A FRESH LOOK AT THE SCISSION CONFIGURATION       Fedir A. Ivanyuk

Institut for Nuclear Research, Kiev, Ukraine

• Shape parameterisations• The variational principle for liquid drop

shapes• Two point boundary problem, the relaxation

method• The scission configuration• Mass-asymmetric shapes• Applications: the barriers of heavy nuclei• Summary and outlook

The shape parameterisations

• Expansion around sphere in terms of spherical harmonics

• (Distorted) Cassinian ovaloids• Koonin-Trentalange parameterisation• (modified) Funny-Hills parameterisation• Two smoothly connected spheroids • The two center shell model

2 ( ) ( / )n n on

y z a P z z

Cassini ovaloids

( ; ) (1 ( ))0R x R P xn nn

Parameteization of Moeller et al

r

a2a1

r R=0.75 (1+ )0 d2/3 r

z =z1 2=0r r r

z

z2a1 a2

b1

b2

a1 | |z1 z2 a2

b2

|zmax1 | zmin

2

| |z1

E0

E

= /E0E

z

V V V

V0

b1

b1 b2

( )a ( )b ( )c

The two center shell model

J. Maruhn and W. Greiner, Z. Phys, 1972

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

1 2 12

212

( ) profile fun

,

0

2

cti n

( )

oLD LD surf Coul

LD

E E y E y E y

E V Ry

R y z z

y y z

dzV

d d

2

1

2

1

2

1

2surf

22

Coul

22 2 2

2 ( ) 1 ( / )

1 ( )( ) ( , ( ))2

3 ( )( , ) ( ) ( ) ( ) ( , ) ( , )4

( , ) ( , ) elliptic integrals of first and se

z

z

z

LD Sz

z

S z

E y z dy dz dz

dy zE x y z z y z dzdz

dy zz y y z y z z z z F a b E a b dzdz

F a b E a b

cond type

2 2 3/ 2

2 2 3/ 21 2

2 2

1 2

1 ( ) 10 ( ) (1 ( ) )

the fissility parameter, ( / ) /( / )( ) the Coulomb potential on the surface

1 ( ) 10 ( ) (1 ( )

( ) ( )

)

LD S

LD LD crit

S

S S

LD Syy y y z

yy y y z x z y

x x Z A Zz

z

x z y

A

z z

d

d

-2 -1 00,0

0,5

z / R0

y(z)

0,75

1,00

S(z)

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

0,0 0,5 1,0 1,50,0

0,1

0,2

0

(2)0

(1)

()

The two point boundary value problem

n n

n n nk k-1 k k-1 k

In the one replaces the ordinary differential equations

dy /dx= g (x; y) with an algebraic equations relating function values at t

relaxation method

y - y = (x -

wo points k; k

x ) g [

- 1

(x

:

+(0)

k k k k k-1

k- k -

k

1 k 1

y = y y ; g(x; y) is expanded with respect to y , ywhat leads to the system of k-1 algebraic equations for ythe missing equation is given by bou

x )/2;

ndary conditionPress

(y

W

+ y )/2]

Numerical Recipes in F.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.

, Vol. 1, Cambridge University Preor sstr , an 77 1986

Optimal shapes

-2 -1 0 1 2

-1

0

1x

LD=0.75

y(z)

/ R

0

z / R0

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

Deformation energy, (R12 )crit = 2.3 R0

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

The scission point: the stiffness with respect to neck is sero

1.0 1.5 2.0 2.50.00

0.01

0.02

0.03

0.04

0.05

xLD

=0.75

Ede

f / E

(0) su

rf

R12

/ R0

U.Brosa, S.Grossmann and A.Muller, Phys. Rep. 197 (1990) 167—262.

Cassini ovaloids

1,0 1,5 2,0 2,5

-0,05

0,00

0,05

0,100.5

0.6

0.7

0.8

xLD

=0.9 "optimal" shapes Cassini ovaloids

E

def

R12

/ R0

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

FH, B-minimization MFH, B-minimization "optimal" shapes

FH: M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky and C. Y. Wong, Rev. Mod. Phys. 44, 320 (1972).MFH: K. Pomorski and J. Bartel, Int. J. Mod. Phys. E 15, 417 (2006).

0,75 1,00 1,25 1,50 1,75 2,00 2,25

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

n=8

n=6

n=4n=2

n=0 xLD=0.75

a n

R12

/ R0

20

20

S. Trentalange, S.E. Koonin, and A.J. Sierk, Phys. Rev. C 22 (1980) 1159

( ) ( / )n nn

y z R a P z z

How unique are the „optimal“ shapes ?

1 2

2 2 2 3/21 2

2 2 3/2

2 2

1 2

3/2

2 21 2

1 2

1 1 1( ) average curvature2

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

1 ( ) 2 ( )[1 ( ) ]

1 ( ) 10 ( ) [1 ( )

10 ( ) 2 ( )

( ) / 0

]

4 1

LD S

LD S

LD

z x z H

H zR R

R y y R y y

yy y

z

z y z

yH z y

yy y y z x z y

x

( ) 2 ( )S z H z

Q2 - constraint

1,0 1,5 2,0 2,5

0,00

0,05

0,10

0.5

0.6

0.7

R12

restriction Q

2 restriction

xLD

=0.8

E

LDde

f / E

(0) su

rf

R12

/ R0

Mass-asymmetric shapes

1 2 12 3

1 22

12 2 3/2

2

2

2

0

2 ( ) 1 / 1 /

[1 (

asymme

) ],

( ) [1 ( ) ]

( )

(

try :

( *)

)

LD

c

R L

m

R L

E V RyH z R R

R y y

R y

V VV V

sign z z

y

dz y zV

z dz y zV

z

d dd

d

d

2 * 2 3/2

1 2

2 * 2 3 21

3

2

*

* /3

sign(1 ( ) 10 ( ) [1 ( ) ]

1 ( ) 10 ( ) [

)

1 )) ]( (

LD S

LD S

yy y y z z x z y

y

z

y y y xzz z

z

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

asym

met

ry

0

0.75 1.0 1.25 1.5 1.75 2.0 2.25

Deformation energy

1.0 1.5 2.0 2.5

0.00

0.02

0.04

0.06

d = 0.8

d = 0.1

xLDM

=0.75

Ede

fLD /

ES(s

ph)

R12

/ R02

dash - shape divided in parts be the neck solid - shape divided by the point of maximal curvature

Deformation energy

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00

0.02

0.04

0.06

d = 0.8

d = 0.1

xLDM

=0.75

Ede

fLD /

ES(s

ph)

Q2 / MR

02

shape is divided in parts by the point of maximal curvature

The scission shapes, Rneck =0.2 R0

-2 -1 0 1 2-2

-1

0

1

2

0.1 < d < 0.9

xLDM

=0.75

y / R

0

z / R0

Optimal/Cassini shapes

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.02

0.04

0.06

optimal shapes Cassini ovaloids, ,

1

d = 0.7

d = 0.1

xLDM=0.75

Ede

fLD /

ES(s

ph)

Q2 / MR

02

shape is divided in parts by the point of maximal curvature

Optimal/Cassini shapes

-2 -1 0 1 2

-1

0

1

optimal shapes Cassini ovaloids, ,

1

xLDM

=0.75, d=0.5, Q2/MR

02=1.5

y / R

0

z / R0

(z-z*)/octupole constraint

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.02

0.04

z-z* constraint octupole constraint

d = 0.5

d = 0.1

xLDM

=0.75

Ede

fLD /

ES(s

ph)

Q2 / MR

02

shape is divided in parts by the point of maximal curvature

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

Businaro-Gallone point

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

=0LD

-bar

rier h

eigh

t / E

S(0)

(MR-M

L)/(M

R+M

L)

The barriers of heavy nuclei, surface curvature energy

Leptodermous expansion:ETF = Evol+ Esurf + Ecurv + EGcurv

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

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

The LSD barrier heights

0.1 0.2 0.3

0

5

10

15

20

25

90

85

80

105100

95

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 52

6 7 8

( ) ,( )

( )

B Z a a Z a Z a ZI Z a a Z

I Z a a Z a Z

20

max( )

( , ) ( )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 CAr 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 h

eigh

t (M

eV)

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).

Summary and outlook

• 1. The relaxation method allows to solve the variational problem for the shapes of contiional eqilibrium with a rather general constraints

• 2. The extension of this method to separated shapes • and account of the• surface diffuseness, attractive interaction• (eventually) shell corrections would result in a very accurate method for the

calculation of the potential energy surface

z

VRV

L

yR(z)

yL(z)

R12