α decay half-lives for Z = 108, 114, 120, 126 isotopes and N = 162, 184 isotones

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2013 J. Phys. G: Nucl. Part. Phys. 40 045103

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IOP PUBLISHING JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS

J. Phys. G: Nucl. Part. Phys. 40 (2013) 045103 (7pp) doi:10.1088/0954-3899/40/4/045103

α decay half-lives for Z = 108, 114, 120, 126 isotopesand N = 162, 184 isotones

J M Wang1, H F Zhang1 and J Q Li2

1 School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,People’s Republic of China2 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,People’s Republic of China

E-mail: zhanghongfei@lzu.edu.cn

Received 25 October 2012Published 6 March 2013Online at stacks.iop.org/JPhysG/40/045103

AbstractWithin the generalized liquid drop model (GLDM) including the proximityeffects between nuclei in the neck and the gap between the nascent fragments,the α decay half-lives of the 108,114,120,126 isotopes and 162,184 isotones areinvestigated in the quasi-molecular shape path. The Qα is obtained from themacroscopic–microscopic method (MMM) or from the experimental data.The calculated results are compared with experimental data, and both revealthat 270Hs is a double magic nucleus. The correct agreement allows us tomake predictions for the α decay half-lives of other still-unknown superheavynuclei(SHN) in the same framework. It is concluded that 298114 may be adouble magic nucleus and N = 184 the neutron magic number for Z = 120and Z = 126, however, further experimental evidence is needed.

(Some figures may appear in colour only in the online journal)

1. Introduction

Studies on the synthesis mechanism and the structure and decay properties of superheavynuclei(SHN) [1–13] have a very important significance for understanding current nucleartheory. So far, the elements Z = 112 to 118 have been synthesized using cold-fusionreactions with the double magic nucleus 208Pb or nearly magic nucleus 209Bi and withhot-fusion reactions with the double magic nucleus 48Ca bombarding actinide nuclei in afusion–evaporation reaction mechanism. The most synthesized SHN are observed via theirα cascade decay [2, 14–20]. Recently the importance of heavy-particle radioactivity hasbeen put forward and investigated [21–23] for the heaviest nuclei. The experimental resultsstimulated new theoretical studies on the α decay. Presently, many theoretical approaches areused to describe the α decay, such as the cluster model [24], the density-dependent M3Y(DDM3Y) effective interaction [25, 26], the generalized liquid drop model (GLDM) [27, 28],the relativistic mean field theory [29] and the Skyrme–Hartree–Fock mean field model [30].

0954-3899/13/045103+07$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. G: Nucl. Part. Phys. 40 (2013) 045103 J M Wang et al

All the above theoretical methods are concentrated on the half-lives of these observed SHN,and the results are reasonably consistent with the experimental data.

Among the mentioned theories, the GLDM is one of the most-successful macroscopicmodels for describing the process of fusion [31], fission [32], the light nucleus [33, 34] andα particle emission [35, 36], as well as for obtaining a precise nuclear radius and mass, andfor investigating charge asymmetry, deformation and the proximity effect. The process of α

decay in quantum mechanics is considered to be a potential barrier-penetration problem, so theWKB method can be used to describe the half-lives of α decay. Two points are important forcorrectly describing the α decay half-life: one is the reasonable height of the potential barrier;the other is the position of the barrier in the potential. In GLDM, the proximity energy reducesthe barrier height by several MeV and moves the position of the barrier top, which correspondsto the two separated fragments being in an unstable equilibrium by the balance between theattractive nuclear proximity force and the repulsive Coulomb force, outwards. Unlike thedensity-dependent cluster model (DDCM) [37], the GLDM is a macroscopic model with aclear physical image and is easily generalized. The microscopic effects are important for alphadecay. In the GLDM, the proximity energy is introduced to take into account the microscopicnucleon–nucleon interaction between the two quasispherical decay products, and the shelleffect is included in the experimental decay energy Q. Here it is used to describe α decayhalf-lives of Z = 108, 114 isotopes and N = 162 isotones by using both the experimentalQα values [38] and those from theoretical macroscopic–microscopic method (MMM) [39]energies and using the WKB approximation, then to compare the results with the experimentaldata. Finally predictions within the GLDM are given for the partial α decay half-lives of thestill unobserved SHN Z = 120, 126 isotopes and N = 184 isotones.

2. Brief introduction of GLDM

For any deformed nucleus, the macroscopic total GLDM energy is defined as [31]:

E(def) = ELDM(def) + EProx(def), (1)

where ELDM(def) and EProx(def) are the liquid droplet energy and proximity energy,respectively. Assuming that the density and volume of a nucleus are not changing, the liquid-droplet energy can be represented as:

ELDM(def) = EV (def) + ES(def) + EC(def). (2)

The one-body volume energy EV (def), surface energy ES(def) and Coulomb energy EC(def)are:

EV (def) = −aV (1 − κV I2)A, (3)

ES(def) = aS(1 − κSI2)A2/3 S

4πR20

, (4)

EC(def) = 0.6 e2 Z2

R0

1

2

∫V (θ )

V0

(R(θ )

R0

)3

sin θ dθ, (5)

where A, Z and I = (N − Z)/A are the mass number, charge number and relative neutronexcesses, respectively. V (θ ) is the electrostatic potential at the surface and V0 is the surfacepotential of the sphere. The volume coefficient and surface coefficient are aV = 15.494and aS = 17.9439, respectively. The volume asymmetry coefficient and surface asymmetriccoefficient are κV = 1.8 and κS = 2.6, respectively. The effective radii R0 are given by

R0 = (1.28A1/3 − 0.76 + 0.8A−1/3). (6)

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 045103 J M Wang et al

After the two bodies are separated, the volume energy EV (def), surface energy ES(def) andCoulomb energy EC(def) are:

EV = −aV[(

1 − κV I21

)A1 + (

1 − κV I22

)A2

], (7)

ES = aS[(

1 − κSI21

)A2/3

1 + (1 − κSI2

2

)A2/3

2

], (8)

EC = 0.6 e2Z21

R1+ 0.6 e2Z2

2

R2+ e2Z1Z2

r, (9)

where Ai, Zi, Ri and Ii are the masses, charges, radii and relative neutron excesses of the twonuclear, respectively. Each coefficient has the same value as that from the one-body situation.

The surface energy results from the effects of surface tension forces in a half space. Whenthere are nucleons in close proximity in a neck or in a gap between separated fragments, anadditional term called proximity energy must be added to take the effects of the nuclear forcesbetween the close surfaces into account. This term is essential to smoothly describe the one-body to two-body transition and to obtain reasonable fusion barrier heights. It moves thebarrier top to an external position and strongly decreases the pure Coulomb barrier:

EProx = 2γ

∫ hmax

hmin

φ

[D(r, h)

b

]2πh dh, (10)

where h is the distance varying from the neck radius or zero to the height of the neck border.D is the distance between the surfaces in regarding and b = 0.99 fm is the surface width. φ

is the proximity function of Feldmeier [40]. The surface parameter γ is the geometric meanbetween the surface parameters of the two nuclei or fragments:

γ = 0.9517√(

1 − 2.6I21

)(1 − 2.6I2

2

). (11)

The combination of the GLDM and of a quasi-molecular shape sequence has allowed us toreproduce the fusion barrier heights and radii, the fission and the α and cluster radioactivitydata.

3. Half-lives of superheavy nuclei

The main decay mode of SHN is α emission, which is a quantum tunnelling through thepotential barrier leading from the mother nucleus to the two emitted fragments: the α particleand the daughter nucleus. The height and position of the potential barrier play an importantrole in calculating the half-life of the α decay, since they directly affect the barrier penetrationprobability, which can be described successfully using the WKB approximation. The barrierpenetrability P is calculated with the action integral:

P = exp

[−2

�

∫ Rout

Rin

√2B(r)[E(r) − Esphere] dr

]. (12)

The deformation energy is small until the rupture point between the fragments [35]. It is worthmentioning that the influence of the deformation-dependent shell correction on the alpha decaypotential was investigated in [34], and one may find out that the shell correction decreasesdeeply with increasing deformation, so that the shell correction only affects the alpha decaypotential before the input point on the barrier, and will thus not affect the α-decay half-lives. Inthe present work we aim to find double magic nuclei in the super-heavy region; the deformationmay not be taken into account. In the equation (3.12), the two following approximations areused:

Rin = Rα + Rd, (13)

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 045103 J M Wang et al

Rout = e2ZDZα/Qα, (14)

and

B(r) = μ

{1 + f (r)

272

15exp

[−128

51(r − Rin)/R0

]}(15)

here,

f (r) =

⎧⎪⎨⎪⎩

(Rcont − r

Rcont − Rin

)2

, r � Rcont

0, r � Rcont

(16)

However the mass inertia from these formulae does not deviate noticeably from B(r) = μ,where μ is the reduced mass of the daughter nucleus and α particle, so B(r) = μ is used inthe work.

The decay constant of the α emitter is simply defined as

λ = Pαν0P, (17)

where ν0 is the assault frequency which has been taken as:

ν0 = 1

2R

√2Eα

M. (18)

An analytic formula is proposed for the preformation factor because of the nuclear shellstructure [41]:

Pα = exp[a + b(Z − Z1)(Z2 − Z) + c(N − N1)(N2 − N) + dA], (19)

where Z, N and A are the charge, neutron and mass number of the parent nucleus, respectively,which provided general guidance for the microscopic study of the particle preformation factorand nuclear structure as studied in [41]. Z1 and Z2 are the proton magic numbers around Z(Z1 < Z � Z2), and N1 and N2 are the neutron magic numbers around N (N1 < N � N2). Soin a different scope of Z or N, Z1, Z2 or N1, N2 are different, and naturally the parameters a,b, c and d are also different because of the microscopic nuclear structure. The correspondingdetails are presented in previous study [41].

The half-life is related to the decay constant λ, which can be calculated by:

T1/2 = ln 2

λ. (20)

The half-life is extremely sensitive to the α decay Q value, such that an uncertainty of 1 MeVin Q value corresponds to an uncertainty of α-decay half-life ranging from 103 to 105 times inthe heavy element region [42]. So it is significant to get a reasonable Q value for the half-livesof α-decay. Recently, a MMM [39] was developed. The root-mean-square (rms) deviation withrespect to 2149 measured nuclear masses is reduced to 0.441 MeV while the correspondingresult of FRDM is 0.656 MeV. The rms deviation of the α-decay energies of the 46 SHN fallsto 0.566 MeV with FRDM but to 0.263 MeV with the MMM [39]. Thus MMM has obtainedone of the best results to date. The α-decay energy can be calculated by the binding energies,and where the binding energies may be obtained by MMM: Qα = ED

b + Eαb − EP

b , where EDb ,

Eαb and EP

b are the binding energies of the daughter nucleus, α particle and parent nucleus,respectively.

In this work, we calculate the partial α-decay half-lives of Z = 108, 114, 120, 126 isotopesand N = 162, 184 isotones within the GLDM using experimental α-decay energies [38] andtheoretical MMM energies [39] for comparison. The calculations are shown in figures 1 and2, respectively. In figure 1, circles indicate the results of GLDM by using experimental α

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 045103 J M Wang et al

(a) (b)

Figure 1. Comparison between experimental α-decay half-lives and those from theoretical resultsusing GLDM with QExp and QMMM for the nuclei Hs, and the same for element Z = 114 nuclei.

(a) (b)

Figure 2. The predictions of calculated α-decay half-lives for Z = 120 and Z = 126 isotopes usingGLDM.

decay energies, and black squares indicate those derived from theoretical MMM energies.The experimental α-decay half-lives are presented by red solid dots for comparison. It canbe seen that the half-lives of these SHN vary from microseconds to years. For Z = 108 (Hs)isotopes, the maximum value of α-decay half-lives lies at the neutron magic number N = 162,which is confirmed by experiment[43]. The long α-decay half-lives beyond the 277Hs do notmeasure the stability of the nuclei, since experimentally it is detected that the 277Hs goes tospontaneous fission with the half life being 11 ms, so naturally the isotope heavier than 277Hshas an even shorter fission half life. The α-decay half-lives of the Z = 114 isotope chainexplicitly display a maximum at the number N = 184, which may imply a neutron magicnumber. With careful observation, it can be found that the calculated α-decay half-lives fromexperimental Qα coincide excellently with the experimental ones, implying that as long as wehave the correct Qα , the presently used method can give precise results for α-decay half-lives.It can also be found that the calculated α-decay half-lives with Qα from MMM are reasonablyconsistent with those from experimental data, which allow us to predict the α-decay half-livesusing the present method. In the following, the α-decay half-lives of SHN, which have notbeen synthesized, are predicted using GLDM. The calculated results are shown in figure 2.It can be seen that there are long-lived isotopes for Z = 120 and Z = 126 nuclei againstα-decay, and the shortest α-decay half-lives are all located at N = 186. From N = 184 toN = 186, there is a sudden drop down for the α-decay half-lives; in this scope the N = 184nuclei looks relatively stable against α-decay. This seems to imply that N = 184 is a shellmagic number for Z = 120 and Z = 126, but it requires further experimental identification.

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 045103 J M Wang et al

(a) (b)

Figure 3. Comparison between experimental α-decay half-lives and those from theoretical resultsof 162 isotones using GLDM with QExp and QMMM, respectively, and the α-decay half-lives forN = 184 isotones by QMMM.

It is worth reiterating that beyond N = 184, spontaneous fission may dominate, and the longα-decay half-lives cannot compete.

Now we use GLDM to study the α-decay half-lives for N = 162 and N = 184 isotonesas a function of the element charge number Z. The α decay half-lives for N = 162 andN = 184 isotones are shown in figure 3, and hopefully this will give us some informationabout the proton magic number in this region. As shown in the panel (a) of figure 3, in general,the α-decay half-lives of the N = 162 isotones decrease with increasing proton number. AtZ = 108, the solid dot stands for the experimental result. The result by using the experimentalQ value(represented by open circles) shows a sudden drop down with increasing Z, suggestingthat Z = 108 is a magic proton number for N = 162 isotones. The α-decay half-lives ofN = 184 isotones by QMMM is shown in panel (b) of figure 3. The sudden drop down ofα-decay half-lives clearly appear at Z = 114, implying that Z = 114 is a proton magic numberfor N = 184 isotones.

4. Conclusion

In conclusion, the half-lives for α-decay have been analysed in the quasi-molecular shape pathwithin a generalized liquid drop model (GLDM) including the proximity effects, the massand charge asymmetry. The results from experimental Qα by using GLDM are in reasonableagreement with those from the published experimental data for the α-decay half-lives ofZ = 108 and Z = 114 isotopes and N = 162 isotones. It is found both experimentally andtheoretically that 270Hs is a double magic nucleus. Predictions are made for the partial α-decayhalf-lives of some unknown superheavy nuclei. The calculation indicates that 298114 may bea double magic nucleus and N = 184 is a magic number for Z = 120 and Z = 126. Currentlyan outstanding aim of nuclear physics is to confirm the existence of the island of stability ofsuperheavy nuclei. Different models have predicted different magic numbers and up to nowthis island of stability has not yet been localized experimentally. GLDM may be useful forseeking the location of the island of stability, and may shed some light on future synthesizingand identification of SHN experimentally.

Acknowledgments

HFZ is grateful to Professor Royer for the valuable discussion on GLDM. The workis supported by the Natural Science Foundation of China (grants 10775061, 10805061,

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 045103 J M Wang et al

11105035, 10975064, 11120101005 and 11175074), the Fundamental Research Funds forthe Central Universities (grants lzujbky-2012-5), by the CAS Knowledge Innovation projectno. KJCX-SYW-N02.

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