and nuclear structure - univerzita karlova · and nuclear structure pavel cejnar institute of...

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


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)


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)


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


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 −+−=


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)


Prolate – oblate transition

Jolie, Linnemann, PRC 68,031301 (2003)

Part 2 of 3



Models describing shape phase Models describing shape phase transitions in nucleitransitions in nuclei


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


W.Greiner… (1971)


[ ] ( ) .....][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


W.Greiner… (1971)

[ ] ( ) .....][5][2

35][5.....][

2

5 2)0()0()2()0()0( +× +×× −× ++× = αααααααππ CBAK

H

γ-soft spherical

1st order


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


Playing with the modelsPlaying with the modelsLearning new physics on quantum phase transitions


Program:

Shape phase transitions in nuclear structure data


> 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.


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)

and nuclear structure - univerzita karlova · and nuclear structure pavel cejnar institute of...

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