1) Institute of Particle and Nuclear Physics, Charles University, - - PowerPoint PPT Presentation

SLIDE 1

Monopole E0 resonance in deformed nuclei

• J. Kvasil 1), V.O. Nesterenko 2), A. Repko 1) ,
• D. Božík 1), W. Kleinig 2,3), P.-G. Reinhard 4) ,

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

SLIDE 2

Motivation

Giant Monopole Resonance (GMR) centroid is connected with finite – nucleus incompressibility by (see e.g. J.Blaizot, Phys. Rep. 64, 171 (1980))

GMR

E

A

K

  

2 2

r m K E

A GMR

 The incompressibility ( together with the nucleus mass and radius) belongs to the bulk properties used for the determination of the energy functional (n-n effective interaction ) parameters GMR is the subject of intensive investigation from 60-s up to now From the point of view of theory the position of is usually obtained by means of moments of energy weighted E0 strength functions

GMR

E

1 m

m EGMR 



 ) ; ( E E S dE m

k k



   

     

  ) ( | | ) ( ˆ | | ) ; (

2 ) (

E E el M E E E S

IS k k

where is the isoscalar E0 transition operator



  



A i i IS

Y r el M

1 00 2 ) (

) ( ) ( ˆ

 

SLIDE 3

Motivation

Using this approach a lot of papers analyzing centroids of GMR appeared:

• some of the latest:
• P. Avogadro, C.A.Bertulani, PRC 88, 044319 (2013)

P.Veselý, J.Toivanen, B.C.Carlsson, J.Dobaczewski, M.Michel, A.Pastore, PRC 86, 024303 (2012) L.Cao, H.Sagawa, G.Coló, PRC 86, 054313 (2012) P.Avogadro, T.Nakatsukasa, PRC 87, 014331 (2013) K.Yoshida, T.Nakatsukasa, PRC88, 034309 (2013)

Analyses performed in these papers ( based on the GMR centroids calculated in terms of the RPA ) showed that the energy – density - functional (EDF) approaches with the incompressibilities MeV give the good agreement with the experimentally determined centroids in 208Pb and 144Sm. However, the experimental data on Sn ( see T.Li, U.Garg, et al., PRL 99, 162503 (2007) ) and Cd (see D.Patel, et al., Phys.Lett. B 718, 447 (2012) ) cannot be reproduced equally well with the same functionals in the comparison with Pb-Sm data.

230 

nm

K

In papers P.Avogadro, et al., PRC88, 044319 (2013) and P.Veselý, , et al., PRC 86, 024303

(2012) the modification of the pairing interaction was used for the explanation of

the problem of the simultaneous reproduction of Sn-Cd and Pb-Sm data.

) ( ) ( 1 ) , ( ) ( ) , ( r r r V r r V r r V r r V

pair pair

                                       



 

• volume pairing

1  

• surface pairing

SLIDE 4

However, these attempts of the solving of the problem of the simultaneous reproduction of Sn-Cd and Pb-Sm data by the new type of the pairing have not helped.

Motivation

In the paper K.Yoshida, T.Nakatsukasa, PRC88, 034309 (2013) microscopical fully self-consistent Skyrme QRPA analyses of the shape evolution of giant resonances of different types (ISGMR including): double-peak structure of the GMR in deformed nuclei is caused by the mixing of E0

and E2 modes (the higher peak is a primal ISGMR and the lower peak is induced by the E2-E0 mixing from ISGQR) in spite of the fact that in this paper the calculated energy distribution of GMR is shown only the comparison of calculated positions (centroids) and widths of the GMR with corresponding experimental values was performed – relatively good agreement for Sm isotopes was obtained

So, in spite of the fact that the experimental energy distributions of the ISGMR are available for 144, 154Sm the comparison with experimental values was done

• nly for positions (centroids) and widths of the ISGMR (theoretical positions

and widths were determined by the fitting of one- (for spherical nuclei) or two- (for deformed nuclei) Lorentzians to the calculated values of the isoscalar E0 excitation probability for individual RPA solutions)

SLIDE 5

Motivation

There are two main groups in the world providing the data on E0 resonance, namely: Texas A&M University (TAMU): D.H.Youngblood, et al., PRC69, 034315 (2004) - 116Sn, 208Pb, 144Sm, 154Sm

D.H.Youngblood, et al., PRC69, 054312 (2004) - 90Zr D.H.Youngblood, et al., PRC88, 021301(R) (2004) - 92Zr, 92Mo, 90Zr, 96Mo, 96Mo, 98Mo, 100Mo

Research Center for Nuclear Physics (RCNP) at Osaka University M.Uchida, et al., PRC69, 051301 (2004) - 90Zr, 116Sn, 208Pb M.Itoh, et al., PRC68, 064602 (2003) - 144Sm, 148Sm, 150Sm, 152Sm, 154Sm T.Li, et al., PRC99, 162503 (2007) - 112-124Sn All these papers give not only GMR centroids but also shapes of the GMR and both experimental groups used reaction for the determination of E0 strength functions. However, in the case when both groups measured E0 strength function for the same nucleus ( 90Zr, 144,154Sm, 208Pb ) one can see substantial differences in the E0 strength functions between both groups (mentioned already in P.Avogadro, et al., PRC88, 044319 (2013) ) .

) , (   

SLIDE 6

Motivation

In spite of the fact that the experimental shapes of E0 strength functions are available for many spherical and also for several deformed nuclei all papers with theoretical analyses have compared only GMR centroids determined by simple expr. or widths (determined by the fitting of Lorentzian to calculated values of the excitation probabilities of individual RPA solutions)

1 m

m EGMR 

The deeper theoretical analyses of the GMRs were done in the paper K. Yoshida,

T.Nakatsukasa, PRC 88, 034309 (2013) with the Skyrme QRPA approach for SkM*,

SLy4 and SkP Skyrme interactions (for Sm isotopes) but the comparison with experimental data was done only for positions (centroids) and widths of GMR

We analyze the shape and position of the GMR from the point of view of the comparison of the experimental values of the ISGMR energy distribution with the calculated values with different Skyrme parametizations for a broad ensemble of Sm, Pb, Sn, Mo isotopes (not only position and width). E0 strength is also determined for some superheavy nuclei. Deformation effect (double peak structure of the GMR) is illustratively demonstrated in terms of the Separable RPA (SRPA) approach Energy distribution of the ISGMR in spherical and deformed nuclei is analyzed from the point of view of different Skyrme parametrizations (with different incompressibility modulus)

SLIDE 7

Theoretical background - SRPA

In this contribution two theoretical approaches are used:

• 1. separable RPA (sRPA) -
• 2. standard RPA (fRPA) -

Separable RPA

SRPA = modification of the RPA based on the Skyrme energy functional for axially deformed nuclei using multi-dimensional response approach

V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, PRC66, 044307 (2002) - formulation of SRPA V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Veselý, P.-G.Reinhard, PRC74, 064306 (2006) - GDR P.Veselý, J.Kvasil, V.O.Neterenko, W.Kleinig, P.-G.Reinhard, V.Yu.Ponomarev, PRC80, 0313012(R) (2009) - M1 giant resonance V.O.Nesterenko, J.Kvasil, P.Veselý, W.Kleinig, P.-G.Reinhard, V.Yu.Ponomarev,

• J. Phys. G37, 064034 (2010) - M1 giant resonance

J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, P.Veselý, PRC84, 034303 (2011) - toroidal and compression E1 modes A.Repko, P.-G.Reinhard, V.O.Nesterenko, J.Kvasil, PRC87, 024305 (2013) - toroidal nature of low-lying E1 modes J.Kvasil, V.O.Nesterenko, W.Kleinig, D.Božík, P.-G.Reinhard, N.Lo Iudice ,

• Eur. Phys. J. A49, 119 (2013) - toroidal, compression E1 modes

2 codes 1 code coupled scheme (spherical nuclei) m- scheme (deformed nuclei)

SLIDE 8

The sRPA starts with the Skyrme energy functional (see Appendix B for details):

coul pair Sk kin

T s j J

    

      ) , , , , , (     Basic idea of the sRPA: nucleus is excited by external s.p. fields:

k k k k k k k k k k k k

Q i P H P T P T P P P i Q H Q T Q T Q Q ˆ ] ˆ , ˆ [ ; ˆ ˆ ; ˆ ˆ ˆ ] ˆ , ˆ [ ; ˆ ˆ ; ˆ ˆ

1 1

        

   

 

K k P Q

k k

, , 1 , ˆ , ˆ  

Theoretical background - sRPA

The optimal set of generators was discussed in:

V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC 74, 064306 (2006) P.Vesely, J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, V.Yu.Ponomarev, PRC. 80, 031302(R) (2009) J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, P.Vesely, PRC 84, 034303 (2011)

) ˆ , ˆ (

k k P

Q

Using linear response theory corresponding Hamiltonian is:

 

  

  

d d d HFB

J r J r d h ˆ ) ( ˆ

3





res HFB

V h H ˆ ˆ ˆ  

 





  

K k k k k k k k k k k res

Y Y X X V

1 ' , ' ' ' '

ˆ ˆ ˆ ˆ 2 1 ˆ  

where are given by the 2-d derivatives of the functional with respect to densities and currents

k k k k k k

Y X ˆ , ˆ ,

,   





no free paramerters except those

• f the Skyrme functional

SLIDE 9

Theoretical background – full RPA

Total Hamiltonian consists from the BCS (HFB) mean field and residual interactions:

: ) ( ˆ ) ( ˆ : ) ( ) ( 2 1 ˆ ˆ

2 3 3

r J r J r J r J r d r d h H

d d d d d d BCS

    

  

 

       

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

    ) ( ) ( ) ( ) ( * *      

c c E E c c A B B A

with the standard full RPA equation

ij kl ij kl d ij d d d d d l l ijkl

dr r r J r J J J A

l k

     

 



, 2 * ; ; 2

) ( ) ( 1 2 ) 1 (    

    

 

dr r r J r J J J B

kl d ij d d d d d l l J T ijkl

l k d

2 * ; ; 2 ) (

) ( ) ( 1 2 ) 1 (

 

    



    

 

  

with

 



     

 

j i i j j i j i j i

c c Q     

   ) ( ) (

) (

The solving of the sRPA or fRPA equations gives the forward and backward amplitudes and of the phonon creation operator

) (   ij

c

) (   ij

c

with corresponding phonon energy



E

SLIDE 10

Theoretical background

Knowing the structure and energies of one-phonon states one can determine the strength function of given transition operator ( in our case the monopole electric operator





  i i is

Y r el M ) ( ) ( ˆ

00 2 ) (  

) ( | | ) ( ˆ | | ) ( ) ; (

2 ) (     

  E E RPA el M E E E S

is k k

   

 



) ( | | ) ( ˆ | | ) ( ) ; (

2 ) (     

  E E RPA el M E E E S

is k k

   

  



Where is the Lorentz weight function

) (



 E E 



4 ) ( 2 1 ) (

2 2

     

  

  E E E E

with the corresponding energy weighted strength function:

SLIDE 11

Dependence of E0 strength function in the spherical nuclei (heavier 208Pb,144Sm; lighter 112, 116, 124Sn) on Skyrme parametrizations with different K

parametrizations with K~230 MeV fits the experimental values for heavier nuclei (Sm, Pb) experimental values for lighter Sn isotopes require parametrizat- ions with the lower values of K (K~200 MeV like SkP  differences in RCNP and TAMU data

SLIDE 12

GMR in the isotopes 144-154 Sm

comparison of SRPA results with TAMU and RCNP exp. data

TAMU data renormalized for absolute units:

1 1 00 2 1 1 4 00 2

) ; ( ) ; (

  

   E EWSR MeV EWSR fraction Y r S MeV fm Y r S

cl

   

2 2 00 2 1

2 ) ( r A m Y r m EWSRcl  

different shapes of

• exp. GMR for TAMU

and RCNP data: much bigger defor- mation effect in TAMU data SRPA calculation: nucleus excited by 5 external fields:

    

00 00 20 2 00 4 00 2

) 6 . ( , ) 4 . ( , , , Y z j Y z j Y r Y r Y r

EWSRcl EWSRRPA

fm4 MeV fm4 MeV

144Sm 23158 24306 148Sm 24335 24650 150Sm 25128 25232 152Sm 26782 26990 154Sm 27162 27458

SLIDE 13

GMR in 154Sm – coupling of E0 and E2 modes

involving the E2 field among the external exciting fields in the SRPA calculation

• f GMR improves the

agreement with the TAMU GMR data



) (

20 2Y

r

the volume and surface pairing give practically the same results

SLIDE 14

      

GMR in isotopes 106-116Cd

Comparison of the sRPA (vithout and with E0-E2 mixing) with the fRPA results and with experimental data

small discrepancies between sRPA and fRPA in 110-116Cd the fRPA agrees better with the experimental data (at least in the positions

• f maxima)

SLIDE 15



Comparison of the GMR in 154Sm calculated with SV-bas and SkP parametrizations in the framework of sRPA (without and with E0-E2 mixing) and fRPA with corresponding experimental data

small discrepancies between sRPA and fRPA the SV-bas Skyrme interaction gives better agreement with experiment

SLIDE 16

E0 and E2 strength functions in Nd isotopes (the comparison with experimental GMR centroids)

in the deformed case the position of the 1-st GMR maximum agrees with the position of the maximum of the E2 K=0 strength

  

SLIDE 17

   

GMR in 172Yb and 238U

the position of the 1-st GMR maximum agrees with the position of the maximum of the E2 K=0 strength

SV-bas

SLIDE 18

     

GMR in superheavy nuclei

the position of the 1-st GMR maximum agrees with the position of the maximum of the E2 K=0 strength

SV-bas

SLIDE 19

Conclusion

Significant deformation effect observed in the TAMU GMR data is in the agreement with the SRPA and fRPA results ( double-peak structure of the GMR in deformed nuclei) Double-peak structure of the GMR for deformed nuclei is caused by the coupling of E2 and E0 modes for nonzero deformation (original idea – U.Garg, et al., PRC29, 93 (1984)) There are discrepancies between TAMU and RCNP experimental data from the point of view of the shape of GMR Volume and surface pairing give the similar results (confirmation of previous results) Theoretical analyses and comparison with exp. values of the GMR cannot be restricted only on resonance centroids (the shape of E0 strength function is important) Positions of the GMR depends on the incompressibility of the nuclear

• matter. The better agreement of calculated and exp. GMR values was
• btained for parametrizations with K ~ 230 MeV for heavier nuclei

(144Sm, 208Pb) while with K ~ 200 MeV for lighter nuclei (112,116,124Sn)

SLIDE 20

Thank you for your attention

SLIDE 21

Appendix A: Spurious mode connected with the number of particles

In the energy interval E > 8 MeV the influence of the spurious mode is very small

J.Li, G.Colo, J.Meng

SLIDE 22

22

Appendix b: Brief formulation of the SRPA approach

(more details)

 



      r d t r t H t t J E

3

) , ( ) ( | ˆ | ) ( ) (  H

 

  ) ( | t

     ) ( | ) ( ˆ | ) ( ) , ( t r J t t r J  

   

  ) ( | t

 |

    

       

 

 

| ˆ ) ( exp ˆ ) ) ( ( exp ) ( |

1 , k k K k k k k p n

P t ip Q q t q i t

     

k k k k k k k k k k k k

Q i P H P T P T P P P i Q H Q T Q T Q Q

           

ˆ ] ˆ , ˆ [ ; ˆ ˆ ; ˆ ˆ ˆ ] ˆ , ˆ [ ; ˆ ˆ ; ˆ ˆ

1 1

        

    k

Q ˆ

k

P



ˆ

SLIDE 23

23

) (t q k



) (t p k



| ˆ | | ˆ | ) ( | ˆ | ) ( ) ( ) ( | ˆ | ) ( ) (                     

k k k k k k k k

P p Q q t P t t p t Q t t q

       

 

    r d r h h r J r J E r h

3 ,

) ( ˆ ˆ ) ( ˆ ) ( ) ( ˆ    

     

) (t q k



) (t p k



  

              



| ˆ ) ( ˆ ) ) ( ( ) ( | ) ( | | ) ( |

     

 

k k k k k k

Q t p P q t q i t t t

SLIDE 24

24

) , ( ) ( ) , ( t r J r J t r J   

     

  

   

 



                  

k k k k k k

r J Q t p r J P q t q i r J t r J t t r J

               

 | ) ( ˆ , ˆ | ) ( | ) ( ˆ , ˆ | ) ) ( ( | ) ( ˆ | ) ( | ) ( ˆ | ) ( ) , (     

) , ( ˆ ) ( ˆ ) , ( ˆ t r h r h t r h

res 

   

) ( ˆ0 r h 

 

    

                 

           

) , ( ˆ ) ( ˆ ) ( ˆ ) ( ) ( ˆ ) ) ( ( ) ( ˆ ) , ( ] ) ( ) ( [ ) , ( ] ) ( ˆ [ ) , ( ˆ

3 2 3

t r h r d t h r Y t p r X q t q r J t r J r J r J E r d t r J r J h t r h

res res k k k k k k res

         

                       

 

SLIDE 25

25

  

     

       

           ) ( ˆ | )] ( ˆ , ˆ [ | ] ) ( ) ( [ ) ( ˆ ) ( ˆ

2 3

r J r J P r J r J E r d i r X r X

k k k

     

               

 

  r d r Y Y r d r X X

k k k k 3 3

) ( ˆ ˆ ) ( ˆ ˆ  

   

with | ] ˆ , ˆ [ | | ] ˆ , ˆ [ | | ] ˆ , ˆ [ |         

     

B A B A B A

where

  

  A T A T ˆ ˆ

1

and

k k k k

Y T Y T X T X T

   

ˆ ˆ ˆ ˆ

1 1

  

 





p n K k , , , 1    





p n K k , , , 1    

  

     

       

           ) ( ˆ | )] ( ˆ , [ | ] ) ( ) ( [ ) ( ˆ ) ( ˆ

2 3

r J r J Q r J r J E r d i r Y r Y

k k k

      

               

where enumerates T- even densities enumerates T- odd densities









p n K k , , , 1    

where enumerates T- even densities enumerates T- odd densities









p n K k , , , 1    

where

SLIDE 26

26

k

X ˆ

k

Y ˆ

           | ˆ | ) ( | ˆ | ) ( ) ( ˆ

k k k

X t X t t X

  





        

   

k k k k k

q t q

    



1 ,

) ) ( (

           | ˆ | ) ( | ˆ | ) ( ) ( ˆ

k k k

Y t Y t t Y

  



1 ,   k k  



1 ,   k k  



    

       

| ] ˆ , ˆ [ |

1 , 1 , k k k k k k

X P i

     

 

          

     

      

  

| )] ( ˆ , ˆ [ | ] ) ( ) ( [ | )] ( ˆ , ˆ [ |

2 3 3

r J P r J r J E r J P r d r d

k k

   

           

          

     

      

  

| )] ( ˆ , ˆ [ | ] ) ( ) ( [ | )] ( ˆ , ˆ [ |

2 3 3

r J Q r J r J E r J Q r d r d

k k

   

           

    

       

| ] ˆ , ˆ [ |

1 , 1 , k k k k k k

P Q i

     

 

 

J ˆ

SLIDE 27

27

  

k k

q t q

 

) ( ) (t p k



) ( 2 1 ) cos( ) (

t i t i k k k k

e e q t q q t q

        





      ) ( 2 1 ) sin( ) (

t i t i k k k

e e p t p t p

       





  

 

   

      

j i j i j i j i j i j i

b t c b t c t

, , ) ( , , ) (

| ) ) ( 1 ( | ) ) ( exp( ) ( |

        

t i t i

e c e c t c

          

 

) ( ) ( ) (

) (

        

  

j i j i j i j i j i j i

b b b      

   j i j i j i

b b b , ,

j j i i j i j i i j j i j j i i j i j i i j j i j j i i j i j i

b b b b b b

                  

                     | ] , [ | | ] , [ | | ] , [ |

with

SLIDE 28

28

 

      ) ( | ] ) ( ˆ ˆ [ ) ( | t t h h t dt d i

res

  k

q

,

  k

p

) , (

k k p

q c c

      

  

   ) (t X k

  

   ) (t Y k

) , (

k k p

q c

   

.

 

 



            

  

k XY k k k k k XX k k k

F p F q

          



) ( , 1 , ) ( ,

 

 



            

  

k k k YY k k k YX k k k

F p F q

          



1 , ) ( , ) ( ,

p n K k k , , , , 1 ,       

SLIDE 29

29

Y

• r

X A A A F

j i j i j i k k AA k k

ˆ ˆ ˆ | | | ˆ |

, , 2 2 ) ( ,

      



                  

     

• the matrix of the eq. system for and is symmetric and

real

• this eq. system has nontrivial solution only if the determinat of its

matrix is zero, - dispersion equation for

  k

q

  k

p

) ( det ) (  

 

  F F





 

       

     

              

    

j i j i j i k k BA k k AB k k

A A F F

, , 2 2 ) ( | , ) ( ,

| | | ˆ | 



 





j i for E E

j i

   j i for E E

j i

   j i for E E

j i

  

SLIDE 30

30

  k

q

  k

p

     

  O O H O O H

RPA RPA

ˆ ] ˆ , ˆ [ ˆ ] ˆ , ˆ [   

 

   



   

] ˆ , ˆ [ O O

) (

ˆ ˆ ˆ

sep res RPA

V h H  

ˆ h

) (

ˆ sep

res

V

 



     

  

k k k k k k k k k k sep res

Y Y X X V

         

 

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

ˆ ˆ ˆ ˆ 2 1 ˆ

) 1 ( ) 1 (

ˆ , ˆ

k k

Y X

 

p-h ( two-qp ) part of corresponding operator

K k k , , 1 ,   

SLIDE 31

31

 



    

 

j i j i j i p n

b b Q

, , ; , ) , ( ) , (             

 

 

Q

                               

                         

   

                        k k k k k k k k k k k k

Y p i X q Y p i X q

* ) ( * ) ( ) , ( ) ( ) ( ) , (

| | | | 4 | | | | 4    







j i for   4 / 1 j i for   4 / 1 j i for   2 / 1

SLIDE 32

• 

 k

q

  k

p

) , (

k k P

Q

k k    



, k k    



,

) , (

k k P

Q