On the formation and alpha decay of superheavy elements

Nuclear Physics A 699 (2002) 479–492 www.elsevier.com/locate/npe

On the formation and alpha decay of superheavy elements G. Royer a,∗ , R.A. Gherghescu b a Laboratoire Subatech, UMR: IN2P3/CNRS–Université–École des Mines, 4 rue A. Kastler,

44307 Nantes cedex 03, France b National Institute for Physics and Nuclear Engineering, PO Box MG-6, RO-76900 Bucharest, Romania

Received 27 July 2001; revised 29 August 2001; accepted 3 September 2001

Abstract The potential energy governing the entrance and α-decay channels for superheavy elements has been determined within a Generalized Liquid Drop Model including the proximity effects, the asymmetry, an accurate nuclear radius, an adjustment to reproduce the experimental Q value and within the asymmetric two-center shell model and the Strutinsky method. In cold fusion paths doublehump fusion barriers stand and incomplete fusion events may occur. Warm fusion reactions lead to one hump potential barriers and to very excited systems cooling down by neutron or even α-particle evaporation. α-decay half-lives of superheavy elements have been calculated and compared with the new available experimental data.  2002 Elsevier Science B.V. All rights reserved. PACS: 25.70.Jj; 21.60.Ev; 23.60.+e Keywords: Superheavy elements; Fusion; Alpha decay; Liquid-drop model; Proximity energy

1. Introduction The synthesis of superheavy elements has apparently strongly advanced recently using both cold (Zn on Pb [1]) and warm (Ca on U, Pu and Cm [2–4]) fusion reactions at Darmstadt, Dubna and Berkeley. These experiments are very delicate since the fusion cross sections are in the picobarn range. For example, the successive experiments performed at GSI, RIKEN and GANIL trying to form a superheavy nucleus of charge Z = 118 in the reaction Kr on Pb have been unsuccessful till now. The experimental data analysis is also discussed [5,6]. The observed decay mode of these superheavy systems is mainly the α emission. * Corresponding author.

E-mail address: [email protected] (G. Royer). 0375-9474/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 2 9 6 - 9

480

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

The purpose of this work is to study, in the entrance channel, the potential barriers against fusion in this very heavy mass range and, in the exit channel, to determine the partial α-decay half-lives and to compare with the new experimental data. The combination of a Generalized Liquid Drop Model and quasimolecular shapes which has allowed most fusion, fission, cluster and α emissions to be reproduced reasonably well [7–11] has been used once again [12,13]. The shell effects have been derived from the asymmetric twocenter shell model and the shell corrections have been calculated with the Strutinsky method [10,14].

2. Generalized Liquid Drop Model and quasimolecular shapes Within this GLDM [8] the macroscopic energy of a deformed nucleus is defined as E = EV + ES + EC + EN .

(1)

For one-body shapes, the volume EV , surface ES and Coulomb EC energies are given by 
EV = −15.494 1 − 1.8I 2 A MeV, (2)  2/3

2 2 S/4πR0 MeV, (3) ES = 17.9439 1 − 2.6I A 


 3 EC = 0.6e2 Z 2 /R0 × 0.5 (4) V (θ )/V0 R(θ )/R0 sin θ dθ. I is the relative neutron excess and S the surface of the one-body deformed nucleus. V (θ ) is the electrostatic potential at the surface and V0 the surface potential of the sphere. When the spherical fragments are separated, 

   EV 12 = −15.494 1 − 1.8I12 A1 + 1 − 1.8I22 A2 MeV, (5)    

2/3 2/3 (6) ES12 = 17.9439 1 − 2.6I12 A1 + 1 − 2.6I22 A2 MeV, EC12 = 0.6e2Z12 /R1 + 0.6e2Z22 /R2 + e2 Z1 Z2 /r,

(7)

where Ai , Zi , Ri and Ii are the masses, charges, radii and relative neutron excesses of the fragments. r is the distance between the mass centers. The radii Ri (i = 1, 2) have been chosen as
1/3 −1/3 fm. (8) Ri = 1.28Ai − 0.76 + 0.8Ai This later formula often used only to determine the proximity energy allows to take 1/3 into account the experimentally observed increase of the ratio ri = Ri /Ai with the mass; for example, r0 = 1.10 fm for 14 C and r0 = 1.18 fm for 248 Cm. The radius of the compound nucleus has been calculated from the radii of the two fragments assuming volume conservation. For comparison, in the RLDM and RFRM approaches [15,16] the reduced radius r0 is respectively 1.225 fm and 1.16 fm with no mass dependence while the surface coefficient as takes on the values 17.94 MeV and 21.13 MeV. The surface energy ES takes into account the surface tension forces in a half space and does not include the contribution due to the attractive nuclear forces between the

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

481

surfaces in regard in a neck or a gap between separated fragments. The nuclear proximity energy term EN allows to take into account these additional surface effects when a neck or a gap appears. This term is essential to describe smoothly the two-body to one-body transition. For example, at the contact point between two spherical 48 Ca and 248 Cm nuclei the proximity energy reaches −39 MeV. h max

  Φ D(r, h)/b 2πh dh,

EN (r) = 2γ

(9)

hmin

where h is the transverse distance varying from the neck radius or zero to the height of the neck border. D is the distance between the opposite surfaces in regard and b the surface width fixed at 0.99 fm. Φ is the proximity function of Feldmeier [17]. The surface parameter γ is the geometric mean between the surface parameters of the two fragments: 

 (10) γ = 0.9517 1 − 2.6I12 1 − 2.6I22 MeV fm−2 . In this GLDM the surface diffuseness is not taken into account and the proximity energy vanishes when there is no neck as for an ellipsoid for example. The deformation path (quasimolecular configurations) previously used to describe the fusion [8], fission [7] and α and cluster emissions [9,11] has been retained again. In the entrance channel this shape sequence allows to describe the rapid formation of a deep neck after the touching point while keeping almost spherical ends. It describes also smoothly the emission of an α particle. The validity of the combination of the GLDM and the shape sequence is warranted by its efficiency to reproduce the fusion barrier heights and radii, the fission and the α and cluster radioactivity data.

3. Fusion barriers in the entrance channel Potential barriers against fusion for the cold fusion reactions 58 Fe, 64 Ni, 70 Zn, 76 Ge, and 86 Kr on 208 Pb and warm fusion reactions 48 Ca on 238U, 244 Pu and 248 Cm are displayed in Figs. 1 and 2. Different hypotheses have been assumed. The dashed line corresponds to the pure macroscopic potential energy given by the GLDM. In ordinary fusion studies, it is often only that barrier which is taken into account. The dashed-dotted line incorporates the shell corrections calculated empirically from the formulas proposed in the Droplet Model [7,18] which implicitly supposes that 114 is a magic number for the charge number. The solid line is the sum of the GLDM macroscopic energy and of the shell effects calculated within the asymmetric two-center shell model [10,14] and adjusted to reproduce the experimental or estimated Q value with a corrected factor beginning at the contact point between the colliding nuclei. This supposes that the nuclear system has had enough time to relax and built its shells and pairs and then that all the microscopic contributions (shells effects, pairing, Wigner term, . . . ) contribute to the total energy. For the reactions for which the Q value is not experimentally known the values predicted

82 Se

482

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

Fig. 1. Fusion barriers versus the mass-centre distance for the 58 Fe, 64 Ni, 70 Zn, 76 Ge, 82 Se and 86 Kr on 208 Pb cold reactions.

by the Thomas–Fermi model [19] have been used since they reproduce nicely the mass decrements from fermium to Z = 112. In all cases the first external top corresponds to two separated sphere configurations; the attractive nuclear forces compensating for the repulsive Coulomb forces. At the contact point the first external top of the barrier is already passed and the approximation starting the fusion process from the contact point is very rough. At this point the pure Coulomb barrier is too high while if the proximity energy is included only at this point then the energy is too low. In the cold fusion reactions a wide macroscopic potential pocket due to the proximity energy appears at large deformations and the energy is almost constant till the spherical compound nucleus. The full microscopic corrections create a plateau for the Ni on Pb

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

Fig. 2. Same as Fig. 1 but for the 48 Ca on 238 U, 244 Pu and 248 Cm warm reactions.

483

484

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

reaction. From the Zn on Pb reaction, double hump fusion barriers appear and the inner peak is the highest for the heaviest systems. In the deep minimum between the two maxima incomplete fusion and fast fission might appear since the neck between the two nuclei is formed and exchanges of nucleons occur. The remaining excitation energy of the composite system is crucial to decide between complete fusion and fast fission. It depends on the pre or post equilibrium nature of the evaporation process of the excess neutron. An open question is whether at large deformations the nucleon shells can take form to stabilize the nuclear system before investigating a peculiar exit channel. In the warm fusion reactions, due to the asymmetry there is no double hump barriers. The barrier against reseparation being high and wide the system will descend toward a quasispherical shape but with an excitation energy (of more than 30 MeV if one assumes a full relaxation) which is much higher than in cold fusion reactions. The emission of several neutrons or even an α particle is energetically possible. The characteristics of these fusion barriers are shown in Table 1. The fusion barrier height derived from the GLDM is systematically higher than the Bass barrier height [20] and the fusion radius is smaller. These two effects lead to smaller cross sections within the GLDM than within the Bass model. Indeed, for most of the incident energies presently used, the reaction is a subbarrier fusion for the GLDM while it is a fusion above the barrier for the Bass model. The external fusion barrier height and position can be reproduced accurately by the following formulas [21]:  
1/3 1/3  Z1 Z2 2.1388Z1Z2 + 59.427 A1 + A2 − 27.07 ln 1/3 1/3 A1 +A2 Hext,fus = −19.38 + ,
1/3 1/3  A1 + A2 (2.97 − 0.12 ln(Z1 Z2 )) (11)


1/3 3.94 1/3  . (12) Rext,fus (fm) = A1 + A2 1.908 − 0.0857 ln(Z1 Z2 ) + Z1 Z2 4. α decay of the heaviest elements The α decay of these very heavy nuclear systems have been studied within this GLDM and adjusted to reproduce the experimental Qα value or the value predicted by the Thomas–Fermi model [19]. As an example the resulting potential barrier for 264 Hs is displayed in Fig. 3. As for more symmetric configurations the proximity forces smooth the unrealistic pure Coulomb barrier, shift the peak to a more external position and lower strongly the barrier height. For this nucleus, the displacement is of 2.1 fm and the lowering of 7.3 MeV. As a result, the approaches using the pure Coulomb barrier to determine the fusion cross sections for the superheavy elements seem very rough. The α-decay half-lives deduced from this model within the WKB barrier penetration probability has been compared with the experimental data [9]. The rms deviation between the theoretical and experimental values of log10 [T1/2(s)] is only 0.63 for a recent data set of 373 α emitters and 0.35 for the subset of the even–even nuclides. The following

Table 1 Characteristics of the macroscopic fusion barriers leading to the heaviest elements. Rint , Hint and Hint,fus are respectively the positions and energies relatively to the sphere and at infinity of the inner maximum of the potential barrier. Rmin , Rext , Hmin and Hext are respectively the positions and energies relatively to the sphere of the external minimum and maximum of the fusion barrier. R12 is the distance between the mass centers at the touching point. Hext,fus is the GLDM external fusion barrier height while RBass and HBass are the barrier radius and height given by the Bass model [20]. Ecm is the mass-centre energy already used in an experiment Experimental

Z1 Z2 Rint

result

Rext

Rext −

Hext

Hext,fus

(MeV)

(fm) (MeV)

(fm)

R12

(MeV)

(MeV)

Hint,fus Rmin

Hmin

Ecm

−Qfus

Rfus

Hfus

(MeV) (MeV) (Bass) (Bass)

? 312 126

3608

–

–

–

12.1 −29.1

12.2

0

−28.5

370.8

–

381.2

13.2

359.2

? 306 122

2496

–

–

–

10.2

−3.7

12.3

0.5

7.6

261.6

–

231.45

13.0

255.7

? 297 122

3552

–

–

–

–

? 302 120

2444

7.0

0.5

247.3

10.1

? 296 120

3116

–

–

–

? 295 119

3034

–

–

–

? 298 118

2112

–

–

–

–

–

–

–

365.5

13.1

356.6

12.4

0.7

9.9

256.7

–

221.46

13.0

250.9

11.3 −16.8

12.1

0

−14.3

324.9

–

321.7

13.1

315.6

11.3 −17.0

12.1

0

−14.4

316.1

–

311.5

13.1

307.2

−0.7

12.5

0.9

12.5

223.4

–

186.8

12.9

218.5

8.4

– −2

–

? 294 118

2952

–

–

–

11.3 −16.8

12.0

0

292 116 + 4n

1920

–

–

–

10.0

−3.5

12.6

1.1

? 290 116

2340

7.2

0.6

237.5

9.8

−1.2

12.4

0.8

? 290 116

2788

–

–

–

11.0 −14.5

11.9

0

−11.1

? 292 114

3240

–

–

–

12.2 −21.5

12.3

0

−21.4

12.1 −20.1

? 292 114

3200

–

–

–

289,8 114 + 3, 4n

1880

–

–

–

? 290 114

2624

–

–

–

287 114 + 3n

1880

–

–

–

? 284 114

2624

–

–

–

? 285 113

1860

–

–

–

−14.1

307.3

–

302.6

13.1

298.8

11.2

203.1

201

168.0

13.0

198.5

10.7

247.6

–

209.37

12.9

242.2

290.85

–

282.1

13.1

283.2

333.15

–

316.6

13.4

322.6

12.25

0

−19.9

329.1

–

322.1

13.4

318.7

12.6

1.1

12.6

199.3

197

161.9

13.0

194.8

10.9 −12.8

11.9

0

−8.8

272.3

–

261.4

13.2

265.3

−3.0

12.5

1.1

11.8

199.6

196

163.5

12.9

195.1

10.6 −11.0

11.8

0

−6.2

275.4

–

261.7

13.0

268.7

−3.2

12.5

0.9

11.4

198.2

–

164.2

12.9

193.9

9.8 9.9 9.9

−2.4

485

104 Ru + 208 Pb 44 82 58 248 26 Fe + 96 Cm 116 Cd + 181 W 48 74 58 Fe + 244 Pu 26 94 88 Sr + 208 Pb 38 82 87 208 Rb + 37 82 Pb 50 Ti + 248 Cm 22 96 86 Kr + 208 Pb 36 82 48 Ca + 248 Cm 20 96 58 232 26 Fe + 90 Th 82 Se + 208 Pb 34 82 142 Xe + 150 Nd 60 54 132 Sn + 160 Gd 50 64 48 244 20 Ca + 94 Pu 82 Ge + 208 Pb 32 82 48 Ca + 242 Pu 20 94 76 Ge + 208 Pb 32 82 48 237 Ca + 20 93 Np

Hint

(fm) (MeV)

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

Reaction

Reaction

486

Table 1 —continued Experimental

Rext

Rext −

Hext

Hext,fus

Ecm

(fm)

(MeV)

(fm)

R12

(MeV)

(MeV)

(MeV)

10.4

−8.7

12.3

0.6

−2.5

263.1

Hint

Hint,fus Rmin

−Qfus

Rfus

Hfus

(MeV) (Bass) (Bass)

? 279 113

2490

–

–

–

283 112 + 3n

1840

–

–

–

9.8

−1.9

12.5

1.1

13.3

195.8

192

159.7

12.9

191.4

277 112 + 1n

2460

–

–

–

10.3

−7.4

12.35

0.7

−0.7

260.0

257

243.8

12.9

254.1

–

247.3

12.9

257.2

? 276 112

2460

–

–

–

10.1

−4.8

12.3

0.6

2.9

261.1

–

241.5

12.8

255.4

272 111 + 1n

2324

–

–

–

10.0

−5.0

12.3

0.8

3.6

247.6

244

227.8

12.8

242.3

273 110 + 5n

1504

–

–

–

–

–

12.2

1.2

32.8

164.4

167

115.0

12.5

161.1

271 110 + 1n

2296

–

–

–

9.9

−3.6

12.3

0.8

5.5

244.7

239

225.1

12.8

239.4

269 110 + 1n

2296

8.2

1.7

237.7

9.6

−0.6

12.2

0.8

10.1

245.9

240

223.2

12.7

240.7

267 110 + 1n

2241

8.3

2.1

231.5

9.5

0.1

12.2

0.8

11.7

241.1

235

214.2

12.7

236.1

266 Mt + 1n 109 267 Hs + 5n 108 265, 4 108 Hs + 1, 2n 264 108 Hs + 1n 262 Bh + 1n 107 262, 1 107 Bh + 1, 2n 261 Bh + 1n 107 260 107 Bh + 1n 266 106 Sg + 4n 263 Sg + 4n 106 261, 0 106 Sg + 1, 2n 259 106 Sg + 3n

2158

8.25

1.2

222.9

9.6

−1.1

12.3

0.9

10.4

232.1

234

208.5

12.7

227.3

1472

–

–

–

12.2

1.3

32.9

161.6

163

110.2

12.5

158.4

2132

8.2

2.1

219.0

9.5

0.25 12.3

2132

8.2

1.7

219.7

9.5

–

–

−0.3

2050

8.3

3.25

208.7

9.2

2.3

1992

8.3

1.45

203.8

9.4

−0.1

2050

8.3

2.8

206.7

9.2

1.7

2050

210.3

8.3

2.4

960

–

–

784

–

–

8.4

2.2

1968

0.9

12.4

229.4

222

205.0

12.7

224.6

12.25

0.9

11.5

229.6

250

205.7

12.7

224.8

12.2

0.9

16.2

221.7

229

194.2

12.6

217.1

12.25

1.0

13.0

215.3

208

189.9

12.6

210.9

12.2

0.9

15.2

221.9

245

194.8

12.6

217.4

9.3

1.1

12.2

0.9

14.2

222.1

245

195.0

12.6

217.6

–

–

–

12.0

1.5

33.2

107.2

107

60.4

12.3

105.2

–

–

–

12.0

1.6

31.2

88.4

46.2

12.2

86.8

9.3

1.0

12.3

1.0

14.8

212.8

210(1n) 187.2

12.6

208.4

200.2

88.6 213(2n) 238(3n)

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

70 209 30 Zn + 83 Bi 48 Ca + 238 U 20 92 70 Zn + 208 Pb 30 82 68 Zn + 208 Pb 30 82 64 209 Ni + 28 83 Bi 34 S + 244 Pu 16 94 64 Ni + 208 Pb 28 82 62 Ni + 208 Pb 28 82 59 209 27 Co + 83 Bi 58 Fe + 209 Bi 26 83 34 S + 238 U 16 92 58 Fe + 208 Pb 26 82 58 207 26 Fe + 82 Pb 55 Mn + 208 Pb 25 82 54 Cr + 209 Bi 24 83 55 Mn + 207 Pb 25 82 55 206 Mn + 25 82 Pb 22 Ne + 248 Cm 96 10 18 O + 249 Cf 98 8 54 Cr + 208 Pb 24 82

Hmin

(fm) (MeV) (MeV)

Z1 Z2 Rint

result

Table 1 —continued Reaction

Experimental

Rext

Rext −

Hext

Hext,fus

Ecm

(fm) (MeV) (MeV)

(fm) (MeV)

(fm)

R12

(MeV)

(MeV)

(MeV)

1968

8.3

1.9

200.9

9.3

0.5

12.3

1.0

14.0

213.0

211

187.6

12.6

208.6

1968

8.25

1.6

201.6

9.4

0.0

12.25

1.0

13.2

213.2

230

187.3

12.6

208.9

1909

8.4

2.6

193.6

9.2

1.8

12.25

1.1

16.5

207.4

201

177.1

12.6

203.3

–

–

–

–

12.0

1.5

32.4

106.5

–

61.4

12.2

104.5

Z1 Z2 Rint

result 260, 59 106 Sg + 1, 2n 259 Sg + 1n 106 258 Sg + 2n 106 261, 0 105 Db + 4, 5n 260 Db + 4n 105 258, 7 105 Db + 1, 2n 256 105 Db + 2n 255 Db + 2n 105 262 Rf + 4n 104 258 Rf + 6n 104 259 104 Rf + 3n 257 Rf + 4n 104 258, 7 104 Rf + 0, 1n 256, 5 104 Rf + 2, 3n

950

–

686

–

Hint,fus Rmin

Hmin

−Qfus

Rfus

Hfus

(MeV) (Bass) (Bass)

–

–

–

–

11.8

1.7

33.5

78.3

–

39.6

12.0

76.9

1826

8.3

1.8

184.7

9.3

0.8

12.2

1.1

15.5

198.4

191

171.9

12.6

194.4

1826

8.4

2.8

184.1

9.2

2.2

12.2

1.1

17.7

199.0

219

168.8

12.5

195.0

1826

8.5

4.5

183.5

9.0

4.2

12.15

1.1

20.2

199.6

220

167.2

12.5

195.6

940

–

–

–

–

–

12.0

1.5

34.1

105.3

105

57.2

12.3

103.3

768

–

–

–

–

–

11.7

1.5

37.3

89.6

–

43.5

12.0

86.2

588

–

–

–

–

–

11.8

1.8

32.75

67.5

–

29.5

11.9

66.3

588

–

–

–

–

–

11.65

1.7

36.1

68.0

–

31.6

11.8

67.0

2.6

181.5

9.2

1.8

12.3

1.2

17.1

196.1

188(1n) 169.7

12.6

192.1

1804

8.4

192(2n) 202(3n)

208 Pb 82 207 Pb 82 208 82 Pb 206 Pb 82 204 Pb 82

256, 5 104 Rf + 1, 2n 256, 5 104 Rf + 1, 2n 255 104 Rf + 1n 254 Rf + 2n 104 253 Rf + 1n 104

1804

8.5

3.7

180.9

9.1

3.3

12.2

1.1

19.4

196.6

219

166.3

12.5

192.7

1804

8.4

2.2

182.1

9.2

1.4

12.2

1.1

16.4

196.3

192.5

169.9

12.6

192.3

1804

8.5

5.1

180.1

8.9

5.0

12.1

1.1

22.2

197.2

210

164.5

12.5

193.3

1804

8.3

1.9

182.2

9.2

1.0

12.2

1.1

15.8

196.5

194

169.5

12.6

192.5

1804

8.3

1.4

183.7

9.3

0.2

12.2

1.1

14.5

196.8

188

169.9

12.5

192.9 487

49 Ti + 22 50 Ti + 22 48 22 Ti + 50 Ti + 22 50 Ti + 22

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

54 Cr + 207 Pb 24 82 54 Cr + 206 Pb 24 82 51 V + 209 Bi 23 83 22 Ne + 243 Am 95 10 15 N + 249 Cf 98 7 50 Ti + 209 Bi 22 83 49 209 Ti + 22 83 Bi 48 Ti + 209 Bi 22 83 22 Ne + 244 Pu 10 94 16 O + 248 Cm 96 8 13 C + 249 Cf 98 6 12 C + 249 Cf 98 6 50 Ti + 208 Pb 22 82

Hint

488

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

Fig. 3. α-decay barrier for 264 Hs. The solid and dashed lines give the deformation energy with and without the contribution of the nuclear proximity energy term.

new formulas have also been extracted, respectively for the even (Z)–even (N ), even–odd, odd–even and odd–odd nuclei wit a rms deviation of 0.285, 0.39, 0.36 and 0.35: √   1.5864Z log10 T1/2 (s) = −25.31 − 1.1629A1/6 Z + √ , Qα

(13)

√   1.5848Z log10 T1/2 (s) = −26.65 − 1.0859A1/6 Z + √ , Qα

(14)

√   1.592Z log10 T1/2 (s) = −25.68 − 1.1423A1/6 Z + √ , Qα

(15)

√   1.6971Z . log10 T1/2 (s) = −29.48 − 1.113A1/6 Z + √ Qα

(16)

As a new checking an α-decay half-life of 0.51 ± 0.16 s has recently been obtained [22] for 259 Db. The formula (15) leads to 0.65 s for Qα = 9.8 MeV [19]. The ranges of the α-decay half-lives extracted from the recently observed α-decay chains [2–4] agree with our predictions (see Table 2) except for 289 114 where the theoretical value is 2 orders of magnitude higher than the experimental data. The explanation of this discrepancy is perhaps that Z = 114 is not a good proton magic number and that the Thomas–Fermi model underestimates Qα for Z = 114 and then overestimates the half-lives. The global predictions for the α-decay half-lives of the heaviest elements within the GLDM and the formulas (13)–(16) are given in Table 3 as functions of the mass and charge numbers. For the even–even nuclei, the difference between the two proposed values varies from 0.4 to 0.7. If these nuclei can be formed and if the α-decay mode is the main decay mode, then their half-lives will vary from microsecond to some days. For all nuclei

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

489

Table 2 Comparison between the experimental α-decay half-lives Texp and the ones Tform and Tgldm predicted by the formulas (13)–(16) and the GLDM. Qα is calculated from the Thomas–Fermi model [19] Z/N 172

173

174

175

A = 292 Q = 11.03 Texp = 18–190 ms Tform = 6.9 ms Tgldm = 1.4 ms

116

A = 287 Q = 9.53 Texp = 5.5 s Tform = 130 s Tgldm = 5.5 s

114

112

176

A = 284 Q = 8.89 MeV Texp = 6–60 s Tform = 8.9 m Tgldm = 2.2 m

A = 288 Q = 9.39 Texp = 1.1–5.2 s Tform = 63 s Tgldm = 14 s

A = 289 Q = 9.08 Texp = 2–23 s Tform = 3000 s Tgldm = 145 s

A = 285 Q = 8.8 Texp = 20–200 m Tform = 84 m Tgldm = 4.5 m

the calculations within the GLDM should give a lower limit of the true value while the extrapolations from the formulas should not be far from the reality.

5. Conclusion The entrance and α-decay channels for the superheavy elements have been studied in the quasimolecular shape path within a generalized liquid-drop model including a proximity energy term and the asymmetry. The microscopic contributions have been incorporated in calculating the shell effects with the asymmetric two-center shell model and in adjusting to the experimental or predicted Q value. The fusion barrier profiles are highly sensitive to the shell effects, the Q values and the asymmetry. Double-hump fusion barriers appear in cold fusion reactions emphasizing the probability of incomplete fusion and fast fission events. Warm fusion reactions investigate one-hump fusion barriers but lead to nuclear systems having enough excitation energy to emit several neutrons or an α particle. The knowledge of the moment of emission of the excess particles which evacuate most of the excitation energy is important to better understand the reaction mechanism. The predicted α-decay half-lives are in agreement with the data obtained recently except for 289 114. The explanation of this discrepancy is perhaps that 114 is not a good proton magic number.

490

Table 3 Predicted log10 [T1/2 (s)] for the heaviest elements versus the charge and mass of the mother nucleus and Qα 285 13.3 −6.91 −5.63 283 12.72 −6.11 −5.11 280 12.65 −6.16 −5.61 278 12.90 −6.85 −5.63 275 12.67 −6.69 −5.46 273 12.31 −6.25 −5.33 270 12.30 −6.41 −5.94

286 13.27 −6.87 −6.35 284 12.64 −5.96 −4.50 281 12.39 −5.67 −4.35 279 12.50 −6.14 −5.21 276 12.55 −6.47 −5.94 274 12.20 −6.05 −4.73 271 12.07 −5.97 −4.79

287 13.11 −6.61 −5.30 285 12.52 −5.74 −4.74 282 11.90 −4.72 −4.04 280 12.24 −5.65 −4.22 277 12.55 −6.48 −5.25 275 12.12 −5.90 −4.97 272 11.82 −5.54 −4.96

288 13.11 −6.63 −6.07 286 12.40 −5.51 −3.99 283 12.03 −5.01 −3.61 281 11.80 −4.74 −3.73 278 12.86 −7.05 −6.60 276 12.11 −5.89 −4.56 273 11.72 −5.34 −4.07

289 12.99 −6.42 −5.09 287 12.31 −5.36 −4.33 284 11.83 −4.59 −3.92 282 11.02 −3.01 −1.25 279 12.27 −5.97 −4.71 277 12.37 −6.41 −5.53 274 11.61 −5.12 −4.53

290 13.04 −6.52 −5.97 288 12.25 −5.25 −3.69 285 11.68 −4.27 −2.86 283 10.30 −1.13 −0.02 280 11.83 −5.09 −4.49 278 12.77 −7.15 −6.04 275 11.70 −5.33 −4.06

291 12.79 −6.06 −4.73 289 12.20 −5.16 −4.12 286 11.65 −4.22 −3.54 284 10.56 −1.85 −0.02 281 11.15 −3.58 −2.22 279 12.83 7.29 −6.50 276 12.05 −6.05 −5.53

292 12.59 −5.69 −5.09 290 12.14 −5.05 −3.47 287 11.52 −3.96 −2.52 285 10.55 −1.83 −0.74 282 10.22 −1.22 −0.54 280 11.67 −5.05 −3.60 277 12.30 −6.55 −5.38

293 12.49 −5.56 −4.14 291 11.94 −4.64 −3.58 288 11.55 −4.04 −3.35 286 10.45 −1.58 0.26 283 9.79 0.00 1.38 281 10.90 −3.29 −2.27 278 12.49 −6.92 −6.48

294 12.51 −5.60 −4.96 292 11.93 −4.62 −3.00 289 11.50 −3.94 −2.51 287 10.48 −1.68 −0.59 284 9.64 0.44 1.10 282 10.00 −0.93 0.87 279 12.03 −6.06 −4.85

295 12.42 −5.45 −4.02 293 11.91 −4.64 −3.55 290 11.34 −3.58 −2.89 288 10.34 −1.32 0.54 285 9.55 0.71 2.07 283 9.56 0.35 1.39 280 11.34 −4.61 −4.03

296 12.52 −5.66 −5.02 294 11.90 −4.62 −2.96 291 11.33 −3.57 −2.14 289 10.24 −1.05 0.03 286 9.61 0.51 1.16 284 9.36 0.96 2.87 281 10.35 −2.20 −0.89

297 12.34 −5.32 −3.88 295 11.80 −4.43 −3.34 292 11.03 −2.85 −2.16 290 10.15 −0.80 1.07 287 9.53 0.74 2.10 285 9.18 1.53 2.54 282 9.62 −0.16 0.46

298 12.73 −6.11 −5.49 296 11.59 −3.98 −2.23 293 11.15 −3.19 −1.74 291 9.88 −0.03 1.03 288 9.39 1.16 1.80 286 9.10 1.78 3.72 283 9.22 1.06 2.36

299 12.87 −6.37 −5.03 297 11.97 −4.83 −3.76 294 11.19 −3.30 −2.60 292 9.75 0.35 2.28 289 9.08 2.16 3.49 287 9.04 1.96 2.97 284 8.89 2.13 2.73

300 12.94 −6.51 −5.96 298 12.16 −5.25 −3.65 295 11.06 −3.00 −1.55 293 9.69 0.49 1.56 290 8.73 3.35 3.95 288 8.97 2.18 4.15 285 8.80 2.43 3.70

301 13.05 −6.72 −5.42 299 12.25 −5.44 −4.41 296 11.33 −3.67 −2.97 294 9.46 1.19 3.20 291 8.66 3.59 4.90 289 8.66 3.24 4.23 286 8.68 2.84 3.41

302 13.07 −6.77 −6.25 300 12.35 −5.65 −4.13 297 11.38 −3.80 −2.36 295 9.87 −0.08 0.99 292 8.47 4.28 4.85 290 9.13 1.63 3.55 287 8.62 3.03 4.29

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

Z = 118 A Qα gldm form Z = 117 A Qα gldm form Z = 116 A Qα gldm form Z = 115 A Qα gldm form Z = 114 A Qα gldm form Z = 113 A Qα gldm form Z = 112 A Qα gldm form

Table 3 —continued 269 11.84 −5.77 −4.90 266 11.99 −6.33 −5.84 264 11.56 −5.72 −4.50 261 11.04 −4.80 −3.67 259 10.07 −2.73 −1.83 256 9.72 −2.07 −1.54 254 9.85 −2.73 −1.40 251 9.72 −2.66 −1.60

270 11.54 −5.16 −3.84 267 11.78 −5.92 −4.76 265 11.33 −5.25 −4.35 262 11.09 −4.96 −4.43 260 10.34 −3.44 −2.09 257 9.59 −1.72 −0.59 255 9.72 −2.38 −1.54 252 9.72 −2.67 −2.20

271 11.21 −4.44 −3.52 268 11.67 −5.71 −5.20 266 11.49 −5.60 −4.38 263 10.67 −4.01 −2.82 261 10.57 −4.02 −3.16 258 9.68 −1.99 −1.47 256 9.48 −1.74 −0.34 253 9.55 −2.23 −1.17

272 10.98 −3.98 −2.47 269 11.28 −4.88 −3.68 267 10.98 −4.49 −3.57 264 10.58 −3.81 −3.26 262 10.22 −3.18 −1.80 259 9.84 −2.45 −1.32 257 9.32 −1.28 −0.44 254 9.38 −1.76 −1.28

273 11.09 −4.25 −3.27 270 11.17 −4.65 −4.10 268 10.70 −3.84 −2.43 265 10.47 −3.55 −2.35 263 10.08 −2.83 −1.94 260 9.93 −2.70 −2.19 258 9.55 −1.96 −0.59 255 9.25 −1.39 −0.34

274 11.07 −4.21 −2.75 271 10.90 −4.05 −2.82 269 10.54 −3.46 −2.53 266 10.21 −2.90 −2.34 264 9.98 −2.58 −1.16 261 9.81 −2.39 −1.28 259 9.65 −2.26 −1.42 256 8.95 −0.51 −0.05

275 11.53 −5.26 −4.33 272 10.77 −3.79 −3.18 270 10.35 −2.99 −1.52 267 10.12 −2.67 −1.48 265 9.77 −2.02 −1.13 262 9.61 −1.86 −1.35 260 9.32 −1.32 0.09 257 9.15 −1.13 −0.09

276 11.73 −5.69 −4.40 273 11.67 −5.82 −4.63 271 10.14 −2.44 −1.52 268 9.90 −2.09 −1.54 266 9.56 −1.44 0.05 263 9.39 −1.24 −0.12 261 9.23 −1.07 −0.25 258 9.33 −1.68 −1.22

277 11.71 −5.66 −4.77 274 10.72 −3.69 −3.10 272 9.62 −1.03 0.58 269 9.74 −1.65 −0.48 267 9.38 −0.92 −0.04 264 9.22 −0.76 −0.25 262 9.00 −0.41 1.07 259 9.12 −1.07 −0.03

278 11.59 −5.42 −4.10 275 10.92 −4.18 −2.94 273 10.02 −2.16 −1.24 270 8.88 0.96 1.46 268 8.48 1.95 3.64 265 8.76 0.67 1.78 263 8.83 0.11 0.95 260 8.90 −0.40 0.03

279 11.12 −4.41 −3.45 276 10.93 −4.22 −3.65 274 10.26 −2.83 −1.34 271 9.02 0.50 1.63 269 8.32 2.50 3.35 266 8.76 0.66 1.14 264 8.67 0.61 2.15 261 8.60 0.55 1.56

280 10.34 −2.48 −0.89 277 10.89 −4.13 −2.90 275 10.26 −2.85 −1.92 272 9.39 −0.70 −0.16 270 8.36 2.35 4.05 267 8.04 3.15 4.22 265 8.49 1.19 2.02 262 8.50 0.82 1.28

281 9.58 −0.34 0.61 278 10.61 −3.46 −2.90 276 10.11 −2.46 −0.95 273 9.61 −1.36 −0.18 271 8.85 0.68 1.52 268 7.77 4.17 4.62 266 7.54 4.70 6.49 263 8.31 1.46 2.49

282 9.00 1.42 3.28 279 9.89 −1.56 −0.33 277 10.04 −2.28 −1.36 274 9.65 −1.49 −0.96 272 9.08 −0.11 1.48 269 7.92 3.59 4.64 267 7.27 5.82 6.61 264 8.14 2.04 2.48

283 8.29 3.89 4.86 280 9.25 0.33 0.87 278 9.58 −0.98 0.60 275 9.58 −1.30 −0.13 273 9.02 0.06 0.94 270 8.35 2.00 2.45 268 7.37 5.39 7.20 265 7.17 5.85 6.84

284 8.22 4.14 6.16 281 8.75 1.95 3.14 279 9.13 0.36 1.26 276 9.52 −1.13 −0.62 274 8.89 0.46 2.08 271 8.59 1.17 2.23 269 7.76 3.80 4.59 266 6.91 7.00 7.38

285 8.09 4.63 5.58 282 7.95 4.81 5.35 280 8.65 1.93 3.70 277 8.93 0.67 1.81 275 9.00 0.10 0.97 272 8.57 1.17 1.66 270 8.02 2.81 4.45 267 7.02 6.49 7.46

491

268 12.10 −6.28 −5.10 265 12.10 −6.53 −5.40 263 11.87 −6.34 −5.50 260 10.85 −4.36 −3.83 258 10.00 −2.53 −1.10 255 10.08 −3.02 −1.89 253 9.97 −3.04 −2.18 250 9.94 −3.23 −2.75

G. Royer, R.A. Gherghescu / Nuclear Physics A 699 (2002) 479–492

Z = 111 A Qα gldm form Z = 110 A Qα gldm form A 109 Mt Qα gldm form A 108 Hs Qα gldm form A 107 Bh Qα gldm form A 106 Sg Qα gldm form A 105 Db Qα gldm form A 104 Rf Qα gldm form

492

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