Abstract: While large-scale shell-model calculations may be useful, perhaps even essential, for reproducing experimental data, insight into the physical underpinnings of many-body phenomena requires a deeper appreciation of the underlying symmetries or near-symmetries of a system. Our program is focused on standard as well as novel algebraic approaches to nuclear structure … many-body shell-model methods within a group theoretical framework.

J. P. Draayer, J. G. Hirsch, Feng Pan, A. I. Gueorguieva

C. Bahri, G. Popa, G. Stoitcheva

V. G. Gueorguiev, K. D. Sviracheva, K. Drumev

H. Ganev, H. Matevosyan, H. Grigoryan

Principals

Students

- End of the Day -

Q Space

Q SpaceP Space

xhyh

postulate: x(~2) + y(~12) = ~12

… sy

mplectic

…

- Ongoing Research Projects -

• Pseudo-SU(3) calculations for heavy nuclei• E2 / M1 (scissors) modes in deformed nuclei• Algebraic solution for non-degenerate pairing• Non-linear modes (surface solitons) of a drop• Boson and Fermion representations of sp(4)• Mixed-mode (oblique) shell-model methods• SU(3) calculations for lower fp-shell nuclei

good

more

WORKING

Protons

ψ ν

θψ π

Neutrons

Castaños / Draayer / HirschDirk Rompf (PhD-1998)

Thomas Beuschel (PhD-1999)Carlos Vargas (PhD-2000)Gabriela Popa (PhD-2001)

Study shell-model dynamics against the background of an extremely

elegant algebraic/geometric picture

Rot(3) SU(3) Macroscopic Microscopic

Reproduce experimental M1 spectra including observed fragmentation and

clustering in a microscopic model

Ann. Phys. 180 (1987) 290Z. Phys. A 354 (1996) 359PRC 57 (1998) 1233; 1703

... PRC 61 (2000) 031301; 62 (2000) 064313 ...

… For the record ...

Enhanced M1 strengths first predicted in 1978 by Lo Iudice and Palumbo within the framework of the so-called phenomenological two-rotor model (TRM) …

[N. Lo Iudice and F. Palumbo, PRL 41 (1978) 1532]

Enhanced M1 strengths first observed in 1984 in 156Gd …[D. Bohle, A. Richter, W. Steffen, A. E. L. Dieperink, N. Lo Iudice, F. Palumbo, and O. Scholten, PL 137 B (1984) 27 & U.E.P. Berg, C. Basing, J. Drexler, R. D. Heil, U. Kneisl, W. Naatz, R. Ratzek, S. Schennach, R, Stock, T. Weber, B. Fischer, H. Hollick, and D. Kollewe, PL 149 B (1984) 59]

Discussed within the context of the pseudo-SU(3) model … [O. Castaños, J. P. Draayer and Y. Leschber, Ann. Phys. 180 (1987) 290]

Considered within the framework of the interacting boson (proton-neutron) model (IBM-2) …

[K. Heyde and C. De Coster, PRC 44 (1991) 2262]

0

0.2

0.4

0.6

0.8

1

1.2

2.00 2.50 3.00 3.50 4.00

M1 transition strength spectrum

Energy [MeV]

B(M1, 0

+

-> 1+) [μ n2 ]

160 Gd ( )Experiment

Typical M1 Spectrum

Clustering

Fragmentation

M1

GS 0+ 1+s2 MeV

4 MeV

*

*Adiabatic representation mixing ...

Basis States: Pseudo-SU(3) (spherical cartesian)

n[f]( μ()

particle number

permutation symmetry

multiplicity label ( μ)

deformation [SU(3)]

K-band [SU(3)]

total orbitaltotal spin

total a.m.a.m. proj.

n[f]( μSMs

Q0 -> z-deformation

xy - orbital a.m.xy - a.m. proj.

total spinspin proj.

cartesian system(Slater Determinants)

sphericalsystem

transform between these yields a bit representation of basis states …

Tricks of the Trade(config geometry algebraic shape)

Invariants InvariantsRot(3) SU(3)Tr(Q2) C2

Tr(Q3) C3

2 ~ 2 + μ + μ + + μ1 = tan1 μ 2 + μ

π μπ ν μν

π + ν μπ + μν π + ν - 2 μπ + μν + 1 π + ν 1 μπ + μν - 2 π + ν - 1 μπ + μν - 12

m,l π+ ν-2m+l μπ+μν+m-2l)k

Direct Product Coupling

Coupling proton and neutron irreps to total (coupled) SU(3):

TWIST

SCISSORS

SCISSORS + TWIST

… Orientation of the π-ν system is quantized with the multiplicity denoted by k = k(m,l)

… multiplicity

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GROUND STATE

Hamiltonian: Symmetry Preserving

SU(3) preserving Hamiltonian:

H = c1(Qπ • Qπ + Qν • Qν) + c2Qπ • Qν

+ c3 L2 + c4 K2 + c5 (Lπ2 + Lν

2)

Using the identities ... Q • Q = C2 - 3 L

2 with π or ν Qπ • Qν = Q • Q - Qπ • Qπ - Qν • Qν]/2

L = Lπ + Lν

l = Lπ - Lν

the Hamiltonian becomes ...H = Hrot + Hint

with rotor & interactions terms … Hrot = a L2 + b K2

Hint = c l2 + d C2 + e (C2π + C2ν)

Hint = c l2 + dC2 + e(C2π + C2ν

kinetic potential

System DynamicsHrot = a L2 + b K2

quantum rotor

It can be shown that:

Hint = cθ lθ2 + dθ θ2

+ c-l-

2 + d--

2

where: - π ν

so that: Eint = hθ (nθ + 1/2)

+ h- (n-

+ 1/2)

where: nθ = m, n- = l

Scissors Mode

Twist Mode

…requires triaxialshape distributions

…always present, axial and triaxial

Recall: dynamics driven by interplay between the interaction and statistics

M1 Operator

M1 = (3/4)1/2 μN gorbit L gspin S

gorbit(proton) 1

gorbit(neutron) 0

gspin(proton)

gspin(neutron) 26

Actual

Values

0

0.5

1

1.5

2

M1 transition strength spectrum

B(M1, 0

+

-> 1+) [μ

n2]16 Gd ( )Experiment

0

0.

1

1.

2.00 2.0 .00 .0 .00

[ ]Energy MeV

conserved ()pseudo SU scissors

+ scissors twist(2,)

(26,)

( 1, 0B M

+

-> 1+) [μ

n2]

Scissors and Twist Mode Examples

0

0.2

0.4

0.6

0.8

1

1.2

M1 transition strength spectrum

�

B(M1, 0

+

-> 1+) [μ

n2]160 Gd ( )Experiment

0

0.

1

1.

2

1.60 2.00 2.0 2.0 .20 .60 .00

[ ]Energy MeV

( 1, 0B M

+

-> 1+) [μ

n2]

(29,6)

(26,9) (2,)1,2

twist

+ scissors twistscissors

()conserved pseudo SU

Axial+

Triaxial

Triaxial+

Triaxial

Example: M1 Modes in 156-160Gd

Particle distributions:

total normal unique μ

π 156-160Gd 14 8 6 (10,4)

ν 156Gd 10 6 4 (18,0) 158Gd 12 6 6 (18,0) 160Gd 14 8 6 (18,4)

Valence space: U(π) U(ν)

total normal unique

π = 32 nπ = 20 (pf-shell) u

π = 12 (h11/2)

ν = 44 nν = 30 (sdg-shell) u

ν = 14 ( i13/2 )

~~

~~~

Hamiltonian: Realistic (Version 1)

H = H0 Oscillator

… mixed irrep shell-model basis with 5 proton and 5 neutron irreps plus all products having S = 0 and J 8

SymmetryPreserving

SymmetryBreaking

+ dπ l2π Proton single-particle l2

+ dν l2ν Neutron single-particle l2

+ gπ Hpπ Proton Pairing

+ gν HpνNeutron Pairing

- a2 C2 Q•Q symmetric

+ a3 C3 Q•(QxQ) asymmetric

+ b K2 K-Band splitting

+ c L2Rotational bands

Results for 156-160 Gd Nuclei

160Gd: Excitation Spectrum

160Gd: M1 Strength Distribution

Fragmentation of M1 Strength in 160Gd

Results for 160-164 Dy Nuclei

M1 Sumrule vs E2 Strength

Triaxial Case: 196 Pt

Sumrule ~ 1.5 μ2 or about one-halfthat of the strongly deformed case

Shape (m,l) Mode B(M1) [μ2N]

a + a (1,0) s

π

2 π

π

+

− 1

a + t (1,0) s

π

μπ

+ μ

+ 2

μπ

+ μ

+ 1

π

π

+

− 1

(1,1) s + t

π

1

μπ

+ μ

+ π

+

μπ

μν

( π

+

)

π

+

+ 1

+

π

ν

( μπ

+ μ

)

μπ

+ μ

+ 1

+ ν

μπ

+ π

μν

⎡

⎣

⎢

⎤

⎦

⎥

t + t (1,0) s

π

μπ

+ μ

+ 2

μπ

+ μ

+ 1

π

π

+

− 1

(0,1) t

π

π

+

+ 2

π

+

+ 1

μπ

μ

μπ

+ μ

− 1

(1,1) (s + ) + (t t + s)

π

1

μπ

+ μ

+ π

+

μπ

μν

( π

+

)

π

+

+ 1

+

π

ν

( μπ

+ μ

)

μπ

+ μ

+ 1

+ ν

μπ

+ π

μν

⎡

⎣

⎢

⎤

⎦

⎥

Analytic Results

Castaños ‘87

Rompf ‘96

Beuschel ‘98

Castaños ‘87

Rompf ‘97

Note: t s … (μ) (μ)

Odd-A Case: 163 Dy

0

0.2

0.4

0.6

0.8

1

Energy Spectrum 163 Dy

Energy [Mev]

exp. theo. exp. theo. exp. theo.

Ground StateBand

7/2-

9/2-

11/2-

13/2-

15/2-

5/2-

17/2-

1/2-

5/2-

3/2-

7/2-

1/2-

3/2-

7/2-

5/2-

5/2-

3/2-

5/2-

3/2-

7/2-

9/2-

11/2-

13/2-

15/2-

5/2-

17/2-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1.50 2.00 2.50 3.00 3.50 4.00 4.50

M1 transition strength 163 Dy

TheoryExperiment

Energy [MeV]

B(M1, 0

+

-> 1+) [μ

n2]

Recent Results: Beuschel/Hirsch

Hamiltonian: Realistic (Version 2)

H = H0 Oscillator

… mixed irrep shell-model basis with ~5 proton and ~5 neutron irreps & all products having S = 0 and J 8

SymmetryPreserving

SymmetryBreaking

+ dπ l2π Proton single-particle l2

+ dν l2ν Neutron single-particle l2

+ gπ Hpπ Proton Pairing

+ gν HpνNeutron Pairing

- (2 Q • Q Q•Q symmetric

+ a3 C3 Q•(QxQ) asymmetric

+ b K2 K-Band splitting

+ c L2Rotational bands

+ asym C2 Intrinsic symmetry

FIXED

FINETUNE

Nucleus Gπ Gν HO a b a asym16Gd 0.00 0.1 0.109 .616 0.00 0.16 9. 10- 0.01Gd 0.006 0.109 0.10 .9 0.001 0.10 1.010- 0.10-

160Gd 0.00 0.106 0.106 .2 0.002 0.10 1.10- 1.910-

160Dy 0.00 0.106 0.106 .2 0.00 0.000 1.10- 0.10-

Hamiltonia n ParametersHamiltonian Parameters

FIXED FINE TUNE

- (2 Q • Q

+ dπ l2π

+ dν l2ν

+ gπ Hpπ

+ gν Hpν

+ a3 C3

+ b K2

+ c L2

+ asym C2

-0.5

0

0.5

1

1.5

2

0+

8+

6+

4+

2+

1+

0+

2+

4+

2+

4+

5+

3+

6+

Exp Th Exp Th Exp Th Exp Th

158 Gd

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2.00 2.50 3.00 3.50 4.00 [ ]Energy MeV

1 Gd

158Gd

160Gd

-0.5

0

0.5

1

1.5

2

0+

8+

6+

4+

2+

2+

4+

6+

8+

0+

2+

4+7+

5+

3+

1+

Exp Th Exp Th Exp Th

Ene

rgy

[MeV

]

160Gd

0

0.2

0.4

0.6

0.8

1

2.0 2.5 3.0 3.5 4.0

ExpTh

[ ]Energy MeV

160 Gd

Some Observations

• Fragmentation can be interpreted as effect of non-collective residual interactions

• Gross structure reproduced by adding a new “twist” to the “scissors” system

• Triaxial proton-neutron configuration gives a physical interpretation of multiplicity

• Analytic results for scissors as well as twist modes and combination now available

• Future research with S=1 mode for odd-A as well as even-even nuclei underway

Scissors

Twist

• Scissors mode (proton-neutron) oscillation always exists in deformed systems

• Extra “twist” mode if triaxial proton or neutron distributions are included

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Conclusions - Part 1: E2 / M1 modes ...

EEXXCCIITTIINNGG(simple)(simple)

PPHHYYSSIICCSS!!

EEXXCCIITTIINNGG(simple)(simple)

PPHHYYSSIICCSS!!

• Scissors mode always present …

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• Twist mode(s) requires triaxiality …• Fragmentation via symmetry mixing ...

• B(E2) values good (effective charge) …• Near perfect excitation spectra …

Mixed-Mode SMC … a new shell-model code (SMC) that integrates the best shell

model methods available:m-scheme spherical shell-model

SU(3) symmetry based shell-model

----- Developers -----Vesselin Gueorguiev

Jerry DraayerErich Ormand Calvin Johnson

(H - Eg)0

The Challenge ...Nuclei display uniqueand distinguishable characteristics:

• Single-particle Featues• Pairing Correlations• Deformation/Rotations

ClosedShell

ClosedShell

Single

Single

RotationsSU(3)

PairingSUQ(2)

PairingSUQ(2)

.

Usual Shell-Model Approach• Select a model space based on a simple approximation (e.g.

Nilsson scheme) for the selection of basis states.

• Diagonalize a realistic Hamiltonian in the model space to obtain the the energy spectrum and eigenstates.

• Evaluate electromagnetic transition strengths (E2, M1, etc.) and compare the results with experimental data.

Dual Basis “Mixed Mode” Scheme• Select a dual (non-orthonormal) basis (e.g. pairing plus

quadrupole scheme) for the interaction Hamiltonian.

• Diagonalize a realistic Hamiltonian in the model space to obtain the the energy spectrum and eigenstates.

• Evaluate electromagnetic transition strengths (E2, M1, etc.) and compare the results with experimental data.

Hamiltonian Matrix

SU(3)

standard jj-couplingshell-model scheme

single-particle___________

pairing driven

o v e r l a p

overlap

Shell-Model Hamiltonian

€

H = εiai+ai

i

∑ + Vijkli, j,k,l

∑ ai+aj

+akal

where ai+ and ai are fermion creation and annihilation operators,

εi and Vijkl are real and Vijkl =Vklij =−Vjikl =−Vijlk

Spherical shell-model basis states are eigenstates of the one-body part of the Hamiltonian - single-particle states.

The two-body part of the Hamiltonian H is dominated by the quadrupole-quadrupole interaction Q·Q ~ C2 of SU(3).

SU(3) basis states - collective states - are eigenstates of H for degenerate single particle energies and a pure Q·Q interaction.

Eigenvalue Problem in an Oblique Basis

Spherical basis states ei SU(3) basis states E

Overlap matrix

€

ˆ g =ei ej ei Eβ

Eα ej Eα Eβ

⎛

⎝ ⎜

⎞

⎠ ⎟ =

1 μ

μ+ 1

⎛

⎝ ⎜ ⎞

⎠ ⎟

€

Hψ =Eψ ⇒ ˆ H ⋅ ˆ ψ =E ˆ g ⋅ ˆ ψ

The eigenvalue problem

Current Evaluation Steps

Matrix elements

( H and g )

i and <j1j2J'T'|V|j3j4J''T''>

m-scheme

SM basis(spherical)

SU(3) basis(cylindrical)

H and gg=UUT (Cholesky)

(U-1)THU-1 (U)=E (U)Eigenstates (Lanczos)

Expectation values and matrix elements of

operators<O> and <E1| O |E2>

O and g |E>

We use the Wildenthal USD interaction and denote the spherical basis by SM(#) where # is the number of nucleons outside the d5/2 shell, the SU(3) basis consists of the leading irrep (8,4) and the next to the leading irrep, (9,2).

Example of an Oblique Basis Calculation: 24Mg

Visualizing the SU(3) space with respect to the SM space using the naturally induced basis in the SU(3) space.

SM(4) & SU3+

SU(3) basis space

SM spaceSM space

SU(3) basis space

SM(2) & SU3+

10064.179.921.570.101.750.450.08%

285031829028294492950012823Dimension(m-scheme)

FullSM(4)SM(2)SM(1)SM(0)GT100SU3+

(8,4) & (9,2)

SU3

(8,4)

Model Space

Ground State Convergence for 24Mg

-94

-91

-88

-85

-82

-79

-76

-2000 4000 10000 16000 22000 28000

Number of Basis States

Ground State Energy (MeV)

SM ground state

SM+1 SU(3) irrep

SM+2 SU(3) irreps

SM(0)

SM(1)

SM(2)

SM(3)

SM(4)SM(5) SM(6)

FULL

3.3 MeV (0.5%) 4.2 MeV(54%)

Better Dimensional Convergence!

-95

-90

-85

-80

-75

-70En

erg

y (

MeV

)

SM(0)

(8,4)

(8,4)(9,2)More

irreps

FULLSM(4)

SM(2)

SM(1)

0

2

2434

Level Structure

-95

-90

-85

-80

-75

-70

Energy (MeV)

SM(0)(8,4)

SM(1)(8,4)

SM(0)(8,4)(9,2)

SM(1)(8,4)(9,2)

SM(2)(8,4)

SM(4)(8,4)

SM(4)

SM(4)(8,4)(9,2)

FULL

SM(2)(8,4)(9,2)

0

2

24

43

Oblique Basis Spectral Results

Oblique basis calculations have the levels in the correct order!

• Spherical basis needs 64.2% of the total space - SM(4)!• SU(3) basis needs 0.4% of the total space - (8,4)&(9,2)!• Oblique basis 1.6% of the total space for SM(1)+(8,4)! • SM(0)+(8,4) is almost right… 0.2%

Overlaps With The Exact Eigenvectors For 24Mg

0

20

40

60

80

100

120

Eigenvectors

Mai

n C

ontr

ibu

tion

%

SM(2) 57.77 53.02 39.78 42.50 42.99 35.92

SU(3)+ 63.02 63.77 71.49 59.46 70.15 54.14

SM(2)+SU(3)+ 91.58 90.95 87.72 89.06 87.35 82.23

SM(4) 93.25 92.81 89.98 92.47 91.10 88.33

SM(4)+SU(3)+ 98.57 98.73 97.92 98.41 98.55 96.59

1 2 3 4 5 6

4 35 45 3 8 4

Oblique basis calculations have better overlap with the exact states!

• Using 10% of the total space in oblique basis SM(2)+(8,4)+(9,2)

as good as

• using 64% of the total space in spherical basis SM(4)

Mixed mode (oblique basis) calculations lead to:

better dimensional convergence

correct level order of the low-lying states

significant overlap with the exact states

Conclusions - Part 2: Mixed modes ...

Further Considerations #1

Traditional configuration shell-model scheme(s) versus

SU(3) and symplectic shell-model approaches

Common “bit arithmetic” (Slater determinants) Both use harmonic oscillator basis functions … Translation invariance okay for both approaches Both use effective reduced matrix element logic One single-particle the other quadrupole driven

Complementary … ???

Further Considerations #2

4h 0h2h

ConfigurationShell-Model (multi-h

horizontalslices

SymplecticShell-Model (multi-h

verticalslices

6hsymplectic

slice

translation invarianceokay for both

SU(3) limit (0h

Further Considerations #3… mixed-mode shell-model …

Q Space

Q SpaceP Space

xhyh

postulate: x(~2) + y(~12) = ~12

… sy

mplectic

…

Abstract: … consider standard as well as novel algebraic approaches to nuclear structure, including use of the Bethe ansatz and quantum groups, that are being used to explore special features of nuclei: pairing correlations, quadrupole collectivity, scissors modes, etc. In each case the underlying physics is linked to symmetries of the system and their group theoretical representation.

• E2 / M1 (scissors) modes in deformed nuclei• Mixed-mode shell-model methods feasible• Possible configuration + symplectic extension

?