a fresh look at the scission configuration fedir a. ivanyuk institut for nuclear research, kiev,...
DESCRIPTION
Cassini ovaloidsTRANSCRIPT
![Page 1: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/1.jpg)
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
![Page 2: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/2.jpg)
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
![Page 3: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/3.jpg)
Cassini ovaloids
![Page 4: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/4.jpg)
( ; ) (1 ( ))0R x R P xn nn
![Page 5: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/5.jpg)
![Page 6: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/6.jpg)
Parameteization of Moeller et al
![Page 7: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/7.jpg)
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
![Page 8: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/8.jpg)
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
![Page 9: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/9.jpg)
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)
![Page 10: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/10.jpg)
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)
()
![Page 11: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/11.jpg)
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
![Page 12: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/12.jpg)
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
![Page 13: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/13.jpg)
Deformation energy, (R12 )crit = 2.3 R0
![Page 14: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/14.jpg)
R.W.Hasse, W.D.Myers, Geometrical Relationships of Macroscopic Nuclear Physics:
![Page 15: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/15.jpg)
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.
![Page 16: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/16.jpg)
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
![Page 17: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/17.jpg)
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).
![Page 18: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/18.jpg)
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
![Page 19: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/19.jpg)
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
![Page 20: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/20.jpg)
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
![Page 21: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/21.jpg)
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
)
![Page 22: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/22.jpg)
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
![Page 23: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/23.jpg)
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
![Page 24: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/24.jpg)
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
![Page 25: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/25.jpg)
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
![Page 26: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/26.jpg)
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
![Page 27: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/27.jpg)
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
![Page 28: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/28.jpg)
(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
![Page 29: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/29.jpg)
K.T.R.Davies and A.J.Sierk, Phys.Rev.C 31 (1985) 915
![Page 30: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/30.jpg)
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)
![Page 31: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/31.jpg)
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
![Page 32: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/32.jpg)
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
![Page 33: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/33.jpg)
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).
![Page 34: A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle](https://reader036.vdocuments.net/reader036/viewer/2022062600/5a4d1ba17f8b9ab0599c7a5d/html5/thumbnails/34.jpg)
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