CrankedCranked--RPA description of RPA description of wobbling motionwobbling motionin in triaxiallytriaxially deformeddeformed nucleinuclei

Takuya Shoji and Yoshifumi R. ShimizuDepartment of Physics, Graduate School of Sciences,

Kyushu University, Fukuoka 812-8581,Japan

axially-sym. def. triaxial def.

Wobbling motion (Wobbling motion (triaxiallytriaxially deformeddeformed))

Rotor Model

1-phonon excited bandE

yrast band

I 2+I

E2 (ΔI=±1)

E2 (ΔI=－2)

2-phonon excited band

(vacuum of phonon)

Bohr – Mottelson vol.II

2

wob 1 1x x

x y z

Iω⎛ ⎞ ⎛ ⎞

= − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

J JJ J J

2

wob( 1) 1( , ) 12 2x

n IE I nI ω+ ⎛ ⎞= + +⎜ ⎟⎝ ⎠J22 2

rot 2 2 2yx z

x y z

II IH = + +J J J

1I

x-軸

Itilting

Body-fixed frame

S. W. Ødegård et al., Phys. Rev. Lett. 85 (2001) 5866

1−→ II

Evidence of wobbling motionEvidence of wobbling motion

B(E2)in : large

2−→ IIE

I 2+I

B(E2)out : large

Rotor Model

B(E2)in

B(E2)out

TSD1

TSD2

TSD3

163 71 Lu

D. R. Jensen et al., Phys. Rev. Lett. 89 (2002) 142503

TriaxialTriaxial deformation in the rotor modeldeformation in the rotor modelTriaxial deformation

2 2 220 (2 )aQ z x y∝ ⟨ − − ⟩∑

2 222 ( )

aQ x y∝ ⟨ − ⟩∑

22

20

2tan QQ

γ = −

Two intrinsic Q-momentsγ

Three moments of inertia

, , x y zJ J J

( )( )wob x y x zω ∝ − −J J J J

irrotational moments of inertia

B(E2)out (ΔI= -1) 0γ >

y x zJ >J >JΔI= +1

ΔI= -1

TSD1 TSD2

in the Lund convention

Microscopic study of wobbling motionMicroscopic study of wobbling motion

Cranked mean field

Random Phase Approximation (RPA)+

E

yrast band

excited band

total angular momentum

† †[ ', ] nn nH X Xω=int' 'H h V= +

def rot' xh h Jω= −

rotω

nω

Microscopic Calculation Rotor Model PictureMicroscopic Calculation Rotor Model Picture(Interpretation)

Microscopic wobbling theory

E. R. Marshalek, Nucl. Phys. A331 (1979) 429

sph.H h P QQ= + +def resH h V= +

Realistic Calculation

M. Matsuzaki, Nucl. Phys. A509 (1990) 269

M. Matsuzaki, Y. R. Shimizu, K. Matsuyanagi, Phys. Rev. C 65,041303(R)Recent Calculation for Lu, Hf nuclei

Y. R. Shimizu, M. Matsuzaki, Nucl. Phys. A588 (1995) 559

Nilsson potential as a mean field

general def.symmetry (J) restoring int.

using

Phys. Rev. C 69,034325(2004)

UUniformlyniformly--RRotating frameotating frame

x-axis

Shape fluctuation

Jω

Angular momentum fluctuationUR

def rot( ) xh t h Jω= −PA

def( () )h t h Jtω= − ⋅

JPPrincipalrincipal--AAxis framexis frame

( ) ( )zx zx xy xzx xy yQQ t QQ tκ κ− −⟨ ⟩ ⟨ ⟩

2

1

5 [ ]16

A

ij i j ij aa

Q x x r δπ =

= −∑

( ) ( ) ( ), , ,( ) ( )n n n

y z y z y zJ t tω⟨ ⟩ = J

Quadrupole tensor

2

( ) ( )1 1x xn n n

x y z

Iω⎛ ⎞⎛ ⎞

= − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

J JJ J J

xJ I⟨ ⟩ ∼ rot rot/ /x xJ Iω ω=⟨ ⟩J ∼

PA 0xyQ⟨ ⟩ =

PA 0zxQ⟨ ⟩ =PA-conditions

MarshalekNucl. Phys. A331

(1979) 429x-axis

moment of inertia

MeanMean--field (field (Woods-Saxon potential))Nilsson potential

is too large due to term2l

Woods-Saxon potential

Skyrme:S3

W.S.:R.Wyss

H.O.2l+

ls+ls+No ls

152 66 Dy

208 82 Pb

208 82 Pb

xJ

good parametrization

ls+

[fm]r

[fm]r

( )rDensity distribution ρρ

ρ

Shape of nuclei at I=53/2Shape of nuclei at I=53/2Cranked Nilsson Strutinsky Lund convention

20~γ60γ =

60γ = −

cos( 30 )oε γ +

163 71 Lu

( , ) ( , 1 / 2)π α = + +proton 13 / 2i

ND0.1

0.1 0.4

0.2

0.3

-0.1

-0.20.2 0.3

0

0.4

TSD

0γ =

2 0.42β =

4 0.034β =18oγ =

Paring gaps (monopole pairing)Paring gaps (monopole pairing)

( )

2

ro t0

0

ro t 2

0 0

ro t

112

2

ωω

ωωω

⎧ ⎡ ⎤⎛ ⎞⎪ ⎢ ⎥Δ − ⎜ ⎟⎪ ⎢ ⎥⎝ ⎠⎪ ⎣ ⎦Δ = ⎨

⎛ ⎞Δ⎪⎜ ⎟⎪⎝ ⎠⎪⎩

rot 0for ω ω<

rot 0for ω ω≥

Self-consistent cal.

[MeV]Δ [MeV]Δ

rot[MeV]ω rot[MeV]ω

00( )

2ω Δ

Δ =

0Δ

0

2Δ

0ωSelf-consistent cal. Used in Cal.

collapse !

RPA interaction (RPA interaction (““minimalminimal””))Separable force

Broken rotational symmetry

RPA equation of motion is determined by a given mean field

Restore broken symmetryin RPA order

2res

,

12 k k

k y zV Fκ

=

= ∑[ ]def , ,k k kh iJ iJκ ⎡ ⎤= ⟨ ⟩⎣ ⎦

[ ]def ,k kF h iJ=

def[ , ] 0kh J ≠def res RPA[ , ] 0kh V J+ =

Bohr-Mottelson vol.II

C. G. Andersson et al.,Nucl. Phys. A361 (1981) 147

No adjustable parameters

Moment of inertiaMoment of inertia

D. R. Jensen et al., Nucl. Phys. A 703 (2002) 3

EXP.CAL.

CAL. (p+Rotor)

nice agreement with experimental data

I. Hamamoto,G. B. Hagemann, Phys. Rev. C 67, 014319 (2003)

EXP.

CAL. (p+Rotor)

rot[MeV]ω

163 71 Lu

rot

xx

Jω⟨ ⟩

=J

rotEI

ω ∂=∂

(R)y 45=J

20oγ =(R)x 48=J

(R)z 17=J

Excitation energy of RPAExcitation energy of RPA--phononphonon

EXP.

CAL. (RPA)

CAL. (p+Rotor)

D. R. Jensen et al., Nucl. Phys. A 703 (2002) 3I. Hamamoto,G. B. Hagemann, Phys. Rev. C 67, 014319 (2003)

EXP.CAL. (p+Rotor)

Wobbling-like RPA solution exists,rot[MeV]ω

163 71 Lu

wob rot 1 1x x

y z

ω ω⎛ ⎞⎛ ⎞

= − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

J JJ J

disappear

even if parameters are changed in a reasonable range.

B(E2) ratioB(E2) ratio

A. Görgen et al., Phys. Rev. C 69, 031301(R) (2004)

EXP.

CAL. (RPA)

CAL. (p+Rotor)IN

OUT

EXP.CAL. (p+Rotor)

Extremely collective transitionsrot[MeV]ω

163 71 Lu

I. Hamamoto,G. B. Hagemann, Phys. Rev. C 67, 014319 (2003)

~ 50 Weisskopf units

Three moments of inertiaThree moments of inertia

is the largest

163 71 Lu

xJ

yJ

zJ

xJ

rot[MeV]ω

(wob),(wob)

, (wob),

y zy z

y z

Jω

⟨ ⟩=J

rot

xx

Jω⟨ ⟩

=J

( )( )wob x y x zω ∝ − −J JJ J

• The excitation energy is small compared with the experimental data (Effects of particle-rotation coupling ?)

• The B(E2) ratio of the in-band and out-of-band is small compared with the experimental data (Definitions of ?)

ConclusionConclusion• We have studied nuclear wobbling motions by using

Cranked mean field plus RPA on microscopic view point.

• Woods-Saxon mean field and symmetry restoring RPA interaction

Wobbling-like RPA solutions exist.

γ

Microscopic justification of the rotor model

Effects of particleEffects of particle--rotation couplingrotation coupling

2 22 22 2 2

rot( )

2 2 2 2 2 2y yx xz z

x y z x y x x

x

z

I II j II I jH I j−= + + + +−+ =

J J JJ J J J J

Particle plus Rotor Model Hamamoto-Hagemann, Phys. Rev. C 67,014319

2

( 1, 1) ( , 0) 1 1 x x

x y xz

IE I n E I n j⎛ ⎞ ⎛ ⎞+ = − = = − − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

J JJ J J J

-1 270 [MeV ]x ∼J132

j ∼

93 [keV]x

jJ∼

rot[MeV]ω

Definitions ofDefinitions of

rot[MeV]ωpotγ

2 2122

dens 2 2 220

2 3tan2

Q y xQ z x y

γ − ⟨ − ⟩= − =

⟨ − − ⟩

out in( 2) / ( 2)B E B E

densγEXP.

γ

if ,y z=J J( 2) / ( 2)out inB E B E

2tan ( 30 )oγ∝ +

in the rotor model

163 71 Lu

• The excitation energy is small compared with the experimental data (Effects of particle-rotation coupling ?)

• The B(E2) ratio of the in-band and out-band is small compared with experiments (Definitions of ?)

ConclusionConclusion• We studied wobbling motion by using Cranked mean field

plus RPA on microscopic view point.

• Woods-Saxon mean field and symmetry restoring RPA interaction

Wobbling-like RPA solutions exist.

γ

Microscopic justification of the rotor model

W.S.:UniversalW.S.:R.Wyss

Skyrme:SKM* Gogny:D1S

Density distribution ofDensity distribution of 208 82 Pb

[fm]r

( )rρ

[fm]r

[fm]r

( )rρ ( )rρ( )rρ

( )rρ ( )rρ

[fm]r

Excitation energy of RPAExcitation energy of RPA--phononphonon

EXP.

CAL. (RPA)D. R. Jensen et al., Nucl. Phys. A 703 (2002) 3EXP.

rot[MeV]ω

163 71 Lu

WoodsWoods--Saxon parameter setSaxon parameter set152 66 Dy

R.Wyss Universal

QuasiQuasi--particle energiesparticle energies

rot[MeV]ω

162 70 Yb

rot[MeV]ω

NEUTRONPROTON

1/ 2α = +1/ 2α = −2 0.42β = 4 0.034β =18oγ =

0.50 [MeV]Δ = 8.8 [MeV]Nλ = − 3.6 [MeV]Pλ = −

InIn--band transitionsband transitions163 71 Lu

rot[MeV]ω

CAL.

EXP.

A. Görgen et al., Phys. Rev. C 69, 031301(R) (2004)EXP.

Woods-Saxon potential

S0.90.5

0.1

Sr0

Vw.s.

Vw.s.2

LS potential

Coulomb potential

Pairing potential