Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
and and Nuclear StructureNuclear Structure
PavelPavel CejnarCejnarInstitute of Particle & Nuclear Physics, Charles University, Institute of Particle & Nuclear Physics, Charles University, PraPrahaha, CZ, CZ
cejnar @@ipnp.troja.mff.cuni.cz
Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008
Program:Program:
> Shape phase transitions in nuclear structure data
> Models describing shape phase transitions in nuclei
> Playing with the models
Part 1 of 3
Shape (phase) transitions in nuclear structure data
Part 1 of 3Part 1 of 3
Shape (phase) transitions in Shape (phase) transitions in nuclear structure datanuclear structure data
Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008
Program:
> Shape phase transitions in nuclear structure data
Models describing shape phase transitions in nuclei
Playing with the models
All nuclei with keV600)2( 1 <+E
GroundGround --state QPT signatures in nucleistate QPT signatures in nuclei
Shape-phase transition!
Casten et al. , PRL 71,227 (1993)
P-factor Casten et al ., PRL 58,658 (1987), McCutchan et al ., PRC 69,024308 (2004)
GroundGround --state QPT signatures in nucleistate QPT signatures in nuclei
Semi-empirical criterion for a spherical-to-deforme d transition:
5≈+
≡np
np
NN
NNP
( Np , Nn = numbers of valence protons, neutrons or the respective holes)
GroundGround --state QPT signatures in nucleistate QPT signatures in nuclei
McCutchan et al ., PRC 69,024308 (2004)
N
2/4R
Zamfir et al. , PRC 66,021304(2002)
)2()4(
1
12/4 +
+
=E
ER
Energy ratio
GroundGround --state QPT signatures in nucleistate QPT signatures in nuclei
Dieperink,Scholten,Iachello, PRL 44,1747(1980)García-Ramos et al ., NPA 688,735(2001)García-Ramos et al ., PRC 68,024307(2003)
2n separation energies
),(2)2,( 22 NZMcmNZMS nn −+−=
N
2/4R
Zamfir et al. , PRC 66,021304(2002)
N
Iachello,Zamfir,PRL92,212501(2002)
)02,E2( 11++ →B
Transition strength
N
)2()4(
1
12/4 +
+
=E
ER
Energy ratio
magic
magicN
2n separation energies
),(2)2,( 22 NZMcmNZMS nn −+−=
GroundGround --state QPT signatures in nucleistate QPT signatures in nuclei
Courtesy: F.Iachello
Dieperink,Scholten,Iachello, PRL 44,1747(1980)García-Ramos et al ., NPA 688,735(2001)García-Ramos et al ., PRC 68,024307(2003)
GroundGround --state QPT signatures in nucleistate QPT signatures in nuclei
Prolate – oblate transition
Jolie, Linnemann, PRC 68,031301 (2003)
Part 2 of 3
Models describing shape phase transitions in nuclei
Part 2 of 3Part 2 of 3
Models describing shape phase Models describing shape phase transitions in nucleitransitions in nuclei
Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008
Program:
Shape phase transitions in nuclear structure data
> Models describing shape phase transitions in nuclei
Playing with the models
B
A
critical
spinodal
spherical
oblate
prolate0
Geometric (Collective) Model (GCM)Geometric (Collective) Model (GCM) A.Bohr (1952)
W.Greiner… (1971)
[ ] ( ) .....][5][2
35][5.....][
2
5 2)0()0()2()0()0( +× +×× −× ++× = αααααααππ CBAK
H
γ-soft
432 3cos βγββ CBAV ++=
α… quadrupole tensor of collective coordinates2 shape parameters: β, γ3 Euler angles
π ... corresponding tensor of momenta
B
A
critical
spinodal
0
oblate
prolate
Geometric (Collective) Model (GCM)Geometric (Collective) Model (GCM) A.Bohr (1952)
W.Greiner… (1971)
432 3cos βγββ CBAV ++=
[ ] ( ) .....][5][2
35][5.....][
2
5 2)0()0()2()0()0( +× +×× −× ++× = αααααααππ CBAK
H
γ-soft spherical
β0>0,γ0=[0,2π) β0=0
β0>0,γ0=π/3
β0>0,γ0=0
B
A
critical
spinodal
oblate
prolate0
Geometric (Collective) Model (GCM)Geometric (Collective) Model (GCM) A.Bohr (1952)
W.Greiner… (1971)
[ ] ( ) .....][5][2
35][5.....][
2
5 2)0()0()2()0()0( +× +×× −× ++× = αααααααππ CBAK
H
γ-soft spherical
1st order
432 3cos βγββ CBAV ++=
1st order2nd order
∑∑∑ =+= +++
iii
lkjilkjiijkl
jijiij GCwbbbbvbbuH ][
,,,,
F.Iachello, A.Arima (1975)Interacting Boson Model (IBM)Interacting Boson Model (IBM)
+s+md
2,...,2 +−=mCasimir invariants of U(6) subgroupsU(5),O(6),SU(3),O(5),O(3)
Phase transitions caused by competing dynamical symmetries
prolate oblate
spherical
SU(3)
O(6)
U(5)
O(6)
deformed
spherical
2ndorder
SU(3)
1storder
General H(7 parameters)
Phase structure Simplified H(2 parameters)
U(6)…spectrum generating(dynamical) algebra
O(3)…invariant symmetry algebra
P.Cejnar,J.Jolie, Prog.Part.Nucl.Phys.(2008) ; arXiv:0807.3467[nucl-th]
∑∑∑ =+= +++
iii
lkjilkjiijkl
jijiij GCwbbbbvbbuH ][
,,,,
F.Iachello, A.Arima (1975)Interacting Boson Model (IBM)Interacting Boson Model (IBM)
+s+md
2,...,2 +−=mCasimir invariants of U(6) subgroupsU(5),O(6),SU(3),O(5),O(3)
Phase transitions caused by competing dynamical symmetries
prolate oblate
spherical
Phase structure Simplified H(2 parameters)
U(6)…spectrum generating(dynamical) algebra
O(3)…invariant symmetry algebra
P.Cejnar,J.Jolie, Prog.Part.Nucl.Phys.(2008) ; arXiv:0807.3467[nucl-th]
( ) 00
Ns+∝Ψ
( ) 0000
Nds ++ +∝Ψ β
( ) 0000
Nds ++ −∝Ψ β
FermionicFermionic ModelsModels
PhenomenologicalPhenomenological
Microscopic Microscopic *
Lipkin model Lipkin,Meshkov,Glick(1965)Fermion Dynamical Symmetry Model Ginocchio,Wu,Zhang,Guidry…(1980,86,87,88)Pairing models Chen,Rowe …(1990…), Volya,Zelevinsky(2003), Clark et al .(2006)……
“Collapse of RPA” Thouless(1960)…Early shell model attempts Federman,Pittel,Campos(1979)…Monte Carlo shell model Shimizu,Otsuka,Mizusaki,Honma PRL86,1171(2001)Relativistic mean-field calculations Nikši ć,Vretenar,Lalazissis,Ring PRL99,092502(2007)
* In the microscopic case the infinite-size limit cannot be performed ⇒ all changes are smoothened byquantum fluctuations.
Part 3 of 3
Playing with the modelsLearning new physics on quantum phase transitions
Part 3 of 3Part 3 of 3
Playing with the modelsPlaying with the modelsLearning new physics on quantum phase transitions
Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008
Program:
Shape phase transitions in nuclear structure data
Models describing shape phase transitions in nuclei
> Playing with the models
New types of symmetries at & around the critical po intNew types of symmetries at & around the critical po int
Dynamical Symmetry I Dynamical Symmetry IIPhase I Phase II
Critical-point dynamical symmetryAnalytical solutions which are approximately valid at the QPT critical pointFirst noted by Ginocchio et al. (1987) in the Fermion Dynamical Symmetry ModelCast in the geometric framework by Iachello(2000,2001),Bonatsos et al. (2004) E(5), X(5), Z(5)…
Quasidynamical Symmetry I Quasidynamical Symmetry IIcriticalpoint
Quasidynamical SymmetryExtensions of approximate dynamical symmetries far away from the corresponding limits QDS “is an expression of the possibility that a subset of physical data may exhibit all the properties that would result if the system had a symmetry which, in fact, it does not have.”Rowe et al. (1998,2004,2005)
Partial dynamical symmetryAt the 1st order critical point: distinct subsets of states retain competing dynamical symmetries The PDS idea originally introduced in quantum chaos Leviatan et al. (1992……2007)
• Proton-neutron variables (IBM-2)Caprio,Iachello (2004,2005)Arias,García-Ramos,Dukelsky (2004)
• Odd fermions (IBFM)Jolie et al . (2004)Iachello (2005)Alonso et al. (2005,2006,2007)
• Higher order interactionsIachello (2004)Jolos (2004)Thiamova,Cejnar (2006)
• Other types of bosonsDevi,Kota (1990)Cejnar,Iachello (2007)
• Configuration mixingFrank,Van Isacker,Iachello (2006)Hellemans et al. (2007)
• External rotationCejnar (2002,2003)
Additional degrees of freedom, IBM extensionsAdditional degrees of freedom, IBM extensions
?
?
?
FiniteFinite --size scaling exponentssize scaling exponents
Calculations beyond the mean fieldDusuel, Vidal, Arias, Dukelsky, García-Ramos (2005, 2006,2007)
Example: Energy gap
3/1c
−∝∆ NaNe−∝∆c
01 EE −=∆
1st order 2 nd order
Lipkin Hamiltonian Vidal et al., PRC73,054305(2006)
Yang, Lee (1952)… Cejnar et al. (2005,2007)
Zeros of Z(T) in complex T Degeneracies of H(λ) in complex λ
Mechanism of the 1Mechanism of the 1 stst x 22ndnd order transitionsorder transitions
Thermodynamic analogy for quantum phase transitions
λRe
λIm
cλ
00)(
00)(
c
c
>==≠
αλαλ
k
k
Q
Q
....]7,3()(
]3,1()(
]1,0()(
c
c
c
4
4
2
2
∈∞=
∈∞=
∈∞=
αλ
αλαλ
C
C
C
dxd
dxd
αλρ )(Im∝
α=0
0<α<1α>1
Imλ
ρ
density near Imλ≈0
1st order
continuous
partition function
free energy
specific heat
latent heat
)(TZ
)(ln)( TZTTF −=
)()( 2
2
TFTTCT∂∂−=
∫+
−+→=
ε
εε
T
TdTTCTQ ')'(lim)(
0
[ ] [ ]Ω
≠
Ω
−= ∏
/1
)(
/1 )()()(ki
kik EED λλλ
)(ln)( 1 λλ kk DU Ω−=
)()( 2
2 λλλ kd
dk UC −=
∫+
−+→=
ελ
ελελλλ ')'(lim)(
0dCQ kk
1−=Ω n
Example: linear arrangement of degeneracies (zeros)
ExcitedExcited --state quantum phase transitions state quantum phase transitions
Example: 1D Cusp Hamiltonian
a b
0=b 1−=a+=π±=π
2nd order 1 st order
Cejnar, Heinze, Macek, Jolie, Dobeš (2006,2007)Caprio, Cejnar, Iachello (2008)Cejnar, Stránský (2008)
Thanks to collaborators:P Stránský, M Macek, J Dobeš [Praha]S Heinze, J Jolie … [Köln]M Caprio, F Iachello … [Notre Dame, Yale]
Thank you for attention.
Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008Marie & Pierre Curie 15 th Nuclear Physics Workshop, Kazimierz 2008
Question: Do quantum shape-phase transitions really exist in nuclei ??Tentative answer: They would exist if nuclei were infinite objects.
Finite nuclei only show QPT precursors.
Real QPTs can be studied in various nuclear models⇒ benefit for both nuclear structure theory && QPT theory(further applications in molecular & mesoscopic physics)