saddle point shapes of nuclei - Frankfurt Institute for Advanced Studies

SADDLE POINT SHAPES OF NUCLEI
Poenaru, Gherghescu, W. Greiner,Complex fission phenomena, Nucl. Phys. A 747 (2005) 182.
Poenaru, W. Greiner, Deformation energy minima at finite mass asymmetry, EPL 64 (2003) 164
Poenaru, W. Greiner, Nagame, Gherghescu, Nuclear shapes in fission phenomena, J. Nucl.
Radiochem. Sci. Japan 3 (2002) 43
Dorin Poenaru

OUTLINE
• Nuclear shapes and mass asymmetry
• Equilibrium and scission configurations
• Examples of shapes (given
parametrization)
• David L. Hill’s method
• Integro-differential equation (no param.)
• Phenomenological shell effects
• Fission into 2, 3, and 4 fragments
• Experimental attempts
• Conclusions
Dorin Poenaru

SCIENTISTS and INSTITUTES
NSTITUTES
Walter GREINER, FIAS
Joseph H. HAMILTON, DPA
Yuichiro NAGAME, ASRC
FIAS = Frankfurt Institute for Advanced Studies, J. W. Goethe University,
Frankfurt am Main, http://www.fias.uni-frankfurt.de
DPA = Department of Physics & Astronomy, Vanderbilt
University, Nashville, TN, USA, http://www.vanderbilt.edu
IFIN-HH = H. Hulubei National Institute of Physics & Nuclear Engineering,
Bucharest, http://www.theory.nipne.ro
ASRC = Advanced Science Research Center, Japan Atomic Energy Research
Institute, Tokai, http://asr.tokai.jaeri.go.jp
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Akunuri V. RAMAYYA, DPA
Radu A. GHERGHESCU, IFIN-HH IFIN HH
Dorin N. POENARU, IFIN-HH IFIN HH

HYSTORICAL MILE-STONES
MILE STONES
• 1929 Fine structure in α−decay
• 1939 Induced fission
• 1940 Spontaneous fission
• 1946 Ternary (particle-accomp.) fission
• 1980 Prediction of cluster Radioactivity
• 1981 Cold binary fission
• 1984 cR experimentally confirmed
• 1998 Cold α and Be accompanied fission
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FISSION FRAGMENT MASS ASYMMETRY
Symmetry for some
Fm, Md, No
isotopes when the
most probable
fragments are very
close to the doubly
magic 132 Sn
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EQUILIBRIUM QUILIBRIUM CONFIGURATIONS
Nuclear gs deformation and fission fragment
mass asymetry are not explained within LDM.
One should add shell corrections to calculate
Potential Energy Surfaces, PES eg
Equilibrium: ∂E
∂E
∂q
1
=
∂q
= 0
• Minima (ground state, shape isomer)
• Sadde-points: maxima on fission valley
2
2
(min.)
∂ E ∂ E
∂E
∂q
1
∂E
=
∂q
2
= 0
2
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2
∂q1
2
∂ E
∂q
∂q
2
1
∂q1∂q
2
∂ E
∂q
E =
2
2
2
( 1, 2)
q q E
< 0

G STATE, VALLEYS, SADDLE POINTS
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POTENTIAL ENERGY SURFACE of 264Fm
Symmetrical mass
distributions
GS=ground state
SI=shape isomer
SP1=saddle point
SP2=saddle point
— Fission valley
at η = 0
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264 Fm

TWO PATHS FROM SADDLE TO SCISSION
Y. Nagame et al.
J. Radioanalyt. and Nucl.
Chemistry 239 (1999)
97.
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THREE HREE DECAY MODES OF 234 234U LIGHT FRAGM.
α
28 Mg
100 Zr
SHAPES ALONG FISS. PATH
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252 Cf
Cf 146 146Ba Ba + 96 96Sr Sr + 10 10Be Be
Given parametrization and fission path
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GIVEN PARAMETRIZATION
Möller, Madland, Sierk & Iwamoto, Nature 409 (2001)
5 dimensional shape coordinates
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∇v r
DAVID HILL’S HILL METHOD (I)
Thesis (Princeton 1951) – large amplitude motion
of an incompressible fluid with irrotational flow
=
0,
∇ × v r
mass density µ under bulk force
=
0
of electrostatic nature and a pressure distr. p 1
r
dv
r
µ = χ − ∇ p
dt
1
H µ
2
1
≅
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−∇
p
r
F = −∇U
∇
2
H = 0
2
= U + p1
+ v p( z ) = U ( z ) + σΚ(
z )
Pseudopressure p, surf. tens. σ, and curvature K

DAVID HILL’S HILL METHOD (II)
D.L. Hill: The Dynamics
of Nuclear Fission,
Proc. of the 2 nd United
Nations International
Conference on the
Peaceful Uses of Atomic
Energy, Vol. 15,Geneva,
1958
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STATIC PATH ON PES
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STATIC PATH ON CONTOUR PLOT
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INTEGRO-DIFFERENTIAL INTEGRO DIFFERENTIAL EQUATION (I)
Κ(
z ) =
1
ℜ
a
+
1
ℜ
b
=
1
ρ(
1 + ρ'
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2
)
1 / 2
−
ρ'
'
2
( 1 + ρ'
ρ = ρ (z) is the surface equation (cylindrical symm.)
minimizing the deformation energy
E def
LDM
s C
( α ) = E + δE
= E + E + δE
with constraints: const. deformation α and
α =
z +
c
L
z
c
R
mass asymmetry η
η
=
A1
− A2
A + A
1
2
)
3 / 2

INTEGRO-DIFFERENTIAL INTEGRO DIFFERENTIAL EQUATION (II)
The pressure at the surface
r r
ρ ( r ) + 2σΚ
( r )
eV
s
is in equilibrium. V s is the electrostatic potential
ρ e is the charge density
2
ρρ'
' −ρ' −[
λ + λ z + ℑ(
ρ,
z )] ρ(
1 + ρ'
1
2
2
)
3 / 2
ℑ = 10 XVs
( z , ρ)
Lagrange multipliers λ1,2 −1
= 0
X = fissility V s is expressed by a double integral
Conditions
ρ ( z1
) = ρ(
z 2 ) = 0
dρ( z1
, 2 )
= ±∞
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INTEGRO-DIFFERENTIAL INTEGRO DIFFERENTIAL EQUATION (III)
z 1,2 are the intercepts with z axis at the 2 tips.
Symmetry: z 2 = z p = - z 1 . New function
Vsd s
2 2
u( v)
= Λ ρ [ z(
v)];
( v)
= z −v
/ Λ
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z p
1 2
u'' = 2 + [ u'
+ ( v − d + Vsd
)( 4u
+ u'
u
5X
= V
2Λ
− a −vb;
2
)
3/
2
u ( 0)
= u'(
vpn)
= 0;
u '( 0)
= 1/
d;
d and n are input parameters related to elongation
& number of necks. Runge-Kutta numerical meth.
]

REFLECTION ASYMMETRICAL SHAPES
Equations for left hand side (L) and right h.s. (R).
Matching ρ L(0) = ρ R(0)
2π
M L = ( 1+
η)
= πΛ
3
−3
L
And similarly for right hand side.
∫
vp
0
n ≠
u
L
,
Alternatively: and continuity of ρ’’(0)
L
1/
2
L
n
R
1/
2
L
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u ( v ) / Λ = u ( v ) / Λ
( v)
dv;
( d −v
) u ( v ) = ( d − v
L
pL
pL
R
p
L
1/
2
R
⎧3
Λ 0 = ⎨ ∫ 2 0 ⎩
p v
L u
pR
) u
1/
2
R
( v
pR
)
L
p
R
⎫
( v)
dv⎬
⎭
1/
3

DEFORMATION ENERGY
Liquid Drop Model (LDM)
E
LDM
X =
0
= E − E
0
0
= Es
( Bs
−1)
+ EC
( BC
0 0
E /( 2E
) - fissility
C
s
Deformation dependent surface energy
y(x) = surface equation with (–1,+1) intercepts on x axis
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−1)
0
2 2/
3
0 2 −1/
3
Es = as(
1−κI
) A ; I = ( N − Z)
/ A;
EC
= aCZ
A
B s
1/
2
2 + 1
2
2
d ⎡
2 1 ⎛ dy ⎞ ⎤ z2
− z
= ∫ ⎢y
+ ⎥ dx;
d =
2 4 ⎜
dx ⎟
2R
−1⎢⎣
⎝ ⎠ ⎥⎦
0
1

B c
COULOMB ENERGY
Assume uniform charge density ρ0e = ρ1e = ρ2e 5 + 1 + 1
5d
= ∫dx∫dx' F(
x,
x')
;
F(
x,
x')
= { yy1[(
K − 2D)
/ 3]
•
8π
−1
−1
2 2
⎡
⎤
2 2
2 3 ⎛ dy ⎞
1 dy
⎢2(
y + y − x − x + x − x ⎜ − ⎟
1 ) ( ')
( ')
⎥ +
⎣
2 ⎝ dx'
dx ⎠⎦
2
2
⎧ 2 2 ⎡ 2 x − x'
dy ⎤⎡
2 x − x'
dy ⎤⎫
1 −1
K − K'
K⎨y
y1
/ 3+
⎢y
− ⎥⎢y
1 − ⎥⎬}
aρ
; D = 2
⎩ ⎣ 2 dx ⎦⎣
2 dx'
⎦⎭
k
K, K’ are complete elliptic integrals of 1st and 2nd
kind. After scission E =
2
e Z Z / R
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C
∑
i≠
j
i
j
ij

PHENOMENOLOGICAL SHELL CORRECTIONS
Adapted after Myers & Swiatecki. Number of nucleons
Ζ , N proportional to the volume.
= ∑ ∑
δE
( δE
+ δEni)
= C [ s(
Zi
) + s(
i)];
i pi N
i
Ni are magic numbers. C=6.2 MeV, c=0.2 from fit to
exp. masses and def. Variation with deformation:
C ⎧
L ⎫
= ⎨∑
i
δE
[ s(
Ni
) + s(
Zi
)] ⎬
2 ⎩ i
Ri
⎭
For magic nb. δE is minimum.
L i are the lengths of fragments.
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s −
2/
3
1/
3
( Z)
= Z F(
Z)
cZ
−
5/
3 5/
3
3 ⎡ Ni
− Ni
5/
3 5/
3⎤
−1 F( n)
= ⎢ ( n − Ni−1)
− n + Ni−1
⎥;
n∈
( Ni−1,
Ni
) is Z or N
5 ⎣ Ni
− Ni−1
⎦

CALCULATED SHAPES for X = 0.60
X = 0.60 (e.g. 170 Yb) η = 0 n L = n R = 2
d L = d R = 1.40, 1.50, 1.70, 1.91(SP) input parameters
α/R 0 = 1.31, 1.64, 2.10, 2.30 deformation
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CALCULATED SHAPES for X = 0.70
X = 0.70 (e.g. 204 Pb) η = 0 n L = n R = 2
d L = d R = 1.40, 1.70, 2.00 , 1.55(SP) input parameters
(α/R 0 ) SP = 1.82 SP deformation
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DEFORMATION and ENERGY vs input parameter
X = 0.60 (e.g. 170 Yb)
and 0.70 (e.g. 204 Pb)
η = 0
n L = n R = 2
Maximum energy at
the Saddle Point
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CALCULATED SADDLE POINT SHAPES
η= 0 n L = n R = 2 d L = d R = d
X = 0.60, e.g. 170 Yb
d=1.91 α/R 0 = 2.30
X = 0.70, e.g. 204 Pb
d=1.55 α/R 0 = 1.82
X = 0.82, e.g. 252 Cf
d=1.38 α/R 0 = 1.16
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η
≠
0
ASYMMETRICAL SHAPES (I)
Two ways to obtain: (1) n L = n R = n
(1) n=2
X = 0.77
e.g. 238 U
d R =1.4
d L =1.4,
1.45,
1.5, 1.6
n ≠
(2) L R
L L
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n
d ≠
L
d ≠
d
d
L

η
≠
0
ASYMMETRICAL SHAPES (II)
(2) n L = 4, n R = 2, d L = 2.47, 2.44, 2.39
X = 0.60, e.g. 170 Yb d R = 1.46, 1.43, 1.40
α/R 0 = 2.06, 1.98, 1.84
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MASS ASYMMETRY AT THE 2 nd SADDLE POINT
The 2 nd barrier height is
lower at some finite
mass asymmetry.
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SP MINIMUM DUE TO SHELL EFFECTS (I)
Binary cold fission. Saddle point energy vs mass
asymmetry with and without shell effects.
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SP MINIMUM DUE TO SHELL EFFECTS (II)
Binary cold fission. Saddle point energy vs mass
asymmetry for Th isotopes
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SP MINIMUM DUE TO SHELL EFFECTS (III)
Binary cold fission. Saddle point energy vs mass
asymmetry for U isotopes
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CONTRIBUTION OF SHELL EFFECTS
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MASS NUMBER OF THE HEAVY FRAGMENT
Mass number of the heavy fragment corresponding
to the minimum of the saddle point energy
Nuclei X d L –d R η A 1
238 U 0.769 0.04 .04988 124.93
232 Th 0.754 0.06 .07517 124.72
228 Th 0.758 0.08 .09512 124.84
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TERNARY FISSION (I)
Binary fissility X = 0.60 e.g. 170 Yb n L = n R = 3
d L = d R = 2.25, 2.80, 7.00 α/R 0 = 1.65, 2.31, 2.73
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TERNARY FISSION (II)
Config. d = 7, E/E s 0 = 0.134 is not far from a
true ternary fission 170 Yb 56 V + 56 V + 58 Cr,
Q=83.64 MeV, (E t -Q)/ E s 0 = 0.239 for spherical
shapes in touch
For 170 Yb 10 Be + 80 As + 80 As
Q=70.86 MeV, (E t -Q)/ E s 0 = 0.147
For 170 Yb 4 He + 83 Se + 83 Se
Q=87.48 MeV, (E t -Q)/ E s 0 = 0.103 — higher Q-value
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and lower fission barrier

QUATERNARY FISSION
Binary fissility X = 0.60 e.g. 170 Yb n L = n R = 4
d L = d R = 2.3, 2.7, 4.00 α/R 0 = 2.14, 3.14, 3.23
Config. d = 4, E/E s 0 = 0.214 is not far from a
true quaternary fission
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170Yb 42Cl+ 42Cl+ 43Ar+ 43Ar Q=53.17 MeV, (Et-Q)/ E 0
s = 0.324 for touching
spheres

EXPERIMENTAL ATTEMPTS
• Theory: R. D. Present (Phys. Rev. 59 (1941) 466): Uranium
tripartition would release about 20 MeV more energy than binary
• Exp. Rosen & Hudson (1950): 235U + nth . Triple gas filled
ionization chamber. Yield of 6.7 per 106 binary fission acts
• Iyer & Coble (1968): 235U + He ions of intermediate energy.
Also optimistic.
• Schall, Heeg, Mutterer & Theobald (1987): spontaneous
fission of 252Cf. Triple coincidences with detectors at 120o Pessimistic: yield under 10-8 per binary.
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CONCLUSIONS
• One can obtain saddle-point shapes by solving
an integro-differential equation
• There is no limitation imposed by the space of
deformation coordinates (no parametrization)
• Fission barriers for true ternary and
quaternary fission are lower than for aligned
spherical fragments in touch
• Mass-asymmetry in binary fission is explained
by adding shell corrections to the LDM energy
• Experimental confirmation of true ternary
fission still needed
Dorin Poenaru