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Masses (Binding energies) and the IBA
Extra structure-dependent binding: energy
depression of the lowest collective state
The dominant features of separation
energies are: the linear behavior along
chains of isotopes, the generally parallel
slopes for different Z’s and the huge drops
after magic numbers.
Specifically structure-dependent effects
are superposed on this secular behavior.
With recent capabilities to measure masses
precisely, we can now access the physics
that these non-linearities point to. How can
we calculate them theoretically?
Classifying Structure -- The Symmetry Triangle
Def.
Sph.
1st order Ph. Tr
17
25
25
Yb
23
23
21
21
16
Er
17
17
15
15
15
S (2n) MeV
S(2n) MeV
MeV
S(2n)
19
19
13
13
11
11
9
Dy
14
Sm
Sm
Hf
Hf
Gd
Ba
Ba
13
Pb
7
Sn
5
52
56
60
64
68
68
12
72
76
80
84
88
88
92
96
100
104
108
108
Neutron
Neutron Number
Number
Ba
Ce
112
Sm
Nd
11
84
86
88
90
92
Neutron Number
94
96
116
120
124
128
128
132
35
30
S(2n) MeV
25
20
15
Ge
10
Ti
Ca
Cr
Ni
Zr
Ru
Pd
Zn
Fe
Mo
Se
Kr
Cd
Sr
5
18
24
30
36
42
48
54
Neutron Number
60
66
72
78
84
Binding energies and separation energies in the
semi-empirical mass formula
BE(A, Z) = avA – asA2/3 – acZ (Z – 1)A-1/3 – aA (A – 2Z)2 A-1
2
Z
4
2
S2n  2 (av - aA) - as A-1/3 +
ac Z (Z - 1) A-4/3 + 8aA
A (A – 2)
3
3
BE’s: linear and quadratic in N, Z or A. Hence, separation
energies are linear.
Reflects shell filling (asymmetry term dominates).
The only practical model to calculate a wide variety of
collective nuclear observables is the IBA. Virtually all
previous efforts with the IBA have focused on
collectivity and its manifestations in energy levels and
electromagnetic moments and transition rates.
However, the IBA can be extended to incorporate mass
predictions, and the opportunities this seems to
provide are quite exciting. Where do masses come into
the IBA and how do we need to modify the usual IBA
calculations to provide masses? (Note: I use masses
and binding energies interchangeably.)
Let’s review group chains and degeneracy-breaking.
Consider a Hamiltonian that is a function ONLY of:
That is:
s†s + d†d
H = a(s†s + d†d) = a (ns + nd ) = aN
Note that such a term does NOT normally appear in the IBA – because
we deal usually with only a single nucleus. The H above can be written
in terms of Casimir operators of U(6). We’ll see now what that means.
In H, the energies depend ONLY on the total number of bosons, that is,
on the total number of valence nucleons.
ALL the states with a given N are degenerate. These states are a
“representation” of the group U(6) with the quantum number N. U(6)
has OTHER representations, corresponding to OTHER values of N, but
THOSE states are in DIFFERENT NUCLEI (numbers of valence nucleons).
Now, add a term to this Hamiltonian:
H’ = H + b d†d = aN + b nd
Now the energies depend not only on N but also on
nd
States of a given nd are now degenerate. They are
“representations” of the group U(5). States with
different nd are not degenerate
2a N + 2
a
H’ = aN + b d†d = a N + b nd
N+1
b
2
1
0
0
2b
N
0
nd
E
U(6)
H’ = aN
U(5)
+
b d† d
Here we are
interested in
masses, or binding
energies.
So, we are
comparing the total
binding of different
nuclei.
So, we are
interested in the
relative energies of
the different
representations of
U(6)
Masses: Expanding the Concept of a Dynamical
Symmetry
N -1
N
N +1
(N)
Above, I discussed what happens with cases
where the Hamiltonian corresponds to a specific
dynamical symmetry but that was just to
illustrate the idea.
The same basic concepts work for calculations
anywhere in the triangle. One needs to
compare total binding energies of two different
nuclei and this will give separation energies.
Where does the binding from collective effects
enter in the IBA? How does the IBA produce
extra binding? What is the BASIC physics ?
Recall:
IBA Hamiltonian
Most general IBA Hamiltonian in terms with up to four boson operators (given N)
These terms CHANGE the
numbers of s and d
bosons: MIX basis states
of the model
Crucial for masses
structure
This lowering of the
lowest state (ground
state) represents
extra BINDING,
changing the mass.
17
Yb
16
Er
15
Dy
14
Gd
13
We can calculate this
with the IBA
12
Sm
Ba
Ce
Nd
11
84
86
88
90
92
94
96
Separation energies in the IBA
S2n (N) = BE (N) – BE (N – 1)
Since we are now dealing with two different nuclei, we need to take
account of the energy differences of different representations of U(6)
or, more generally, of the ground state energies for two adjacent eveneven nuclei. That is, we need to incorporate NEW terms in the IBA
Hamiltonian corresponding to the linear and quadratic Casimir
operators of U(6).
This sounds fancy but isn’t. They simply give terms in the BE that are
linear and quadratic in boson number and therefore reflect the same
physics as the semi-empirical mass formula. BUT: This linear
dependence does NOT include correlations or collective effects !!
This is precisely where the IBA, as a model of nuclear STRUCTURE,
can contribute. It gives a way to calculate those effects. It turns out, as
I discovered only yesterday, that this has some very significant
consequences way beyond the obvious ones !!!
Separation energies in the IBA
So, the energies in the IBA will be given by those
two linear and quadratic terms PLUS the
contribution from the collective correlations
E′ = E0 + A’N + B’/2 N (N – 1) + BE(IBA)
S2n (N) = (A’ – B’/2) + B’N + [BEIBA (N) – BEIBA (N – 1)]
S2n (N) = A + BN + [BEIBA (N) – BEIBA (N – 1)]
S2n (N) = A + BN + S2n (IBA) (N)
So, how do we calculate structure in the IBA? Recall:
Binding energies and separation energies in
the IBA for the dynamical symmetries
Recall Spectrum of U(5)
H   nd
No mixing of states of
given number of dbosons
What is spectrum? Equally spaced levels defined by number of d bosons
3
6 +, 4 +, 3 +, 2 +, 0 +
2
4 +, 2 +, 0 +
1
2+
0
0+
nd
S2(Nv) = Av + BvNv
U(5)
NO collective correlations, hence separation energies are
purely linear
Recall:
IBA Hamiltonian
Most general IBA Hamiltonian in terms with up to four boson operators (given N)
These terms CHANGE the
numbers of s and d
bosons: MIX basis states
of the model
Separation energies in the IBA for the
dynamical symmetries
S2(Nv) = Av + BvNv
U(5)
NO collective correlations. Linear
S2(Nv) = Av + BvNv + 8𝜅 (N + Nv) + 10𝜅
SU(3)
Linear dependence PLUS
LINEAR collective correlations if
kappa is CONSTANT
O(6) Similar to SU(3) -- I don’t have the formula at hand.
It is also linear if kappa is constant
So, if a series of nuclei are described by a
dynamical symmetry AND the coefficients on
the Hamiltonian are CONSTANT, the separation
energies are linear.
This seldom happens.
If the coefficients of the symmetry vary or, more
commonly, if the structure of the nuclei varies,
then there will be non-linear contributions to
the separation energies
How do we calculate BE’s with the IBA for nuclei
away from the dynamical symmetriesin practice?
• First, establish the parameters for the IBA calculation –
epsilon and chi, using the OCC technique described earlier,
with, possibly, some fine tuning ( use Kappa = 0.02 here).
• For masses, we need to worry, however, about the
ABSOLUTE energies. So, we cannot just get relative
energies but need to fit the overall scale of the IBA
calculations to the excitation energies, especially E(2).
• Then, we use: S2n (N) = A + BN + S2n (IBA) (N)
• We write this:
S2n (N)exp - S2n (IBA) (N) = A + BN
• We then fit A and B to the separation energy data
corrected for collective contributions, add back the IBA
part . See if the deviations from linearity are reproduced.
Examples of calculations of BE’s or
S2n values in the IBA
IBA with constant parameters
30
(IBA calculation)
70
60
50
40
30
20
20
S2n
Absolute IBA Binding Energy
80
10
10
4
6
8
Boson Number
10
12
6
8
10
Boson Number
12
Realistic IBA calculations for Gd and Os – the collective
contribution from the IBA
0.3
3.0
0.5
Os
(IBA calculation)
(IBA calculation)
S2n
Gd
2.5
2.0
1.5
0.1
S2n
1.0
0.2
0.0
0.0
150
152
154
156
A
158
160
162
174
176
178
180
A
NORMALIZED
17
Yb
16
Er
S (2n) MeV
15
Dy
14
Gd
13
12
Sm
Ba
Ce
Nd
11
84
86
88
90
92
Neutron Number
94
96
Now, here’s the shocker:
5
S2n (IBA calculation)

4
Er


3
2
1
0
-1
152 154 156 158 160 162 164 166 168 170
A
5
S2n (IBA calculation)

4
3
Er


+
No 21 normalization
2
1
0
-1
-2
156
158
160
162
A
164
166
168