Role of nuclear deformations and proximity interactions in

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Eur. Phys. J. A (2014) 50: 175
DOI 10.1140/epja/i2014-14175-9
Role of nuclear deformations and proximity
interactions in heavy particle radioactivity
Gudveen Sawhney, Kirandeep Sandhu, Manoj K. Sharma and Raj K. Gupta
Eur. Phys. J. A (2014) 50: 175
DOI 10.1140/epja/i2014-14175-9
THE EUROPEAN
PHYSICAL JOURNAL A
Regular Article – Theoretical Physics
Role of nuclear deformations and proximity interactions in heavy
particle radioactivity
Gudveen Sawhney1,a , Kirandeep Sandhu2 , Manoj K. Sharma2 , and Raj K. Gupta1
1
2
Department of Physics, Panjab University, Chandigarh-160014, India
School of Physics and Materials Science, Thapar University, Patiala-147004, India
Received: 26 September 2014 / Revised: 7 November 2014
c Società Italiana di Fisica / Springer-Verlag 2014
Published online: 27 November 2014 – Communicated by M. Hjorth-Jensen
Abstract. Based on the preformed cluster model (PCM), we have extended our earlier study on cluster
decays of heavy parent nuclei to analyze the effects of different nuclear proximity potentials in the groundstate clusterization of superheavy nuclei with Z = 113, 115 and 117. In order to look for the possible role
of deformations, calculations are performed for spherical as well as β2 -deformed choices of fragmentation.
The relevance of “hot compact” over “cold elongated” configurations due to orientations is also explored,
in addition to the role of Q value and angular momentum ℓ effects. As the PCM is based on collective
clusterization picture, the preformation and penetration probabilities get modified considerably, and hence
do so the decay constants and half-lives of the clusters, with the use of different nuclear proximity potentials.
The comparative importance of nuclear proximity potentials Prox-1977 and Prox-2000 is analyzed and
the calculated decay half-lives in the framework of PCM are compared with the recent predictions of
the analytical super-asymmetric fission model (ASAFM). The possible role of shell corrections is also
investigated for understanding the dynamics of heavy particle radioactivity. Finally, the potential energy
surfaces are compared for different proton and neutron magic numbers in superheavy mass region.
1 Introduction
Apart from the three basic decay modes (α-decay, β-decay
and γ-emission), the phenomenon of cluster radioactivity, predicted theoretically in 1980’s [1], is now well established. The first observation was the detection of 14 C
emitted from 223 Ra [2], which happens to have the highest
branching ratio 10−10 with respect to α-decay. Since then,
a number of cluster radioactive decays from heavy parent
nuclei with Z = 87 to 96 were observed, leading to 14 C,
18,20
O, 23 F, 22,24–26 Ne, 28,30 Mg, and 32,34 Si cluster emissions with their respective half-lives measured. The daughter nucleus for all these cluster decays in trans-lead region
is always 208 Pb (a doubly magic closed shell nucleus) or
its neighboring nucleus. This observation has been verified in almost all experimental and theoretical studies,
and hence consequently one may presume that the shell
structure of daughter nucleus or the Q-value [= B.E.parent (B.E.daughter + B.E.cluster )] plays a very significant role in
the cluster radioactivity process.
In order to understand any physical phenomenon or
process, one needs to perform a detailed theoretical anala
e-mail: [email protected]
ysis. In view of this, different approaches, namely i) the
unified fission model (UFM), such as the analytic super
asymmetric fission model (ASAFM) [1, 3] and ii) the preformed cluster model (PCM) [4–7] have been advanced
for the possible explanations of cluster dynamics. These
models are found equally successful in reproducing the
data on observed exotic cluster decays. The knowledge of
interaction potential acts as an essential ingredient in all
models, which consists of long-range plus short-range potentials. The long-range part of the interaction potential is
determined by the Coulomb and centrifugal interactions,
whereas the short-range nuclear part is not fully understood as yet. Therefore, an appropriate choice of nuclear
potential is extremely desirable for a better understanding
of this rare nuclear phenomenon.
Because of the instability of nuclei in the superheavy
element (SHE) region, no measurement of cluster decays
is available to date. Stressing the importance of shell effects, Gupta and collaborators [8, 9] were the first to look
for the possible branching of the α-decay to some (theoretically) most probable heavy cluster decays referring to
some new or known magic daughters (or clusters) in the
exotic cluster radioactivity process. This study, using the
Page 2 of 11
PCM, was carried out for even 110 < Z < 118 superheavy
nuclei [8, 9], indicating the interesting possibility of 14 C
decay of 281
112 Cn giving rise to a deformed (nearly) magic
Z = 106, N = 161 267 Sg daughter (N = 162 is established
as a deformed magic shell) or a Z = 20 magic 48–51 Ca cluster from any parent nucleus in the α-decay chain of a SHE.
More recently, the ASAFM is also used [10, 11] to explore
the possibility of cluster emission in SHE mass region. The
authors of [10, 11] explored the emergence of heavy particle radioactivity (HPR) from SHE’s with Z > 110 via
spontaneous emission of heavy clusters of Z > 28 with
corresponding daughter nuclei around the doubly magic
208
Pb. With an aim to address the HPR within collective clusterization process in PCM, in the following, we
have investigated the cluster decay in the ground states
of 278 113, 287−289 115 and 293,294 117 isotopes, leading to
doubly magic 208 Pb or its neighboring nuclei. These nuclei have been observed [12–14] in the evaporation channels produced in a cold fusion reaction using a beam of
70
Zn on 209 Bi target and hot fusion reactions induced by
48
Ca projectile with actinide targets 243 Am and 249 Bk,
respectively.
In our recent work with PCM [15, 16], the role of deformations and orientations is studied for the first time
in cluster decays of various radioactive nuclei from the
trans-lead region, particularly for those decaying to doubly closed spherical shell 208 Pb daughter nucleus and
then extended this study to the parent nuclei resulting
in daughters other than 208 Pb. Except for 14 C decays, the
effect of deformations up to quadrupole (β2 ) alone and
“optimum” orientations [17] are found enough to fit all
the experimental data on cluster decay half-lives. Later
it was observed in ref. [18] that the inclusion of higher
multipole deformations up to hexadecapole (β2 , β3 , β4 )
together with cold “compact” orientations [19] are essential for the decay of 14 C clusters. It is to be noted that the
nuclear proximity potential of Blocki et al. [20] is used in
these studies to understand the cluster dynamics. As already stated in the last paragraph above, based on the
above-mentioned PCM calculations [8, 9, 15, 16, 18] that
provide reasonable estimates for the observed cluster decay life times of heavy (Z > 87) parents, we have explored
in this paper, the phenomenon of clustering in SHE nuclei
with Z = 113, 115, and 117. The PCM finds its basis in
the well-known quantum mechanical fragmentation theory
where clusters of different sizes are considered to be preformed in the parent nucleus with different probabilities.
This inclusion of preformation probability enables us to
make significant observations about the nuclear structure
effects of the parent nucleus as well as its decay products.
Thus, the aim of the present work is to investigate the
most probable heavy cluster emissions, and their corresponding decay half-lives, from odd Z 278 113, 287−289 115
and 293,294 117 parents, taking proton magic number Z =
126 and neutron number N = 184 for the SHE region. Since the deformations and orientations of nuclei are
known to play a significant role in the context of cluster
decay process, these are included here up to quadrupole
deformations β2 and with “optimum” orientations θopt. of
Eur. Phys. J. A (2014) 50: 175
nuclei [17]. The calculations are done with in the framework of PCM, using two different versions of nuclear proximity potential, the Prox-1977 [20] and Prox-2000 [21],
having different isospin and asymmetry dependent parameters. A comparison of our calculated half-lives is made
with the predictions of ASAFM [10, 11], obtained by using different mass tables AME11 (Atomic Mass Evaluation), LiMaZe01 (Liran-Marinov-Zeldes) and KTUY05
(Koura-Tachibana-Uno-Yamada), emphasizing the applicability and validity of our formalism. To explore the
structural aspects for odd-Z nuclei in the SHE mass region, the effect of using different magic shells on heavy
particle radioactivity is also studied. In other words, a
comparative analysis of the possible shell closures with
Z = 114, 120, 126 and N = 184 is checked via the calculation of the fragmentation potentials, and hence the
preformation factor.
The paper is organized as follows: Section 2 gives a
brief account of the preformed cluster model (PCM) and
different versions of the proximity potential used. The calculations and results for the ground state decays of the
chosen parent nuclei are presented in sect. 3. Finally, the
conclusions drawn are discussed in sect. 4.
2 The preformed cluster model
The PCM has been described in detail in refs. [15, 16, 18],
and we give here only some relevant details. Following the
quantum mechanical fragmentation theory [22, 23], it is
worked out in terms of the collective coordinates of mass
and charge asymmetries,
η=
A1 − A2
A1 + A2
and
ηZ =
Z1 − Z2
,
Z1 + Z2
the relative separation R, and the multipole deformations
βλi and orientations θi (i = 1, 2) of daughter and cluster
nuclei. The decay constant λ or the decay half-life T1/2 in
PCM is defined as
λ=
ln 2
= P0 ν0 P.
T1/2
(1)
Here P0 is the cluster (and daughter) preformation
probability and P the barrier penetrability which refer, respectively, to the η and R-motions. ν0 is the barrier assault
frequency, which remain almost constant ∼ 1021 s−1 for all
the heavy cluster decays studied in the present work. P0
is the solution of stationary Schrödinger equation in η, at
a fixed R = Ra ,
∂
∂
1
h̄2
− + VR (η) ψ ω (η) = E ω ψ ω (η),
2 Bηη ∂η Bηη ∂η
(2)
which on proper normalization gives
P0 = Bηη |ψ[η(Ai )]|2 (2/A),
(3)
with i = 1 or 2 and ω = 0, 1, 2, 3, . . . . We are interested
here only in the ground-state solution (ω = 0).
Eur. Phys. J. A (2014) 50: 175
Page 3 of 11
for deformed and oriented nuclei (for details, refer to
refs. [24, 25]). Note that shell effects enter here mainly
through the ground state binding energies Bi (Ai , Zi ) [34,
35]. The deformation parameters of nuclei, βλi are taken
from the tables of Möller et al. [34], with the “optimum”
orientations taken from table 1 of ref. [17]. In the above
eqs. (4) and (5), αi is the angle between the symmetry axis
and the radius vector of the colliding nucleus, measured
in the clockwise direction from the symmetry axis, and
θi is the orientation angle between the nuclear symmetry
axis and the collision Z-axis, measured in the anticlockwise direction (see, e.g., fig. 1 of ref. [17]). For ground-state
decays, ℓ=0 is a good approximation [36].
The binding energy of a nucleus has been defined
within the Strutinsky renormalization procedure, as a sum
of liquid-drop energy (VLDM ) and shell corrections (δU ),
i.e., B(A, Z) = VLDM + δU . However the role of taking
δU = 0 is also analyzed (see the next section). The shell
corrections, according to the “empirical” formula of Myers
and Swiatecki [37], are given by
δU = C
Fig. 1. The scattering potential V (R) for the decay 294 117 →
Pb + 86 Br, showing barrier penetration in PCM, using both
Prox-1977 (solid line) and Prox-2000 (dotted line), for 86 Br
considered as deformed nucleus with “optimum” hot orientation angles θiopt. from table 1 of ref. [17].
208
= Rt (α, η) + ΔR,
(4)
where the η dependence of Ra is contained in Rt , and
ΔR is a parameter, assimilating the neck formation effects of two-centre shell model shape. This method of introducing the neck length parameter ΔR is also used in
the dynamical cluster decay model (DCM) of Gupta and
collaborators [24–29] and in the scission-point [30] and
saddle-point [31, 32] (statistical) fission models for decay
of a hot and rotating compound nucleus. The mass parameters Bηη , representing the kinetic energy part in eq. (2)
are the classical hydrodynamical masses [33].
The structure information of the decaying nucleus is
contained in P0 via the fragmentation potential VR (η) in
eq. (2), calculated as
VR (η) = −
2
[Bi (Ai , Zi )] + VC (R, Zi , βλi , θi )
2
(A/2) 3
− cA
1
3
,
(6)
3
F (X) =
5
−
5
5
3
Mi3 − Mi−1
(X − Mi−1 )
Mi − Mi−1
5
3
5
3
,
X 3 − Mi−1
5
(7)
with X = N or Z, Mi−1 < X < Mi and Mi as the magic
numbers 2, 8, 14 (or 20), 28, 50, 82, 126 (or 120 and 114)
for protons and 2, 8, 14 (or 20), 28, 50, 82, 126 and 184
for neutrons. The constants C = 5.8 MeV and c = 0.26.
The VLDM is taken from the semi-empirical mass formula of Seeger [38], defined as
2
VLDM (A, Z) = α(0)A + β(0)A 3
2
I +2|I |
η(0)
+ γ(0) − 1
A
A3
2.29
Z2
0.7636
+
−
1
−
1
2
1
R0 A 3
Z3
[R0 A 3 ]2
+δ(0)
f (Z, A)
3
A4
,
(8)
where
i=1
+VP (R, Ai , βλi , θi ) + Vℓ (R, Ai , βλi , θi ).
F (N ) + F (Z)
where
For Ra , the first turning point of the penetration path,
shown in fig. 1 for the decay 294 117 → 208 Pb + 86 Br, it is
postulated that [6]
Ra (η) = R1 (α1 ) + R2 (α2 ) + ΔR
I = aa (Z − N ),
aa = 1,
(5)
VC , VP , and Vℓ are, respectively, the Coulomb, nuclear
proximity, and angular momentum dependent potentials
and f (Z, A) = (−1, 0, 1), respectively, for even-even, evenodd, and odd-odd nuclei. For T = 0, Seeger [38] obtained the constants, by fitting all even-even nuclei and
Page 4 of 11
Eur. Phys. J. A (2014) 50: 175
488 odd-A nuclei available at that time, as
α(0) = −16.11 MeV,
β(0) = 20.21 MeV,
γ(0) = 20.65 MeV,
η(0) = 48.00 MeV,
with the pairing energy term from ref. [39],
δ(0) = 33.0 MeV.
The constants of VLDM (T = 0), particularly the bulk
constant α(0) and the proton-neutron asymmetry constant aa were adjusted by some of us [25,40] to get the experimental binding energies [35, 41] with shell corrections
determined from the “empirical” formula (6) [37]. Wherever the experimental data were not available, the theoretically estimated binding energies of Möller et al. [34] were
used. In the present calculations, the VLDM and δU for addressing the heavy particle radioactivity is worked out by
taking Z = 126 and N = 184 as possible magic numbers
in SHE region. Evidently, the δU changes if the magicity
at Z = 126 is changed to Z = 114 or 120 and hence accordingly the constants of VLDM have been re-fitted [42]
and used here to compare the preformation probability for
all the three cases of Z = 114, 120 and 126, N = 184.
The penetrability P in eq. (1) is the WKB integral
between the Ra and Rb , the first and second turning
points, respectively (see fig. 1). In other words, the tunneling begins at R = Ra and terminates at R = Rb , with
V (Rb ) = Q value for ground state decay. Thus, as per
fig. 1, the transmission probability P is divided in to three
processes (a) the penetrability Pi from Ra to Ri , b) the
(inner) de-excitation probability Wi at Ri , taken as unity,
i.e., Wi = 1 for heavy cluster-decays [43], and c) the penetrability Pb from Ri to Rb , giving
P = Pi Wi Pb ,
(9)
where Pi and Pb in the WKB approximation are defined
below:
1/2
2 Ri 2μ[V (R) − V (Ri )]
dR , (10)
Pi = exp −
h̄ Ra
2
Pb = exp −
h̄
Rb
Ri
1/2
2μ[V (R) − Q]
dR .
(11)
In eq. (5), the proximity potential VP for deformed and
oriented nuclei is given by
Vp (s0 ) = 4π R̄γbΦ(s0 ),
(12)
where s0 = R − R1 − R2 , the nuclear surface thickness
b = 0.99, and R̄ is the mean curvature radius (for details,
see ref. [44]). Φ in eq. (12) is the universal function, independent of the shapes of nuclei or the geometry of the
nuclear system, but depends on the minimum separation
Fig. 2. Fragmentation potential for the parent nucleus 294 117
using proximity potentials Prox-1977 and 2000 for β2 deformed
choice of nuclei with “optimum” θiopt. forming hot compact
(filled symbols) and cold elongated (open symbols) configuration, taking into account all possible fragments.
distance s0 . A brief details of the different nuclear proximity potentials used in the present calculations are as
follows:
a) Prox-1977: The universal function is given by Blocki
et al. [20] for this proximity potential,
⎧
1
⎪
⎨ − (s0 − 2.54)2 − 0.0852(s0 − 2.54)3 ,
2
Φ(s0 ) =
⎪
⎩ −3.437 exp − s0
0.75
(13)
respectively, for s0 ≤ 1.2511 and s0 ≥ 1.2511. The
surface energy constant used for this potential is
2 N −Z
γ = γ0 1 − ks
MeV fm−2 .
(14)
A
Here, N and Z are the total neutrons and protons
numbers, and the coefficients γ0 and ks were taken to
be 0.9517 MeV/fm2 and 1.7826, respectively.
b) Prox-2000: In this version of proximity potential, the
universal function is taken from Myers and Swiatecki [21], as
⎧
5
⎪
⎪
⎪
−0.1353
+
[cn /(n + 1)](2.5 − s0 )n+1
⎪
⎪
⎪
⎪
n=0
⎨
for 0 < s0 ≤ 2.5,
Φ(s0 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪ −0.09551 exp[(2.75 − s0 )/0.7176]
⎩
for s0 ≥ 2.5.
(15)
Eur. Phys. J. A (2014) 50: 175
Page 5 of 11
Fig. 3. Preformation probability P0 for the decay of 278 113, 289 115 and 294 117 nuclei using nuclear potentials Prox-1977 and
Prox-2000 for spherical (panels (a), (c), and (e)) and deformed (panels (b), (d) and (f)) choices of nuclei. The most probable
cluster corresponding to Pb daughter is pointed out with solid lines.
The values of different constants cn are c0 = −0.1886,
c1 = −0.2628, c2 = −0.15216, c3 = −0.04562, c4 =
0.069136, and c5 = −0.011454. For further details of
surface energy coefficient and nuclear charge radius,
etc., see ref. [21].
3 Calculations and discussion
In this section, the analysis of our previous cluster decay calculations, considered with the atomic number 2 <
Zcluster ≤ 20 within the PCM, is extended to heavier
max
clusters with Zcluster
= Zparent − 82, to learn about the
most probable cluster emitted across doubly magic daughter 208 Pb for the island of superheavy elements. The effects of deformations are considered up to quadrupole (β2 )
only in reference to “optimum” orientation. Different versions of proximity potentials are employed in this work to
study the heavy particle radioactivity (HPR) of 278 113,
287−289
115 and 293,294 117 systems. It is important to note
that, in reference to our earlier work on PCM [8, 9], the
magic numbers for the SHE region are taken as Z = 126,
N = 184. In order to estimate the cluster decay half lives
of various SHE nuclei, we started our calculations using
the nuclear proximity potential Prox-1977 for spherical as
well as quadrupole deformed choices of nuclei. Our calculated T1/2 values compare nicely with the predictions of
Poenaru et al. [10,11] for majority of the systems studied,
within β2 -deformation approach only; the same is not true
for spherical choice. Beside this, the choice of other proximity potential Prox-2000 is checked and interestingly we
are able to account for Poenaru et al. estimates of half lives
reasonably well within PCM using both spherical and deformed choices of fragmentation. The above observations
indicate that deformation effects are highly desirable to
account for the HPR emission within both proximity approaches. To investigate further the importance of deformations, in the following, we have carried out a comparative analysis of fragmentation path VR (η), preformation
probability P0 , and barrier penetrability P for the use of
both proximity potentials. In addition, the possible role of
different magic shells for SHE region is also investigated
in order to reveal useful information about the dynamics
involved in the context of heavy cluster emission process.
First of all, we look at the behavior of fragmentation
potentials illustrated in fig. 2 for the parent nucleus 294 117,
calculated for the fragments with quadrupole deformations (β2i ) and “optimum” orientations θiopt. of ref. [17].
The role of “hot compact” and “cold elongated” configurations of orientations are also worked out in this figure.
Page 6 of 11
Note that the orientations are uniquely fixed (optimized)
on the basis of the signs of β2i alone which manifests
in the form of “hot compact” and “cold elongated” configuration. However, for investigating the role of higherorder deformations “compact” orientations [19] should be
used instead of optimum orientations [17]. The calculations are done using nuclear proximities Prox-1977 and
Prox-2000 at internuclear separation distances (equivalently, first turning point of penetration) Ra = Rt +1.1 and
Ra = Rt + 0.9 fm, respectively. While the “hot compact”
configuration refers to smallest interaction radius and
highest barrier, the “cold elongate” corresponds to largest
interaction radius and lowest barrier. Despite the use of
different nuclear proximity potentials, our analysis of fig. 2
clearly shows that 86 Br is the most probable cluster (showing strongest minimum in potential energy surface) with
corresponding 208 Pb daughter for the choice of “hot” in
comparison to “cold” configuration. In other words, the
region for heavy particle radioactivity is more favorable,
and hence show a clear preference in the fragmentation
potential, for “hot compact” in comparison to “cold elongated” configuration. Following this prescription, in this
paper, further calculations are done by considering “optimum” orientations of “compact hot” configuration only.
The scattering potential in fig. 1 is illustrated for the
decay 294 117 → 208 Pb + 86 Br using nuclear potentials
Prox-1977 (solid lines) and Prox-2000 (dotted lines), plotted for the case of deformed choice of cluster and daughter nuclei. Both the potentials have different isospin and
asymmetry dependence. The choice of heavy cluster 86 Br
in fig. 1 is based on the minima in the fragmentation potential VR (η) and thus for the case of largest preformation
factor P0 , illustrated in fig. 3. In fig. 1 one can clearly see
significant changes in the barrier characteristics for the
use of different proximity potentials. The relative comparison of Prox-1977 and Prox-2000 suggests that the barrier height VB gets reduced and position RB increases in
the earlier Prox-1977 case, thereby affecting the tunneling
probability. Note that the calculated decay half-life T1/2
in PCM, defined by eq. (1), depends on penetrability P ,
and hence considerably on the type of interaction potential used.
Knowing that, in PCM, T1/2 is a combined effect of
both P0 and P , fig. 3 presents the calculated preformation
probability P0 as a function of fragment mass number (Ai ,
i = 1, 2) for the parent nuclei 278 113, 289 115, and 294 117
using both the proximity potentials. The specific role of
deformations is studied in this figure, where the calculations have been done using spherical consideration as well
as for β2 -deformed choice of fragments. In order to look
for the exclusive role of interaction potentials, the calculations have been made at the same neck length parameter ΔR, chosen in reference to Prox-2000 and is clearly
depicted in fig. 3 for 278 113, 289 115, and 294 117 nuclei.
However for the other parents 287 115, 288 115, and 293 117,
the ΔR values used are 0.45, 0.15, 0.28 fm and 0.96, 0.95,
1.005 fm, respectively, for spherical and deformed choices
of nuclei. It is clearly evident from fig. 3 that the inclusion
of deformation and orientation effects of the decay fragments change the potential energy surface (PES) which
Eur. Phys. J. A (2014) 50: 175
Fig. 4. Difference of the potential V (Ra ) at first turning point
Ra and the Q-value plotted as a function of parent nucleus
mass (A Z) using Prox-1977 and Prox-2000. Calculations are
done at a fixed value of ΔR taken in reference to fig. 3. The
emitted clusters are also mentioned together with the parent
mass.
in turn affects the decay constant and half life time accordingly. A solid vertical line is drawn in order to point
out the most probable 73 Ga, 83 As and 86 Br clusters and
their respective 205 Pb, 206 Pb and 208 Pb daughters emitted from 278 113, 289 115 and 294 117 parent nuclei. Interestingly, the emergence of 86 Kr cluster and its complement
192
Ir seems to be more prominent in the fragmentation of
278
113, irrespective of spherical or deformed configuration
(see figs. 3(a) and (b)). However, if one refers only to Pb
radioactivity, then 73 Ga may be considered as the most
preferred cluster. It is relevant to note here that the maximum preformation probability P0 corresponds to minima
in the fragmentation potential, illustrated as an example
for the 86 Br cluster emitted from 294 117 parent nucleus
in fig. 2. We notice three important results from fig. 3:
i) Except for the change in the magnitude, no noticeable
change in the structure of PES is observed while going
from Prox-1977 to Prox-2000 with spherical as well as deformed fragments; ii) the status of the preferred cluster
remains intact, independent of the choice of nuclear proximity potential and deformation used; iii) the favorable
clusters shift toward the heavier mass region with the increase in mass number of parent nuclei.
Eur. Phys. J. A (2014) 50: 175
Page 7 of 11
Fig. 5. The decay half-lives calculated on the basis of PCM for the most favored cluster decays of the parents with Z = 113,
115 and 117, taking the nuclei as (a) spherical and (b) with deformations β2i alone, compared with those calculated on the basis
of ASAFM [10, 11].
Figure 4 shows a comparison of the difference of potentials at first turning point i.e., V (Ra ), and the Q-values
calculated by using binding energies from refs. [34, 35] as
a function of masses of parent nuclei. It is observed that
the potential V (Ra ), calculated by using Prox-1977, comes
out to be less than the Q-value of the decay for majority
of the chosen nuclei, which means to indicate that relatively larger value of ΔR are required for Prox-1977 to
address the T1/2 data. One can clearly see that the difference is positive for Prox-2000, irrespective of the spherical
or deformed consideration. For Prox-1977, fig. 4(a) clearly
depicts that this difference is negative except for 278 113
nucleus for the spherical choice of nuclei. With the inclusion of deformations of the decaying fragments (fig. 4(b)),
the deviation of V (Ra ) from the Q-value improves significantly for all nuclei except for Z = 117 isotopes. Hence,
in order to calculate the half-lives using Prox-1977 potential, the entrance point has been modified (by including
neck of ∼ 1 fm) for those nuclei where V (Ra ) is less than
Q-value.
Figures 5(a) and (b) show the calculated logarithms
of half-life times log10 T1/2 for the best preformed clusters emitted from 278 113, 287−289 115, and 293,294 117 nuclei using nuclear proximities Prox-1977 and Prox-2000
for spherical as well as quadrupole deformed choices of
nuclei. The emitted clusters are also marked in the figure.
The Q-value of each cluster decay in PCM is calculated
by using the binding energies of Audi et al. [35]. In other
words, essentially the experimental binding energies Bexpt
are used. Wherever Bexpt were not available, the theoretical estimates of Möller Nix [34] are used. Alternatively, the
binding energies could also be calculated by using experimental atomic mass evaluation (AME12) mass tables [45]
and/or semi-empirical nuclear mass formula based on the
Table 1. Comparison of standard rms deviation (σ) of
PCM calculated cluster decay half lives from the results of
ASAFM [10, 11] obtained by using different AME11 [48], LiMaZe01 [49,50] and KTUY05 [51] mass tables. Calculations are
made using nuclear proximity potentials Prox-1977 and Prox2000, for cases of spherical and deformation β2i alone with “hot
compact” orientations θiopt. .
Spherical
Deformed(β2i , θiopt. )
Prox-1977
Prox-2000
Prox-1977
Prox-2000
AME11
19.94
0.1856
12.405
0.1305
LiMaZe01
17.308
3.084
10.080
3.150
KTUY05
20.139
0.7502
12.672
0.8057
macroscopic-microscopic method [46,47] where the isospin
and mass dependence of the parameters is investigated
with the Skyrme energy density functional. However, the
comparison of our PCM calculations are made with the
recent results of Poenaru et al. [10, 11] based on the
ASAFM with the binding energies for Q-values taken from
the Atomic Mass Evaluation (AME11) [48] experimental
mass tables as well as the calculated Liran-Marinov-Zeldes
(LiMaZe01) [49, 50] and Koura-Tachibana-Uno-Yamada
(KTUY05) [51] mass tables. Note that ΔR is the only
parameter of the model given by eq. (4) and depicted in
fig. 7 (discussed later) for the best possible estimates of
decay half-lives w.r.t. the predictions given by ASAFM.
It is clearly evident from fig. 5 that PCM calculated halflives with deformed choices of fragments are in good agreement with the ASAFM predictions for both Prox-1977 and
Prox-2000. However, the same is not true for spherical
case. For the sake of comparison, the standard rms devi-
Page 8 of 11
Eur. Phys. J. A (2014) 50: 175
Table 2. PCM calculated preformation probability P0 and penetrability P for the most favorable clusters emitted from various
isotopes of odd Z = 113, 115 and 117 nuclei, for cases of spherical and β2i deformations alone with hot “optimum” orientations
of nuclei. The assault frequency ν0 ∼ 1021 s−1 for each case.
Parent
Preformation probability P0
Prox-1977
Penetration probability P
Prox-2000
Prox-1977
Prox-2000
Sph.
β2i
Sph.
β2i
Sph.
β2i
Sph.
β2i
278
113
1.92 × 10−17
3.41 × 10−29
3.05 × 10−28
1.87 × 10−27
4.06 × 10−5
6.62 × 10−7
3.13 × 10−7
1.58 × 10−7
287
115
5.51 × 10−9
2.05 × 10−25
4.41 × 10−24
2.60 × 10−25
2.39 × 10−1
9.62 × 10−3
2.64 × 10−5
4.59 × 10−4
288
115
2.70 × 10−8
3.19 × 10−28
1.42 × 10−23
1.15 × 10−26
2.44 × 10−1
6.11 × 10−3
2.58 × 10−7
3.15 × 10−4
289
115
4.93 × 10−8
6.19 × 10−22
1.40 × 10−21
4.09 × 10−24
2.59 × 10−1
1.72 × 10−2
1.50 × 10−6
5.71 × 10−4
293
117
1.68 × 10−6
2.94 × 10−4
2.78 × 10−18
1.78 × 10−20
3.80 × 10−1
3.14 × 10−1
1.32 × 10−5
1.70 × 10−3
294
117
1.15 × 10−7
3.28 × 10−6
2.26 × 10−22
1.00 × 10−23
3.60 × 10−1
2.19 × 10−1
4.92 × 10−5
7.22 × 10−4
ation is also calculated and listed in table 1 which comes
out to be minimum for Prox-2000 with both spherical and
deformed configurations. However, the nuclear proximity
Prox-1977 deviates in a big way in reference to the results
of ASAFM [10, 11] obtained by using different mass tables. Furthermore, we notice in fig. 5 that the calculated
numbers for both the models present an interesting result,
i.e., the cluster decay half life for 278 113 is very high which
means that this parent nucleus is very stable against 73 Ga
cluster decay. However, other heavier clusters 83 As and
85,86
Br are predicted to decay with comparatively smaller
half-lives and hence present themselves as further interesting cases for cluster decay measurements.
Table 2 gives the relevant details of cluster preformation (P0 ) and penetration (P ) probabilities for the preferred cluster decays of the considered parents in SHE
region using spherical and deformed choices of fragmentation path. It is clearly observed from table 2 that the
choice of nuclear potential (Prox-1977 or Prox-2000) affects both the preformation probability and penetrability
significantly. The strong dependence of the half-life time
on the Q-value is further illustrated in fig. 6(a) where the
calculated Q-values are plotted as a function of parent
nuclear masses A Z. One can clearly see from the comparison of fig. 5 and fig. 6(a) that when the Q-value is
small, the log10 T1/2 is large and vice versa for all the
predicted heavy clusters emitted with various isotopes of
Pb daughter. This implies that Q-values of decay fragments play a important role in deciding the clusterization
process in SHE region which, in turn, seem to suggest
that heavy particle radioactivity behaves similar to normal cluster emission process. Since smaller Q-value should
also mean a relative decrease in the penetrability P , the
same is shown to be the case with our calculations presented in fig. 6.
Fig. 6. (a) The PCM calculated Q-values plotted against cluster configurations from various parent nuclei in the ground
state. (b) Penetration probability P using Prox-1977 and Prox2000 potentials for deformed (up to β2 ) choice of cluster and
daughter nuclei.
Eur. Phys. J. A (2014) 50: 175
Page 9 of 11
Fig. 7. PCM predicted decay half-lives for the most probable cluster emitted from 278 113,
over a range of neck-length parameter ΔR for the use of β2 -deformed decay products.
It is to be noted that PCM calculations are sensitive
to the choice of neck-length parameter ΔR which in turn,
plays an important role in the calculation of decay halflife. Therefore, an attempt have been made in fig. 7 to
predict a range of cluster decay half-lives for the considered parents with effects of deformations and orientations
included. Also shown in this figure are the predictions of
Poenaru et al. [10,11] based on ASAFM, obtained by using
different (AME11, LiMaZe01 and KTUY05) mass tables.
We notice that half-life time is influenced by the variation
in ΔR; the only parameter of the model which decides
the entry point of barrier penetration as well as of the
cluster’s preformation. At a given ΔR value, log10 T1/2 is
relatively larger for the choice of Prox-2000 in comparison
to Prox-1977. Apart from providing nice agreement with
the results of ASAFM, the PCM predicted half-lives over
a wide range of neck values could provide a testing ground
for the future experiments on cluster decay studies.
Although we are considering here the ground-state decays of heavy clusters from various radioactive parent nuclei and the role of angular momentum is anticipated to be
minimal. However, in view of the fact that the deformed
cluster and daughter are considered to be preborn within
the PCM, we have investigated in fig. 8 the role of the
287−289
115 and
293,294
117 isotopes
Fig. 8. The fragmentation potential V (η) calculated for different ℓ-values using Prox-2000 for the decaying parent nucleus 289 115, taking in to account the β2i deformations and
hot “compact” orientations.
Page 10 of 11
Eur. Phys. J. A (2014) 50: 175
Fig. 9. (a) Role of shell corrections δU , illustrated for 289 115 by plotting the preformation factor P0 as a function of both the
cluster and daughter masses, using Z = 126 and N = 184. (b) The same as (a), taking the magic numbers for the SHE region,
respectively, as Z = 126, 120 or 114 and N = 184 for the use of Prox-2000. ΔR = 0.98 fm, having kept the same in all the three
cases for comparison.
angular momentum in the decay of 289 115. As expected,
ℓ effects up to 12h̄ do not seem to influence the fragmentation potential and hence a non-dependence on angular
momentum seems to be working here despite the fact that
choice of deformation is essential for the heavy particle radioactivity.
Finally, we have investigated the possible role of shell
corrections for understanding the cluster dynamics in the
SHE region, taking 289 115 → A1 + A2 as an example, with
deformations upto β2 and for the case of Prox-2000 only.
We consider in fig. 9(a) the dependance of P0 on-shell corrections, by taking δU = 0 and compare it with the case of
δU = 0. Figure 9(a) clearly shows that the maximum yield
(P0 ) is obtained for 83 As fragment, and its complementary
heavy residue 206 Pb, only when the shell corrections δU
are added to the liquid drop part of the binding energy
term. The above observation clearly suggest that shell corrections play an important role and hence are essential to
make concrete and explicit predictions of the clusterization process in SHE region also. Therefore, it is of interest
to see what differences could arise as a result of proton
closed shell being taken as Z = 114 or 120, instead of the
above used Z = 126, with N = 184.
In order to investigate the above result further, the
variation of P0 is shown in fig. 9(b) for all the three choices
of magic numbers (Z = 114, 120 or 126, N = 184), using the deformed choice of fragments. In each case, the
shell corrections are obtained by using the “empirical”
formula [37], with the corresponding liquid drop energies
adjusted [42] to give the experimental binding energies.
For comparison, the neck length parameter ΔR is kept
same for all the three cases and refers to the best possible value obtained for the Z = 126, N = 184 case to the
data on cluster decay half-life (here Poenaru et al. [10, 11]
calculations). Apparently, the emergence of heavy clus-
ter 83 As and the corresponding 206 Pb daughter is found
equally favored, i.e., nearly indistinguishable for the three
magic pairs of Z, N which in turn, seem to suggest that
HPR emission does not depend on the choice of proton
magicity in SHE region.
4 Summary
In this paper, an attempt is made to examine the possibility of heavy cluster emission in the ground-state decays
of 278 113, 287−289 115 and 293,294 117 isotopes resulting in
a doubly magic 208 Pb daughter or its neighboring nuclei.
The calculations are performed with in the framework of
preformed cluster model (PCM) with choices of spherical
as well as quadrupole deformed (β2 ) with hot “compact”
orientations θiopt. of decay products. The effect of different
nuclear proximity potentials on cluster decay half-lives
(T1/2 ), preformation probability (P0 ) and penetration
probability P , along with the possible role of magic shells
is investigated in order to extract a clear picture of the
dynamics involved. The standard rms deviation is also
calculated to check the accuracy of different proximity potentials with respect to the predictions of analytical superasymmetric fission model (ASAFM). The calculations are
further shown to be extremely sensitive to the choice of
neck-length parameter ΔR and hence an effort is made
to predict a range of T1/2 values which could possibly
provide new directions for cluster decay measurements.
The present study points out the importance of deformation effects, together with the choice of proximity
potential in deciding the clusterization process in SHE region. It will be of further interest to investigate the role of
higher order deformations for complete and comprehensive knowledge of heavy cluster dynamics in SHE mass
region.
Eur. Phys. J. A (2014) 50: 175
The financial support from University Grants Commission
(UGC), New Delhi in the form of Dr. D.S. Kothari Scheme
and Research Project is gratefully acknowledged.
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