THE SHAPE TRANSITION IN ROTATING NUCLEI

         F. Ivanyuk1, K. Pomorski2 and J.Bartel3

1Institut for Nuclear Research, Kiev, Ukraine2Theoretical Physics Division, UMCS, Lublin, Poland3Institute Pluridisciplinaire Hubert Curien, Strasbourg University, Strasbourg, France

• Introduction• The Strutinsky‘s procedure for the shape of

fissioning nuclei• The shape transition in axially symmetric

rotating charged liquid drop• The shape of triaxial rotating drop• Summary and outlook

Strutinsky‘s optimal shapes

( ) profile function

LD LD surf Coul curvE E E E

z

E

2

1

2

1

2surf

22

Coul

2 ( ) 1 ( / )

1 ( )( ) ( , ( ))

2

z

z

z

LD Sz

E z d dz dz

d zE x z z z dz

dz

21 2 12 12

20, ( )LDE V R R z z dz

V

2 2 3/2

1 2

(0) (0) 2the fissility parameter, / 2 ( / ) / 49

( ) the Coulomb pot

1 ( ) 10 ( ) (

ential on the surf

1 ( )

ce

)

a

LD S

LD LD C S

S

x x E E

x

z

z

A

z

Z

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

(z)

z

Optimal shapes of fissioning drop

-2 -1 0 1 2

-1

0

1

xLD

=0.75

(z)

/R0

z / R0

The 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

E

def /

E(0

) surf

R12

/ R0

F. Ivanyuk, Int. J. Mod. Phys. E18 (2009) 130

The shape of axially symmetric rotating drop

1 2 12var .principle : 0 0LD LD rotE V R E V E

2 2 2

:

/ 2 , ( / ) rigid-bodymoment of i

theenerg

ert

y

n ia

RLD surf Coul curv rot

rot

E E E E E

E L J J M V r dV

h

J. Bartel, F. Ivanyuk and K.Pomorski, Int. Jour. Mod. Phys. E19 (2010) 601

2 2 2 3/2

(0) (0) (0)

2 2

2

/ , ( ) / , /

( ) 2 , for the perpendicular rotation

2 ( ) , for the par

151 ( )

d

10 ( )

allel rotat

( ) (1 ( ) )

iff.equ

ion

4

ation

LD rot S rot rot rot rot

LD S

rot

rot LD rot

y E E z

x z B y z

B B E E

z z

z

( ) {rot z

The energy and shape transition in axially symmetric rotating drop

( 1) 2 ( 1) ( 1)RLD surf LD Coul LD rotB B x B y B

0.0 0.2 0.4 0.6-0.15

-0.10

-0.05

0.00

'perpendicular' rotation 'parallel' rotation

xLD

=0

BR

LD

0.0 0.1 0.2 0.3 0.4-0.10

-0.05

0.00

xLD

=0.25

0.00 0.05 0.10 0.15 0.20

-0.04

-0.02

0.00 xLD

=0.50

BR

LD

yLD

0.00 0.01 0.02 0.03 0.04 0.05

-0.004

-0.002

0.000x

LD=0.75

yLD

The deformation energy of rotating drop

0.5 1.0 1.5 2.0

0.00

0.01

0.02

0.03

0.04

0.05

yLD

=0.04

yLD

=0

xLD

=0.75E

defLD

M+

Ero

t

R12

/ R0

The energy and shape transition in axially symmetric rotating drop

( 1) 2 ( 1) ( 1)RLD surf LD Coul LD rotB B x B y B

0.0 0.2 0.4 0.6-0.15

-0.10

-0.05

0.00

'perpendicular' rotation 'parallel' rotation

xLD

=0

BR

LD

0.0 0.1 0.2 0.3 0.4-0.10

-0.05

0.00

xLD

=0.25

0.00 0.05 0.10 0.15 0.20

-0.04

-0.02

0.00 xLD

=0.50

BR

LD

yLD

0.00 0.01 0.02 0.03 0.04 0.05

-0.004

-0.002

0.000x

LD=0.75

yLD

The shape transition in rotating drop

-1 0 10.0

0.5

1.0

z / R0

'perpendicular' rotation 'paralel' rotation

xLD

=0.5, yLD

=0.124

(z)

/ R

0

-1 0 10.0

0.5

1.0

xLD

=0.25, yLD

=0.225

(z)

/ R

0

The energy and shape transition in axially symmetric rotating drop

( 1) 2 ( 1) ( 1)RLD surf LD Coul LD rotB B x B y B

0.0 0.2 0.4 0.6-0.15

-0.10

-0.05

0.00

'perpendicular' rotation 'parallel' rotation

xLD

=0

BR

LD

0.0 0.1 0.2 0.3 0.4-0.10

-0.05

0.00

xLD

=0.25

0.00 0.05 0.10 0.15 0.20

-0.04

-0.02

0.00 xLD

=0.50

BR

LD

yLD

0.00 0.01 0.02 0.03 0.04 0.05

-0.004

-0.002

0.000x

LD=0.75

yLD

The critical rotational velocities

-1 0 1

-1

0

1

(

z) /

R0

z / R0

0.0 0.5 1.00

1

2

3

0.612

ycr

it

x

The non-axial rotating drops

K. Pomorski, F. Ivanyuk and J. Bartel, Acta Phys. Polon. B42 (2011) 455 A. May, K. Mazurek, J. Dudec, M. Kmiecik and D. Rouvel, Int. J. Mod. Phys. E19 (2010) 532

2

2

22

2 22

( ) ( , ), ( , , ; , )

Euler-Lagrange equation: 0

Fourier series: ( , ) ( )cos( )

Approximations: ( , ) ( )[1 ( )cos(2 )]

(1 / ) 1( , )

1 cos(2( ,

)

RLD z

z

n

n

z z E dz d z

d d

dz d

z z n

z

z

z z

z cz

2( ) 1 ( ))

1 ( ) cos(2 )

z z

z

The equations for ( ) and ( )z z

2 2 2 21 2 3 4 5 6

23 2

2 2 2 21 2 3 4 5 6

2

2

15 51 (1 )[2 ] ( ) 0,

216

2

15 51 (1 )( ) 0

2 216

LD

LD

LDSa

rot

LDSb

rot

a a a a a a

y xz z

B

b b b b b b

y xz

B

2 2

20 0

( , ) [ ( ) cos(2 )] ( , )( ) , ( )

2 1 ( )cos(2 ) 2 [1 ( )cos(2 )]S S

Sa Sbz z zd d

z zz z

21 , ( , ) theCoulombpotentialon thesurfaceS z

The shape of axially non-symmetric rotating drop

-1 0 10.0

0.5

1.0

xLD

=0.75

yLD

=0.043

yLD

=0

(z

) / R

02

-1 0 10.0

0.5

1.0

yLD

=0.044

yLD

=0.01

-1 0 1

0.0

0.1

0.2

(z)

z / R0

-1 0 1

0.0

0.1

0.2

z / R0

The energy of axially non-symmetric rotating drop

0.00 0.05 0.10 0.15 0.20-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

'perpendicular' rotation 'parallel' rotation nonaxial 'perp.' rotation

xLD

=0.5

B

RLD

yLD

The energy of axially non-symmetric rotating drop

0.00 0.01 0.02 0.03 0.04 0.05-0.004

-0.003

-0.002

-0.001

0.000

0.001

'perpendicular' rotation 'parallel' rotation nonaxial 'perp.' rotation

xLD

=0.75

B

RLD

yLD

The comparison with ellipsoidal shapes

0.0 0.1 0.2

1

2 xLD

=0.5

sem

i-axe

s

yLD

22 2 22

2 2 2

( ) 1 ( )1(dash), ( , ) (solid)

1 ( ) cos(2 )

z zx y zz

za b c

The effective ellipsoids

0.0 0.1 0.2

1

2

3

xLD

=0.5

sem

i-axe

s

yLD

2 220 20

.

2 222 22

.

( , ) ( , ) ,

( , ) ( , )

ellips optimal

ellips optimal

r Y r Y

r Y r Y

The effective ellipsoids, the energy

2 220 20

.

2 222 22

.

( , ) ( , ) ,

( , ) ( , )

ellips optimal

ellips optimal

r Y r Y

r Y r Y

0.00 0.05 0.10 0.15 0.20-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

xLD

=0.5

BR

LD

yLD

triaxial ellipsoides nonaxial 'perpendicular' rotation

The limiting values of rotational velocity

0.00 0.25 0.50 0.75 1.000.0

0.2

0.4

0.6

0.8

ymax, fission

yJac.

xLD

y max

0 100 200 3000

25

50

75

100

ymax, fission, axial

L max

A

Summary and outlook

• Within a liqiud drop model we have developed a method for the calculation of optimal shape of the surface within a broad region of rotational velocitiy and fissility parameter

• We have found out that sharp Jacobi transition from oblate to triaxiall ellipsoides is the result of limitation imposed by the assumption of the ellipsoidal shape

• For the more flexible shape parmeterisation the Jacobi transition gets smoothed

• For the drops with the fissility parameter xLD>0.612 the fission takes place before the Jacobi transition

• The investigation of Poincare instability will be the subject of future studies

Thank you for attention

The barrier heights, topographical 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 )

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