Transcript Slide 1

Coulomb excitation with
radioactive ion beams
• Motivation and introduction
• Theoretical aspects of Coulomb excitation
• Experimental considerations, set-ups and
analysis techniques
• Recent highlights and future perspectives
Lecture given at the
Euroschool 2009 in Leuven
Wolfram KORTEN
CEA Saclay
Wolfram KORTEN
Euroschool Leuven – September 2009
1
Shell structure of atomic nuclei
126
82
stable double-magic nuclei
4He, 16O, 40Ca, 48Ca, 208Pb
protons
50
28
82
20
radioactive: 56Ni, 132Sn,
magic ? 48Ni, 78Ni, 100Sn
50
8
2
28
20
2 8
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no 28O !
48Ca
neutrons
32Mg
42Si
70Ca
?
40Ca
Euroschool Leuven – September 2009
2
Changes in the nuclear shell structure
126
82
Only observable in heavy nuciei:
(Z,N) > 40
Spin-orbit term of the nuclear int. ?
132Sn
82
protons
50
N/Z
110Zr
70
“drip line”
stabil
f7/2
f7/2
28
82
20
82
h11/2
d3/2
s1/2
g7/2
50
8
2
28
20
2 8
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neutrons
h11/2
70
d5/2
50
g9/2
Euroschool Leuven – September
p1/2 2009
g7/2
d3/2
s1/2
d5/2
g9/2
40
p31/2
Changes in the nuclear shell structure
Observables:
N=20
E(2+) [MeV]
Masses and separation energies
Properties of 2+ states
16
20
3
2
20Ca
28
1
Ca
K
2 104
12Mg
Ar
Cl
16S
0
S
1,5 104
P
12
Si
16 20
N
24
Al
1 104
Mg
48
Ar
47
24
O
5000
44
42
20
C
17
0
N
Ne
Mg
Na
35
F
15
P S
20
25
F
O
C
40
Al
B
30
Neutron number
Example: “Island of inversion” around
 deformed ground state
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Ne
N
37
32
29
B
10
Si
23
45
Cl
Na
B(E2) [e2fm4]
S2N (keV)
4
400
N=20
sd+fp
200
sd
32Mg
40Ca 38Ar
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36S
34Si 32Mg 30Ne
N/Z
4
Shapes of atomic nuclei
126
Z, N = magic numbers
protons
50
28
82
20
50
8
2
28
20
2 8
50
sngle particle enegies
Closed shell = spherical shape 82
Nilssondiagramm
28
20
Oblate
8
Prolate
neutrons
Spherical
2
The vast majority of all nuclei shows
aWolfram
non-spherical
mass distribution
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Euroschool Leuven – September 2009
elongation
Deformed
5
Quadrupole deformation of nuclei
M. Girod, CEA
Coulomb excitation can, in principal, map the shape of all atomic nuclei
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Euroschool Leuven – September 2009
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Quadrupole deformation of nuclei
N=Z
Oblate
Pb & Bi
N~Z
Fission
fragments
Prolate
M. Girod, CEA
Oblate deformed nuclei are far less abundant than prolate nuclei
Shape coexistence possible for certain regions of N & Z
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Shape variations and intruder orbitals
152Dy 108Cd
single-particle energy (Woods-Saxon)
 i13/2235U
N+3 shell
N+2 shell
N+1 shell
N shell
Z=48
Energy
Fermi level
Deformation
ND
SD
quadrupole deformation
HD
 (N+1) intruder
 normal deformed, e.g. 235U
 (N+2) super-intruder
 Superdeformation, e.g. 152Dy,
80Zr
 (N+3) hyper-intruder
 Hyperdeformation in 108Cd, ?
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Exotic shapes and “deformation” parameters

Generic nuclear shapes can be described
R(t )  R0 1 
by a development of spherical harmonics

quadrupole
a20  2 cos
a22  a2 2 
1
 2 sin 
2

a (t ) Y ( ,  )
 



a:deformation parameters
Tetrahedral
Triaxial Y22
Y32
deformation
deformation
octupole
Lund
convention
oblate
non-collective

spherical
hexadecapole
2 : elongation
Static  rotation
Dynamic vibration
: triaxiality
prolate
non-collective
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prolate
collective
oblate
collective
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Coulomb excitation – an introduction
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10
Rutherford scattering – some reminders
target
b
projectile
r(t)
x2 y2
 2 1
2
a
b
a  beam energy
b = impact parameter
• Elastic scattering of charged particles (point-like 
monopoles) under the influence of the Coulomb field
FC = Z1Z2e2/r2 with r(t) = |r1(t) – r2(t)|
 hyperbolic relative motion of the reaction partners
• Rutherford cross section
ds/dq  Z1Z2e2/Ecm2 sin-4(qcm/2)
valid as longas Ecm  m0 v 2 
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m P  mT 2
v  Vc  Z1Z2e2 /R int
m P  mT
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Coulomb excitation – some basics
Nuclear excitation by the electromagnetic interaction
acting between two colliding nuclei.
target
b
projectile
Target and projectile excitation possible
often heavy, magic nucleus (e.g. 208Pb)
 as projectile  target Coulomb excitation
 as target  projectile Coulomb excitation
(important technique for radioactive beams)
small impact parameter
 back scattering
 close approach
 strong EM field
large impact parameter
 forward scattering
 large distance
 weak EM field
Coulomb trajectories only if the colliding nuclei do not reach the “Coulomb barrier” 
purely electromagnetic process, no nuclear interaction, calculable with high precision
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Reactions below the Coulomb barrier
VC= Z1Z2e2/rb (1-a/rb)
rb = 1.07(A11/3+A21/3) + 2.72 fm
a=0.63 fm (surface diffuseness)
sexp/sRuth
 Beam energy per nucleon (EB/A)
rather constant, e.g. for 208Pb target
5.6 to 6 A.MeV
Pure Coulomb excitation requires a
much larger distance between the nuclei
”safe energy” requirement
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 Inelastic scattering (e.m/nucl.)
 Transfer reactions
 Fusion
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„Safe“ energy requirement
target
b
projectile
Dmin
q
Ecm
Z P ZT e

Dmin
2
• Rutherford scattering only if the distance of closest
approach is large compared to nuclear radii + surfaces:
„Simple“ approach using the liquid-drop model
Dmin  rs = [1.25 (A11/3 + A21/3) + 5] fm
• More realistic approach using the half-density radius of a
Fermi mass distribution of the nuclei :
Ci = Ri(1-Ri-2) with R = 1.28 A1/3 - 0.76 + 0.8 A-1/3
 Dmin  rs = [ C1 + C2 + S ] fm
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„Safe“ energy requirement
sexp/sRuth
Dmin

Empirical data on surface distance S as function of half-density radii Ci
require distance of closest approach S > 5 - 8 fm
 choose adequate beam energy (D > Dmin for all q)
low-energy Coulomb excitation
 limit scattering angle, i.e. select impact parameter b > Dmin,
high-energy Coulomb excitation
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Validity of classical Coulomb trajectories
projectile
b=0
target
D=2a
Sommerfeld parameter
a Z P ZT e 2
η 
 1

v 
Pb
 >> 1 requirement for a semi classical
treatment of equations of motion
 measures the strength of the
monopole-monopole interaction
 equivalent to the number of
exchanged photons needed to force
the nuclei on a hyperbolic orbit
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Ar
O
He
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Coulomb trajectories – some more definitions
target
b
projectile
D(q)
q
r (w) = a (e sinh w + 1)
t (w) = a/v (e cosh w + w)
a = Zp Zt e2 E-1
Principal assumption >>1  classical description of the relative motion
of the center-of-mass of the two nuclei  hyperbolic trajectories
1


θ


cm
 distance of closest approach (for w=0):
D (θcm )  a (1  ε )  a 1  sin 
 
2

 

θ


 impact parameter:
b  D2 - 2aD  a  cot  cm 
 2 
q 
 angular momentum :
L   e 2  1   cot cm 
 2 
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Euroschool Leuven – September 2009
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Coulomb excitation – the principal process
target
b
vi
projectile
vf
ΔE  Ei  Ef  12 m0 v i2  v f2  with m0 
m P  mT
m P  mT
Inelastic scattering: kinetic energy is transformed into nuclear excitation energy
e.g.
rotation
vibration
Excitation probability (or
cross section) is a measure
of the collectivity of the
nuclear state of interest
 complementary to, e.g.,
transfer reactions
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Coulomb excitation – “sudden impact”
Excitation occurs only if nuclear time scale is long compared to the collision time:
„sudden impact“ if nucl >>  coll ~ a/v  10 fm / 0.1c  2-310-22 s
coll ~ nucl ~ ћ/E adiabatic limit for (single-step) excitations
ΔE
ΔE a
Z1Z2e2  1 1 
  
ξ
 τ coll 


 v
  vf vi 
: adiabacity paramater
sometimes also (q) with D(q) instead of a
 v
 ΔE max (ξ  1) 
a
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Limitation in the excitation energy E
for single-step excitations in particular
for low-energy reactions (v<c)
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Coulomb excitation – first conclusions
Maximal transferable excitation energy and spin in heavy-ion collisions
=1
c v i
for v i /c  1
a c
c
ΔE max (ξ  1) 
βγ
a
ΔE max (ξ  1) 
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Z1e2Qrot
Lmax 
(1  cosθcm )
2
v  D
 θgr 
Lmax  ηcot 
ECM  Vc
 2 
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Coulomb excitation – the different energy regimes
Low-energy regime
(< 5 MeV/u)
Energy cut-off ΔE max 
Spin cut-off:
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 v
 2 MeV
aε
L up to 30ћ
High-energy regime
(>>5 MeV/u)
ΔE max  c
βγ
 10MeV( β  0.4)
aε
L~ n with n~1
Euroschool Leuven – September 2009
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Summary
• Coulomb excitation is a purely electro-magnetic excitation
process of nuclear states due to the Coulomb field of two
colliding nuclei.
• Coulomb excitation is a very precise tool to measure the
collectivity of nuclear excitations and in particular nuclear
shapes.
• Coulomb excitation appears in all nuclear reactions (at
least in the incoming channel) and can lead to doorway
states for other excitations.
• Pure electro-magnetic interaction (which can be readily
calculated without the knowledge of optical potentials etc.)
requires “safe” distance between the partners at all times.
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Euroschool Leuven – September 2009
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