Masses and Fission Barriers of Atomic Nuclei

Masses and Fission Barriers
of Atomic Nuclei
K. Pomorski
Department of Theoretical Physics
Marie Curie-Sklodowska University, Lublin, Poland
Workshop on Level Density and Gamma Strength in Contiunuum
Oslo, May 21-24,2007
Masses and Fission Barriers of Atomic Nuclei – p.1/33

My co-workers:
Jerzy Dudek and Johann Bartel
from IPHC and ULP, Strasburg, France
Bożena Nerlo-Pomorska and Artur Dobrowolski
from UMCS Lublin, Poland
Masses and Fission Barriers of Atomic Nuclei – p.2/33

Program:
• Introduction
• Leptodermus expansion of energy functional
• Role of different components of the liquid drop
model
• Lublin – Strasbourg Drop
• Parameterization of nuclear shapes
• Potential energy surfaces of fissioning nuclei
• Fission barriers and saddle point masses
• Shell and pairing energies
• Summary and conclusions
Masses and Fission Barriers of Atomic Nuclei – p.3/33

Liquid drop models:
• C.F. von Weizsäcker, Z. Physik 96, 431 (1935).
• H.A. Bethe, R.F. Bacher, Rev. Mod. Phys. 8, 82 (1936).
• L. Meitner, O.R. Frisch, Nature 143, 239 (1939).
• N. Bohr, J.A. Wheeler, Phys. Rev. 56, 426 (1939).
• D.L. Hill, J.A. Wheeler, Phys. Rev. 89, 1102 (1953).
• W.D. Myers, W.J. Świa¸tecki, Nucl. Phys. 81, 1 (1966).
• H. von Groote, E. Hilf, Nucl. Phys. A129, 513 (1969).
• K. Pomorski , J. Dudek, Phys. Rev. C67, 044316 (2003).
Other models: droplet, Yukawa+exp., FRDM, TF, ETFSI
Masses and Fission Barriers of Atomic Nuclei – p.4/33

Chart of known isotopes
114
Antony 2002
Audi-Wapstra 1993
Green’s β stability line
82
Z
50
28
20
8
2
2 8 20 28 50 82 126 184
N
Audi-Wapstra – 1654, Antony – 2766 isotopes with Z ≥ 8 and N ≥ 8.
Masses and Fission Barriers of Atomic Nuclei – p.5/33

Nuclear energy functional
The one-body density of a nucleus can be expressed as
∫ ∫ ∫
ρ = A . . . Ψ ⋆ Ψ dτ 2 · · · dτ A
and the corresponding energy-density as
∫ ∫ ∫
η = . . . Ψ ⋆ ĤΨ dτ 2 · · · dτ A .
The total energy of the nucleus is
∫
B = η d 3 r
V
Masses and Fission Barriers of Atomic Nuclei – p.6/33

Leptodermous expansion
The total energy integral can be decomposed around diffused
around a nuclear surface Σ:
∮
B = b vol A + dσ
∫ ∞
(η − b vol ρ) dr ⊥
Σ
= b vol A + γ (0) ∮
0
dσ + γ ′ κ a ∮
κ dσ
+ 1 2 γ ′′
κκ a 2 ∮
Σ
Σ
κ 2 dσ + γ ′ Γ a 2 ∮
Σ
Σ
Γ dσ + . . . ,
where κ = 1 R 1
+ 1 R 2
and Γ = 1
R 1·R 2
are the 1 st and the 2 nd
order (Gauss) curvatures and R 1 and R 2 are the local main radii
of the surface.
Masses and Fission Barriers of Atomic Nuclei – p.7/33

Leptodermous expansion of the Skyrme energy
functional:
100 Sn Force: SkM * b vol =-15.387 MeV
20
1.4
b sur
r 0
b curG
b [MeV]
10
0
1.3
1.2
r 0 [fm]
-10
0 10 20 30 1.1
b cur [MeV]
Masses and Fission Barriers of Atomic Nuclei – p.8/33

Fit of the liquid drop parameters to experimental
binding energies of all known isotopes as function
of the charge radius:
30
1
δB
b sur
b [MeV]
15
0
-15
b curG
b cur
b vol
0.5
0
-0.5
< δB(Z,A) > [MeV]
-30
1.18 1.2 1.22 1.24 1.26 -1
ch
r 0 [fm]
Masses and Fission Barriers of Atomic Nuclei – p.9/33

Lublin – Strasburg Drop model:
M(Z, N; def) = ZM H + NM n − b elec Z 2.39
+ b vol (1 − κ vol I 2 ) A
+ b surf (1 − κ surf I 2 ) A 2/3 B surf (def)
+ b cur (1 − κ cur I 2 ) A 1/3 B cur (def)
+ 3 5 e2 Z 2
Z
B
r0 ch Coul (def) − C 2
4 A1/3 A
+ E micr (Z, N; def) + E cong (Z, N)
Masses and Fission Barriers of Atomic Nuclei – p.10/33

Fit to the 2766 experimental masses:
Term Units LDM LSD
b vol MeV -15.8484 -15.4920
κ vol - 1.8475 1.8601
b surf MeV 19.3859 16.9707
κ surf - 1.9830 2.2938
b cur MeV 0 3.8602
κ cur - 0 -2.3764
r 0 fm 1.18995 1.21725
C 4 MeV 1.1995 0.9181
δM MeV 0.732 0.698
δV BZ>70 MeV 5.58 0.88
M H =7.289034 MeV; M n =8.071431 MeV;
K. Pomorski , J. Dudek, Phys. Rev. C67, 044316 (2003).
b elec =1.433 eV
Masses and Fission Barriers of Atomic Nuclei – p.11/33

LSD+YF mass deviation:
M LSD - M exp [MeV]
4
0
-4
0 40 80 120 160
N
Masses and Fission Barriers of Atomic Nuclei – p.12/33

Masses of isotopes:
120
120
100
M LSD - M exp [MeV]
100
M TF - M exp [MeV]
Z
80
60
40
-1 01234
-2
-3
Z
80
60
40
-1 01234
-2
-3
-4
20
0
r.m.s. dev.= 0.698 MeV
0 20 40 60 80 100 120 140 160 180
N
20
0
r.m.s. dev.= 0.757 MeV
0 20 40 60 80 100 120 140 160 180
N
exp: The mass deviations are plotted for 2766 isotopes taken from the
Chart of Nuclides by M.S. Antony, Strasbourg, 2002.
LSD: K. Pomorski, J. Dudek, Phys. Rev. C67, 044316 (2003).
TF: P. Möller, J.R. Nix, W.D. Myers, W.J. Świa¸tecki, At. Data Nucl.
Data Tables 59, 185 (1995).
Masses and Fission Barriers of Atomic Nuclei – p.13/33

LSD masses:
120
100
M LSD - M exp [MeV]
Z
80
60
40
-1 01234
-2
-3
20
0
r.m.s. dev.= 0.698 MeV
0 20 40 60 80 100 120 140 160 180
N
K. Pomorski , J. Dudek, Phys. Rev. C67, 044316 (2003).
Masses and Fission Barriers of Atomic Nuclei – p.14/33

TF and LSD mass differences:
140
120
E TF - E LSD = X 3 [MeV]
100
Z
80
60
40
20
0
X
-2 -1 0 1
0 50 100 150 200 250
N
Data plotted for 8757 isotopes taken from the mass table of P. Möller et al., At. Data Nucl. Data
Masses and Fission Barriers of Atomic Nuclei – p.15/33
Tables 59, 185 (1995).

TF and LSD mass differences:
M LSD - M TF [MeV]
2
1
0
Sn
M LSD - M TF [MeV]
2
1
0
Pb
-1
90 100 110 120 130 140
A
-1
170 180 190 200 210 220 230
A
Masses and Fission Barriers of Atomic Nuclei – p.16/33

Predictive power of mass formulae
Root means square deviations (δM in MeV) of theoretical and
experimental binding energies of 1654 isotopes taken from the
Audi-Wapstra tables and 2766 isotopes from the Antony chart:
Model
δM(2766) δM(1654)
LSD(2766) 0.698 0.610
LSD(1654) 0.711 0.600
TF-MS(1654) 0.757 0.655
ETFSI(1888) 0.828 0.738
ETFSI: S. Goriely, J.M. Pearson, and F. Tondeur, At. Data and Nucl. Data Tables 77, 311 (2001).
Masses and Fission Barriers of Atomic Nuclei – p.17/33

Spherical harmonics expansion:
6
2
4
V LD [MeV]
2
0
-2
-4
232 Th
LSD
λ max
4
6
8
-6
-8
-0.5 0 0.5 1 1.5 2 2.5 3 3.5
β 2
10
12
14
R(θ) = R 0
∞∑
λ=0
β λ P λ (cos θ)
Masses and Fission Barriers of Atomic Nuclei – p.18/33

Modiffied Funny-Hills:
232 Th LSD
5
4
Gauss
F-Hills
β, λ max =14
V LD [MeV]
3
2
1
0
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
R 12 /R 0
˜ρ 2 s (z) = R2 0
c f(a,B) (1 − u2 ) (1 + αu − B e −a2 u 2 ) ,
where u = (z − z sh )/(c R 0 ). Masses and Fission Barriers of Atomic Nuclei – p.19/33

MFH shapes examples:
α=0.00
c=2.0
a=2.00
B=+0.00
B=-0.60
B=-0.90
B=-0.99
B=-1.00
α=0.25
c=1.6
α=0.00
B=+0.30
B=+0.00
B=-0.30
B=-0.60
B=-0.90
c=1.2
α=0.00
B=+0.60
B=+0.30
B=+0.00
B=-0.30
B=-0.60
Masses and Fission Barriers of Atomic Nuclei – p.20/33

PES of 240 Pu in the MFH space:
-1809
-1806
-1803
-1800
-1803
-1800
-1797
-1794
0.20
-1812
-1806
-1791
-1791
-1809
-1803
0.15
-1794
-1797
-1806
-1803
-1806
-1800
-1803
0.10
-1806
-1800
-1809
-1803
0.05
-1809
-1809
-1806
-1803
0.00
-1803
-1806
-1806
-1800
-1809
-1812
-1806
-1800
-1803
-0.05
-1800
-1803
-1806
-1803
-1812
-1809
-0.10
-1797
-1800
-1800
-1803
-0.15
-1809
-1806
-1797
-1800
-1800
-1800
-1794
-0.20
-1797
-1803
-1806
-1794
-1791
-0.25
240
94
Pu
146
a=5-2c, B=h(3c-1)
E_tot
0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
c
E [MeV]
-1790.0
-1791.0
-1792.0
-1793.0
-1794.0
-1795.0
-1796.0
-1797.0
-1798.0
-1799.0
-1800.0
-1801.0
-1802.0
-1803.0
-1804.0
-1805.0
-1806.0
-1807.0
-1808.0
-1809.0
-1810.0
-1811.0
-1812.0
-1813.0
h
LSD with shell and pairing corrections obtained with the Yukawa-folded single particle potential.
Masses and Fission Barriers of Atomic Nuclei – p.21/33
A. Dobrowolski, K. Pomorski, J. Bartel, Phys. Rev. C75(2007) 024613.

Fission barrier heights:
16
δV B (LSD)=1.74 MeV
LSD
V B
th -VB
exp [MeV]
12
8
4
0
-4
15 20 25 30 35 40
Z 2 /A
Deformation dependent congruence energy term is included here according to:
W.D. Myers, .J. Świa¸ tecki, Nucl. Phys. A612 (1997) 249.
K. Pomorski, J. Dudek, Int. Journ. Mod. Phys. E13 (2004) 107.
Masses and Fission Barriers of Atomic Nuclei – p.22/33

Saddle point masses:
80
M exp.
M LSD
Cm
70
Cf
M sadd. [MeV]
60
50
Th
U
Pu
40
r.m.s. dev. = 0.382 MeV
230 235 240 245 250
A
Masses and Fission Barriers of Atomic Nuclei – p.23/33

Stiffness at the saddle point:
45
45
V [MeV]
40
35
172 Yb
β 2 = 2.0
MS-LD
LSD
V [MeV]
40
35
172 Yb
β 2 = 2.0
MS-LD
LSD
30
YF
30
YF
25
25
0.2 0.4 0.6 0.8 1
β 4
-0.4 -0.2 0 0.2
β 6
Masses and Fission Barriers of Atomic Nuclei – p.24/33

Shell and pairing corrections:
E tot = E macro + δE shell + δE pair
δE shell = ∑ occ
2e ν − 〈 ∑ occ
2e ν 〉
E pair = E BCS − ∑ occ
2e ν
δE pair = E pair − 〈 E pair 〉
δE shell + δE pair = E BCS − 〈 E BCS 〉
Masses and Fission Barriers of Atomic Nuclei – p.25/33

Harmonic oscillator s.p. energy sum
700
600
E
10
Σ N i=1 e i [/hω 0 ]
500
400
300
200
100
Ē=1.0878 N 4/3 /hω 0
5
0
-5
Harmonic oscillator
-10
∆E [/hω 0 ]
0
0 100 200 300 400 500 600
N 4/3
0 100 200 300 400 500
N
10
10
5
5
∆E [/hω 0 ]
0
∆E [/hω 0 ]
0
-5
-5
-10
-10
1 2 3 4 5 6 7 8
N 1/3
2 4 6 8 10
e i [/hω 0 ]
K. Pomorski, Phys. Rev. C70 (2004) 044306. Masses and Fission Barriers of Atomic Nuclei – p.26/33

Gogny mean-field Hamiltonian:
Σ Z i=1 (e i - V 0 ) [MeV]
7000
6000
5000
4000
3000
2000
1000
0
40
Gogny
a Z 4/3 20
0
V 0 = -71.25 MeV
-20
a= 11.119 MeV
-40
0 100 200 300 400 500 600
Z 4/3
Σ Z i=1 (e i - V 0 ) - a Z 4/3 [MeV]
protons
0 50 100 150 200
Z
Σ N i=1 (e i - V 0 ) [MeV]
8000
6000
4000
2000
0
40
Gogny
a N 4/3 20
0
V 0 = -80.91 MeV
-20
a= 11.238 MeV
-40
0 200 400 600 800 1000
N 4/3
Σ N i=1 (e i - V 0 ) - a N 4/3 [MeV]
neutrons
0 50 100 150 200
N
K. Pomorski, Phys. Rev. C70 (2004) 044306. Masses and Fission Barriers of Atomic Nuclei – p.27/33

N -folding of the s.p. energy sum
E sum (n) =
n∑
ν=1
The energy sum smoothed in N 1/3 -space is given by:
( )
N∑
max
2 E sum (n) N 1/3 − n 1/3
〈 E sum (N) 〉 =
j
3 n 2/3 γ
N min
e ν
where γ ≈ 0.7 is the smearing width and
j(u) = 1 √ π
e −u2 ( 35
16 − 35 8 u2 + 7 4 u4 − 1 6 u6 )
.
K. Pomorski, Phys. Rev. C70 (2004) 044306.
Masses and Fission Barriers of Atomic Nuclei – p.28/33

Strutinsky and new shell energy
5
5
0
0
E shell [MeV]
-5
-10
208 Pb
E shell [MeV]
-5
-10
208 Pb
-15
new
protons
old
diff.
-20
0.6 0.8 1 1.2 1.4 1.6
c
-15
new
neutrons
old
diff.
-20
0.6 0.8 1 1.2 1.4 1.6
c
10
10
E shell [MeV]
5
0
-5
-10
protons
232 Th
new
old
diff.
E shell [MeV]
5
0
-5
-10
neutrons
232 Th
new
old
diff.
0.6 0.8 1 1.2 1.4 1.6
c
0.6 0.8 1 1.2 1.4 1.6
c
Masses and Fission Barriers of Atomic Nuclei – p.29/33

Plateau condition:
2.6
new
p
n
208 Pb
c=1.2
2.6
old
208 Pb
c=1.2
E shell [MeV]
2.4
2.2
2
E shell [MeV]
2.4
2.2
2
1.8
1.8
0.6 0.7 0.8 0.9 1
γ
0.9 1 1.1 1.2 1.3 1.4 1.5
γ S [/hω 0 ]
K. Pomorski, Phys. Rev. C70 (2004) 044306.
Masses and Fission Barriers of Atomic Nuclei – p.30/33

N -folding of the BCS energy:
E BCS (n) = 2 ∑ ν
v 2 ν e ν − G ∑ ν
v 4 ν − ∆2 /G
The BCS energy smoothed in N 1/3 -space is given by:
〈 E BCS (N) 〉 =
N∑
max
n=N min
2 E BCS (n)
3 n 2/3 j
(
N 1/3 − n 1/3
γ
where
j(u) = 1 √ π
e −u2 ( 35
16 − 35 8 u2 + 7 4 u4 − 1 6 u6 )
.
K. Pomorski, B. Nerlo-Pomorska, Physica Scripta T125 (2006) 21. Masses and Fission Barriers of Atomic Nuclei – p.31/33

Smooth pairing energy
0
-0.1
N
e i [/hω 0 ]
E BCS (N ) - Σ
i=1
-0.2
-0.3
-0.4
-0.5
-0.6
ε = 0.45
G = 0.285/N 2/3 [/hω 0 ]
-0.7
20 40 60 80 100 120 140 160 180 200
N
K. Pomorski, B. Nerlo-Pomorska, Physica Scripta T125 (2006) 21.
B. Nerlo-Pomorska, K. Pomorski, Int. Journ. Mod. Phys. E16 (2007) 328.
Masses and Fission Barriers of Atomic Nuclei – p.32/33

Summary and conclusions
• Lublin Strasburg Drop, which apart traditional
liquid drop terms contains the curvature energy,
describes well the masses of known isotopes.
• The fission barriers as well as the saddle point
masses are reproduced well by the LSD model.
• New method of evaluation of the Strutinsky
smoothed energy by folding in the particle
number space is proposed.
• The N -folding method can be applied to obtain
the average pairing energy.
• Large scale calculation of the nuclear mass table
is planned.
Masses and Fission Barriers of Atomic Nuclei – p.33/33