Symmetries in Nuclei. Piet Van Isacker Grand Accélérateur National d Ions Lourds, Caen, France

Transcription

1 Symmetries in Nuclei Piet Van Isacker Grand Accélérateur National d Ions Lourds, Caen, France

2 Symmetries in Nuclei Symmetry, mathematics and physics Examples of symmetries in quantum mechanics Symmetries of the nuclear shell model Symmetries of the interacting boson model

3 What is symmetry? Oxford Dictionary of English: (beauty resulting from the) right correspondence of parts; quality of harmony or balance (in size, design) between parts Examples: disposition of a French garden; harmony of themes in a symphony.

4 Etymology Ancient Greek roots: sun means with, together metron means measure For the ancient Greeks symmetry was closely related to harmony, beauty and unity. Aristotle: The chief forms of beauty are orderly arrangement [taxis], proportion [symmetria] and definiteness [horismenon].

5 Origin For the ancient Greeks symmetry implied the notion of proportion. 17 th century: Symmetry starts to imply also a relation of equality of elements that are opposed (eg. between left and right). 19 th century: Definition of symmetry via the notion of invariance under transformations such as translations, rotations, reflections. Introduction of the notion of a group of transformations.

6 Group theory Group theory is the mathematical theory of symmetry. Group theory was invented (discovered?) by Evariste Galois in Group theory became one of the pillars of mathematics (cfr. Klein s Erlangen programme). Group theory has become of central importance in physics, especially in quantum physics.

7 The birth of group theory Are all equations solvable algebraically? Example of quadratic equation: ax 2 + bx + c = 0 x 1,2 = b ± b2 4ac 2a Babylonians (from 2000 BC) knew how to solve quadratic equations in words but avoided cases with negative or no solutions. Indian mathematicians (eg. Brahmagupta ) did interpret negative solutions as `depths. Full solution was given in 12 th century by the Spanish Jewish mathematician Abraham bar Hiyya Ha-nasi.

8 The birth of group theory No solution of higher equations until dal Ferro, Tartaglia, Cardano and Ferrari solve the cubic and quartic equations in the 16 th century. ax 3 + bx 2 + cx + d = 0 & ax 4 + bx 3 + cx 2 + dx + e = 0 Europe s finest mathematicians (eg. Euler, Lagrange, Gauss, Cauchy) attack the quintic equation but no solution is found. 1799: proof of non-existence of an algebraic solution of the quintic equation by Ruffini?

9 The birth of group theory 1824: Niels Abel shows that general quintic and higher-order equations have no algebraic solution. 1831: Evariste Galois answers the solvability question: whether a given equation of degree n is algebraically solvable depends on the symmetry profile of its roots which can be defined in terms of a subgroup of the group of permutations S n.

10 The insolvability of the quintic

11 The axioms of group theory A set G of elements (transformations) with an operation which satisfies: 1. Closure. If g 1 and g 2 belong to G, then g 1 g 2 also belongs to G. 2. Associativity. We always have (g 1 g 2 ) g 3 =g 1 (g 2 g 3 ). 3. Existence of identity element. An element 1 exists such that g 1=1 g=g for all elements g of G. 4. Existence of inverse element. For each element g of G, an inverse element g -1 exists such that g g -1 =g -1 g=1. This simple set of axioms leads to an amazingly rich mathematical structure.

12 Example: equilateral triangle Symmetry transformations are - Identity - Rotation over 2π/3 and 4π/3 around e z - Reflection with respect to planes (u 1,e z ), (u 2,e z ), (u 3,e z ) Symmetry group: C 3h.

13 Groups and algebras 1873: Sophus Lie introduces the notion of the algebra of a continuous group with the aim of devising a theory of the solvability of differential equations. 1887: Wilhelm Killing classifies all Lie algebras. 1894: Elie Cartan re-derives Killing s classification and notices two exceptional Lie algebras to be equivalent.

14 Lie groups A Lie group contains an infinite number of elements characterized by a set of continuous variables. Additional conditions: Connection to the identity element. Analytic multiplication function. Example: rotations in 2 dimensions, SO(2). ˆ g α ()= cosα sinα sinα cosα

15 Lie algebras Idea: to obtain properties of the infinite number of elements g of a Lie group in terms of those of a finite number of elements g i (called generators) of a Lie algebra.

16 Lie algebras All properties of a Lie algebra follow from the commutation relations between its generators: [ g ˆ i, g ˆ j ] g ˆ i g ˆ j g ˆ j g ˆ i = r k=1 c ijk ˆ g k Generators satisfy the Jacobi identity: [ ] [ g ˆ i,[ g ˆ j, g ˆ ]]+ [ g ˆ k j, g ˆ k, g ˆ i ]+ [ g ˆ k,[ g ˆ i, g ˆ ]]= 0 j Definition of the metric tensor or Killing form: g ij = r k,l=1 c l k ik c jl

17 Classification of Lie groups Symmetry groups over R, C and H (quaternions) preserving a specified metric: { x R[]: n xx =1, det =1} SO n () { x C[]: n xx =1, det =1} SU n { x H[]: n xx =1} Sp n () () The five exceptional groups G 2, F 4, E 6, E 7 and E 8 are similar constructs over the normed division algebra of the octonions, O.

18 Rotations in 2 dimensions, SO(2) Matrix representation of finite elements: cosα sinα g ˆ ()= α sinα cosα Infinitesimal element and generator: limg ˆ () α α 0 1 e ˆ + αˆ g 1 α Exponentiation leads back to finite elements: g ˆ ()= α lim e ˆ + α n g n n ˆ 1 = exp( e ˆ + αˆ g 1 )= exp 1 α = g ˆ () α α 1

19 Rotations in 3 dimensions, SO(3) Matrix representation of finite elements: cosα 3 sinα 3 0 cosα 2 0 sinα sinα 3 cosα cosα 1 sinα sinα 2 0 cosα 2 0 sinα 1 cosα 1 Infinitesimal elements and associated generators: g ˆ 1 = 0 0 1, g ˆ 2 = 0 0 0, g ˆ 3 =

20 Rotations in 3 dimensions, SO(3) Structure constants from matrix multiplication: 3 [ g ˆ k, g ˆ l ]= c m kl g ˆ m with c m kl = ε klm m=1 ( Levi Civita) Exponentiation leads back to finite elements: 3 g ˆ ( α 1,α 2,α 3 )= exp e ˆ + α k g ˆ k k=1 Relation with angular momentum operators: l ˆ k = iˆ g k l ˆ k,ˆ 3 [ l l ]= i ε klmˆ l m m=1

21 Casimir operators Definition: The Casimir operators C n [G] of a Lie algebra G commute with all generators of G. The quadratic Casimir operator (n=2): r C ˆ 2 [ G]= g ij g ˆ i g ˆ j with g ij g jk = δ ik. Ex: C ˆ 2 SO 3 i, j =1 r i, j =1 [ ()]= ˆ The number of independent Casimir operators (rank) equals the number of quantum numbers needed to characterize any (irreducible) representation of G. 3 2 l k k =1

22 Symmetry in physics Geometrical symmetries Example: C 3h symmetry (X 3 molecules, 12 C nucleus). Permutational symmetries Example: S A symmetry of an A-body hamiltonian: ˆ H = A p k 2 + V ˆ 2 r k r k 2m k k=1 A 1 k< k ( )

23 Symmetry in physics Space-time (kinematical) symmetries Rotational invariance, SO(3): x k x k = Lorentz invariance, SO(3,1): x μ x μ = Parity: x k x k = x k Time reversal: t t = t Euclidian invariance, E(3): x k x k = l Poincaré invariance, E(3,1): x μ x μ = Dilatation symmetry: x k x k = D k x k l ν A kl x l A kl x l A μν x ν A μν x ν ν + a k + a μ

24 Symmetry in physics Quantum-mechanical symmetries, U(n) n ψ k ψ k = B kl ψ l inv. ψ 2 l l l=1 Internal symmetries. Example: p n, isospin SU(2) u d s, flavour SU(3) Gauge symmetries. Example: Maxwell equations, U(1) Dynamical symmetries. Example: Coulomb problem, SO(4)

25 Symmetry in quantum mechanics Assume a hamiltonian H which commutes with operators g i that form a Lie algebra G: ˆ g i G : [ H ˆ, g ˆ i ]= 0 H has symmetry G or is invariant under G. Lie algebra: a set of (infinitesimal) operators that closes under commutation.

26 Consequences of symmetry Degeneracy structure: If γ is an eigenstate of H with energy E, so is g i γ : H ˆ γ = E γ H ˆ g ˆ i γ = g ˆ i H ˆ γ = Eˆ g i γ Degeneracy structure and labels of eigenstates of H are determined by algebra G: ˆ H Γγ = E Γ ()Γγ ; Γ g ˆ i Γγ = a Casimir operators of G commute with all g i : ˆ H = m μ m ˆ C m [ G] γ γ γ ()Γ i γ

27 Symmetry rules Since its introduction by Galois in 1831, group theory has become central to the field of mathematics. Group theory remains an active field of research, (eg. the recent classification of all groups leading to the Monster.) Symmetry has acquired a central role in all domains of physics.

28 Symmetries in Nuclei Symmetry, mathematics and physics Examples of symmetries in quantum mechanics Symmetries of the nuclear shell model Symmetries of the interacting boson model

29 Examples of symmetries in quantum mechanics Symmetry in quantum mechanics The hydrogen atom The harmonic oscillator Isospin symmetry in nuclei Dynamical symmetry

30 Symmetry in quantum mechanics Assume a hamiltonian H which commutes with operators g i that form a Lie algebra G: ˆ g i G : [ H ˆ, g ˆ i ]= 0 H has symmetry G or is invariant under G. Lie algebra: a set of (infinitesimal) operators that closes under commutation.

31 Consequences of symmetry Degeneracy structure: If γ is an eigenstate of H with energy E, so is g i γ : H ˆ γ = E γ H ˆ g ˆ i γ = g ˆ ˆ i H γ = Eˆ g i γ Degeneracy structure and labels of eigenstates of H are determined by algebra G: H ˆ Γγ = E Γ ()Γγ ; Γ g ˆ i Γγ = a Casimir operators of G commute with all g i : ˆ H = m μ m ˆ C m [ G] γ γ γ ()Γ i γ

32 The hydrogen atom The hamiltonian of the hydrogen atom is H ˆ = p2 2M α r Standard wave quantum mechanics gives H ˆ Ψ nlm ( r,θ,ϕ)= Mα 2 ( r,θ,ϕ) 2h 2 n Ψ 2 nlm with n =1,2,K; l = 0,1,K,n 1; m = l,k,+l Degeneracy in m originates from rotational symmetry. What is the origin of l-degeneracy?

33 Degeneracies of the H atom

34 Classical Kepler problem Conserved quantities: Energy (a=length semi-major axis): E = α 2a Angular momentum (ε=eccentricity): L = r p, L 2 = Mα a( 1 ε 2 ) Runge-Lenz vector: R = p L M α r r, R2 = 2E M L2 + α 2 Newtonian potential gives rise to closed orbits with constant direction of major axis.

35 Classical Kepler problem

36 Quantization of operators From p -ih : H ˆ = h2 2M 2 + α r ˆ L = ih r ( ) R ˆ = h2 2M [ ( r ) ( r ) ] α r r Some useful commutators & relations: [,r k ]= kr k 2 r, [ 2,r]= 2, 2,r k R ˆ 2 = 2 H ˆ M ˆ L 2 + h 2 ( )+ α 2 [ ] [ ]= kr k 2 ( k +1)+ 2r

37 Conservation of angular momentum The angular momentum operators L commute with the hydrogen hamiltonian: [ H ˆ, L ˆ ] 2 + 2κ r 1,r [ ] ( κ = Mα /h 2 ) = [ 2,r] +2κ r [ r 1, ]= L operators generate SO(3) algebra: [ L ˆ, L ˆ j k ]= ihε ˆ jkl L l, j,k,l = x, y,z H has SO(3) symmetry m-degeneracy.

38 Conserved Runge-Lenz vector The Runge-Lenz vector R commutes with H: [ H ˆ, R ˆ ] 2 + 2κ r 1, ( r ) ( r ) +2κ r 1 r [ ] = [ 2,2κ r 1 r]+ [ 2κ r 1, ( r ) ( r ) ]= 0 R does not commute with the kinetic and potential parts of H separately: [ 2M 2, R ˆ ]= α [ r 1, R ˆ ]= h2 α 1 M r r r 3 h2 Hydrogen atom has a dynamical symmetry. ( 1+ r )

39 SO(4) symmetry L and R (almost) close under commutation: [ L ˆ j, L ˆ k ]= ihε ˆ jkl L l, [ L ˆ, R ˆ j k ]= ihε ˆ jkl R l, [ R ˆ, R ˆ 2H j k ]= ihε ˆ jkl M j,k,l = x, y,z j,k,l = x, y,z ˆ L l, j,k,l = x, y,z H is time-independent and commutes with L and R choose a subspace with given E. L and R' (-M/2H) 1/2 R form an algebra SO(4) corresponding to rotations in four dimensions.

40 Energy spectrum of the H atom Isomorphism of SO(4) and SO(3) SO(3): ˆ F ± = 1 2 ( L ˆ ± R ˆ ) [ F ˆ ± j, F ˆ ± k ]= ihε ˆ jkl F ± l, Since L and R' are orthogonal: F ˆ + F ˆ + = F ˆ F ˆ = j( j+1)h 2 [ F ˆ + j, F ˆ k ]= 0 The quadratic Casimir operator of SO(4) and H are related: C ˆ 2 SO() 4 F + F ˆ + + F ˆ ˆ L ˆ 2 M R ˆ 2 = Mα 2 [ ]= ˆ C ˆ 2 SO 4 () [ ] = 2 j j+1 F = 1 2 2H Mα ( 2 )h 2 E = 22j +1 4H 1 2 h2 ( ) 2 h 2, j = 0, 1 2,1, 3 2,K W. Pauli, Z. Phys. 36 (1926) 336

41 The (3D) harmonic oscillator The hamiltonian of the harmonic oscillator is H ˆ = p2 2M Mω 2 r 2 Standard wave quantum mechanics gives ˆ H Ψ nlm ( r,θ,ϕ )= 2n + l hωψ nlm ( r,θ,ϕ) with n = 0,1,K ; l = 0,1,K ; m = l,k,+l Degeneracy in m originates from rotational symmetry. Additional degeneracy for all (n,l) combinations with 2n+l=N. What is the origin of this degeneracy?

42 Degeneracies of the 3D HO

43 Raising and lowering operators Introduce the raising and lowering operators η x = 1 2 ξ x = 1 2 The 3D HO hamiltonian becomes ˆ H = p2 x, η x y = 1 2 x +, ξ x y = 1 2 with x = x /l, y = y /l, z = z /l; l = 2M Mω 2 r 2 = η i ξ i + 1 hω 2 i=x,y,z y, η y z = 1 2 y +, ξ y z = 1 2 h Mω z z z + z M. Moshinsky, The Harmonic Oscillator in Modern Physics

44 U(3) symmetry of the 3D HO The raising and lowering operators satisfy ξ i = η i +, [ ξ i,ξ j ]= 0, [ η i,η j ]= 0, [ ξ i,η j ]= δ ij The bilinear combinations u ij commute with H: { } u ˆ ij η i ξ j [ u ˆ ij, H ˆ ]= 0, i, j x, y,z The nine operators u ij generate the algebra U(3): [ u ˆ ij, u ˆ kl ]= u ˆ il δ jk u ˆ kj δ il The symmetry of the harmonic oscillator in 3 dimensions is U(3).

45 The U(3)=U(1) SU(3) algebra The generators u ij of U(3) can be written as H η x ξ x +η y ξ y +η z ξ z = ˆ hω 3 2 L ˆ z = ih x y y = ihη x x ξ y η y ξ x Q ˆ 0 = h( 2η z ξ z η x ξ x η y ξ ) y ( ) + cyclic Q ˆ m1 = h 3 ( 2 ±η zξ x ± η x ξ z iη y ξ z iη z ξ ) y Q ˆ m2 = h 3 ( 2 η xξ x η y ξ y miη x ξ y miη y ξ ) x

46 Many particles in the 3D HO Define operators for each particle k=1,2,,a: η k x = 1 x k, η k 2 x y = 1 y k, η k k 2 y z = 1 z k k 2 z k ξ k x = 1 x k +, ξ k 2 y = 1 y k +, ξ k 2 z = 1 z k + 2 x k The total U(3) algebra is generated by A k η ik ξ j, i, j x,y,z k =1 { } y k z k

47 Many particles in the 3D HO Many-body hamiltonian with U(3) symmetry: ˆ H = H ˆ, A η xk ξ k x +η yk ξ k k y +η zk ξ z + V ˆ k,l k =1 A k =1 k η ik ξ j, A k< l =1 i, j { x, y,z} ( ) This property is valid if the interaction equals C ˆ [ 2 SU() 3 ]= 1 2 L L + 1 A 6 Q Q = 1 2 L k k<l=1 () L()+ l 1 6 Q() k Q() l J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562

48 Dynamical symmetry Two algebras G 1 G 2 and a hamiltonian ˆ H = μ ˆ m C m [ G 1 ] + ν ˆ n C n G 2 m n [ ] H has symmetry G 2 but not G 1! Eigenstates are independent of parameters μ m and ν n in H. Dynamical symmetry breaking splits but does not admix eigenstates.

49 Isospin symmetry in nuclei Empirical observations: About equal masses of n(eutron) and p(roton). n and p have spin 1/2. Equal (to about 1%) nn, np, pp strong forces. This suggests an isospin SU(2) symmetry of the nuclear hamiltonian: n: t = 1, m 2 t =+ 1 ; p: t = 1, m 2 2 t = 1 2 t ˆ + n = 0, t ˆ + p = n, t ˆ n = p, t ˆ p = 0, ˆ t z n = 1 2 n, ˆ t z p = 1 2 p W. Heisenberg, Z. Phys. 77 (1932) 1 E.P. Wigner, Phys. Rev. 51 (1937) 106

50 Isospin SU(2) symmetry Isospin operators form an SU(2) algebra: [ t ˆ z, t ˆ ± ]=±ˆ t ±, [ t ˆ +,ˆ t ]= 2ˆ Assume the nuclear hamiltonian satisfies [ H ˆ, T ˆ nucl ν ]= 0, T ˆ ν = A k=1 ˆ t z t ν () k H nucl has SU(2) symmetry with degenerate states belonging to isobaric multiplets: ηtm T, M T = T, T +1,K,+T

51 Isospin symmetry breaking: A=49 Empirical evidence from isobaric multiplets. Example: T=1/2 doublet of A=49 nuclei. O Leary et al., Phys. Rev. Lett. 79 (1997) 4349

52 Isospin symmetry breaking: A=51

53 Isospin SU(2) dynamical symmetry Coulomb interaction can be approximated as H ˆ Coul κ 0 + κ ˆ 1 T z + κ ˆ 2 T 2 z H nucl +H Coul has SU(2) dynamical symmetry and SO(2) symmetry. M T -degeneracy is lifted according to Summary of labelling: [ H ˆ, T ˆ Coul z ]= 0, [ H ˆ Coul, T ˆ ± ] 0 H ˆ Coul ηtm T = ( 2 κ 0 + κ 1 M T + κ 2 M T )ηtm T SU 2 T ( ) SO( 2) M T

54 Isobaric multiplet mass equation Isobaric multiplet mass equation: ( )= κ η,t 2 E ηtm T ( ) +κ 1 M T +κ 2 M T Example: T=3/2 multiplet for A=13 nuclei. E.P. Wigner, Proc. Welch Found. Conf. (1958) 88

55 Isospin selection rules Internal E1 transition operator is isovector: ˆ T μ E1 = A e k r μ k=1 k ()= e 2 k=1 A A r μ () k + 2 t ˆ z () k r μ () k k= CM motion isovector Selection rule for N=Z (M T =0) nuclei: No E1 transitions are allowed between states with the same isospin. L.E.H. Trainor, Phys. Rev. 85 (1952) 962 L.A. Radicati, Phys. Rev. 87 (1952) 521

56 E1 transitions and isospin mixing B(E1;5-4 + ) in 64 Ge from: lifetime of 5 - level; δ(e1/m2) mixing ratio of transition; relative intensities of transitions from 5 -. Estimate of minimum isospin mixing: PT= ( 1, 5 ) PT= ( 1, 4 + ) 2.5% E.Farnea et al., Phys. Lett. B 551 (2003) 56

57 Dynamical algebra Take a generic many-body hamiltonian: H ˆ = ε i c + i c i + 1 υ ijkl c + i c + j c l c k +L 4 i ijkl Rewrite H as (bosons: q=0; fermions: q=1) ˆ H = ε i δ il () q 1 υ ijlk 4 u ˆ il + () q 1 υ ijlk u ˆ ik u ˆ jl +L il j 4 ijkl Operators u ij generate the dynamical algebra U(n) for bosons and for fermions (q=0,1): u ˆ ij c + i c j [ u ˆ ij, u ˆ kl ]= u ˆ il δ jk u ˆ kj δ il

58 Dynamical symmetry (DS) With each chain of nested algebras U n is associated a particular class of many-body hamiltonian ˆ H = ()= G dyn = G 1 G 2 L G sym μ ˆ m C m [ G 1 ] + ν ˆ n C n [ G 2 ] +L m n Since H is a sum of commuting operators m,n,a,b : [ C ˆ m [ G a ], C ˆ n [ G b ]]= 0 it can be solved analytically!

59 DS in nuclear physics

60 Symmetries in Nuclei Symmetry, mathematics and physics Examples of symmetries in quantum mechanics Symmetries of the nuclear shell model Symmetries of the interacting boson model

61 Symmetries of the nuclear shell model The nuclear shell model Racah s pairing model and seniority Wigner s supermultiplet model Elliott s SU(3) model and extensions

62 The nuclear many-body problem The nucleus is a system of A interacting nucleons described by the hamiltonian ˆ H = A 2 p k1 + V ˆ 2 ξ k1,ξ k2 + V ˆ 3 ξ k1,ξ k2,ξ k3 +L k 1 =1 2m k1 A 1 k 1 <k 2 ( ) [ ξ ki { r ki, s ˆ ki,ˆ t }, p ki ki = ih ] ki 1 k 1 <k 2 <k 3 ( ) Separation in mean field + residual interaction: ˆ H = A k 1 =1 2 p k1 + V ˆ ( ξ ) k1 2m k mean field A A A + V ˆ ( 2 ξ k1,ξ ) k2 +L V ˆ ( ξ ) k1 1 k 1 <k 2 k 1 = residual interaction

63 The nuclear shell model Hamiltonian with one-body term (mean field) and two-body interactions: ˆ H SM = A U ˆ ξ k1 + W ˆ 2 ξ k1,ξ k2 k 1 =1 ( ) A 1 k 1 <k 2 ( ) Entirely equivalent form of the same hamiltonian in second quantization: H ˆ SM = ε i a + i a i + 1 υ ijlk a + i a + j a k a l 4 i ijkl ε, υ: single-particle energies & interactions ijkl: single-particle quantum numbers

64 Boson and fermion statistics Fermions have half-integer spin and obey Fermi- Dirac statistics: { + a i,a j } a i a + j + a + j a i = δ ij, { a i,a j }={ a + + i,a j }= 0 Bosons have integer spin and obey Bose-Einstein statistics: [ + b i,b j ] b i b + j b + j b i = δ ij, [ b i,b j ]= [ b + + i,b j ]= 0 Matter is carried by fermions. Interactions are carried by bosons. Composite matter particles can be fermions or bosons.

65 Boson/fermion systems Denote particle (boson or fermion) creation and annihilation operators by c + and c. Introduce (q=0 for bosons; q=1 for fermions): [ u ˆ, v ˆ } u ˆ v ˆ ( ) q v ˆ u ˆ q Boson/fermion statistics is summarized by [ c i,c } = δ, [ ci+ j+ ij,c } = [ c,c j} = 0 j+ i q q q Many-particle state: () + c n i i o, c i o = 0 n i! i

66 Bosons and fermions

67 Algebraic structure Introduce bilinear operators u ˆ ij = c + i c j [ u ˆ ij, u ˆ kl ]= u ˆ il δ jk u ˆ kj δ il U Ω Rewrite many-body hamiltonian H ˆ = E 0 + ε i c + i c i i = E 0 + ε i δ il il ijkl () q 1 4 υ ijlk c i + c j + c k c l υ ijlk j ˆ u il ( ) + () q 1 4 ijkl υ ijlk ˆ u ik ˆ u jl

68 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: ˆ H = A k=1 A 2 p k 2m mω 2 r 2 k ζ ˆ ls l k s ˆ k ζ ˆ ll + W ˆ 2 ξ k1,ξ k2 1 k 1 <k 2 ( ) l k 2

69 Racah s SU(2) pairing model Assume pairing interaction in a single-j shell: j 2 JM J Spectrum 210 Pb: V ˆ pairing j 2 JM J = 1 2 j +1 2 ( )g 0, J = 0 0, J 0 G. Racah, Phys. Rev. 63 (1943) 367

70 Pairing SU(2) dynamical symmetry The pairing hamiltonian, H ˆ = g ˆ 0 S + S ˆ, ˆ S + = 1 2, S ˆ = ˆ a + + jm a jm has a quasi-spin SU(2) algebraic structure: [ S ˆ +, S ˆ ]= 1 2 n ˆ 2 j 1 2 ( ) 2S ˆ z, H has SU(2) SO(2) dynamical symmetry: g ˆ 0 S + S ˆ = g 0 Eigensolutions of pairing hamiltonian: m S ˆ 2 S ˆ 2 z + ˆ ( S ) z ( S ) + + [ S ˆ, S ˆ z ± ]=±ˆ g ˆ 0 S + S ˆ SM S = g ( 0 SS+1 ( ) M ( S M S 1) )SM S S ± A. Kerman, Ann. Phys. (NY) 12 (1961) 300

71 Interpretation of pairing solution Quasi-spin labels S and M S are related to nucleon number n and seniority υ: S = 1 4 ( 2 j υ +1), M S = 1 4 ( 2n 2 j 1) Energy eigenvalues in terms of n, j and υ: j n υjm J g 0 S ˆ + S ˆ j n 1 υjm J = g n υ 0 4 Eigenstates have an S-pair character: j n υjm J ( ) n υ ˆ S + ( )/2 j υ υjm J ( ) ( ) 2 j n +υ + 3 Seniority υ is the number of nucleons not in S pairs (pairs coupled to J=0).

72 Pairing between identical nucleons Analytic solution of the pairing hamiltonian based on SU(2) symmetry. E.g. energies: j n υj n 1 k<l ˆ V pairing ( k,l) j n 1 υj = g n υ 0 4 ( )2 j n υ + 3 ( ) Seniority υ (number of nucleons not in pairs coupled to J=0) is a good quantum number. Correlated ground-state solution (cf. BCS). G. Racah, Phys. Rev. 63 (1943) 367 G. Racah, L. Farkas Memorial Volume (1952) p. 294 B.H. Flowers, Proc. Roy. Soc. (London) A 212 (1952) 248 A.K. Kerman, Ann. Phys. (NY) 12 (1961) 300

73 Nuclear superfluidity Ground states of pairing hamiltonian have the following correlated character: Even-even nucleus (υ=0): Odd-mass nucleus (υ=1): Nuclear superfluidity leads to Constant energy of first 2 + in even-even nuclei. ˆ ( S ) n /2 + o, ˆ + a ˆ mb S + ( ) n /2 o S + = a + + m a m Odd-even staggering in masses. Smooth variation of two-nucleon separation energies with nucleon number. Two-particle (2n or 2p) transfer enhancement. m

74 Two-nucleon separation energies Two-nucleon separation energies S 2n : (a) Shell splitting dominates over interaction. (b) Interaction dominates over shell splitting. (c) S 2n in tin isotopes.

75 Pairing gap in semi-magic nuclei Even-even nuclei: Ground state: υ=0. First-excited state: υ=2. Pairing produces constant excitation energy: E ()= + x j ( )g 0 Example of Sn isotopes:

76 Algebraic definition of seniority For a system of n identical bosons with spin j ( ) SO( 2 j +1) L SO( 3) U2j +1 [] n υ J For a system of n identical fermions with spin j ( ) Sp( 2 j +1) L SO( 3) U2j +1 [] 1 n υ J Alternative definition with quasi-spin algebras.

77 Conservation of seniority Seniority υ is the number of particles not in pairs coupled to J=0 (Racah). Conditions for the conservation of seniority by a given (two-body) interaction V can be derived from the analysis of a 3-particle system. Any interaction between identical fermions with spin j conserves seniority if j 7/2. Any interaction between identical bosons with spin j conserves seniority if j 2. G. Racah, Phys. Rev. 63 (1943) 367 I. Talmi, Simple Models of Complex Nuclei

78 Conservation of seniority Necessary and sufficient conditions for a two-body interaction ν λ to conserve seniority: 2λ +1 a λ v ji = 0, I = 2,4,K,2 j λ λ ν λ j 2 ;λ ˆ V j 2 ;λ a λ ji = δ λi + 2 ( 2λ +1)2I +1, 4 ( 2λ +1)2I ( +1) j j I ( 2 j +1)2 j +1+ 2σ ( ) j j λ For fermions σ = -1; for bosons σ = +1. ( ) I. Talmi, Simple Models of Complex Nuclei

79 Conservation of seniority Bosons: j = 3:11v 2 18v 4 + 7v 6 = 0, j = 4:65v 2 30v 4 91v v 8 = 0, j = 5 : 3230v v v v 10 = 0, Fermions: j = 9/2:65v 2 315v v 6 153v 8 = 0, j =11/2 :1020v v 4 637v v v 10 = 0, j =13/2 :1615v v v v v v 12 = 0,

80 Is seniority conserved in nuclei? The interaction between nucleons is short range. A δ interaction is therefore a reasonable approximation to the nucleon two-body force. A pairing interaction is a further approximation. Both δ and pairing interaction between identical nucleons conserve seniority. In semi-magic nuclei seniority is conserved to a good approximation.

81 Partial conservation of seniority Question: Can we construct interactions for which some but not all of the eigenstates have good seniority? A non-trivial solution occurs for four identical fermions with j=9/2 and J=4 and J=6. These states are solvable for any interaction in the j=9/2 shell. They have a wave function which is independent of the interactions ν J. This finding has relevance for the existence of seniority isomers in nuclei. P. Van Isacker & S. Heinze, Phys. Rev. Lett. 100 (2008) L. Zamick & P. Van Isacker, Phys. Rev. C 78 (2008)

82 Energy matrix for (9/2) 4 J=4 a V ˆ a = 3 5 v v v v v, 8 a V ˆ 14Δ b = , a V ˆ c = 2 170Δ , Δ= 65ν ν 4 403ν ν 8 b V ˆ b = v v v v 8, ( ) b V ˆ c = ν 9ν 13ν + 9ν c V ˆ c = v v v v. 8

83 Energy matrix for (9/2) 4 J=6 a V ˆ a = 3 5 v v v v v, 8 a V ˆ b = 5Δ , a V ˆ c = Δ , Δ= 65ν ν 4 403ν ν 8 b V ˆ b = v v v v 8, ( ) b V ˆ c = ν 9ν 13ν + 9ν c V ˆ c = v v v v. 8

84 Energies Analytic energy expressions: E[ ( 9/2) 4,υ = 4,J = 4] = v + v v v, 8 E 9/2 ( ) 4,υ = 4,J = 6 [ ] = v v 4 + v v 8,

85 E2 transition rates Analytic E2 transition rate: ( ( ) 4,υ = 4,J = 6 ( 9/2) 4,υ = 4,J = 4) B E2; 9/2 = ( ( ) 2,J = 2 ( 9/2) 2,J = 0) B E2; 9/2

86 N=50 isotones

87 Nickel (Z=28) isotopes

88 Seniority isomers in the g 9/2 shell

89 Wigner s SU(4) symmetry Assume the nuclear hamiltonian is invariant under spin and isospin rotations: [ H ˆ nucl, S ˆ μ ]= [ H ˆ nucl, T ˆ ν ]= [ H ˆ nucl, Y ˆ μν ]= 0 ˆ S μ = A k=1 ˆ s μ (), k ˆ T ν = A t ˆ ν () k, Y ˆ μν = k=1 A k=1 Since {S μ,t ν,y μν } form an SU(4) algebra: H nucl has SU(4) symmetry. Total spin S, total orbital angular momentum L, total isospin T and SU(4) labels (λ,μ,ν) are conserved quantum numbers. ˆ s μ () k ˆ t ν () k E.P. Wigner, Phys. Rev. 51 (1937) 106 F. Hund, Z. Phys. 105 (1937) 202

90 Physical origin of SU(4) symmetry SU(4) labels specify the separate spatial and spinisospin symmetry of the wave function. Nuclear interaction is short-range attractive and hence favours maximal spatial symmetry.

91 Wigner energy Extra binding energy of N=Z nuclei (cusp). Wigner energy B W is decomposed in two parts: B W = W A ()N Z da ()δ N,Z π np W(A) and d(a) can be fixed empirically from binding energies. P. Möller & R. Nix, Nucl. Phys. A 536 (1992) 20 J.-Y. Zhang et al., Phys. Lett. B 227 (1989) 1 W. Satula et al., Phys. Lett. B 407 (1997) 103

92 Connection with SU(4) model Wigner s explanation of the kinks in the mass defect curve was based on SU(4) symmetry. Symmetry contribution to the nuclear binding energy is K A ()g λ,μ,ν ( )= KA () N Z [( ) N Z + 8δ N,Z π np + 6 δ pairing ] SU(4) symmetry is broken by spin-orbit term. Effects of SU(4) mixing must be included. D.D. Warner et al., Nature Physics 2 (2006) 311

93 Pairing with neutrons and protons For neutrons and protons two pairs and hence two pairing interactions are possible: 1 S 0 isovector or spin singlet (S=0,T=1): ˆ S + = m>0 a + + m a m 3 S 1 isoscalar or spin triplet (S=1,T=0): ˆ P + = m>0 a + + m a m

94 Neutron-proton pairing hamiltonian The nuclear hamiltonian has two pairing interactions V ˆ pairing = g ˆ 0 S + S ˆ g ˆ 1 P + ˆ P SO(8) algebraic structure. Integrable and solvable for g 0 =0, g 1 =0 and g 0 =g 1. B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673

95 Quartetting in N=Z nuclei Pairing ground state of an N=Z nucleus: ( P ) n /4 + o cosθ S ˆ + S ˆ + sinθ P ˆ + ˆ Condensate of α-like objects. Observations: Isoscalar component in condensate survives only in N Z nuclei, if anywhere at all. Spin-orbit term reduces isoscalar component. J. Dobes & S. Pittel, Phys. Rev. C 57 (1998) 6883

96 (d,α) and (p, 3 He) transfer

97 Elliott s SU(3) model of rotation Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type: A 2 H ˆ p = k 2m mω 2 2 r k g 2 Q ˆ Q ˆ, k=1 A k=1 Q ˆ μ r 2 k Y 2μ ˆ A k=1 () r k p k + p k 2 Y 2μ ˆ ( ) J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562

98 Importance & limitations of SU(3) Historical importance: Bridge between the spherical shell model and the liquiddrop model through mixing of orbits. Spectrum generating algebra of Wigner s SU(4) model. Limitations: LS (Russell-Saunders) coupling, not jj coupling (no spinorbit splitting) (beginning of) sd shell. Q is the algebraic quadrupole operator no major-shell mixing.

99 Breaking of SU(4) symmetry SU(4) symmetry breaking as a consequence of Spin-orbit term in nuclear mean field. Coulomb interaction. Spin-dependence of the nuclear interaction. Evidence for SU(4) symmetry breaking from masses and from Gamow-Teller β decay.

100 SU(4) breaking from masses Double binding energy difference δv np δv np ( N,Z)= 1 4 [ BN,Z ( ) BN ( 2,Z) BN,Z ( 2)+ BN ( 2,Z 2) ] δv np in sd-shell nuclei: P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607

101 SU(4) breaking from β decay Gamow-Teller decay into odd-odd or even-even N=Z nuclei. P. Halse & B.R. Barrett, Ann. Phys. (NY) 192 (1989) 204

102 Pseudo-spin symmetry Apply a helicity transformation to the spin-orbit + orbit-orbit nuclear mean field: 1 u ˆ ( k ζ l ˆ k s ˆ k + κ l ˆ k l ˆ k )ˆ u k = ( 4ζ κ)ˆ l k s ˆ k + κ l ˆ k l ˆ k + c te s u ˆ k = 2i ˆ k p k p k Degeneracies for 4ζ=κ. occur K.T. Hecht & A. Adler, Nucl. Phys. A 137 (1969) 129 A. Arima et al., Phys. Lett. B 30 (1969) 517 R.D. Ratna et al., Nucl. Phys. A 202 (1973) 433 J.N. Ginocchio, Phys. Rev. Lett. 78 (1998) 436

103 Pseudo-SU(4) symmetry Assume the nuclear hamiltonian is invariant under pseudo-spin and isospin rotations: [ H ˆ nucl, S ˆ ] = [ ˆ μ H nucl, T ˆ ν ]= H ˆ nucl, ˆ ˆ S μ = A k=1 ˆ s μ (), k ˆ T ν = A [ Y ] = 0 μν t ˆ ν () k, ˆ Y μν = k=1 A k=1 Consequences: Hamiltonian has pseudo-su(4) symmetry. Total pseudo-spin, total pseudo-orbital angular momentum, total isospin and pseudo-su(4) labels are conserved quantum numbers. ˆ s μ () k ˆ t ν () k D. Strottman, Nucl. Phys. A 188 (1971) 488

104 Test of pseudo-su(4) symmetry Shell-model test of pseudo-su(4). Realistic interaction in pf 5/2 g 9/2 space. Example: 58 Cu. XL-ELAF, México DF, August P. Van Isacker et al., Phys. Rev. Lett. 82 (1999)

105 Pseudo-SU(4) and β decay Pseudo-spin transformed Gamow-Teller operator is deformation dependent: s ˆ μˆ t ν u ˆ 1ˆ s μˆ t ν u ˆ = 1 s 3 ˆ μˆ t ν Test: β decay of 18 Ne vs. 58 Zn. 1 r 2 [( r r) () 2 s ˆ ] 1 μ ()ˆ t ν A. Jokinen et al., Eur. Phys. A. 3 (1998) 271

106 Three faces of the shell model

107 Symmetries in Nuclei Symmetry, mathematics and physics Examples of symmetries in quantum mechanics Symmetries of the nuclear shell model Symmetries of the interacting boson model

108 Symmetries of the interacting boson model The interacting boson model Algebraic structure Geometric interpretation Partial dynamical symmetries Extensions of the IBM

109 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: [ S ˆ, S ˆ n + ]=1 ˆ 1while [ s,s+ ]=1 Ω

110 Microscopy of IBM In a boson mapping, fermion pairs are represented as bosons: ( ) 0 0 s + S ˆ + α j a + + j a j, d + μ D ˆ + μ β jj' a + + j a j' Mapping of operators (such as hamiltonian) should take account of Pauli effects. Two different methods by j requiring same commutation relations; associating state vectors. ( ) jj' ( ) ( ) μ 2 1 T. Otsuka et al., Nucl. Phys. A 309 (1978)

111 The interacting boson model Describe the nucleus as a system of N interacting s and d bosons. Hamiltonian: ˆ H IBM = Justification from 6 ε i b + i b i i=1 6 ijkl=1 υ ijlk b i + b j + b k b l 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.

112 Dimensions Assume Ω available 1-fermion states. Number of N- fermion states is Ω Ω! = N N! ( Ω N)! Assume Ω available 1-boson states. Number of N- boson states is Ω + N 1 N = ( Ω + N 1 )! N! ( Ω 1)! Example: 162 Dy 96 with 14 neutrons (Ω=44) and 16 protons (Ω=32) ( 132 Sn 82 inert core). SM dimension: IBM-1 dimension: 15504

113 U(6) algebra and symmetry Introduce 6 creation & annihilation operators: { b + i,i =1,K,6}= { s +,d + 2,d + 1,d + 0,d ,d +2 }, b i = ( + b ) + i The hamiltonian (and other operators) can be written in terms of generators of U(6): [ b + i b j,b + k b l ]= b + i b l δ jk b + k b j δ il The harmonic hamiltonian has U(6) symmetry ˆ H U(6) = E b + i b i [ H ˆ U(6),b + i b j ]= 0 i=1 Additional terms break U(6) symmetry.

114 The IBM hamiltonian Rotational invariant hamiltonian with up to N-body interactions (usually up to 2): H ˆ IBM = E 0 + ε s n ˆ s + ε d n ˆ d + L υ l1 l 2 l 1 For what choice of single-boson energies ε and boson-boson interactions υ is the IBM hamiltonian solvable? This problem is equivalent to the enumeration of all algebras G satisfying U6 () G SO 3 () l 1 l 2 l 1 l 2,L l 2 ( b + + l1 b ) L l2 { L ˆ = 10( d + d ) ( 1) } μ μ ( ) ( b b ) L l 1 l 2 ( ) A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069

115 Dynamical symmetries of the IBM U(6) has the following subalgebras: U5 ()= d + d ( ), d + d ( ), d + d SU 3 SO 6 {( ) 0 ( ) 1 ( ) ( 2), ( d + d ) ( 3), ( d + d ) ( 4) } μ μ μ μ μ {( ) () 1, ( s + d + d + s ) () 2 7 ( d + d ) () 2 } μ μ 4 μ ()= d + d {( ) () 1, ( s + d + d + s ) () 2, ( d + d ) () 3 } μ μ μ ()= d + d { (), d + () } SO()= 5 ( d + d ) 1 ( d ) 3 μ μ Three solvable limits are found: U5 () SO( 5) U6 () SU() 3 SO() 3 SO() 6 SO() 5 O. Castaños et al., J. Math. Phys. A 20 (1979) 35

116 The U(5) vibrational limit U(5) Hamiltonian: H ˆ U5 () = ε n ˆ d + 1 c L d + d + L= 0,2,4 Energy eigenvalues: 2 ( ) L ( ) d d ( ) L En ( d,υ,l)= ε n d + κ 1 n d ( n d + 4)+ κ 4 υ( υ + 3)+ κ 5 LL+1 ( ) with κ 1 = 1 12 c 0 κ 4 = 1 10 c c c 4 κ 5 = 1 14 c c 4 ( )

117 The U(5) vibrational limit Anharmonic vibration spectrum associated with the quadrupole oscillations of a spherical surface. Conserved quantum numbers: n d, υ, L. A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253 D. Brink et al., Phys. Lett. 19 (1965) 413

118 The SU(3) rotational limit SU(3) Hamiltonian: H ˆ SU () = a ˆ 3 Q Q ˆ + bl ˆ L ˆ Energy eigenvalues: E( λ,μ,l)= κ ( 2 λ 2 + μ 2 + 3λ + 3μ + λμ)+ κ 5 LL+1 ( ) with κ 2 = 1 2 a κ 5 = b 3 8 a

119 The SU(3) rotational limit Rotation-vibration spectrum of quadrupole oscillations of a spheroidal surface. Conserved quantum numbers: (λ,μ), 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

120 The SO(6) γ-unstable limit SO(6) Hamiltonian: H ˆ SO () = a ˆ 6 P + P ˆ + bt ˆ 3 T ˆ 3 + cl ˆ L ˆ Energy eigenvalues: E( σ,υ,l)= κ [ 3 NN+ ( 4) σ( σ + 4) ]+ κ 4 υ( υ + 3)+ κ 5 LL+1 with κ 3 = 1 4 a κ 4 = 1 2 b κ 5 = 1 10 b + c ( )

121 The SO(6) γ-unstable limit Rotation-vibration spectrum of quadrupole oscillations of a γ-unstable spheroidal surface. Conserved quantum numbers: σ, υ, L. A. Arima & F. Iachello, Ann. Phys. (NY) 123 (1979) 468 L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788

122 The IBM symmetries Three analytic solutions: U(5), SU(3) & SO(6).

123 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.

124 Applications of IBM

125 The ratio R 42

126 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

127 Vibration about a spherical shape Vibrations are characterized by a multipole quantum number λ in surface parametrization: + λ R( θ,ϕ)= R α λμ Y * λμ ( θ,ϕ) λ μ = λ λ=0: compression (high energy) λ=1: translation (not an intrinsic excitation) λ=2: quadrupole vibration

128 Vibration about a spheroidal shape The vibration of a shape with axial symmetry is characterized by a λν. Quadrupolar oscillations: ν=0: along the axis of symmetry (β) ν=±1: spurious rotation ν=±2: perpendicular to axis of symmetry (γ) γ c β γ c β

129 Classical limit of IBM For large boson number N, a coherent (or intrinsic) state is an approximate eigenstate, ( ) N o μ H ˆ IBM N;α μ EN;α μ, N;α μ s α μ d μ The real parameters α μ are related to the three Euler angles and shape variables β and γ. Any IBM hamiltonian yields energy surface: N;α ˆ μ H IBM N;α μ = N;βγ H ˆ IBM N;βγ V( β,γ) 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

130 Geometry of IBM A simplified, much used IBM hamiltonian: H ˆ CQF = ε d n ˆ d κq ˆ χ Q ˆ χ, Q ˆ χ μ = s + d μ + d + μ s + χ d + d H CQF can acquire the three IBM symmetries. H CQF has the following classical limit: V CQF ( β,γ) N;βγ H ˆ CQF N;βγ = ε d N β 2 1+ β κn 5 + ( 1+ χ 2 )β β 2 κ NN 1 ( ) 1+ β 2 7 χ 2 β χβ 3 cos3γ + 4β 2 ( ) 2 2 ( ) ( ) μ 2

131 Phase diagram of IBM E. López-Moreno and O. Castaños, Phys. Rev. C 54 (1996) 2374 J. Jolie et al., Phys. Rev. Lett. 87 (2001)

132 Partial dynamical symmetries (PDS) Hamiltonians with dynamical symmetry: G dyn L G break L G sym [ ] Σ Λ h N Dynamical symmetries can be partial: Type 1: All labels Σ, Σ remain good quantum numbers for some eigenstates. Type 2: Some of the labels Σ, Σ 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

133 How to construct a type-1 PDS? Starting point: A DS classification of eigenstates and tensor operators. For a given representation Σ 0, find n-particle annihilation operators T such that ˆ T hn [ ]σλ h N [ ]Σ 0 Λ = 0, Λ Σ 0 Any interaction written in terms of T and T + preserves solvability of [h N ] Σ=Σ 0 Λ states. Task: Check whether any state in [h N-n ] belongs to the Kronecker product Σ 0 σ. J.E. García-Ramos et al., Phys. Rev. Lett. 102 (2009)

134 SO(6) dynamical symmetry Classification scheme: U6 () SO( 6) SO 5 N ( ) SO( 3) [ ] Σ τ ν Δ L Hamiltonian with SO(6) DS: H ˆ DS = κ 1 P + P + κ 2 C ˆ [ 2 SO( 5) ]+ κ 3 C ˆ [ 3 SO( 3) ], P = ss d d All states [N]ΣτL are solvable with energy 1 E = κ 1 ( 4 N Σ )( N +Σ+4)+ κ 2 ττ+ ( 3)+ κ 3 LL+1 ( )

135 SO(6) partial dynamical symmetry Classification scheme: U6 () SO( 6) SO 5 [ N] Σ τ ν Δ L ( ) SO( 3) Hamiltonian with SO(6) PDS: H ˆ PDS = H ˆ DS + η 1 P ˆ + V ˆ 1 P ˆ + η 2 P ˆ + V ˆ 2 P ˆ +L Only states [N]Σ=NτL are solvable with energy E = κ 2 τ( τ + 3)+ κ 3 LL+1 ( ) J.E. García-Ramos et al., Phys. Rev. Lett. 102 (2009)

136 Example: 196 Pt

137 Example: 196 Pt

138 Anharmonic vibrations The SO(6) limit yields a spectrum of vibrational bands for a γ-unstable deformed rotor, with L π =0 + band heads with Σ=N-2υ, υ=0,1,2, Anharmonicity of the vibration quantified by R=E x (0 +,υ=2)/e x (0 +,υ=1)-2. Comparison with observed anharmonicity: In SO(6) DS: R=-2/(N+1)=-0.29 In SO(6) PDS: R=-0.63 In 196 Pt: R=-0.70

139 E2 transitions rates in 196 Pt

140 Example of type-2 PDS P. Van Isacker, Phys. Rev. Lett. 83 (1999) 4269

141 Extensions of the IBM Neutron and proton degrees freedom (IBM-2): F-spin multiplets (N ν +N π =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.

142 Scissors excitations Collective displacement modes between neutrons and protons: Linear displacement (giant dipole resonance): R ν -R π E1 excitation. Angular displacement (scissors resonance): L ν -L π M1 excitation. D. Bohle et al., Phys. Lett. B 137 (1984) 27

143 SO(6) (mixed-)symmetry in 94 Mo 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 ν and e π. N. Pietralla et al., Phys. Rev. Lett. 83 (1999) 1303

144 Bosons + fermions Odd-mass nuclei are fermions. Describe an odd-mass nucleus as N bosons + 1 fermion mutually interacting. Hamiltonian: ˆ H IBFM = ˆ H IBM + Ω ε j a + j a j + υ i1 j 1 i 2 j 2 b + i1 a + j1 b i2 a j2 j=1 6 i 1 i 2 =1 Ω j 1 j 2 =1 Algebra: U6 () U( Ω)= b + i 1 b i2 a + a j1 j2 Many-body problem is solved analytically for certain energies ε and interactions υ.

145 Example: 195 Pt 117

146 Example: 195 Pt 117 (new data)

147 Nuclear supersymmetry Up to now: separate description of even-even and odd-mass nuclei with the algebra U6 () U( Ω)= b + i 1 b i2 a + a j1 j2 Simultaneous description of even-even and oddmass nuclei with the superalgebra U6/Ω ( )= b + i 1 b i2 b + i1 a j2 a + j1 b i2 a + j1 a j2 F. Iachello, Phys. Rev. Lett. 44 (1980) 777

148 U(6/12) supermultiplet

149 Example: 194 Pt 116 & 195 Pt 117

150 Quartet supersymmetry A simultaneous description of even-even, even-odd, odd-even and odd-odd nuclei (quartets). Example of 194 Pt, 195 Pt, 195 Au & 196 Au: a + π b π 195 Pt b π+ a π Au 117 a ν + b ν b ν + a ν a ν + b ν b ν + a ν a π 194 Pt b π b π+ a π Au 116 P. Van Isacker et al., Phys. Rev. Lett. 54 (1985) 653

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

152 Example of 196 Au

153 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 (M T =-1) and pp (M T =+1) pairs. IBM-3: full isospin T=1 triplet of nn (M T =-1), np (M T =0) and pp (M T =+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.

154 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.

155 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.

156 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 γ-unstable nuclei with s,d-boson IBM.