Date post: | 30-Dec-2015 |
Category: |
Documents |
Upload: | poppy-harrell |
View: | 276 times |
Download: | 1 times |
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.
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)
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.
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