Vorticity in atomic nuclei V.O. Nesterenko J. Kvasil, A. Repko - - PowerPoint PPT Presentation

SLIDE 1

Vorticity in atomic nuclei

BLTP, JINR, 05.12.2012

V.O. Nesterenko

• J. Kvasil, A. Repko

Institute of Particle and Nuclear Physics, Charles University, Praha , Czech Republic

• W. Kleinig

Laboratory of Theoretical Physics, JINR; Technical University of Dresden, Institute for Analysis, Dresden, Germany P.-G. Reinhard

Institute of Theoretical Physics II, University of Erlangen,

Erlangen, Germany

SLIDE 2

Introduction 1

Nuclei demonstrate both

• irrotational flow

examples: most of electric giant resonances (GR)

• vortical flow

examples:

• nuclear rotation of deformed nuclei,
• s-p excitations,
• toroidal E1 GR
• twist M2 GR

( ) ( ) w r v r    ( ) ( ) w r v r   

• rotation-like oscillations

Toroidal E1 Twist M2

SLIDE 3

Introduction 2

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 r

Vorticity is related to the exotic E1 modes of high interest:

• toroidal
• compression
• pygmy

nuc

j    

2

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

 

       Beyond long-wave approximation:

second-order GR ISGDR

• N. Paar, D. Vretenar, E. Kyan,
• G. Colo, RPP, 70 691 (2007).

Leading dipole modes in T=0 channel !!!

Manifestation of nuclear elasticity (toroidal, twist, …) May exist in other systems (atomic clusters, …)

VON et al, PRL, 85 , 3141 (2000)

SLIDE 4

Interplay of pygmy, toroidal, and compression flows in the PDR region?

• A. Repko, P.G. Reinhard, VON, J. Kvasil,

to be submitted

Skyrme-RPA calculations

MeV

SLIDE 5

Theoretical studies:

Many publications on toroidal and compressional (ISGDR) modes and 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).

• X. Roca-Maza et al, PRC 85, 024601 (2012).

……….

• N. Paar, D. Vretenar, E. Kyan, G. Colo, Rep. Prog. Phys. 70 691 (2007).

Recent review

• J. Kvasil, VON, W. Kleinig, P.-G. Reinhard, P. Vesely, PRC, 84, 034303 (2011)

SLIDE 6

Content

• Two definitions of vorticity in nuclear physics:
• hydrodynamical (HD)
• Rawenhall –Wambach (RW)
• Theory: derivation of operators for toroidal and

compression modes

• Numerical results for vortical, toroidal and compression responses
• strength functions, transition densities, velocity fields
• role of convection and magnetization (spin) nuclear current
• Perspective to observe the vortical E1 GR in experiment?
• Twist M2 GR

  



  ˆ ˆ ˆ ( ) ( ) ( )

j vor tor com

M E M E M E

• J. Kvasil, VON, W. Kleinig, P.-G. Reinhard,
• P. Vesely, PRC, 84, 034303 (2011)

P.G. Reinhard, V.O. Nesterenko, J. Kvasl, A. Repko, to be submitted

SLIDE 7

May we use, in analogy to HD,

( ) j r 

as a measure of the nuclear vorticity? No, because:

( ) v r 

1) and are very different values     ( ) ( ) ( )

nuc

j r v r r

2

( ) ( ) ( ) ( ) ( ) ( )

nuc nuc

r j r r j r v r r            

( ) j r 

( ) ( ) ( ) ( ) j r v r j r v r      

2)

1

(2 1)!! ˆ ( ) ( ) 1 ˆ [ ( ) ]

nuc

M Ek dr j kr Y c j k r

  

   



     



Multipole electric operator If to use as a measure of vorticity, then all the electric modes would be vortical. But this contradicts numerous exper. data:, e.g. for electric GR.

( ) j r 

SLIDE 8

Two conceptions of vorticity in nuclear theory:

• 1. Hydrodynamical vorticity:

( ) ( ) w r v r  

    ( ) ( ) ( )

nuc

j r v r r

( ) ( )( ) ( ) ( )

nuc

j r v r w r        

( )

( ) * ( ) * 1 1 1 1

( | ) ˆ ( ) | ( )| [ ( ) ( ) ] 2 1

fi

fi fi i i f f f f nuc i i f

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

      

 

   

   



• 2. Wambach vorticity vorticity



    

     

1

* * 10 10 12 12

ˆ ( ) | ( )| 0 [ ( ) ( ) ] 3

nuc

i j r j r j r Y j r Y

• current transition

density

nuc

j    

• continuity equation

  ( )

j r



j



j

• independent part of charge-current distribution, decoupled to CE
• may be the measure of the vorticity

HD and j+ prescriptions give opposite conclusions

• n CM vorticity!

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



j

SLIDE 9

( )

( ) * ( ) * 1 1 1 1

( | ) ˆ ( ) | ( )| [ ( ) ( ) ] 2 1

fi

fi fi i i f f f f nuc i i f

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

      

 

   

   



Definition 2 (Ravenhall and Wambach)

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

( )

( ) *

( | ) ˆ ( ) | ( )| ( ) 2 1

fi

fi i i f f f f nuc i i f

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

  

      



nuc

j    

fi fi

i j     

1 1

1 1 2 ( ) ( ) ( ) 2 1 2 1 d d i r j r j r dr dr

  

        

 

                    

2 1 1

( ) (2 1) ( ) dr r r dr r j r

    

    

    

  

 

 

1 1 1 1

( ) lim ( )

r

d dr r j r r j r dr

         





So just

• is decoupled to CE
• has to be chosen as

measure of vorticity

( ) 1( ) fi

j r



SLIDE 10

( )

( ) *

( | ) ( ) ( ) 2 1

fi

fi i i f f f

j m j m j r T r Y j

  

    



1 1

1 1 2 ( ) ( ) ( ) 2 1 2 1 d d T r j r j r dr dr

  

       

 

                    

2

1 ( ) dr r T r

  

  

 

 



2

( ) dr r r

  

  

 

 

How to cure to make it indeed vortical? How to decouple from the CE?

( ) fi

j  

( ) fi

j  

( ) ( )

( ) ( )

fi fi

r j r S     

( )

( ) *

( | ) ( ) ( ) 2 1

fi

fi i i f f f

j m j m r r Y j

  

    



( )

( ) *

( | ) 1 ( ) ( ) 2 1

fi

fi i i f f f

j m j m S r r Y j

  

      



( ) ( )

1 ( ) ( ) ( )

fi fi

r T r r

  

      

Thus we get the vorticity transition density

2

( ) dr r r

 



 





SLIDE 11

One may construct the vorticity transition density and strength

D.G.Raventhall, J.Wambach, NPA 475, 468 (1987). ( ) 4 ( )

( )

fi fi

r r dr

  

 

 

 

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

( ) 1( ) fi

j r



nuc

j    

So, is an independent part of charge-current distribution. Definition 2 (Ravenhall and Wambach)

( ) 1( ) fi

j r

 ( ) 1( ) fi

j r



1

1 ( 2 ( ) ) 2 d j r r dr r

  

   



         

This approach does not use the vortical operator. However, such operator could be useful for the comparison of vortical, toroidal and compression flows.

SLIDE 12

3 2 11

5 1 ˆ ˆ ( 1 ) [( ) ( ] ( ) [ ] 10 2 ) 3

nuc tor

r r M r j r E dr Y r c



      



3 2 1

5 ˆ ˆ ' ( 1 ) ( ) [ ] 3

com

M E dr r r r r Y       



3 2 1

1 ˆ ˆ ( 1 ) ( ) [ ( ) ] 2 5 ( 3 )

tor nuc

M E dr j r r Y r r r c



       



How to get the relevant RW vortical operator and relate it to toroidal and compression operators? The toroidal and compression operators are related!

3 1

( ) ( )

com

v r r Y   

• irrotational motion of CR

3 1

( ) ( )( )

tor

v r r r Y    

• vortical flow

( )

com

v r  



12

( )

tor

v r rY



 

SLIDE 13

1

(2 1)!! ˆ ( ) ( ) 1 ˆ [ ( ) ]

nuc

M Ek dr j kr Y c j k r

  

   



     



Multipole electric operator

ˆ ˆ ( ) ( ) ( ) ( )

nuc

kc r v r j r i r r         

Then

1

(2 1)!! ˆ ( ) [ ( ) ( , )] 1

vor

M Ek dr j kr Y ck

  

     



    

ˆ ˆ ( ˆ ˆ ˆ ( ) ( ) ( ) )

S nuc

M Ek M k i j r kc r r                    

) ( ˆ r jnuc     

Main idea:

1 ˆ ˆ ˆ ˆ [ , ] ,

if i f

v r r H ikcr i E E kc       

truly vortical

Derivation of the vortical operator (Wambach)

( ) ( ) ( ) ( ) ( )

nuc nuc

r v r j r r v r          

( ) ( ) ( )

nuc

j r r v r    

SLIDE 14

Long-wave approximation:

2

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

 

       The second order term gives:

• toroidal operator
• compression operator
• vortical operator

SLIDE 15

ˆ ˆ ˆ ( ) ( ) ( )

tor

M Ek M E kM E     

ˆ ( ) ( ) M E dr r r Y

 

   

1 1 1

2 ˆ ˆ ( ) ( ) ( ) 2 2 1 1 2 3

tor nuc

i M E dr j r r Y Y c

    

     

  

      



ˆ ˆ ˆ ( ) ( ) ( )

S com

M Ek M E kM E     

1 1 1

1 2 ˆ ˆ ( ) ( ) ( ) 2 2 1 2 3

com nuc

i M E dr j r r Y Y c

    

     

  

     



ˆ ˆ ( ) ' ( )

com com

M E kM E    

2

1 ˆ ˆ ' ( ) ( ) 2(2 3)

com

M E dr r r Y

 

  



 



ˆ ˆ ˆ ˆ ˆ ( ) ( ) ( ) ( ) ( )

vor S tor com

M Ek M Ek M Ek k M E M E             

1 1

1 ˆ ˆ ( ) ( ) (2 3) 2 1

vor nuc

i M E dr j Y r r c

  

   

 

     

ˆ ˆ ˆ ( ) ( ) ( )

vor tor com

M E M E M E     

SLIDE 16

2 2 2 2

1 2 2 ˆ / ( 1 )/0 [ ( ) ] 6 5 ( ) ( )

com

j M E dr r r r r r с j r

 

 





       



2 2 2 2

1 2 ˆ / ( 1 )/0 [ ( ) ] 6 5 ( ) ( )

tor

j M E dr r r r r r с j r

 

 





       





 





   



4

1 ˆ / ( 1 )/0 5 ( ) 2

j vor

M E dr r с j r

, 3 2 12

1 ˆ | ( 1 ) 5 ( ) | 0 2

nu v r c j

• M

E dr r c j Y r

 

 

 

    





           



3 3 2 11

1 5 ˆ | ( 1 )| 0 [ ] ˆ [ ( ) 3 1 2 ]

tor nuc

M E dr r r r Y c j r



          



3 3 2 1

1 5 ˆ | ( 1 )| 0 [ ] 3 10 ˆ [ ( ) ] 2

c n

• m

uc

M E dr r r r Y c j r

E1(T=0)

 

           



3 3 2 11

1 5 ˆ | ( 1 )| 0 [ ] 3 10 [ ( ) ] 2

HD vor

M E dr r v r Y r r c

SLIDE 17

Presence of is decisive to make the flow vortical

Wambach vorticity

3 2 1

1 5 ˆ ˆ ' ( 1 ) ( )[ ] 10 3

com

M E dr r r r r Y       



12 10 2 2 2

2 2 ˆ ˆ ( 1 ) ( ) [ ( ) ] 5 2 3

com nuc

i M E dr j r r r r c Y Y

 

      



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

12

j



12

Y



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 r

 



Thus an essential difference between HD and Wambach vorticity.

( ) 1( ) fi

j r



SLIDE 18

• self-consistent Skyrme RPA ,
• SLy6 force

208Pb

Strength function

  

   



    



2 1

ˆ ( 1 ; ) | | | 0 | ( )

E

S E M

2 2

1 ( ) 2 [( ) ] 4

 

            with the Lorentz weight

1 MeV  

Toroidal, compressional, vortical operators

The model for numerical calculations:

SLIDE 19

Comparison of VM, TM, and CM

• Purely vortical VM does not coincide

with partly vortical TM, especially at HE. TM was previously considered as a typical example of the vortical flow.

• 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).

1 1 2 2

12.7 MeV, 3.5 MeV 23.0 MeV, 10.3 MeV E E      

Uchida et al., 2003:

exp exp

• J. Kvasil, VON, W. Kleinig, P.-G. Reinhard,
• P. Vesely, PRC, 84, 034303 (2011)

SLIDE 20

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

,

ˆ ˆ ˆ ˆ ˆ ( ) ( ) ( ) ( ( ) ( ))

q q nuc con mag com mag q n p

e j r j r j r j r j r m



   



 

ˆ ( ) ( ( ))

q q con eff k k k k k q

j r ie r r r r  



      



ˆ ˆ ( ) ( ) 2

q s mag k qk k k q

g j r s r r 



   



SLIDE 21

Vortical, toroidal, and compressional T=0 strength SLy6

1MeV  

• dominant contribution of

to VM and TM

con

j

1( ) 5.58 , 3.82 0.88 2

T p n s s p n s s s

g g g g g   

 

    

Small T=0 g-factors!

• no contribution to:
• CM
• HE strength

mag

j

• J. Kvasil, VON, W. Kleinig, P.-G. Reinhard,
• P. Vesely, PRC, 84, 034303 (2011)

SLIDE 22

Vortical, toroidal, and compressional T=1 strength

SLy6

1MeV  

VM and TM:

• dominant contribution of !!

mag

j

Large T=1 g-factors!

1

5.58 , 3.82 , 4 1 ) 2 .7 (

T p n s s n s p s s

g g g g g   



     

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

mag

j

• J. Kvasil, VON, W. Kleinig, P.-G. Reinhard,
• P. Vesely, PRC, 84, 034303 (2011)

SLIDE 23

• A. Repko, P.G. Reinhard, VON, J. Kvasil,

to be submitted

GDR E1 compression

6-9 MeV

MeV

SLIDE 24

Is it possible to observe the vortical GR in experiment?

( , ')  

D.Y. Youngblood et al, 1977 H.P. Morsch et al, 1980 G.S. Adams et al, 1986 B.A. Devis et al, 1997 H.L. Clark et al, 2001 D.Y. Youngblood et al, 2004 M.Uchida et al, PLB 557, 12 (2003), PRC 69, 051301(R) (2004)

(toroidal) (compression) LE HE

2 3 10

( 1) ( ) B E dr r r Y r    



• radial factors make reaction very peripheral
• flow in the interior and thus the vorticity becomes

invisible

SLIDE 25

Basics of E1 (T=0) toroidal and compression modes

V.M. Dubovik (1975) S.F. Semenko (1981)

TM

vorticity:

• no, irrotational mode

CM

dependence on nuclear incompress.: energy:

yes no

• yes, vortical mode

M.N. Harakeh (1977)

• S. Stringari (1982)
• S. Misiku, PRC, 73,

024301 (2006) 



1/ 3

68 E A MeV





1/ 3

132 E A MeV

SLIDE 26

Is it possible to observe the vortical electric GR in experiment?

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

( , ')   (e,e’)

( , ' ')   

• most promising!

SLIDE 27

M2 twist GR (IV,IS)

• predicted: G. Holzwarth and G. Eckard, NPA, v.325, 1 (1979).
• observed:
• characterized by strong M2 transitions to the ground state
• orbital magnetic flow
• manifestation of nuclear elasticity
• may exist in other systems: atomic clusters, trapped fermi-gas, …

( , ,0) v yz xz  

1 2 1 2

2 ˆ ˆ ( 2 ) 10 [ { } { } ] 3

b s l

F M r g Y s g Y l

 

   

10

( )

z z

zl r Y l 

• velocity field
• external field to generate twist
• P. von Neumann-Cosel et al., PRL v.82, 1105 (1999).: (e,e’), back scatt.

SLIDE 28

Conclusions

The vorticity is a fundamental property of nuclear motion. The toroidal E1 resonance:

• one of the dominant modes in E1(T=0) channel,
• is one of the most interesting manifestations of the vorticity,
• very hot topic nowadays (correlation to CM, pygmy)

Still there are open problems:

• various definitions of vorticity,
• challenge to experiment to observe unambiguously a vortical

flow in GR

SLIDE 29

Thank you for attention!