a fresh look at the scission configuration fedir a. ivanyuk institut for nuclear research, kiev,...
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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
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