symmetries of the nuclear shell modelific.uv.es/~euschool/lec_isacker02.pdf · symmetries of the...

Symmetries in N~Z nuclei (II), Valencia, September 2003

Symmetries of thenuclear shell model

P. Van Isacker, GANIL, France

Racah’s SU(2) pairing modelWigner’s SU(4) symmetryElliott’s SU(3) model of rotationGeneralized pairing models*Generalized SU(3) models*


Nuclear shell model• Many-body quantum mechanical problem:

• Independent-particle assumption. Choose Vand neglect residual interaction:

H =pk

2

2mkk =1

A

Â + W xk,xl( )k < l

A

Â xk ≡r r k,

r s k ,r t k{ }, r p k = -ih

r — k[ ]

=pk

2

2mk

+ V xk( )È

Î Í ˘

˚ ˙ k =1

A

Âmean field

1 2 4 4 4 3 4 4 4

+ W xk ,xl( )k <l

A

Â - V xk( )k =1

A

ÂÈ

Î Í ˘

˚ ˙

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

H ª HIP =pk

2

2mk

+ V xk( )È

Î Í ˘

˚ ˙ k =1

A

Â


Independent-particle shell model• Solution for one particle:

• Solution for many particles:

pk2

2mk

+ V xk( )È

Î Í ˘

˚ ˙ f i k( ) = Eif i k( ) f i k( ) ≡ fir r k ,r s k ,

r t k( )[ ]

F i1i2Ki A1,2,K, A( ) = fik

k( )k = 1

A

’

HIPFi1i2 Ki A1, 2,K, A( ) = Eik

k =1

A

ÂÊ

Ë Á ˆ

¯ Fi 1i2 KiA

1, 2,K, A( )


Independent-particle shell model• Antisymmetric solution for many fermions

(Slater determinant):

• Example for A=2 fermions:

Yi1i2Ki A1,2,K, A( ) =

1A!

f i11( ) fi 1

2( ) K f i1 A( )fi 2

1( ) fi 22( ) K fi 2

A( )M M O M

fiA 1( ) f iA2( ) K fiA A( )

Yi1i2 1,2( ) =12

f i1 1( )fi 22( ) -f i1

2( )fi2 1( )[ ]


Hartree-Fock approximation• Vary fi (i.e. V) to minize the expectation value

of H in a Slater determinant:

• Application requires choice of H. Many globalparametrizations (Skyrme, Gogny,…) havebeen developed.

dYi1i2 KiA

* 1, 2,K, A( )HYi1i2Ki A1,2,K,A( )dx1dx2 KdxAÚ

Yi1i 2KiA

* 1,2,K, A( )Yi1i2 KiA1, 2,K, A( )dx1dx2 KdxAÚ

= 0


Poor man’s Hartree-Fock• Choose a simple, analytically solvable V that

approximates the microscopic HF potential:

• Contains– Harmonic oscillator potential with constant w.– Spin-orbit term with strength zls.– Orbit-orbit term with strength zll.

• Adjust w, zls and zll to best reproduce HF.

HIP =

pk2

2m+

12

mw 2rk2 -z ls

r l k ⋅

r s k -z ll lk2È

Î Í ˘

˚ ˙ k = 1

A

Â


Symmetries of the shell model• Three bench-mark solutions:

– No residual interaction fi IP shell model.– Pairing (in jj coupling) fi Racah’s SU(2).– Quadrupole (in LS coupling) fi Elliott’s SU(3).

• Symmetry triangle:

HIP =pk

2

2m+

12

mw 2rk2 -z ls

r l k ⋅

r s k -z ll lk2È

Î Í ˘

˚ ˙ k = 1

A

Â

+ VRI xk,xl( )k <l

A

Â


Racah’s SU(2) pairing model• Assume large spin-orbit splitting zls which

implies a jj coupling scheme.• Assume pairing interaction in a single-j shell:

• Spectrum of 210Pb:

j2 JMJ Vpairing j2JMJ =- 1

22 j + 1( )g, J = 0

0, J ≠ 0Ï Ì Ó


SU(2) quasi-spin formalism• The pairing hamiltonian,

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

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

• Eigensolutions of pairing hamiltonian:

H = E0 - gS+ ⋅ S- , S+ = 12 2j + 1 aj

+ ¥ aj+( )

0

0( ), S- = S+( )+

S+ ,S-[ ] = 12 2nj - 2j - 1( ) ≡ -2Sz, Sz ,S±[ ] = ±S±

-gS+ ⋅ S- = -g S2 - Sz2 + Sz( )

-gS+ ⋅ 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 nj and seniority u:

• Energy eigenvalues in terms of nj and u:

• Eigenstates have an S-pair character:

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

S = 14 2j - u + 1( ), MS = 1

4 2nj - 2j - 1( )

j njuJMJ - gS+ ⋅ S- jnj uJMJ = - 14 g nj -u( ) 2j - nj +u + 3( )

j njuJMJ µ S+( ) nj -u( ) / 2 juuJMJ

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


Pairing and superfluidity• Ground states of a pairing hamiltonian have

superfluid character:– Even-even nucleus (u=0):– Odd-mass nucleus (u=1):

• Nuclear superfluidity leads to– Constant energy of first 2+ in even-even nuclei.– Odd-even staggering in masses.– Two-particle (2n or 2p) transfer enhancement.

S+( )n j / 2 o

aj+ S+( )nj / 2 o


Superfluidity in semi-magic nuclei• Even-even nuclei:

– Ground state hasu=0.

– First-excited state hasu=2.

– Pairing producesconstant energy gap:

• Example of Snnuclei:

Ex 21+( ) = 1

2 2j + 1( )g


Shell structure of nuclei• Direct evidence for nuclear shell structure is

obtained from Ex(2+) (here scaled by A1/3):


Superfluidity versus magicity• Two-nucleon separation energies S2n:

(a) Shell splitting dominates over interaction.(b) Interaction dominates over shell splitting.(c) S2n in tin isotopes.


Generalized pairing models• Trivial generalization from a single-j shell to

several degenerate j shells:

• Pairing with neutrons and protons:– T=1 pairing: SO(5).– T=0 & T=1 pairing: SO(8).

• Non-degenerate shells:– Talmi’s generalized seniority.– Richardson’s integrable pairing model.

S+ µ 12 2j + 1 aj

+¥ a j

+( )0

0( )

jÂ


Pairing with neutrons and protons• For neutrons and protons two pairs and hence

two pairing interactions are possible:– Isoscalar (S=1,T=0):

– Isovector (S=0,T=1):-S+

01 ⋅ S-01 , S+

01 = l + 12 a

l 12

12

+ ¥ al 1

212

+Ê Ë

ˆ ¯

001( )

, S-01 = S+

01( )+

-S+10 ⋅ S-

10 , S+10 = l + 1

2 al 1

212

+ ¥ al 1

212

+Ê Ë

ˆ ¯

010( )

, S-10 = S+

10( )+


Neutron-proton pairing hamiltonian• A hamiltonian with two pairing terms,

• …has an SO(8) algebraic structure.• H is solvable (or has dynamical symmetries)

for g0=0, g1=0 and g0=g1.

H = -g0S+10 ⋅ S-

10 - g1S+01 ⋅S-

01


SO(8) ‘quasi-spin’ formalism• A closed algebra is obtained with the pair

operators S± with in addition

• This set of 28 operators forms the Lie algebraSO(8) with subalgebras

n = 2 2l + 1 al 1

212

+ ¥ al 1

212

+Ê Ë

ˆ ¯ 000

000( )

, Ymn = 2l + 1 al 1

212

+ ¥ al 1

212

+Ê Ë

ˆ ¯ 0 mn

011( )

Sm = 2l + 1 al 1

212

+ ¥ al 1

212

+Ê Ë

ˆ ¯

0m 0

010( )

, Tn = 2l + 1 al 1

212

+ ¥ al 1

212

+Ê Ë

ˆ ¯

00n

001( )

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

SO 6( ) ª SU 4( ) = S,T,Y{ }, SOS 5( ) = n,S,S±10{ },

SOT 5( ) = n,T,S±01{ }, SOS 3( ) = S{ }, SOT 3( ) = T{ }


Solvable limits of the SO(8) model• Pairing interactions can expressed as follows:

• Symmetry lattice of the SO(8) model:

• fiAnalytic solutions for g0=0, g1=0 and g0=g1.

S+01 ⋅ S-

01 = 12C2 SOT 5( )[ ] - 1

2C2 SOT 3( )[ ] - 1

8(2l - n + 1)(2l - n +7)

S+01 ⋅ S-

01 + S+10 ⋅ S-

10 = 12 C2 SO 8( )[ ] - 1

2 C2 SO 6( )[ ] - 18 (2l - n + 1)(2l - n + 13)

S+10 ⋅ S-

10 = 12 C2 SOS 5( )[ ] - 1

2 C2 SOS 3( )[ ]- 18 (2l - n + 1)(2l - n + 7)

†

SO(8) …

SOS 5( ) ƒ SOT 3( )SO(6) ª SU 4( )

SOT 5( ) ƒ SOS 3( )

Ï

Ì Ô

Ó Ô

¸

˝ Ô

˛ Ô

… SOS 3( ) ƒ SOT 3( )


Superfluidity of N=Z nuclei• Ground state of a T=1 pairing hamiltonian for

identical nucleons is superfluid, (S+)n/2oÒ.• Ground state of a T=0 & T=1 pairing

hamiltonian with equal number of neutronsand protons has different superfluid character:

• fi Condensate of a’s (q depends on g0/g1).• Observations:

– Isoscalar component in condensate survives onlyin N~Z nuclei, if anywhere at all.

– Spin-orbit term reduces isoscalar component.

cosq S+10 ⋅ S+

10 - sinq S+01 ⋅ S+

01( )n / 4o


Superfluidity versus magicity

12e j - eaj

Â S+ j( )Ê

Ë Á

ˆ

¯ ˜

a= 1

n / 2

’ o

1 - g 2j + 12e j - ea

- 4g 1ea - eb

= 0, a = 1,2,K,n/ 2b ≠a( )Â

jÂ

R.W. Richardson, Phys. Lett. 5 (1963) 82

• Superfluid ground state for degenerate shells:

• ‘Superfluid’ ground state for non-degenerateshells (Richardson-Gaudin-Dukelsky):

S+ j( )j

ÂÊ

Ë Á

ˆ

¯ ˜

n / 2

o


Wigner’s SU(4) symmetry• Assume the nuclear hamiltonian is invariant

under spin and isospin rotations:

• Since {Sm,Tn,Ymn} 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 (lmn) areconserved quantum numbers.

Hnucl ,Sm[ ] = Hnucl ,Tn[ ] = Hnucl ,Ymn[ ] = 0

Sm = sm k( ),k = 1

A

Â Tn = tn k( )k =1

A

Â , Ymn = sm k( )k = 1

A

Â tn 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 wavefunction:

• Nuclear interaction is short-range attractiveand hence favours maximal spatial symmetry.


Breaking of SU(4) symmetry• Non-dynamical breaking of SU(4) symmetry

as a consequence of– Spin-orbit term in nuclear mean field.– Coulomb interaction.– Spin-dependence of residual interaction.

• Evidence for SU(4) symmetry breaking from– Masses: rough estimate of nuclear BE from

– b decay: Gamow-Teller operator Ym,±1 is agenerator of SU(4) fi selection rule in (lmn).

B N, Z( ) µ a + bg lmn( ) = a + b lmn C2 SU 4( )[ ] lmn


SU(4) breaking from masses• Double binding energy difference dVnp

• dVnp in sd-shell nuclei:dVnp 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 b decay• Gamow-Teller decay into odd-odd or even-

even N=Z nuclei:

P. Halse & B.R. Barrett, Ann. Phys. (NY) 192 (1989) 204


Elliott’s SU(3) model of rotation• Harmonic oscillator mean field (no spin-orbit)

with residual interaction of quadrupole type:

• State labelling in LS coupling:

H =pk

2

2m+

12

mw 2rk2È

Î Í ˘

˚ ˙ k =1

A

Â -kQ ⋅Q,

Qm =4p5

rk2Y2m ˆ r k( )

k = 1

A

Â + pk2Y2m ˆ p k( )

k = 1

A

ÂÊ

Ë Á ˆ

¯

U 4w( )Ø

1M[ ]…

U L w( )Ø

˜ l ̃ m ̃ n ( )

…SUL 3( )Ø

l m ( )

…SOL 3( )Ø

L

Ê

Ë

Á Á

ˆ

¯

˜ ˜

ƒ

SUST 4( )Ø

lmn( )

…SU S 2( )Ø

S

ƒ SUT 2( )Ø

T

Ê

Ë

Á Á

ˆ

¯

˜ ˜

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


Importance and limitations of SU(3)• Historical importance:

– Bridge between the spherical shell model and theliquid droplet model through mixing of orbits.

– Spectrum generating algebra of Wigner’s SU(4)supermultiplet.

• Limitations:– LS (Russell-Saunders) coupling, not jj coupling

(zero spin-orbit splitting) fi beginning of sd shell.– Q is the algebraic quadrupole operator fi no

major-shell mixing.


Generalized SU(3) models• How to obtain rotational features in a jj-

coupling limit of the nuclear shell model?• Several efforts since Elliott:

– Pseudo-spin symmetry.– Quasi-SU(3) symmetry (Zuker).– Effective symmetries (Rowe).– FDSM: fermion dynamical symmetry model.– ...


Pseudo-spin symmetry• Apply a helicity transformation to the spin-

orbit + orbit-orbit nuclear mean field:

• Degeneraciesoccur for 4zll=zls.

uk-1 z ls

r l k ⋅

r s k +z ll lk2( )uk = 4z ll -z ls( )

r l k ⋅

r s k +z ll lk2 + cte

uk = 2ir s k ⋅

r r krk

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) symmetry• Assume 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) labelsare conserved quantum numbers.

Hnucl , ˜ S m[ ] = Hnucl ,Tn[ ] = Hnucl , ˜ Y mn[ ] = 0

˜ S m = ˜ s m k( ),k = 1

A

Â Tn = tn k( )k =1

A

Â , ˜ Y mn = ˜ s m k( )k= 1

A

Â tn 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

pf5/2g9/2 space.• Example: 58Cu.

P. Van Isacker et al., Phys. Rev. Lett. 82 (1999) 2060


Pseudo-SU(4) and b decay• Pseudo-spin transformed Gamow-Teller

operator is deformation dependent:

• Test: b decay of18Ne vs. 58Zn.

˜ s mtn ≡ u-1smtnu = -

13

smtn +203

1r 2

r r ¥ r r ( ) 2( )¥

r s [ ]m

1( )tn

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


Quasi-SU(3) symmetry• Similar matrix elements of Q in LS and jj

coupling lead to approximate decoupling ofj=l-1/2 and j=l+1/2 spaces.

A. Zuker et al., Phys. Rev. C 52 (1995) R1741


Fermion dynamical symmetry model• Construct shell-model hamiltonian in terms of

S and D pairs such that the S,D-spacedecouples from the full shell-model space.

• Generalization of pseudo-spin (j=k+i):

• Many possible combinations of single-particleorbits via an appropriate choice of k and i.

• Limitation: structure of pairs is algebraicallyimposed.

akm k imi

+ = kmk imi jmj al 1

2, jm j

+

jm j

Â

J.N. Ginocchio, Ann. Phys. (NY) 126 (1980) 234C.L. Wu et al., Phys. Rev. C 36 (1987) 1157

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