Nuclear vorticity and general treatment of vortical, toroidal, and compression modes

J. Kvasil 1), V.O. Nesterenko 2), W. Kleinig 2,3), P.-G. Reinhard 4) , and P. Vesely 1,5)

1) Institute of Particle and Nuclear Physics, Charles University, CZ-18000 Praha 8, Czech Republic

2) Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia

3) Technical University of Dresden, Institute for Analysis,D-01062, Dresden, Germany

4) Institute of Theoretical Physics II, University of Erlangen, D-91058, Erlangen, Germany

5) Depart. of Physics, Univ. of Jyvaskyla, FI-40014, Finland

KLFTPKLFTP--BLTP Joint Workshop on Nuclear PhysicsBLTP Joint Workshop on Nuclear Physics, , 66--8. 8. 09.11, Beijing, China09.11, Beijing, China

Motivation

Nuclei demonstrate both- irrotational flow (most of electric GR) - vortical flow (toroidal GR, s-p excitations) ( ) ( ) 0w r v r= ∇× ≠

rr rr r( ) ( ) 0w r v r= ∇× =rr rr r

Vorticity is a fundamental quantity:- does not contribute to the continuity equation,- represents an independent part of charge-current distribution beyond the continuity equation.

( )w rrr

0nucjρ +∇ ⋅ =rr

&

distribution beyond the continuity equation.

Vorticity is related to the exotic modes:- toroidal E1 mode (TM) ,- compression E1 mode (CM),

which are now of a keen interest .

Theoretical studies:Many publications on toroidal and compressional (ISGDR) modesand manifestations of vorticity:V.M. Dubovik and A.A. Cheshkov, SJPN 5, 318 (1975).M.N. Harakeh et al, PRL 38, 676 (1977).S.F. Semenko, SJNP 34 356 (1981).J. Heisenberg, Adv. Nucl. Phys. 12, 61 (1981). S. Stringari, PLB 108, 232 (1982).E. Wust et al, NPA 406, 285 (1983).E.E. Serr, T.S. Dumitrescu, T.Suzuki, NPA 404 359 (1983). D.G.Raventhall, J.Wambach, NPA 475, 468 (1987).E.B. Balbutsev and I.N. Mikhailov, JPG 14, 545 (1988).S.I. Bastrukov, S. Misicu, A. Sushkov, NPA 562, 191 (1993)., , , , ( )I. Hamamoto, H.Sagawa, X.Z. Zang, PRC 53 765 (1996).E.C.Caparelli, E.J.V.de Passos, JPG 25, 537 (1999).N.Ryezayeva et al, PRL 89, 272502 (2002). G.Colo, N.Van Giai, P.Bortignon, M.R.Quaglia, PLB 485, 362 (2000).D. Vretenar, N. Paar, P. Ring,T. Nikshich, PRC 65, 021301(R) (2002).V.Yu. Ponomarev, A.Richter, A.Shevchenko, S.Volz, J.Wambach, PRL 89, 272502 (2002).J. Kvasil, N. Lo Iudice, Ch. Stoyanov, P. Alexa, JPG 29, 753 (2003).A. Richter, NPA 731, 59 (2004).……….N. N. PaarPaar, D. , D. VretenarVretenar, E. , E. KyanKyan, G. , G. ColoColo, Rep. , Rep. ProgProg. Phys. . Phys. 7070 691 (2007). 691 (2007). Recent

review

Operators of the modes:

Toroidal mode E1(T=0):

2 2 2 2 01 0

3

1

20 1

5( )

1 ˆˆ ( 1 ) ( ) [ ( ) ]20

2ˆ ( ) [ (

3

) ]52 3

tor nuc

nuc

M E dr j r rc

idr j r r Y r r Y

c

r r r Y

μ

μ

μ

μ = ⋅ ∇× ×∇

= − ⋅ + − < >

− < >∫

∫

r r rr r r

r r rr r

C i d E1(T 0)

V.M. Dubovik and A.A. Cheshkov, SJPN 5, 318 (1975).

S.F. Semenko,

cmc

cmc

Compression mode E1(T=0):

3 20 1

1ˆ ˆ' ( 1 ) ( )10

5[ ]

3com r r r YM E dr r μμ ρ − < >= ∫r r The TM and CM

operators are related.

S.F. Semenko, SJNP 34 356 (1981).

Vortical mode E1(T=0): NO yet OPERATOR

Observation of ISGDR : CM and perhaps TM:

( , ')α αD.Y. Youngblood et al, 1977H.P. Morsch et al, 1980 G.S. Adams et al, 1986B.A. Devis et al, 1997 H.L. Clark et al, 2001D Y Youngblood et al 2004D.Y. Youngblood et al, 2004

M.Uchida et al, PLB 557, 12 (2003), PRC 69, 051301(R) (2004)

N.Ryezayeva et al, PRL 89, 272502 (2002).

( , ')γ γ

- definition of nuclear vorticity (HD vs Wambach),

- IS (T=0) and IV(T=1) branches of the modes,

- role of magnetization (spin) nuclear current,

- there is no the VM operator, VM vs TM/CM,

different conclusions on CM vorticity

J. Kvasil, V.O. Nesterenko, W. Kleinig, P.-G. Reinhard, P. Vesely, PRC, 84, n.3, 2011arXiv: 1105.0837[nucl-th]

Open problems

We will show that the VM operator may be derived and related to TM and CM operators

ˆ ˆ ˆ( ) ( ) ( )vor tor comM E M E M Eλμ λμ λμ= +

This relation allows to understand better the connectionbetween VM, TM, and CM

How to introduce the vorticity for nuclei?

( ) ( ) 0w r v r= ∇× ≠rr rr r

The vorticity cannot be measured by the current curl

( )nucj r∇×rr r

Nuclear quantum theory deals with the nuclear current ( )nucj rr r

In HD the vorticity is defined as

( )0

0

ˆ( ) ( ) ( )( )

( )nucj r r v r

v rr

δ ρρ

∇× − ∇ ×∇× =

rr rr r rrr rr

r

Then, how to introduce the nuclear vorticity?

0

( )( )

( )nucj r

v rr

δρ

=r r

rrr ˆˆ ( ) ~ ( ) [....]nucM E dr jλμ ∇×∫

rrr

( )nucj

because

Two definitions of vorticity in nuclear theory:

Definition 1 (from hydrodynamics):

( ) ( )w r v r= ∇×rr rr r

0

( )( )

( )nucj r

v rr

δρ

=r r

rrr

(ˆ| ( ) | 0 [....])nucM E d jr νν δλμ< > ∇×= ∫rrr

CM is irrotational:

0ˆ| ( ) | 0 [..( ) ..]vorM E d vr νν λμ ρ< >= ∇×∫

r r r

3 20 1

5( 1 ) [ ]

3comv E r r r Y μμ ∝ ∇ − < >rr

( ) 0comv r∇× =r rr

By using expansion of the current transition

( )( ) * ( ) *

1 1 1 1

( | )ˆ( ) | ( ) | [ ( ) ( ) ]2 1

fifi fii i f f

f f nuc i i

f

j m j mj r j m j r j m j r Y j r Y

jλλ λλ μ λλ λλ μ

λμ

λμδ − − + += = ++∑

r r r rr r

one may construct the vorticity transition density

D.G.Raventhall, J.Wambach, NPA 475, 468 (1987).

Definition 2 (Ravenhall and Wambach)

1

1(

2( ) )

2 dw j rr

dr r λλλ λλ λλ ++ +⎛ ⎞= +⎜ ⎟

⎝ ⎠and strength

∫∞

+=0

)(4)( )( drrwr fifiλλ

λλν

expressed through the particular transverse current multipole ,which, unlike , does not contribute to the continuity equation ( )

1( )fij rλλ−

0nucjρ +∇ ⋅ =rr

&

So, is an independent part of charge-current distribution.

( )1( )fij rλλ+

( )1( )fij rλλ+

This approach does not use the vortical operator.

The mode is vortical if it is unconstrained by CE and has

Wambach vorticity

3 20 1

1 5ˆ ˆ' ( 1 ) ( )[ ]10 3comM E dr r r r r Y μμ ρ= − < >∫

r r

12 102 2 2

0

2 2ˆˆ ( 1 ) ( ) [ ( ) ]52 3

com nuc

iM E dr j r r r r

cY Yμ μμ = − + − < >∫r rrr r

Difference between HD and Wambach vorticity.

( )1( )fij rλλ+

3

CM involves and so . Hence CM is vortical despiteits gradient flow !?

12j μ

r12Y μ

r

The reason of contradiction: The Wambach vorticity was introduced mainly

as a quantity fully unconstrained by the CE rather than the purely vortical value in the HD sense.

1( ) ( )w r j rλλ λλ+∝

3( )comv r Yλμ∝ ∇rr

1

(2 1)!!ˆ ( ) ( )1

ˆ[ ( ) ]nucM Ek dr j kr Yc

jk

rλ λλμλλ λλμ

λ+= ⋅ ∇++

×∫rrr rr

Multipole electric operator

0 0ˆ ˆ( ) ( ) ( ) ( )nuc

kcr v r j r i r rρ ρ

λ∇× = ∇× − ∇ ×

rr r rr r r r rr)(

ˆrjnuc

rrr×∇

Main idea:

Derivation of the vortical operator (Wambach)

Then

1

(2 1)!!ˆ ( ) [ ( ) ( , )]1vorM Ek dr j kr Y

ck λ λλμλλ λλμ ϑ ϕ

λ+

+= ⋅+ ∫

rr

ˆ ˆ(

ˆ ˆˆ( ) (

) ( )

)

S

nuc

M Ek M k

ij r kc r r

μ λ

ρλ

λ μ

⎡ ⎤∇× − ∇ ×⎢ ⎥⎣⋅ =

= −⎦

rr rr r r1ˆ ˆˆ ˆ[ , ] ,

if i f

v r r H ikcri

E E kcω

= = =

= − =

r r rr&

h

h h

truly vortical

Long-wave approximation:2( ) ( )

( ) [1 ](2 1)!! 2(2 3)

kr krj kr

λ

λ λ λ= − +

+ +K

The second order term gives:- toroidal operator- compression operator

vortical operator- vortical operator

ˆ ˆ ˆ( ) ( ) ( )torM Ek M E kM Eλμ λμ λμ= + ˆ ( ) ( )M E dr r r Yλ λμλμ ρ= ∫r r

11 1

2ˆˆ ( ) ( ) ( )2 2 1 1 2 3tor nuc

iM E dr j r r Y Y

cλ

λλ μ λλ μλ λλμλ λ λ

+− += − ⋅ +

+ + +∫r r rr r

ˆ ˆ ˆ( ) ( ) ( )S comM Ek M E kM Eλμ λμ λμ= −

11 1

1 2ˆˆ ( ) ( ) ( )com nuc

iM E dr j r r Y Yλ

λλ μ λλ μλ λλμ +

++= ⋅ −∫

r r rr r1 1( ) ( ) ( )

2 2 1 2 3com nucM E dr j r r Y Yc λλ μ λλ μλμ

λ λ λ− ++ +∫

ˆ ˆ( ) ' ( )com comM E kM Eλμ λμ= −21ˆ ˆ' ( ) ( )

2(2 3)comM E dr r r Yλλμλμ ρ

λ+=

+ ∫r r

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )vor S tor comM Ek M Ek M Ek k M E M Eλμ λμ λμ λμ λμ⎡ ⎤= − = +⎣ ⎦

11

1 ˆˆ ( ) ( )(2 3) 2 1vor nuc

iM E dr j Yr r

c λλλ

μλλμ

λ λ+

++= −

+ + ∫rr r r

ˆ ˆ ˆ( ) ( ) ( )vor tor comM E M E M Eλμ λμ λμ= +

ˆ ˆ ˆ( ) ( ) ( )vor tor comM E M E M Eλμ λμ λμ= +

11 1

2

2ˆˆ ( ) ( ) ( )2 2 1 1 2 3

1 12 1 2 3

ˆ[ ( ) ]

tor c

c

n

nu

u

iM E dr j r r Y Y

c

dr r Yc

j r

λλλ μ λλ μ

λλλμ

λ λλμλ λ λλ

λ λ

+− +

+

= − ⋅ ++ + +

×= −+ +

∇

∫

∫

r r

r

r

r

r

r

r

rr

11

1ˆ ( )(2 3) 2 1

ˆ ( )nucvor

iM E d r Y

cjr rλ

λλ μλλμ

λ λ+

++= −

+ + ∫r rrr

11 1

2

1 2ˆˆ ( ) ( ) ( )2 2 1 2 3

2 2 3ˆ[ ( ) ]

com nuc

nuc

iM E dr j r r Y Y

ci

dr r j rYc

λλλ μ λλ μ

λλμ

λ λλμλ λ λλλ

+− +

+

+= ⋅ −+ +

=+

∇⋅

∫

∫r

r rr

r r

rr

r

The numerical results: - fully self-consistent Skyrme separable RPA (SRPA) ,- SLy6 force- 208Pb Phys. Rev. C: 66, 044307 (2002);

74, 064306 (2006);78, 044313 (2008);

IJMP(E): 16, 624 (2007); 17, 89 (2008); 18, 975 (2009). SRPA :

Electric GR

Magnetic spin-flip GR

PRC, 80, 031302(R) (2009);JPG, 37, 064034 (2010);IJMP (E), 19, n.4, 558 (2010).p p ( ), , , ( )

Strength function

2

0

ˆ( 1; ) | | | 0 | ( )ES E Mν λ νν

ω ς ω ω≠

= < Ψ > −∑

22

1( )

2[( ) ]

4

ν

ν

ς ω ωπ ω ω

Δ− =Δ− +

with the Lorentz weight

1 M e VΔ =

Toroidal, compressional, vortical operator

Comparison of VM, TM, and CM

- Broad low-energy (LE) and high-energy (HE) bumps for VM, TM, and CM.

- LE strength is dominated by VM and TM - HE strength is dominated by VM and CM

- General agreement for TM and CM with previous studies.

- Poor agreement with exper. of Ichida(like in previous studies).

- Purely vortical VM does not coincidewith partly vortical TM, especially at HE.

1 1

2 2

12.7 MeV, 3.5 MeV

23.0 MeV, 10.3 MeV

E

E

= Γ == Γ =

Uchida et al., 2003:

First direct comparison of VM, TM and CM!

- Convection and magnetization (spin) parts of nuclear current,- T=0 and T=1 channels

,

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ( ) ( ))q qnuc con mag com mag

q n p

ej r j r j r j r j r

m =

= + = +∑r r r r rr r r r rh

( )ˆ ( ) ( ( ))q qcon eff k k k k

k q

j r ie r r r rδ δ∋

= − − ∇ −∇ −∑r r sr r r r r

ˆ ˆ( ) ( )2

q smag k qk k

k q

gj r s r rδ

∋= ∇ × −∑

r rr r r r

T=0:, 1

1, ( 0) ( ) , 0.60. 42

88n p n p n peff eff s s se e g T g g ςς= = = = + = =

( ), 11, 1 ( ) , 0.64

24.70n p n p n p

eff eff s s se e g T g g ςς= − = = = − = =−T=1:

, ,| ( 0)| | ( 1) |n p n ps sg T g T= << =

Vortical, toroidal, and compressional T=0 strength SLy6

1MeVΔ =

- dominant contribution ofto VM and TM

conj

-no contribution to: - CM - HE strength

magj

0 1( )

5.58 , 3.82

0.882

T p ns s

p ns

s

sg

g g g

g ς ς

ς= = +

= = −

=

Small T=0 g-factors!

Vortical, toroidal, and compressional T=1 strength

SLy6

1MeVΔ =VM and TM:

- dominant contribution of !!magj

1

5.58 , 3.82 ,

41

)2

.7(T p ns s

n

s

ps s

g g g

g gς ς

ς=

= = −

== −

Large T=1 g-factors!

2

VM and TM in T=1 channelare suitable to see the effect of in electric excitations. magj

Wambach vs HD vorticity

208PbSLy6 force

HD is modest

W ~ CMHD is small

W ~ HD ~Tvortical region

irrotational region

Conclusions

-The (Wambach) vortical operator has been derived.This allows to treat and compare VM,TM, and CM on the same theor. ground.

Th d i t l f

- Both T=0 and T=1 VM,TM, and CM were considered

-The difference between Wambach and HD voirticities was discussed.

- The dominant role of:- convection nuclear current in isoscalar VM and TM- magnetization nuclear current in isovector VM and TM- E1(T=1) VM and TM: remarkable example of strong effect in electric GR.

magj

Outlook:- more comparison of HD and Wambach vorticity- dependence of Skyrme forces- proposals for (e,e’) and hadron reactions

Thank you for your attention!

1

(2 1)!!ˆ ( ) ( 1) ( )( 1

ˆ[ ( ) ]) nucM Ek i dr j kr Y

cj r

k λ λλμλλλμ λ λ

λ+

+= ×+

∇+ ⋅∫rr rrr

HD vortical operator

The HD vortical operator (matrix element) may be also constructed:

ˆ ( )nucj r∇×rr r

ˆ ˆˆˆ ˆ( ) ( ) ( ) ( )nuc

kcr v r j r i r rρ ρ

λ∇ × = ∇ × − ∇ ×

rr r rr r r r rrWambach:

HD

0

( )( )

( )j r

v rr

νν

δρ

=r r

rrr 0 0

0

1( ) ( ) ( ) ( ) ( )

( )r v r j r r j r

rν νρ δ ρ δρ

∇× = ∇× − ∇ ×r rr r rr r r r rr

r

0 00

1ˆ ˆˆ( ) ( ) ( ) ( ) ( )( )nuc nucr v r j r r j rr

ρ ρρ

∇× = ∇× − ∇ ×r rr r rr r r r rr

r

SRPA vs 1ph strength

- Collective down-shifts:~ 1-3 MeV for LE bump~ 4 MeV for HE bump

- Quite collective RPA states:

t t t 8 3 M Vstate at 8.3 MeV:

- In LE bump the structure of VM, TM, and CM responses is mainly of 1ph origin

The 1ph origin of the vorticity ?D.G.Raventhall, J.Wambach, NPA 475, 468 (1987).

SRPA vs 1ph strength - Collective up-shifts:~ 1-2 MeV for LE bump~ 4 MeV for HE bump

- In LE bump the structure of VM, TM, and CM responses is mainly of 1ph origin

The 1ph origin of the vorticity ?D.G.Raventhall, J.Wambach, NPA 475, 468 (1987).

Velocity fields for T=0 vortical state at 8.3 MeV

s-p manifestation of vorticity

nn[2g7/2 – 2f5/2] 36% pp[1i13/2-1h11/2] 18%

Wambach does not use the vortical operatorthough it would be useful for comparison of VM, TM, and CM.

We will derive the vortical operator and relate it with TM and CM ones:

ˆ ˆ ˆ( ) ( ) ( )vor tor comM E M E M Eλμ λμ λμ= +

2ˆi

where for we have 1λ =

2 21

202 10

2ˆ ( 1 ) ( ) [ ( ) ]52 3

tor nuc Yi

M E dr j r r rc

Yrμ μμ = − + − < >∫r rr rr

12 102 2 2

0

2 2ˆˆ ( 1 ) ( ) [ ( ) ]52 3

com nuc

iM E dr j r r rY r Y

cμ μμ = − − < >−∫

r rrr r

212

3 ˆˆ ( 1 ) ( )5 2vor nuc

iM E dr j r r Y

c μμ = − ∫rrr r - involves

- no c.m.c. 12Y μ

r

E1(T=1) GR SLy6

1MeVΔ =

exper

Center of mass corrections

ˆ ( ) ( )

( ) ( ) 0

O dr r o r

dr j r o r

δ δρ

δ

=

= ⋅∇ =

∫∫

r r r

r rr r r

1 1

1ˆ ˆ( )A A

k kk k

O o r O zA= =

= → =∑ ∑r

translation invariance: perturbation does not changez-coordinate of the c.m.

δρ

0

1ˆ ˆ0 | ( ) | 0 | | ( ) ( )2

j r F r f rmiν

ν ν ρ= ∇∑r rr r r 10 100( ) ( ) 3o r rY Y∇ = ∇ =

rr rr

0

1ˆˆ0 | ( ) | 0 | | [ ( ) ( )]2

r F r f rmν

νω ρ ν ν ρ= − ∇ ⋅ ∇∑

r rr r rν

0 0

0 0

ˆ( ) 0 | ( ) | ( ) ( ) ( ) ( )

ˆ( ) 0 | ( ) | [ ( ) ( )] [ ( ) ( )]

j r j r r f r r v r

r r r f r r v r

ν

ν

δ ν ρ ρ

δρ ρ ν ρ ρ

= ∝ ∇ ∝

= ∝∇⋅ ∇ ∝∇⋅

r r rr r r r r rr

r r rr r r r r rr

212 10

2 212 10

2 212 10

2( )

5

2( )

5

vor

tor

com

v r Y Y

v r Y r Y

v r Y r Y

μ μ

μ μ

μ μ

η

η

η

= +

= + −

= − −

r rr

r rr

r rr

2

0

2

0

0

5'

3

vor

tor com

com

r

r

η

η η

η

=

= =

=

Wambach vs HD vorticity -1

Toroidal, compressional, and vortical operators for :

2 212 10 0 10

1 2ˆˆ ( 1 ) ( ) [ ( ) ]2 3 5tor nuc

iM E dr j r r Y Y r Y

c μ μ μμ = − + − < >∫r r r rr r

1λ =

3 20 1

5ˆ ˆ( 1 ) ( ) [ ]3comM E dr r r r r Y μμ ρ= − < >∫

r r

212

3 ˆˆ ( 1 ) ( )5 2vor nuc

iM E dr j r r Y

c μμ = − ∫r rr r

to be shown later:

- no c.m.c. for the vortical operator

Wambach vs HD vorticity -2

Motivation

Nuclei demonstrate both- irrotational flow (most of electric GR) - vortical flow (toroidal GR) ( ) ( ) 0w r v r= ∇× ≠

rr rr r( ) ( ) 0w r v r= ∇× =

rr rr r

Vorticity is a fundamental quantity:- does not contribute to the continuity equation,- represents an independent part of charge-current distribution beyond the continuity equation.

( )w rrr

beyond the continuity equation.

Vorticity is related to the exotic modes:- toroidal E1 mode (TM) ,- compression E1 mode (CM),

which are now of a keen interest .

Open points:- definition of nuclear vorticity (HD vs Wambach),- the vortical mode (VM) and its operator field, - relation between VM and TM/CM,- IS (T=0) and IV(T=1) branches of the modes,- role of magnetization (spin) nuclear current.

different conclusions on CM vorticity

J. Kvasil, V.O. Nesterenko, W. Kleinig, P.-G. Reinhard, P. Vesely, subm. to PRC, arXiv: 1105.0837[nucl-th]

SRPA (1)

( ( , )) ,E J r t Hα = Ψ Ψr

( , ) { ( , ), ( , ), }J r t r t j r tα ρ∈rr r r

K

( , ) ( ) ( , )J r t J r J r tα α αδ= +r r r

Time-dependent formulation:

ˆ( , ) | |α α=< Ψ Ψ >rJ r t J T-even and T-odd

densities

Linear regime: small time-dependent perturbation

M fi ld h ilt i

E. Lipparini and S. Stringari, NPA, 371, 430 (1981).

0

2

' '

( , ) ( ) ( , )

ˆ ˆ[ ] ( ) [ ] ( , ) ( )α α αα ααα α α

δ δ δδ δ δ= =

= +

= +∑ ∑

r r r

r r r

res

J J J J

h r t h r h r t

E EJ r J r t J r

J J J

Mean field hamiltonian: static g.s. + time-dependent response

Now we have to specify the perturbed many-body wave function ( )tΨ

( ) ( ) | | ( ) 0 | | 0α α αδ = Ψ Ψ −J t t J t J The only unknowns

SRPA (2)

Macroscopic step:' ' '

, ' 1

ˆ ˆ ˆ ˆ{ }κ η=

+⇒ ∑K

res kk k k kk k kk k

V X X Y Y

Perturbed w.f. via scaling:

1

1

ˆ ˆ ˆ ˆˆ ˆ,

ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ[ , ] ,

k k k k

k k ph k k k

Q Q TQ T Q

P i H Q P TP T P

+ −

+ −

= =

= = = −( ) cos( )

( ) sin( )k k

k k

q t q t

p t p t

ωω

==

ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) { ( ) ( ) } 1/ 2 { ( ) ( ) }κ δ η δ= + = +∑ ∑h t q t X p t Y X t X Y t Y

1ˆˆ( ) exp{ ( ) }exp{ ( ) }| 0 ,

K

k k k kkt q t P p t Q

=Ψ = − −∏

both ( ), 0Ψ t are Slater determinants

'kk 'kk

Microscopic step:ˆ( ) {1 ( ) )} 0Th ph ph

ph

t c t A+Ψ = +∑( ) i t i t

ph ph phc t c e c eω ω+ − −= +

Perturbed w.f. via Thouless theorem:

Merging step:

' ''

( ) { ( ) ( ) } 1/ 2 { ( ) ( ) }κ δ η δ= − + = +∑ ∑res k k k k kk k k kk k kk kk

h t q t X p t Y X t X Y t Y

ˆ ˆ ˆ ˆ( )| ( )| , ( )| ( )|δ δ δ δ= =k sc k Th k sc k ThX t X t Y t Y tBoth scaling and Thouless w.f.( )tΨ must give equal variations:

Perturbed w. f. by Thouless theorem

∑ Ψ+=Ψ +

phphph Atct 0)ˆ)(1()(

Time-dependent HF equation )())(()( 0 tthhtdt

di res Ψ+=Ψh

Harmonic oscillations

( ) ,

1( ) cos( ) ( ),

2

i t i tph ph ph

i t i tk k k

c t c e c e

q t q t q e e

ω ω

ω ωω

+ − −

−

= +

= = +

ˆ ˆ ˆ( ) { ( ) ( ) }= − +∑res k k k kk

h t q t X p t YResponse Hamiltonian

SRPA (3)

ˆ ˆ{ | | 0 | | 0 }12 ε ω

± = −±

∑ mk k k kk

phph

q ph X i p ph Yc

1( ) cos( ) ( )

2i t i t

k k kp t p t p e ei

ω ωω −= = −

TDHF gives the coupling

,± ↔ph k kc q p

which allows to reduce a largeset of 1ph apmlitudes to a few unknowns

±phc

,k kq pSo high-rank RPA problem can be reducedto low-rank one! However, still remain to be unknow,k kq p

SRPA (4)

ˆ ˆ ˆ ˆ( ) | ( ) | , ( ) | ( ) |δ δ δ δ= =k sc k Th k sc k ThX t X t Y t Y t

Physical requirement:

Variations of operators of the residual interaction must be the same for both macroscopic (scaling) and microscopic (Thouless) many-body wave functions.

1ˆ ( ) | ( )k sc k k kk

qX t tδ κ −′ ′

′= ∑

1' ' '

1' ' '

{ ( ( ) ) ( )} 0

{ ( ) ( ( ) )} 0

kk kkk k

k k

kkk

kk kk kkk

d XX d XY

d YX d

q p

q p YY

κ

η

−

−

− + =

+ − =

∑

∑

*

'

'

*

'ˆ ˆ| | 0 | | 0

( )( )

ˆ ˆ| | 0 | | 0

( )[ ]k k

kk

ph ph

k k

ph

ph X ph Yd XY

ph X ph Y

ε ω ε ω=

− ++∑

final RPA equations for

,k kq p

ˆ ˆ ˆ( ) | ( ( ) | | 0 ( ) 0 | | )k Th ph k ph kph

c cX t t ph X t X phδ ∗ ∗= < > + < >∑k

SRPA (3)

Final RPA equations:

1' ' '

1' ' '

{ ( ( ) ) ( )} 0

{ ( ) ( ( ) )} 0

kk kkk k

k k

kkk

kk kk kkk

d XX d XY

d YX d

q p

q p YY

κ

η

−

−

− + =

+ − =

∑

∑

where e.g.*

'

'

*

'ˆ ˆ| | 0 | | 0

( )( )

ˆ ˆ| | 0 | | 0

( )[ ]k k

kk

ph ph

k k

ph

ph X ph Yd XY

ph X ph Y

ε ω ε ω=

− ++∑

det[ ] 0jω =

2

0 ' ' ''

ˆ ˆ ˆ ˆ1/ 2 { }kk k k kk k kkk

H h X X Y Yκ η= + +∑

T-even T-odd

RPA spectrum

ˆ ˆ{ | | 0 | | 0 }12

k kk

phph

k kph X i ph Yc

q p

ε ω± = −

±

∑ m

[ ]+ − + + += −∑ ph p h ph h pph

C c a a c a a

2

'' '

2

'' '

ˆ ˆ ˆ( ) [ ] 0 |[ , ] | 0

ˆˆ ˆ( ) [ ] 0 |[ , ] | 0

α ααα α α

α ααα α α

δδ δδ

δ δ

=

=

∑

∑

r

r

k k

k k

EX r i J P J

J J

EY r i J Q J

J J

1' '

1' '

ˆ ˆ0 | [ , ] | 0

ˆˆ0 | [ , ] | 0

κ

η

−

−

=

=

kk k k

kk k k

i X P

i Y Q

RPA phonon - Rank of RPA matrix is 4K. For giant resonancesusually K=2 is enough.

Very low rank!

2

' '' ' ' '

' ' ' ' ' '' ' ' '

' ' '

ˆ( , ) [ ] ( , ) ( )

ˆ ˆ ˆ ˆ ˆ ˆ{ ( ) ( ) } { ( ) ( ) }

α ααα α α

δ δδ δ

κ δ η δ

==

= + = +

∑ ∑

∑ ∑ ∑ ∑

r r rres s sJ J

ss s s

s s s k s kqs sk qs sk sk sk s k sk sk s k

ss k ss kk

Eh r t J r t J r

J J

q t X p t Y X t X Y t Y

2'

' 'ˆ ˆ ˆ ˆ ˆ[ ] [ ]s

k k k

EX X i P J J

δ= = ⟨ ⟩∑ ∑

SRPA (5): detailed expressions with isospin indices s={n,p}

ˆˆ( ) { ( ) [ , ] ( ) [ , ] }α α α α αδ = ⟨ ⟩ + ⟨ ⟩∑s s sk s s sk sk

J t i q t P J p t Q J

' '' ' ' ' '

[ ] [ , ]sk sk sk s ss s s s

X X i P J JJ J α α

αα α αδ δ⟨ ⟩∑ ∑

2'

' '' ' ' ' '

ˆˆ ˆ ˆ ˆ[ ] [ , ]ssk sk sk s s

s s s s

EY Y i Q J J

J J α ααα α α

δδ δ

= = ⟨ ⟩∑ ∑

ˆ ˆ ˆ ˆ[ , ] [ , ] 0

ˆ ˆ[ , ] 0

A B A B

A B

+ + − −

+ −

⟨ ⟩ = ⟨ ⟩ =

⟨ ⟩ ≠

† 1ˆ ˆ ˆ ˆ ˆ ˆ,A A T A T A−± ±= = ±If then

1 ' ' 1 '' ' ' '

ˆ ˆ[ ] [ ] [ , ]s k sk ssk s k s k ski P Xκ κ− −= = − ⟨ ⟩

1 ' ' 1 '' ' ' '

ˆ ˆ[ ] [ ] [ , ]s k sk ssk s k s k ski Q Yη η− −= = − ⟨ ⟩

ˆ ˆ ˆ ˆ ˆ ˆ[ , ] { | | | | | | | | }ph

A B A ph ph B B ph ph A⟨ ⟩ = ⟨ ⟩⟨ ⟩ −⟨ ⟩⟨ ⟩∑Calculation of average commutators via s-p matrix elements

SRPA (6): strength function

2ˆ( , ) | | 0 ( )L X XLS D Dλμ λμ

νν νω ν ξ ω ωω= < > − =∑

22 )2()(21

)( Δ+−Δ=−

νν ωωπ

ωωξLorentz weight

L=0,1,3

' ''

( ) ( ) ( )

( )det ( )

Re [ ]L

F z A z A z

z zzF

sz

ν ν νββ β β

ββν ν

νζ ω ων

= − = ±

∑∑

/ 2Re Re[...] [...]ph

z i z phs sω ε= ± Δ =±≈ +∑

heavily calculated

easily calculated

2 2

ˆ ˆ| | 0 | | 0ph

ph ph

ph X ph DA

z

βνβ

εε

=−∑

skβ τ=

= XYτ

' '' 2

/ 2

( ) ( ) ( )1 ˆ[ ] | | 0 ( )

( )

L

Lz i ph X

ph

z F z A z A z

ph DF z

ββ β βββ

ω λμ νε ξ ω ωπ = + Δ= ℑ + < > −

∑∑

Contribution of residual inter. Unperturbed 2qp strength

Justification: sum of all residues at the complex plane is ZERO. Then

( ) ( ) ( ) ( ) ( /2Re Re Re Re Re

Re Re

)

( /2) ( )

phph

phph

s sf z f z f z f z fs s z i

f z i f z

s

s s

ν νν ν

ω ω ε ω

ω ε

= =− = ±∞ − =− − =± − = ± Δ

≈− = ± Δ − =±

∑ ∑ ∑

∑

Comparison of VM, TM, and CM

- Broad low-energy (LE) and high-energy (HE) bumps for VM, TM, and CM.

- LE strength is dominated by VM and TM - HE strength is dominated by VM and CM

- General agreement for TM and CM with previous studies.

- Poor agreement with exper. of Ichida(like in previous studies).

exp exp

- Purely vortical VM does not coincidewith partly vortical TM, especially at HE.

TM was previously considered as a typical example of the vortical flow.

1 1

2 2

12.7 MeV, 3.5 MeV

23.0 MeV, 10.3 MeV

E

E

= Γ == Γ =

Uchida et al., 2003:

Vortical, toroidal, and compressional T=0 strength SLy6

1MeVΔ =

- dominant contribution ofto VM and TM

conj

-no contribution to: - CM - HE strength

magj

0 1( )

5.58 , 3.82

0.882

T p ns s

p ns

s

sg

g g g

g ς ς

ς= = +

= = −

=

Small T=0 g-factors!

Vortical, toroidal, and compressional T=1 strength

SLy6

1MeVΔ =VM and TM:

- dominant contribution of !!magj

1

5.58 , 3.82 ,

41

)2

.7(T p ns s

n

s

ps s

g g g

g gς ς

ς=

= = −

== −

Large T=1 g-factors!

2

Vortical and toroidal modes in the T=1 channel are suitable to see the effect of in electric modes. magj