Nuclear Collective Motions

Nuclear Collective Motions Takashi Nakatsukasa Theoretical Nuclear Physics Laboratory, RIKEN Nishina Center

2009.1.19-28 20th Chris Engelbrecht Summer School in Theoretical Physics “Nuclei and Nucleonic Systems: Exotic nuclei, halos, nuclear synthesis and more” @National institute for Theoretical Physics at Stellenbosch Institute for Advanced Study, Stellenbosch, Western Cape, South Africa

Contents • Liquid-drop, shell, unified models, cranking model • Nuclear structure at high spin and large deformation • Sum-rule approaches to giant resonances • Basic theorem for the Time-dependent densityfunctional theory (TDDFT) • Linearized TDDFT (RPA) and elementary modes of nuclear excitation • Theories of large-amplitude collective motion • Anharmonic vibrations, shape coexistence phenomena

Quarks, Nucleons, Nuclei, Atoms, Molecules atom

nucleon

nucleus

q q

N

e

molecule

N

q α

α

Strong Binding

clustering deformation rotation vibration

“Strong” Binding

rare gas

cluster matter “Weak” binding

“Weak” binding

Separation energy (Ionization potential) Ionization potential of atom

O– O O+ O2+ O3+ O4+ O5+ O6+ O7+

1.47 eV 13.61 35.15 54.93 77.39 113.87 138.08 739.08 871.12

Rapid decrease

Neutron separation energy Abbas, Mod Phys Lett A 20, (2005), 2553

Gradual decrease (roughly constant) → Nuclear clustering and deformation

Saturation Binding energy B/A ≈ 8 MeV Density ρ ≈ 0.14 fm-3

d ≈ 2 fm

Liquid drop model Bethe-Weizsäcker mass formula

B( N , Z ) = aV A − aS A2 / 3 ( N − Z )2 − asym A Z2 − aC 1/ 3 + δ ( A) A

Bohr & Mottelson, Nuclear Structure Vol.1

Liquid drop picture

0+ , 2+, 4+ 2+ 0+

• Liquid drop vibrator and rotor – Surface vibrations in spherical nuclei  1  1 H = ∑  π + C α  2  2B  – 5-dim. quadrupole oscillator 4 λµ

2 λµ

λ

2 λµ

4+

λ

+

I k' H= ∑ k = 1, 2 , 3 2ℑ k ( β , γ )

2+ 0+

g-band  1  1 ∂ ∂ 1 ∂ ∂ 4   + V ( β , γ ) − β + 2 sin 3γ 4 2 B  2β ∂ β ∂ β β sin 3γ ∂ γ ∂γ 

– Nuclear fission (Bohr-Wheeler) EC Z2 x= ~ 2 ES A

Nucleus is liquid?

2+ 0+

β-band

4+ 3+ 2+

γ-band

Electronic single-particle motion in atom V(r)

r

Single-particle levels in Coulomb pot. “Magic number”

2s,2p

Ionization Potential

1s

Electrons as “free gas” trapped by the Coulomb potential

Bohr & Mottelson, Nuclear Structure Vol.1

Nucleonic single-particle motion in nucleus Neutron Separation energy

Neutron #　N Bohr & Mottelson, Nuclear Structure Vol.1

Proton Separation energy

Bohr & Mottelson, Nuclear Structure Vol.1

Proton #　Z

Mayer-Jensen’s Shell Model Harmonic oscillator potential + spin-orbit force

  1 2 2 2 V (r ) = Mω r + vll  + vls  ⋅ s 2 → Correct magic numbers. Nucleons as “free gas” in the potential. Nucleus is gas?

λ free > > R0

Collective (Unified) Model by Bohr & Mottelson

• Nucleons are independently moving in a potential that slowly changes. – Collective motion induces oscillation/rotation of the potential. – The fluctuation of the potential changes the nucleonic single-particle motion. H = H vib + H part + H coupl H vib =

 1  π  2 Bλ

∑

λµ

2 λµ

+

1 Cλ α 2

2 λµ

  

In modern approaches, microscopic construction

 p i2  H part = ∑  + V0 ( ri )  i  2M  H coupl = ∑ { V ( ri ; α λ µ ) − V0 ( ri )} ≈ − ∑ k (ri )∑ α i

i

λµ

* λµ

Yλ µ (ϑ i , ϕ i )

Weak coupling case • 3− phonon + particle (h9/2 proton) • Hcoupl is small → Multiplet (3/2+, …, 15/2+)

Hamamoto, PR10 (1974) 63.

Bi

209

Pb

208

“Weak” coupling case • 2+ phonon + (quasi-)particle (g9/2 proton) Shibata, Itahashi, Wakatsuki, NPA237 (1975) 382.

Precursor phenomenon of the quadrupole instability Anomalous coupling states 45

Rh

9/2+ 7/2+

“Intermediate” coupling • More protons make coupling of 2+ phonon to the intruder g9/2 proton, more important → Anomalous coupling • Phonon+1QP → Dressed 3QP mode (“3QP-QRPA”) • Pauli effects favor I=j-1 state (Different sign for 6j coef.) Kuriyama, Marumori, Matsuyanagi, Prog. Theor. Phys. Suppl. 58 (1975).

u1v2+v1u2 u1u2−v1v2

Higher order effects, such as mode-mode coupling and anharmonicity, demand a systematic treatment. → “Nuclear Field Theory” (D.Bes, Prog. Theor. Phys. Suppl. 74&75 (1983) 1.)

Toward strong coupling Effective potential

spherical

transitional

deformed

V(β)

β Quadrupole deformation (order parameter)

Spontaneous Symmetry Breaking (SSB)

Nambu-Goldstone modes SSB → Degenerate “ground state”

[J , H ] = 0 ,

e iε J 0 ≠ 0

≠

Then, zero-energy modes of excitation, which connect different “ground” states exist. (The deformation defines the orientation, then, the rotational motion is generated.) In finite systems, this is intimately related to the separation of “collective” d.o.f.

H = H coll (q, p ) + H ' (ξ , π ) , E = Ecoll + E ' ,

Ψ = Φ

(H

⊗ Φ coll

'

= H − H coll

'

The “ground” state of H’ is degenerate.

P2 H = H− 2 AM

Φ

J2 H = H− 2ℑ

Φ

'

'

)

loc

=

∑

cK K

⊗ Φ coll

∑

cJ J

⊗ Φ coll

K

def

=

J

' 0 ' 0

Strong coupling Hamiltonian (Axially symmetric quadrupole deformation) def H = H rot + H Coriolis + H rec + H part + H vib + H rot - vib + H part - vib

H rot =

[

1 I 2 − I 32 2ℑ ( β 0 )

H Coriolis = − H rec = H

def part

=

1 1 j⊥ ⋅ I ⊥ = − [ I + j− + I − j+ ] ℑ 2ℑ

1 j2⊥ 2ℑ ( β 0 )

∑ i

] Rotation-particle coupling

Renormalized in the Hpart : → Coriolis attenuation

 p i2  ( ) + V r ; β 0 i 0  ≈   2M 

∑ i

 p i2  ˆ ( ) + V r − k ( r ) β Y ( r ) 0 i i 0 20 i    2M 

Case that Hvib, Hrot-vib, and Hpart-vib are not present. → Particle-Rotor Model

Strong coupling limit In case that the Coriolis effect is weak, the K-quantum number is approximately a good quantum number. R Strong-coupling wave function (with R-symmetry)

IKM ∝ Φ

I ( Θ ) + (− 1) I + K Φ ⊗ DMK

nK

nK

⊗ DMI − K ( Θ

)

K

Energy spectrum with ΔI = 1

EnK ( I ) = ε nK +

1 ( I ( I + 1) − K 2 ) 2ℑ

Energy spectrum for K=1/2 band with ΔI = 1

EnK = 1/ 2 ( I ) = ε nK + a= − Φ

nK = 1 / 2

1 2ℑ

j+ Φ

 1 1 I + 1/ 2  I ( I + 1 ) − + ( − 1 ) a I +    4 2  

nK = 1 / 2

Decoupling parameter

j

239

Pu

Decoupling limit (Aligned band) In the opposite limit to the strong coupling case that the Coriolis force is much stronger than the energy splitting in different K states. R

H Coliolis

1 1 = − [ I ⊥ ⋅ j⊥ ] ≈ − I ⋅ i ℑ ℑ

i

j

R ≈ I − i : must be even integer Stephens et al, PRL29 (1972) 438. Energy spectrum with ΔI= 2

1 [ I ( I + 1) − 2 I ⋅ i ] 2ℑ 1 = const. + R( R + 1) 2ℑ

E ( I ; i ) = const. +

Rotational spectrum of a favored band (maximally aligned with i=j ) shows a striking similarity to that of the neighboring even nucleus.

j = 11 / 2, (h11/ 2 neutron) ⇒

Coriolis attenuation and cranking model Analysis with the particle-rotor model reveals that we need an additional factor to weaken the Coriolis interaction.

H Coriolis → ρ H Coriolis , ρ = 0.4 ~ 0.8 The recoil term

1 j2⊥ 2ℑ ( β 0 )

H rec =

Effect of the recoil term

H rot + H Coriolis + H rec = =

ω=

R I⊥  i ≈  1−  ℑ ℑ  I

is proportional to j2 − K2 : K-dependence

[

]

1 1 I 2 − j2 − R ⋅ j 2ℑ ( β 0 ) ℑ

[

]

1 I 2 − j2 − ω ⋅ j 2ℑ ( β 0 )

⇒

ρ = 1−

i I

The attenuation is important at low spin.

The Cranking Model automatically takes account of the attenuation factor.

H

def part

=

∑ i

 p i2  ( ) + V r ; β − ω ⋅ j 0 i 0 c    2M 

Uniform rotation → ωc : c-number

Cranking model Picture in the rotating frame Time-dependent Schrödinger equation

i

∂ Ψ (t ) = H Ψ (t ) ∂t

ωrot

In the uniformly rotating frame with the rotational frequency ω

Ψ (t ) = e − iω t ⋅ J Φ (t )

i

∂ Φ (t ) = ( H − ω ⋅ J ) Φ (t ) ∂t

Choose the rotational axis as x-axis:

H ' ≡ H − ω⋅ J ⇒

“1-dim. cranking”

H ' = H − ω rot J x

Cranking violates the time-reversal symmetry. However, in case of quadrupole deformation, it conserves the parity and signature symmetry:

Rˆ x = e − iπ J x

r = e − iπ α

± i, α =  1 2  , r=  0 ± 1 , α =   1 

Experimentally, often defined as

α = I

(mod 2)

Collective and non-collective rotations Non-collective rotation

Collective rotation

Cranking model is applicable to both cases.

Cranked shell model Single-particle motion in the rotating frame Cranked shell model Hamiltonian ' def

h

=

∑ i

 p i2  + V0 ( ri ; β 0 ) − ωrot ( j x ) i   2 M  

The signature quantum number

Eigenvalues of h’ ' ( ω rot ) ϕ = e' ( ω rot ) ϕ hdef

As a function of ω called “routhian diagram”

r = e − iπ α r = ± i for α =  1

2

Alignment

de ' i ≡ ϕ jx ϕ = − dω rot Cranked Nilsson s.p. spectra (ε2=0.26) Solid → (π,α)=(+,+1/2) Dotted → (π,α)=(+,−1/2) Dashed → (π,α)=(−,+1/2) Dashdotted → (π,α)=(+,+1/2)

signature splitting

Toward the high spin Inglis (1954) introduced the cranking Hamiltonian to treat the cranking term as a perturbation (low-spin limit).

I (ω rot ) = 2∑ ω→ 0 ω ph rot

ℑ Inglis ≡ lim

Φ

n

Jx Φ

2 0

En − E0

Inglis’ moment of inertia

Including the pairing effect, this is qualitatively consistent with experimental data. However, the cranking model becomes the most valuable tool to investigate the high-spin states: Back-bending phenomena Quasi-particle routhians Superdeformed rotational bands

ℑ irr < ℑ exp < ℑ rig

Rotating objects in the universe Nucleus is one of the fastest rotating many-body system.

R0ω rot 2ℑ super 2ℑ normal This is expected to occur around I=30~40 in the rare-earth region However, the quantal size effects (QSE) are important in nuclei: For instance, the size of the Cooper pair

ξ0 =

v F π∆

k F− 1 ~ 0.7 fm , ξ 0 ~ 17 fm > R0 ; Epair ~ − 2 MeV , kTc =

2 ∆ ~ 0.5 − 0.8 MeV 3.54

Fluctuation is important: “Beyond mean field” 4qp

The back-bending provides an example of stepwise phase transition (super to normal).

s-band (2qp) g-band (0qp)

Microscopic structure of collective modes of excitations The ground state is assumed to be a Slater determinant.

Φ

0

= c1+  c A+ 0

In case of free particles (no interaction), excited states are

Ω

+ ph

Ω

+ pp 'hh '

+ p h

= c c ,

Φ

ph

= c +p c +p 'ch ch ' ,

= Ω Φ

+ ph

Φ

pp 'hh '

Φ

ph

= Ω

+ ph

Φ

0

= Ω

+ pp 'hh '

Φ

0

However, there is no “collective” state. All the excitations are “individual”, or not “coherent”. For the collective excitations, the (residual) interaction plays an essential role.

0

Problem 1:

Prove the followings:

The Hamiltonian

H = H 0 + Vres , Vres = κ Q ⋅ Q Suppose all the 1p-1h excitations have the same excitation energy and that all the matrix elements among 1p-1h states are the same:

H0 Φ Φ

p 'h '

ph

= ε Φ

Vres Φ

ph

ε = e p − eh

ph

= v ( = κ q02 )

Diagonalize the Hamitonian matrix in the subspace of 1p-1h states. Then, the following state has an eigenenergy of

Φ

coll

=

∑

ph

1 Φ N ph

Ecoll = ε + N ph v

ph

and the all the rests have Enon -coll = ε If v0, high-lying collective states, such as (isovector) giant resonances

Problem 2:

With the same Hamitonian

H = H 0 + Vres , Vres = κ Q ⋅ Q Suppose all the 1p-1h states have the same matrix element of Q

Φ

ph

QΦ

0

= q0

Then, show that the collective state

Φ

coll

=

∑

ph

1 Φ N ph

ph

has a transition matrix element:

Φ

coll

and all the rests have zero transitions:

QΦ Φ

2 0

= N ph q02

non -coll

QΦ

2 0

= 0

The state is called “collective” because the state has the “coherence” to the operator Q. The “collectivity” should be defined by a certain (one-body) operator Q. The number of 1p-1h states contained in the state is sometimes misleading.

Φ

non -coll

=

N ph / 2

∑

ph

1 Φ N ph

ph

−

N ph / 2

∑

p 'h '

1 Φ N ph

p 'h '

Φ

coll

=

∑

C ph Φ

ph

ph

+

∑

C php 'h ' Φ

php 'h '

+

php 'h '

The random-phase approximation (RPA) includes a part of these higher-order terms

Ω

+ n

=

∑ {X ph

n

}

( ph)c +p ch + Yn ( ph)ch+ c p ,

Φ

n

= Ω

+ n

Φ

0

This can be regarded as a harmonic approximation around the ground state.

Φ

ph

= Ω

+ ph

Φ

0

Collective states under rapid rotation At high spin, each p-h pair is aligned by the Coriolis force, to produce an At even higher spin, one of aligned phonon. the p-h pairs is completely aligned, by escaping from the λ-coupling.

At low spin, many p-h pairs of spin λ contribute to the collective state. λ=even

J=2jmax

J=λ Jz=λ J=λJ z=K

λ=odd

J=jmaxj+j’max

Φ Φ

coll

 ≈  ∑ α  ph

i ≈ 10

ph

(

 c +p ch λ K  Φ 

)

0

aligned coll

 ≈  ∑ α  ph

ph

(c c )

+ p h λλ

  Φ  

0

i ≈ 5 ~ 10 a A+ aB+ Φ

HFB

Band crossing between collective bands and non-collective 2qp bands E’ 2Δ

E2 qp ≈ 2∆

The lowest mode of excitation:

Ecor ≈ 2∆ − Ecoll

Collective surface vibration → aligned 2qp band

Ecoll < 2∆  2 for quadrupole icoll ≤ λ =   3 for octupole i2 qp ≈ 2 jmax ~ 10 ωc

ωc ≈

E cor 2∆ − Ecoll ~ i2qp − iλ i2 qp − λ

ωrot

Quasi-particle random-phase approximation in the rotating frame Marshalek, NPA266 (1976) 317 Shimizu & Matsuyanagi, Prog. Theor. Phys. 79 (1983) 144 T.N., Matsuyanagi, Mizutori, Shimizu, PRC53 (1996) 2213

• Collective vibrational states are calculated with the QRPA based on the cranked shell model H = hdef + Γ pair − ω H res = −

rot

J x + H res

1 '' '' κ Q ⋅ Q ∑ λ λ λ 2 λ = 2,3

[H , Ω ]

+ n QRPA

= ω n Ω

+ n

Octupole vibrations in 164Yb T.N., Act. Phys. Pol. B27 1996) 59

K=0 K=2 K=1

∑

K

n Q3 K

 > 200 fm 3 2 0  > 100 fm 3  < 100 fm 3 

Octupole vibrations in 238U T.N., Act. Phys. Pol. B27 1996) 59 K=1

K=0

∑

K

n Q3 K

 > 300 fm 3 2 0  > 150 fm 3  < 150 fm 3 

In 1986, the first superdeformed band was discovered in 152Dy.

γ

I+6 I+4 I+2 I

１９８６　@Daresbury, UK P.J. Twin et al, PRL 57 (1986) 811 J.D. Garret et al, Nature 323 (1986) 395.

•Large moment of inertia •Large intraband B(E2) 　B(E2)≈2000　W.u. Major : Minor axes ~ 2:1

We have studied elementary modes of excitation in SD states. S.Mizutori, Y.R.Shimizu, K.Matsuyanagi, Prog. Theor. Phys. 83 (1990) 666; 85 (1991) 559; 86 (1991) 131. TN, SM, YRS, KM, Prog. Theor. Phys. 87 (1992) 607. TN, KM, SM, W.Nazarewicz, Phys. Lett. B 343 (1995) 19. TN, KM, SM, YRS, Phys. Rev. C 53 (1996) 2213.

• Single-particle orbitals with different Nosc • Quadrupole collectivity is weak. • Octupole modes of excitation are dominant in low-lying spectra.

Excitation energy in the rotating frame relative to the ground-state SD band.

RPA routhians for SD 152Dy

RPA in the rotating shell model predicted an excited SD band in 152Dy is the K=0 octupole vibrational band. T.N., Mizutori, Matsuyanagi, Nazarewicz, PLB343 (1995) 19

Octupole band with K=0 (Y30)

MOI

Exp: Dagnall et al, PLB 335 (1994)

Determination of spin and parity Spin and parity of SD bands have been determined by observation of decay gamma rays. 194

Hg, 192,194Pb, 152Dy, etc.

Lauritsen, PRL 88 (2002) 042501

Ex SD6

SD1

Decay

I

Exp RPA B(E1)≈10-4 W.u.

In A=190 region, we assign most of excited SD bands as Y32 octupole vibrational bands. QRPA in the rotating shell model predicts the K=2 (Y32) octupole vibration as the lowest excitation modes for even-even SD nuclei. This assignment systematically reproduces moments of inertia. TN, KM, SM, YRS, Phys. Rev. C 53 (1996) 2213

Confirmed by experiments

Linking transitions B(E1)≈10-5 W.u.

Octupole dominance has been confirmed by many experiments.

1994 Crowell et al (PLB 333, 320; PRC 51, R1599) 1996 Wilson et al (PRC 54, 559) 1997 Hackman et al (PRL 79, 4100) 1997 Amro et al (PLB 413, 15) 1997 Bouneau et al (ZPA 358, 179) 2001 Rossbach et al (PLB 513, 9) 2001 Korichi et al (PRL 86, 2746) 2001 Prévost et al (EPJA 10, 13) 2002 Lauritsen et al (PRL 89, 042501)

Strong octupole correlation is established in SD states. Then, what else can we expect to see? Freq. ratio

Fission isomer

A≈190

Single-particle energy levels

A≈15 0

If we can extend the exploring area of SD states with radioactive beam, deformation

Anything new in open-shell SD states?

Shape phase transition phenomena (sph. to def.) for open-shell nuclei are well known in the ground states. spherical

transitional

deformed

Effective potential V(β)

Increasing neutrons β Quadrupole deformation (order parameter) Spontaneous symmetry breaking (SSB)

What happens if we add neutrons to “closed-shell” SD nuclei? Effective potential V(β) (I+7)− (I+6)

+

(I+5)− (I+4)

+

(I+2)

+

β

I

+

SD 152Dy What kind of parameter?

(I+3)− (I+1)−

(I+6)+ (I+4)+

？

(I+2)+

？

I+ 162

Dy

Qualitative analysis with harmonic oscillator potential T.N., S.M., K.M., Prog. Theor. Phys. 87 (1992) 607. Freq. ratio

Possible p-h excitations Spherical nuclei

unocc occ

Single-particle energy levels

r3Y30

r3Y31 r3Y32 r3Y33

K=1 (Y31) mode of excitation is analogous to quadrupole modes in spherical nuclei. Driving to shape transition in open-shell configuration. deformation

cf) Bohr & Mottelson’s textbook (1975).

r2Y20,22

Banana-(Y31-type) shape phase transition in open-shell SD states T.N., S.M., K.M., Prog. Theor. Phys. 87 (1992) 607.

Increase of valence nucleons ⇓

Z=80+2, N=80+Nval

HO+QRPA

Strong pairing　　　　Weak pairing

Banana-superdeformation

∆p=1.3 MeV

∆p=1.1 MeV

SSB towards banana shape

More realistic calculation Nilsson+BCS+QRPA

Ex [ MeV ]

Superdeformed Dy isotopes

ε=0.59, Δ=0.5 MeV, κ=1.05κHO A

Shape transition to banana-super-deformation

Where are they? Increasing (decreasing) valence neutrons (protons) by 8-10 leads to regions near beta-stable line A≈190

A≈15 0

162

Dy

Fusion reaction with radioactive beam might populate these high-spin SD states near beta-stable line.