THE FISSION BARRIERS OF SUPERHEAVY

AND EXOTIC NUCLEI         

Fedir A. Ivanyuk1 and Krzysztof Pomorski2

1Institut for Nuclear Research, Kiev, Ukraine2Theoretical Physics Division, UMCS, Lublin, Poland

• The variational principle for liquid drop shapes

• The account of curvature energy• The fission barriers within LSDrop model• The semi-analytical expression for the barrier

heights based on the topological theorem• Numerical results, mass-asymmetric shapes• Summary and outlook

2

0max

( )( , ) ( )exp

( )LSD

I I ZB Z I B Z

I Z

28 30 32 34 36 380

10

20

30

barr

ier

heig

ht (

MeV

)

Z2 / A

Bexp

BLSD

-Emicr

(gs)

B LSD micrV = + E ,B

• For Emicr see P. Moeller, J. R. Nix,

• W. D. Myers and W. J. Swiatecki, • At. Data and Nucl. Data Tables, • 59, 249 (1995).

2 4max 0 1 2 3

0 4 5

26 7 8

( ) ,

( )

( )

B Z a a Z a Z a Z

I Z a a Z

I Z a a Z a Z

• The rms-deviation is 1.113 MeV for known nuclei withZ>70

V.M.Strutinsky et al, Nucl. Phys. 46, 659 (1963)

1 2 12

212

( ) profile function

0

2( )

LD LD surf Coul curv

LD

E E y E y E y E y

E V Ry

R y z z

y y

z

z

dV

2

1

2

1

2surf

22

Coul

2 2 3/2

( )

1 2

0) (0 2

1 ( ) 10

2 ( ) 1 ( / )

1 ( )( ) ( , ( ))

2

the fissility parameter, / 2 ( / ) / 49

( ) the

( ) (1 ( )

Coulomb pot ti

)

en

L

z

z

z

LD Sz

LD LD C S

S

S

D

E y z dy dz dz

dy zE x y z z y

yy y y

z dzdz

x x E E Z

x

z

z z y

A

al on the surface

Optimal shapes

-2 -1 0 1 2

-1

0

1

xLD

=0.75

y(z)

/R0

z / R0

Deformation energy

xLD

=1.0

xLD

=0.0

1.0 1.5 2.0 2.5-0.1

0.0

0.1

0.2

0.3

Ede

f / E

(0) su

rf

R12

/ R0

R.W.Hasse, W.D.Myers, Geometrical Relationships of Macroscopic Nuclear Physics:

Modified Funny Hills shape parametrization

0.5 1.0 1.5 2.0 2.5

0.000

0.005

0.010

0.015

xLD

=0.75

Ede

fLD /

Esu

rf

(0)

R12

/ R0

MFH, B-minimization "optimal" shapes

2 22 2

0

2

2 3 4

1( ) (1 )(1 ),

( , )

( ) / ,

3( , ) 1 ( 1/2 ) Erf ( )4 2

1,

( ) 8.4977 24.893 25.566 10.836 1.6624

a u

sh

y z u u Becf a B

u z z cR

B af a B e a a aa

a

B c c c c c

• deviation < 150 keV

Surface curvature energy

Leptodermous expansion:

ETF = Evol+ Esurf + Ecurv + EGcurv

1.0 1.5 2.0 2.5

0.00

0.05

0.10

0.15

0.20

0.25

0.75

0.65

0.5

0.3xLD

=0.15

/R0= 0.05

Ede

fLD/ E

(0) su

rf

R12

/ R0

2 2 2 3/21 2

(0)

0

1 2

(0)

20

(0)

20

(0) (0)

( )4

1 1 1( )

2

4

(

[1 ( ) ], ( ) [1 ( ) ]

1 )4

/

curvcurv

SS

SS curv

curv S

EE H z dS

R

H zR R

EE dS

R

EE E H dS

R

E E

R y y R y y

3/22 2 2 21 2(1 / ( ) ) 1 ( ) 10 1 ( )LD Syy y yy y y z x yy

The LSD barrier heights

0.1 0.2 0.3

0

5

10

15

20

25

90

85

80

105

10095

Z=75

BLS

D /

MeV

(N-Z)/A

0.0 0.1 0.2 0.3

20

30

40

50

60

40

50

60

7065

55

45

Z=35

BLS

D /

MeV

(N-Z)/A

2 4max 0 1 2 3

0 4 5

26 7 8

( ) ,

( )

( )

B Z a a Z a Z a Z

I Z a a Z

I Z a a Z a Z

2

0max

( )( , ) ( )exp

( )LSD

I I ZB Z I B Z

I Z

F.A.Ivanyuk and K.Pomorski, Phys: Rev. C 79, 054327 (2009)

2

2 2/3

2 1/3

2 2 2

41/30

(1 )

(1 ) ( )

(1 ) ( )

3( )

5

LSD vol vol

surf surf S

curv curv K

Cch

E b I A

b I A B def

b I A B def

Z e ZB def C

Ar A

K.Pomorski and J. Dudek, Phys. Rev. C 67, 044316 (2003)

The rms dev.for 35<Z< 105, 0<I< 0.3 is 150 keV

The barrier heights, topological theorem

28 30 32 34 36 380

10

20

30

barr

ier

heig

ht (

MeV

)

Z2 / A

Bexp

BLSD

-Emicr

(gs)

(saddle) (saddle) (g.s.) (g.s)B LSD LSDV = E +δE -E +δE

W. D.Myers and W. J. Swiatecki, Nucl. Phys. A601, 141 (1996): the “barrierwill be determined by a path that avoids positive shell effects and has no use for negative shell effects. Hence the saddle point energy will be close to what it would have been in the absence of shell effects, i.e., close to the value given by the macroscopic theory!”

(saddle)B LSD micr

(g.s) (g.s.) (sph)micr LSD LSD

V = V + E ,

E =δE +( E -E )

• For Emicr see P. Moeller, J. R. Nix, W. D. Myers and W. J. Swiatecki,

At. Data and Nucl. Data Tables, 59, 249 (1995).

Barriers of light nuclei

10 12 14 16 18 20 22 24 26 28 30 32 34 36 380

10

20

30

40

50

60

barr

ier

heig

ht (

MeV

)

Z2 / A

Bexp

BLSD

-Emicr

(gs)

75Br35 , 90Mo42 , 94 Mo42 , 98Mo42

Mass asymmetry

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

ELD

=EC+E

S

0.7

0.6

0.5

0.4

0.3

0.1

0.2

xLD

=0L

D-b

arr

ier

he

igh

t /

ES

(0)

(MR-M

L)/(M

R+M

L)

The temperature dependence of the LSD-barriers

B. Nerlo-Pomorska, K. Pomorski and J. Bartel, Phys. Rev. C 74, 034327 (2006)

1.0 1.5 2.0-4

-2

0

2

4

232ThT=0

5

4

3

2

1

Ede

f(LS

D)

(MeV

)

R12

/ R0

2

2/3

1/3 2 1/3

( , ) ( , ) ( , ), ( , ) 2

( , ) ln (1 )ln(1 )

( , ) ( , ), ( , ) ( , 0)

/ 0.092 0.036 ( )

0.275 ( ) 0.00146 / (

Yukawa folded mean field

k kk

k k k kk

surf

curv Coul

F N T E N T T S N T E N T n

S N T n n n n

F N T F N T F N T F N T aT

a MeV A A B def

A B def Z A B def

% % %

2

)

( , , ) ( , 0, ) ( )LSD LSDF N T def F N T def a def T

Summary• The optimal shapes of fissioning nuclei are studied within the Lublin-

Strasbourg drop model

• The liquid drop fission barriers were calculated for isotopic chains in the range 35< Z < 110 and a simple approximation for the liquid drop fission barriers heights containing only Z and (N-Z)/A is obtained.

• The topological theorem by Myers and Swiatecki was used to express the fission barrier height as the sum of the macroscopic barrier height and the ground-state shell correction. This provides a very simple method to calculate fission barrier heights. In addition to the ground-state shell

• correction, which should be taken from the tables, one only has to calculate simple analytical estimate for the liquid drop part of the barrier height.

• The rms deviation of calculated versus experimental values of fission-barrier heights for known nuclei with Z < 70 is 1.113 MeV, a value that is comparable with the experimental uncertainties.

• Dear colleagues,• This is the first circular of the 3-rd International Conference on Current Problems• in Nuclear Physics and Atomic Energy (NPAE-Kyiv2010), which will be held from• June 7 to June 12, 2010 in Kyiv, Ukraine.

• The NPAE-Kyiv2010 conference will cover the following topics:• Collective processes in atomic nuclei• Nuclear reactions• Nuclear structure and decay processes• Rare nuclear processes• Neutron and reactor physics, nuclear data• Problems of atomic energy• Applied nuclear physics in medicine and industry• Experimental facilities and detection techniques

Mass-asymmetric shapes

1 2 12 3

1 2

21

2 2 3/ 22

0

1 1 1( )

2

as

[1 ( ) ],

(

ymmetry

) [1 ( ) ]

: R L

R L

LD

V

E V Ry

H zR R

R y y

R y y

V

V V

2 * 2 3/ 21 2

*3sign1 ( ) 10( ) ( ) [1 ( ) ]LD Szyy y y z z x z yz

-2 -1 0 1 20,0

0,5

1,0

y(z)

z / R0

-2 -1 0 1 20

1

2

xLD

=0.75

H(z

)

Mass asymmetric shapes, x = 0.75

0.9

0.6

0.3

R12

/R0

asy

mm

etr

y

0

0.75 1.0 1.25 1.5 1.75 2.0 2.25

K.T.R.Davies and A.J.Sierk, Phys.Rev.C 31 (1985) 915