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Symmetries in Nuclei. Symmetry and its mathematical description The role of symmetry in physics Symmetries of the nuclear shell model Symmetries of the interacting boson model. Interacting boson approximation.

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Symmetries in Nuclei

- Symmetry and its mathematical description
- The role of symmetry in physics
- Symmetries of the nuclear shell model
- Symmetries of the interacting boson model

Symmetries in Nuclei, Tokyo, 2008

Interacting boson approximation

- Dominant interaction between nucleons has pairing character two nucleons form a pair with angular momentum J=0 (S pair).
- Next important interaction between nucleons with angular momentum J=2 (D pair).
- Approximation: Replace S and D fermion pairs by s and d bosons. Argument:

Symmetries in Nuclei, Tokyo, 2008

Microscopy of IBM

- In a boson mapping, fermion pairs are represented as bosons:
- Mapping of operators (such as hamiltonian) should take account of Pauli effects.
- Two different methods by
- requiring same commutation relations;
- associating state vectors.

T. Otsuka et al., Nucl. Phys. A 309 (1978) 1

Symmetries in Nuclei, Tokyo, 2008

The interacting boson model

- Describe the nucleus as a system of N interacting s and d bosons. Hamiltonian:
- 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

Symmetries in Nuclei, Tokyo, 2008

Dimensions

- Assume available 1-fermion states. Number of N-fermion states is
- Assume available 1-boson states. Number of N-boson states is
- Example: 162Dy96 with 14 neutrons (=44) and 16 protons (=32) (132Sn82 inert core).
- SM dimension: 7·1019
- IBM dimension: 15504

Symmetries in Nuclei, Tokyo, 2008

U(6) algebra and symmetry

- Introduce 6 creation & annihilation operators:
- The hamiltonian (and other operators) can be written in terms of generators of U(6):
- The harmonic hamiltonian has U(6) symmetry
- Additional terms break U(6) symmetry.

Symmetries in Nuclei, Tokyo, 2008

The IBM hamiltonian

- Rotational invariant hamiltonian with up to N-body interactions (usually up to 2):
- For what choice of single-boson energies and boson-boson interactions is the IBM hamiltonian solvable?
- This problem is equivalent to the enumeration of all algebras G satisfying

Symmetries in Nuclei, Tokyo, 2008

Dynamical symmetries of the IBM

- U(6) has the following subalgebras:
- Three solvable limits are found:

Symmetries in Nuclei, Tokyo, 2008

Dynamical symmetries of the IBM

- The general IBM hamiltonian is
- An entirely equivalent form of HIBM is
- The coefficients and are certain combinations of the coefficients and .

Symmetries in Nuclei, Tokyo, 2008

The solvable IBM hamiltonians

- Excitation spectrum of HIBM is determined by
- If certain coefficients are zero, HIBM can be written as a sum of commuting operators:

Symmetries in Nuclei, Tokyo, 2008

The U(5) vibrational limit

- Anharmonic vibration spectrum associated with the quadrupole oscillations of a spherical surface.
- Conserved quantum numbers: nd, , L.

A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253

D. Brink et al., Phys. Lett. 19 (1965) 413

Symmetries in Nuclei, Tokyo, 2008

The SU(3) rotational limit

- Rotation-vibration spectrum of quadrupole oscillations of a spheroidal surface.
- Conserved quantum numbers: (,), L.

A. Arima & F. Iachello,

Ann. Phys. (NY) 111 (1978) 201

A. Bohr & B.R. Mottelson, Dan. Vid.

Selsk. Mat.-Fys. Medd. 27 (1953) No 16

Symmetries in Nuclei, Tokyo, 2008

The SO(6) -unstable limit

- Rotation-vibration spectrum of quadrupole oscillations of a -unstable spheroidal surface.
- Conserved quantum numbers: , , L.

A. Arima & F. Iachello, Ann. Phys. (NY) 123 (1979) 468

L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788

Symmetries in Nuclei, Tokyo, 2008

Synopsis of IBM symmetries

- Three standard solutions: U(5), SU(3), SO(6).
- Solution for the entire U(5) SO(6) transition via the SU(1,1) Richardson-Gaudin algebra.
- Hidden symmetries because of parameter transformations: SU±(3) and SO±(6).
- Partial dynamical symmetries.

Symmetries in Nuclei, Tokyo, 2008

Applications of IBM

Symmetries in Nuclei, Tokyo, 2008

The ratio R42

Symmetries in Nuclei, Tokyo, 2008

Partial dynamical symmetries

- Solvable models with dynamical symmetry:
- Dynamical symmetries can be partial
- Type 1: All labels , ’… remain good quantum numbers for some eigenstates.
- Type 2: Some of the labels , ’… remain good quantum numbers for all eigenstates.

Y. Alhassid and A. Leviatan, J. Phys. A 25 (1995) L1285

A. Leviatan et al., Phys. Lett. B 172 (1986) 144

Symmetries in Nuclei, Tokyo, 2008

Example of type-2 PDS

P. Van Isacker, Phys. Rev. Lett. 83 (1999) 4269

Symmetries in Nuclei, Tokyo, 2008

Modes of nuclear vibration

- Nucleus is considered as a droplet of nuclear matter with an equilibrium shape. Vibrations are modes of excitation around that shape.
- Character of vibrations depends on symmetry of equilibrium shape. Two important cases in nuclei:
- Spherical equilibrium shape
- Spheroidal equilibrium shape

Symmetries in Nuclei, Tokyo, 2008

Vibration about a spherical shape

- Vibrations are characterized by a multipole quantum number in surface parametrization:
- =0: compression (high energy)
- =1: translation (not an intrinsic excitation)
- =2: quadrupole vibration

Symmetries in Nuclei, Tokyo, 2008

Vibration about a spheroidal shape

- The vibration of a shape with axial symmetry is characterized by a.
- Quadrupolar oscillations:
- =0: along the axis of symmetry ()
- =1: spurious rotation
- =2: perpendicular to axis of symmetry ()

Symmetries in Nuclei, Tokyo, 2008

Classical limit of IBM

- For large boson number N, a coherent (or intrinsic) state is an approximate eigenstate,
- The real parameters are related to the three Euler angles and shape variables and .
- Any IBM hamiltonian yields energy surface:

J.N. Ginocchio & M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744.

A.E.L. Dieperink et al., Phys. Rev. Lett. 44 (1980) 1747.

A. Bohr & B.R. Mottelson, Phys. Scripta 22 (1980) 468.

Symmetries in Nuclei, Tokyo, 2008

Classical limit of IBM

- For large boson number N the minimum of V()=N;H approaches the exact ground-state energy:

Symmetries in Nuclei, Tokyo, 2008

Geometry of IBM

- A simplified, much used IBM hamiltonian:
- HCQF can acquire the three IBM symmetries.
- HCQF has the following classical limit:

Symmetries in Nuclei, Tokyo, 2008

Phase diagram of IBM

J. Jolie et al. , Phys. Rev. Lett. 87 (2001) 162501.

Symmetries in Nuclei, Tokyo, 2008

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.

Symmetries in Nuclei, Tokyo, 2008

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.

D. Bohle et al., Phys. Lett. B 137 (1984) 27

Symmetries in Nuclei, Tokyo, 2008

SO(6) (mixed-)symmetry in 94Mo

- Analytic calculation in SO(6) limit of IBM-2.
- Complex spectrum with mixed-symmetry states.
- E2 and M1 transition rates reproduced with two effective boson charges e and e.

N. Pietralla et al., Phys. Rev. Lett. 83 (1999) 1303

Symmetries in Nuclei, Tokyo, 2008

Bosons + fermions

- Odd-mass nuclei are fermions.
- Describe an odd-mass nucleus as N bosons + 1 fermion mutually interacting. Hamiltonian:
- Algebra:
- Many-body problem is solved analytically for certain energies and interactions .

Symmetries in Nuclei, Tokyo, 2008

Example: 195Pt117

Symmetries in Nuclei, Tokyo, 2008

Example: 195Pt117 (new data)

Symmetries in Nuclei, Tokyo, 2008

Nuclear supersymmetry

- Up to now: separate description of even-even and odd-mass nuclei with the algebra
- Simultaneous description of even-even and odd-mass nuclei with the superalgebra

F. Iachello, Phys. Rev. Lett. 44 (1980) 777

Symmetries in Nuclei, Tokyo, 2008

U(6/12) supermultiplet

Symmetries in Nuclei, Tokyo, 2008

Example: 194Pt116 &195Pt117

Symmetries in Nuclei, Tokyo, 2008

Quartet supersymmetry

- A simultaneous description of even-even, even-odd, odd-even and odd-odd nuclei (quartets).
- Example of 194Pt, 195Pt, 195Au & 196Au:

P. Van Isacker et al., Phys. Rev. Lett. 54 (1985) 653

Symmetries in Nuclei, Tokyo, 2008

Quartet supersymmetry

A. Metz et al., Phys. Rev. Lett. 83 (1999) 1542

Symmetries in Nuclei, Tokyo, 2008

Example of 196Au

Symmetries in Nuclei, Tokyo, 2008

Isospin invariant boson models

- Several versions of IBM depending on the fermion pairs that correspond to the bosons:
- IBM-1: single type of pair.
- IBM-2: T=1 nn (MT=-1) and pp (MT=+1) pairs.
- IBM-3: full isospin T=1 triplet of nn (MT=-1), np (MT=0) and pp (MT=+1) pairs.
- IBM-4: full isospin T=1 triplet and T=0 np pair (with S=1).

- Schematic IBM-k has only s (L=0) bosons, full IBM-k has s (L=0) and d (L=2) bosons.

Symmetries in Nuclei, Tokyo, 2008

Symmetry rules

- Symmetry is a universal concept relevant in mathematics, physics, chemistry, biology, art…
- In science in particular it enables
- To describe invariance properties in a rigorous manner.
- To predict properties before any detailed calculation.
- To simplify the solution of many problems and to clarify their results.
- To classify physical systems and establish analogies.
- To unify knowledge.

Symmetries in Nuclei, Tokyo, 2008

Symmetry in physics

- Fundamental symmetries: Laws of physics are invariant under a translation in time or space, a rotation, a change of inertial frame and under a CPT transformation.
- Noether’s theorem (1918): Every (continuous) global symmetry gives rise to a conservation law.
- Approximate symmetries: Countless physical systems obey approximate invariances which enable an understanding of their spectral properties.

Symmetries in Nuclei, Tokyo, 2008

Symmetry in nuclear physics

- The integrability of quantum many-body (bosons and/or fermions) systems 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.

Symmetries in Nuclei, Tokyo, 2008

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