Discovery of a Quasi Dynamical Symmetry and Study of a possible Giant

Pairing Vibration

R.F. CastenWNSL, 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?

Discovery of the first non-trivial Quasi Dynamical Symmetry

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

Group Structure of the IBAU(5)vibratorSU(3)rotor

O(6)γ-soft

U(6)

Magical group theory

stuff happens here

Symmetry Triangle of the IBA

Sph.

Def. R4/2= 2.0 R4/2= 3.33

R4/2= 2.5

6-Dim. problem

Three Dynamic symmetries, nuclear shapes

Arc of RegularityOrder 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

0+

2+

4+,2+,0+

6+,4+,3+,2+,0+

nd

1

2

3

0

U(5) – Vibrator – Order and regularity

SU(3) O(3)

SU(3) – Rotor – Order and regularity

Whelan, Alhassid, ca 1989

What happens inside the triangle?

U(5) SU(3)

O(6)

+2.9

+2.0

+1.4

+0.4+0.1

-0.1

-0.4

-1

-2.0 -3.0

)2(

)2()0(

1

2

E

EE

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

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).

N

N + 1

N + 2

nd

1 2

0

a

2a

E

0 0

b

2b

H = aN + b d†d = a N + b nd

U(6) U(5)

U(6) U(5)

H = aN + b d†d

Etc. with

furth

er term

s

Example of a nuclear dynamical symmetry -- O(6)

N

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

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.

Degeneracies along the Arc of RegularityAll of them persist as well as the analytic

ratios of the 0+ “bandheads”

First example of a non-trivial

QDS in nuclei

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

Pairing in nucleiPair 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.

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

Pairing Vibrations 0+ states

132Sn (p,t) 130Sn

Pair Removal mode(different cross section

than pair addition)

Concept of Pairing Vibrations(analogy to geometrical vibrational phonon model

except there are two modes – pair creation and pair removal)

A

2A

3A

2B

3B

B

BA

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?

Predicted GPV wave function in 210-Po

• 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) suggeststhat 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 muchcloser to the optimum.

Why has the GPV never been observed?

• pp RPA calculations on 208Pb.

• Two-neutron transfer form factors from collective model

• DWBA (Ptolemy) calculation of sGPV

L. Fortunato et al., somewhereL. Fortunato, Yad. Fiz. 66 (2003) 1491

Some Recent Calculations

Recent searches for the GPV

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

2-Level mixing

Normal mixing of

bound levels

Mixing of bound

level with unbound

level

Lower level (GPV) gets width and is lowered LESS

GPV

Width of upper unperturbed level

(Constant mixing matrix element)

Mixing of bound and unbound levels

Message to Rick:

Don’t forget to stop here and show movie

Principal Collaborators

QDS

Dennis Bonatsos

Libby McCutchanJan JolieRobert Casperson

GPV

Augusto MacchiavelliRod ClarkMichael Laskin

With huge help fromPeter von Brentano and Hans Weidenmuller who actually understand the

mixing of bound and unbound levels

Summary: QDS + GPV THANKS !

Backup slides

What happens inside the triangle?

Whelan, Alhassid, ca 1989

• 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 representthe 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 morefamiliar Giant Shape Vibrations based on ph excitations.

~ 20 G

~ 60 70MeV /A1/ 3

Investigating the Giant Pairing Vibration

6He beams with EBEAM~4-7 MeV/A at intensities of I≥106 particles/sec.

Experimental set-up is simple:

6He

Thin Aufoil

RutherfordCounter

a

208Pb target~10 mg/cm2

Annular SiDE-E detectors

Rate estimate:

sGPV = 3mb Target = 10 mg/cm2

IBEAM = 106 p/s eDET = 25%2000 counts/day in GPV

Experimental Considerations

L=0 transfer favors forward scattering angles

Dynamical Symmetries

6+, 4+, 3+, 2+, 0+

4+, 2+, 0+

2+

0+

3

2

1

0

nd U(5) -- vibrator

SU(3) -- Deformed rotor

Note regularities, degeneracies