Fission barriers of heavy and superheavy nuclei analyzed in multidimensional deformation space

I. IntroductionII. Method III. Deformation space IV. Results and discussion V. Conclusions

XIII Nuclear Physics WorkshopKazimierz Dolny, 27. 09 - 1.10. 2006

M. Kowal, L. Shvedov and A. Sobiczewski

Sołtan Institute for Nuclear Studies, Warsaw, Poland

I. Introduction

1. Two main problems with heaviest nuclei (HN):

cross sections (~1 pb ~50 fb) Bfst

half-lives

2. Present state of HN (f1,f1a)

3. Role of Bfst (f2)

sensitivity of to Bfst

a need for a large accuracy of Bfst

98

99

100

101

102

103

104

Db 267Db 267

115 287115 287 115 288115 288

113 283113 283 113 284113 284

111 279111 279 111 280111 280

Mt 275Mt 275 Mt 276Mt 276

Bh 271Bh 271 Bh 272Bh 272

120120120120

119119119119

118118118118

117117117117

116116116116

115115115115

114114114114

113113113113

112112112112

111111111111

Hs 264Hs 264

Ds 267Ds 267

Mt 266Mt 266

111 272111 272

Mt 268Mt 268

Hs 266Hs 266 Hs 267Hs 267

Bh 262Bh 262 Bh 264Bh 264

Sg 263Sg 263

Bh 267Bh 267Bh 266

Db 263Db 263

Rf 263Rf 263

Es 248 Es 249 Es 250 Es 251 Es 252 Es 253 Es 254 Es 255 Es 256Es 256

Fm 249 Fm 250 Fm 251 Fm 252 Fm 253 Fm 254 Fm 255 Fm 256

Md 250 Md 251 Md 252 Md 253 Md 254 Md 255 Md 256 Md 257

Fm 257 Fm 258Fm 258 Fm 259Fm 259

Md 258 Md 259 Md 260Md 260

No 251 No 253 No 254 No 255 No 256 No 257 No 258

Lr 252 Lr 253 Lr 254 Lr 255 Lr 256 Lr 257 Lr 258 Lr 259

No 259 No 260No 260

Lr 260 Lr 261 Lr 262Lr 262

Rf 253 Rf 254 Rf 256 Rf 257 Rf 258 Rf 259 Rf 260

Db 257 Db 258 Db 259

Rf 261 Rf 262 Rf 267Rf 267 Rf 268Rf 268

Db 262 Db 268

Sg 258 Sg 259 Sg 260 Sg 261 Sg 262

Bh 261

Sg 265Sg 265 Sg 266Sg 266 Sg 271Sg 271

Hs 265 Hs 270Hs 270 Hs 275Hs 275

Ds 269 Ds 270Ds 270 Ds 271Ds 271 Ds 279Ds 279 Ds 281Ds 281

112 282112 282 112 283112 283 112 284112 284 112 285112 285

114 286114 286 114 287114 287 114 288114 288 114 289114 289

116 290116 290 116 291116 291 116 292116 292 116 293116 293

118 294118 294

CfCfCfCf

EsEs

FmFm

MdMd

NoNo

LrLr

RfRf

DbDbSgSg

BhBhHsHs

MtMtDsDs

No 262No 262

Cf 247Cf 247 Cf 248Cf 248 Cf 249Cf 249 Cf 250Cf 250 Cf 251Cf 251 Cf 252Cf 252 Cf 253Cf 253 Cf 254Cf 254 Cf 255Cf 255 Cf 256Cf 256

Rf 255

Db 260 Db 261

No 252

Db 256

160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 149 150 151 152 153 154 155 156 157 158 159

105

106

107

108

109

110

113 278113 278

112 277112 277

111 274111 274

Ds 273Ds 273

Mt 270Mt 270

Hs 269Hs 269

Reaktionen und Compoundkerne

II. Method

Macro-micro (same as used for description of many properties of HN)

III. Deformation space

1. As large as possible

2. Larger space, better description of the properties

(e.g. mass, especially Tsf)

3. Specification of the space: axial, non-axial and reflection-asymmetric

shapes included

A large, 10-dimensional spaceOne to one correspondence between values of parameters and shape

IV. Results1. Axial symmetry - example: 278112 (f4)

- dependence on max (f5)

2. Quadrupole non-axiality ( =2) (f6-8)

- mechanism of decreasing Bfst by non-axial shapes

3. Hexadecapole non-axiality ( =4) (f9-9a)

- also a discussion by M. Kowal

4. Comparison with exp. (f10)

5. Reflexion asymmetry (f11)

The barrier: thin but high, created totally by shell effects

Effect of total hexadecapole deformation

Effect of non-axial hexadecapole deformations

Effect of non-axiality parameter

2 4 6 80,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

5,5

6,0

6,5

7,0

7,5

8,0

2 4

AxialB

fst (

MeV

)

max

250Cf Non-axial

Effect of reflection-asymmetric deformations

-0,50

-1,0

-0,50

-1,0

-1,5-2,0

-2,5

-0,50

-3,0-3,5

-4,0-4,5 -5,0

-1,5

-2,0

-2,5

0,0 0,2 0,4 0,6 0,8 1,0 1,2

-0,1

0,0

0,1

0,2

0,3

0,4

(3,6)

250Cf

(-4,7)

Conclusions

1. Barriers of HN are totally created by shell effects. They are thin, but high.

2. Their height Bfst strongly depends on the deformation space, in

which they are calculated.

3. An increase of the dimension of the space results in an increase of Bf

st for deformed nuclei, and in a decrease of it for spherical ones, in the case of axial symmetry.

4. Non-axial shapes are important for Bfst . They may decrease it by up

to about 2 MeV. This is again due to shell effects, because macr. part of the energy is stiff against non-axiality. Only after the inclusion of non-axiality, calculated Bf

st well reproduces exp. value of it.

5. Reflexion-asymmetric shapes do not contribute to Bfst for heaviest

nuclei.

BF=4 MeV

3,5

2,5

1,5

0,50

-0,50-0,50

-1,51,5

-2,5

-1,5

0,50

-3,5

-0,50

0,50

0,0 0,1 0,2 0,3 0,4 0,5 0,6

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

= 6

0o

= 30

o

262106 2

sin

( 2)

2cos(

2)

(-0,2)

(-5,3)

-0,50

-1,0

-1,5

-2,0

-0,50

-0,50

-1,0

-2,5

-0,50

-1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,60,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

= 30

o = 6

0o

2si

n( 2

)

2cos(

2)

262106

3,0

2,0

1,0 2,0

1,0

0

3,0

0

-1,0

1,0

-2,0

-1,02,0-3,0

-2,0

-4,0

0,0 0,1 0,2 0,3 0,4 0,5 0,60,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

2 =

60

o

2 =

30o

262106

2si

n( 2

)

2cos(

2)

-0,10

-0,10

-0,20

0,0 0,1 0,2 0,3 0,4 0,5 0,60,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

= 30

o

= 6

0o

262106

2si

n( 2

)

2cos(

2)

-0,20

-0,40-0,60-0,80

-1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,60,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

= 30o

= 6

0o

262106 2si

n( 2

)

2cos(

2)