Post on 02-Apr-2015
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