Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 201–203, 2002

201

Alpha-decay Half-lives and Fission Barriers for Superheavy Nuclei Predicted by a Nuclear
Mass Formula
H. Koura∗,a,ba
RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japanb
Advanced Research Institute for Science and Engineering, Waseda University, Okubo 3-4-1, Shinjukuku, Tokyo 169-8555, Japan
Received: December 23, 2001; In Final Form: December 23, 2001
We estimate α-decay half-lives from Qα values with phenomenological formulas. The parameter values of theformulas are adjusted by using experimental half-lives and Q values of the Evaluated Nuclear Structure Data File(ENSDF) for a wide nuclidic region. The half-lives Tα for unknown nuclei are estimated by this phenomenological formula with the use of Qα values taken from KUTY mass formula. In addition to the half-lives, we estimate
spontaneous fission barriers calculated by the method to obtain the shell energies of KUTY formula. In the superheavy region, the barrier heights of the nuclei near the nucleus 304 122 are about 8 MeV and their widths are fairlywide. Therefore these spontaneous-fission half-lives are expected to be very long. On the other hand, there are someneutron-deficient nuclei whose spontaneous-fission half-lives expected to be rather short because their fission barrierheights are small and the widths of them are narrow.

1. Introduction
The productions of superheavy nuclei are recently well performed.1 In this region, α decay and spontaneous fission are themain decay modes.
Our group has recently constructed a nuclear mass formula,2, 3
which we refer to as KUTY formula, composed of a gross partand a shell part. For the shell part, we first calculate protonand neutron spherical shell energies by using modified WoodsSaxon type potentials, which we have newly constructed.4 A
notable feature of our mass formula is a new method of obtaining shell energies of deformed nuclei. The shell energy ofa deformed nucleus is expressed as an appropriate mixture ofspherical shell energies added to an average deformation energy.
This mass formula gives ground-state masses and shapes for thenuclei ranging from 4 He to the superheavy nuclei. The standard deviation of the calculated masses from the experimental
masses of the 1995 Mass Evaluation5 is about 680 keV. Using
this formula, we estimate α-decay Q values and spontaneous fission barriers.6 As for the estimation of the α-decay half-lives,
we take some phenomenological formulas with some adjustableparameters. As for the spontaneous fission barriers, we calculate
the potential energy surface by the method to obtain the shellenergies of KUTY formula and then obtain the fission-barrierheight.
In sect. 2 we estimate the α-decay half-lives, and in sect. 3
we show the spontaneous fission barrier heights.
2. Alpha-decay Half-life
2.1. Phenomenological Formulas of α-decay Half-lives.
We first estimate the α-decay half-lives Tα (s) from experimental Qα values (MeV) with a phenomenological relation in a widenuclidic region. The α-decay half-life is written as
Tα = loge 2/(Ncoll × P) ,(1)

where Ncoll is collision frequency of an α particle to a potential
wall and P is penetration probability. In the WKB approximation, the probability P for a spherical nucleus is approximatelywritten asZmα m f2 b{2(V (r) − Qα )}1/2 dr ,(2)P = exp −h̄ Rmα + m f

where b and R are outer and inner radii of α-particle potential
V (r) penetrated by α particle with Qα , and mα and m f are massesof an α particle and a daughter nucleus, respectively.
Here we consider two phenomenological formulas. One isthe Viola-Seaborg formula7 with an even-odd hindrance term has
Formula (A)log10 Tα = (aZ + b)/ Qα + (cZ + d) + h ,(3)where a, b, c, d are parameters.
Another formula is deduced from the penetration probability
neglecting higher order terms asFormula (B)A−4Z D / QαAA−4 √− 1.2901
RZDA
A−4 3/2 1/2+ 0.07466
R /ZD QαA− log10 Ncoll − 1.59175 + h ,log10 Tα = 1.7195(4)

whereR = r0 A1/3
D + d0 ,r0 = 1.08 fm ,(5)where the decimals in eq 4 are obtained when we take a spherical
Coulomb potential, the subscript D indicates a daughter nucleus,
and Ncoll and d0 are adjustable parameters.
These parameter values are adjusted with the use of the experimental Tα and Qα of the Evaluated Nuclear Structure Data
File (ENSDF), 2000 August version.8 We remove the data estimated by systematics or calculations, the data having only upperor lower limits, and the data of 8 Be from input data. The Qα of180
Pb is evaluated as 5.851 MeV in the ENSDF. However, sincethe original experimental α-particle energy Eα is 7.23 MeV,9we take 7.39 MeV [ = Eα × A/(A − 4)] as Qα for 180 Pb. We firstadjust the parameters for even-even nuclei. As for odd-A andodd-odd nuclei, we use the same parameters for even-even nuclei and then we determine h so as to reproduce the reasonablehalf-lives for odd-A and odd-odd nuclei. As a result, we choose
a simple expression of h as∗Corresponding author. E-mail: koura@postman.riken.go.jp.
FAX: +81-(0)48-462-4707.

h = h0 δeo ,c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002(6)

Koura1Log(Tαcal./Tαexp.)

108
6
4
20-1
odd-A-26080

80

1001201401 Formula (B)01401602h0

10-1-2odd-odd-360

even-even6080

100120

Neutron number Nfor even-evenfor odd-A
for odd-odd.

100120140160

Neutron number N

Figure 1. Experimental Qα (upper). Differences between experimental
and estimated log Tα of Formula (A) (middle) and Formula (B) (lower).
All data are for even-even nuclei.

 0δeo =1
280

140

Figure 2. Differences between experimental and estimated Tα for oddA nuclei (upper) and for odd-odd nuclei (lower). The even-odd hindrance factors h0 and 2h0 are also seen as dashed lines.
Tα (exp.)
Tα (Formula (B))
Tα (Formula (A))

1000
10010(7)Here, h0 is taken from the average of differences of experimental
half-lives from estimated ones (as h = 0) for odd-A nuclei.
The results for two formulas are in the following.
Formula (A)The values of parameters are a = 1.55261, b = 0.73247,
c = −0.21669, d = −31.9949, and h0 = 0.56718 for Qα and Tα in
MeV and second, respectively. The root-mean square (RMS)
deviation of log Tα from experimental ones of 120 even-evennuclei is 0.3625. The RMS deviation is 0.7708 for 151 oddA nuclei and is 0.9845 for 63 odd-odd nuclei. Although 10−droughly corresponds to the collision frequency Ncoll of the α particle which should be about 1020−22 , the above absolute value ofd seems to be too large. In Figure 1 (the upper and middle parts),
we show the experimental Qα and the differences between theexperimental and estimated Tα with the use of Formula (A).
Formula (B)The values of fitted parameters are Ncoll = 1020.05 , d0 = 2.0 fm,
and h0 = 0.61410 for Qα and Tα in MeV and second, respectively.
The values of Ncoll and d0 are within reasonable values. The
RMS deviation of log Tα for 120 even-even nuclei is 0.3512. In
Figure 1 (the lower part), we show the differences between theexperimental and estimated Tα with the use of Formula (B). In
the region 126 ≤ N ≤ 142, the discrepancy of Formula (B) is reduced in comparison with one of Formula (A). Both of the middle and lower figures show distinct discontinuities at N = 126
because of the magicity. At N = 102 (174 Hf102 ), large discrepancies are also seen. This nucleus is located on the vicinity ofβ-stability line and is isolated from the other even-even nuclei
on the N-Z plane and have relatively larger deformation thanthe others. We show the differences between the experimental
and estimated Tα for odd-A and odd-odd nuclei in Figure 2. The
RMS deviation is 0.7500 for 151 odd-A nuclei, and is 0.9802 for
63 odd-odd nuclei.
2.2. Estimation in the Superheavy Region. In order tocompare the above two formulas, we show the experimental andTα [s]

-1

120

2even-even

Log(Tαcal./Tαexp.)

-1

100

Neutron number N

Neutron number N
Log(Tαcal./Tαexp.)

h00060

with1

Formula (A)

10.1

0.01

285

0.001 284
0.0001112173287

114173 289114175

284

292

288

112172116176114174286

288290Mass number A

292

294

Figure 3. Estimated and experimental α-decay half-lives Tα in the
superheavy nuclidic region. Dotted lines connect α-decay chains.

20Tα from KUTY(Z=even)N − Z =70

N − Z =80

Z=114
Z=106 Z=110

LogTαLog(Tαth./Tαexp.)

Qαexp [MeV]202 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

10 ydm0 s

Z=102−10160

Z=118 Z=122 Z=126
164

168

N − Z =50

172
176
180
Neutron number N

N − Z =60

184

188

Figure 4. Tα of superheavy nuclei by KUTY formula2, 3 for even Z. We
use Formula (B) to estimate Tα . The solid lines connect isotopes anddotted lines connect α-decay chains.

estimated Tα for the superheavy nuclei in Figure 3. In this estimation, experimental Qα are taken from Reference 1. These
nuclei are not input data for parametrization because these werelacking or estimated data in the ENSDF file. This figure showsAlpha-decay Half-lives and Fission Barriers

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

280112
6.0

0.10

4.0

0.0

2.0

6.0
4.0
2.0
0.0

0.0

0.05

α4-2.0

0.00

-2.0

-4.0
0.0
2.0
4.0

0.0

-0.1

-6.0

-2.0

-4.0
-2.0

-0.10
-0.2

-8.0

-4.0

-2.0

-4.0

-4.0

-0.05

-10.0

-0.0

0.1

0.2

α20.30.40.5

Figure 5. Calculated energy surface of 280 112. The ground-state shapeof this nucleus is at about α2 = 0.11 and α4 = −0.06.
300120α40.10 6.0

4.0
2.0

8.0

0.05

4.0

0.00

2.0 0.0-2.0MeV-2.0-8.0 -9.0

-4.0

-0.05
-0.10
-0.2

-10.0

-14.0

0.0
-12.0

-6.0
-8.0

0.0

-6.0

-0.1

-6.0
-4.0

2.0
4.0

-0.0

0.1

0.2

α20.3

0.4

0.5

Figure 6. Calculated energy surface of 300 120. The ground-state shapeof this nucleus is at about α2 = α4 = 0.0.

130

140

150

1.00.0160

170

Proton number Z

130120

1.0MeV

190

200

2.0

1.0

3.0

7.0
8.0
6.0

120

1.01102.0

4.01001006.0 5.0
7.090
9.0

130

14090

8.0150 160 170 180
Neutron number N

are defined as the highest saddle points from the ground-stateshell energies towards the prolate shapes. In this report we takethe α2 , α4 , α6 deformations in the range −0.2 < α2 < 0.5.
We show the energy surfaces against the nuclear deformation
for two superheavy nuclei in Figures 5 and 6. For the nucleus
280
112, the height of the fission barrier is only about 2 MeV and
its width is relatively narrow. The spontaneous-fission half-life
is consequently expected to be rather short for this nucleus. On
the contrary, for the nucleus 300 120, the fission barrier height
is about 8 MeV, and this width is fairly wide. Therefore, thespontaneous fission of this nucleus is expected to have a very
long partial half-life, much longer than the α-decay half-life.
We show the fission barrier heights in Figure 7 for even-evennuclei in the range 84 ≤ Z ≤ 130 and 126 ≤ N ≤ 200. The nuclei which locate below the solid line may have a higher saddle
point in the region α2 > 0.5 because we limited the range on thepresent calculation.
This figure shows the “hill” of the barrier heights of the nuclei near 304 122. These barrier heights are about 8 MeV or more.
Therefore these spontaneous-fission half-lives are expected to bevery long. On the contrary, the “basin” of the barrier heights of
the nuclei near 278 110 is also seen. These heights are about 2MeV. There are also other neutron-deficient nuclei having relatively small fission barrier heights whose spontaneous-fission
half-lives expected to be rather short.

1305.0
4.0

2.0

110

180

203

190

200Figure 7. Fission-barrier heights for even-even nuclei. The dashedline is the proton-drip line of KUTY formula (even-Z). The neutron-richnuclei located below the solid line may have the higher saddle point inthe region α2 > 0.5.

that the estimated Tα are smaller than the measured ones. Thevalues from Formula (B) are relatively larger than those from
Formula (A) for the nuclei with large mass numbers.
With the use of Formula (B), we systematically calculate theTα for superheavy nuclei. In order to estimate the Qα of superheavy nuclei, we use the KUTY mass formula.2, 3 The result isshown in Figure 4. In this figure our α-decay half-lives present
a feature of magicity at Z = 114 and at Z = 126 as relatively widegaps between isotope lines, while a similar figure with the useof FRDM mass formula10 has a larger gap only at Z = 114, andthat with the use of ETFSI mass formula11 shows no gap. (Theresults of FRDM and ETFSI are not shown in the figures.) The
magicity at N = 184 is also seen as steep decreasing of isotope
lines just beyond N = 184. The oscillations of the isotope lines
are seen because of the even-odd hindrance effect.
3. Spontaneous Fission
Although our mass formula is constructed by considering
only the equilibrium nuclear shapes, the potential energy surface
for spontaneous fission can be calculated by the same method as
used for obtaining the shell energies. The fission barrier heights

Acknowledgment. The author thanks Prof. M. Yamada forvaluable comments. The numerical calculations were mainly
done with the computer VX at Media Network Center (MNC),
Waseda University and with the VPP700 at Computer and Information Division, RIKEN. This work was financially supported
by MNC of Waseda University as Specific Studies A in 2001.
References(1) Yu. Ts. Oganessian, Nucl. Phys. A 685, 17c (2001); Yu.
Ts. Oganessian, Tours Symposium on Nuclear Physics IV(TOURS2000), Tours, Sep. 4–7, 2000, AIP Conf. Proc. 561(AIP, New York, 2001), p. 303.
(2) H. Koura, M. Uno, T. Tachibana, and M. Yamada, Nucl.
Phys. A 674, 44 (2000).
(3) H. Koura, M. Uno, T. Tachibana, and M. Yamada, RIKENAF-NP-394 (The Institute of Physical and Chemical Research (RIKEN), 2001).
(4) H. Koura and M. Yamada, Nucl. Phys. A 671, 96 (2000).
(5) G. Audi and A. H. Wapstra, Nucl. Phys. A 595, 409 (1995).
(6) Some early results for superheavy nuclei appeared in:
H. Koura, Tours Symposium on Nuclear Physics IV(TOURS2000), Tours, Sep. 4–7, 2000, AIP Conf. Proc. 561(AIP, New York, 2001), p. 388; H. Koura, T. Tachibana, andT. Yoshida, International Conference on Nuclear Data for
Science and Engineering (ND2001), Tsukuba, Oct. 7–12,2001.
(7) V. E. Viola, Jr. and G. T. Seaborg, J. Inorg. Nucl. Chem. 28,741 (1966).
(8) Evaluated Nuclear Structure Data File (ENSDF), 2000 August version, communicated through Nuclear Data Center,
Japan Atomic Energy Research Institute.
(9) K. S. Toth, J. C. Batchelder, D. M. Moltz, and J. D.
Robertson, Z. Phys. A 355, 225 (1996).
(10) P. Möller, J. R. Nix, W. D. Myers, and J. Swiatecki, At.
Data Nucl. Data Tables 59, 185 (1995).
(11) Y. Aboussir, J. M. Pearson, A. K. Dutta, and F. Tondeur, At.
Data Nucl. Data Tables 59, 409 (1995).