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PHYSICAL REVIEW C 91, 047301 (2015)
Microscopic investigations of α emitters close to the N = Z line
J. P. Maharana,1,* A. Bhagwat,2 and Y. K. Gambhir3,4
1
Computer Centre, IIT-Bombay, Powai, Mumbai-400076, India
2
UM-DAE Centre for Excellence in Basic Sciences, Mumbai 400 098, India
3
Department of Physics, IIT-Bombay, Powai, Mumbai-400076, India
4
Manipal Centre for Natural Sciences, Manipal University, Manipal 576104, Karnataka, India
(Received 19 January 2015; revised manuscript received 5 March 2015; published 23 April 2015)
The results of fully microscopic calculations for the α-decay half-lives of light α emitters close to the N = Z
line are presented. The ground-state properties are calculated in the relativistic mean field (RMF) approach which
reproduce the experiment well, as expected. The α-daughter nucleus potential is then calculated in the double
folding model using the density dependent M3Y nucleon-nucleon interaction and the RMF nuclear density
distributions. This potential in turn is used to calculate the α-decay half-lives in the WKB approximation. The
results closely agree with the corresponding values obtained by using the available phenomenological expressions
as well as with the experiment. The α-decay half-lives of a few Te, I, Xe, Cs, and Ba isotopes have also been
predicted.
DOI: 10.1103/PhysRevC.91.047301
PACS number(s): 21.60.−n, 23.60.+e, 25.85.Ca, 27.90.+b
α-decay studies play a crucial role in the nuclear structure
investigations. Natural α decays are observed in nuclei lying
in the medium, heavy, and superheavy nuclear regions. Nuclei
far away from the stability line are being produced and studied
using the existing radioactive ion beam (RIB) facilities. Some
of these nuclei may lie close to the drip lines. So far, the
doubly magic 100 Sn is the heaviest N = Z nucleus produced
and studied. The isotopes of the neighboring elements Te, I,
Xe, etc., are interesting because their lightest isotopes decay
through α emission. The Q value and the half-life of 105
52 Te
(being the lightest observed α emitter) and of 109
Xe
have
also
54
been measured and reported [1,2]. These cases are interesting
and unique because they lie very close to the N = Z line.
In this Brief Report we present a fully microscopic
calculation for the α-decay half-lives (τ1/2 ) of Te, I, Xe, Cs,
and Ba isotopes starting from their respective N = Z isotopes.
For completeness, similar results for the other relevant nuclei
(lying between Z = 50 and 82) are also presented.
The calculations proceed in three steps: (1) determination
of ground-state properties, (2) calculation of double folded
nuclear and Coulomb potentials, and (3) evaluation of halflives within the WKB approximation. As the first step,
the highly successful and well-tested relativistic mean field
(RMF) theory [3–5] is used to calculate the ground-state
properties such as binding energies, deformation parameters
(β), radii, and one- and two-nucleon separation energies of the
relevant nuclei. The explicit calculations require a Lagrangian
parameter set. Here, we employ the most frequently used
Lagrangian parameter set NL3 [6]. Further details of the
calculations (treatment of pairing, choice of basis, etc.) can
be found in Ref. [7].
Next, the α- daughter nucleus potential is
calculated using the double folding procedure (tρρ
*
[email protected]
0556-2813/2015/91(4)/047301(4)
approximation):
D (rD )d rα d rD .
VαD (R) = ρα (rα )v(|rD − rα + R|)ρ
(1)
Here, R is the distance between the centers of masses of α and
the daughter nucleus (D) and ρα (ρD ) is the density distribution
of α (daughter). The interaction v is the nucleon-nucleon
interaction, which is taken to be the density dependent M3Y
interaction [8–10] for the nuclear potential (VN ), whereas the
usual Coulomb interaction is used to calculate the Coulomb
potential (Vc ). The deformation effects are expected to play
an important role in the penetration factor. The groundstate neutron and proton density distributions obtained in
the RMF framework are deformed, in general. Here, the
L = 0 (spherical) components of the density distributions
are projected out, to be used in the computation of double
folding potentials and hence the half-lives. This simplified
procedure seems to work well, as is evident from the agreement
achieved between the calculated and experimental half-lives
of superheavy nuclei against α decay [7,11,12]. The density
distribution of the α particle used here is of conventional
Gaussian shape.
Finally, the α-decay half-lives are calculated in the WKB
approximation using the total α-daughter nucleus potential
obtained in the second step. The half-life of the parent nucleus
against α decay is given by
τ1/2 =
ln 2
(1 + eK ),
νo
(2)
where νo is the assault frequency. The exponent K is the action
integral, expressed as
2 Rb K=
2μ(VN (r) + Vc (r) − Q) dr.
(3)
Ra
The classical turning points Ra and Rb are obtained by
requiring that the integrand in the expression for K vanishes.
The assault frequency used in the present work is given
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©2015 American Physical Society
BRIEF REPORTS
PHYSICAL REVIEW C 91, 047301 (2015)
FIG. 1. Neutron skin thickness (rn − rp ) and charge radius (rc ) in
fm of the α emitters near the N = Z line.
by [7,12]
1
νo =
2R
2E
,
Mα
(4)
with R = 1.2A1/3 , E is the energy of α corrected for recoil,
and Mα is the mass of α. For comparison, the decay halflives are also calculated using the available phenomenological
expressions [13,14].
As expected, the calculated binding energies (Eb ), deformation parameters β, radii, one- and two-nucleon separation
energies, and the α-decay Q values agree well with the
corresponding experimental values where available.
The calculated Eb agree with the corresponding experimental values [15] within 0.25% and the charge radii [16] are
reproduced up to the second decimal place of a Fermi. The
results are shown in Figs. 1 and 4 for the nuclei close to the
N = Z line and for the other relevant nuclei in this region,
FIG. 2. Deformation parameter β of the α emitters near the N =
Z line.
respectively. The calculated neutron skin (the difference
between the neutron and proton rms radii, rn − rp ) are also
plotted (lower panel) in the respective figures. The calculated
deformations are found to be small, the maximum value being
around 0.2. These calculated values of β, as is evident from
Figs. 2 and 5, are very close to the corresponding values
obtained by M¨oller and Nix [17]. The calculated one- and twonucleon separation energies and the α-decay Q values are also
very well reproduced. This is gratifying, as these quantities are
the differences between large numbers. An error in any one
can easily be detrimental to the agreement achieved.
This accuracy in the calculated α-decay Q values is not
enough for the calculation of the α-decay half-lives (τ1/2 ).
It is known that a small (a few hundred keV) variation in
the α-decay Q value used in WKB method can cause an
order of magnitude difference in the calculated τ1/2 . Since
we intend to compare the calculated half-lives with the
corresponding values obtained by using the phenomenological
expressions [13,14] where only experimental Q values are
used, the experimental Q values (listed in [18]) have also been
used in all our calculations. The calculated α-decay half-lives
are shown in the Fig. 3 for the nuclei close to the N = Z line
along with the corresponding values obtained by using the
phenomenological expressions of Brown [13] and Dasgupta
and collaborators [14]. The experimental (Expt.) values (where
available) [18] are also shown in the same figure for comparison. It is clear from the figure that the calculated values of
half-lives are very close to the corresponding values obtained
with both the phenomenological expressions and reproduce the
experiment remarkably well. The phenomenological values of
Dasgupta agree with the experiment relatively better than those
of Brown. It is found that the predicted half-lives of 107 I as well
as the Cs and Ba isotopes near the N = Z line are more than 1
ms, which are long enough to be measured. The half-lives for
114,115
Xe turn out to be very long, hinting toward the possibility
of a small α-decay branching ratio.
Similar results for the decay half-lives for the relevant
heavier nuclei in this region are shown in Fig. 6. Clearly the
calculated and the phenomenological values are very close to
each other and both reproduce the experiment rather well. The
half-lives predicted by the Dasgupta formula agree relatively
better than those of Brown.
In summary, the α-decay properties of nuclei in the Sn
region have been studied systematically, within the framework
of RMF theory. The half-lives have been computed by
employing the double folded α-daughter interaction potential
within the WKB approximation. For comparison, the half-lives
are also computed by using two successful phenomenological
formulas [13,14]. The calculations are found to be close to the
experimental data. At a finer level, it is found that the present
calculations and the half-lives predicted by the Dasgupta
formula [14] are in tune with each other. The present work
predicts half-lives of some of the neutron-deficient I, Cs, and
Ba isotopes to be more than 1 ms. It is hoped that future
experiments may confirm some of these predictions.
The authors wish to thank M. Gupta, P. Ring, and R. Wyss
for their interest in this work.
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BRIEF REPORTS
PHYSICAL REVIEW C 91, 047301 (2015)
FIG. 3. Half-life τ1/2 of the α emitters near the N = Z line.
FIG. 4. Neutron skin thickness (rn − rp ) and charge radius (rc ) in fm of the representative cases of known heavy α emitters.
FIG. 5. Deformation parameter β of the representative cases of known heavy α emitters.
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BRIEF REPORTS
PHYSICAL REVIEW C 91, 047301 (2015)
FIG. 6. Half-life τ1/2 of the representative cases of known heavy α emitters.
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