symmetries in nuclei, tokyo, 2008 symmetries in nuclei symmetry and its mathematical description the...

Symmetries in Nuclei, Tokyo, 2008

Symmetries in Nuclei

Symmetry and its mathematical descriptionThe role of symmetry in physicsSymmetries of the nuclear shell modelSymmetries of the interacting boson model


Symmetry in physics

Geometrical symmetriesExample: C3 symmetry of O3, nucleus 12C, nucleon

(3q).

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

€

ˆ H =pk

2

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 x ll∑

€

xμ → ′ x μ = Aμν xνν∑

€

xk → ′ x k = −xk

€

t → ′ t = −t

€

xk → ′ x k = Akl x ll∑ + ak

€

xμ → ′ x μ = Aμν xνν∑ + aμ

€

xk → ′ x k = Dk xk


Symmetry in physics

Quantum-mechanical symmetries, U(n)

Internal (global gauge) symmetries. Examples:pn, isospin SU(2)uds, flavour SU(3)

(Local) gauge symmetries. Example: Maxwell equations, U(1).

Dynamical symmetries. Example: the Coulomb problem, SO(4).

€

ψk → ′ ψ k = Bklψ l

l=1

n

∑ inv. ψ ll∑

2


Symmetry in quantum mechanicsAssume 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 close under commutation.€

∀ˆ g i ∈ G : ˆ H , ˆ g i[ ] = 0


Consequences of symmetry

Degeneracy structure: If is an eigenstate of H with energy E, so is gi:

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

∑


Hydrogen atom

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−

α

r

€

ˆ H Ψnlm r,θ,ϕ( ) = −Mα 2

2h2n2Ψnlm r,θ,ϕ( )

with n =1,2,K ; l = 0,1,K ,n −1; m = −l,K ,+l

E. Schrödinger, Ann. Phys. 79 (1926) 361


Energy spectrum


Classical Kepler problem

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

Angular momentum (=eccentricity):

Runge-Lentz vector:

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

€

E = −α

2a

€

L = r∧p, L2 = Mα a 1−ε 2( )

€

R =p∧L

M−α

r

r, R2 =

2H

ML2 + α 2


Classical Kepler problem


Quantization of operators

From p-ih:

Some useful commutators & relations:

€

ˆ H = −h2

2M∇ 2 +

α

r

⎛

⎝ ⎜

⎞

⎠ ⎟

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

Mˆ L 2 + h2

( ) + α 2


Conservation of angular momentumThe 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 jklˆ L l , j,k, l = x,y,z


Conserved Runge-Lentz vector

The Runge-Lentz 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 [ ] =

h2α

M

1

r∇ −

r

r31+ 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/2E)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 jkl

2 ˆ H

Mˆ L l , j,k, l = x, y,z


Energy spectrum

Isomorphism of SO(4) and SO(3)SO(3):

Since L and A' are orthogonal:

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

€

ˆ F ± =1

2ˆ 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 − =1

2ˆ L 2 −

M

2Hˆ R 2

⎛

⎝ ⎜

⎞

⎠ ⎟= −

Mα 2

4H−

1

2h2

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

2 2 j +1( )2h2

, j = 0, 1

2,1, 3

2,

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


Dynamical symmetry

Two algebras G1 G2 and a hamiltonian

H has symmetry G2 but not G1!

Eigenstates are independent of parameters m and n in H.

Dynamical symmetry breaking “splits but does not admix eigenstates”.

€

ˆ H = μmˆ C m G1[ ]

m

∑ + ν nˆ 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 = 1

2, mt = + 1

2; p : t = 1

2, mt = − 1

2

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

2n, ˆ t z p = − 1

2p

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=49Empirical 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 symmetryCoulomb interaction can be approximated as

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

MT-degeneracy is lifted according to

Summary of labelling:

€

ˆ H Coul ≈ κ 0 + κ1ˆ T z + κ 2

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

€

ˆ H Coul ηTMT = κ 0 + κ1MT + κ 2MT2

( ) ηTMT

SU 2( ) ⊃ SO 2( )

↓ ↓

T MT


Isobaric multiplet mass equationIsobaric multiplet mass equation:

Example: T=3/2 multiplet for A=13 nuclei.E ηTMT( )=κ η,T( ) +κ1MT +κ2MT

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( ) =e

2 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 mixing

B(E1;5-4+) in 64Ge from:

lifetime of 5- level;(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%


Pairing SU(2) dynamical symmetryThe pairing hamiltonian,

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

H has SU(2) SO(2) dynamical symmetry:

Eigensolutions of pairing hamiltonian:

€

ˆ H = −g ˆ S + ⋅ ˆ S −, ˆ S + = 1

2a jm

+ a jm +

m∑ , ˆ S − = ˆ S +( )

+

€

ˆ S +, ˆ S −[ ] = 1

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

€

−g ˆ S + ⋅ ˆ S − = −g ˆ S 2 − ˆ S z2 + ˆ S z( )

€

−g ˆ S + ⋅ ˆ S − SMS = −g 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 :

Energy eigenvalues in terms of n, j and :

Eigenstates have an S-pair character:

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

€

S = 1

42 j −υ +1( ), MS = 1

42n − 2 j −1( )

€

j nυ JMJ − g ˆ S + ⋅ ˆ S − j nυJMJ = − 1

4g n −υ( ) 2 j − n + υ + 3( )

€

j nυJMJ ∝ ˆ S +( )n−υ( ) / 2

jυ υJMJ

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


Quantal many-body systems

Take a generic many-body hamiltonian:

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

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

€

ˆ H = ε ic i+c i + υ ijklc i

+c j+ckc l

ijkl

∑i

∑ +L

€

ˆ H = ε iδ il − −( )q

υ ijkl

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ˆ u il + −( )

qυ ijkl

ˆ u ik ˆ u jlijkl

∑il

∑ +L

€

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


Quantal many-body systems

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 nˆ 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

symmetries in nuclei, tokyo, 2008 symmetries in nuclei symmetry and its mathematical description the...

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