even-evennuclei

odd-evennuclei

odd-oddnuclei

3.1 The interacting boson-fermion model

s,d,aj

N bosons1 fermion

e-o nucleus

s,d

N+1 bosons

IBFA

e-e nucleus

fermions cj

M valence nucleons

A nucleons

L = 0 and 2 pairs

nucleon pairs

A. Arima, F. Iachello, T. Otsuka, O. Scholten, I Talmi

Odd-A nuclei: the interacting boson-fermion approximation

0,, jmjmjmjm aaaa

jmjm aa ,´´, mllm bb

´,´,´´, mmjjmjjm aa

0,, ´´´´ mllmmllm bbbb ´´´´, mmllmllm bb

0,,,, ´´´´´´´´ mjlmmjlmmjlmmjlm abababab

and

and

BFFB VHHH

jj

jF nH ˆ

0

0´´´´´´

~~

Jjljl

J

mjlm

J

jlJ

jjlBF ababvV

HB the IBM-1 Hamiltonian

the single particle Hamiltonian with the energies j.

the boson-fermion interaction

The most general hamiltonian contains much too many parameter andis replaced by a simpler one based on shell model considerations and BCS.

0

0

)0()0( ~~

jjjj

BF aaddVmonopole

It has three terms

00´

)2(

´´~

jjjj

Bjj

BF aaQVquadrupole

:~~:

0

0´´´

´´)(´´)(´´´

jjj

j

j

j

jjjj

BF adadVexchange

BCS calculation gives: quasiparticle energies Ej and occupation numbers uj and vj as afunction of j and .

)(1´´2

´||´´||)(´´||||)(10

´||||)(

,

´´

2´´´´´´2´´´´´´´

2´´´

jj

jjjjjjjjjjj

jjjjjj

j

EEj

jYjuvvujYjuvvuBFE

jYjvvuuBFQ

BFM

O. Scholten PhD + ODDA code

jj

jF nEH ˆ

mit the excitation energy of the first 2+ state in the corresponding semimagical nucleus we now have n single particle energies, the gap and three parameters + six for the boson part.

Example: odd Rhodium isotopes (J.Jolie et al. Nucl.Phys. A438 (1985)15

HB from fit of Pd isotopes by Van Isacker et al.

=+ =-

=1.5 MeV

N s,d bosons: U(6) symmetry for model space

222

22120212222

21012

....

.....

.....

.....

ddsd

ddddddddddsd

dsdsdsdsdsss

36 generatorsN s,d bosons+ j fermion: UB(6)xUF(2j+1) Bose-Fermi symmetry

jj aa

bb

0

0

36 boson generators + (2j+1)2 fermion generators, which both couple to integer total spin and fullfil the standard Lie algebra conditions.

3.2 Bose-Fermi symmetries

Bose-Fermi symmetries

Two types of Bose-Fermi symmetries: spinor and pseudo spin types

Spinor type: uses isomorphism between bosonic and fermionic groupsSpin(3): SOB(3) ~ SUF(2)Spin(5): SOB(5) ~ SpF(4)Spin(6): SOB(6) ~ SUF(4)

Exemple: SO(6) core and j=3/2 fermionBalantekin, Bars, Iachello, Nucl. Phys. A370 (1981) 284.

UB(6)xUF(4) SOB(6)xSUF(4) Spin(6) Spin(5) Spin(3) ¦ ¦ ¦ ¦ ¦ ¦ ¦ [N] [1] < ><1/2,1/2,1/2> <1,2 ,3 > (1 ,2) J

H= A’ C2[SOB(6)] + A C2[Spin(6)] + B C2[Spin(5)]+ C C2[Spin(3)]

E = A’ ((+4)) + A(1(1+4) + 2(2+2) + 32 ) + B(1(1+3) +2(2+1)) + CJ(J+1)

Pseudo Spin type: uses pseudo-spins to couple bosonic and fermionic groups

Example: j= 1/2, 3/2, 5/2 for a fermion: P. Van Isacker, A. Frank, H.Z. Sun, Ann. Phys. A370 (1981) 284.

UB(6)xUF(12) UB(6)xUF(6)xUF(2)

UB+F(5)xUF(2)... UB+F(6)xUF(2) SUB+F(3)xUF(2)... SOB+F(3)xUF(2) Spin (3) SOB+F(6)xUF(2)...

H= B0 + A1 C2[UB+F(6)] + A C1[UB+F(5)] + A´ C2[UB+F(5)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SUB+F(3)] + E C2[SOB+F(3)] + F Spin(3)

L=2

L=0 1/2

3/25/2 L=2

L=0x x

L=0

L=2x S= 1/2

UB(6) x UF(12) UB(6) x UF(6) x UF(2)

This hamiltonian has analytic solutions, but also describes transitional situations.

Example: the SO(6) limit

H= B0 + A1 C2[UB+F(6)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + EC2[SOB+F(3)] + FSpin(3)

E= A(N1(N1 +5)+ N2(N2 +3)) + B(1(1 +4)+ 2(2 +2)) + C(1(1 +3)+ 2(2 +1))+EL(L+1) + F J(J+1)

Can we connect atomic nuclei using supersymmetry?

s,d,aj

fermions cj

s,d

N+1 bosons N bosons1 fermionM valence nucleons

A nucleons

IBFAL = 0 und 2 pairsNucleon pairs

e-e nucleus odd-A nucleus

SUSY

ajbl

F. Iachello, Phys. Rev. Lett. 44 (1980) 672

jjj

j

aaba

abbb

(6+ 2j+1)2 generators of bosonic or fermionic type

Supersymmetrie: U(6/2j+1) symmetry

Note: graded Lie algebras U(6/m) are no Lie algebras.Their generators fullfil a mixture of commutation and anticommutation relations!By removing the mixed generators one finds that theBose-Fermi symmetry is always a subalgebra of thegraded Lie algebra:

FB

NB

FB

NNNwith

NN

mUUmU

F

:

]1[][}[

)()6()/6(

[N}

[N]

[N-1]x[1] <N-1>x[1]

<N-3>x[1]

<N-5>x[1]

<N>

<N -2>

<N-4>

<N -2>

<N -4 >

<N>

<N+1/2,1/2,1/2>

<N-1/2,3/2,1/2>

<N-1/2,1/2,1/2>

(0)(1)

(2)

0+

2+

2+4+

(1/2,1/2)

(3/2,1/2)

(5/2,1/2)

3/2+1/2+5/2+7/2+

U(6/4) UB(6)xUF(4) SOB(6)xSUF(4) Spin(6) Spin(5) Spin(3) ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ [N} [N] [1m] < ><1/2,1/2,1/2> <1,2 ,3 > (1 ,2) J

In the case of a dynamical supersymmetry the same parameter set describes states in both nuclei.

E = A’ ((+4)) + A(1(1+4) + 2(2+2) + 32 )+ B(1(1+3) +2(2+1)) + CJ(J+1)

d3/2

s1/2

h13/2

d5/2

g7/2

191Ir190Os

A’=-18.3 keV, A= -27.3 keV, B= 32.3 keV, C= 9.5 keV

H = B0 + A1 C2[UB+F(6)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SOB+F(3)] + E C2[Spin(3)] E = B0 + A (1(1+6) + 2(2+4)) + B (1(1+4) + 2(2+2)) + C (1(1+3) +2(2+1)) + D L(L+1) + E J(J+1)

f5/2

p1/2

i13 /2

h9/2

f7/2

p3/2

SO(6) limit and j = 1/2, 3/2, 5/2

Example: SO(6) limit of U(6/12)

Eo-e = A (1(1+6) + 2(2+4)) + B (1(1+4) + 2(2+2)) + C (1(1+3) +2(2+1)) + D L(L+1) + E J(J+1)Ee-e = A (+6) + B (1(1+4) + 2(2+2)) + C (1(1+3) +2(2+1)) + (D+E) L(L+1)

3.5 A case study: 195Pt and the SO(6) Limit of U(6/12)

A. Mauthofer et al., Phys. Rev. C 34 (1986) 1958.

Electromagnetic transition rates

B(E2) values

B(M1) values

One particle transfer reactions (pick-up):

New results for 195Pt

DBWAlj

ljlj d

dG

dd

)()( Angular distribitions:

Spectroscopic strenghts:

12

ˆ2

22

i

ilj

fljlj J

JTJCSCG

Q3D Spectrometer at accelerator laboratory (TUM-LMUGarching)

/ . .

Q3D Spectrometer

Particle detector

196Pt (p,d)

Fribourg/Bonn/Munich

Detailed studies of 195Pt and 196Au were performed in parallel

The angular distributions reveal the parity and orbital angular momentum of the transferred neutron. Model space relevant information.The spin cannot be uniquely determined.p: 1/2 or 3/2 f: 5/2 or 7/2

0+

196Pt (d,t)

196Pt

Unique spin assignments can be obtained from polarised transfer. Then the cross sections become sensitive to the orientation of the spin of thetransferred particle.

yAlj 195Ptj with =(-1)l

New result for 195Pt

A =46.7, B+B´= -42.2 C= 52.3, D = 5.6 E = 3.4 (keV)A. Metz, Y. Eisermann, A. Gollwitzer, R. Hertenberger, B.D. Valnion, G. Graw,J. Jolie, Phys. Rev. C61 (2000) 064313

Comparison of the transfer strenghts with theory

Microscopic transfer operator:

J. Barea, C.E. Alonso, J.M. Arias, J. Jolie Phys. Rev. C71 (2005) 014314

3.5 Supersymmetry without dynamical symmetry

U(6/12) UB(6)xUF(12) UB(6)xUF(6)xUF(2) UB+F(6)xUF(2)

UB+F(5)xUF(2)... SUB+F(3)xUF(2)... SOB+F(3)xUF(2) Spin (3) SOB+F(6)xUF(2)...

H= B0 + A1 C2[UB+F(6)] + A C1[UB+F(5)] + A ´C2[UB+F(5)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SUB+F(3)] + E C2[SOB+F(3)] + F Spin(3)

This hamiltonian has analytic solutions, but also describes transitional situations,in even-even and odd-A nuclei.

Example: The Ru-Rh isotopes A. Frank, P. Van Isacker, D.D. Warner Phys. Lett. B197(1987)474

H= (7N-42) C2[UB+F(6)] + (841-54N) C1[UB+F(5)] -23.3 C2[SOB+F(6)] + 30.8 C2[SOB+F(5)] -9.5 C2[SOB+F(3)] + 15 Spin(3) (all in keV)

Even-even Ru Odd proton Rh

Proton pick-up reactions on Palladium isotopes:

2/5,2/52/3,2/32/1,2/1 2/52/32/1 jjjlj avavavT

1/23/25/2

Phase transitions in odd-A nuclei:changing single particle orbits, AND finding a simple hamiltonian

Partial solution: use the U(6/12) supersymmetry

U(6/12): U(5), O(6) and SU(3) limits + j=1/2,3/2,5/2

An extension of the Casten triangle for odd-A nuclei was proposed:D.D. Warner, P. Van Isacker, J. Jolie, A.M. Bruce, Phys. Rev. Lett. 54 (1985)1365.

Here we apply the very simple Hamiltonian

FBFBFB QQ

NUCaH

ˆ.ˆ)1()5(ˆˆ 1

with the quadrupole operator of UB+F(6). FBQ

ˆ

P. Van Isacker, A.Frank, H.Z. Sun, Ann. of Phys. 157 (1984) 183.

SU(3)-SU(3) with 10 bosons SU(3)-SU(3) with 10 bosons and one fermion

(J= 1/2 states)(J= 0+ states)

)3(FBSU )6(FBSO )3(FBSU )3(SU )6(SO )3(SU

A phase transition at as expected. )6(FBSO

Groundstate energies in SU(3) to SU(3) transitions.

0+-states 1/2-states

Similar for the U(5)-SU(3) first order phase transition

The extended Casten triangle for odd-A nuclei becomes:

Fig 1

J. Jolie, S. Heinze, P. Van Isacker, R.F. Casten, Phys. Rev. C 70 (2004) 011305(R).

But is everything so normal and expected?

)3(SU )6(SO )3(SU )3(FBSU )6(FBSO )3(FBSU

No crossings except at symmetries. Additional crossings occur!

)6(FBSO )3(FBSU

)3()2()3(

)...6(

)...3(

)...5(

)2()6(

)2()6()6()12()6()12/6(

SpinSUO

O

SU

U

SUU

UUUUUU

FFB

FB

FB

FB

FFB

FFBFB

L 1/2 J

Conserved quantities allow real crossings

FBFBFB QQ

NUCaH

ˆ.ˆ)1()5(ˆˆ

1 231

ˆ.ˆ)1()5(ˆˆ

FBFBFB QQ

NUCaH

0.2 0.3 0.4

0.20.0 0.4

But there is even more to the story !

)3()2()3()3(

)....6()6(

)...3()3(

)...5()5(

)2()6()6()12()6()12/6(

SpinSUOO

OO

SUSU

UU

UUUUUU

FFB

FB

FB

FB

FFBFB

0.2 0.3 0.4

25

,231

ˆ.ˆ)1()5(ˆˆ

FBFBFB QQ

NUCaH

0.2 0.3 0.4

231

ˆ.ˆ)1()5(ˆˆ

FBFBFB QQ

NUCaH

Applications to real nuclei:

there are no symmetry related constraintsneeded are dominant j = 1/2,3/2, 5/2 orbits

D.D. Warner, P. Van Isacker, J. Jolie, A.M. Bruce, Phys. Rev. Lett. 54 (1985)1365.

W,Pt

Ru,RhA. Frank, P. Van Isacker, D.D. Warner, Phys. Lett. B197 (1987)474.

Se,AsA. Algora et al.Z. f. Phys. A352 (1995) 25

Odd- neutron nuclei in the W-Pt region

J. Jolie, S. Heinze, P. Van Isacker, R.F. Casten, Phys. Rev. C 70 (2004) 011305(R).

)]3([ˆ)]6([ˆˆ.ˆˆ22 SpinCBUCAQQ

Na

H FBFBFB

a= -47 keVA= 52 keVB=3.4 keV