1. R. F. Casten Yale and MSU-FRIB SSNET17, Paris, Nov.6-10, 2017
A new approximate symmetry for heavy deformed nuclei:
Proxy-SU(3)
R. F. Casten
Yale and MSU-FRIB
SSNET17, Paris, Nov.6-10, 2017
Kerman’s warning to RFC (1981) about experimentalists talking about theory:
“Experimentalists should not dabble in thought”
(This advice should be squared when it comes to group theory!)

2. Motivation and Goal
A simple approximation, a new scheme, Proxy – SU(3) -- present a Nilsson calculation that validates this new scheme
What is the starting idea of this scheme?
How does the model work?
Predicting nuclear shapes -- b and g.
Experimental evidence for a prolate-to-oblate shape/phase transition in the rare earth region around neutron number N = 116, in agreement with predictions by microscopic calculations.
Consider in the framework of the proxy-SU(3) model, a new approximate symmetry scheme applicable in medium-mass and heavy deformed even-even nuclei
The dominance of prolate (American football) over oblate (disk) shapes in the ground states of even-even nuclei is still a puzzling open problem in nuclear structure.

3. Evolutionary patterns of Structure
Regularity suggests some form of symmetry at work
Courtesy, R. B. Cakirli

4. The Path to Symmetry Regularity out of chaos  Patterns
Simple interpretations
Geometry
 Symmetries – Quantum numbers
 Algebra

5. Fermionic Symmetries in the Shell Model
Key idea: shell model is based on a harmonic oscillator -- with modifications (such as squaring off of the potential and the spin-orbit force). The harmonic oscillator can be described using group theory in terms of the SU(3) group. The requirement is:
A set of single particle shell model levels consisting of orbital angular momenta (of a given parity) spanning from l = 0 (or 1 for odd parity) to some maximum value and that all the single particle total angular momenta, j = l + ½ and j = l- ½, are present.
2s-1d shell in light nuclei: l =
j = ½ /2, 5/2

6. Fermionic Symmetries in the Shell Model
1950’s: Elliott used this idea to interpret the properties of many light nuclei in the Mg-Ar region in terms of SU(3) in this 2s-1d shell: 2s1/2,1d3/2,1d5/2
Problem in heavier nuclei due to the intruder UPO

7. Here is the problem H.O H.O Heavier nuclei: 2 features break SU(3)
Unique parity “intruder” orbit descends into next lower shell. This gives an extra orbit in that shell and the loss of the highest j orbit from the parent shell. Hence the s.p. levels no longer satisfy the requirements of a harmonic oscillator or SU(3). Example:
50-82: l =
j = ½ 3/2 5/2 7/2 + UPO
NO j = 9/2 and extra UPO orbit
H.O
H.O

8. Broken symmetries are common in nature
Question is: Can some form of SU(3) symmetry (approximate) be recovered in heavy nuclei??
Pseudo-SU(3) is a well known and much studied example that has had success.
We want to propose another, called Proxy-SU(3).

9. Part 1 What is the Proxy symmetry and its motivation? Is it any good?

10. UPO h11/2 orbit and loss of g 9/2
The Proxy-SU(3) Scheme 82
sdg shell
h11/2 (UPO)  from the next pfh shell
g9/2 lowered to the previous sdg shell
UPO h11/2 orbit and loss of g 9/2
ruins SU(3)
Solution:
replace h11/2 with g9/2
Lose only one orbital
UPO 50 40

11. NOW have full set of sdg l, j l = 0 2 4 J = 1/2, 3/2, 5/2 7/2, 9/2
Normal
Proxy
UPO
NOW have full set of sdg l, j
l =
J = 1/2, /2, 5/ /2, 9/2
Recovers H.O.  SU(3). Lose one orbit: 30/32 particles
Big problem: Parity issue Spurious matrix elements.

12. Short answer: Inspired by work on p-n interactions (from masses)
Why make this choice?
Short answer: Inspired by work on p-n interactions (from masses)
Note peaks: largest p-n ints
Nilsson orbits K[N, nz, L]
168 Er: p 7/2 [523]; n 7/2 [633]
172 Yb: p 1/2 [411]; n 1/2 [521]
178 Hf: p 7/2 [404]; n 7/2 [514]
dK[dN, dnz, dL] =
0[110]
Highly overlapping
Cakirli et al, PRC 82, (R) (2010).

13. Spatial overlaps (yp2 yn2) of Nils. wave functions
0 [110]
Overlap: 0.804
½ [521]
½ [411]
0 [330]
Overlap: 0.616
½ [631]
½ [301]

14. Inspired the idea to focus on proton-proton and neutron-neutron orbits related by 0[110] as well

15. The Proxy Approximation - look at Nilsson orbits, quantum numbers, for these two h11/2 and g9/2 orbits
Original
shell model orbits
Proxy-SU(3)
orbits
0[110]
dK[dN, dnz, dL]
0[110]
No partner  minor role
A. Martinou

16. Spurious matrix elements: number and size
Proxy Nilsson Hamiltonian matrix
Spurious matrix elements: number and size
Small off-diag and few:
~ 3- 5%
So, maybe not so bad. We will see.

17. How to test if the spurious matrix elements are a real problem
How to test if the spurious matrix elements are a real problem? Calculate a Nilsson diagram and see how different it is from the normal one.

18. How Good ? Fermionic SU(3) –Valid for deformed nuclei
Nilsson diagrams for the N = shell in normal and Proxy shells
Similarity is clear
Greater curvatures  g9/2 orbitals interact with normal parity ones while
h11/2 orbitals do not. Spurious result of approx scheme
Similarity suggests proxy scheme is not too bad. Better the bigger the shell – heavy nuclei. So we will use it to predict some nuclear properties.
For shell with N proxy particles, relevant group is U(N/2) which has sub-group SU(3). e.g., for 50-82, Proxy-SU(3) accounts for 30 particles, so
U(15) > SU(3)
For other shells  same similarity
Bigger the shell, better the approx  heavy nuclei

19. Now lets look at how to predict things with this scheme.
Now that we did this, we see that the Proxy substitution is not so bad. So, forget all the Nilsson stuff – that was just to vet the idea.
Now lets look at how to predict things with this scheme.
Start with ground state properties. Fermion Proxy SU(3) symmetry – need SU(3) quantum numbers. Called (l, m). These categorize sets of states.
Analogy, a bosonic SU(3)
Part 2

20. Typical Bosonic SU(3) Scheme
(l, m)
K bands in (l,m ) : K = 0, 2, 4, m
Series of bands labelled by SU (3) Irreps. Note: Ground state band is (2N,0) where 2N is the number of valence nucleons. Need to get analogous irreps for Proxy SU(3).

21. Magical group theory stuff happens here
Proxy-SU(3):
For shell with N particles, group is U(N/2) with sub-group SU(3). Identify ground state with highest weight irreps labeled by SU(3) quantum numbers: (l, m)
Magical group theory stuff happens here
p -h asymm comes from Pauli Principle:
asymmetric predictions of b, g
(see Talk by Cakirli)

22. Combined (proton + neutron) Proxy-SU(3) irreps

23. Relation of (l, m) to (b, g)
Relation obtained through a linear mapping between the eigenvalues of the invariant operators of the collective model, β2 and β3 cos 3γ, and the eigenvalues of the invariants of SU(3). The basic idea behind this approach is that if the invariant quantities of two theories are used to describe the same physical phenomena, their measures must agree. The derivation is based on the fact that SU(3) contracts to the quantum rotor algebra for low values of angular momentum and large values of the second order Casimir operator of SU(3). Large values of C2 occur for large values of λ and/or μ. The SU(3) irreps appearing in the Tables have λ and/or μ large, thus the use of the contraction limit is justified in these cases.

24. The next talk, by R. Burcu Cakirli, will start from here and discuss practical applications of this model for predicting nuclear shapes, and their comparison with the data.
See also:
Phys. Rev. C95, (2017): Proxy-SU(3) symmetry in heavy deformed nuclei
Phys. Rev. C95, (2017): Analytic predictions for nuclear shapes, prolate dominance, and the prolate-oblate shape transition in the proxy-SU(3) model

25. Summary
A new shell model scheme, based on an physics-motivated orbit substitution, recovers an approx SU(3) symmetry applicable to heavy deformed nuclei
The Proxy Scheme is extremely simple but also has inherent flaws whose significance is assessed
The result is a set of analytic parameter-free predictions for nuclear ground state shapes and shape changes.
Needed improvements (e.g., incorporation of pairing) and extensions (e.g., higher intrinsic excitations)

26. THANKS Collaborators Dennis Bonatsos Nikolay Minkov R. Burcu Cakirli
Klaus Blaum
Andriana Martinou
I.E. Assimakus
S. Sarantopoulou
THANKS