Magnetic and Antimagnetic rotation in Covariant …2-D Cranking: Olbratowski, Dobaczewski, Dudek, Rzaca- Urban, Marcinkowska, Lieder, APPB 33, 389(2002) Outline 19 2011-9-21 Introduction

Magnetic and Antimagnetic rotation in Covariant Density Functional Theory

Jie Meng 孟杰

北京航空航天大学物理与核能工程学院 School of Physics and Nuclear Energy Engineering

Beihang University (BUAA)

北京大学物理学院 School of Physics Peking University (PKU)

Outline

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Introduction

Theoretical framework

Magnetic rotation

Antimagnetic rotation

Summary & Perspectives

Angular Momentum World of Nucleus

3

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Electric Rotation

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Substantial quadrupole deformation

Strong electric quadrupole (E2) transitions

Coherent collective rotation of many nucleons Bohr PR1951

Twin PRL1986

Electric & Magnetic Rotation 2011-9-21

∆I = 2

E2 Transitions

∆I = 1

M1 Transitions

Hübel PPNP2005

Twin PRL1986

Magnetic rotation

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2011-9-21

near spherical or weakly deformed nuclei

strong M1 and very weak E2 transitions

rotational bands with ∆I = 1

shears mechanism

∆I=1 regular bands

First attempt in verify MR

ΔI = 1 Enhanced magnetic dipole transition

How does B(M1) change with spin I ?

Good agreement between TAC and PRM

Good agreement with prediction for BM1 versus I

Experiment: MR

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2011-9-21

Experiment: MR

Magnetic rotation: 78 nuclei

A~60

Antimagnetic Rotation (AMR) Magnetic rotation Ferromagnet rotational bands with ΔI = 1

near spherical nuclei; weak E2 transitions

strong M1 transitions

B(M1) decrease with spin

shears mechanism

Antimagnetic rotation Antiferromagnet rotational bands with ΔI = 2

near spherical nuclei; weak E2 transitions

no M1 transitions

B(E2) decrease with spin

two “shears-like” mechanism

Experiment: AMR Antimagnetic rotation: 3 nuclei

Other mass regions

Simons PRL2003; Simons PRC2005

Small B(E2) Decrease tendency

Large J(2)/B(E2) Increase tendency

Theory

Semiclassical particle plus rotor model Clark ARNPS2000

simple geometry for the energies and transition probabilities

2

2 2( )( ) ( )

2I j jE I V Pπ ν θ− −

= +ℑ

jπ

jν

I

R

Core Rotor Particle shears mechanism

Theory



Pairing-plus-quadrupole tilted axis cranking (TAC) model Frauendorf NPA1993; Frauendorf NPA2000

semi-phenomenological Hamiltonian

H H Jω′ = − ⋅

2

22sphH H Q Q GP P Nµ µµ

χ λ+ +

=−

= − − −∑

Theory



Pairing-plus-quadrupole tilted axis cranking (TAC) model Frauendorf NPA1993; Frauendorf NPA2000

semi-phenomenological Hamiltonian

Phenomenological investigations

polarization effects are neglected or only partially considered

nuclear currents are treated without self-consistency

adjusted to show MR/AMR in some way or another

A fully self-consistent microscopic investigation?

DFT: Cranking version

TAC based on Covariant Density Functional Theory Meson exchange version:

3-D Cranking: Madokoro, Meng, Matsuzaki, Yamaji, PRC 62, 061301 (2000)

2-D Cranking: Peng, Meng, Ring, Zhang, PRC 78, 024313 (2008)

Point coupling version: Simple and more suitable for systematic investigations

2-D Cranking: Zhao, Zhang, Peng, Liang, Ring, Meng, PLB 699, 181 (2011)

TAC based on Skyrme Density Functional Theory

Fully self-consistent microscopic investigations

fully taken into account polarization effects

self-consistently treated the nuclear currents

without any adjustable parameters for rotational excitations

3-D Cranking: Olbratowski, Dobaczewski, Dudek, Płóciennik, PRL 93, 052501(2004)

2-D Cranking: Olbratowski, Dobaczewski, Dudek, Rzaca-Urban, Marcinkowska, Lieder, APPB 33, 389(2002)

Outline

19

2011-9-21

Introduction


Magnetic rotation



ˆ

δδEh =

ρ iiih ϕεϕ =ˆMean field: Eigenfunctions:

ˆ

2

δδδ EV =ρ ρ

Interaction:

Skyrme Gogny

Extensions: Pairing correlations, Covariance Relativistic Hartree Bogoliubov (RHB) theory

Density functional theory in nuclear physics

Starting point of CDFT

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2011-9-21

Nucleons are coupled by exchange of mesons via an effective Lagrangian with all relativistic symmetries, used in a mean field concept and no-sea approximation

σ ω ρ

π

meson Jπ T π 0− 1 σ 0+ 0 ω 1− 0 ρ 1− 1

22

Brief introduction of CDFT 2011-9-21

3152

2 2 2

2 2

[ ( ) ]

1 1 1 1 1 2 2 4 2 41 1 1 1 2 2 2 4

fmL i M g g g e A

m m R R

m m F F

π

π

τµ µ µµ σ ω µ ρ µ µ µ

µ µν µ µνµ σ µν ω µ µν

µ µ µνρ µ µ π µν

ψ γ σ γ ω τ ρ γ γ π τ ψ

σ σ σ ω ω

ρ ρ π π π π

−= ∂ − − − + • + − ∂ •

+ ∂ ∂ − − Ω Ω + − •

+ + ∂ • ∂ − • −

RF A A

µν µ ν ν µ

µν µ ν ν µ

µν µ ν ν µ

ω ω

ρ ρ

Ω = ∂ − ∂

= ∂ − ∂

= ∂ − ∂

4

, , , ,

1( ) ( ) ( ) ( , ) ( ) ( )2

i ii A

H i M d y x y D x y y x

T Vσ ω ρ π

ψ ψ ψ ψ ψ ψ=

= − • ∇ + + Γ

= +

∑∫γ

1 2

1 2 5 1 5 2

2

em 3 1 3 2

(1, 2) (1) (2), (1, 2) ( ) ( ) ,

(1, 2) ( ) ( ) , (1, 2) ( ) ( )

(1,2) ( (1 )) ( (1 ))4

f fm m

g g g g

g g

e

π π

π π

µσ σ σ ρ ρ µ ρ

µ νω ω µ ω µ π µ ν

µµ

γ τ γ τ

γ γ τγ γ τγ γ

γ τ γ τ

Γ ≡ − Γ ≡ +

Γ ≡ + Γ ≡ − ∂ ∂

Γ ≡ + − −

Hamiltonian:

Lagrangian:

Brief introduction of CDFT

23

2011-9-21

'

'† ††

†

†

( )( ) [ ( ) ]

( ) ([ ( ) ])

i

i

i

i

i

i

i ti i

i

i t

i ti i

i tii

i

x f e c

x

g e d

ef e dc g

εε

ε ε

ψ

ψ −

−= +

= +

∑

∑

xx

xx

†0 0cα

α

Φ = ∏

0 0 0 0 0 0, , , ,

emem

ii A

D D D Dk

E E E E

E H T V

E E E E EE E E E Eσ ω ρ π

σ ω ρσ ω ρ π

=

= Φ Φ = Φ Φ + Φ Φ

= + + + + + + + + +

∑

Energy density functional:

, , , ,i

i AH T V

σ ω ρ π=

= + ∑

†

††1 2 ' ' ' '

; ' '

( ) ,

1 (1) (2) (1,2) (1,2) (2) (1)2i i i

T d f i M f c c

V d d c c c c f f D f f

α β α βαβ

α β β α α β β ααβ α β

γ= − ⋅∇ +

= Γ

∑∫

∑∫

x

x xHartree

Fock

Equations of motion in RMF theory

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2011-9-21

Same footing for

Deformation Rotation Pairing (RHB,BCS,SLAP)

…

For system with time invariance:

( )( ) ( ) i i iV M Sα β ψ ε ψ⋅ + + + = p r r

33

1( ) ( ) ( ) ( )2

( ) ( )

V g g e A

S r g

ω ρ

σ

τω τ ρ

σ

− = + + =

r r r r

r

( ) ( )

2 2 32 3

2 33

2 33

s

b

n pb b

m g g g

m g c

m g d

σ σ

ω ω

ρ ρ

σ ρ σ σ

ω ρ ω

ρ ρ ρ ρ

−∆ + = − − − −∆ + = −

−∆ + = − −

1

1

3 31

31

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1( ) ( ) ( )2

i

i

i

As i ii

Av ii

Aii

Ac ii

ρ ψ ψ

ρ ψ ψ

ρ ψ τ ψ

τρ ψ ψ

=

+=

+=

+=

= = = − =

∑∑∑

∑

r r r

r r r

r r r

r r r

RMF theory with Point-Coupling interaction

25

2011-9-21

( )

2 2 2 3 42 3

0 30

14

1 1 1((

12

) )2 3 4

12

ii i i

v

s

H i M F F

m g g g

gg

νν

σ σ

ρω

ψ γ ψ

σ σ σρ σ σ

ω ρ ρρ

= − • ∇ + +

+ ∇ + + +

++

+

ω

( )

2 3 4

22

14

1 1 1 12 2 3 4

1 12 2

1 12 2

ii i i

S S S S S S

V

S S S

TV V TV TV TV V V VV

H i M F Fνν

α ρ

ψ γ ψ

α ρ δ ρ ρ β ρ γ ρ

α ρ δ ρ ρδ ρ ρ

= − ∇ + +

+ + ∆ + +

+ +∆ ∆++

ω

2 2

2

2

2 2 4

11 / v v v vv v v

ggm m

g gm m

ω

ωω ω

ω

ω

ωωω ρ ρ ρ ρα ρδ= = + ∆ + ≈ + ∆

− ∆

Equations of motion in RMF-PC theory

26

2011-9-21

For system with time invariance:

( )( ) ( ) i i iV M Sα β ψ ε ψ⋅ + + + = p r r

3 33 3

2 3

1( ) ( ) ( ) ( ) ( ) ( ) ( )2

( )

V V V V V V TV TV TV TV

S S S S S S S S

V e A

S r

τα ρ γ ρ δ ρ τ α ρ τ δ ρ

α ρ β ρ γ ρ δ ρ

− = + + ∆ + + ∆ + = + + + ∆

r r r r r r r

1

1

3 31

31

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1( ) ( ) ( )2

i

i

i

As i ii

Av ii

Aii

Ac ii

ρ ψ ψ

ρ ψ ψ

ρ ψ τ ψ

τρ ψ ψ

=

+=

+=

+=

= = = − =

∑∑∑

∑

r r r

r r r

r r r

r r r

Without Klein-Gordon equation

Tilted axis cranking CDFT

27

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3 4 2

3

( )1 1 1( )( ) ( )( ) ( )( )2 2 21 1 1( ) ( ) [( )( )]3 4 41 1 1( ) ( ) ( ) ( ) ( ) ( )2 2 21 1

2 4

S V TV

S S V

S V TV

L i m

e A F F

µµ

µ µµ µ

µµ

ν ν µ νν ν µ ν µ µ

µ µνµ µν

ψ γ ψ

α ψψ ψψ α ψγ ψ ψγ ψ α ψτγ ψ ψτγ ψ

β ψψ γ ψψ γ ψγ ψ ψγ ψ

δ ψψ ψψ δ ψγ ψ ψγ ψ δ ψτγ ψ ψτγ ψ

τ ψγ ψ

= ∂ −

− − −

− − −

− ∂ ∂ − ∂ ∂ − ∂ ∂

−− −

General Lagrangian density

Transformed to the frame rotating with the uniform velocity

Koepf NPA1989; Kaneko PLB1993; Madokoro PRC1997

( ,0, ) ( cos ,0, sin )x z θ θΩ ΩΩ = Ω Ω = Ω Ω

1 00

t ttx xα α

= → = = x R xx

TAC RMF:equations of motion

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Dirac Equation

Potential

Spatial components of vector field are involved due to the time-reversal invariance broken

( )( ) ( )( ) ( ) i i ii M S VV Jα β ψ ε ψΩ⋅ ⋅ − ∇ − + + + − =

rr r

2 3

3 33 3

( )1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2

S S S S S S S S

V V V V V V TV TV TV TV

S r

V j j j j j e Aµ µ µ µ µ µ µ

α ρ β ρ γ ρ δ ρτα γ δ τ α τ δ

= + + + ∆ −

= + + ∆ + + ∆ +r r r r r r r

Observables 2011-9-21

3 2tot

1

3 4 2

00

1

1 1 1( ) ( )2 2 2

2 3 3 1 1( ( ) ) ( )3 4 4 2 21 1( ) | |2 2

A

k S S V V V TV TV TVk

S S S S V V V S S S V V V

A

TV TV TV pk

E d r j j j j

j j j j

j j ej A k J k

µ µµ µ

µ µµ µ

µµ

α ρ α α

β ρ γ ρ γ δ ρ ρ δ

δ

=

=

= − + +

+ + + + ∆ + ∆

+ ∆ + + ⟨ Ω ⟩

∑ ∫

∑

Binding energy

Angular momentum 2ˆ ( 1)J I I⟨ ⟩ = +

Quadrupole moments and magnetic moments

2 2 2 220 22

5 153 ,16 32

Q z r Q x yπ π

= ⟨ − ⟩ = ⟨ − ⟩

23 ?†[ ( ) ( ) ( ) ( )]

A

i i i i i ii

mcd r Q r r r r rc

µ ψ αψ κψ β ψ= × + Σ∑∫

Where 1, 0, 1.793, 1.913p n p nQ Q κ κ= = = = −

Observables 2011-9-21

3 2tot

1

3 4 2

00

1

1 1 1( ) ( )2 2 2

2 3 3 1 1( ( ) ) ( )3 4 4 2 21 1( ) | |2 2

A

k S S V V V TV TV TVk

S S S S V V V S S S V V V

A

TV TV TV pk

E d r j j j j

j j j j

j j ej A k J k

µ µµ µ

µ µµ µ

µµ

α ρ α α

β ρ γ ρ γ δ ρ ρ δ

δ

=

=

= − + +

+ + + + ∆ + ∆

+ ∆ + + ⟨ Ω ⟩

∑ ∫

∑

Binding energy

Angular momentum 2ˆ ( 1)J I I⟨ ⟩ = +

Quadrupole moments and magnetic moments

2 23 3( 1) ( sin cos )8 8 x J z JB M µ µ θ µ θπ π⊥= = −

B(M1) and B(E2) transition probabilites

2

2 220 22

3 2( 2) cos (1 sin )8 3J JB E Q Qθ θ

= + +

RMF parameterizations

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2011-9-21

Meson Exchange

2

NL3, NLSH, TM1, TM2, PK1, …

Density dependent parameterizations:

2 3 3 3, , , , , , , , , ,M m m m g g g g g c dσ ω ρ σ ω ρ

Point Coupling

, , , , , , , , ,S V TV S V TV S S VM α α α δ δ δ β γ γ

PC-LA, PC-F1, PC-PK1 …

Density dependent parameterizations:

Nonlinear parameterizations: Nonlinear parameterizations:

TW99, DD-ME1, DD-ME2, PKDD, … DD-PC1, …

, , , , ( ), ( ), ( )M m m m g g gσ ω ρ σ ω ρρ ρ ρ , , ( ), ( ), ( )S S V TVM δ α ρ α ρ α ρ

Parameterizations: PC-PK1

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2011-9-21

Zhao, Li, Yao, Meng, PRC 82, 054319 (2010)

Spherical nuclei

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2011-9-21


Deformed nuclei

34

2011-9-21


Nuclear Mass

35

2011-9-21

Exp value for 2149 nuclei from Audi et al. NPA2003

Outline

36

2011-9-21

Introduction


Magnetic rotation



MR: 60Ni lightest nucleus with magnetic rotation

Torres PRC 2008

MR: 60Ni

Harmonic oscillator shells: Nf = 10

Parameter set: PC-PK1

Configurations:

Zhao, Zhang, Peng, Liang, Ring, Meng, PLB 699, 181 (2011)

Numerical Details

(Py=PRy(π))

P PxT Px PzT Pz PyT Py

Jx √ × √ √ × √ ×

Jz √ √ × × √ √ ×

Identification of the energy levels

Symmetry

|nx ny nz ns⟩ → |nljmz⟩ → |nljmx⟩

Spherical basis Cartesian basis

Constrained intrinsic framework

( )22 1 2 1

12

H H C Q a− −′ = + −

Enforce principal axes to be identical with the x, y, and z axis.

2 1 0a − =

Numerical Details

Parallel transport principle

(δ) ( ) 1j iφ φΩ + Ω Ω ≈

Ω : Φi(Ω) Ω+δΩ : Φj(Ω+δΩ)

MR: 60Ni


MR: 60Ni


MR: 60Ni

Shears mechanism

Magnetic Rotation

Electric Rotation


MR: 60Ni

Electromagnetic transition properties Zhao, Zhang, Peng, Liang, Ring, Meng, PLB 699, 181 (2011)

Outline

45

2011-9-21

Introduction


Magnetic rotation



AMR: 105Cd first odd-A nucleus with antimagnetic rotation Choudhury PRC2010

AMR: 105Cd Harmonic oscillator shells: Nf = 10

Parameter set: PC-PK1

Configurations:

Polarizations:

107Sn 107Sn + π[(g9/2)-2] 105Cd Zhao, Peng, Liang, Ring, Meng PRL 177, 122501(2011)

Choudhury PRC2010

2011-9-21

AMR: Single particle routhians

Neutron Proton

• Time reversal symmetry broken energy splitting

• For proton, two holes in the top of g9/2 shell

• For neutron, one particle in the bottom of h11/2 shell, the other six are

distributed over the (gd) shell with strong mixing

• This configuration is similar to , but not exactly

AMR: 105Cd

Zhao, Peng, Liang, Ring, Meng PRL 177, 122501(2011)

AMR: 105Cd

Polarization effects play important roles in the description of AMR, especially

for E2 transitions.


AMR: 105Cd

Two “shears-like” mechanism


AMR: 105Cd

In the microscopic point of view, increasing angular momentum results from

the alignment of the proton holes and the mixing within the neutron orbitals.


Outline

53

2011-9-21

Introduction


Magnetic rotation



Summary and Perspectives 2011-9-21

Covariant density functional theory has been extended to describe rotational excitations including MR and AMR.

PC-PK1: fitted with BE & Rc for 60 nuclei provide better description for the isospin dependence of BE 60Ni: MR reproduce well the E, I, B(M1), and B(E2) values, inclduing the transition from electric rotation to magnetic rotation

105Cd: AMR reproduce well the AMR pictures, E, I, and B(E2) values in a fully self-consistent microscopic way for the first time and find that polarization effects play important roles

Summary and Perspectives 2011-9-21

? Pairing effects

? Transition from the electric to magnetic rotation

? Anti-magnetic rotation in other mass regions

? High-K isomer

Thank You!

Magnetic and Antimagnetic rotation in Covariant …2-D Cranking: Olbratowski, Dobaczewski, Dudek, Rzaca- Urban, Marcinkowska, Lieder, APPB 33, 389(2002) Outline 19 2011-9-21 Introduction

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