Shell effects in atomic nuclei

1. Shell effects in atomic nuclei
Part 2: shapes and superheavy elements
Laurent Gaudefroy1, Alexandre Obertelli2
1CEA DAM, DIF, France
2CEA Saclay, IRFU, France

2. protons
neutrons
82
50
28
28
50
82
20
8
2
2
8
20
126
Changes in the nuclear shell structure
Lecture (part 1) given by Laurent Gaudefroy

3. Shapes of atomic nuclei
protons
neutrons
82
50
28
28
50
82
20
8
2
2
8
20
126
The vast majority of all nuclei shows
a non-spherical mass distribution
Z, N = magic numbers
Closed shell = spherical shape
Deformed
Spherical
8
20
28
50
2
sngle
particle
enegies
elongation
Nilsson diagram
Oblate Prolate

4. Nuclear structure description framework
[Addendum to yesterday’s lecture]
1- Shell-model:
• nucleus described in the laboratory frame
• the nucleus is described as a superposition of spherical configurations
• « intrinsic deformation » is implicitely contained in correlations
2- Mean-field like description:
• nucleus described in its intrinsic frame
• « angular momentum » is not a good quantum number
• intrinsic deformation is explicit
In this lecture, the deformed mean-field approach will be followed

5. Nilsson diagram
nlj=1f7/2 Kp=1/2-
Kp=3/2-
Kp=5/2-
Kp=7/2-
b
0
• core + single particle
• short range & attractive int.
• Pauli : orbit repulsion

6. Shapes and “deformation” parameters







  








 
 )
,
(
)
(
1
)
( 0 Y
t
a
R
t
R
quadrupole
octupole
hexadecapole

b cos
2
20 
a 
b sin
2
1
2
2
2
22 
 
a
a
oblate
non-collective
prolate
collective
b2 : elongation

prolate
non-collective
Lund
convention
spherical
oblate
collective
: triaxiality
Generic nuclear shapes can be described
by a development of spherical harmonics
a:deformation parameters
Tetrahedral Y32 deformation
Dynamic vibration
Static  rotation
Triaxial Y22 deformation

7. Shapes and “deformation” from experiment
quadrupole
 Quadrupole moments via low-energy Coulomb excitation
 Reorientation effect
projectile
target
Intrinsic quadrupole moment IK
E
M
IK
eQ )
2
(
5
16
2
/
1
0 






p
Jp=0+
Jp=2+
Jp=4+
Jp=6+
Jp=8+
even-even
)
1
(
2
)
(
2


 J
J
I
E

)
(
1
)
2
(
)
2
(
)
(
)
(
)
1
(
J
E
J
J
J
E
J
E
J
E
J
E
J
















Moment of inertia via rotational-band spectroscopy / model dependent
Coulomb field
*
excitation
de-excitation
photon

8. M. Girod, CEA
N=Z
Oblate deformed nuclei are far less abundant than prolate nuclei
Shape coexistence possible for certain regions of N & Z
Prolate
Quadrupole deformation of nuclei
Oblate Pb & Bi
N~Z
Fission
fragments
N~28 n-rich
actinides

9. Shape coexistence
oblate prolate
74Kr
b

M. Girod
M. Bender et al., PRC 74, 024312 (2006)
0+
0+
2+
2+
4+
4+
6+
6+
8+
Configuration mixing:












obl
pro
2
obl
pro
1
0
cos
0
sin
0
0
sin
0
cos
0




electric monopole (E0) transition
)
(
cos
sin
0
)
0
(
0 2
obl
2
pro
1
2 b
b

 



E
M

10. Shape coexistence in light Krypton isotopes
0+
0+
2+
2+
4+
4+
6+
6+
8+

11. SPIRAL beams
76Kr 5105 pps
74Kr 104 pps
4.7 MeV/u
[24°, 55°] [55°, 74°] [67°, 97°] [97°, 145°]
74Kr
Shape coexistence in light Krypton isotopes
Coulomb excitation of 74,76Kr
78Kr
68.5 MeV/u
1012 pps
74Kr
4.7 MeV/u
104 pps
ECRIS
SPIRAL
target
CIME
78Kr source
CSS1
CSS2

12. Shape coexistence in light Krypton isotopes
Quadrupole moments
24
.
0
23
.
0
53
.
0 



s
Q
4
.
0
2
.
0
8
.
0 



s
Q
3
.
0
5
.
0
3
.
1 



s
Q
21
.
0
17
.
0
24
.
0 



s
Q
9
.
0
3
.
0
3
.
0 



s
Q
)
2
(
)
4
(
1
1




I
I
Fit matrix elements
(transitional and diagonal)
to reproduce experimental
-ray yields (as function of )
 14 B(E2) values
 5 quadrupole moments
E. Clément et al., PRC 75, 054313 (2007)
first reorientation measurement
with radioactive beam
SPIRAL1, GANIL (France), 2005

13. prolate oblate
Qs<0
prolate
Qs>0
oblate
experimental B(E2;) [e2fm4]
Comparison with ‘beyond-mean-field’ theory
K=2
 vibration
E. Clément et al.,
PRC 75, 054313 (2007)
GCM (GOA) calculation
q0, q2: triaxial deformation
Gogny D1S
M. Girod et al.
prolate oblate
GCM calculation
axial deformation
Skyrme SLy6
M. Bender et al.
PRC 74, 024312 (2006)

14. Extreme shapes and intruder orbitals
single-particle
energy
(Woods-Saxon)
quadrupole deformation
ND
235U
SD
152Dy
Z=48
HD
108Cd
p i13/2
 (N+1) intruder
 normal deformed, e.g. 235U
 (N+2) super-intruder
 Superdeformation, e.g. 152Dy, 80Zr
 (N+3) hyper-intruder
 Hyperdeformation in 108Cd, ?
N+2 shell
N+3 shell
N shell
N+1 shell
Fermi level
Energy
Deformation

15. The quest for high-spin superdeformation: 152Dy
 first discrete superdeformed band
 energy spacing: E = 47 keV
TESSA3 (12 detectors), Daresbury (UK)
P. Twin et al., Phys. Rev. Lett. 57, 811 (1986)
TESSA Ge array
Extracted moment of inertia
0+
2+
4+
6+
8+
even-even
)
1
(
2
)
(
2


 J
J
J
E


16. 20 years later
Argonne National Lab.
Gammasphere
108 Ge detectors
T. Lauritsen et al., Phys. Rev. Lett. 88, 042501 (2002)
The quest for high-spin superdeformation: 152Dy
 Properties of the superdeformed band firmly established

17. Pushing the limits:
The quest for nuclear hyperdeformation
Hyperdeformation favored at high-spin
 Competition with fission
Fission barrier vs. High spin
stable beam
n-rich beam
 Need for intense neutron-rich beams
 Spiral2 : intense Kr and Sn neutron-rich beams

18. The AGATA germanium array
• 180 large volume 36-fold segmented Ge crystals in 60 triple-clusters
• Digital electronics and sophisticated signal processing algorithms (PSA)
• Operation of Ge detectors in position sensitive mode  -ray tracking
> Efficiency ~ 40 %
Huge gain in γγ, γγγ, … efficiency
> Cristal rate up to 50 kHz
 Allow larger beam intensity
http://www-w2k.gsi.de/agata/
New generation gamma-detection array
based on the tracking method

19. Existence and structure
of heavy elements
208Pb
238U
~4.5 109 y
Limits of stability ?
Shell structure ?
Next magic number ?
Chart from http://www.nndc.bnl.gov/chart/

20. Synthesis of heavy elements in the universe
B. Pfeiffer et al., NPA (2001)
Cassiopea A supernova
Why SHE do not exist on earth ?
1- not stable
2- not formed during the r-process

21. Upper limit of stability : positron emission
Nuclei for Z larger than 173 become unstable against positron emission.
The most deeply bound electrons from the 1s1/2 shell reach an energy of -511 keV
W. Pieper, W. Greiner Z. Phys. A 218 (1968) 327
J. Reinhardt et al, Z. Phys. A 303 (1981) 173

22. Limits of stability : fission
• B(A,Z) = av A volume – nuclear attractive force
- as A2/3 less binding at the surface
- ac Z2/A1/3 Coulomb – proton repulsion
- aa (A-2Z)2/A asymmetry
+δ A-1/3 pairing
R a
b
V= 4/3pR3
S=4pR2
a=R(1+)
b=R(1+)-1/2
V=4/3pab2
S=4pR2(1+2/52+…)

  1
b2
a2
Surface prefers spherical nuclei  Coulomb favours deformation
If BE(ε) -BE(ε=0)> 0: gain in energy with deformation  fission

23. Fission barrier – liquid drop
Deformation β
Liquid
drop
energy
(MeV/A)

24. Limits of stability from liquid drop model
Stability = balance
between surface and coulomb
• Fissility parameter
x = Ecoulomb/ 2 Esurface
• ~ 1/50 Z2 / A
• scaling of the fission barrier
• x > 0.8 : no survival
• Possible definitions of SHE :
No macroscopic fission barrier
Bf < 1 MeV
x > 0.8

25. State of the art
Superheavy elements synthesized in laboratory
Shell effects balance fission and
are responsible for the existence of superheavies!
Superheavy elements Z 104

26. H
1
Li
3
Be
4
Na
11
Mg
12
Fr
87
Ra
88
119 120
K
19
Ca
20
Rb
37
Sr
38
Cs
55
Ba
56
Sc
21
Ti
22
Y
39
Zr
40
La
57
Hf
72
V
23
Cr
24
Nb
41
Mo
42
Ta
73
W
74
Mn
25
Fe
26
Tc
43
Ru
44
Re
75
Os
76
Co
27
Ni
28
Rh
45
Pd
46
Ir
77
Pt
78
Cu
29
Zn
30
Ag
47
Cd
48
Au
79
Hg
80
Ds Rg 112
Ga
31
Ge
32
In
49
Sn
50
Tl
81
Pb
82
113 114 115
As
33
Se
34
Sb
51
Te
52
Bi
83
Po
84
Br
35
Kr
36
I
53
Xe
54
At
85
Rn
86
116 117 118
F
9
Ne
10
Cl
17
Ar
18
N
7
O
8
P
15
S
16
B
5
C
6
Al
13
Si
14
He
2
Ce
58
Th
90
Pr
59
Pa
91
Nd
60
U
92
Pm
61
Np
93
Sm
62
Pu
94
Eu
63
Am
95
Gd
64
Cm
96
Tb
65
Bk
97
Dy
66
Cf
98
Ho
67
Es
99
Er
68
Fm
100
Tm
69
Md
101
Yb
70
No
102
Lu
71
Lr
103
Lanthanides
Actinides
Ac
89
Rf
104
Db
105
Sg
106
Bh
107
Hs
108
Mt
109 110 111
Point of view of chemist :
Actinides 90  Z  103
Transactinides 104  Z  121 (?)
Arbitrary point of view :
Superheavies: existence due to shell effects
Cn (2010)
copernicium
Chemist point of view

27. 238U
~4.5 109 y
238U
Peninsula vs island of stability
Deformed 254No, 270Hs
Spherical 298114
LDM
LDM
LDM
LDM
LDM
LDM
162
184
152

28. M. Bender et al . PL B515 (2001) 42
Z N
W.S 114 184
HFB 126 184
RMF 120 172
Note 1 :Up to 208Pb : proton and neutron magic numbers identical.
Note 2 : Models rely on extrapolations –parameters are adjusted on
known cases
Modern-theory predictions

29. Theoretical challenges
Level density increases with A, Z
M.
Bender
et
al.,
Phys.
Lett.
B
515
(2001)
42
132Sn :
Large gap
Super-heavies :
Gap function of models
and not marked

30. Why is it so difficult to get information on SHE?
times needed to observe on
average 1 event
present sensitivity:
limit  1 pbarn
beam dose:
1.51018 projectiles
10 days
1 minute
1 hour
1 day
1 second

31. known
CN
277112
273110
269Hs
265Sg
261Rf
257No
11.45 MeV
280 s
11.08 MeV
110  s
9.23 MeV
19.7 s
4.60 MeV (escape)
7.4 s
8.52 MeV
4.7 s
253Fm
8.34 MeV
15.0 s
Date: 09-Feb-1996
Time: 22:37 h
277112
70Zn 208Pb 277112
n
kinematic separation
in flight identification
by a-a correlations
to known nuclides
Synthesis and Identification of SHE

32. JINR/FLNR
Dubna, Russia
GSI
State-of-the-art worldwhile
294118: Yu. Oganessian et al., J. Phys. G R165 (2007)
294117: Yu. Oganessian et al., Phys. Rev. Lett. 104, 142502 (2010)
RIKEN
Tokyo, Japan

33. Spectroscopy of Transfermium elements
Access to high j deformed orbitals :
probe of higher lying spherical orbitals
R.-D. Herzberg et al., Nature 442, 896-899 (2006)
S.K. Tandel et al., PRL 97, 082502 (2006)
(courtesy of P.-H. Hennen)
Prompt and/or decay spectroscopy

34. M Block et al., Nature 463, 785-788 (2010)
Cyclotron resonance curve of 253No2.
Bridging the gap from heavies to superheavies
253,254,255No
mass measurement

35. The S3 spectrometer at SPIRAL2
A spectrometer for the high intensity stable ion beams of SPIRAL2 (from 2012)
Isotopic exploration
40-48Ca+238U275-283112+3,4n
S3 (I=20pµA)  40evt/week/pb
New elements
54Cr+248Cm299120+3n
S3 (I=10pµA)
 1evt/month@σest~0.01pb
?
Closed-shell deformed nucleus ???
40Ar+238U274Ds (+4n)  270Hs + α
S3 (I=50pµA) 190evt/week@σth=2pb

36. Summary
 superheavy elements exist only because of shell effects
 theory predicts deformed + spherical shell gaps
 next proton magic number still to be discovered
 very low production cross sections
 direct production and undirect experimental techniques
 SPIRAL2 and S3 spectrometer
shape coexistence: interplay between shell effects and macroscopic properties
essential to constrain collective nuclear models
 Very large deformations encoutered at high spin
 superdeformation evidenced / hyperdeformation still to be discovered
 AGATA high-resolution germanium array
 most nuclei are deformed
 prolate quadrupole deformation are the most common

37. Key questions and shell effects in nuclei
• What is the shape of a nucleus, how large can be nuclear deformation?
hyperdeformation, shape-coexistence
• Is there any island of stability for superheavy elements?
Next proton magic number, stabilizing deformed shell gaps
• Next-generation facilities and innovative detectors worldwhile built this decade
• How does shell structure evolve away from stability?
magic numbers, shell-model, spin-orbit, tensor
• How do nuclear clusters and molecules form?
few-body systems, halos, clusters