symmetry approach to nuclear collective motion ii
DESCRIPTION
Symmetry Approach to Nuclear Collective Motion II. Symmetry and dynamical symmetry Symmetry in nuclear physics: Nuclear shell model Interacting boson model. P. Van Isacker, GANIL, France. The three faces of the shell model. Symmetries of the shell model. Three bench-mark solutions: - PowerPoint PPT PresentationTRANSCRIPT
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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
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Nuclear Collective Dynamics II, Istanbul, July 2004
The three faces of the shell model
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
∑
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Nuclear Collective Dynamics II, Istanbul, July 2004
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?
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Nuclear Collective Dynamics II, Istanbul, July 2004
Shell structure from Ex(21)
• High Ex(21) indicates stable shell structure:
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
⎧ ⎨ ⎩
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Nuclear Collective Dynamics II, Istanbul, July 2004
Shell structure from masses
• Deviations from Weizsäcker mass formula:
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
⎧ ⎨ ⎩
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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.
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Nuclear Collective Dynamics II, Istanbul, July 2004
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( )
+
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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{ }
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Nuclear Collective Dynamics II, Istanbul, July 2004
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( )
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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.
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
∑
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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.
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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.
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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 ט b l( )
L( )
ijklJ∑ +L
U 6( )⊃ G⊃SO 3( ) ≡ Lμ = 10d+ ט d ( )μ
1( )⎧ ⎨ ⎩
⎫ ⎬ ⎭
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Nuclear Collective Dynamics II, Istanbul, July 2004
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 ט 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( )[ ]
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Nuclear Collective Dynamics II, Istanbul, July 2004
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( )[ ]
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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?
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Nuclear Collective Dynamics II, Istanbul, July 2004
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.
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
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
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Nuclear Collective Dynamics II, Istanbul, July 2004
Example of 195Pt
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Nuclear Collective Dynamics II, Istanbul, July 2004
Example of 196Au
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Nuclear Collective Dynamics II, Istanbul, July 2004
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( )
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Nuclear Collective Dynamics II, Istanbul, July 2004
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