1. Discovery of a Quasi Dynamical Symmetry and Study of a possible Giant Pairing Vibration
R.F. Casten
WNSL, Yale
May 11,2011
Evidence for a Quasi Dynamical Symmetry along the arc of regularity with an intro to the relevant Group Theory of dynamical symmetries
What and where is the Giant Pairing Vibration and why?

2. Discovery of the first non-trivial Quasi Dynamical Symmetry

3. Symmetries, degeneracies, and Group theory Illustrate using the IBA
Valence nucleons, in pairs as bosons. Number of bosons is half number of valence nucleons – fixed for a given nucleus.
Only certain configurations. Only pairs of
nucleons coupled to angular momentum
0(s) and 2(d). N = ns + nd
Simple Hamiltonian in terms of s an d boson creation, destruction operators – simple interactions
Group theoretical underpinning 3

4. Group Structure of the IBA
vibrator
6-Dim. problem
U(6)
SU(3)
rotor
O(6)
γ-soft
Magical group theory stuff happens here
R4/2= 2.5
Three Dynamic symmetries,
nuclear shapes
Symmetry Triangle of the IBA
Def.
R4/2= 2.0
Sph.
R4/2= 3.33

5. Arc of Regularity Order amidst chaos
An isolated region of regularity inside the triangle
The boundary distinguishing different structural regions
Degeneracies along the arc
The first example of an SU(3) Quasi-dynamical symmetry in nuclei

6. U(5) – Vibrator – Order and regularity0+ 2+
4+,2+,0+
6+,4+,3+,2+,0+ nd 1 2 3

7. SU(3) – Rotor – Order and regularity
SU(3) O(3)

8. What happens inside the triangle?
Whelan, Alhassid, ca 1989 8

9. +2.9
+2.0
+1.4
+0.4
+0.1
-0.1
-0.4 -1
-2.0
-3.0

12. Let’s illustrate group chains and degeneracy-breaking.
Consider a Hamiltonian that is a function ONLY of: s†s + d†d
That is: H = a(s†s + d†d) = a (ns + nd ) = aN
H “counts” the numbers of bosons and multiplies by a boson energy, a. The energies depend ONLY on total number of bosons -- the total number of valence nucleons. The states with given N are degenerate and constitute 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.
Of course, a nucleus with all levels degenerate is not realistic (!!!) and suggests that we should add more terms to the Hamiltonian. I use this example to illustrate the idea of successive steps of degeneracy breaking related to different groups and the quantum numbers they conserve.
Note that s†s = ns and d†d = nd and that ns + nd = N = ½ val nucleons

13. H = H + b d†d = aN + b nd
Now, add a term to this Hamiltonian
Now the energies depend not only on N but also on nd
States of different nd are no longer degenerate. They are “representations” of the group U(5).

14. H = aN + b d†d = a N + b nd U(6) U(5) 2 1 nd H = aN + b d†d N + 2 2a
Etc. with further terms 1 b N nd E
U(6) U(5)
H = aN b d†d

15. Example of a nuclear dynamical symmetry -- O(6)
Spectrum generating algebra
Each successive term:
Introduces a new sub-group
A new quantum number to label the states described by that group
Adds an eigenvalue term that is a function of the new quantum number, and thus
Breaks a previous degeneracy N 15

16. Quasi-dynamical symmetries
Dynamical symmetries where some or all of the degeneracies and quantum numbers of the symmetry are preserved despite large changes in the wave functions from those of the symmetry itself.
Are there such QDS in the “triangle”?
Long standing question for 20 years.

17. First example of a non-trivial QDS in nuclei
Degeneracies along the Arc of Regularity All of them persist as well as the analytic ratios of the 0+ “bandheads”
First example of a non-trivial QDS in nuclei

19. Search for the Giant Pairing Vibration
What is the GPV
Where should it be and why don’t we see it
The experimental situation and hopes
Why it may not be where we think it should be
a little known feature of mixing of bound and unbound states

20. Pairing in nuclei
Pair correlations in nucleonic motion have provided a key to understanding the excitation spectra of even-A nuclei, including the famous pairing gap, compression of energies in odd-A nuclei, odd-even mass differences, rotational moments of inertia, and other phenomena. Pairing “vibrations” were predicted and discovered around 1970, confirming simple models of these 2-particle states.
The Giant Pairing Vibration – a pairing mode in the next higher major shell – should also exist, roughly at ~ 14 MeV in heavy nuclei, and be strongly populated in Q-matched two-nucleon transfer reactions.
As pairing is such a fundamental feature of nuclei, searches for the GPV are extremely important. However, they have never been seen, despite extensive searches.

21. Pairing Vibrations 0+ states
132Sn(t,p) 134Sn
Collective mode: The two nucleons occupy many final 0+ levels with coherent wave functions
Strongly populated in 2-nucleon transfer reactions like (p,t) and (t,p). This PAIR transfer is exactly analogous to the quadrupole phonon creation in the GEOMETRIC VIBRATOR model. Multi-phonon states.
Extensively studied in 1970’s. Model verified

22. Pairing Vibrations 0+ states
132Sn (p,t) 130Sn
Pair Removal mode
(different cross section than pair addition)

23. Concept of Pairing Vibrations (analogy to geometrical vibrational phonon model except there are two modes – pair creation and pair removal) 3B 3A A B 2B 2A B A

24. Giant Pairing Vibration
90 Zr(t,p) 92Zr
Collective mode TWO shells up: The two nucleons occupy many final 0+ levels with coherent wave functions. Predicted at energies from ~ 7 to ~14 Mev by different models
Should be strongly populated in 2-nucleon transfer reactions like (p,t) and (t,p).
Never found – why?

25. Predicted GPV wave function in 210-Po

26. Why has the GPV never been observed?
Despite efforts using conventional pair transfer reactions, such as (t, p) and (p,t) (G. M. Crawley et al., Phys. Rev. Lett. 39 (1977) 1451), the GPV has never been identified.
Fortunato et al. (Eur. Phys. J. A14 (2002) 37) suggests
that beams, such as t or 14C, do not favor excitation of high-energy collective pairing modes due to a large energy mismatch.
Q-values in a stripping reaction involving weakly bound 6He are much
closer to the optimum.

27. Some Recent Calculations
L. Fortunato et al., somewhere
L. Fortunato, Yad. Fiz. 66 (2003) 1491
pp RPA calculations on 208Pb.
Two-neutron transfer form factors
from collective model
DWBA (Ptolemy) calculation of sGPV

28. Recent searches for the GPV

29. What we did and why (progress report, not final story)
Basic idea is that mixing with UNbound levels with widths, has two effects:
Gives width to GPV, making it hard to see –spread out in energy
Increases its energy relative to mixing with bound levels

30. Mixing of bound level with unbound level Normal mixing of bound levels
2-Level mixing
Mixing of bound level with unbound level
Normal mixing of bound levels
GPV
Lower level (GPV) gets width and is lowered LESS

31. Width of upper unperturbed level
(Constant mixing matrix element)

33. Mixing of bound and unbound levels

35. Message to Rick: Don’t forget to stop here and show movie

36. Principal Collaborators
QDS
Dennis Bonatsos
Libby McCutchan
Jan Jolie
Robert Casperson
GPV
Augusto Macchiavelli
Rod Clark
Michael Laskin
With huge help from
Peter von Brentano and Hans Weidenmuller who actually understand the mixing of bound and unbound levels
THANKS !
Summary: QDS + GPV

37. Backup slides

38. What happens inside the triangle?
Whelan, Alhassid, ca 1989 38

39. Investigating the Giant Pairing Vibration
Fundamental excitation mode of the nucleus predicted nearly 30 years ago
(R. A. Broglia and D. R. Bès, Phys. Lett. B69 (1977) 129), but never seen.
Schematic of the dispersion relation. The two bunches of vertical lines represent
the unperturbed energy of a pair of particles placed in a given potential. The GPV
is the collective state relative to the second major shell.
The GPV is a coherent superposition of pp excitations analogous to the more
familiar Giant Shape Vibrations based on ph excitations.

40. Experimental Considerations
6He beams with EBEAM~4-7 MeV/A at intensities of I≥106 particles/sec.
Experimental set-up is simple:
6He
Thin Au
foil
Rutherford
Counter a
208Pb target
~10 mg/cm2
Annular Si
DE-E detectors
L=0 transfer favors forward scattering angles
Rate estimate:
sGPV = 3mb
Target = 10 mg/cm2
2000 counts/day in GPV
IBEAM = 106 p/s
eDET = 25%

43. Note regularities, degeneracies
Dynamical Symmetries
Note regularities, degeneracies
6+, 4+, 3+, 2+, 0+
4+, 2+, 0+ 2+ 0+ 3 2 1 nd
U(5) -- vibrator
SU(3) -- Deformed rotor