Nuclei at high spins and Finite Temperature
1. Heavy Ions Fusion reactions 2. Compound nucleus decay
3. Level Density
4. Angular Momentum Degree of Freedom 5. Temperature Degree of Freedom
6. Experimental Setups: γ-ray and particles detection
Heavy Ion Fusion Reactions
(A>4 up to
238U)
Study of
• equation of state
(relationship among
pressure, density and temperature in nuclear matter→ neutron stars…)
• nuclear structure
at finite Temperature
Nuclear scattering Direct reactions
Compound NucleusFusion
Coulomb Scattering
Rt b
10-22s
10-16-10-18s
The
compound nucleus
formed by heavy ions fusion reactions has Density, Temperature and Angular MomentumNot reachible with light ions
l l
l l l
i l T
k
kb pb
vb l
i
i
) 1 2 (
1
2 +
=
=
=
=
∑
+=
=
σ π
µ h
h
)
max (
max kb k Rp Rt
l = h = h +
h
(2l+1)
nuclear interactions
Compound Nucleus Reactions
If b < Rtarget 1. scattering of incident nucleons with target nucleons 2. target nucleons collide with each others
3. ripartition of incident energy among nucleons of combined system (projectile + target)
evaporation
: statistical probability that a nucleon gains enough energy to escape(analogous: evaporation of molecule from warm liquid)
The intermediate state is called compound nucleus state
A+X → C
*→ Y+b
decay
(more than 1
and of different types)
projectile
target
∆t≈10-16-10-18s
compound nucleus
The reaction is a two step process 1. formation
2. decay
Decay probability to a given final state is indipendent of formation
of compound nucleus
(α,xn)
Projectile Energy
Cross section
Evaporation:
Giving more energy to CN more particles are evaporated
Gaussian shapes
Isotropic angular distribution
(p,α)
Ep = 44.3 MeV
pre-equilibrium CN
1/ 2 1/ 2 3/ 2
( ) 2 .exp
M
M M
E E
N E
T T
π
= −
Maxwellian distribution of emitted particles:
CN decay is a statistical process from an equilibrated system
Maxwellian Distribution at given temperature T
The energy distribution of the emitted particles is used as an indicator of equilibrated compound nucleus formation
The compound nucleus model works well
at low energies (≤ 10 MeV/u) and for medium-heavy nuclei
103 104 105
20 40 60 80
103 104 105 106
20 40 60 80
Counts [a.u.]
74°
60°
47°
35°
Counts [a.u.]
Eα [MeV]
74°
60°
47°
35°
64Ni + 68Zn Ebeam = 500 MeV
Eα [MeV]
16O + 116Sn Ebeam = 250 MeV
Increasing incident energy
Pre-equilibrium component increases
20
Ne +
165Ho →
184Ir+n
132Ce* at E*=200MeV
16O+116Sn
Ebeam=250MeV
64Ni+68Zn
Ebeam=500MeV
Heavy Ion reactions populate nuclear states at high spins
Effective potential
acting between the two ions ) ( )
( )
( )
(r V r V r V r V = N + C + l
+ −
= −
a R r r V
VN
exp 1 )
( 0
Nuclear
(Wood-Saxon) Coulomb Centrifugal
2 2
2
) 1 (
cb cb
cm R
l E l
E µ
+ +
≥ h
2 / 1 max Rcb[2 (Ecm Ecb)]
l = µ −
) ( 44 . ) 1
(
5 . 0 ) (
36 . 1
2 1 2
2 1
3 / 1 2 3 / 1 1
fm R
Z Z R
e Z MeV Z
E
fm A
A R
cb cb
cb cb
=
=
+ +
=
r
)( ) (
0 .
1 A11/3 A12/3 R1 R2 RF ≈ + < +
incident
energy in CM ECM
16O+120Sn
the pocket occurs at the same distance RF
Angular momentum limit
r e Z r Z
VC
2 2
) 1
( =
2 2
2 ) 1 ) (
( r
l r l
Vl
µ
= h +
) ( 11/3 12/3
0 2
1 R R A A
R
R= + = +
r V(r) VC(r)
) (r Vl
) (r VN
lmax≈70h for A≈160
Angular momentum limits
(from liquid drop calculations)
Radioactive Beams
(Neutron Rich)
population of larger angular momenta:
fission barrier increases with neutron number
Yb
stable
∼
10h RIBA
L max(B f~S n)
Swiatecki-Myers
Bn=8 MeV
48Ca + 124Sn Yb (Z=70)
132Sn + 48Ca I=70 h
B f[MeV]
A
2 ) 1 5 (
2 2 2/3 2
A Z a A a
a B
s c s
s
f = δ −
A l(h)
Bf=8 MeV Bf= 0
triaxial
oblate
Stable Beams
fission limits
maximum angular momentum
≈ 1021 rotations/sec
