Electric and Vorticity Strengths in Heavier Nuclei

J. Kvasil 1) , V.O. Nesterenko 3) ,

W. Kleinig 2), P.-G. Reinhard 4) , P. Vesely 1)

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

2) Technical Universiy of Dresden, Institute for Analysis, D-01062, Dresden, Germany

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

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

1. Motivation and brief formulation of the Separable RPA approach

2. Different Skyrme parametrizations are analyzed from the point of view of the photoabsorption cross section

3. Photoabsorption cross section in the Pigmy region is discussed

4. M1 resonance is dicussed from the point of view of different Skyrme parametrizations

4. Vorticity multipole operator strength function is introduced as a measure of the irrotationality of a nuclear matter

5. Some preliminary results of the vorticity strength is presented

Effective n-n interactions (Skyrme , Gogny, relativistic mean field) are widely used for the description of the static characteristics of spherical and deformed nuclei

Dynamics of small amplitude vibrations is mainly described by the RPA. However, for heavy nuclei the standard RPA method requires the construction and diagonalization of huge matrices. RPA problem becomes simpler if the residual two-body interaction in the nuclear Hamiltonian is factorized as a product of two s.p. operators (see e.g. P.Ring, P.Schuck, The Nuclear Many-Body Problem, Springer N.Y. (1980)).

k k

kkkkkkkkres YYXXV )1()1()1()1( ˆˆˆˆ21ˆ Kk ,,1

ij

jikk aajXiX |ˆ|ˆkk XTXT ˆˆ 1

ij

jikk aajYiY |ˆ|ˆkk YTYT ˆˆ 1

where and are two-quasiparticle parts of s.p. operators )1(ˆ

kX )1(ˆkY

resHFB VhH ˆˆˆ

We developed a general self-consistent separable RPA (SRPA) approach applicable to any density- and current- dependent functional - see e.g. sperical nuclei:

deformed nuclei:

V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, Phys.Rev. C66, 044307 (2002)V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard Phys.Rev. C74, 064306 (2006)

for the determination of operators and by fully self-consistentmethod starting from the general energy functional

)1(ˆkX

)1(ˆkY

3| | ( ( ))E HFB H HFB J r d r

Hwith

( ) | ( ) |J r HFB J r HFB some are time-even and

some are time-odd

ˆ ( )J r

Basic idea of the SRPA method:nucleus is excited by external s.p. fields :)ˆ,ˆ( kk PQ Kk ,,1

kkkkkk

kkkkkk

QiPHPTPTPP

PiQHQTQTQQ

ˆ]ˆ,ˆ[;ˆˆ;ˆˆ

ˆ]ˆ,ˆ[;ˆˆ;ˆˆ

1

1

)'(ˆ|)](ˆ,ˆ[|])()(

[ˆ'

'

233 rJrJQ

rJrJE

rdrdiY kk

K

kkkkkkkkkk

sepres YYXXV

1',''''

)( ˆˆˆˆ21ˆ

JTJTJTJT ˆˆ,ˆˆ 11kkkk YTYTXTXT ˆˆ,ˆˆ 11

|)](ˆ,ˆ[|])()(

[|)](ˆ,ˆ[|2

33' rJP

rJrJE

rJPrdrd kkkk

|)](ˆ,ˆ[|])()(

[|)](ˆ,ˆ[|2

33' rJQ

rJrJE

rJQrdrd kkkk

where strength constant matrixes are1

1

Using TDHFB with the linear response theory we obtain :resHFB VhH ˆˆˆ

i

lik iYrQ )(ˆ

for electric typeexcitation

il

lik YrP ][ˆ for magnetic type

excitation

)(ˆ])(

[ˆ 3 rJrJ

ErdhHFB

)'(ˆ|)](ˆ,ˆ[|])()(

[ˆ'

'

233 rJrJP

rJrJE

rdrdiX kk

RPA equations:

],[],[],[ OOOOHOOH

gives energies, forward and backward amplitudes of phonon operator

ijijijijij bbO )()(

RPA equations with the separable residual interactions can be transferred into the homogeneous system of algebraic equations.Dimension of the matrix of this system is given by the number ofs.p. operators and in the residual interaction. Detaileddescription of our SRPA method can be found in the papers: W.Kleinig, V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, P.Vesely, PRC78, 044315 (2008) V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC74, 064306 (2006)

kX kY

Knowing the structure of phonons we can calculate el.mg. reduced probability from the RPA ground state to one-phonon state with the energy

RPA| RPAO |

RPAMORPARPAZB Z |],[||)||,(

.,. mgelZ transition multipolarity ZM transition multipole operator

Then the energy weighted strength function is:

)()||;();(

ERPAZBEZS LL

)()||;(

ERPAZB L

This quantity can be determined even without the solving the RPA equations for each individual phonon state usingthe Cauchy theorem and the substitution

see V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC74, 064306 (2006)

RPAO ||

22 )2/()(1

21

)()(

EEE

or in more detail:

)(402.0)( 11 ESEE EE

)(10089.3)( 337

2 ESEE EE

)(10199.1)( 3513

3 ESEE EE

)(10437.4)( 13

1 ESEE MM

2)( fminE122)( MeVfmeinESE

MeVinE

1222)( MeVfminES NM

)()()()()( 1321 EEEEE MEEE

)(2 ESE

1

]!)!12[(1

)()(8

)( 222

12

2

3

cE

eE

)()|0|;()|0|;( EEEBEB

where1371

fmmce

N 105.0~2

Knowing the reduced probability or strength function we can determinethe photoabsorption cross section:

We use the Skyrme energy density for the energy functional- see e.g. J.Dobaczewski, J.Dudek, Phys.Rev. C52, 1827 (1995):

rdrTjsE 3)(),,,,,,(

Hwith )()()()()( rrrrr CoulpairSkkin

HHHHH

)(2

)(2

rm

rkin

H

3

431

232

)]([3

43

)(||

1)(

2)( rer

rrrrd

er pppCoul

H

])(1[41

)(,

)0(2

nmpnt

ttpair Vr

H

)()()( )(2)()()(2)()(tttttttttttt

event CCCCCr

H

)()()( )(2)()()(2)()(tt

jtt

jttt

Ttt

stt

st

oddt jsCjCTsCssCsCr

H

1,0 1,0

)()()(t t

oddt

eventSk r HHH

,,,,,,

,,,,)(,)()0()()()()()(

)()()()()(

tj

tj

tTt

stt

ttts

tt

VCCCCC

CCCCC

interactionparameters

gauge invariance

)()(

)()(

)()(

tj

t

tTt

tj

t

CC

CC

CC

The dependence of the energy density on goes through the following densities and currents:

)(r

H r

density

jiij

ji aarrr

)()()(ˆ

spin-orbit current

ij

jijiji aarrrri

r )()()()(2

)(ˆ

jiij

i aarrr

)()()(ˆ

kinetic energy density

ijjijiji aarrrr

irj )()()()(

2)(

ˆ current

ij

jiji aarrrs )()()(ˆ

spin-current

jijij

i aarrrT

)()()(ˆ

kinetic energy – spin current

pairing density)(ˆ r

)()()( iiiiii

i aaaarr

Comparison of experimental photoabsorption cross-section with calculated values for different Skyrme parametrizations

photoabsorption cross section gives possibility to test different parametrizations

exp. taken fromP.Carlos et al.NPA 172, 437(1971)

similar results andsimilar agreementwere obtained alsofor Mo, Sm, Snizotopes

236U

5 10 15 20

E1

stre

ng

th f

un

ctio

n [

arb

. un

its]

234U

5 10 15 20 25

[MeV]

238U

232Th

W. Kleinig, V.O. Nesterenko, J. Kvasil, P.-G. Reinhard and P. Vesely,Phys. Rev. C, 78, 044313 (2008)

166Er

5 10 15 20

E1

stre

ng

th f

un

ctio

n [

arb

. un

its]

160Gd

5 10 15 20 25

[MeV]

168Er

156Gd

- Z=102, 114, 120; isotopic chains -deformations: good agreement with Lublin-Strasbourg drop model;

- energy trend:

Skyrme-RPA description of E1(T=1) GR in rare-earth, actinide and superheavy nuclei

-1/3E=81A MeV---1/ 3 1/ 6(31.2 20.6 ) MeVE A A --x--

SLy6

6 8 10 12E [MeV]

4 6 8 10 12

51015

51015

51015

51015 A=100 b=0.30

A= 98 4b=0.0A= 96 b=0.0A= 94 b=0.0A= 92 0b=0.

SkT6

A=100 b=0.30A= 98 b=0.19A= 96 b=0.05A= 94 b=0.0A= 92 0b=0.

SkM*

A=100 b=0.27A= 98 b=0.22A= 96 b=0.05A= 94 b=0.0A= 92 0b=0.

SLy6

b=0.42

A= 98 b=0.22A= 96 b=0.05A= 94 b=0.0A= 92 0b=0.

SkI3b=0.0

b=-0.24A=100

Experiment

WS+QRPA

Talys

Cumulative integral photoabsorption cross section in the low-energy(Pigmy) region (4 - 13 MeV)

for bigger deformation steeperincrease of the cumulative cross section starting from some energy (see 100Mo)

this starting energy depends on deformation splitting of strength funtion- see next pages

)(1 ESE

][)()||;1()( 12

1 MeVfmEEEBESE

][)(01.4)( 11 mbESEE EE

E

E MeVmbEEd4

1 ][)(

][60 MeVmbAZN

TRK

92-100Mo

Cumulative integral photoabsorption crossection in the low-energy(Pigmy) region (4 - 13 MeV)

)(1 ESE

-see alsoD.P.Arteaga, E.Khan, P.Ring, PRC 79, 034311 (2009)

for bigger deformation steeperincreas of the cumulative cross section starting from some energy

this starting energy depends on deformation splitting of strength funtion- see next pages

E

E MeVmbEEd4

1 ][)(

][60 MeVmbAZN

TRK

E1 excitation strength function

1,0

12211 ][)()(

MeVfmeESES EE

)()||;1(

)(1

EEEB

ESE

•exciting operators:

33

13

1 ;; YrYrrY

eAZ

neute

eA

Nprote

eff

eff

)(

)(

one can see splitting and broadening of E1 resonancewith increasing b

HFB total energy for neutron rich Sn isotopes

In the paper D.P.Arteaga, E.Khan, P.Ring, PRC 79, 034311 (2009) neutron very rich Sn isotopeswere analysed in the framework of RMF approach - we tried to compare with different Skyrme parametrizations

144-160Sn nuclei aredeformed but soft

b b

b

Comparison of equilibrium neutron, proton and total deformations fordifferent Skyrme parametrizations with the RMF results (see D.P.Arteaga, E.Khan, P.Ring, PRC 79, 034311 (2009))

Energy weighted E1 strength function for selected neutron rich Snisotopes for Pygmy energy interval

S(E

1, E

) [

efm

]1

22

with the increasingnumber of neutronsE1 strength goes downin the Pygmy region

Strength function for 142-152Nd isotopes),1(),1(1,0

EMSEMS

orbital (scissor) part contributes onlyfor deformed systems

A

i

ieffsi

ieffliN sglgMM

1

)(),()(),(

43

)1(

)(),( 7.0 freei

effsi gg

),( efflig

0

1

neutrons

protons

time-odd densities andcurrents should beinvolved in the Skyrmefunctional

so called high-energyM1 mode (inducedby E2) - see:

I.Hamamoto, W.Nazarewicz Phys.Lett. B297, 25 (1992)

J.Kvasil, N.Lo Iudice, F.Andreozzi, F.Knapp, A.Porrino, Phys.Rev. C73, 034502 (2006)

22

12

1 ,, Yrsrs

excited by operators:

Comparison of M1 strength functions calculated withSkI3, SkM*, SkT6, SLy6 Skyrme parametrizations.

),1( EMS

relatively big differenciesof different parametrizations

possibility to testparametrizations bya comparison withexperimental dataon M1 strength

22

12

1 ,, Yrsrs

excited by operators:

Comparison of M1 strength functions calculated with different Skyrme parametrizations with experimental values.

),11( EMS

experimental values from: P.Sarriguren, et al.,PRC 54, 690 (1996)

none of used parametrizationsdescribes M1 strength for allinvestigated nuclei

122

112

11 ,, Yrsrs

exciting operators:

deformed nuclei - two peaksstructure of the spin-flip resonance

Comparison of M1 strength functions calculated with different Skyrme parametrizations with experimental values.

),11( EMS

experimental values from: Ca - S.K.Nanda, et al., PRC 29, 660 (1984)) Pb - R.M.Laszewski, et al., PRL 61, 1710 (1988)

none of used parametrizationsdescribes M1 strength for allinvestigated nuclei

exciting operators:

122

112

11 ,, Yrsrs

sperical nuclei - one peakstructure of the spin-flipresonance

RPA shifts and spin-orbital neutron-proton splitting

the shape (one-peak or two-peaks structure) of the spin-flip resonance is a result of the interplay between the RPA energy shift and spin-orbital splitting

HFBRPA EEE - energy shift of the centroid of the spin-flip resonance caused by the switching on the residual interaction

)()( , nso

pso EE - centroids of proton and neutron spin-orbital splittings

2 2

,1 1[ ] [ ]

n p

sTb b s T

- M1 vs M1(T=1)- impact of tensor interaction

Essential difference between M1 and M1(T=1) strength:

tensor termin Skyrmefunctional

G. Colo, H. Sagawa, S. Fracasso, and P.F. Bortignon,Phys. Lett. B, v.646, 227 (2007)

no tensor

with tensor

SV-bas - strike effect of tensor interaction! - principle possibility to get both 1- and 2-bump structure- importance of refitting of Skyrme parameters

Vorticity

One of the basic questions of all hydrodynamical nuclear models:irrotationality of nuclear matter (with or without whirls?)

irrotationality: velocity fieldoperator :

condition does not guarantee because:

)(

)(ˆ

)()(ˆ)()(ˆ)()(ˆ

r

rjrrvrrvrrj

nuc

nucnucnucnucnuc

)(ˆ)()(ˆ

)(ˆ)( rvrrjrvr nucnucnuc

One can expect that if the nuclear matter is irrotational then:

0)(ˆ)( rvrnuc

The question:how to investigate the irrotationalityof nuclear matter in practice?

But this is a problem because andare coupled by charge-current conservation:

)(ˆ rnuc

)(

ˆrjnuc

irjfirfi nucnuc |)(ˆ||)(ˆ|

0)( rv

0)(ˆ

rjnuc

)(

)(ˆ

)(ˆr

rjrv

nuc

nuc

In papers: D.G.Raventhall, J.Wambach, NPA 475, 468 (1987).E.C.Caparelli, E.J.V.de Passos, J.Phys.G 25, 537 (1999).N.Ryezayeva, T.Hartmann, Y.Kalmykov, H.Lenske, P.von Neumann-Cosel,V.Yu.Ponomarev, A.Richter, A.Shevchenko, S.Volz, J.Wambach, PRL 89, 272502 (2002).

so called transitional vorticity strength is defined – the idea isfolllowing:

)( fi

)(ˆ

)(ˆ

)(ˆ

rjrjrj vortirrotnuc

0)(ˆ

rjvort

])(ˆ,ˆ[)(ˆ

rHci

rjirrot

vorticity operator:

or:

if nuclear matter is irrotational then all matrix elements of the vorticity operator are zero

for the first sight (see in the previous slide) it seems:but it is not so because ofthe charge-current conservationgives uncertainty what isand what is

)(ˆ rnuc

)(ˆ

rjnuc

)(ˆ)()(ˆ

rvrrj nucirrot

)(ˆ)()(ˆ

rvrrj nucvort

irjfirjfirwf irrotnuc |)(ˆ||)(

ˆ||)(ˆ|

)(

ˆ)(

ˆ)(

ˆ)(ˆ rjrjrjrw irrotnucvort

In the paper D.G.Raventhall, J.Wambach, NPA 475, 468 (1987)decomposition into the spherical vectors is done:

ll

fil

f

ffiiiinucff Yrj

j

mjmjmjrjmj ),()(

12

|(|)(

ˆ| *)()

),( lY

ll

fil

f

ffiiiiff Yrw

j

mjmjmjrwmj ),()(

12

|(|)(ˆ| *)()

and it was shown (using the charge-current conservation) that

with

all information about the transitional vorticity isgiven by the radial transitional component of thenuclear charge current

it was also shown that )(ˆ)(1

)(ˆ

rvrrj nucvort

)()( rw fi

)()(1 rj fi

),()(

12

|(|)(ˆ| *)() Yrw

j

mjmjmjrwmj fi

f

ffiiiiff

( ) ( )1

2 1 2( ) ( )fi fid

w r j rdr r

In papers:

D.G.Raventhall, J.Wambach, NPA 475, 468 (1987).E.C.Caparelli, E.J.V.de Passos, J.Phys.G 25, 537 (1999).N.Ryezayeva, T.Hartmann, Y.Kalmykov, H.Lenske, P.von Neumann-Cosel,V.Yu.Ponomarev, A.Richter, A.Shevchenko, S.Volz, J.Wambach, PRL 89, 272502 (2002).

so called transition vorticity strength : )( fi

0

)(4)( )( drrwr fifi

was introduced as a measure of the irrotationality of the nuclearmatter (usually ).

It was shown that the vorticity strength is significant for the transitions from the ground state to states in the Pigmy region and that these states have a toroidal character (for some lighterspherical nuclei).

fRPAi ||| f|

1

we introduced another quantity as a measure of the irrotationality- vorticity multipole operator

vorticity multipole operator is directly connected with the long-wavedecomposition of the standard electric multipole operator:

)](

ˆ[)],()([)1(

)1(!)!12(

)(ˆ 31 rjYkrjrd

kckM nucE

using Bessel functiondecomposition

)0(ˆ)0(ˆ kMkkM torE

where is the transition energy and

toroidal multipole operator see e.g.

D.Vretenar, N.Paar, P.Ring,T.Niksic, PRC 65, 021301 (2002)

S.F.Semenko, Yad.Fiz. 34, 639 (1981) (nonstandard normalization of el.mg. multipoles)

),(

322

1),()(

ˆ

122)0(ˆ

1113

YYrrjrdci

kM nuctor

),()(ˆ)0(ˆ 3

YrrrdkM nucE

kc

3221

!)!12()(

)(

22

rkkr

krj

Vorticity multipole operator is obtained from (see exp. )by the following substitution:

)(ˆ kME

)(ˆ

)(ˆ

)(ˆ

)(ˆ rjrjrjrw irrotnucvort

)(

ˆrjnuc

)(ˆ)(1

)(ˆ

rvrrj nucnuc

Then

)],()([)1(

)1(!)!12(

)(ˆ 31

Ykrjrd

kckM vor

)(ˆ)(

1)(

ˆrvrrj nucnuc

Bessel functiondecomposition

)0(ˆ kMk vor

with long-wave limit of the vorticity multipole operator:

1

13 ),()(ˆ

112

321

)0(ˆ

rYrjrd

ci

kM nucvor

k

The nonzero value of all matrix elements of all vorticity multipoleoperators can serve as a measure of the irrotationality of the nuclearmatter. We restrict ourselves for and we calculate the strengthfunction of :

1,0

)1();1(

vorSEvorS

)(||])0(ˆ,[|| 21

ERPAkMORPA vor

Dipole vorticity strength function can be compared with the dipoletoroidal strength function:

1,0

)1();1(

torSEtorS

)(||])0(ˆ,[|| 21

ERPAkMORPA tor

where dipole toroidal operator (involving the corrections to the C.o M.motion) is:

1)0(ˆ

1 kM vor

)(ˆ

31

2)0(ˆ 3

1 rjrdci

kM nuctor

),(),(52

),( 102

12102 YrYYr

Dipole vorticity strength function can be also compared with the squeezed dipole electric (or isoscalar E1) strength function:

1,0

)1();1(

EsqSEEsqS

)(||])0(ˆ,[|| 21

ERPAkMORPA Esq

with the squeezed dipole E1 transition operator:

),(

35

)(ˆ)0(ˆ1

2331 YrrrrrdkM Esq

Connection between squeezed dipole E1 operator and the dipole toroidal operator is discussed in J.Kvasil, N.Lo Iudice, Ch.Stoyanov, P.Alexa, J.Phys G 29, 753 (2003)

jijpn ij

ieffnuc aarreer

)()()(,

)(

Nuclear charge density operator:

Nuclear charge current operator consists from convectional andmagnetization parts:

)(ˆ

)(ˆ

)(ˆ

rjrjrj magconnuc

( )

,

ˆ ( ) ( ) ( ) ( ) ( )2con eff i j i j i j

n p ij

i ej r e r r r r a a

m

( )

,

ˆ ( ) ( ) ( )2mag eff i j i j

n p ij

ej r g r r a a

m

where effective charges and gyromagnetic ratios depend on the processof excitation (see M.N.Harakeh, A.van der Woude, Giant Resonances, Clarendon 2001)

58.5)( psping

82.3)( psping

7.0

el.mag. isoscalar T=0 isovector T=1

01 )()( neff

peff ee

)()( pspin

peff gg

)()( nspin

neff gg

11 )()( neff

peff ee

2/)( )()()( nspin

pspin

peff ggg

2/)( )()()( nspin

pspin

neff ggg

11 )()( neff

peff ee

2/)( )()()( nspin

pspin

peff ggg

2/)( )()()( nspin

pspin

neff ggg

• isoscalar vorticity strength is mainly formed by convective part of the nuclear charge current• isovector vorticity strength is mainly formed by magnetization part of the nuclear charge current

T=0

vorticity – exc. operators:

toroidal– exc. operators:

squeezed E1– exc. operators:

2123 )(

ˆYrrjrd con

1

33 )(ˆ Yrrrd

2123 )(

ˆYrrjrd mag

133 )(ˆ Yrrrd

2101

23

52

)(ˆ

YYrrjrd con

2101

23

52

)(ˆ

YYrrjrd mag

2101

23

52

)(ˆ

YYrrjrd con

1

33 )(ˆ Yrrrd

similarity of the basic structure ofthe vorticity, toroidal and sqeezeddipole resonance

][ fm

][ fm

Velocity field for the RPA state 8.3049E MeV

definition of the velocity field:

| |i Q RPA

| j RPA

with the density and current operator:

jiij

ji aarrr

)()()(ˆ

ijjijiji aarrrr

irj )()()()(

2)(

ˆ

ˆ( ) | ( ) |r HFB r HFB

ˆ( ) | ( ) |ijj r i j r j

( )( )

( )ijj r

v rr

][ fm

][ fm

in the figure the velocity projection onto the plane is plotted ( )( , )z 0

Conclusions

• SRPA – effective method for the investigation of excited states in heavy nuclei

• different Skyrme parametrisations (SkI3, SkM*, SkT6, SLy6) give very similar and good agreement with experimental photoabsorption cross section (not so for M1 giant resonance)

• for bigger deformation steeper increase of the cumulative integral photoabsorption cross section with the increasing excitation energy is observed for energies above the particle emission threshold. Below this threshold this increase is not so conclusive

• significant vorticity dipole strength is observed in the excitation energy intervals (for 208Pb): in these energy intervals one can

expect a significant irrotationalityof the nuclear matter in positiveparity excited states

• isoscalar dipole vorticity strength is mainly formed by the convective charge current while the isovector dipole vorticity strength (low energy part) is mainly caused by the magnetization charge current

MeVEMeV

MeVEMeV

3727

207