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PHYSICAL REVIEW C 85, 054612 (2012)

Systematic study of various proximity potentials in 208 Pb-daughter cluster radioactivity Raj Kumar* and Manoj K. Sharma School of Physics and Materials Science, Thapar University, Patiala 147004, India (Received 28 March 2012; revised manuscript received 18 April 2012; published 8 May 2012) Various types of nuclear proximity potentials are employed to study the cluster decay of radioactive nuclei, particularly those decaying to a doubly closed shell 208 Pb-daughter nucleus using the preformed cluster-decay model (PCM). The deformation effect is included up to quadrupole (β2 ) with “optimum” cold orientations. The use of different proximity potentials modifies the potential barrier characteristics (i.e., barrier height, position, and frequency), which in turn change the preformation probability P0 and tunneling probability P , and hence the decay half-life considerably. The analysis of 208 Pb-daughter cluster radioactivity is worked out at touching as well as at an elongated neck configuration by taking the neck-length parameter R = 0.5 fm. A wide range of barrier characteristics is covered by using various nuclear proximity potentials. We observe that Prox 1977 and Prox 1988 can reproduce the experimental half-lives very well at R = 0.5 fm; however, the use of the mod-Prox 1988 potential seems more reliable for 14 C cluster decay. The relevance of barrier characteristics of other nuclear proximity potentials is also explored in the context of 208 Pb cluster radioactivity. DOI: 10.1103/PhysRevC.85.054612

PACS number(s): 21.60.Gx, 21.10.Tg, 23.60.+e, 23.70.+j

I. INTRODUCTION

Cluster radioactivity or heavy-ion radioactivity is an intermediate process between alpha decay and nuclear fission. In this process clusters heavier than alpha particles but lighter than fission fragments are produced. In 1984, Rose and Jones [1] first observed the emission of 14 C from 223 Ra. However Sandulescu, Poenaru, and Greiner had already made the prediction of cluster radioactivity as an intermediate process between alpha decay and spontaneous fission in 1980 [2]. Since the theoretical prediction [2] and experimental observation [1], cluster radioactivity is now a well known and widely studied phenomena. During last three decades, a number of cluster radioactive decays from 221 Fr to 242 Cm parent nuclei were observed leading to 12,14 C, 15 N, 18,20 O, 23 F, 22,24–26 Ne, 28,30 Mg, and 32,34 Si cluster emissions, and their respective half-lives have been measured. Recently the concept of heavy-particle radioactivity is further explored by Poenaru et al. [3]. Many theoretical models such as the superasymmetric fission model [4], unified fission model (UFM) [4–12], and preformed cluster-decay model (PCM) [13–16] were proposed for the possible explanation of this phenomenon. The UFM is a fission model in which the cluster formation probability is calculated as an internal barrier penetration, whereas in PCM the cluster preformation is calculated by solving the Schrödinger equation for the dynamic flow of charges and masses. Cluster decay is a quantum tunneling process, in which the cluster has to penetrate the potential barrier in order to come out from the parent nucleus. In both the models, i.e., UFM and PCM, the cluster is supposed to penetrate the potential barrier with available Q value, which also plays a vital role in calculating the half-lives of the emitted clusters [17]. Thus in both types of model the preformation probability varies from cluster to cluster. Alternatively, some models use phenomenological formulas for the preformation

*

[email protected]

0556-2813/2012/85(5)/054612(11)

probability [18–20]. The main ingredient of all models is the interaction potential, which consists of long-range and short-range potentials. The short-range nuclear potential is the backbone of all the theoretical models. An appropriate knowledge of the potential barrier is extremely desirable for the overall understanding of this phenomenon, as the preformation and penetration probabilities depend considerably on the type of interaction potential used. The potential barrier also depends upon the deformations and orientations of the clusters, and hence such effects should be directly incorporated in any model-dependent calculations. Recently one of us and collaborators [17,21] studied the role of deformations and orientation of the nuclei in cluster decays of various radioactive nuclei, particularly for those decaying to doubly closed shell 208 Pb-daughter nuclei for the first time, and then extended the study to the parent nuclei resulting in daughters other than 208 Pb. The higher multipole deformations were included up to hexadecapole (β2 , β3 , β4 ), with the “optimum” orientation of the cold decay process. It may be noted that the optimum cold orientations are good for deformations up to β2 alone and it was concluded in Refs. [17,21] that, except for 14 C decays, the deformations up to β2 are enough in order to fit the experimental data through the only parameter, namely the neck-length parameter R, of the model. However, in the later work [22], it was observed that the deformations up to hexadecapole with use of compact orientations of the cold elongated process are required, particularly for fitting the half-life of a 14 C cluster on the basis of PCM. The possible roles of the Q value and angular momentum were also explored in Ref. [22]. In the present work, we have included the deformation effect up to quadrupole (β2 ) with in the optimum orientation approach [23]. It may be relevant to mention here that the measured β4 values are not available in general, so the higher multipole deformations [24] should be used carefully, specially for 16 A 26 cluster emissions. It is important to note that so far only the nuclear proximity potential of Blocki et al. [25] is used in the framework

054612-1

©2012 American Physical Society

RAJ KUMAR AND MANOJ K. SHARMA


of PCM to understand the cluster decay process. Recently one of us used various nuclear proximity potentials to study the fusion reaction cross section above as well as below the Coulomb barrier, and suggested a modified version of the proximity potential [26]. In the present work we are using the preformed cluster-decay model (PCM) with various versions of proximity potentials having different isospin and asymmetry dependent parameters, at touching configuration and at a fixed neck-length parameter R. For the trans-lead nuclei the decay rates are larger for clusters with corresponding daughter nuclei having magic or near-magic configurations in the neighborhood of 208 Pb (a doubly closed shell spherical nucleus with Z = 82, N = 126). Our present study is confined to the clusters emitted across 208 Pb-daughter nuclei. The main aim of this work is to understand the effect of different nuclear potentials on cluster half-lives (T1/2 ), preformation probability (P0 ), and penetration probability (P ), along with the possible role of β2 deformation in 208 Pb-daughter cluster radioactivity. The experimental data on cluster decay half-lives are taken from Ref. [27]. The paper is organized as follows. Sec. II gives the details of the preformed cluster-decay model and various versions of the proximity potential [25,26,28,29]. The deformations are included up to β2 with optimum cold orientations. Calculations and results are presented in Sec. III. Finally, the conclusions drawn are discussed in Sec. IV.

II. THE PREFORMED CLUSTER MODEL (PCM)

In PCM, the decay constant, and hence the decay half-life time, is defined as ln 2 λ = ν0 P0 P , T1/2 = , (1) λ which means that in PCM the clusters are taken to be preborn in the parent nucleus with preformation probability P0 , hit the barrier with impinging frequency ν0 , and penetrate it with penetrability P . If R0 is the radius of the parent nucleus and E2 = 12 μv 2 is the kinetic energy of the emitted cluster, then ν0 is given by ν0 =

v (2E2 /μ)1/2 = . R0 R0

(2)

ν0 is nearly constant ∼1021 s−1 for all the observed cluster decays. In terms of the (positive) Q value of decay, since both the emitted cluster and daughter nuclei are produced in the ground state, the entire Q value is the total kinetic energy (Q = E1 + E2 ) available for the decay process, which is shared between the two fragments such that, for the emitted cluster (1 and 2 stand, respectively, for daughter and cluster), A1 Q (3) A and E1 (= Q − E2 ) is the recoil energy of the daughter nucleus. P0 and P are calculated within the well-known quantum mechanical fragmentation theory (QMFT) [30,31], with effects of deformation and orientation degrees of freedom included. E2 =

FIG. 1. Scattering potential for the decay of 236 Pu→ Pb + 28 Mg, showing steps of barrier penetration, with both the daughter and cluster taken as a spherical (dashed line) and deformed (solid line) nucleus with quadrupole deformations β2i alone, and with opt. optimum cold orientation angles θi of Table I in Ref. [23]. The path of tunneling is also shown. 208

The QMFT 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 orientation angle θi is measured anticlockwise from the decay Z axis up to the symmetry axis; see Fig. 1 of Ref. [23] for details). Apparently, the coordinates η and R refer, respectively, to the nucleon division (or exchange) between the daughter and cluster, and the transfer of positive Q value to the total kinetic energy (E1 + E2 ) of two nuclei, determining P0 and P , respectively. The preformation probability P0 (Ai ) (≡ |ψ(η(Ai ))|2 , i = 1 or 2) is the solution of the stationary Schrödinger equation in η, at fixed R = Ra , 1 ∂ ∂ h ¯2 − √ + V (R, η, T ) ψ ω (η) √ 2 B ηη ∂η B ηη ∂η = E ω ψ ω (η),

(4)

with ω = 0, 1, 2, 3, . . . referring to ground-state (ω = 0) and excited-state and for a Boltzmann-like function solutions, ω 2 ω |ψ|2 = ∞ |ψ | exp(−E /T ). Note that here we are ω=0 interested only in the ground-state solution (ω = 0, T = 0). Ra is the first turning point of the penetration path used for calculating the penetrability P , illustrated in Fig. 1 for

054612-2

SYSTEMATIC STUDY OF VARIOUS PROXIMITY . . .


Pu→208 Pb + 28 Mg decay where the daughter and cluster emissions are worked out using quadrupole deformation β2i opt. along with the optimum cold orientation θi approach [23]. In the present study we have kept the first turning point at touching configuration or a nearby higher value (Ra RT ). This means that, in the present cluster decay calculations, the potential V (R < Ra ) does not come in to play, and hence the shell effects inside the barrier are not very significant, though these could very well be calculated using the two-center shell model [15]. The structure information of the decaying nucleus is contained in P0 via the fragmentation potential, defined for R = Ra ( Rt ), 236

VR (η) = −

2

or, equivalently, the neck-length parameter R, is the only parameter of the model. This method of introducing the necklength parameter R is also used in the dynamical clusterdecay model (DCM) of Gupta and collaborators [35–37] and in the scission-point [38] and saddle-point [39,40] (statistical) fission models for decay of a hot and rotating compound nucleus. In the present calculations, in order to see the effect of use of various nuclear proximity potentials, a fixed value of the neck-length parameter R is used. Here, the radius vector is defined as (0) Ri (αi ) = R0i 1 + βλi Yλ (αi ) (8) λ

with the R0i given by

[Bi (Ai , Zi )] + VC (R, Zi , βλi , θi )

1/3

R0i = 1.28Ai

i=1

+ VP (R, Ai , βλi , θi ) + V (R, Ai , βλi , θi ), (5) used in the stationary Schrödinger Eq. (4). Note that the shell effects enter here mainly through the ground-state binding energies Bi (Ai , Zi ) [24]. The deformation parameters of nuclei βλi are taken from the tables of Möller et al. [24], with the optimum orientations for the cold decay process taken from Table I of Gupta et al. [23]. Thus, the dynamical modifications of βλi are also neglected, which are again of interest more for the overlap region [15]. In Eq. (5), VC , VP , and V are, respectively, the Coulomb, nuclear proximity, and angularmomentum dependent potentials for deformed and oriented nuclei. For ground-state decays, = 0 is a good approximation [32]. The details of nuclear potentials are discussed later. The Coulomb potential for a multipole-multipole interaction and two nonoverlapping charge distributions [33] is given by VC =

Riλ (αi ) Z1 Z2 e2 + 3Z1 Z2 e2 R (2λ + 1)R λ+1 λ,i=1,2 4 2 (0) (0) × Yλ (θi ) βλi + βλi Yλ (θi ) . 7

.

(7)

where the dependence of Ra is contained in Rt , and R is a parameter assimilating the neck formation effects of the two-center shell model [15]. The first turning point Ra

(9)

Thus, the tunneling begins at R = Ra and terminates at R = Rb , with V (Rb ) = Q for ground-state decay. Thus, as per Fig. 1, the transmission probability P consists of the following three contributions [13,14]: 1. the penetrability Pi from Ra to Ri , 2. the (inner) deexcitation probability Wi at Ri , and 3. the penetrability Pb from Ri to Rb , giving the penetration probability as P = Pi Wi Pb .

(10)

The shifting of the first turning point from Ra to R0 , the compound nucleus radius, gives a penetrability P similar to that of Shi and Swiatecki [6] for spherical nuclei, which is known not to fit the experimental data without the adjustment of assault frequency ν0 . Following the excitation model of Greiner and Scheid [41], we take the deexcitation probability Wi = 1 for a heavy cluster decays, which reduces Eq. (10) to the following: P = Pi Pb ,

(6)

Here the orientation angle θi is the angle between the nuclear symmetry axis and the collision Z axis, measured in the anticlockwise direction, and angle 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 (see, e.g., Fig. 1 of Ref. [23]). The mass parameters Bηη (η), entering the stationary Schrödinger Eq. (4) via the kinetic energy term, are the smooth classical hydrodynamical masses of Kröger and Scheid [34], whose predictions match (on the average) the microscopic cranking model calculations. The penetrability P in Eq. (1) is the WKB integral between the two turning points Ra and Rb , respectively. For the first turning point Ra , we use the postulate Ra (η) = R1 (α1 ) + R2 (α2 ) + R = Rt (α, η) + R,

−1/3

− 0.76 + 0.8Ai

where Pi and Pb in the WKB approximation are

2 Ri Pi = exp − {2μ[V (R) − V (Ri )]}1/2 dR h ¯ Ra and

2 Rb 1/2 Pb = exp − {2μ[V (R) − Q]} dR . h ¯ Ri

(11)

(12)

(13)

In the following, we present the details of various nuclear proximity potentials used in the present calculations. The nuclear proximity potential for deformed, oriented nuclei [42] is given by ¯ b(s0 ), VP (s0 ) = 4π Rγ

(14)

where b = 0.99 is the nuclear surface thickness and R¯ is the mean curvature radius (for details see Ref. [42]). in Eq. (14) is the universal function, independent of the shapes of nuclei or the geometry of the nuclear system, but depends on the minimum separation distance s0 (see Fig. 2 in Ref. [42]). Various versions of nuclear proximity potentials used in the present work are as follows.

054612-3



A. Proximity 1977 (Prox 1977)

For this proximity potential the universal function is given by [25] − 12 (s0 − 2.54)2 − 0.0852(s0 − 2.54)3 , (15) (s0 ) = s0 , −3.437 exp − 0.75 respectively, for s0 1.2511 and s0 1.2511. The surface energy constant used for this potential is

N −Z 2 γ = γ0 1 − ks MeV fm−2 . A Here N and Z are the total numbers of neutrons and protons. In the present version, the coefficients γ0 and ks were taken to be 0.9517 MeV/fm2 and 1.7826, respectively. This potential is referred to as Prox 1977.

B. Proximity 1988 (Prox 1988)

Möller and Nix [43] improved the mass formula later and, because of this improvement, the value of coefficients γ0 and ks were changed respectively to 1.2496 MeV fm−2 and 2.3, respectively. Reisdorf [28] labeled this modified version “Proximity 1988”. It is to be noted that this set of coefficients gives stronger attraction, i.e., a deeper pocket compared to Prox 1977. Even a more recent compilation by Möller and Nix yields similar results. We labeled this potential Prox 1988.

C. Proximity 2000 (Prox 2000)

Using the droplet model concept, Myers and Swiatecki [29] modified Eq. (14) by using available knowledge of nuclear radii and surface tension coefficients. Using the droplet model [44], matter radius Ci was calculated as Ci = Ri +

Ni ti , Ai

Using the droplet model [44], neutron skin ti reads as −1/3 1 J Ii − 12 c1 Zi Ai 3 ti = r 0 (i = 1, 2). −1/3 2 Q + 9 J Ai

(19)

4

Here r0 is 1.14 fm, the value of the nuclear symmetric energy coefficient J = 32.65 MeV, and c1 = 3e2 /5r0 = 0.757895 MeV. The neutron skin stiffness coefficient Q was taken to be 35.4 MeV. The nuclear surface energy coefficient γ in terms of neutron skin was given as t12 + t22 1 18.63(MeV) − Q , (20) γ = 4π r02 2r02 where t1 and t2 were calculated using Eq. (19). The universal function for this is given by ⎧ 5 ⎪ ⎪ ⎪ ⎪ −0.1353 + [cn /(n + 1)](2.5 − ξ )n+1 ⎪ ⎪ ⎨ n=0 (21) (ξ ) = for 0 < ξ 2.5, ⎪ ⎪ ⎪ ⎪ − exp[(2.75 − ξ )/0.7176] ⎪ ⎪ ⎩ for ξ 2.5, where ξ = R − C1 − C2 . The values of different constants cn were c0 = −0.1886, c1 = −0.2628, c2 = −0.15216, c3 = −0.04562, c4 = 0.069136, and c5 = −0.011454. This potential is labeled Prox 2000.

D. Modified Proximity 1988 (mod-Prox 1988)

Recently one of us modified the value of coefficient γ0 of Prox 1988 in order to fit the fusion cross section for the various reactions simultaneously within the extended Wong model [26]. This modified version is denoted mod-Prox 1988. The value of coefficient γ0 used for mod-Prox 1988 is 1.65 MeV fm−2 , and the rest are same as the Prox 1988 including the universal function.

(16) E. Bass potential (Bass 1980)

where Ri is given by Eq. (8). For this potential R0i in Eq. (8) denotes the half-density radii of the charge distribution and ti is the neutron skin of the nucleus. To calculate R0i , these authors [29] used the two-parameter Fermi function values given in Ref. [45], and the remaining cases were handled with the help of the parametrization of charge distribution described below. The nuclear charge radius (denoted R00 in Ref. [46]), is given by the relation

1.646 Ai − 2Zi 1/3 1+ fm, − 0.191 R00i = 1.24Ai Ai Ai (17) where i = 1, 2. The half-density radius ci was obtained from the relation

7 b2 49 b4 R0i = R00i + 1 − − + · · · (i = 1, 2). 2 4 2 R00i 8 R00i (18)

The universal function for this is given by [28] s −1 s + 0.007 exp , (s) = 0.033 exp 3.5 0.65 with the central radius R0i of Eq. (8) as

0.98 (i = 1, 2), R0i = Rs 1 − 2 Rs

(22)

(23)

where Rs is same as given by Eq. (9) with Ii = Ni − Zi /Ai (i = 1, 2). We denote this potential Bass 1980.

F. Christensen and Winther 1976 (CW 1976)

Christensen and Winther [47] derived the nucleus-nucleus interaction potential by analyzing heavy-ion elastic scattering data, based on semiclassical arguments and the recognition that optical-model analysis of elastic scattering determines the real part of the interaction potential. The nuclear part of the

054612-4



empirical potential is given as ¯ VP (r) = −50R(R − R1 − R2 ),

(24)

1/3

where R¯ is the mean curvature radius. Here radius parameters are given by 1/3

R0i = 1.233Ai

−1/3

− 0.978Ai

fm (i = 1, 2).

(25)

The universal function (s = R − R1 − R2 ) is given by (s) = exp(−s/0.63).

(26)

This potential is denoted CW 1976.

G. Broglia and Winther (BW 1991)

Broglia and Winther [28] also obtained a revised version of the CW 1976 potential by taking a Woods-Saxon parametrization with the subsidiary condition of being compatible with the value of the maximum nuclear force predicted by the proximity potential Prox 1977. This refined potential is given as VN (R) = −

V0 MeV, 0 1 + exp R−R 0.63

(27)

with V0 = 16π

R 1 R2 γ a, R1 + R2

where Rip is taken from the work of Royer and Rousseau [49] and is given by Rip = 1.2332Ai

−2/3

+ 2.8961Ai

1/3

− 0.18688Ai Ii . (31)

Here Ii = Ni − Zi /Ai (i = 1, 2). The universal function (s = r − R1 − R2 − 2.65) is given by the following complex form: ⎧ 1 − s/0.7881663 + 1.229218s 2 ⎪ ⎪ ⎪ ⎪ ⎪ − 0.2234277s 3 − 0.1038769s 4 ⎪ ⎪ ⎪ ⎪ R2 ⎪ − RR11+R (0.1844935s 2 + 0.07570101s 3 ) ⎪ ⎪ 2 ⎪ ⎪ 2 3 ⎪ ⎪ ⎨ + (I1 + I2 )(0.04470645s + 0.03346870s ) for − 5.65 s 0, (ξ ) = (32) ⎪ ⎪ R R s ⎪ 1 − s 2 0.05410106 1 2 exp − ⎪ ⎪ R1 +R2 1.760580 ⎪ ⎪ s ⎪ ⎪ − 0.5395420(I + I ) exp − 1 2 ⎪ 2.424408 ⎪ ⎪ s ⎪ ⎪ × exp − 0.7881663 ⎪ ⎪ ⎩ for s 0. The above form of the universal function not only depends on the separation distance s but also has complex dependence on the mass as well as on the relative neutron excess content. This potential is labeled Denisov 2002.

(28) III. CALCULATIONS AND RESULTS

where a = 0.63 fm and R0 = R1 + R2 + 0.29 fm. The surface energy constant used for this potential is

N1 − Z1 N2 − Z2 MeV fm−2 , γ = γ0 1 − k s A1 A2 Here N and Z are the total numbers of neutrons and protons. In the present version, γ0 and ks were taken to be 0.95 MeV/fm2 and 1.8, respectively. This potential is denoted BW 1991.

H. Denisov potential (Denisov 2002)

Denisov performed numerical calculations and parameterized the potential based on the 7140 pair within a semimicroscopic approximation [48]. A simple analytical expression for the nuclear part of the interaction potential VN (R) between two nuclei is presented as R1 R2 R1 + R2 (r − R1 − R2 − 2.65) VP (r) = −1.989843

A1 A2 3/2 × 1 + 0.003525139 + A2 A1 − 0.4113263(I1 + I2 ) . (29) The effective nuclear radius R0i is given as

3.413817 2 R0i = Rip 1 − Rip

0.4Ai (i = 1, 2), (30) +1.284589 Ii − Ai + 200

Figure 1 shows the scattering potential using nuclear proximity interaction Prox 1977 for the decay of 236 Pu→ 208 Pb + 28 Mg plotted for the cases of spherical and deformed (up to β2 ) choice of cluster and daughter nuclei. It is clear from this figure that the barrier characteristics, i.e., barrier height (VB ), position (RB ), and frequency (¯hω), change considerably (for more detail see Refs. [17,21,50]). Thus with the inclusion of the deformation effect the preformation probability (P0 ) and barrier penetration probability (P ) also get modified. This change in preformation probability and barrier penetration affects the decay constant (λ) and half-life (T1/2 ) as given by Eq. (1). This figure also shows the two steps of the decay process in the PCM. It is important here to note that, for a one-dimensional barrier penetration model, such as PCM, the main ingredient is the interaction potential, and a slight change in its magnitude can change the decay constants and decay half-lives significantly. Figure 2 shows the fragmentation potential V (A2 ) for the decay of 232 U using nuclear proximity potentials Prox 1977, Prox 1988, and Prox 2000 (having different dependences on isospin and asymmetry of the colliding nuclei), for both spherical [Figs. 2(a) and 2(c)] and β2 -deformed [Figs. 2(b) and 2(d)] choice of fragmentation. The calculations are done at internuclear separation distances (also denoted the first turning point of penetration in PCM) Ra = RT = R1 + R2 and Ra = RT + 0.5 fm. It is clear from Fig. 2 that the fragmentation potential changes significantly with the use of different proximity interactions. A solid vertical line is drawn in order to point out the 24 Ne cluster emitted from the 232 U parent nucleus. The first turning point Ra is shifted to R1 + R2 + 0.5 fm, i.e., a neck length of 0.5 fm is added

054612-5



FIG. 2. Fragmentation potential for the parent nucleus 232 U using the proximity potentials Prox 1977, 1988, and 2000 at Ra = RT = opt. R1 (α1 ) + R2 (α2 ) for (a) spherical and (b) deformed choice of nuclei, with quadrupole deformation β2 and cold “optimum” orientation θi of Table I in Ref. [23]. Parts (c) and (d) are same as (a) and (b) but at Ra = RT + 0.5 fm. The solid vertical line points out the fragmentation potential of the most probable 24 Ne cluster emitted from the 232 U.

to the touching point because the potential V (Ra ), calculated using nuclear proximity potential Prox 1988, comes out to be less than the Q value [see Figs. 3(a) and 3(b)]. Although the magnitudes of fragmentation potential are quite different for Prox 1977, Prox 1988, and Prox 2000 potentials, their potential energy surface (PES) behavior is similar and shows α-nucleus structure for fragments up to A2 = 50, independent of the barrier entrance point or deformations used. The clear preference for the 24 Ne cluster in the decay of the 232 U nucleus is evident in Fig. 2, independent of internuclear separation and/or deformation effects. In other words 24 Ne emerges as the preferred cluster from the 232 U parent nucleus independent of the barrier penetration path and deformation used. The deformations are included here up to quadrupole (β2 ) with optimum cold orientations from Table 1 of Ref. [23]. The optimum orientations are good only for deformations up to β2 , so higher-order deformations are not included in the present analysis. However if one is interested in investigating the role of higher-order deformations, then “compact” orientations of the cold elongated process [51] should be preferred instead of optimum orientations. Figure 3 shows a comparison of the difference of the potentials at first turning point, i.e., V (Ra ), and the Q values from the table of Moller and Nix [24] as a function of parent-nuclei mass number A. It is clear from Figs. 3(a) and 3(b) that this difference is positive for Prox 1977 and Prox 2000 at Ra = RT and negative for Prox 1988 except at a few parent nuclei mass numbers. However, Figs. 3(c) and 3(d) show that, when a fixed neck length of 0.5 fm is added to the touching point, this difference increases, i.e., become

positive for Prox 1988 and increases further for the remaining two proximity potentials. The deviation of V (Ra ) from the Q value plays an important role in the calculation of half-life (T1/2 ). This effect is evident from a discussion of Fig. 4. In this figure the experimentally observed decay half-lives of various clusters with 208 Pb as the daughter product are compared, with the ones calculated in a framework of PCM using nuclear proximities Prox 1977, Prox 1988, and Prox 2000 for spherical as well as quadrupole deformed choice of nuclei at fixed Ra = RT and Ra = RT + 0.5 fm. At Ra = RT the comparison of calculations with β2 is better for Prox 1977 or Prox 1988, whereas the same is not true for Prox 2000. Rather, Prox 2000 overestimates the half-life on a large scale. Since the potential at first turning point using Prox 1988 is less than the Q value, in order to calculate half-lives using this potential we have modified the entrance point Ra as RT + 0.5 fm. It is clear from the comparison of Figs. 4(c) and 4(d) that Prox 1977 and Prox 1988 are close to the experimental data at Ra = RT + 0.5 fm, whereas Prox 2000 is still bad here. We have also calculated the standard rms deviation from experimental data and found that, for the best two potentials, i.e., Prox 1977 and Prox 1988, it comes out to be respectively 3.70 and 3.72 for spherical and 2.96 and 4.34 for deformed choice of fragmentation. It may be noted that only the lower limits of half-life are known in the cases of four clusters (15 N, 18 O, 22 Ne, and 23 F) among the eleven cases investigated here. These clusters are identified by upper arrows in Fig. 4. In case the precise half-lives of these clusters are made available, the standard rms deviation reported above can be further improved.

054612-6



FIG. 3. Difference of the potential V (Ra ) at first turning point Ra and the Q value from Ref. [24] is plotted as a function of parent nuclei mass number A, using the nuclear proximity potentials Prox 1977, Prox 1988, and Prox 2000 at Ra = RT = R1 (α1 ) + R2 (α2 ) for (a) spherical and (b) deformed choice of nuclei. Parts (c) and (d) are same as parts (a) and (b) but at a different first turning point Ra = RT + 0.5 fm.

FIG. 4. Comparison of the decay half-lives experimentally observed [27] and the ones which are calculated with PCM, for various clusters with 208 Pb as the daughter product, using nuclear proximity potentials Prox 1977, Prox 1988, and Prox 2000 at Ra = RT = R1 (α1 ) + R2 (α2 ) for (a) spherical and (b) deformed choice of nuclei. Parts (c) and (d) are same as parts (a) and (b) but at different first turning point Ra = RT + 0.5 fm. Arrows pointing upward show that the measured half-lives are greater than the marked points. The emitted clusters are mentioned in parts (c) and (d). 054612-7



FIG. 5. Preformation probability P0 for the decay of 232 U using nuclear potentials Prox 1977, Prox 1988 and Prox 2000 at Ra = RT +0.5 fm, for deformed (up to β2 ) choice of fragmentation.

In Fig. 4, all the clusters (14 C to 34 Si) emitted along with 208 Pb daughters are considered and are labeled in Figs. 4(c) and 4(d). It is clear from Fig. 4 that only deformed Prox 1977 is better if the cluster dynamics is worked out at touching configuration. However Prox 1977 and 1988 seem to compete with each other at Ra = RT + 0.5 fm for spherical as well as deformed choice of fragmentation. It seems that, by using neck-length parameter R = 0.5 fm, the deformation effect is included to a fair extent and hence the comparison improves even for the spherical choice of fragmentation. One point to be made clear here is that, because in PCM the decay constant λ is directly proportional to P0 and P , with the increase of neck-length parameter R the penetration probability increases. However, the preformation probability may or may not increase, as the preformation of the desired cluster not only depends on its potential at first turning point but also on the potential of other fragments considered in the decay of the parent ucleus. Figure 5 shows the preformation probability P0 for the decay of 232 U using nuclear potentials Prox 1977, Prox 1988, and Prox 2000 at fixed Ra = RT + 0.5 fm, with deformations up to β2 . The preformation is different for different clusters. A solid vertical line in Fig. 5 points out P0 for the 24 Ne cluster. The preference for the 24 Ne cluster in the decay of 232 U is further explored and the enhanced preformation factor (P0 ) for 24 Ne cluster ensures that it is the most preferred decay independent of the choice of nuclear proximity force. Figure 6 shows the comparison of preformation and penetration probability calculated using PCM with deformations up to β2 using nuclear proximity potentials Prox 1977, Prox 1988, and Prox 2000 for all the parent nuclei considered. It is clear from Fig. 6(a) that the preformation probability is maximum for Prox 1988 and minimum for Prox 2000, with Prox 1977 being just below Prox 1988. A similar overall trend is observed for the penetration probability in Fig. 6(b), except for the fact that Prox 1977 competes with Prox 1988. This systematic behavior of P0 and P for different proximity interactions can provide important information for the overall understanding of cluster dynamics from a variety of parent nuclei. The relevant details of P0 and P for spherical fragmentation at Ra = RT + 0.5 fm are given in Table I.

FIG. 6. (a) Preformation and (b) penetration probability for the parent nuclei using nuclear potentials Prox 1977, Prox 1988, and Prox 2000 at Ra = RT + 0.5 fm, for deformed (up to β2 ) choice of fragmentation.

The effects of other nuclear proximity potentials [26,28, 47,48] such as Bass 1980, CW 1976, BW 1991, Denisov 2002, and mod-Prox 1988 are also studied in our calculations. Figure 7 shows the interaction potentials calculated for the opt. decay of 232 U→208 Pb + 24 Ne at optimum orientations θ1 = opt. = 900 , using various versions of nuclear sph. and θ2 proximity potentials. It is clear from Fig. 7 that the potentials whose barrier characteristics are closer to Prox 1977 and Prox 1988 are expected to behave similarly to these potentials. Since interaction potential is the main recipe of all the theoretical models, we have covered a wide range of barriers characteristics, as is clear from Fig. 7, where Prox 2000 gives the highest barrier and mod-Prox 1988 gives the lowest barrier and the barriers corresponding to other potentials lie in between these two interaction limits. The barrier characteristics of CW 1976, BW 1991, and Bass 1980 behave similarly to Prox 1977 and Prox 1988 and contrarily to Prox 2000 and Denisov 2002, and hence the first five of them are expected to behave in a similar way in the cluster emission process. Also the barrier is much lower for mod-Prox 1988, so it would be of further interest to see application of this interaction in the cluster decay process. This point is explored in Fig. 8. Another point to be discussed here is that the half-life of 14 C could not be fitted with any of Prox 1977, Prox 1988, and Prox 2000 (see Fig. 4). In one of our earlier works [22], it was observed that hexadecapole deformations are necessary in order to fit the half-life for 14 C. This point is explored further

054612-8



TABLE I. Preformation probability P0 and penetration probability P of spherical fragmentation at Ra = RT + 0.5 fm. PCM (spherical) Parent

222

Ra Ac 226 Th 228 Th 231 Pa 230 U 232 U 234 U 236 Pu 238 Pu 242 Cm 223

Cluster

14

C O 18 O 20 O 23 F 22 Ne 24 Ne 26 Ne 28 Mg 30 Mg 34 Si 18

Preformation probability P0 Prox 1988

Prox 2000

Prox 1977

Prox 1988

Prox 2000

8.41 × 10−22 2.39 × 10−23 1.11 × 10−24 4.29 × 10−24 2.15 × 10−27 9.33 × 10−28 1.06 × 10−25 3.58 × 10−28 2.23 × 10−27 2.69 × 10−28 6.39 × 10−27

1.82 × 10−17 1.31 × 10−20 1.60 × 10−20 5.99 × 10−20 5.01 × 10−23 2.12 × 10−23 2.35 × 10−21 9.78 × 10−24 1.35 × 10−22 2.69 × 10−23 7.24 × 10−22

3.78 × 10−37 7.12 × 10−41 8.45 × 10−44 1.87 × 10−40 3.18 × 10−43 1.78 × 10−45 9.38 × 10−43 4.74 × 10−44 4.53 × 10−48 9.56 × 10−46 6.91 × 10−46

4.51 × 10−19 2.15 × 10−19 2.41 × 10−18 1.06 × 10−20 1.61 × 10−23 3.49 × 10−19 3.84 × 10−19 6.69 × 10−23 7.06 × 10−18 1.42 × 10−19 7.69 × 10−17

4.09 × 10−23 2.93 × 10−21 1.46 × 10−20 3.83 × 10−23 5.64 × 10−25 2.75 × 10−20 6.81 × 10−21 2.69 × 10−24 5.79 × 10−19 9.82 × 10−21 7.22 × 10−18

3.75 × 10−21 6.46 × 10−22 5.97 × 10−21 1.89 × 10−23 9.29 × 10−27 5.35 × 10−22 5.68 × 10−22 4.07 × 10−26 1.30 × 10−20 1.56 × 10−22 1.88 × 10−19

in this work as well. It is to be reminded here that the mod-Prox 1988 nuclear proximity was found to be more suitable for the reactions exhibiting fusion hindrance phenomena at energies below the Coulomb barrier (for more details see Ref. [26]). So we tried the application of this potential in ground-state cluster radioactivity. It may be noted from Fig. 7 that the interaction barrier of mod-Prox 1988 is lowest, and also this potential has strong dependence on isospin and asymmetry of the colliding nuclei [26]. So we have seen the effect of mod-Prox 1988 on the half-lives of the clusters using PCM with the first turning point fixed at Ra = RT + 1.0 fm, because of the same reason

FIG. 7. Interaction potentials calculated for the decay of opt. U→208 Pb + 24 Ne at fixed optimum orientations θ1 = sph. and opt. ◦ θ2 = 90 , using various versions of nuclear proximity potentials. 232

Penetration probability P

Prox 1977

for which we changed Ra for Prox 1988 from RT to RT + 0.5 fm. Figure 8 shows the comparison of experimental and calculated half-lives using PCM, for spherical and β2 -deformed choice of fragmentation. With the use of mod-Prox 1988 at fixed Ra = RT + 1.0 fm, we are able to fit the experimental half-life of the 14 C cluster, whereas fitting for the other clusters has gone slightly off as shown in Fig. 8. One may assume that this may be the effect of introducing the additional neck length in the internuclear separation, so other proximities are also tested with the use of larger neck length up to 1 fm, but still the half-life of the 14 C cluster could not be fitted. Thus mod-Prox 1988 is performing well when compared to experimental data of the 14 C cluster. The corresponding preformation and penetration probabilities for the mod-Prox 1988 proximity potential are shown in Table II for spherical and deformed choice of fragmentation path.

FIG. 8. Comparison of the decay half-lives of experimentally observed clusters [27] with the ones that are calculated with PCM using nuclear proximity potentials mod-Prox 1988 at Ra = RT = R1 (α1 ) + R2 (α2 ) + 1.0 fm, for spherical and deformed choice of fragmentation.

054612-9



TABLE II. Calculated preformation and penetration probability, i.e., P0 and P , using PCM for various clusters with 208 Pb as daughter nuclei for spherical and quadrupole deformed (β2 ) nuclei with optimum orientations, using the nuclear proximity potential mod-Prox 1988 at Ra = RT + 1.0 fm. PCM Parent

222

Ra Ac 226 Th 228 Th 231 Pa 230 U 232 U 234 U 236 Pu 238 Pu 242 Cm 223

Cluster

14

C O 18 O 20 O 23 F 22 Ne 24 Ne 26 Ne 28 Mg 30 Mg 34 Si 18

P0

P

Sph.

β2

Sph.

β2

7.81 × 10−16 1.01 × 10−18 1.34 × 10−18 2.37 × 10−18 2.76 × 10−21 3.25 × 10−21 1.81 × 10−19 9.09 × 10−22 4.74 × 10−20 6.46 × 10−21 3.60 × 10−19

5.23 × 10−15 3.13 × 10−19 6.81 × 10−17 1.96 × 10−20 8.57 × 10−21 9.93 × 10−16 1.25 × 10−17 2.85 × 10−21 2.08 × 10−14 1.26 × 10−17 4.92 × 10−16

1.17 × 10−19 7.51 × 10−19 6.30 × 10−18 3.12 × 10−20 2.68 × 10−22 4.29 × 10−18 2.40 × 10−18 1.23 × 10−21 5.59 × 10−17 1.35 × 10−18 6.80 × 10−16

9.12 × 10−19 1.65 × 10−19 1.79 × 10−17 2.00 × 10−20 1.55 × 10−21 1.23 × 10−16 2.87 × 10−17 5.48 × 10−21 3.37 × 10−14 1.42 × 10−17 4.85 × 10−16

IV. CONCLUSIONS

We conclude that the use of different proximity interactions changes the preformation and penetration probabilities considerably, and hence the decay constants and half-lives of the clusters emitted along with complimentary 208 Pb daughter nucleus get modified accordingly. The proximities Prox 1977 and Prox 1988 perform well at fixed neck length R = 0.5 fm except for the 14 C cluster. The more recent mod-Prox 1988, having strong dependence on isospin and asymmetry of the colliding nuclei, is able to account for the experimental halflives, particularly of the 14 C cluster with reasonable response to other clusters. Besides this the barrier characteristics suggests that Bass 1980, CW 1976, and BW 1991 may respond equally well for the study of cluster dynamics. So one can also use these nuclear proximity potentials for understanding the cluster decay process involving 208 Pb as a daughter nucleus. It is also observed that the Q value is an important parameter in the calculations for predicting half-lives of the clusters. The

[1] H. J. Rose and G. A. Jones, Nature (London) 307, 245 (1984). [2] A. S˘andulescu, D. N. Poenaru, and W. Greiner, Sov. J. Part. Nucl. 11, 528 (1980). [3] D. N. Poenaru, R. A. Gherghescu, and W. Greiner, Phys. Rev. Lett. 107, 062503 (2011); Phys. Rev. C 85, 034615 (2012). [4] D. N. Poenaru, M. Ivascu, A. S˘andulescu, and W. Greiner, Phys. Rev. C 32, 572 (1985). [5] D. N. Poenaru, W. Greiner, K. Depta, M. Ivascu, D. Mazilu, and A. S˘andulescu, At. Data Nucl. Data Tables 34, 423 (1986). [6] Y. J. Shi and W. J. Swiatecki, Phys. Rev. Lett. 54, 300 (1985). [7] G. A. Pik-Pichak, Fiz. Elem. Chastits At. Yadra 44, 1421 (1986); Sov. J. Part. Nucl. 44, 923 (1986). [8] G. Shanmugam and B. K. Kamalharan, Phys. Rev. C 38, 1377 (1988). [9] B. Buck and A. C. Merchant, J. Phys. G 15, 615 (1989). [10] A. S˘andulescu, R. K. Gupta, W. Greiner, F. Carstoiu, and M. Horoi, Int. J. Mod. Phys. E 1, 379 (1992).

relevant role of β2 deformation is explored in the context of different proximity interactions in the cluster emission process. Interestingly, the status of preferred cluster (say, 24 Ne in 232 U) remains intact, independent of choice of nuclear proximity potential used. It will be of further interest to investigate the role of different proximity potentials for the half-lives of the other clusters involving non-208 Pb-daughter nuclei. For comprehensive knowledge of cluster dynamics, the role of higher-order deformations with cold-elongated orientations can also be investigated in reference to different proximity potentials.

ACKNOWLEDGMENTS

Financial support from the Council of Scientific and Industrial Research (CSIR), New Delhi is duly acknowledged. One of us (R.K.) is thankful to UGC for financial support under Dr. D. S. Kothari grant.

[11] G. Royer, R. K. Gupta, and V. Yu. Denisov, Nucl. Phys. A 632, 275 (1988). [12] D. N. Poenaru and W. Greiner, in Clusters in Nuclei, Vol. I, edited by C. Beck, Lecture Notes in Physics No. 818 (Springer, Berlin, 2010), Chap. 1, pp. 1–56. [13] R. K. Gupta, in Proceedings of the 5th International Conference on Nuclear Reaction Mechanisms, edited by E. Gadioli (Ricerca Scientifica ed Educazione Permanente, Milan, 1988), p. 416. [14] S. S. Malik and R. K. Gupta, Phys. Rev. C 39, 1992 (1989). [15] S. Kumar and R. K. Gupta, Phys. Rev. C 55, 218 (1997). [16] R. K. Gupta, in Heavy Elements and Related New Phenomena, Vol. II, edited by W. Greiner and R. K. Gupta (World Scientific, Singapore, 1999), Chap. 18, p. 731. [17] S. K. Arun, R. K. Gupta, B. B. Singh, S. Kanwar, and M. K. Sharma, Phys. Rev. C 79, 064616 (2009). [18] R. Blendowske and H. Walliser, Phys. Rev. Lett. 61, 1930 (1988). [19] Z. Ren, C. Xu, and Z. Wang, Phys. Rev. C 70, 034304 (2004); D. Ni and Z. Ren, ibid. 82, 024311 (2010).

054612-10



[20] B. B. Singh, S. K. Patra, and R. K. Gupta, Phys. Rev. C 82, 014607 (2010); Int. J. Mod. Phys. E 20, 1003 (2011). [21] S. K. Arun, R. K. Gupta, S. Kanwar, B. B. Singh, and M. K. Sharma, Phys. Rev. C 80, 034317 (2009). [22] G. Sawhney, M. K. Sharma, and R. K. Gupta, Phys. Rev. C 83, 064610 (2011). [23] R. K. Gupta, M. Balasubramaniam, R. Kumar, N. Singh, M. Manhas, and W. Greiner, J. Phys. G: Nucl. Part. Phys. 31, 631 (2005). [24] P. Möller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, At. Data Nucl. Data Tables 59, 185 (1995). [25] J. Blocki, J. Randrup, W. J. Swiatecki, and C. F. Tsang, Ann. Phys. (N.Y.) 105, 427 (1977). [26] R. Kumar, Phys. Rev. C 84, 044613 (2011). [27] R. Bonetti and A. Guglielmetti, Romanian Reports in Physics 59, 301 (2007). [28] W. Reisdorf, J. Phys. G: Nucl. Part. Phys. 20, 1297 (1994). ´ [29] W. D. Myers and W. J. Swiatecki, Phys. Rev. C 62, 044610 (2000). [30] J. Maruhn and W. Greiner, Phys. Rev. Lett. 32, 548 (1974). [31] H. J. Fink, J. Maruhn, W. Scheid, and W. Greiner, Z. Phys. 268, 321 (1974). [32] R. K. Gupta and W. Greiner, Int. J. Mod. Phys. E 3, 335 (1994), Suppl. [33] C. Y. Wong, Phys. Rev. Lett. 31, 766 (1973). [34] H. Kröger and W. Scheid, J. Phys. G 6, L85 (1980). [35] R. K. Gupta, in Clusters in Nuclei, Vol. I, edited by C. Beck, Lecture Notes in Physics No. 818 (Springer, Berlin, 2010), p. 223. [36] M. Kaur, R. Kumar, and M. K. Sharma, Phys. Rev. C 85, 014609 (2012); K. Sandhu, M. K. Sharma, and R. K. Gupta, ibid. 85,

[37] [38] [39]

[40] [41] [42] [43] [44] [45]

[46] [47] [48] [49] [50] [51]

054612-11

024604 (2012); M. K. Sharma, S. Kanwar, G. Sawhney, R. K. Gupta, and W. Greiner, J. Phys. G: Nucl. Phys. 38, 055104 (2011). D. Jain, R. Kumar, M. K. Sharma, and R. K. Gupta, Phys. Rev. C 85, 024615 (2012). T. Matsuse, C. Beck, R. Nouicer, and D. Mahboub, Phys. Rev. C 55, 1380 (1997). S. J. Sanders, D. G. Kovar, B. B. Back, C. Beck, D. J. Henderson, R. V. F. Janssens, T. F. Wang, and B. D. Wilkins, Phys. Rev. C 40, 2091 (1989). S. J. Sanders, Phys. Rev. C 44, 2676 (1991). M. Greiner and W. Scheid, J. Phys. G: Nucl. Part. Phys. 12, L229 (1986). R. K. Gupta, N. Singh, and M. Manhas, Phys. Rev. C 70, 034608 (2004). P. Möller and J. R. Nix, Nucl. Phys. A 361, 117 (1981). ´ W. D. Myers and W. J. Swiatecki, Ann. Phys. 55, 395 (1969); Nucl. Phys. A 336, 267 (1980). C. W. de Jager, H. de Vries, and C. de Vries, At. Data Nucl. Data Tables 14, 479 (1974); H. de Vries, C. W. de Jager, and C. de Vries, ibid. 36, 495 (1987). K. A. Brueckner, J. R. Buchler, and M. Kelly, Phys. Rev. 173, 944 (1968). P. R. Christensen and A. Winther, Phys. Lett. B 65, 19 (1976). V. Y. Denisov, Phys. Lett. B 526, 315 (2002). G. Royer and R. Rousseau, Eur. Phys. J. A 42, 541 (2009). R. Kumar, M. Bansal, S. K. Arun, and R. K. Gupta, Phys. Rev. C 80, 034618 (2009). R. K. Gupta, M. Manhas, and W. Greiner, Phys. Rev. C 73, 054307 (2006).