symmetry approach to nuclear collective motion ii

Nuclear Collective Dynamics II, Istanbul, July 2004

Symmetry Approach toNuclear Collective Motion II

P. Van Isacker, GANIL, France

Symmetry and dynamical symmetry

Symmetry in nuclear physics:Nuclear shell model

Interacting boson model


The three faces of the shell model


Symmetries of the shell model

• Three bench-mark solutions:– No residual interaction IP shell model.– Pairing (in jj coupling) Racah’s SU(2).– Quadrupole (in LS coupling) Elliott’s SU(3).

• Symmetry triangle:

€

H =pk

2

2m+

1

2mω2rk

2 −ζ ls lk ⋅sk −ζ ll lk2

⎡

⎣ ⎢

⎤

⎦ ⎥

k=1

A

∑

+ VRI rk,rl( )k<l

A

∑


Evidence for shell structure

• Evidence for nuclear shell structure from– 2+ in even-even nuclei [Ex, B(E2)].

– Nucleon-separation energies & nuclear masses.– Nuclear level densities.– Reaction cross sections.

• Is nuclear shell structure modified away from the line of stability?


Shell structure from Ex(21)

• High Ex(21) indicates stable shell structure:


Weizsäcker mass formula

• Total nuclear binding energy:

• For 2149 nuclei (N,Z≥8) in AME03: aV16, aS18, aI7.3, aC0.71, aP13 rms2.5 MeV.

€

BW N,Z( ) = aVA − aSA2 / 3( ) 1− aI

Tz Tz +1( )A2

⎡

⎣ ⎢

⎤

⎦ ⎥

− aC

Z Z −1( )A1/ 3

+ aP

1

A1/ 2×

+1 pair−pair

0 impair

−1 impair−impair

⎧ ⎨ ⎩


Shell structure from masses

• Deviations from Weizsäcker mass formula:


Racah’s SU(2) pairing model

• Assume large spin-orbit splitting ls which implies a jj coupling scheme.

• Assume pairing interaction in a single-j shell:

• Spectrum of 210Pb:

€

j 2JMJ Vpairing r1,r2( ) j 2JMJ =− 1

22 j +1( )g, J = 0

0, J ≠ 0

⎧ ⎨ ⎩


Solution of pairing hamiltonian

• Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell:

• Seniority (number of nucleons not in pairs coupled to J=0) is a good quantum number.

• Correlated ground-state solution (cfr. super-fluidity in solid-state physics).

€

j nυ J Vpairing rk,rl( )k<l

n

∑ j nυJ = − 1

4G n −υ( ) 2 j − n −υ + 3( )

G. Racah, Phys. Rev. 63 (1943) 367


Pairing and superfluidity

• Ground states of a pairing hamiltonian have a superfluid character:– Even-even nucleus (=0):– Odd-mass nucleus (=1):

• Nuclear superfluidity leads to– Constant energy of first 2+ in even-even nuclei.– Odd-even staggering in masses.– Smooth variation of two-nucleon separation

energies with nucleon number.– Two-particle (2n or 2p) transfer enhancement.

S+( )n/ 2

o

aj+ S+( )

n/ 2o


Superfluidity in semi-magic nuclei• Even-even nuclei:

– Ground state has =0.

– First-excited state has =2.

– Pairing produces constant energy gap:

• Example of Sn nuclei:

Ex 21+

( )=12 2j +1( )g


Two-nucleon separation energies

• Two-nucleon separation energies S2n:

(a) Shell splitting dominates over interaction.

(b) Interaction dominates over shell splitting.

(c) S2n in tin isotopes.


Pairing with neutrons and protons

• For neutrons and protons two pairs and hence two pairing interactions are possible:– Isoscalar (S=1,T=0):

– Isovector (S=0,T=1):−S+

01⋅S−01, S+

01= l +12 a

l 12

12

+ ×al 12

12

+⎛ ⎝

⎞ ⎠

001( )

, S−01 = S+

01( )

+

−S+10⋅S−

10, S+10 = l +1

2 al 1

212

+ ×al 1

212

+⎛ ⎝

⎞ ⎠

010( )

, S−10 = S+

10( )

+


Neutron-proton pairing hamiltonian

• A hamiltonian with two pairing terms,

• …has an SO(8) algebraic structure.

• H is solvable (or has dynamical symmetries) for g0=0, g1=0 and g0=g1.

H =−g0S+10⋅S−

10 −g1S+01⋅S−

01


SO(8) ‘quasi-spin’ formalism

• A closed algebra is obtained with the pair operators S± with in addition

• This set of 28 operators forms the Lie algebra SO(8) with subalgebras

n=2 2l +1 al 1

2

1

2

+ ×al 1

2

1

2

+⎛ ⎝

⎞ ⎠ 000

000( )

, Yμν = 2l +1 al 1

2

1

2

+ ×al 1

2

1

2

+⎛ ⎝

⎞ ⎠ 0μν

011( )

Sμ = 2l +1 al 1

212

+ ×al 12

12

+⎛ ⎝

⎞ ⎠

0μ0

010( )

, Tν = 2l +1 al 12

12

+ ×al 1

212

+⎛ ⎝

⎞ ⎠

00ν

001( )

B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673

SO 6( ) ≈SU 4( ) = S,T,Y{ }, SOS 5( )= n,S,S±10

{ },

SOT 5( ) = n,T,S±01

{ }, SOS 3( )= S{ }, SOT 3( ) = T{ }


Solvable limits of SO(8) model• Pairing interactions can expressed as follows:

• Symmetry lattice of the SO(8) model:

Analytic solutions for g0=0, g1=0 and g0=g1.

S+01⋅S−

01 =12C2 SOT 5( )[ ]−1

2C2 SOT 3( )[ ]−1

8(2l −n+1)(2l −n+7)

S+01⋅S−

01+S+10⋅S−

10 =12C2 SO 8( )[ ]−1

2 C2 SO 6( )[ ]−18 (2l −n+1)(2l −n+13)

S+10⋅S−

10 =12C2 SOS 5( )[ ]−1

2 C2 SOS 3( )[ ]−18 (2l −n+1)(2l −n+7)

€

SO(8)⊃

SOS 5( )⊗SOT 3( )

SO(6) ≈ SU 4( )

SOT 5( )⊗SOS 3( )

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪⊃SOS 3( )⊗SOT 3( )


Superfluidity of N=Z nuclei

• T=0 & T=1 pairing has quartet superfluid character with SO(8) symmetry. Pairing ground state of an N=Z nucleus:

Condensate of ’s ( depends on g01/g10).

• Observations:– Isoscalar component in condensate survives only

in N~Z nuclei, if anywhere at all.– Spin-orbit term reduces isoscalar component.

cosθS+10⋅S+

10 −sinθS+01⋅S+

01( )

n/4o


Deuteron transfer in N=Z nuclei

• Deuteron intensity cT2

calculated in schematic model based on SO(8).

• Parameter ratio b/a fixed from masses.

• In lower half of 28-50 shell: b/a5.


Symmetries of the shell model

• Three bench-mark solutions:– No residual interaction IP shell model.– Pairing (in jj coupling) Racah’s SU(2).– Quadrupole (in LS coupling) Elliott’s SU(3).

• Symmetry triangle:

€

H =pk

2

2m+

1

2mω2rk

2 −ζ ls lk ⋅sk −ζ ll lk2

⎡

⎣ ⎢

⎤

⎦ ⎥

k=1

A

∑

+ VRI rk,rl( )k<l

A

∑


Wigner’s SU(4) symmetry• Assume the nuclear hamiltonian is invariant

under spin and isospin rotations:

• Since {S,T,Y} form an SU(4) algebra:– Hnucl has SU(4) symmetry.– Total spin S, total orbital angular momentum L,

total isospin T and SU(4) labels () are conserved quantum numbers.

Hnucl,Sμ[ ]= Hnucl,Tν[ ] = Hnucl,Yμν[ ] =0

Sμ = sμ k( ),k=1

A

∑ Tν = tν k( )k=1

A

∑ , Yμν = sμ k( )k=1

A

∑ tν k( )

E.P. Wigner, Phys. Rev. 51 (1937) 106F. Hund, Z. Phys. 105 (1937) 202


Physical origin of SU(4) symmetry

• SU(4) labels specify the separate spatial and spin-isospin symmetry of the wave function:

• Nuclear interaction is short-range attractive and hence favours maximal spatial symmetry.


Breaking of SU(4) symmetry

• Non-dynamical breaking of SU(4) symmetry as a consequence of– Spin-orbit term in nuclear mean field.– Coulomb interaction.– Spin-dependence of residual interaction.

• Evidence for SU(4) symmetry breaking from– Masses: rough estimate of nuclear BE from

decay: Gamow-Teller operator Y,1 is a generator of SU(4) selection rule in ().

B N,Z( )∝ a+bgλμν( ) =a+b λμν C2 SU 4( )[ ]λμν


SU(4) breaking from masses

• Double binding energy difference Vnp

Vnp in sd-shell nuclei:

δVnp N,Z( )=14 B N,Z( )−B N −2,Z( )−B N,Z−2( )+B N −2,Z−2( )[ ]

P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607


SU(4) breaking from decay

• Gamow-Teller decay into odd-odd or even-even N=Z nuclei:

P. Halse & B.R. Barrett, Ann. Phys. (NY) 192 (1989) 204


Elliott’s SU(3) model of rotation

• Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type:

• State labelling in LS coupling:

H =pk

2

2m+

12

mω2rk2⎡

⎣ ⎢ ⎤

⎦ ⎥ k=1

A

∑ −κQ⋅Q,

Qμ =4π5

rk2Y2μ ˆ r k( )

k=1

A

∑ + pk2Y2μ ˆ p k( )

k=1

A

∑⎛ ⎝ ⎜ ⎞

⎠

U 4ω( )

↓

1M[ ]

⊃

U L ω( )

↓˜ λ ̃ μ ̃ ν ( )

⊃SUL 3( )

↓

λ μ ( )

⊃SOL 3( )

↓

L

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⊗

SUST 4( )

↓

λμν( )

⊃SUS 2( )

↓

S

⊗SUT 2( )

↓

T

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562


Importance/limitations of SU(3)

• Historical importance:– Bridge between the spherical shell model and the

liquid droplet model through mixing of orbits.– Spectrum generating algebra of Wigner’s SU(4)

supermultiplet.

• Limitations:– LS (Russell-Saunders) coupling, not jj coupling

(zero spin-orbit splitting) beginning of sd shell.– Q is the algebraic quadrupole operator no

major-shell mixing.


Tripartite classification of nuclei

• Evidence for seniority-type, vibrational- and rotational-like nuclei:

• Need for model of vibrational nuclei. N.V. Zamfir et al., Phys. Rev. Lett. 72 (1994) 3480


The interacting boson model

• Spectrum generating algebra for the nucleus is U(6). All physical observables (hamiltonian, transition operators,…) are expressed in terms of s and d bosons.

• Justification from– Shell model: s and d bosons are associated with S

and D fermion (Cooper) pairs.– Geometric model: for large boson number the IBM

reduces to a liquid-drop hamiltonian.

A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253; 111 (1978) 201; 123 (1979) 468


Algebraic structure of the IBM

• The U(6) algebra consists of the generators

• The harmonic oscillator in 6 dimensions,

• …has U(6) symmetry since

• Can the U(6) symmetry be lifted while preserving the rotational SO(3) symmetry?

U 6( ) = s+s,s+dm,dm+s,dm

+dm'{ }, m,m'=−2,K ,+2

H =ns +nd =s+s+ dm+dm

m=−2

+2

∑ =C1 U 6( )[ ] ≡N

∀gi ∈U 6( ) : H,gi[ ]=0


The IBM hamiltonian

• Rotational invariant hamiltonian with up to N-body interactions (usually up to 2):

• For what choice of single-boson energies s and d and boson-boson interactions L

ijkl is the IBM hamiltonian solvable?

• This problem is equivalent to the enumeration of all algebras G that satisfy

HIBM =εsns +εdnd + υijkl

L bi+×bj

+( )

L( )⋅ ˜ b k ×˜ b l( )

L( )

ijklJ∑ +L

U 6( )⊃ G⊃SO 3( ) ≡ Lμ = 10d+ ×˜ d ( )μ

1( )⎧ ⎨ ⎩

⎫ ⎬ ⎭


Dynamical symmetries of the IBM

• The general IBM hamiltonian is

• An entirely equivalent form of HIBM is

• The coefficients i and j are certain combinations of the coefficients i and L

ijkl.

HIBM =εsns +εdnd + υijklL bi

+×bj+

( )L( )

⋅ ˜ b k ×˜ b l( )L( )

ijklJ∑

HIBM =η1C1 U 6( )[ ]+η2C1 U 5( )[ ]+κ 0C1 U 6( )[ ]C1 U 5( )[ ]

+κ1C2 U 6( )[ ]+κ2C2 U 5( )[ ]+κ3C2 SU 3( )[ ]

+κ 4C2 SO 6( )[ ]+κ5C2 SO 5( )[ ]+κ6C2 SO 3( )[ ]


The solvable IBM hamiltonians

• Without N-dependent terms in the hamiltonian (which are always diagonal)

• If certain coefficients are zero, HIBM can be written as a sum of commuting operators:

HIBM =η1C1 U 5( )[ ]+κ1C2 U 5( )[ ]+κ2C2 SU 3( )[ ]

+κ 3C2 SO 6( )[ ]+κ4C2 SO 5( )[ ]+κ 5C2 SO 3( )[ ]

HU 5( ) =η1C1 U 5( )[ ]+κ1C2 U 5( )[ ]+κ 4C2 SO 5( )[ ]+κ5C2 SO 3( )[ ]

HSU 3( ) =κ 2C2 SU 3( )[ ]+κ5C2 SO 3( )[ ]

HSO 6( ) =κ3C2 SO 6( )[ ]+κ4C2 SO 5( )[ ]+κ 5C2 SO 3( )[ ]


The U(5) vibrational limit

• Spectrum of an anharmonic oscillator in 5 dimensions associated with the quadrupole oscillations of a droplet’s surface.

• Conserved quantum numbers: nd, , L.

A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253D. Brink et al., Phys. Lett. 19 (1965) 413


The SU(3) rotational limit

• Rotation-vibration spectrum with - and -vibrational bands.

• Conserved quantum numbers: (,), L.

A. Arima & F. Iachello, Ann. Phys. (NY) 111 (1978) 201A. Bohr & B.R. Mottelson, Dan. Vid. Selsk. Mat.-Fys. Medd. 27 (1953) No 16


The SO(6) -unstable limit

• Rotation-vibration spectrum of a -unstable body.

• Conserved quantum numbers: , , L.

A. Arima & F. Iachello, Ann. Phys. (NY) 123 (1979) 468L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788


Synopsis of IBM symmetries

• Symmetry triangle of the IBM:– Three standard solutions: U(5), SU(3), SO(6).– SU(1,1) analytic solution for U(5) SO(6).– Hidden symmetries (parameter transformations).– Deformed-spherical coexistent phase.– Partial dynamical symmetries.– Critical-point symmetries?


Extensions of the IBM

• Neutron and proton degrees freedom (IBM-2):– F-spin multiplets (N+N=constant).

– Scissors excitations.

• Fermion degrees of freedom (IBFM):– Odd-mass nuclei.– Supersymmetry (doublets & quartets).

• Other boson degrees of freedom:– Isospin T=0 & T=1 pairs (IBM-3 & IBM-4).– Higher multipole (g,…) pairs.


Scissors excitations• Collective displacement

modes between neutrons and protons:– Linear displacement

(giant dipole resonance): R-R E1 excitation.

– Angular displacement (scissors resonance): L-L M1 excitation.

N. Lo Iudice & F. Palumbo, Phys. Rev. Lett. 41 (1978) 1532F. Iachello, Phys. Rev. Lett. 53 (1984) 1427D. Bohle et al., Phys. Lett. B 137 (1984) 27


Supersymmetry

• A simultaneous description of even- and odd-mass nuclei (doublets) or of even-even, even-odd, odd-even and odd-odd nuclei (quartets).

• Example of 194Pt, 195Pt, 195Au & 196Au:

F. Iachello, Phys. Rev. Lett. 44 (1980) 772P. Van Isacker et al., Phys. Rev. Lett. 54 (1985) 653A. Metz et al., Phys. Rev. Lett. 83 (1999) 1542


Example of 195Pt


Example of 196Au


Algebraic many-body models

• The integrability of any quantum many-body (bosons and/or fermions) system can be analyzed with algebraic methods.

• Two nuclear examples:– Pairing vs. quadrupole interaction in the nuclear

shell model.– Spherical, deformed and -unstable nuclei with

s,d-boson IBM.

U 6( )⊃

U 5( ) ⊃SO 5( )

SU 3( )

SO 6( )⊃SO 5( )

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

⊃SO 3( )


Other fields of physics• Molecular physics:

– U(4) vibron model with s,p-bosons.

– Coupling of many SU(2) algebras for polyatomic molecules.

• Similar applications in hadronic, atomic, solid-state, polymer physics, quantum dots…

• Use of non-compact groups and algebras for scattering problems.

U 4( )⊃U 3( )

SO 4( )

⎧ ⎨ ⎩

⎫ ⎬ ⎭

⊃SO 3( )

F. Iachello, 1975 to now

symmetry approach to nuclear collective motion ii

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