics of barrier heights as deduced from fission probability measurements. The implications of a possible octupole deformation at the second barrier for fragment mass distributions are also discussed, including the size of the mass asymmetry and recent experiments on the competition between symmetric and asymmetric fission as a function of excitation energy.
The consequences of spheroidal deformation of nuclei on the barrier transmission in alpha decay are considered. A set of coupled differential equations is derived relating the amplitudes of the various groups of alpha particles emitted from a nucleus described by the Bohr-Mottelson model. The cases of the decay of Th228 and Cm242 were studied numerically and from them information regarding the probability distribution of alpha particles on the nuclear spheroidal surface is obtained. It is found that the one-body model of an alpha particle in a well does not yield these distributions, and it is thus concluded that "alpha-particle clusters" have a short mean free path in nuclear matter. The shift in the surface distributions of Th228 and Cm242 may be explained qualitatively in terms of the order of nucleon orbital filling.The over-all penetration factors for the spheroidal case are compared with those for the spherical case, and it is found that the resultant enhancement due to the deformation is not early as large as that predicted by Hill and Wheeler on the basis of a one-dimensional approximation.
Fine structure in the ground-state proton radioactive decay of highly deformed ¹³¹Eu has been identified. In addition to the previously observed ground-state line, measured here with a proton energy of 932(7) keV, a second proton peak with energy 811(7) keV was observed. We interpret this line as proton decay from the ¹³¹Eu ground state to the first excited 2{sup +} state of the daughter nucleus ¹³°Sm . Comparing the measured branching ratio with calculations enables the ground-state configuration of ¹³¹Eu to be unambiguously assigned to the 3/2{sup +} [411] Nilsson configuration. {copyright} {ital 1999} {ital The American Physical Society }
Using triple-coincidence events of prompt fission gamma rays from 1995 Gammasphere data on spontaneous fission of ²âµÂ²Cf, we made a careful analysis of the yield matrix of coincident pairs of barium (Z=56) and molybdenum (Z=42) fission fragments. Branching from gamma bands (K=2) and octupole bands (K=0) are presented for even-even partners, where observable. From this reanalysis the previously proposed ''extra-hot-fission mode'' (up to ten neutrons evaporated) as determined by twofold coincidences of 1993 Gammasphere data is much weaker, but not excluded. This finding is in agreement with a recently published similar study from the Legnaro gamma-detector array.
The most probable decays by spontaneous emission of heavy ions are listed for nuclides with Z = 47--106 and total half-lives>1 ..mu..sec. Partial half-lives, branching ratios relative to ..cap alpha.. decay, kinetic energies, and Q values are estimated by using the analytical superasymmetric fission model, a semiempirical formula for those ..cap alpha..-decay lifetimes which have not been measured, and the new Wapstra--Audi mass tables. Numerous ''stable'' nuclides with Z>40 are found to be metastable with respect to the new decay modes. The current experimental status is briefly reviewed.
A possible way to remove the discrepancy between calculated and measured alpha-widths is discussed. The decay rates of 212Po and 210Po are computed with the help of shell-model wave functions for parent and daughter nucleus.
Elastic alpha-scattering on the isotopes 40,42,44,48Ca has been studied microscopically within the framework of the generator coordinate method. For energies above the Coulomb barrier the backangle enhancement of the cross section in elastic alpha-40Ca scattering is explained by several overlapping barrier resonances, while for energies below the barrier the backangle rise is caused by individual long-living resonances which - in agreement with recent experimental data - confirm the concept of an ``alpha-40Ca quasimolecule''. The isotope effect showing up in a different behaviour of the cross section at backangles is due to a different absorptive strength of the isotopes. ``Neutron blocking'' has been ruled out as a possible explanation of the isotope effect.
The sequential emission and then decay of particle unstable resonances (such as 2He) from compound nuclei is discussed. Calculations (based on a statistical model) of yields and energy spectra for the final stable products are presented. The effect of this sequential decay process on coincidence measurements is discussed.
The energy spectrum and α-spectroscopic factors of40Ca are calculated by using an α+36Ar orthogonality condition model. Overall agreement is obtained with the experimental data. The model successfully produces the parity-doubletK
π=O
+
and O− α+36Ar cluster bands. It is shown that the coexistence and interference of α-cluster states and shell-model states are indispensable to understand the structure of40Ca.
With the use of a schematic40Ca*+α model, the low-lying positiveK=O2+ and 2+ bands in44Ti, which start from the band-head energyE
x
=1.90 MeV and 2.88 MeV, respectively, are shown to be well interpreted by an α-cluster structure with an excited40Ca core, i.e.,36Ar+α+α structure.
The α decay of mass-separated202Rn is studied at the ISOLDE separator. Time sequential α-singles spectra together with α-X-t and α-e-t coincidence events are collected. Fine structure in the α decay is observed and feeding of a low-lying 0
2
+
state at 816 keV in198Po is evidenced. This 0+ state can be interpreted as the bandhead of an intruder-state based deformed band, coexisting with the spherical groundstate band. Mixing between normal and intruder states is discussed. A preliminary α-decay study of200Rn did not yet reveal any fine structure.
The theoretical development of the microscopic (shell-model) theory of α decay of deformed nuclei is reviewed, and the detailed formulas applying to even- and odd-mass nuclei are presented. Results are presented from new calculations on ground-rotational-band α-decay patterns for even-even nuclei from atomic numbers 90-106. These calculations represent a refinement over previously published work in that particle-number-conserving variational wave functions were used instead of Bardeen-Cooper-Schrieffer (BCS) wave functions. The new results for even nuclei are not, however, very different from the earlier BCS results. The main contribution of this paper is the tabulation of several hundred theoretical α amplitudes for odd-mass nuclei from elements 92-101. The theoretical intensities derived from these tables are compared with experiment for a representative sampling of α emitters. The main factors governing hindrance for unfavored transitions are discussed in terms of loss of the pairing correlation enhancement and in terms of the Nilsson functions of the odd-nucleon wave function.
Approximate analytic methods are developed for calculating the amplitudes of alpha partial waves at the surface of deformed even-even nuclei. A two-term expansion modifying the ordinary Coulomb function to account for nuclear quadrupole coupling is applied. The amplitudes of alpha partial waves at the nuclear surface are tabulated for eight choices of phase and three values of the intrinsic nuclear quadrupole moment. A detailed comparison is made between this treatment, that of Fröman, and the numerical integration of Rasmussen and Hansen for Cm242. An approximate method of calculating phase shifts induced by the nuclear quadrupole moment is developed and compared with the results of Rasmussen and Hansen for Cm242.
In the first five sections the dispersion theory is developed with an internal region V whose boundary S is quite close to the nuclear surface. Two types of quantities then occur: those like the derivative matrix R, which connect the values and derivatives of the wave function at S, and those like the collision matrix U which give the asymptotic behavior of the wave function. These latter are, of course, independent of the position of S. Because of the proximity of S to the nuclear surface, the wave function in the closed channels and its derivative remain appreciable at S. They may, however, be eliminated from the formalism, and this is done in Sec. II, leading to a reduced R which connects the values and derivatives of the wave function in the open channels only. In the next section this reduced R is used to obtain expressions for the S-independent quantities, in particular, for the collision matrix. These expressions are given more explicit, but approximate form in Sec. IV. The development shows that the quantity, which in the usual formulas for the cross sections is interpreted as the nuclear radius, need not be equal to this at all, but is the distance at which two opposite effects compensate. The fifth section gives exact expressions for the poles and residues of some S-independent quantities, which are then compared with the poles and residues of the approximate expressions of the previous chapter. In this way criteria are derived for the accuracy of these approximate expressions.
In α-decay, after the particle has been emitted, the asymmetrical electric field of the residual nucleus can cause the α-particle to alter its total energy in conjunction with a change in the state of the nucleus. Since the ability of a particle to penetrate the potential barrier increases rapidly with its energy, the number of particles of each possible energy finally escaping will not be in accord with the decay constants predicted by the Geiger-Nuttall law.
A number of authors have developed the theory of α-radioactivity in the case of zero angular momentum of the escaping particle. For l≠0, formulae have also been given but it is shown that these are inaccurate. In order to discuss the case of non-zero values of l, the one-body theory with a rectangular potential well is treated rigorously. A suitable integral representation of the confluent hypergeometric function is developed by the method of steepest descent into an asymptotic series which represents the solution of the wave equation outside the nucleus for l=0. Solutions for l≠0 are obtained from this by recursion operators. Boundary conditions at the nuclear radius give two equations linking the radius of the nucleus and the depth of its potential hole with the decay constant and the energy of the emitted α-particles. The usefulness and reliability for this problem of the one-body model are examined (Section 6). A re-evaluation of nuclear radii has been performed and the results are tabulated. The possibility that the decay constant, for fixed energy, is not a monotonic function of l but has an initial rise is discussed. A quantitative study is made of the λ−Eα relationships in a few α-ray spectra with "fine structure."
Using the collective model in the strong coupling scheme, it is ; established that the reduced width of a favored alpha transition with single ; angular momentum for an odd-mass nucleus of mass number A is proportional to the ; reduced width of the favored alpha transition with the same angular momentum ; for the even-even nucleus A-1, if both nuclei have the same deformation. This ; makes it possible to express the alpha intensities of odd-mass nuclei in terms ; of the reduced widths of the alpha transitions in even-even nuclei. Values of ; the latter widths extracted by Rasmussen from experimental data were used to ; calculate intensities of favored alpha transitions of odd-mass nuclei and ; relative intensities of their compenents of given angular momentum. The ; theoretical results are in good agreement with experimental data. (auth);
DOI:https://doi.org/10.1103/RevModPhys.37.336
The emission of 10Be in ternary cold neutronless spontaneous fission of 252Cf is observed with the Gammasphere consisting of 72 detectors. The gamma ray corresponding to the decay of the first 2+ state in 10Be is observed in coincidence with the gamma rays of the fission partners of 96Sr and 146Ba. The yield to the first excited state of 10Be in the 96Sr-146Ba split is the order of 4.0×10-4 per 100 fission events.
The eigenstates of a Woods-Saxon axially deformed potential are calculated by solving the corresponding coupled channel Schr{umlt o}dinger equation with outgoing boundary condition at large distances, thus providing both the bound states and the outgoing resonances. The level energies and conditions for bound states to become narrow resonances, and vice versa, are investigated as a function of the deformation. These studies may be important to interpret future measurements with nuclei close to the nucleon drip line. {copyright} {ital 1997} {ital The American Physical Society}
α decay through a deformed potential barrier produces significant mixing of angular momenta when mapped from the nuclear interior to the outside. Using experimental branching ratios and either semiclassical or coupled-channels transmission matrices, we have found that there is a set of internal amplitudes which is essentially constant for all even-even actinide nuclei. These same amplitudes also give good results for the known anisotropic α-particle emission of the favored decays of odd nuclei in the same mass region.
A microscopic description of heavy-cluster decay using a single particle basis suited to describe the continuum part of the spectrum is presented. The role of pairing correlations and quadrupole deformations is analyzed. The stability of the results as a function of the number of major shells included in the calculation is attested. {copyright} {ital 1997} {ital The American Physical Society}
Hindrance factors corresponding to {alpha} decay from two quasiparticle isomeric high K states are evaluated in superheavy nuclei. We found that the hindrance factors are very sensitive to the deformations and, therefore, they may constitute a powerful tool to extract spectroscopic information in these nuclei. The hindrance factors turn out to be very large, specially for nonaligned configurations. This indicates that if one of such states is reached the parent nucleus may become isomeric. It is also possible that {alpha} decay may not proceed through ground state to ground state chains but rather through excited states.
Using a single-particle basis with good angular momentum, α decay processes from deformed nuclei are described microscopically. The influence of the quadrupole deformation upon transitions to the ground state and to excited states is studied. It is found that the long-range proton-neutron interaction strongly affects the hindrance factor corresponding to the decay to excited states. {copyright} {ital 1997} {ital The American Physical Society}
The recently discovered cluster radioactivities, intermediate decays between alpha decay and fission, i.e. spontaneous emission of carbon, oxygen, neon, magnesium and silicon, are discussed in terms of two extreme approaches: as alpha-decay-like or fission-like theories. It is shown that the phenomenological formulations, which reproduce the present experimental data well, give little information on nuclear structure but are very useful in predicting new decay modes. Microscopic descriptions are limited only to very simple cases. A new interpretation of cluster decays as a new large-amplitude collective motion is presented, which leads to a new coexistence model consisting of the usual shell model and a cluster-like model describing a soliton moving on the nuclear surface. The corresponding amplitudes describe the experimental spectroscopic factors excellently. In addition, cold fission as the emission of deformed fragments and cold fusion as the inverse process of decay for the synthesis of new elements are reviewed. A new road to the true spherical superheavy elements is indicated.
The derivation of the width for ..cap alpha..-decay is examined with particular emphasis on methods which do not involve arbitrary channel radii. A new method of treating the initial decaying states is introduced and the use of ambiguous phenomenological potentials is avoided. This method yields consistent and acceptable quantitative results for the g.s. to g.s. (ground state) transitions in even polonium isotopes and for the branching ratio for the decay of the isomeric state ²¹²/sup m/Po.
Total energies are calculated for 138 even-even nuclei with proton numbers Z = 58-74 and for spins varying between I = 0 (ground states) and the maximum value corresponding to the fission limit induced by rotation. Calculations are based on the Strutinsky method and the cranking approximation. The results are presented in the form of two-dimensional maps of the total nuclear energy as a function of the quadrupole deformations, βâ and γ, and spin. At each spin value and at each (βâ, γ) point the energies are minimized with respect to the hexadecapole deformation, βâ. The macroscopic energy term in the Strutinsky formula corresponds to the folded-Yukawa plus exponential formulation proposed by Moeller and nix. The microscopic energy term was calculated using the universal Woods-Saxon approach. Both terms were tested to ascertain that they reproduce the observed high-spin behavior of nuclei in the mass range considered.
A liquid-drop-type mass formula with deformation and shell corrections and few free parameters is presented. The shell correction is a simple analytical expression of the equilibrium deformation of the nucleus. The formula is applied to nuclei having proton and neutron numbers greater than 50. In the fit to all experimental masses the root-mean-square deviation is 0.55 milli mass units (mu). copyright 1988 Academic Press, Inc.
A boson model for the major collective bands based on the coherent state formalism is applied to the shape transitional Pt isotopes 190 and 192. A very reasonable description of both spectra and the electromagnetic quadrupole transition rates is obtained.
We formulate a microscopic description of charge-changing excitations, including dynamical mixture of one- and two-phonon states. The present formalism generalizes our previous microscopic anharmonic vibrator approach (MAVA), designed to treat charge-conserving two- and four-quasiparticle degrees of freedom and their mixing. In the present formalism, the proton–neutron MAVA, pnMAVA, the main building blocks are the quasiparticle random-phase approximation (QRPA) phonons in the charge-conserving channels, and the proton–neutron QRPA (pnQRPA) phonons in the charge-changing channels. The QRPA and pnQRPA equations are directly used in deriving the equations of motion for two-phonon states. The beta-decay Gamow–Teller matrix elements, connecting the 1+ states of a double-odd nucleus with two-phonon 0+ and 2+ states of the neighboring double-even nucleus, contain contributions from the metric matrix and scattering terms of the beta-decay operator. The pnMAVA has been applied to predict β−-decay and β+-decay amplitudes for transitions feeding the one- and two-phonon states in the palladium isotopes 106,108Pd. A fair agreement with the data has been achieved.
Theoretical approaches to deformed proton emitters are briefly reviewed. The newly developed non-adiabatic method based on the coupled-channel Schrodinger equation with Gamow states is used to study the competition between proton emission and rotation. Calculations are performed for several experimentally seen, non-spherical nuclei beyond the proton dripline. By comparing theory and experiment, we are able to characterize the angular momentum content of the observed narrow resonance.
The most commonly used implementation of the WKB method for calculating spherical decay rates is shown to be unsatisfactory, and some improvements are suggested. Also, the smaller than expected experimental spectroscopic factor for the 1d(3/2) state in Lu-151 can be explained by particle-core coupling to a weakly deformed oblate shape. The Oh(11/2) State in Lu-151 is also briefly considered.