symmetries in nuclei - unambijker/elaf2010/isacker.pdf · symmetries of the nuclear shell model....

XL-ELAF, México DF, August 2010

Symmetries in Nuclei

Piet Van IsackerGrand Accélérateur National d’Ions Lourds,

Caen, France



Symmetry, mathematics and physicsExamples of symmetries in quantum mechanicsSymmetries of the nuclear shell modelSymmetries of the interacting boson model


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.


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].”


Origin

For the ancient Greeks symmetry implied the notion of proportion.

17th century: Symmetry starts to imply also a relation of equality of elements that are opposed (eg. between left and right).

19th 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.


Group theory

Group theory is the mathematical theory of symmetry.Group theory was invented (discovered?) by Evariste

Galois in 1831.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.


The birth of group theory

Are all equations solvable algebraically?Example of quadratic equation:

Babylonians (from 2000 BC) knew how to solve quadratic equations in words but avoided cases with negative or no solutions.

Indian mathematicians (eg. Brahmagupta 598-670) did interpret negative solutions as `depths’.

Full solution was given in 12th century by the Spanish Jewish mathematician Abraham bar Hiyya Ha-nasi.

ax 2 + bx + c = 0 ⇒ x1,2 =−b ± b2 − 4ac

2a



No solution of higher equations until dal Ferro, Tartaglia, Cardano and Ferrari solve the cubic and quartic equations in the 16th century.

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?

ax 3 + bx 2 + cx + d = 0 & ax 4 + bx 3 + cx 2 + dx + e = 0



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 Sn .


The insolvability of the quintic


The axioms of group theory

A set G of elements (transformations) with an operation ×

which satisfies:

1. Closure. If g1 and g2 belong to G, then g1 ×g2 also belongs to G.

2. Associativity. We always have (g1 ×g2 )×g3 =g1 ×(g2 ×g3 ).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.


Example: equilateral triangle

Symmetry transformations are

- Identity- Rotation over 2π/3 and 4π/3 around ez

- Reflection with respect to planes (u1 ,ez ), (u2 ,ez ), (u3 ,ez )

Symmetry group: C3h .


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.


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

⎡

⎣ ⎢

⎤

⎦ ⎥


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 gi (called generators) of a Lie algebra.


Lie algebras

All properties of a Lie algebra follow from the commutation relations between its generators:

Generators satisfy the Jacobi identity:

Definition of the metric tensor or Killing form:

ˆ g i, ˆ g j[ ]≡ ˆ g i × ˆ g j − ˆ g j × ˆ g i = cijk ˆ g k

k=1

r

∑

ˆ g i, ˆ g j , ˆ g k[ ][ ]+ ˆ g j , ˆ g k , ˆ g i[ ][ ]+ ˆ g k, ˆ g i, ˆ g j[ ][ ]= 0

gij = cikl c jl

k

k,l=1

r

∑


Classification of Lie groups

Symmetry groups over RR, CC and H (quaternions) preserving a specified metric:

The five exceptional groups G2 , F4 , E6 , E7 and E8 are similar constructs over the normed division algebra of the octonions, OO.

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( )


Rotations in 2 dimensions, SO(2)

Matrix representation of finite elements:

Infinitesimal element and generator:

Exponentiation leads back to finite elements:

limα →0

ˆ g α( ) ≈1 00 1

⎡

⎣ ⎢

⎤

⎦ ⎥ + α

0 1−1 0

⎡

⎣ ⎢

⎤

⎦ ⎥ ≡ ˆ e + α ˆ g 1

ˆ g α( )= limn →∞

ˆ e + αn

ˆ g 1⎛ ⎝ ⎜

⎞ ⎠ ⎟

n

= exp ˆ e + α ˆ g 1( )= exp1 α

−α 1⎡

⎣ ⎢

⎤

⎦ ⎥ = ˆ g α( )

ˆ g α( )=cosα sinα−sinα cosα

⎡

⎣ ⎢

⎤

⎦ ⎥



Matrix representation of finite elements:

Infinitesimal elements and associated generators:

ˆ g 1 =0 0 00 0 −10 1 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ , ˆ g 2 =

0 0 10 0 0−1 0 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ , ˆ g 3 =

0 −1 01 0 00 0 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

cosα3 −sinα3 0sinα3 cosα3 0

0 0 1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

cosα2 0 sinα2

0 1 0−sinα2 0 cosα2

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

1 0 00 cosα1 −sinα1

0 sinα1 cosα1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥



Structure constants from matrix multiplication:

Exponentiation leads back to finite elements:

Relation with angular momentum operators:

ˆ g α1,α2,α3( )= exp ˆ e + αk ˆ g kk=1

3

∑⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ g k, ˆ g l[ ]= cklm ˆ g m

m=1

3

∑ with cklm = εklm Levi − Civita( )

ˆ l k = i ˆ g k ⇒ ˆ l k, ˆ l l[ ]= i εklmˆ l m

m=1

3

∑


Casimir operators

Definition: The Casimir operators Cn [G] of a Lie algebra G commute with all generators of G.

The quadratic Casimir operator (n=2):

The number of independent Casimir operators (rank) equals the number of quantum numbers needed to characterize any (irreducible) representation of G.

ˆ C 2 G[ ] = gij ˆ g i ˆ g ji, j =1

r

∑ with gijgjk = δ ik

i, j =1

r

∑ . Ex : ˆ C 2 SO 3( )[ ]= ˆ l k2

k =1

3

∑


Symmetry in physics

Geometrical symmetriesExample: C3h symmetry (X3 molecules, 12C nucleus).

Permutational symmetriesExample: SA symmetry of an A-body hamiltonian:

ˆ H = pk2

2mkk=1

A

∑ + ˆ V 2 rk − r ′ k ( )1≤k< ′ k

A

∑


Symmetry in physics

Space-time (kinematical) symmetriesRotational invariance, SO(3):Lorentz invariance, SO(3,1):Parity:Time reversal:Euclidian invariance, E(3):Poincaré invariance, E(3,1):Dilatation symmetry:

xk → ′ x k = Akl xll∑xμ → ′ x μ = Aμν xνν∑

xk → ′ x k = −xk

t → ′ t = −txk → ′ x k = Akl xll∑ + ak

xμ → ′ x μ = Aμν xνν∑ + aμ

xk → ′ x k = Dk xk


Symmetry in physics

Quantum-mechanical symmetries, U(n)

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)

ψk → ′ ψ k = Bklψ ll=1

n

∑ inv. ψll∑ 2


Symmetry in quantum mechanics

Assume a hamiltonian H which commutes with operators gi that form a Lie algebra G:

∴

H has symmetry G or is invariant under G.Lie algebra: a set of (infinitesimal) operators that closes

under commutation.

∀ ˆ g i ∈ G : ˆ H , ˆ g i[ ]= 0


Consequences of symmetry

Degeneracy structure: If ⏐γ⟩

is an eigenstate of H with energy E, so is gi ⏐γ⟩:

Degeneracy structure and labels of eigenstates of H are determined by algebra G:

Casimir operators of G commute with all gi :

ˆ H γ = E γ ⇒ ˆ H g i γ = ˆ g i ˆ H γ = Eˆ g i γ

ˆ H Γγ = E Γ( )Γγ ; ˆ g i Γγ = a ′ γ γΓ i( )Γ ′ γ

′ γ ∑

ˆ H = μmˆ C m G[ ]

m∑


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.



Symmetry, mathematics and physicsExamples of symmetries in quantum

mechanicsSymmetries of the nuclear shell modelSymmetries of the interacting boson model


Examples of symmetries in quantum mechanics

Symmetry in quantum mechanicsThe hydrogen atomThe harmonic oscillatorIsospin symmetry in nucleiDynamical symmetry


Symmetry in quantum mechanics

Assume a hamiltonian H which commutes with operators gi that form a Lie algebra G:

∴

H has symmetry G or is invariant under G.Lie algebra: a set of (infinitesimal) operators that

closes under commutation.

∀ ˆ g i ∈ G : ˆ H , ˆ g i[ ]= 0


Consequences of symmetry

Degeneracy structure: If ⏐γ⟩

is an eigenstate of H with energy E, so is gi ⏐γ⟩:

Degeneracy structure and labels of eigenstates of H are determined by algebra G:

Casimir operators of G commute with all gi :

ˆ H γ = E γ ⇒ ˆ H g i γ = ˆ g i ˆ H γ = Eˆ g i γ

ˆ H Γγ = E Γ( )Γγ ; ˆ g i Γγ = a ′ γ γΓ i( )Γ ′ γ

′ γ ∑

ˆ H = μmˆ C m G[ ]

m∑


The hydrogen atom

The hamiltonian of the hydrogen atom is

Standard wave quantum mechanics gives

Degeneracy in m originates from rotational symmetry. What is the origin of l-degeneracy?

ˆ H = p2

2M−

αr

ˆ H Ψnlm r,θ,ϕ( )= −Mα 2

2h2n2 Ψnlm r,θ,ϕ( )with n =1,2,K; l = 0,1,K,n −1; m = −l,K,+l


Degeneracies of the H atom


Classical Kepler problem

Conserved quantities:Energy (a=length semi-major axis):

Angular momentum (ε=eccentricity):

Runge-Lenz vector:

Newtonian potential gives rise to closed orbits with constant direction of major axis.

E = −α2a

L = r ∧ p, L2 = Mα a 1−ε2( )

R =p ∧ L

M−α r

r, R2 =

2EM

L2 + α 2


Classical Kepler problem


Quantization of operators

From p→-ih∇:

Some useful commutators & relations:

ˆ H = −h2

2M∇2 +

αr

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ L = −ih r ∧∇( )

ˆ R = − h2

2M∇ ∧ r ∧∇( )− r ∧∇( )∧∇[ ]−α r

r

∇,rk[ ]= k rk−2r, ∇2,r[ ]= 2∇, ∇2,rk[ ]= k rk−2 k +1( )+ 2r ⋅ ∇[ ]ˆ R 2 =

2 ˆ H M

ˆ L 2 + h2( )+ α 2


Conservation of angular momentum

The angular momentum operators L commute with the hydrogen hamiltonian:

L operators generate SO(3) algebra:

H has SO(3) symmetry ⇒ m-degeneracy.

ˆ H , ˆ L [ ]∝ ∇2 + 2κ r−1,r ∧∇[ ] κ = Mα /h2( )= ∇2,r[ ]∧∇ + 2κ r ∧ r−1,∇[ ]= 0 + 0

ˆ L j , ˆ L k[ ]= ihε jkl

ˆ L l , j,k,l = x, y,z


Conserved Runge-Lenz vector

The Runge-Lenz vector R commutes with H:

R does not commute with the kinetic and potential parts of H separately:

Hydrogen atom has a dynamical symmetry.

ˆ H , ˆ R [ ]∝ ∇2 + 2κ r−1,∇ ∧ r ∧∇( )− r ∧∇( )∧∇ + 2κ r−1r[ ]= ∇2,2κ r−1r[ ]+ 2κ r−1,∇ ∧ r ∧∇( )− r ∧∇( )∧∇[ ]= 0

−

h2

2M∇2, ˆ R [ ]= − −α r−1, ˆ R [ ]=

h2αM

1r

∇ −rr3 1+ r ⋅ ∇( )

⎡ ⎣ ⎢

⎤⎦⎥


SO(4) symmetry

L and R (almost) close under commutation:

H is time-independent and commutes with L and R ⇒ choose a subspace with given E.

L and R'≡(-M/2H)1/2R form an algebra SO(4) corresponding to rotations in four dimensions.

ˆ L j , ˆ L k[ ]= ihε jklˆ L l , j,k,l = x,y,z

ˆ L j , ˆ R k[ ]= ihε jklˆ R l , j,k,l = x,y,z

ˆ R j , ˆ R k[ ]= −ihε jkl2 ˆ H M

ˆ L l , j,k, l = x,y,z


Energy spectrum of the H atom

Isomorphism of SO(4) and SO(3)⊕SO(3):

Since L and R' are orthogonal:

The quadratic Casimir operator of SO(4) and H are related:

ˆ F ± =

12

ˆ L ± ˆ ′ R ( )⇒ ˆ F j±, ˆ F k

±[ ]= ihε jklˆ F l

±, ˆ F j+, ˆ F k

−[ ]= 0

ˆ F + ⋅ ˆ F + = ˆ F − ⋅ ˆ F − = j j +1( )h2

ˆ C 2 SO 4( )[ ]= ˆ F + ⋅ ˆ F + + ˆ F − ⋅ ˆ F − =12

ˆ L 2 −M2H

ˆ R 2⎛ ⎝ ⎜

⎞ ⎠ ⎟ = −

Mα 2

4H−

12

h2

ˆ C 2 SO 4( )[ ] = 2 j j +1( )h2 ⇒ E = −Mα 2

2 2 j +1( )2h2, j = 0, 1

2 ,1, 32 ,K

W. Pauli, Z. Phys. 36 (1926) 336


The (3D) harmonic oscillator

The hamiltonian of the harmonic oscillator is

Standard wave quantum mechanics gives

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?

ˆ H Ψnlm r,θ,ϕ( )= 2n + l +32

⎛ ⎝ ⎜

⎞ ⎠ ⎟ hωΨnlm r,θ,ϕ( )

with n = 0,1,K ; l = 0,1,K ; m = −l,K ,+l

ˆ H =p2

2M+

12

Mω 2r2


Degeneracies of the 3D HO


Raising and lowering operators

Introduce the raising and lowering operators

The 3D HO hamiltonian becomes

ˆ H =p2

2M+

12

Mω 2r2 = ηiξi +12

⎛ ⎝ ⎜

⎞ ⎠ ⎟

i=x,y,z∑ hω

ηx =12

′ x −∂

∂ ′ x ⎛ ⎝ ⎜

⎞ ⎠ ⎟ , ηy =

12

′ y −∂

∂ ′ y ⎛

⎝ ⎜

⎞

⎠ ⎟ , ηz =

12

′ z −∂

∂ ′ z ⎛ ⎝ ⎜

⎞ ⎠ ⎟

ξx =12

′ x +∂

∂ ′ x ⎛ ⎝ ⎜

⎞ ⎠ ⎟ , ξy =

12

′ y +∂

∂ ′ y ⎛

⎝ ⎜

⎞

⎠ ⎟ , ξz =

12

′ z +∂

∂ ′ z ⎛ ⎝ ⎜

⎞ ⎠ ⎟

with ′ x = x / l, ′ y = y / l, ′ z = z / l; l =h

Mω

M. Moshinsky, The Harmonic Oscillator in Modern Physics


U(3) symmetry of the 3D HO

The raising and lowering operators satisfy

The bilinear combinations uij commute with H:

The nine operators uij generate the algebra U(3):

⇒ The symmetry of the harmonic oscillator in 3 dimensions is U(3).

ξi = ηi+, ξi,ξ j[ ]= 0, ηi,η j[ ]= 0, ξi,η j[ ]= δ ij

ˆ u ij ≡ ηiξ j ⇒ ˆ u ij , ˆ H [ ]= 0, ∀i, j ∈ x,y,z{ }

ˆ u ij , ˆ u kl[ ]= ˆ u ilδ jk − ˆ u kjδ il


The U(3)=U(1)⊕SU(3) algebraThe generators uij of U(3) can be written as

ηxξx +ηy ξy +ηzξz =ˆ H

hω−

32

ˆ L z = −ih x∂∂y

− y∂∂x

⎛

⎝ ⎜

⎞

⎠ ⎟ = −ih ηxξy −ηyξx( ) + cyclic

ˆ Q 0 = h 2ηzξz −ηxξx −ηyξy( )ˆ Q m1 = h 3

2±ηzξx ± ηxξz − iηyξz − iηzξy( )

ˆ Q m2 = h32

ηxξx −ηyξy miηxξy miηyξx( )


Many particles in the 3D HO

Define operators for each particle k=1,2,…,A:

The total U(3) algebra is generated by

ηxk =

12

′ x k −∂

∂ ′ x k

⎛

⎝ ⎜

⎞

⎠ ⎟ , ηy

k =12

′ y k −∂

∂ ′ y k

⎛

⎝ ⎜

⎞

⎠ ⎟ , ηz

k =12

′ z k −∂

∂ ′ z k

⎛

⎝ ⎜

⎞

⎠ ⎟

ξxk =

12

′ x k +∂

∂ ′ x k

⎛

⎝ ⎜

⎞

⎠ ⎟ , ξy

k =12

′ y k +∂

∂ ′ y k

⎛

⎝ ⎜

⎞

⎠ ⎟ , ξz

k =12

′ z k +∂

∂ ′ z k

⎛

⎝ ⎜

⎞

⎠ ⎟

ηikξ j

k

k =1

A

∑ , i, j ∈ x,y,z{ }


Many particles in the 3D HO

Many-body hamiltonian with U(3) symmetry:

This property is valid if the interaction equals

ˆ H = ηxkξx

k +ηykξy

k +ηzkξz

k

k =1

A

∑ + ˆ V k,l( )k< l =1

A

∑

ˆ H , ηikξ j

k

k =1

A

∑⎡

⎣ ⎢

⎤

⎦ ⎥ , ∀i, j ∈ x,y,z{ }

ˆ C 2 SU 3( )[ ]=12

L⋅ L +16

Q⋅ Q =12

L k( )⋅ L l( )+16

Q k( )⋅ Q l( )⎛ ⎝ ⎜

⎞ ⎠ ⎟

k<l =1

A

∑

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


Dynamical symmetry

Two algebras G1 ⊃

G2 and a hamiltonian

∴

H has symmetry G2 but not G1 !Eigenstates are independent of parameters μm and

νn in H.Dynamical symmetry breaking “splits but does not

admix eigenstates”.

ˆ H = μmˆ C m G1[ ]

m∑ + ν n

ˆ C n G2[ ]n∑


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:

Isospin symmetry in nuclei

n : t = 12 , mt = + 1

2 ; p : t = 12 , mt = − 1

2

⇒ ˆ t +n = 0, ˆ t +p = n, ˆ t −n = p, ˆ t −p = 0, ˆ t zn = 12 n, ˆ t z p = − 1

2 p

W. Heisenberg, Z. Phys. 77 (1932) 1E.P. Wigner, Phys. Rev. 51 (1937) 106


Isospin SU(2) symmetry

Isospin operators form an SU(2) algebra:

Assume the nuclear hamiltonian satisfies

∴

Hnucl has SU(2) symmetry with degenerate states belonging to isobaric multiplets:

ˆ t z, ˆ t ±[ ]= ±ˆ t ±, ˆ t +, ˆ t −[ ]= 2ˆ t z

ˆ H nucl, ˆ T ν[ ]= 0, ˆ T ν = ˆ t ν k( )k=1

A

∑

ηTMT , MT = −T,−T +1,K ,+T


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


Isospin symmetry breaking: A=51


Isospin SU(2) dynamical symmetry

Coulomb interaction can be approximated as

∴

Hnucl +HCoul has SU(2) dynamical symmetry and SO(2) symmetry.

MT -degeneracy is lifted according to

Summary of labelling:

ˆ H Coul ≈ κ0 + κ1ˆ T z + κ2

ˆ T z2 ⇒ ˆ H Coul, ˆ T z[ ]= 0, ˆ H Coul, ˆ T ±[ ]≠ 0

ˆ H Coul ηTMT = κ0 + κ1MT + κ2MT2( )ηTMT

SU 2( ) ⊃ SO 2( )↓ ↓T MT


Isobaric multiplet mass equation

Isobaric multiplet mass equation:

Example: T=3/2 multiplet for A=13 nuclei.E ηTMT( )= κ η,T( ) +κ 1MT +κ 2MT

2

E.P. Wigner, Proc. Welch Found. Conf. (1958) 88


Isospin selection rules

Internal E1 transition operator is isovector:

Selection rule for N=Z (MT =0) nuclei: No E1 transitions are allowed between states with the same isospin.

ˆ T μE1 = ek

k=1

A

∑ rμ k( )=e2 k=1

A

∑ rμ k( )⎛

⎝ ⎜

CM motion1 2 4 3 4

+ 2 ˆ t z k( )k=1

A

∑ rμ k( )⎞

⎠ ⎟

isovector1 2 4 4 3 4 4

L.E.H. Trainor, Phys. Rev. 85 (1952) 962L.A. Radicati, Phys. Rev. 87 (1952) 521


E1 transitions and isospin mixingB(E1;5-→4+) in 64Ge from:

lifetime of 5- level;δ(E1/M2) mixing ratio of 5-→4+ transition;relative intensities of transitions from 5-.

Estimate of minimum isospin mixing:

E.Farnea et al., Phys. Lett. B 551 (2003) 56

P T = 1, 5−( )≈ P T = 1,4+( )≈ 2.5%


Dynamical algebra

Take a generic many-body hamiltonian:

Rewrite H as (bosons: q=0; fermions: q=1)

Operators uij generate the dynamical algebra U(n) for bosons and for fermions (q=0,1):

ˆ H = εici+ci +

14

υ ijklci+c j

+clckijkl∑

i∑ +L

ˆ H = εiδil − −( )q 14

υ ijlkj

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ u il + −( )q 1

4υ ijlk ˆ u ik ˆ u jl

ijkl∑

il∑ +L

ˆ u ij ≡ ci+c j ⇒ ˆ u ij , ˆ u kl[ ]= ˆ u ilδ jk − ˆ u kjδ il


Dynamical symmetry (DS)With each chain of nested algebras

…is associated a particular class of many-body hamiltonian

Since H is a sum of commuting operators

…it can be solved analytically!

ˆ H = μmˆ C m G1[ ]

m∑ + ν n

ˆ C n G2[ ]n∑ +L

U n( )= Gdyn = G1 ⊃ G2 ⊃ L ⊃ Gsym

∀m,n,a,b : ˆ C m Ga[ ], ˆ C n Gb[ ][ ]= 0


DS in nuclear physics


Symmetries of the nuclear shell model

The nuclear shell modelRacah’s pairing model and seniorityWigner’s supermultiplet modelElliott’s SU(3) model and extensions


The nuclear many-body problem

The nucleus is a system of A interacting nucleons described by the hamiltonian

Separation in mean field + residual interaction:

ˆ H =pk1

2

2mk1k1 =1

A

∑ + ˆ V 2 ξk1,ξk2( )

1≤k1 <k2

A

∑ + ˆ V 3 ξk1,ξk2

,ξk3( )1≤k1 <k2 <k3

A

∑ +L

ξki≡ rki

, ˆ s ki, ˆ t ki{ }, pki

= −ih∇ki[ ]

ˆ H =pk1

2

2mk1

+ ˆ V ξk1( )⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ k1 =1

A

∑mean field

1 2 4 4 4 3 4 4 4

+ ˆ V 2 ξk1,ξk2( )

1≤k1 <k2

A

∑ +L− ˆ V ξk1( )k1 =1

A

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

residual interaction1 2 4 4 4 4 4 3 4 4 4 4 4


The nuclear shell modelHamiltonian with one-body term (mean field) and

two-body interactions:

Entirely equivalent form of the same hamiltonian in second quantization:

ε, υ: single-particle energies & interactionsijkl: single-particle quantum numbers

ˆ H SM = ˆ U ξk1( )k1 =1

A

∑ + ˆ W 2 ξk1,ξk2( )

1≤k1 <k2

A

∑

ˆ H SM = εiai+ai +

14

υ ijlkai+a j

+akalijkl∑

i∑


Boson and fermion statistics

Fermions have half-integer spin and obey Fermi- Dirac statistics:

Bosons have integer spin and obey Bose-Einstein statistics:

Matter is carried by fermions. Interactions are carried by bosons. Composite matter particles can be fermions or bosons.

ai,a j+{ }≡ aia j

+ + a j+ai = δij , ai,a j{ }= ai

+,a j+{ }= 0

bi,b j+[ ]≡ bib j

+ − bj+bi = δij , bi,b j[ ]= bi

+,b j+[ ]= 0


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):

Boson/fermion statistics is summarized by

Many-particle state:

ˆ u , ˆ v }[ q≡ ˆ u v − −( )q ˆ v u

ci+( )ni

ni!o

i∏ , ci o = 0

ci,c j+}[ q

= δij , ci+,c j

+}[ q= ci,c j}[ q

= 0


Bosons and fermions


Algebraic structure

Introduce bilinear operators

Rewrite many-body hamiltonianˆ u ij = ci

+c j ⇒ ˆ u ij , ˆ u kl[ ]= ˆ u ilδ jk − ˆ u kjδil ⇒ U Ω( )

ˆ H = E0 + εici+ci +

14

υ ijlkci+c j

+ckclijkl∑

i∑

= E0 + εiδil − −( )q 14

υ ijlkj

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ u il

il∑ + −( )q 1

4υ ijlk ˆ u ik ˆ u jl

ijkl∑


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 = pk2

2m+

12

mω 2rk2 −ζ ls

ˆ l k ⋅ ˆ s k −ζ llˆ l k

2⎡

⎣ ⎢

⎤

⎦ ⎥

k=1

A

∑

+ ˆ W 2 ξk1,ξk2( )

1≤k1 <k2

A

∑


Racah’s SU(2) pairing model

Assume pairing interaction in a single-j shell:

Spectrum 210Pb:

j 2JMJˆ V pairing j 2JMJ =

− 12 2 j +1( )g0, J = 0

0, J ≠ 0

⎧ ⎨ ⎩

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


Pairing SU(2) dynamical symmetry

The pairing hamiltonian,

…has a quasi-spin SU(2) algebraic structure:

H has SU(2) ⊃

SO(2) dynamical symmetry:

Eigensolutions of pairing hamiltonian:

ˆ H = −g0ˆ S + ⋅ ˆ S −, ˆ S + = 1

2 a jm+ a jm

+

m∑ , ˆ S − = ˆ S +( )+

ˆ S +, ˆ S −[ ]= 12 2 ˆ n − 2 j −1( )≡ −2 ˆ S z, ˆ S z, ˆ S ±[ ]= ± ˆ S ±

−g0ˆ S + ⋅ ˆ S − = −g0

ˆ S 2 − ˆ S z2 + ˆ S z( )

−g0ˆ S + ⋅ ˆ S − SMS = −g0 S S +1( )− MS MS −1( )( )SMS

A. Kerman, Ann. Phys. (NY) 12 (1961) 300


Interpretation of pairing solution

Quasi-spin labels S and MS are related to nucleon number n and seniority υ:

Energy eigenvalues in terms of n, j and υ:

Eigenstates have an S-pair character:

Seniority υ is the number of nucleons not in S pairs (pairs coupled to J=0).

S = 14 2 j −υ +1( ), MS = 1

4 2n − 2 j −1( )

j nυJMJ − g0ˆ S + ⋅ ˆ S − j nυJMJ = −g0

14 n −υ( ) 2 j − n +υ + 3( )

j nυJMJ ∝ ˆ S +( )n−υ( )/ 2jυυJMJ


Pairing between identical nucleons

Analytic solution of the pairing hamiltonian based on SU(2) symmetry. E.g. energies:

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

Correlated ground-state solution (cf. BCS).

j nυJ ˆ V pairing1≤k< l

n

∑ k,l( ) j nυJ = −g014 n −υ( ) 2 j − n −υ + 3( )

G. Racah, Phys. Rev. 63 (1943) 367G. Racah, L. Farkas Memorial Volume (1952) p. 294B.H. Flowers, Proc. Roy. Soc. (London) A 212 (1952) 248A.K. Kerman, Ann. Phys. (NY) 12 (1961) 300


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 toConstant 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.

ˆ S +( )n / 2o , ˆ S + = am↓

+ am ↑+

m∑

amb

+ ˆ S +( )n / 2o


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 gap in semi-magic nucleiEven-even nuclei:

Ground state: υ=0.First-excited state: υ=2.Pairing produces constant excitation energy:

Example of Sn isotopes:

Ex 21+( )= 1

2 2 j +1( )g0


Algebraic definition of seniority

For a system of n identical bosons with spin j

For a system of n identical fermions with spin j

Alternative definition with quasi-spin algebras.

U 2 j +1( ) ⊃ SO 2 j +1( ) ⊃ L ⊃ SO 3( )↓ ↓ ↓n[ ] υ J

U 2 j +1( ) ⊃ Sp 2 j +1( ) ⊃ L ⊃ SO 3( )↓ ↓ ↓1n[ ] υ J


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) 367I. Talmi, Simple Models of Complex Nuclei



Necessary and sufficient conditions for a two-body interaction νλ

to conserve seniority:

For fermions σ = -1; for bosons σ = +1.

2λ +1 a jIλ vλ

λ∑ = 0, I = 2,4,K,2 j⎣ ⎦,

ν λ ≡ j 2;λ ˆ V j 2;λ

a jIλ = δλI + 2 2λ +1( ) 2I +1( )

j j λj j I

⎧ ⎨ ⎩

⎫ ⎬ ⎭

−4 2λ +1( ) 2I +1( )2 j +1( ) 2 j +1+ 2σ( )

I. Talmi, Simple Models of Complex Nuclei



Bosons:

Fermions:

j = 3 :11v2 −18v4 + 7v6 = 0,j = 4 : 65v2 − 30v4 − 91v6 + 56v8 = 0,j = 5 : 3230v2 − 2717v6 − 3978v8 + 3465v10 = 0,

j = 9 /2 : 65v2 − 315v4 + 403v6 −153v8 = 0,j =11/2 :1020v2 − 3519v4 − 637v6 + 4403v8 − 2541v10 = 0,j =13/2 :1615v2 − 4275v4 −1456v6 + 3196v8 − 5145v10

− 4225v12 = 0,


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.


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 νJ .

This finding has relevance for the existence of seniority isomers in nuclei.

P. Van Isacker & S. Heinze, Phys. Rev. Lett. 100 (2008) 052501L. Zamick & P. Van Isacker, Phys. Rev. C 78 (2008) 044327


Energy matrix for (9/2)4 J=4a ˆ V a =

35

v0 +6799

v2 +746715

v4 +1186495

v6 +918715

v8,

a ˆ V b =14Δ

495 2119, a ˆ V c =

2 170Δ429 489

,

Δ = −65ν 2 + 315ν 4 − 403ν 6 +153ν 8

b ˆ V b =3316116137

v2 +18001793

v4 +7038280685

v6 +185478965

v8,

b ˆ V c =−10 595 13ν 2 − 9ν 4 −13ν 6 + 9ν 8( )

5379 39

c ˆ V c =25845379

v2 +4880923309

v4 +6580926895

v6 +114066116545

v8.


Energy matrix for (9/2)4 J=6a ˆ V a =

35

v0 +3499

v2 +1186715

v4 +658495

v6 +1479715

v8,

a ˆ V b =− 5Δ

1287 97, a ˆ V c =

2 2261Δ2145 291

,

Δ = −65ν 2 + 315ν 4 − 403ν 6 +153ν 8

b ˆ V b =3304919206

v2 +2573327742

v4 +1933119206

v6 +6505927742

v8,

b ˆ V c =5 11305 13ν 2 − 9ν 4 −13ν 6 + 9ν 8( )

41613 3

c ˆ V c =10073201

v2 +2637013871

v4 +77233201

v6 +1902613871

v8 .


Energies

Analytic energy expressions:

E 9 /2( )4,υ = 4,J = 4[ ]=6833

v2 + v4 +1315

v6 +11455

v8,

E 9 /2( )4,υ = 4,J = 6[ ]=1911

v2 +1213

v4 + v6 +336143

v8,


E2 transition rates

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

=209475176468

B E2; 9 /2( )2,J = 2 → 9 /2( )2,J = 0( )


N=50 isotones


Nickel (Z=28) isotopes


Seniority isomers in the g9/2 shell


Wigner’s SU(4) symmetryAssume the nuclear hamiltonian is invariant under

spin and isospin rotations:

Since {Sμ

,Tν

,Yμν

} 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 (λ,μ,ν) are conserved quantum numbers.

ˆ H nucl, ˆ S μ[ ]= ˆ H nucl, ˆ T ν[ ]= ˆ H nucl, ˆ Y μν[ ]= 0

ˆ S μ = ˆ s μ k( ),k=1

A

∑ ˆ T ν = ˆ t ν k( )k=1

A

∑ , ˆ Y μν = ˆ s μ k( )k=1

A

∑ ˆ t ν 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.


Wigner energyExtra binding energy of N=Z nuclei (cusp).Wigner energy BW is decomposed in two parts:

W(A) and d(A) can be fixed empirically from binding energies.

BW = −W A( )N − Z

− d A( )δN ,Zπ np

P. Möller & R. Nix, Nucl. Phys. A 536 (1992) 20J.-Y. Zhang et al., Phys. Lett. B 227 (1989) 1W. Satula et al., Phys. Lett. B 407 (1997) 103


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

SU(4) symmetry is broken by spin-orbit term. Effects of SU(4) mixing must be included.

−K A( )g λ,μ,ν( )= K A( ) N − Z( )2 + 8 N − Z + 8δN ,Zπ np + 6 ′ δ pairing[ ]

D.D. Warner et al., Nature Physics 2 (2006) 311


Pairing with neutrons and protons

For neutrons and protons two pairs and hence two pairing interactions are possible:

1S0 isovector or spin singlet (S=0,T=1):

3S1 isoscalar or spin triplet (S=1,T=0):

ˆ S + = am↓+ am ↑

+

m>0∑

ˆ P + = am↑+ am ↑

+

m>0∑


Neutron-proton pairing hamiltonian

The nuclear hamiltonian has two pairing interactions

SO(8) algebraic structure.Integrable and solvable for g0 =0, g1 =0 and g0 =g1 .

ˆ V pairing = −g0ˆ S + ⋅ ˆ S − − g1

ˆ P + ⋅ ˆ P −

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


Quartetting in N=Z nuclei

Pairing ground state of an N=Z nucleus:

⇒ 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.

cosθ ˆ S + ⋅ ˆ S + − sinθ ˆ P + ⋅ ˆ P +( )n / 4o

J. Dobes & S. Pittel, Phys. Rev. C 57 (1998) 6883


(d,α) and (p,3He) transfer


Elliott’s SU(3) model of rotation

Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type:ˆ H = pk

2

2m+

12

mω 2rk2

⎡

⎣ ⎢

⎤

⎦ ⎥

k=1

A

∑ − g2ˆ Q ⋅ ˆ Q ,

ˆ Q μ ∝ rk2Y2μ ˆ r k( )

k=1

A

∑

+ pk2Y2μ ˆ p k( )

k=1

A

∑

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-

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

Limitations:LS (Russell-Saunders) coupling, not jj coupling (no spin-

orbit splitting) ⇒ (beginning of) sd shell.Q is the algebraic quadrupole operator ⇒ no major-shell

mixing.


Breaking of SU(4) symmetry

SU(4) symmetry breaking as a consequence ofSpin-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.


SU(4) breaking from masses

Double binding energy difference δVnp

δVnp in sd-shell nuclei:δVnp N,Z( )= 1

4 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 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


Pseudo-spin symmetry

Apply a helicity transformation to the spin-orbit + orbit-orbit nuclear mean field:

Degeneracies occur for 4ζ=κ.

ˆ u k−1 ζ ˆ l k ⋅ ˆ s k + κ ˆ l k ⋅ ˆ l k( ) u k = 4ζ −κ( )ˆ l k ⋅ ˆ s k + κ ˆ l k ⋅ ˆ l k + cte

ˆ u k = 2iˆ s k ⋅ pk

pk

K.T. Hecht & A. Adler, Nucl. Phys. A 137 (1969) 129A. Arima et al., Phys. Lett. B 30 (1969) 517R.D. Ratna et al., Nucl. Phys. A 202 (1973) 433J.N. Ginocchio, Phys. Rev. Lett. 78 (1998) 436


Pseudo-SU(4) symmetryAssume the nuclear hamiltonian is invariant under

pseudo-spin and isospin rotations:

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 H nucl, ˜ ˆ S μ[ ]= ˆ H nucl, ˆ T ν[ ]= ˆ H nucl, ˜ ˆ Y μν[ ]= 0

˜ ˆ S μ = ˜ ˆ s μ k( ),k=1

A

∑ ˆ T ν = ˆ t ν k( )k=1

A

∑ , ˜ ˆ Y μν = ˜ ˆ s μ k( )k=1

A

∑ ˆ t ν k( )

D. Strottman, Nucl. Phys. A 188 (1971) 488


Test of pseudo-SU(4) symmetry

Shell-model test of pseudo-SU(4).Realistic interaction in pf5/2 g9/2 space.Example: 58Cu.



Pseudo-SU(4) and β decayPseudo-spin transformed Gamow-Teller operator is

deformation dependent:

Test: β decay of 18Ne vs.58Zn.

˜ ˆ s μ ˆ t ν ≡ ˆ u −1ˆ s μ ˆ t ν ˆ u = −13

ˆ s μ ˆ t ν +203

1r2 r × r( ) 2( ) × ˆ s [ ]

μ

1( ) ˆ t ν

A. Jokinen et al., Eur. Phys. A. 3 (1998) 271


Three faces of the shell model


Symmetries of the interacting boson model

The interacting boson modelAlgebraic structureGeometric interpretationPartial dynamical symmetriesExtensions of the IBM


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:

ˆ S , ˆ S +[ ]=1−ˆ n

Ω≈1 while s,s+[ ]=1


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 byrequiring same commutation relations;associating state vectors.

s+ ⇔ ˆ S + ≡ α j a j+ × a j

+( )0

0( )

j∑ , dμ+ ⇔ ˆ D μ

+ ≡ β jj ' a j+ × a j '

+( )μ

2( )

jj '∑

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


The interacting boson model

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

Justification fromShell 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.

ˆ H IBM = εibi+bi

i=1

6

∑ +14

υ ijlkbi+b j

+bkblijkl=1

6

∑


DimensionsAssume Ω

available 1-fermion states. Number of N-

fermion states is

Assume Ω

available 1-boson states. Number of N- boson states is

Example: 162Dy96 with 14 neutrons (Ω=44) and 16 protons (Ω=32) (132Sn82 inert core).SM dimension: 7·1019

IBM-1 dimension: 15504

ΩN

⎛

⎝ ⎜

⎞

⎠ ⎟ =

Ω!N! Ω − N( )!

Ω + N −1N

⎛

⎝ ⎜

⎞

⎠ ⎟ =

Ω + N −1( )!N! Ω −1( )!


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.

bi

+,i =1,K,6{ }= s+,d−2+ ,d−1

+ ,d0+,d+1

+ ,d+2+{ }, bi = bi

+( )+

bi+b j ,bk

+bl[ ]= bi+blδ jk − bk

+bjδil

ˆ H U(6) = E0 + bi+bi

i=1

6

∑ ⇒ ˆ H U(6),bi+b j[ ]= 0


The IBM hamiltonian

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

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

ˆ H IBM = E0 + εs ˆ n s + εd ˆ n d + ˜ υ l1l2 ′ l 1 ′ l 2L bl1

+ × bl2

+( )L( )⋅ ˜ b ′ l 1

× ˜ b ′ l 2( )L( )

l1l2 ′ l 1 ′ l 2 ,L∑

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

1( ){ }A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069


Dynamical symmetries of the IBMU(6) has the following subalgebras:

Three solvable limits are found:

U 5( )= d+ × ˜ d ( )μ

0( ), d+ × ˜ d ( )μ

1( ), d+ × ˜ d ( )μ

2( ), d+ × ˜ d ( )μ

3( ), d+ × ˜ d ( )μ

4( ){ }SU 3( )= d+ × ˜ d ( )μ

1( ), s+ × ˜ d + d+ × ˜ s ( )μ

2( )− 7

4d+ × ˜ d ( )μ

2( ){ }SO 6( )= d+ × ˜ d ( )μ

1( ), s+ × ˜ d + d+ × ˜ s ( )μ

2( ), d+ × ˜ d ( )μ

3( ){ }SO 5( )= d+ × ˜ d ( )μ

1( ), d+ × ˜ d ( )μ

3( ){ }

U 6( )⊃U 5( )⊃ SO 5( )

SU 3( )SO 6( )⊃ SO 5( )

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

⊃ SO 3( )

O. Castaños et al., J. Math. Phys. A 20 (1979) 35


The U(5) vibrational limit

U(5) Hamiltonian:

Energy eigenvalues:

ˆ H U 5( ) = ε ˆ n d + cL12

L= 0,2,4∑ d+ × d+( )L( )

⋅ ˜ d × ˜ d ( )L( )

E nd ,υ,L( )= ε nd + κ1nd nd + 4( )+ κ4υ υ + 3( )+ κ5L L +1( )with

κ1 = 112 c0

κ4 = − 110 c0 + 1

7 c2 − 370 c4

κ5 = − 114 c2 + 1

14 c4


The U(5) vibrational limit

Anharmonic vibration spectrum associated with the quadrupole oscillations of a spherical surface.

Conserved quantum numbers: nd , υ, L.

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


The SU(3) rotational limit

SU(3) Hamiltonian:

Energy eigenvalues:

ˆ H SU 3( ) = a ˆ Q ⋅ ˆ Q + b ˆ L ⋅ ˆ L

E λ,μ,L( )= κ2 λ2 + μ2 + 3λ + 3μ + λμ( )+ κ5L L +1( )with

κ2 = 12 a

κ5 = b − 38 a


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


The SO(6) γ-unstable limitSO(6) Hamiltonian:

Energy eigenvalues:

E σ,υ,L( )= κ3 N N + 4( )−σ σ + 4( )[ ]+ κ4υ υ + 3( )+ κ5L L +1( )with

κ3 = 14 a

κ4 = 12 b

κ5 = − 110 b + c

ˆ H SO 6( ) = a ˆ P + ⋅ ˆ P + b ˆ T 3 ⋅ ˆ T 3 + c ˆ L ⋅ ˆ L


The SO(6) γ-unstable limitRotation-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) 468L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788


The IBM symmetries

Three analytic solutions: U(5), SU(3) & SO(6).


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.


Applications of IBM


The ratio R42


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 shapeSpheroidal equilibrium shape


Vibration about a spherical shape

Vibrations are characterized by a multipole quantum number λ in surface parametrization:

λ=0: compression (high energy)λ=1: translation (not an intrinsic excitation)λ=2: quadrupole vibration

R θ,ϕ( ) = R0 1 + αλμYλμ* θ,ϕ( )

μ =− λ

+ λ

∑λ∑

⎛

⎝ ⎜ ⎞

⎠ ⎟

⇔ ⇔


Vibration about a spheroidal shape

The vibration of a shape with axial symmetry is characterized by aλν

.Quadrupolar oscillations:

ν=0: along the axis of symmetry (β)

ν=±1: spurious rotation ν=±2: perpendicular to axis of

symmetry (γ)

c β

c β

⇔⇔γγ


Classical limit of IBMFor large boson number N, a coherent (or intrinsic)

state is an approximate eigenstate,

The real parameters αμ

are related to the three Euler angles and shape variables β and γ.

Any IBM hamiltonian yields energy surface:

ˆ H IBM N;αμ ≈ E N;αμ , N;αμ ∝ s+ + αμdμ+

μ∑( )N

o

J.N. Ginocchio & M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744A.E.L. Dieperink et al., Phys. Rev. Lett. 44 (1980) 1747A. Bohr & B.R. Mottelson, Phys. Scripta 22 (1980) 468

N;αμˆ H IBM N;αμ = N;βγ ˆ H IBM N;βγ ≡ V β,γ( )


Geometry of IBM

A simplified, much used IBM hamiltonian:

HCQF can acquire the three IBM symmetries.HCQF has the following classical limit:

ˆ H CQF = εd ˆ n d −κ ˆ Q χ ⋅ ˆ Q χ , ˆ Q μχ = s+ ˜ d μ + dμ

+s + χ d+ × ˜ d ( )μ

2( )

VCQF β,γ( )≡ N;βγ ˆ H CQF N;βγ

= εd N β 2

1+ β 2 −κN5 + 1+ χ 2( )β 2

1+ β 2

−κN N −1( )1+ β 2( )2

27

χ 2β 4 − 4 27

χβ 3 cos3γ + 4β 2⎛

⎝ ⎜

⎞

⎠ ⎟


Phase diagram of IBM

E. López-Moreno and O. Castaños, Phys. Rev. C 54 (1996) 2374J. Jolie et al., Phys. Rev. Lett. 87 (2001) 162501


Partial dynamical symmetries (PDS)

Hamiltonians with dynamical symmetry:

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) L1285A. Leviatan et al., Phys. Lett. B 172 (1986) 144

Gdyn ⊃ L Gbreak L ⊃ Gsym

↓ ↓ ↓hN[ ] Σ Λ


How to construct a type-1 PDS?

Starting point: A DS classification of eigenstates and tensor operators.

For a given representation Σ0 , find n-particle annihilation operators T such that

∴Any interaction written in terms of T and T+

preserves solvability of ⎢[hN ] Σ=Σ0 Λ⟩

states.Task: Check whether any state in [hN-n ] belongs to

the Kronecker product Σ0 ⊗ σ.

ˆ T hn[ ]σλ hN[ ]Σ0Λ = 0, ∀Λ ∈ Σ0

J.E. García-Ramos et al., Phys. Rev. Lett. 102 (2009) 112502


SO(6) dynamical symmetry

Classification scheme:

Hamiltonian with SO(6) DS:

All states ⎪[N]ΣτL⟩

are solvable with energy

U 6( ) ⊃ SO 6( ) ⊃ SO 5( ) ⊃ SO 3( )↓ ↓ ↓ ↓ ↓N[ ] Σ τ ν Δ L

ˆ H DS = κ1P+P− + κ2ˆ C 2 SO 5( )[ ]+ κ3

ˆ C 3 SO 3( )[ ], P− = ss − d ⋅ d

E = κ114

N − Σ( ) N + Σ + 4( )+ κ2τ τ + 3( )+ κ3L L +1( )


SO(6) partial dynamical symmetry

Classification scheme:

Hamiltonian with SO(6) PDS:

Only states ⎪[N]Σ=NτL⟩

are solvable with energy

U 6( ) ⊃ SO 6( ) ⊃ SO 5( ) ⊃ SO 3( )↓ ↓ ↓ ↓ ↓N[ ] Σ τ ν Δ L

H PDS = ˆ H DS + η1

ˆ P + ˆ V 1 ˆ P − + η2ˆ P + ˆ V 2 ˆ P − +L

E = κ2τ τ + 3( )+ κ3L L +1( )

J.E. García-Ramos et al., Phys. Rev. Lett. 102 (2009) 112502


Example: 196Pt


Anharmonic vibrations

The SO(6) limit yields a spectrum of vibrational bands for a γ-unstable deformed rotor, with Lπ=0+ band heads with Σ=N-2υ, υ=0,1,2,…

Anharmonicity of the vibration quantified by R=Ex (0+,υ=2)/Ex (0+,υ=1)-2.

Comparison with observed anharmonicity:In SO(6) DS: R=-2/(N+1)=-0.29In SO(6) PDS: R=-0.63In 196Pt: R=-0.70


E2 transitions rates in 196Pt


Example of type-2 PDS

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


Extensions of the IBM

Neutron and proton degrees freedom (IBM-2):F-spin multiplets (Nν

+Nπ

=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): Rν

-Rπ

⇒ E1 excitation.Angular displacement (scissors resonance): Lν

-Lπ

⇒ M1 excitation.

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


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 eν

and eπ

.

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


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 ε and interactions υ.

ˆ H IBFM = ˆ H IBM + ε ja j+a j

j=1

Ω

∑ + υ i1 j1i2 j2bi1

+a j1+ bi2

a j2j1 j2 =1

Ω

∑i1i2 =1

6

∑

U 6( )⊕ U Ω( )=bi1

+bi2

a j1+ a j2

⎧ ⎨ ⎩

⎫ ⎬ ⎭


Example: 195Pt117


Example: 195Pt117 (new data)


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

U 6( )⊕ U Ω( )=bi1

+bi2

a j1+ a j2

⎧ ⎨ ⎩

⎫ ⎬ ⎭

U 6 /Ω( )=bi1

+bi2bi1

+a j2

a j1+ bi2

a j1+ a j2

⎧ ⎨ ⎩

⎫ ⎬ ⎭

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


U(6/12) supermultiplet


Example: 194Pt116 &195Pt117


Quartet supersymmetry

A simultaneous description of even-even, even-odd, odd-even and odd-odd nuclei (quartets).

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


78195Pt117

aπ+bπ⎯ → ⎯ ⎯

bπ+aπ

← ⎯ ⎯ ⎯ 79196 Au117

aν+bν ↑ ↓bν

+aν aν+bν ↑ ↓bν

+aν

78194 Pt116

aπ+bπ⎯ → ⎯ ⎯

bπ+aπ

← ⎯ ⎯ ⎯ 79195 Au116


Quartet supersymmetry

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


Example of 196Au


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.


Symmetry rules

Symmetry is a universal concept relevant in mathematics, physics, chemistry, biology, art…

In science in particular it enablesTo 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.


Symmetry in physicsFundamental 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.


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.

symmetries in nuclei - unambijker/elaf2010/isacker.pdf · symmetries of the nuclear shell model....

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