nuclear level densities edwards accelerator laboratory steven m. grimes ohio university athens, ohio
TRANSCRIPT
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Nuclear Level Densities
Edwards Accelerator
Laboratory
Steven M. Grimes
Ohio University
Athens, Ohio
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Nuclear Level DensitiesNuclear Level Densities
Fermi Gas Assumptions:
1.) Non-interacting fermions
2.) Equi-distant single
particle spacing(U) exp∝ [2√au] / U3/2
generally successful
Most tests at U < 20 MeV
For nuclei near the bottom of
the valley of stability
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Nuclear Level DensitiesNuclear Level Densities
DATA:
(1) Neutron resonances
U≈ 7 MeV
near valley of stability
(2) Evaporation spectra
U ≈ 3 – 15 MeV
near valley of stability
(3) Ericson Fluctuations
U ≈ 15 – 24 MeV
near valley of stability
(4) Resolved Levels
U ≤ 4 – 5 MeV
Most points for nuclei in
bottom of valley of stability
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Nuclear Level DensitiesNuclear Level Densities
Study of Al-Quraishi, et al.
20 ≤ A ≤ 110
Found a = A exp[-(Z-Z0)2]
≈ 0.11 ≈ 0.04
Z -- Z of nucleus
Z0 -- Z of stable
nucleus of that Z
Reduction in a negligible
if Z-Z∣0
≤ 1∣
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Nuclear Level DensitiesNuclear Level Densities
Study of Al-Quraishi, et al.
Significant effect for ∣Z-Z
0 = 2∣
Large effect for| Z-Z
0 | ≥ 3
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Nuclear Level DensitiesNuclear Level Densities
Isospin: (N-Z)/2 = TZ
T ≥ TZ
At higher energies, have T=2
multiplets 16C, 16N, 16O, 16F,16Ne(T
Z=0) > (T
Z=1) > (T
Z=2)
predicts a = A exp( (N-Z)2 )
16N 16O 16F
T = 0
T = 1
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Nuclear Level DensitiesNuclear Level Densities
CONCLUSION
Only one analysis finds
a lower off of stability line
Limited data is
the central problem
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Nuclear Level DensitiesNuclear Level Densities
Bulk of Level Density information
comes from neutron resonances
Example: n + 32S → 33S*
At low energy (neutrons) find
1/2+ levels at about 7 MeV
of excitation
Not feasible for unstable targets Only get density at one energy
Need ( = J⟨z2⟩1/2 ) to get
total level density
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Nuclear Level DensitiesNuclear Level Densities
PREDICTIONS
A compound nucleus state must
have a width which is narrow
compared to single particle width
This indicates compound levels
will not be present once
occupancy of unbound single
particle states is substantial.
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Nuclear Level DensitiesNuclear Level Densities
Assume we can use Boltzmann
distribution as approximation to
Fermi-Dirac distribution
Since U = a2
= √U/a is temperature
a is level density parameter
U is excitation Energy
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Nuclear Level DensitiesNuclear Level DensitiesIf occupancy of state at excitation
energy B is less than 0.1
exp[ -B/ ] = exp[ -B√a/U ] ≤ 0.1
a ≈ A/8
exp[ -B √A/8U ] ≤ 0.1
so, -B √A/8U ≤ -2.3
UC ≤ ( AB2 / 42.3 )
Above this energy
we will lose states
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Nuclear Level DensitiesNuclear Level Densities
A B(MeV) U(MeV)
20 8 30.3
200 8 303.
20 6 17.04
200 6 170.0
20 4 7.58
200 4 75.8
20 2 1.9
200 2 19
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Nuclear Level DensitiesNuclear Level Densities
For nucleosynthesis processes, we will
frequently have B ~ 4 MeV
for proton or neutron
For A ~ 20 level density is
substantially reduced
by 10 MeV if B = 4
For A ~ 20 level density
limit is ≈ B if B = 2 MeV
Substantial reduction
in compound resonances
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Nuclear Level DensitiesNuclear Level Densities
Will also reduce level density
for final nucleus in capture
We also must consider parity
In fp shell even-even nuclei
have more levels of + parity
at low U than - parity
Even-odd or odd-even have
mostly negative parity states
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Nuclear Level DensitiesNuclear Level Densities
Thus, compound nucleus states
not only have reduced (U)
but also suppress
s-wave absorption
because of
parity mismatch
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Nuclear Level DensitiesNuclear Level Densities
Also have angular momentum
restrictions
Could have even-even target
near drip-line where
g9/2
orbit is filling
Need 1/2+ levels
in compound nucleus
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Nuclear Level DensitiesNuclear Level Densities
Low-lying states are
0+ g⊗9/2
9/2+
or
2+ g⊗9/2
5/2+, 7/2+, 9/2+,
11/2+, 13/2+
At low energy 1/2+ states
may be missing
No s-wave compound nuclear
(p,) (or (n,)) reactions
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Nuclear Level DensitiesNuclear Level Densities
EXPERIMENTAL TESTS
Measure:55Mn(d,n)56Fe
58Fe(3He,p)60Co58Fe(3He,d)57Fe58Fe(3He,n)60Ni
58Ni(3He,p)60Cu58Ni(3He,)57Ni58Ni(3He,n)60Zn
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Nuclear Level DensitiesNuclear Level Densities
Result: lower a → higher average E→lower multiplicity
If compound nucleus is proton rich
Al Quraishi term will further
inhibit neutron decay
Change in a a (a-a)
Ep
(Ep)
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level Densities
Span range of Z-Z0 values
up to ~ 2.5
Find evidence
that a /A decreases
with Z-Z∣0
increasing∣
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Nuclear Level DensitiesNuclear Level Densities
Currently calculating these effects
Woods-Saxon basis
Find single particle state
energies and widths
Compare with all sp states with
including only bound or
quasibound orbits
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Nuclear Level DensitiesNuclear Level Densities
Expect reduction in a
if Z-Z∣0
≈ 2, 3∣
As Z-Z∣0
increases, new form ∣
with peak at 5-10 MeV may
emerge (i.e. (20) ≈ 0)
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Nuclear Level DensitiesNuclear Level Densities
Calculations and measurements
underway
1.) Hope to determine whether off
of stability line a drops
2.) is form a = A exp[-(Z-Z0)2]
appropriate?
3.) As drip line is approached, we
may have to abandon Bethe
form and switch to Gaussian
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Nuclear Level DensitiesNuclear Level Densities
HEAVY ION REACTIONS
Can get to 30-40 MeV of
excitation with lower pre-
equilibrium component
Cannot use projectile with A
about equal to target (quasi-
fission)
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Nuclear Level DensitiesNuclear Level Densities
HEAVY ION REACTIONS
Recent Argonne measurements60Ni + 92Mo60Ni + 100Mo
got only 30-35% compound
nuclear reactions
Jcontact
too high
Not enough compound levels
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Nuclear Level DensitiesNuclear Level Densities
Projectiles with A < half of
target A are better
Want compound nuclei with|Z-Z
0| ≥ 2 to compare
with |Z-Z0| ≈ 0
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Nuclear Level DensitiesNuclear Level Densities
REACTIONS
24Mg + 58Fe 82Sr*24Mg + 58Ni 82Zr*18O + 64Ni 82Kr*
Excitation energy 60 MeV
82Kr has Z = Z0
82Sr has Z = Z0 + 282Zr has Z = Z0 + 4
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Nuclear Level DensitiesNuclear Level Densities
Compare:
Rohr
a A
Al Quraishi
a A exp[ (Z-Z0)2 ]
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Nuclear Level DensitiesNuclear Level Densities
Decay Fractions
Rohr
.96 n
.04 p
.005
Al-Quraishi
.93 n
.06 p
.01
82Kr
.67 n
.25 p
.05
.02 d
.61 n
.33 p
.05
.01 d
82Sr
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Nuclear Level DensitiesNuclear Level Densities
Decay Fractions
Rohr
.43 n
.49 p
.06
.02 d
Al-Quraishi
.36 n
.55 p
.07
.02 d
82Zr
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Nuclear Level DensitiesNuclear Level Densities
Rohr: Higher a for Z-Z0 2
Al-Quraishi: Lower a
for Z-Z0 2
Get softer spectrum with Rohr
More 4-6 particle emissions than
Al-Quraishi
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Nuclear Level DensitiesNuclear Level Densities
Calculations
Solve for single particle
energies in a single particle
(Woods-Saxon) potential
Compare level density including
all single particle states with
level density including only
those with < 500 keV
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Nuclear Level DensitiesNuclear Level Densities
Calculations
Small effects for Z Z0
Large effects for Z-Z∣0
≥ 4∣
Looking at including two body
effects in these calculations with
moment method expansions
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level Densities
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Nuclear Level DensitiesNuclear Level DensitiesCONCLUSIONS
Astrophysics has need for
level densities off of the
stability line for A ≤ 100
Data base in this region
( Z-Z∣0
≥ 2 ) is poor∣
Model predicts that a decreases
with |Z-Z0|
Need more reaction data for nuclei
in the region of |Z-Z0| 2