Nuclear Sciences
β^{+} decay properties of A=100 isobars
Propiedades de la desintegración β^{+} de isóbaros con A = 100
Fatima Benrachi^{1
}^{*
}, Nadjet Laouet^{1
}
^{1}Laboratoire Physique Mathématique et Subatomique, Université Frères Mentouri Constantine 1 (Algeria).
Abstract
The estimation of spectroscopic properties of neutrondeficient nuclei in the A=100 tin mass region is needed for the understanding of the rpprocess path and the experimental exploration of the nuclear landscape. In order to evaluate some spectroscopic properties of the GamowTeller
β+
decay of neutron deficient isobars of A=100, we have performed shell model calculations by means of Oxbash nuclear structure code. The jj45pn valence space used consists of nine proton and neutron orbitals. The calculations included few valence holeproton and particleneutronin
π
g_{9/2} and vg_{7/2} orbitals respectively, in ^{100}Sn doubly magic core. Effective interaction deduced from CDBonn one is introduced taking into account the nuclear monopole effect in this mass region. The results are then compared with the available experimental data.
Key words: nuclear structure; strontium 100; monopoles; 0 codes; betaplus decay; neutrondeficient isotopes
Resumen
La estimación de las propiedadesespectroscópicas de los núcleosdeficientes en neutrones en la región de masas de estaño A = 100 es necesaria para la comprensióndelcaminodelproceso rp y la exploraciónexperimental de laestructura interna de los núcleos. Con el fin de evaluar algunas propiedades espectroscópicas de la desintegración
β
^{+}de GamowTeller en los isótopos de estañodeficientes en neutrones con A = 100, hemosrealizadocálculosdelmodelo de capas mediante el código de estructuranuclear de Oxbash. El espacio de valencia jj45pn utilizado consiste en nueve orbitales de protones y neutrones. Los cálculosincluyeronpocoshuecoprotóny partículaneutrón de valencia en orbitales
π
g_{9/2} y vg_{7/2}respectivamente, en un núcleo^{100}Sn doblementemágico. La interacciónefectivadeducida de CDBonn se introduceteniendo en cuenta el efectomonopolarnuclear en estaregión de masa. Los resultados se comparanluego con los datosexperimentales disponibles.
Palabrasclave: estructura nuclear; estroncio 100; monopolos; códigos 0; desintegración beta positiva; isótopos deficientes en neutrones
INTRODUCTION
The study of the nuclei with few hole protons and particle neutrons near the heaviest nucleus ^{100}Sn (N=Z) was the main of several theoretical and experimental works that aimed to give a global description of nuclear structure. These isobars are some of the best candidates for giving us opportunity to develop our understanding of nuclear structure due to the proximity of the magic numbers, the proton dripline and the end of the rpprocess.One of the most interesting aspects of the protonrich nuclei is show in some of the unusual features of their GamowTeller beta decays. Very scarce experimental informationin this region are available [^{1}]. The nuclear shape changes rapidly in this region and gives rise to longlived isomers with small excitation energies [^{2},^{3}]. With the development of advanced experimental tools such as isotopeseparation online (ISOL) and inbeam spectroscopy with large
γ
arrays including ancillary particle detectors and by exploiting spallation and heavy ion induced fusionevaporation reactions,many of the nuclei along the Z, N = 4850 isotonic and isotopic chains were be produced with relatively high rates [^{2},^{4},^{5}]. Investigations of testing effective interaction, exploring the evolution of the shell structure and beta decays have been performed in some works [^{3},^{6}^{10}].
The present analysis deals with three neutrondeﬁcient nuclei ^{100}In, ^{100}Cd and ^{100}Ag neighbours of ^{100}Sn core. Shell model calculations are been performed in the full model space that include the proton and neutron orbits below and above the closed shells Z= N = 50 respectively. This model space enables one to reproduce the shape of the GT strength distribution. In these nuclei, protonspartly fill the g_{9/2} orbit and the spinﬂip transformation
π
g_{9/2} → vg_{7/2} allowed by Pauli principle forms the GamowTeller (GT_{+}) states in the daughter nucleus. The main part of the strength turns out to lie within the Q_{EC} window, thus becoming accessible to the β^{+} decay [^{10}].
The shell model calculations of the holeparticlemultiplets in these nuclei were performed with code OXBASH [^{11}] supplementing available experimental data.
THEORETICAL FRAMEWORK
Nowadays, studies of radioactive decays are an integral part of investigations atomic nucleus in order to better understand the physical phenomena governing its behavior. The β decay process plays a major role in determining fundamental quantities such as the period, the mass or the energy of the excited levels helping to understand nuclear interaction and the characterization of populated states. Beta decay is a weak interaction process mediated by the well understood
τ
and
σ
τ
operators that govern the observed Fermi (F) and GamowTeller (GT) transitions, respectively. Under some experimental conditions, Charge Exchange (CE) reactions proceed by F and GT transitions, although mediated by the strong interaction. Consequently,
β
decay and CE processes can be very similar. The main advantage is that
β
decay provides absolute B(GT) strength values, whilst the CE reactions extend our knowledge to excitation energy regions above the Q_{β} value. Also, CE reactions can only be studied readily on stable target nuclei, whereas by deﬁnition
β
decay is from unstable nuclei. Beta decays of nuclei “southeast" of ^{100}Sn are characterized by large decay energies (Q_{EC}) and pure GamowTeller (GT) transitions transforming a g_{9/2} proton to a g_{7/2} neutron. These features make studies of these decays especially important to test the nuclear shell model in general and its predictions of the GT strength in particular. The experimentally derived quantity which can be directly compared to the theoretical predictions, is the
β
strength function. GamowTeller betadecay strengths provide relatively clear information about the structure of nuclear wave functions because the associated operators are simple and selective, coupling only a few singlenucleon orbits to each other. In addition, these operators do not couple to the firstorder admixtures of "excluded" configurations into the "active" space [^{12}].
The theoretically predicted GT strength is defined as the squared matrix element of the free
σ
τ
operator acting between the wave functions of the initial │i>and the final │f> states:
B(GT)=gA22Ji+1〈fGT^i〉2
(1)
GT±^=∑nσ^τ^±
The factor
gA=1.26
is the axialvector coupling constant of the weak interaction and J_{i} is the angular momentum of the initial state.
The quantity ft_{1/2} value of an allowed betadecay transition isrelated to GamowTeller force B(GT) (1). f is a phasespace integral that contains the lepton kinematics [^{13}] and t_{1/2} is the halflife.
The focus on the evolution of shell structure in nuclei was increased, in order to understand the appearance of new magic numbers. The eﬀect of the addition of nucleons on the single particle states can lead tospectroscopic properties nuclei that the realistic interactions derived from the NN force fail to reproduce. The interactions between the supposed inert core and the adding nucleons were be solved by the consideration of the monopole effect introduced by Poves and Zuker [^{14}]. Those were assumed to need drastic revisions of the realistic twobody potentials and proposed to separate the Hamiltonian of the system into two parts:
H=Hm+HM
(2)
where H_{M} denote the multipolepart of the Hamiltonian and
Hm
the monopole one.
Hm
is expressed in term of the average energies over the conﬁgurations of s and t orbits with T = 1 for protonproton and neutronneutron, and T = 0; 1 for protonneutron parts, see [^{15}^{16}] for more details.
VstT=∑J(2J+1)〈jsjtVstjsjt〉JT[1−(−1)J+Tδst]∑J(2J+1)[1−(−1)J+Tδst]
(3)
The two body matrix element
〈jsjtVstjsjt〉J
arisen from the interaction between the particles in the orbits s and t. Itcan be extracted from the proton and/or the neutron separating energies of neighbouring nuclei [^{17}]. The
VstT
(3) defined the diagonal 2b part of the monopole Hamiltonianis associated to a function of the average two body matrix elements (TBMEs).
The main aim of this paper is to present some calculations on nuclear properties of ^{100}In, ^{100}Cd and ^{100}Ag isobars, focusing attention on the levels schemes and the
β
^{+} decay properties. This study isrealized in the framework of the nuclear shell model by means of Oxbash nuclear structure code [^{11}].
SPECTROSCOPIC CALCULATIONS AND DISCUSSION
We have used the ^{100}Sn core and the full model space where proton holes are allowed to occupy the
π
(0f_{5/2}
^{1}, 1p_{3/2}
^{1}, 1p_{1/2}
^{1} and0g_{9/2}
^{1}
)
^{Z28}orbitals and the neutron particles occupy the v(0g_{7/2}, 1d_{5/2}, 1d_{3/2},2s_{1/2} and 1h_{11/2}) ^{N50} orbitals.The experimental single particle energy (pSHE and nSPE) values are takenfrom ^{99}In for protons and ^{101}Sn for neutrons [^{18}], respectively.
Considered the dependence mass factor (78/100)^{0.3}, the two body matrix elements (TBMEs) of the original interaction jj45apn from ^{
78
}
Ni mass region [^{19},^{20}] are scaled. The resulting TBMEs are used in order to calculate the monopole terms. Therefore
V1g9/22d5/20≈−430 keV
;
V1g9/21g9/21≈−110 keV
and
V2d5/22d5/21≈−20 keV
are used to modify
πυ(1g9/22d5/2)J=2,7T=0
;
ππ(1g9/21g9/2)J=0,8T=1
;
υυ(2d5/22d5/2)J=0,4T=1
TBMEs, respectively. These TBMEs are chosen basing on the energetic sequence of the single particle space. Using thenew interaction jj45m, some calculations are carried out in order to reproduce the nuclear properties of the three A=100 isobars cited above. The obtained levels schemes are showing on figure 1. The jj45m interaction reproduce the sequence of levels for eveneven ^{100}Cd nucleus. A good agreement between calculated energies and experiment ones are observed. This interaction cannot reproduce the spin of the experimental ground state for ^{100}In(6^{+}) and ^{100}Ag(5^{+}) isobars, it gives 5^{+} and 2^{+} for this state, respectively. Also, it can’t reproduce the sequence of levels for ^{100}Ag nucleus.
We concentrate on some GamowTeller(GT)
β
^{+} decayproperties between the states (
Δ
J =1) of these nucleiin the vicinity of ^{100}Sn, from (6^{+}) of ^{100}In to (7^{+} or 5^{+}) of ^{100}Cd and from (0^{+}) of ^{100}Cd to (1^{+}) of ^{100}Ag.
We have evaluated B(GT_{+}) strengths (Figure 2), halflives (Table 1) and theoretical occupancy changes (Figure 3) related to the
β
decays of ^{100}In into ^{100}Cd and ^{100}In into ^{100}Ag, with the standard quenching factor of 0.77 for
σ
τ
operator measuring the occupancy of the active particles (model space) in the exact wavefunctions.
The shape of the GT strength distribution (fig. 2(a)) up 9400 keV of excitation energyin ^{100}Cd is located in a large Q_{E.C} window (9880 keV value) and presented a broad symmetric peak centered around 6 MeV. In addition, a small peak at about 9 MeVcan be seen. The calculations were performed over thirty states. Then, the complete distribution of B(GT) up 3586 keV of excitation energy of the (1^{+}) first ten states in ^{100}Ag is located in two narrow peaks centered at about 800 and 2800 MeV and limited by 3963 keV Q_{E.C.}small value.The total GT strengths 8.64 and 9.93are practically the same in both decays.While, the experimental value of ^{100}In is 3.9(9) [^{10}]. But, the calculated T_{1/2} values are far from the experiment onesby a factor ten.
Table l T_{1/2} β^{+} decay calculated of ^{100}In and ^{100}Cd nuclei andthe experimental ones
Nucleus 
T_{1/2} (s) Exp 
T_{1/2} (s) Cal 
Q_{
β+
} (MeV) 
∑B(GT)

^{
100
}
In

5,8 ± 0,2 
0,57 
8,86 
8,64061 
^{
100
}
Cd

49,1 ± 0,5 
2,61 
2,921 
9,9277 
Figure 3 shows that theoretical occupancy changes (with
Δ
J =1) of valence nucleons in the states of parent and daughter nuclei are maximum of proton hole1g_{9/2} and neutron particle1g_{7/2} orbitals respectively. A small contribution of the proton orbitals 2p_{3/2} and 2p_{1/2} is obtained in the case of ^{100}Cd decay. The GT decay towards ^{100}Cd will populate the two quasi particle (2qp) configuration vg_{7/2}².
SUMMARY
This study is based on the energetic spectra and GamowTeller
β
^{+} decay properties calculations for nuclei near ^{100}Sn, with few hole protons and particle neutrons in their valence spaces. The calculations are realized in the framework of the nuclear shell model, by means of Oxbash nuclear structure code. Using the jj45apn original interaction of the code, we carried out some modifications based on the monopole effect to get jj45m new interaction.This interaction reproduce the energetic spectrum for ^{100}Cd nucleus, it can’t reproduce theground statefor oddodd ^{100}In and ^{100}Ag isobars and the sequence of levels for ^{100}Ag nucleus.The complete distributions of B(GT) strengths of excitation states in daughter nuclei are located in centered peaks and limited by Q_{E.C}. values.Their values are practically the same in both decaysand different from experimental value for^{100}In. The calculated T_{1/2} valuesobtained with the standard quenching factor of 0.77 for
σ
τ
operator related to the
β
^{+} decay of ^{100}In and ^{100}Cd are far from the experiment ones by a factor tenand depend substantially on the model used.
Acknowledgement
Authors would like to thank the organizers of LASNPA&WONPNURT 2017 for the organization and the support provided during the symposium.
Special thanks are owed to B. A. Brown, for his help in providing us the Oxbash code (Windows Version), and to M. H. Jensen, for the documents and the information provided about the interaction jj45apn.
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