α-α Folding Cluster Model for α-radioactivity

A. Soylu†, and O. Bayrak‡

†Department of Physics, Nigde University, 51240, Nigde, Turkey and

‡Department of Physics, Akdeniz University, 07058, Antalya, Turkey

Abstract

The α-decay half-lives are calculated for heavy and superheavy nuclei for 52 ≤ Z ≤ 112 and

108 ≤ A ≤ 285 from the ground state to ground state α transitions within the framework of

the Wentzel-Kramers-Brillouin (WKB) method and the Bohr-Sommerfeld quantization. In the

calculations, the α-α single folding cluster potential obtained with the folded integral of the α-α

potential with the α-cluster density distributions is used in order to model the nuclear interaction

between the α-particle and core nucleus. While the results are very good agreement with the

experimental ones in the heavy nuclei region especially for even-even nuclei, the smaller values

than the experimental ones are obtained for superheavy nuclei. As both the density of the core

and the interaction term in the folding integral include the α-clustering effects and in this way all

cluster effects are taken into account in the model, the results of calculations are more physical and

reasonable than the calculations done in the other models. The present method could be applied

to light nuclei with different type nuclear densities.

PACS numbers: 21.60.-n, 21.60.Gx, 21.10.Tg, 23.60.+e, 27.70.+q, 27.80.+w, 27.90.+b

Keywords: α decay, Cluster model, Double folding, Half-life, Superheavy nuclei, WKB method

1

I. INTRODUCTION

In recent years, the α-decays of nuclei have been very useful tool for investigating nuclear

structures. One of these sort of studies is that the α-decay half-life measurements in the

neutron-deficient region and the results of these might give unique shell model information

and show some evidence for shape coexistence [1, 2]. On the other hand, the α-decay is

still being used as a suitable way to identify new superheavy elements studied at accelerator

centers such as Berkeley, GSI, Dubna, GANIL, RIKEN [3, 4]. Furthermore, the α-particle

measurements play an important role in the search for exotic nuclear molecular states in

light nuclei [5]. Even though the α-decay is very helpful for the experimental nuclear physics

studies, the development on the quantitative description of the α-decay in theoretical side

is slow a bit due to the complexity of both the nuclear potential and the nuclear many-body

problem. In order to be able to understand the α-decay mechanism for different heavy and

superheavy nuclei, numerous studies have been done in the literature [6–18]. Furthermore,

investigating the influence of the deformations on the half-lives by charge instability has

attracted a great deal of interest more recently [19–23].

Theoretically, the α-decay is considered conventionally in the framework of the Gamow

model in which the α-particle might be assumed as tunneling through a potential barrier

between the cluster and the daughter nucleus [24]. In the model, the α-particle is already

preformed in the parent nucleus and can penetrate the Coulomb barrier. Therefore the α-

decay width is correspondingly formulated by the product between frequency of the collision

with the potential walls and the barrier penetrability P which can be calculated by using the

semiclassical WKB model[24–26]. In order to be able to determine the interaction between

the α-particle and daughter nucleus, one needs to know the shape of the effective interaction

potential. The effective potential includes the nuclear, the Coulomb and the Langer-modified

centrifugal potentials. In this potential, the Coulomb and the centrifugal potential are known

very well but as one does not write any mathematical equation for the nuclear force we do

not know the proper form of the nuclear potential. Therefore some models and ideas have

to be used in order to determine the nuclear potential. In the literature, there are many

phenomenological and microscopic potentials which are used to model the α-core interaction.

These potentials are Cosh potential [27], optical model potential form in which the potential

parameters are obtained from the elastic scattering experiments [28], the density-dependent,

2

the double folding, potential that consists of the density of the α-particle and the core and

also the nucleon-nucleon interaction [29, 30]. Authors have lately proposed the mean field

potential which could also be applied to such kind of the system [31] and then this potential

was used to investigate the nuclear molecular structure in the light nuclei [32].

More recently, in particular the density-dependent potentials have been applied to light,

heavy and superheavy systems in order to obtain different observable such as the decay

widths, the half-lives and the elastic scattering cross-sections [33–40]. In these type cal-

culations with density-dependent potentials, the effective nuclear potential between the α

and daughter is derived from a double folded integral of the renormalized M3Y potential

with the density distributions of the α-particle and the daughter nucleus. The popular M3Y

potential is derived by Bertsch et al. [41] where it comes from the fitting of the G matrix

element of the Reid potential. The parameterized form of the M3Y potential proposed by

Satchler and Love [42] can be used for the calculations of the α-decay half-lives. In general,

in these calculations, the nucleon-nucleon interaction term with the M3Y potential and the

densities are formulated with the Gaussian form.

In this paper, we consider that the interaction potential is in the α-α single folding

cluster model. Therefore we use the α-α interaction potential of Buck et al. [43] instead of

the nucleon-nucleon term in the model and we get also the densities of the daughter nuclei

within the nuclear matter density code [44]. It should be noted that since both the density

of the daughter and the interaction term in the folding integral include the α-clustering

effects, it allows us the present method is more physical than the others and in this sense

all possible α-cluster effects are considered in the model. For the first time, we have shown

that the α-α single folding cluster model could be applied to the calculations of the α-decay

half-lives of heavy and superheavy nuclei. By using this model, we have calculated log10 T1/2

values for 52 ≤ Z ≤ 112 nuclei. Furthermore, we have calculated the α-decay half-lives for

superheavy, 108 ≤ A ≤ 285, nuclei.

In Sec. II, we introduce the α-α single folding cluster model (SFC) and how to calculate

the α-decay half-lives. We show the results of this study and discuss the results in Sec. III.

Section IV is devoted to a summary.

3

II. MODEL

In the binary cluster model, the total interaction potential between the α-particle and

the daughter nucleus Veff (r) includes,

Veff (r) = Vnuclear(r) + VCoulomb(r) + Vcentrifugal(r), (1)

where r is the separation radius between the center of mass of the α-particle and the daughter

(core) nucleus. An uniform charge distribution is assumed for the Coulomb potential between

the α and daughter nucleus. Hence the Coulomb potential is [45, 46],

VCoulomb(r) =ZαZde

2

r, r ≥ RC ,

= ZαZde2

2RC(3− r2

R2C), r < RC ,

(2)

where RC is the Coulomb radius and RC = 1.2(A1/3α +A

1/3d ). Aα and Ad are the mass number

of the α-particle and daughter nucleus, respectively. Zα and Zd denote the charge numbers

of the α and the daughter nuclei, respectively [45, 46]. The Langer-modified centrifugal

potential is as follows,

Vcentrifugal(r) =~2

2µ

(L+ 12)2

r2, (3)

where L is the angular momentum of the α-particle and µ is the reduced mass of the α-

daughter system. As for the nuclear potential, in this study we consider the nuclear potential

as the α-α single folding cluster model potential in which the α-α effective interaction is

used instead of the nucleon-nucleon effective interaction and the densities are obtained by

using the α-cluster model and matter densities. Therefore, in the α−α single folding cluster

potential, an α-α effective interaction is folded with the α-core density distributions and the

α-α SFC model potential can be formulated as [47],

VSFC(R) = λ

∫ρcd(r)Vα−α(|R− r|)dr, (4)

where λ, ρcd and Vα−α are the renormalized factor, α cluster distributions for the daughter

nuclei and the effective α-α interaction, respectively [42, 47, 48]. The Vα−α interaction

potential between the core and the cluster nuclei can be obtained by using the nucleon-

nucleon double folding potential based on the M3Y effective interaction involving a suitable

exchange term and Gaussian shape of the nucleon-nucleon densities as [42, 43, 47],

Vα−α(r) = −122.6225 exp(−0.202r2). (5)

4

In order to calculate the core density distribution, the nuclear matter density distribution

of the daughter nuclei is defined as [47],

ρMd(r) =

∫ρcd(r′)ρα(|r − r′|)dr′, (6)

The α density distributions can be parameterized with a form of Gaussian shape as follows

ρα(r) = ρ0e−βr2 , (7)

where ρ0 and β are the depth and the shape parameter of the density respectively. Their

values are ρ0 = 0.4229 and β = 0.7024 [49].

In order to determine the α-cluster density distributions for the daughter nuclei, the

nuclear matter density code [44] can be used for each daughter nucleus. By using this

code, neutron and proton distributions for each daughter nuclei can be predicted within the

Hartree-Fock-Bogolubov (HFB) method based on the BSk2 Skyrme force and the sum of

these densities gives the total densities ρMd(r). In order to calculate the α cluster density

distribution for the daughter nuclei, ρcd(r), we use Fourier transformation technique [42] in

Eq.(6). After obtaining the α cluster density distribution for the daughter nuclei in terms

of Eq.(6) and taking the α density distribution as Gaussian shape, we can calculate the α-α

single folding nuclear potential by using Eq.(4). In Fig.1, we have shown that several nuclear

potentials which are obtained from the α-α single folding cluster model.

The renormalized factor λ in Eq.(4) can be calculated for every single decay by using the

Bohr-Sommerfeld quantization rule,

r2∫r1

√2µ

~2(Q− Veff (r))dr = (G− L+ 1)

π

2, (8)

where G numbers, that comes fromWildermuth conditions [50], are global quantum numbers

and their values can be used as follows [26, 30],

G = 20 (N > 126),

G = 18 (82 < N ≤ 126),

G = 16 (N ≤ 82).

(9)

In this paper, we focused on the favored alpha decays where the variation of angular mo-

mentum and parity between parent nucleus and daughter nucleus 0+ that means l = 0 in

5

Eq.(3) and we considered some l = 0 cases in the calculations as well. In the semiclassi-

cal approximation, there are three classical turning points which are r1, r2 and r3 in order

of increasing distance from the origin and they are obtained by the numerical solutions of

equation of Veff (r) = Q where Q is the α-decay energy for specific decays as described in

Refs. [51, 52].

According to the semiclassical theory, the α-decay width Γ can be formulated as,

Γ = PαF~2

4µexp(−2

r3∫r2

k(r)dr), (10)

where Pα is the preformation probability of the α particle in a parent nuclei [25, 30, 51, 52].

In Eq.(10), the normalization factor is given as,

F = 1/

r2∫r1

1

k(r)dr cos2(

r∫r1

k(r′)dr′ − π

4), (11)

where the squared cosine term might be replaced by 0.5 without significant loss of accuracy

[25, 30, 51, 52]. In Eqs. (10) and (11), the wave number k(r) is given by,

k(r) =

√2µ

~2|Q− Veff (r)|. (12)

The relation between the α-decay half-life and the decay width can be formulated by the

following equation,

T1/2 = ~ln 2

Γ. (13)

In the half-lives calculations, in accordance with other previous similar studies, the prefor-

mation probabilities have been chosen as Pα = 1.0 for even-even nuclei, Pα= 0.6 for odd-A

nuclei and Pα = 0.35 for odd-odd nuclei [51, 53].

III. RESULTS AND DISCUSSIONS

In Table I, II and III, we have shown that the α-decay half-lives, log10 T1/2, for 52 ≤

Z ≤ 107 nuclei for l=0 case within the framework of the α-α single folding cluster model

and experimental ones for them in order to compare the results. In these tables, the first

column denotes mass number A and the second one is proton number Z for the parent

nuclei. Q-values are given in column 3. The experimental and calculated log10 T1/2 values

6

in Ref.[54] as well as calculated log10 T1/2 values in present study are listed in columns 4,5

and 6, respectively. In Table I, log10 T1/2 values can be seen for 108 ≤ A ≤ 188 nuclei, in

Table II, log10 T1/2 values are given for 192 ≤ A ≤ 228 nuclei and Table III shows log10 T1/2

values for 232 ≤ A ≤ 261 nuclei. In Table IV, the calculated log10 T1/2 values can be seen

for different nuclei for lmin =0 case. It is seen from Table I, II, III and IV that log10 T1/2

values for 52 ≤ Z ≤ 107 nuclei are very close to the experimental ones which are taken from

Ref.[54]. Thus, we would say that our results are very good agreement with experimental

ones, especially for even-even nuclei. It should be emphasized that the α-α SFC potential

model produces the α-decay half-lives for the favored α-decays very well. However, when

the values are compared for l = 0 and l = 0, the values for l = 0 cases are better than the

values for l = 0.

In Table V and VI, we list that the α-decay half-lives, T1/2, for superheavy nuclei (261 ≤

A ≤ 281 ) and (270 ≤ A ≤ 285) in the framework of the α-α SFC model and the experimental

ones for them, respectively. In Table V and VI, the first column denotes mass number A and

the second one is proton number Z for the parent nuclei. Experimental Q-values are given in

column 3. The experimental, calculated values for T1/2 in other studies and calculated values

in present study are listed in columns 4, 5 and 6, respectively. While the data in Table V are

taken from Ref.[34], the data in Table VI are taken from Ref.[30]. It is seen from Table V-VI

that the calculated half-lives, T1/2, for 261 ≤ A ≤ 281 and 270 ≤ A ≤ 285 nuclei in the α-α

SFC model are smaller than experimental ones for the favored α-decays. It could be said

that the reason of this would come from the matter densities for superheavy nuclei. Since

the matter density calculation code for them does not give the realistic densities for higher

Z and A numbers very well, the obtained values are a bit different from the experimental

values. So it should be mentioned that if one uses new and different type matter densities

for the daughter nuclei for superheavy, the better results can be obtained by using the α-α

SFC model. Moreover, this would show that the alpha-cluster effects in heavy nuclei region

are more important than superheavy nuclei region.

In order to make a detailed comparison the calculated values with the experimental data,

we plot the log10 T1/2 for the α-decay of 108 ≤ A ≤ 261 nuclei. In Fig.2, the comparison

between the experimental and calculated log10 T1/2 values for the α-decay of 108 ≤ A ≤ 261

nuclei is shown. Here, the experimental data are taken from Ref.[54]. As can be seen

in Fig.2, the experimental and obtained log10 T1/2 are very close to each other. Further-

7

more, in order to be able to show the agreement the experimental and calculated ones a

hindrance factor(HF) can be defined. The hindrance factor is HF=T exp1/2 /T

calc1/2 and helpful

tool for the α-decay. Fig.3 shows the variation of logarithmic value of the hindrance factor,

log10(Texp1/2 /T

cal1/2), for all related nuclei. As shown in Fig.3 that the logarithmic values of

the hindrance factor have very small values for many nuclei. Moreover, as seen in Fig.3

that log10(Texp1/2 /T

cal1/2) values for 261 ≤ A ≤ 285 are bigger than log10(T

exp1/2 /T

cal1/2) values for

108 ≤ A ≤ 261 nuclei. Thus, this shows that the calculations for 108 ≤ A ≤ 261 nuclei are

much better than the calculations done for superheavy nuclei.

Finally, in order to check the reliability our results with the experimental data and com-

pare them with unified model for alpha-decay and alpha-capture (UMADAC) [54], we have

calculated rms errors of the decimal logarithmic values by using the following equation,

δ =

√√√√ 1

N − 1

N∑k=1

[log10(Tcal1/2)− log10(T

exp1/2 )]

2, (14)

where N denotes the number of the related nuclei[54]. The calculated rms values for different

models can be seen in Table VII. In unified model for alpha-decay and alpha-capture

(UMADAC) calculations, rms errors of the decimal logarithmic values have been obtained

for 344 total, 136 even-even and 48 odd-odd α emitters for the α transition half-lives[54].

Even if the related number of nuclei in our calculations are smaller than the others, the rms

values will give some idea about the model used to calculate half-lives for total number, even-

even and odd-odd nuclei. As seen in Table VII, our rms values are bigger than the results

of Denisov et al.[54]. This shows that the predictions in Denisov et al.[54] are better than

the calculations of present model to describe half-lives of heavy and superheavy nuclei. It

should be noted that α-α single folding cluster model for α-radioactivity is reliable for even-

even heavy nuclei, but the model cannot explain the alpha-decay half-lives of superheavy

nuclei as seen in Fig.3 or odd-odd nuclei in Table VII. In order to improve the results in the

model, we would take into account the clustered and unclustered structures of the nuclei

separately since superheavy nuclei or odd-odd nuclei have the uncluster structure due to

excess neutrons. We would use the single folding model for clustered part of the nuclei and

would propose a potential model for the unclustered part of the nuclei. This proposal is in

progress. Moreover, in calculation, we should suggest a more realistic preformation factor

such as Q-dependent form. The present study is first application of alpha-alpha SFC model

to obtain half-lives in heavy and superheavy nuclei. Although we could not obtain the best

8

results for superheavy and odd-odd nuclei, we showed that the model would be successfully

applied to this sort of the system for even-even nuclei.

IV. SUMMARY

In summary, for the first time we have used the well-known α-α single folding cluster

model in order to calculate the α-decay half-lives of heavy and superheavy nuclei for the

favored α-decay cases. Using this model, we have obtained log10 T1/2 values for 52 ≤ Z ≤ 112

nuclei and our results are very good agreement with experimental results for even-even

heavy nuclei. Besides, we have applied this model to obtain the α-decay half-lives for

superheavy nuclei which have 261 ≤ A ≤ 285 nuclei. Our results are smaller than the

experimental results. As a results α-α single folding cluster model is successful in order

to calculate half-lives of the even-even heavy nuclei but there is need some modification in

the model in explaining of the half-lives of superheavy and odd-odd nuclei. Our conclusion

on modifications in model for odd-odd and superheavy nuclei is that; 1) we would take

into account more realistic nuclear densities[55]. 2) we would consider nuclear densities as

the clustered and unclustered parts in the single folding calculations. 3) we would take an

energy-dependent preformation factor in the calculations. The α-α SFC model would be

crucial important to take into account all cluster effects in the calculations and this method

would be applied to light nuclei in order to make predictions for experimental studies in this

field.

ACKNOWLEDGMENTS

This work was supported by the Turkish Science and Research Council (TUBITAK),

Grant No. 113F225. Authors would also like to thank Nigde and Akdeniz universities for

the their financial supports as well as Prof. I. Boztosun for stimulating discussions and

useful comments on the manuscript. We are grateful to the anonymous referees for their

illuminating criticism and suggestions.

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12

0 5 10 15r (fm)

-300

-250

-200

-150

-100

-50

0

50

100V

(r)

(MeV

)

Veff

(r)104

Sn+α180

Hg+α200

Po+α232

U+α261

Bh+α270

110+α

FIG. 1. The nuclear potentials calculated from the folding integral versus radius for several α-decay

systems. Here, Veff (r) interaction potential versus radius is shown for 108Te→ 104Sn+ α.

13

100 120 140 160 180 200 220 240 260A(mass number)

0

10

20

30

log 10

T1/

2(s)

ExperimentCalculations

FIG. 2. Comparison between the experimental and obtained log10 T1/2 values for the α-decays of

108 ≤ A ≤ 261 nuclei. The experimental data are taken from Ref.[54].

14

A Z Q(MeV ) log10 Texp1/2 (s) [54] log10 T

cal1/2(s) [54] log10 T

cal1/2(s)(present)

108 52 3.445 0.49 -0.19 0.10

112 54 3.33 2.53 2.26 2.15

144 60 1.905 22.86 23.44 22.91

148 62 1.986 23.34 23.75 23.09

148 63 2.694 14.70 14.96 14.28

148 64 3.271 9.37 9.36 9.08

152 64 2.203 21.53 21.90 21.25

152 66 3.726 6.93 6.93 6.48

152 67 4.507 3.13 2.92 2.56

152 68 4.934 1.06 0.79 0.69

156 69 4.344 5.12 5.18 4.62

156 70 4.811 2.42 2.61 2.09

156 72 6.029 -1.63 -2.00 -2.08

160 72 4.902 2.77 3.20 2.65

160 74 6.065 -0.99 -1.25 -1.60

160 75 6.714 -2.02 -2.68 -3.03

164 74 5.278 2.38 2.29 1.85

168 76 5.818 0.62 0.77 0.23

168 78 6.997 -2.70 -2.73 -3.28

172 76 5.227 3.98 3.38 2.92

176 77 5.237 2.60 4.40 3.79

176 78 5.885 1.22 1.26 0.78

176 80 6.897 -1.69 -1.63 -2.20

180 74 2.509 25.75 25.60 24.95

180 78 5.237 4.24 4.27 3.80

180 80 6.258 0.73 0.67 0.12

184 80 5.662 3.44 3.11 2.67

188 80 4.705 8.72 8.46 7.85

188 82 6.109 2.06 2.22 1.58

TABLE I. Comparison of the experimental and calculated log10 T1/2 values for 108 ≤ A ≤ 188

nuclei. The experimental data are taken from Ref.[54].

15

A Z Q(MeV ) log10 Texp1/2 (s) [54] log10 T

cal1/2(s) [54] log10 T

cal1/2 (s)(present)

192 82 5.221 6.57 6.23 5.86

192 84 7.319 -1.48 -1.52 -2.17

196 84 6.657 0.77 0.86 0.22

196 85 7.198 -0.57 -0.38 -0.94

200 84 5.982 3.66 3.71 2.94

200 85 6.596 1.88 1.87 1.23

204 84 5.484 6.28 6.27 5.33

204 85 6.070 4.15 4.37 3.42

204 86 6.545 2.01 1.79 1.38

204 87 7.171 0.39 0.68 -0.11

208 85 5.751 6.04 5.80 4.87

208 86 6.261 3.37 3.01 2.53

208 87 6.785 1.82 1.94 1.27

208 89 7.727 -1.01 -1.06 -1.24

212 84 8.954 -6.52 -6.67 -7.15

212 86 6.385 3.16 2.79 1.93

212 88 7.032 1.18 1.16 0.25

216 84 6.906 -0.84 -0.63 -1.38

216 86 8.200 -4.35 -4.03 -4.73

216 87 9.175 -6.15 -5.88 -6.46

216 88 9.526 -6.74 -6.73 -7.15

216 90 8.070 -1.57 -1.52 -2.50

220 86 6.404 1.75 2.10 1.42

220 88 7.592 -1.74 -1.34 -2.11

220 90 8.953 -5.01 -4.61 -5.33

224 88 5.789 5.52 6.07 5.23

224 90 7.298 0.12 0.59 -0.33

228 90 5.520 7.93 8.13 7.59

228 92 6.804 2.90 2.97 2.40

TABLE II. (continued) Comparison of the experimental and calculated log10 T1/2 values for 192 ≤

A ≤ 228 nuclei. The experimental data are taken from Ref.[54].

16

A Z Q(MeV ) log10 Texp1/2 (s) [54] log10 T

cal1/2 (s) [54] log10 T

cal1/2 (s)(present)

232 90 4.081 17.76 18.11 17.66

232 92 5.414 9.50 9.53 9.19

232 94 6.716 4.13 4.09 3.62

236 92 4.573 15.00 15.03 14.77

236 94 5.867 8.11 7.76 7.62

240 94 5.256 11.45 11.29 11.10

240 96 6.397 6.52 6.29 5.86

240 98 7.719 2.03 1.75 1.27

244 94 4.665 15.50 15.80 15.12

244 96 5.902 8.87 8.57 8.30

248 96 5.161 13.16 13.14 12.68

248 98 6.360 7.56 7.19 6.87

248 100 8.002 1.66 1.55 0.94

252 98 6.217 8.01 8.08 7.51

252 100 7.152 5.04 4.97 4.13

252 102 8.549 0.74 0.59 -0.15

256 100 7.027 5.14 5.27 4.59

256 102 8.582 0.53 0.54 -0.33

260 106 9.921 -2.04 -1.84 -2.81

261 107 10.562 -1.47 -2.79 -3.89

TABLE III. (continued) Comparison of the experimental and calculated log10 T1/2 values for 232 ≤

A ≤ 261 nuclei. The experimental data are taken from Ref.[54].

17

A Z Q(MeV ) log10 Texp1/2 (s) [54] log10 T

cal1/2 (s) [54] log10 T

cal1/2 (s)(present) lmin

163 74 5.520 1.90 0.83 1.15 2

171 76 5.371 3.47 2.69 2.73 2

173 78 6.354 -0.36 0.18 -0.57 2

175 77 5.396 3.02 3.52 3.06 2

181 79 5.751 3.39 2.82 2.30 2

185 82 6.695 2.32 0.25 -0.23 2

191 83 6.779 2.85 2.39 0.88 5

193 83 6.304 4.50 4.29 2.73 5

195 83 5.832 6.79 6.42 4.81 5

210 87 6.649 2.43 2.52 2.04 2

213 88 6.862 2.66 2.70 1.37 2

219 86 6.946 0.70 0.63 -0.22 2

221 87 6.457 2.55 2.58 2.14 2

223 89 6.783 2.60 2.58 1.68 2

224 89 6.327 5.73 6.36 3.66 1

225 89 5.935 6.23 6.22 5.42 2

226 89 5.536 9.25 8.32 7.49 2

230 91 5.440 11.31 9.76 9.25 2

235 95 6.612 5.17 7.09 4.79 1

237 93 4.958 16.19 15.19 12.86 1

239 95 5.922 11.11 10.41 8.09 1

241 95 5.638 12.60 12.02 9.62 1

243 95 5.439 14.16 13.18 10.76 1

245 97 6.454 9.37 6.94 6.45 2

247 96 5.353 15.55 14.66 12.30 1

249 97 5.525 13.61 12.10 11.37 2

251 98 6.176 12.04 11.30 9.18 5

255 102 8.422 4.20 3.62 1.57 5

TABLE IV. Comparison of the experimental and calculated log10 T1/2 values for 163 ≤ A ≤ 255

nuclei. The experimental data are taken from Ref.[54].18

A Z Q(MeV ) T exp1/2 [30] T cal

1/2 [30] T calc1/2 (present)

261 106 9.773+0.020−0.020 72 ms 34 ms 6.04 ms

263 106 9.447+0.020−0.020 117 ms 266 ms 45.65 ms

265 106 8.949+0.020−0.020 24.1 s 8.0 s 1.32 s

266 106 8.836+0.020−0.020 25.7 s 10.6 s 1.75 s

264 107 9.671+0.020−0.020 440+600

−160 ms 237 ms 40.5 ms

266 107 9.477+0.030−0.030 ∼ 1 s 1 s 0.13 s

267 107 9.009+0.030−0.030 17+14

−6 s 12 s 1.94 s

264 108 10.590+0.050−0.050 0.54+0.3

−0.3 ms 0.71 ms 0.12 ms

265 108 10.7776+0.020−0.020 583 µs 401 µs 70.6 µs

266 108 10.381+0.020−0.020 2.3+1.3

−0.6 ms 2.2 ms 0.38 ms

267 108 10.076+0.020−0.020 74 ms 22 ms 3.76 ms

269 108 9.354+0.020−0.020 7.1 s 2.3 s 0.37 s

268 109 10.299+0.020−0.020 70+100

−30 ms 22 ms 3.63 ms

269 110 11.345+0.020−0.020 270+1300

−120 µs 79 µs 13.4 µs

270 110 11.242+0.050−0.050 100+140

−40 µs 78 µs 13 µs

271 110 10.958+0.020−0.020 0.62 ms 0.58 ms 0.09 ms

273 110 11.291+0.020−0.020 110 µs 93 µs 15 µs

281 110 9.004+0.020−0.020 1.6 min 2.0 min 0.28 min

272 111 11.029+0.020−0.020 1.5+2.0

−0.5 ms 1.4 ms 0.23 ms

TABLE V. Comparison of the experimental and the calculated T1/2 values for superheavy nuclei

(261 ≤ A ≤ 281). The data are taken from Ref. [30].

19

A Z Q(MeV ) T exp1/2 [34] T cal

1/2 [34] T calc1/2 (present)

271 106 8.67(8) 2.7+3.4−0.9 min 1.4 min 0.13 min

270 107 9.06(8) 61+292−28 s 19 s 2.02 s

272 107 9.15(6) 9.8+11.7−3.5 s 9.3 s 0.98 s

274 107 8.93(10) 1.3a min 0.8 min 0.07 min

275 108 9.44(6) 0.19+0.22−0.07 s 1.67 s 0.16 s

274 109 9.90(10) 445+810−176 ms 310 ms 33.34 ms

275 109 10.48(9) 9.7+46−4.4 ms 5.4 ms 0.55 ms

276 109 9.85(6) 0.72+0.87−0.25 s 0.42 s 0.04 s

278 109 9.69(19) 11a s 1.1 s 0.11 s

279 110 9.84(6) 2.0+0.5−0.4 s 0.56 s 0.05 s

278 111 10.85(8) 4.2+7.5−1.7 ms 4.7 ms 0.49 ms

279 111 10.52(16) 170+810−80 ms 19 ms 1.86 ms

280 111 9.87(6) 3.6+4.3−1.3 s 1.7 s 0.17 s

282 111 9.13(10) 0.74a s 283.73 s 26.7 s

283 112 9.67(6) 3.8+1.2−0.7 s 8.0 s 0.78 s

285 112 9.28(5) 29+13−7 s 125 s 11 s

TABLE VI. Comparison of the experimental and calculated T1/2 values for superheavy nuclei

(271 ≤ A ≤ 285). The data are taken from Ref.[34]. a shows the value is deduced from the

observed 117 decay chain (only one event).

rms error

No. Present Denisov[54]

All 106 1.1979 0.6008

even-even 73 0.9018 0.4210

odd-odd 33 1.6966 0.8865

TABLE VII. The rms errors of the present model and Denisov model[54]. Here, second column

shows nucleon numbers for 108 ≤ A ≤ 261 nuclei and third column shows the results of the α-α

folding cluster model and Denisov[54].

20

100 150 200 250 300A(mass number)

-2

-1

0

1

2

3

4

log 10

(Tex

p 1/2/T

cal 1/

2)

FIG. 3. Deviation between the measured and calculated half-lives log10(Texp1/2 /T

cal1/2) versus the

atomic mass number A for 108 ≤ A ≤ 261 nuclei and superheavy nuclei, 261 ≤ A ≤ 285.

21