Determining Modern Energy Functional for Nuclei And The Status of The Equation of State of Nuclear Matter

Shalom Shlomo Texas A&M University

Outline 1.  Introduction. Background, Energy Density Functional, Equation of State,

Collective States

2. Energy Density Functional. Hartree-Fock Equations (HF), Skyrme Interaction Simulated Annealing Method, Data and Constraint 3.  Results and Discussion. 4.  HF-based Random-Phase-Approximation (RPA). Fully Self Consitent HF-RPA, Hadron Excitation of Giant

Resonances, Compression Modes and the NM EOS, Symmetry Energy Density

5. Results and Discussion. 6. Conclusions.

INTRODUCTION Nuclear physics:

Study of structure, interactions and properties of nuclei. Aim to quantitatively understand and relate vast amount of properties of nuclei in terms of few constituents, elementary laws and processes.

Current situation:

Active area of research. We have a certain picture (understanding) obtained through 70 years of phenomenological research, qualitative consideration and application of laws quantum mechanics. Q. M. is very successful in describing properties of nuclei. There is no evidence contradicting Q. M.

Recent emphasize:

Properties of nuclei under extreme conditions of excitation energy (temperature), angular momentum and N-Z (asymmetry).

Relation to other areas:

Astrophysics: source of energy stars, structure and evolution of stars, origin of the elements, neutron stars and supernova. Other systems: atomic clusters, metal clusters, trapped ions, and mesoscopic systems.

(a) Charge density distribution ρc(r) for doubly –magic nuclei 16O, 40Ca, 90Zr, 132Sn, and 208Pb. The theoretical curves are compared with the experimental data points

(units are ρc(efm-3) and r(fm)).

(b) Nuclear matter density distributions ρm(fm-3) for the magic nuclei.

aRr

er

0

1)( 0

−

+

=ρ

ρ

3/100 ArR = fmr 1.10 ≈

fma 55.0=

Matter and charge density distributions

Nuclear Binding (Weizsacker formula)

( )2/1

2

43/1

2

3

3/221),(

AAZNa

AZa

AaAaNZBE

δ+

−++

+=

568.151 =a

The contribution to B/A. Note that the surface, asymmetry and Coulomb terms all subtract from the bulk term.

226.172 −=a

698.03 −=a

7.234 −=a

MeV (Volume)

MeV (Surface)

MeV (Symmetry)

MeV (Coulomb)

(pairing) =δ

2.11

0

2.11− MeV (odd-odd)

(even-odd)

MeV (even-even)

Mean-field approximation

The many-body Schrödinger equation Hψ = E ψ is difficult to solve. In the mean-field approximation each particle moves independently from other nucleons in a single particle potential, representing its interactions with all other nucleons.

∑ ∑<

+=i ji

iji

i VmpH2

2

resi

ii

i HrUmpH +⎥

⎦

⎤⎢⎣

⎡+=∑ )(

2

2

,0 ∑=i

ihH )(2

2

ii

ii rU

mph

+=

)()...1( AAi φφΑ=Φ

Approximation:

iiii Eh φφ =

≡Α Antisymmetrization operator (fermions) or symmetrization operator (bosons)

Spherical symmetry

( ) )(121)(1.)(67.01)( ..0 rU

drrdf

rlUrfU

AZNrU CoulZosZ τστ −+⎥⎦

⎤⎢⎣

⎡ +⎥⎦

⎤⎢⎣

⎡ −−=

Wood-Saxon potential popular:

τχφθφ ),()()( ljmYrrur =

,27.1 3/1 fmARR c == 510 −=U 0.. 22.0 UU os −=,67.0 fmd =

Spherical symmetry

)()()(2

22

rrrUm

εφφ =⎥

⎦

⎤⎢⎣

⎡+∇− is reduced to 0)()(2

2

=+− rursdrud

where ⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−= 2

2

2 2)1()(2)(

mrllrUmrs

ε

1

exp1)(−

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −+=

dRrrf

MeV,

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

22

32 cc R

rRZe

rZe2

cRr ≤

cRr >=)(rUCoul

Single-particle spectrum up to N=5. the various contributions to the full orbital and spin orbit splitting are presented. Partial and accumulated nucleon numbers are also given.

The spin-orbit coupling: describe real nuclei

For the harmonic oscillation + spin orbit interaction, the energy eigenvalues become

δ = -l or l +1 if j = l +1/2 or l-1/2

( ) [ ] ωε 2/3)1(2 ++−= lnjlsn

)(0 jU αδ+−

Where,

Map of the existing nuclei. The black squares in the central zone are stable nuclei, the broken inner lines show the status of known unstable nuclei as of 1986 and the outer lines are the assessed proton and neutron drip lines (Hansen 1991).

1.  Important task: Develop a modern Energy Density Functional (EDF), E = E[ρ], with enhanced predictive power for properties of rare nuclei.

2.  We start from EDF obtained from the Skyrme N-N interaction.

3.  The effective Skyrme interaction has been used in mean-field models for several decades. Many different parameterizations of the interaction have been realized to better reproduce nuclear masses, radii, and various other data. Today, there is more experimental data of nuclei far from the stability line. It is time to improve the parameters of Skyrme interactions. We fit our mean-field results to an extensive set of experimental data and obtain the parameters of the Skyrme type effective interaction for nuclei at and far from the stability line.

OBJECTIVE

Equation of state and nuclear matter compressibility

2

181][][ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+=

o

oo KEE

ρρρ

ρρ

The symmetric nuclear matter (N=Z and no Coulomb) incompressibility coefficient, K, is a important physical quantity in the study of nuclei, supernova collapse, neutron stars, and heavy-ion collisions, since it is directly related to the curvature of the nuclear matter (NM) equation of state (EOS), E = E(ρ).

ofod

AEddk

AEdkKkf

fρ

ρρ 2

22

2

22 )/(9)/(

==

ρ [fm-3]

ρ = 0.16 fm-3

E/A

[MeV

]

E/A = -16 MeV

2

(()((

181]([]([ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

)

)−)+)=)

βρβρρ

βρβρβρo

ooooANM KEE

2][]([ βρβρ JEE oo +=)

2v]([ ββρ τKKK o +=)

AZN /)( −=β

][ oSYMEJ ρ=

AZy /=

2/1,2

2 )/(81)(

=

=y

SYMdy

AEdEρ

ρ

Macroscopic picture of giant resonance

L = 0 L = 1 L = 2

Modern Energy Density Functional

)(),()(

),,( τχστσφτ

αmjlmi rY

rrR

r i=

Within the HF approximation: the ground state wave function Φ

),,(...),,(),,(

),,(...),,(),,(),,(...),,(),,(

!1

21

22222222221

11111121111

AAAAAAAAAA

A

A

rrr

rrrrrr

Aτσφτσφτσφ

τσφτσφτσφ

τσφτσφτσφ

=Φ

In spherical case

ΦΦ= totalHE ˆHF equations: minimize

The total Hamiltonian of the nucleus

( )∑∑=〈=

+=+=11

2

,2

ˆji

ji

A

i i

itotal rrV

mpVTH

.),( Coulij

NNijji VVrrV +=

where

The total energy

')()'()',()'()(

')'()()',()'()(

)()(2

ˆ

**

**

1

*2

rdrdrrrrVrr

rdrdrrrrVrr

rdrrm

HE

jiji

jiji

ii

A

ji

A

ji

A

itotal

αααα

αααα

αα

φφφφ

φφφφ

φφ

∑∫

∑∫

∑∫

〈

〈

=

−

+

Δ−=ΦΦ=

,4 1,

22

∑= −

+−=

A

ji ji

ijijCoulij rr

eV ττ

jiij τττ +=

,))((

)(2

)1(61)()1(

])()()[1(21)()1(

0

3322

221100

ijjijiij

jiji

ijijjiijij

ijjijiijijjiijNN

krrkiW

rrrr

PxtkrrkPxt

krrrrkPxtrrPxtVij

σσδ

δρδ

δδδ

ασσ

σσ

+−

+−⎟⎟⎠

⎞⎜⎜⎝

⎛ +++−+

+−+−++−+=

we adopt the standard Skyrme type interaction NNijV

For the nucleon-nucleon interaction

Skyrme interaction

0,,, Wxt ii α

are 10 Skyrme parameters.

.),( Coulij

NNijji VVrrV +=

The total energy

∫=Φ++Φ=ΦΦ= rdrHVVTHE Coulombtotal)(ˆ

12

where

)(2

)(2

)(22

rm

rm

rH nn

pp

Kinetic

ττ +=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

−= ∫∫ '

')',(

'')'()(

2)(

22

rdrrrr

rdrrrrerH chch

chCoulomb

ρρρ

)()()()( rHrHrHrH SkyrmeCoulombKinetic

++=

sgsofineffSkyrme HHHHHΗrH +++++= 30)(

∑=τ

τρρ )()( rr ),,(),,()(1

τσφτσφρσ

τ rrr i

A

ii

∑∑=

∗=

∑∑=

∇=A

ii rr

1

2),,()(

στ τσφτ

)()( rr

∑=τ

τττ

[ ]∑∑=

×∇−=A

iii rrirJ

1 ,

''*

'

),,(),,()(σσ

τ σσστσφτσφ∑=

ττ )()( rJrJ

∑=',,

''* )21,,()

21,,()',(

σσ

σφσφρi

iich rrrr

.

Now we apply the variation principle to derive the Hartree-Fock equations. We minimize the Energy E, given in terms of the energy density functional

( )ˆtotalE H H r dr= Φ Φ = ∫

r r

(*)0,

,

,,

,

=⎥⎦

⎤⎢⎣

⎡

−=⎥⎦

⎤⎢⎣

⎡−

∑ ∫∑ ∫

τσ

τσ

τστσ

τσ δρ

ρεδ

δρδ

ρεδρδ i

i

ii

rdErdE

∑∫⎥⎥⎦

⎤

⎢⎢⎣

⎡++=

τσσττσττστ

τ

δδρδτδ,

*

2

)()()()()()(2

rdrJrWrrUrrm

E

where

∑=',

* ),',(),',(σ

στ τσδφτσφδρi

ii rr

∑ ∇∇=',

* ),',(),',()(σ

στ τσδφτσφδτi

ii rrr

[ ]∑ ×∇−='',',

* "'),'',(),',()(σσ

στ σσστσφτσδφδi

ii rrirJ

After carrying out the minimization of energy, we obtain the HF equations:

)(

)()(43)1()1(

)(21)(

)()(2

)()1()()(2

*

2

'*

2

2"

*

2

rR

rRrWr

lljj

rmdrd

rrU

rRrmdr

drRrllrR

rm

αα

ατ

αααα

ττ

ατ

ααα

ατ

ε=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡⎥⎦

⎤⎢⎣

⎡ −+−++⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎥⎦

⎤⎢⎣

⎡ ++−

where , , and are the effective mass, the potential and the spin orbit potential. They are given in terms of the Skyrme parameters and the nuclear densities.

)(* rmτ )(rUτ)(rWτ

)()21()

21(

41)()

211()

211(

41

2)(2 22112211

2

*

2

rxtxtrxtxtmrm

τ

ττ

ρρ ⎥⎦

⎤⎢⎣

⎡ +−+−⎥⎦

⎤⎢⎣

⎡ ++++=

( )

[ ] ,')'(

')()(21

)()21()

21(3

81)()

211()

211(3

81

)()21(

61)()()()

21(

12

)()211(

122)()

21()

21(

41

)()211()

211(

41)()

21()()

211()(

.20

22211

22211

33221

33

1332211

22110000

21,∫ −

+∇+∇−

∇⎥⎦

⎤⎢⎣

⎡ ++++∇⎥⎦

⎤⎢⎣

⎡ +−+−

+−++−

++

+⎥⎦

⎤⎢⎣

⎡ +−+−

⎥⎦

⎤⎢⎣

⎡ +++++−+=

−−

+

rrrrderJrJW

rxtxtrxtxt

rxtrrrxt

rxtrxtxt

rxtxtrxtrxtrU

ch

ρδ

ρρ

ρρρρρσ

ρα

τ

τρρ

ττ

τ

τα

ττα

ατ

ττ

[ ] )(][81)()(

81)()(

21)( 2211210 rJxtxtrJttrrWrW

−−−+∇+∇= τττ ρρ

With an initial guess of the single-particle wave functions (example; harmonic oscillator wave functions), we can determine m*, U(r), and W(r) and solve the HF equation to get a set of new single-particle wave functions; then one can proceed in this way until reaching convergence.

NOTES:

1.  One should start close to the solution.

2.  Accuracy and convergence in three dimension

3 Convergence of HFB equations in three dimension?

Infinite Nuclear Matter It’s important to note that the proton and neutron densities are constants.

So that

The EOS of asymmetric NM, with

with

Note that σ = α

Approximations

Coulomb energy

Note that here the direct term and exchange Coulomb terms each include the spurious self-interaction term.

)()(43)()(

21 rrVrrVH p

exCoulp

dirCoulCoulomb ρρ +=

∫ −=

'')'( 3

2

rrrdr

eV pdirCoul

ρ31

2 )(3⎟⎟⎠

⎞⎜⎜⎝

⎛−=

π

ρ reV pex

Coul

Interaction: KDE0, KDE0v1 neglect exchange term KDE, include exchange term

KDEX, include contributions of g.s. correlations

Determining the Skyrme interaction using the HF approach

Center of mass correction

a). Correction to the total binding energy:

We use the harmonic oscillator approximation. The CM energy is taken as

ω43

=oscCMK but ω Is determined by using the mass mean-square radii 2r

∑ ⎥⎦

⎤⎢⎣

⎡ +=i

iNrmA 23

2

2ω

b). Correction to the charge rms radii chr

The charge mean-square radius to be fitted to the experimental data is obtained as

ljnpHFpch ljnljmcZ

rZNr

Arr ∑ +⎟

⎠

⎞⎜⎝

⎛+++−= .)12(123 2222 σµτυ τ

Simulated Annealing Method (SAM)

We use the SAM to determine the values of the Skyrme parameters by searching the global minimum for the chi-square function

The SAM is a method for optimization problems of large scale, in particular, where a desired global extremum is hidden among many local extrema.

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

−=

dN

i i

thii

pd

MMNN 1

2exp2 1

σχ

Nd is the number of experimental data points.

Np is the number of parameters to be fitted.

and are the experimental and the corresponding theoretical values of the physical quantities.

is the adopted uncertainty.

expiM

is the adopted uncertainty.

thiM

iσ

Implementing the SAM to search the global minimum of function:

3. Calculate for a given set of experimental data and the corresponding

2χ

+ Use this modified vector to generate a new set of Skyrme parameters.

0,,, Wxt ii α are written in term of 1. ,...,,/ nmnmKAB ρ

),,,,,,/*,,,/( 0'0 WGLJEmmKABv snmnm κρ

.

2. Define

HF results (using an initial guess for Skyrme parameters).

4. Determine a new set of Skyrme parameters by the following steps:

+ Use a random number to select a component of vector rv

v

+ Use another random number to get a new value of rvη

ηdvv rr +→

2oldχ

v

5. Go back to HF and calculate 2newχ

6. The new set of Skyrme parameters is accepted only if

βχχ

χ >⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

TP newold

222 exp)(

10 << β

7. Starting with an initial value of , we repeat steps 4 - 6 for a large number of loops.

iTT =

8. Reduce the parameter T as and repeat steps 1 – 7.

9. Repeat this until hopefully reaching global minimum of

kTT i=

2χ

Fitted data - The binding energies for 14 nuclei ranging from normal to the exotic (proton or neutron) ones: 16O, 24O, 34Si, 40Ca, 48Ca, 48Ni, 56Ni, 68Ni, 78Ni, 88Sr, 90Zr, 100Sn, 132Sn, and 208Pb.

- Charge rms radii for 7 nuclei: 16O, 40Ca, 48Ca, 56Ni, 88Sr, 90Zr, 208Pb.

-  The spin-orbit splittings for 2p proton and neutron orbits for 56Ni ε(2p1/2) - ε(2p3/2) = 1.88 MeV (neutron) ε(2p1/2) - ε(2p3/2) = 1.83 MeV (proton).

- Rms radii for the valence neutron:

in the 1d5/2 orbit for 17O fmdrn 36.3)1( 2/5 =

in the 1f7/2 orbit for 41Ca fmfrn 99.3)1( 2/7 =

- The breathing mode energy for 4 nuclei: 90Zr (17.81 MeV), 116Sn (15.9 MeV), 144Sm (15.25 MeV), and 208Pb (14.18 MeV).

00 32 ρρρ <<cr1. The critical density

Constraints

( ) ( )212121'

2121' rrGGFFV

lllll

Landauhp

−+++=∑− δττσσσσττ

Landau stability condition: )12(,,, '' +−> lGGFF llll

Example: ( )( )3/1

12

61

022

FF

mkK F

++=

2. The Landau parameter '0G should be positive at 0ρρ =

3. The quantity ρ

ρddSP 3= must be positive for densities up to 03ρ

4. The IVGDR enhancement factor 5.025.0 << κ

)1(2

)(2

11 κ+=∫ == A

NZm

dEEESTL

Self-consistent calculation within constrained HF

5.0 150.0 100.0 120.0 W0 (MeV fm5)

0.10 0.40 0.00 0.08 G’0

0.1 0.5 0.1 0.25 Kappa

10.0 80.0 20.0 47.0 L (MeV)

4.0 40.0 25.0 32.0 J (MeV)

0.3 19.0 17.0 18.0 Es (MeV)

0.04 0.90 0.60 0.70 m*/m

0.005 0.170 0.150 0.160 ρnm (fm-3)

20.0 300.0 200.0 230.0 Knm (MeV)

0.4 15.0 17.0 16.0 B/A (MeV)

d v1 v0 v

Variation of the average value of

the control parameter T for the KDE0 interaction for the two different choices of the starting parameter.

T

2χ as a function of the inverse of

Parameter KDE0 KDE0v1 KDEX

t0 (MeV fm3) -2526.5110 -2553.0843 -1419.8304 t1 (MeV fm5) 430.9418 411.6963 309.1373

t2 (MeV fm5) -398.3775 -419.8712 -172.9562

t3(MeVfm3(1+α)) 14235.5193 14603.6069 10465.3523 x0 0.7583 0.6483 0.1474 x1 -0.3087 -0.3472 -0.0853 x2 -0.9495 -0.9268 -0.6144 x3 1.1445 0.9475 0.0220 W0(MeV fm5) 128.9649 124.4100 98.8973 α 0.1676 0.1673 0.4989 B/A (MeV) 16.11 16.23 15.96 K (MeV) 228.82 227.54 274.20 ρ0 (fm-3) 0.161 0.165 0.155 m*/m 0.72 0.74 0.81 J (MeV) 33.00 34.58 32.76 L (MeV) 45.22 54.69 63.70 κ 0.30 0.23 0.33 G'0 0.05 0.00 0.41

Values of the Skyrme parameters and the corresponding physical quantities of

nuclear matter for the KDE0 and KDE0v1 and KDEX interactions.

G. Audi et al, Nucl. P h y s . A 7 2 9 , 3 3 7 (2003)

-5.584 0.945 -1636.430 208Pb 1.752 -0.422 -1102.850 132Sn 0.180 -3.664 -824.800 100Sn 0.913 -0.127 -783.892 90Zr 2.985 0.826 -768.468 88Sr 2.597 -0.252 -641.940 78Ni 1.532 0.169 -590.408 68Ni 1.853 1.091 -483.991 56Ni 4.946 -1.437 -347.136 48Ni 2.529 0.188 -415.990 48Ca 0.699 0.005 -342.050 40Ca

2.868 -0.656 -283.427 34Si 4.582 -0.581 -168.384 24O 3.202 0.394 -127.620 16O

KDEX KDE0 Bexp Nuclei

ΔB = Bexp -Bth

Binding Energies (MeV)

E. W. Otten, in Treatise onn Heavy-Ion Science, Vol 8 (1989).

H. D. Vries et al, At. Data Nucl. Tables 36, 495 (1987).

F. Le Blanc et al, Phys. Rev. C 72, 034305 (2005).

5.499 5.489 5.500 208Pb

4.717 4.710 4.709 132Sn

4.261 4.266 4.258 90Zr

4.213 4.221 4.219 88Sr

3.848 3.768 3.75 56Ni

3.485 3.501 3.48 48Ca

3.456 3.490 3.49 40Ca

2.713 2.771 2.73 16O

KDEX KDE0 Experiment Nuclei

Charge RMS Radii (fm)

Single-Particle Energies (in MeV) for 40Ca

Orbits Expt. KDE0* Orbits Expt. KDE0*

1s1/2 -50+11 -38.21 1s1/2 - -47.771p3/2 - -26.42 1p3/2 - -34.901p1/2 -34+6 -22.34 1p1/2 - -30.78 1d5/2 - -14.51 1d5/2 - -22.08 2s1/2 -10.9 -9.66 2s1/2 -18.1 -17.00 1d3/2 -8.3 -7.53 1d3/2 -15.6 -14.97

1f7/2 -1.4 -2.76 1f7/2 -8.3 -9.60 2p3/2 -6.2 -4.98

Protons Neutrons

*TAMU Skyrme Interaction: B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 72, 014310 (2005).

GIANT RESONANCES

•  Hadron Scattering

•  HF-Based RPA

•  Results

Equation of state and nuclear matter compressibility

2

181][][ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+=

o

oo KEE

ρρρ

ρρ

The symmetric nuclear matter (N=Z and no Coulomb) incompressibility coefficient, K, is a important physical quantity in the study of nuclei, supernova collapse, neutron stars, and heavy-ion collisions, since it is directly related to the curvature of the nuclear matter (NM) equation of state (EOS), E = E(ρ).

ofod

AEddk

AEdkKkf

fρ

ρρ 2

22

2

22 )/(9)/(

==

ρ [fm-3]

ρ = 0.16 fm-3

E/A

[MeV

]

E/A = -16 MeV

2

(()((

181]([]([ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

)

)−)+)=)

βρβρρ

βρβρβρo

ooooANM KEE

2][]([ βρβρ JEE oo +=)

2v]([ ββρ τKKK o +=)

AZN /)( −=β

][ oSYMEJ ρ=

AZy /=

2/1,2

2 )/(81)(

=

=y

SYMdy

AEdEρ

ρ

The total photoabsorption cross-section for 197Au, illustrating the absorption of photons on a giant resonating electric dipole state. The solid curve show a Breit-Wigner shape. (Bohr and Mottelson, Nuclear Structure, vol. 2, 1975).

The isovector giant dipole resonance

Macroscopic picture of giant resonance

L = 0 L = 1 L = 2

Theorists: calculate transition strength S(E) within HF-RPA using a simple scattering operator F ~ rLYLM:

Experimentalists: calculate cross sections within Distorted Wave Born Approximation (DWBA):

or using folding model.

Hadron excitation of giant resonances

Nucleus α

χi

χf

Ψi

Ψf

VαN

DWBA-Folding model description

EWSR = energy weighted sum rule ⎥⎦

⎤⎢⎣

⎡∫∞

0

)( dEEES

Elastic angular distributions for 240 MeV alpha particle. Filled squares represent the experimental data. Solid lines are fit to the experimental data using the folding model DWBA with nucleon-alpha interaction.

Since α particles have S = 0, T = 0, they are ideal for studying electric (∆S = 0) and isoscalar (∆T = 0) Giant Resonances.

History A.  ISOSCALAR GIANT MONOPOLE RESONANCE (ISGMR):

1977 – DISCOVERY OF THE CENTROID ENERGY OF THE ISGMR IN 208Pb

E0 ~ 13.5 MeV (TAMU) •  This led to modification of commonly used effective nucleon-nucleon interactions.

Hartree-Fock (HF) plus Random Phase Approximation (RPA) calculations, with effective interactions (Skyrme and others) which reproduce data on masses, radii and the ISGMR energies have:

K∞ = 210 ± 20 MeV (J.P. BLAIZOT, 1980).

A.  ISOSCALAR GIANT DIPOLE RESONANCE (ISGDR):

1980 – EXPERIMENTAL CENTROID ENERGY IN 208Pb AT

E1 ~ 21.3 MeV (Jülich), PRL 45 (1980) 337; ~ 19 MeV, PRC 63 (2001) 031301

•  HF-RPA with interactions reproducing E0 predicted E1 ~ 25 MeV.

K∞ ~ 170 MeV from ISGDR ?

T.S. Dimitrescu and F.E. Serr [PRC 27 (1983) 211] pointed out “If further measurement confirm the value of 21.3 MeV for this mode, the discrepancy may be significant”.

→ Relativistic mean field (RMF) plus RPA with NL3 interaction predict K∞=270 MeV from the ISGMR [N. Van Giai et al., NPA 687 (2001) 449].

Hartree-Fock (HF) - Random Phase Approximation (RPA)

4. Carry out RPA calculations of strength function, transition density etc.

In fully self-consistent calculations:

1. Assume a form for the Skyrme parametrization (δ-type).

2. Carry out HF calculations for ground states and determine the Skyrme parameters by a fit to binding energies and radii.

3. Determine the residual p-h interaction

In the Green’s Function formulation of RPA, one starts with the RPA-Green’s function which is given by

1)1( −+= opho GVGG

where Vph is the particle-hole interaction and the free particle-hole Green’s function is defined as

)'(11)(*),',( rrrr iioioi

io EhEhEG ϕ

εεϕ ⎥

⎦

⎤⎢⎣

⎡

+−+

−−−= ∑

where φi is the single-particle wave function, єi is the single-particle energy, and ho is the single-particle Hamiltonian.

Green’s Function Formulation of RPA

The continuum effects, such as particle escape width, can be taken into account using

1 2 20

1 2 ( ) ( ) /mr r U r V r Wh Z 〈 〉=− h

where r< and r> are the lesser and greater of r1 and r2 respectively, U and V are the regular and irregular solution of (H0-Z)ψ = 0, with the appropriate boundary conditions, and W is the Wronskian.

NOTE the two terms in the free particle-hole greens

function

We use the scattering operator F )(1∑=

=A

iifF r

to obtain the strength function

)](Im[1)(0)(2

fGfTrEEnFESn

n ⋅⋅=−=∑ πδ

and the transition density

')],',(Im1[)'()(

),( 3rrrrr dEGfEES

EEtRPA ∫ ⋅⋅

Δ⋅

Δ==

πρδρ

is consistent with the strength in RPAδρ 2/EE Δ±

ErdrfErES RPA Δ= ∫2

)(),()( δρ

The steps involved in the relativistic mean field based RPA calculations are analogous to those for the non-relativistic HF-RPA approach. The nucleon-nucleon interaction is generated through the exchange of various effective mesons. An effective Lagrangian which represents a system of interacting nucleons looks like

It contains nucleons (ψ) with mass M; σ, ω, ρ mesons; the electromagnetic field; non linear self-interactions for the σ (and possibly ω) field.

Values of the parameters for the most widely used NL3 interaction are mσ=508.194 MeV, mω=782.501 MeV, mρ=763.000 MeV, gσ=10.217, gω=12.868, gρ=4.474, g2=-10.431 fm-1 and g3=-28.885 (in this case there is no self-interaction for the ω meson).

NL3: K∞=271.76 MeV, G.A.Lalazissis et al., PRC 55 (1997) 540.

RMF-RPA: J. Piekarewicz PRC 62 (2000) 051304; Z.Y. Ma et al., NPA 686 (2001) 173.

Relativistic Mean Field + Random Phase Approximation

Self-consistent calculation within constrained HF

Isoscalar strength functions of 208Pb for L = 0 - 3 multipolarities are displayed. The SC (full line) corresponds to the fully self-consistent calculation where LS (dashed line) and CO (open circle) represent the calculations without the ph spin-orbit and Coulomb interaction in the RPA, respectively. The Skyrme interaction SGII [Phys. Lett. B 106, 379 (1981)] was used.

Isovector strength functions of 208Pb for L = 0 - 3 multipolarities are displayed. SC (full line) corresponds to the fully self-consistent calculation where LS (dashed line) and CO (open circle) represent the calculations without the ph spin-orbit and Coulomb interaction in the RPA, respectively. The Skyrme interaction SGII [Phys. Lett. B 106, 379 (1981)] was used.

E (MeV)

S(E

) (fm

4 /MeV

)

Isoscalar monopole strength function

90Zr

116Sn

144Sm

208Pb

S. Shlomo and A.I. Sanzhur, Phys. Rev. C 65, 044310 (2002) ISGDR MYrrrf 1

23

35

⎟⎠

⎞⎜⎝

⎛ −=

SL1 interaction, K = 230 MeV, Eα = 240 MeV

)(35310 0

022 rdrdrrrcoll ρρ

ρ ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −+=

Reconstruction of the ISGDR EWSR in 116Sn from the inelastic α-particle cross sections. The middle panel: maximum double differential cross section obtained from ρt (RPA). The lower panel: maximum cross section obtained with ρcoll (dashed line) and ρt (solid line) normalized to 100% of the EWSR. Upper panel: The solid line (calculated using RPA) and the dashed line are the ratios of the middle panel curve with the solid and dashed lines of the lower panel, respectively.

33.0 26.8 37.4 37.4 J (MeV) 229 215 255 272 K (MeV)

13.8 13.6 14.4 13.96±0.30 10-35

13.8 13.6 14.3 14.2 0-60 208Pb

15.5 15.2 16.2 15.40±0.40 10-35

15.5 15.3 16.2 16.1 0-60 144Sm

16.6 16.4 17.3 15.85±0.20 10-35

16.6 16.4 17.3 17.1 0-60 116Sn 18.0 17.9 18.9 17.81±0.30 10-35

18.0 17.9 18.9 18.7 0-60 90Zr KDE0 SGII SK255 NL3 Expt. ω1-ω2 Nucleus

Fully self-consistent HF-RPA results for ISGMR centroid energy (in MeV) with the Skyrme interaction SK255, SGII and KDE0 are compared with the RRPA results using the NL3 interaction. Note the corresponding values of the nuclear matter incompressibility, K, and the symmetry energy , J, coefficients. ω1-ω2 is the range of excitation energy. The experimental data are from TAMU.

18 19 20 21 22 23 24

40Ca E C

EN  (M

eV)  

18

19

20

21

22

23

E CEN

(MeV

)

48Ca

-2

-1

0

1

2 48Ca - 40Ca

-2

-1

0

1

2

200 210 220 230 240 250 260

48Ca - 40Ca

ΔE C

EN (M

eV)

ΔE C

EN (M

eV)

KNM (MeV)

ISGMR (T0 E0)

48Ca – 40Ca > 0 for all the interactions which goes against the trend of decreasing ISGMR with increasing A

Note that for not self-consistent RPA calculations, which neglect the Coulomb and Spin-Orbit parts. Some interactions would fall in the correct 48Ca – 40Ca range.

None of the interactions fall in the Experimental range for 40Ca

€

ECEN =ES(E)dE∫S(E)dE∫

C = 0.95

C = 0.85

C is the Pearson correlation coefficient

C = 0.13

22

24

26

28

30

32

34

36 40Ca

E CEN  (M

eV)  

ISGDR (T0 E1)

27 28 29 30 31 32 33 34 35

E CEN

(MeV

)

48Ca

-1

0

1

2

3

4

5

200 210 220 230 240 250 260

48Ca - 40Ca

ΔE C

EN (M

eV)

KNM (MeV)

22

24

26

28

30

32

34

36 40Ca

E CEN  (M

eV)  

ISGDR (T0 E1)

27 28 29 30 31 32 33 34 35

E CEN

(MeV

) 48Ca

-1

0

1

2

3

4

5

0.5 0.6 0.7 0.8 0.9 1.0 1.1

48Ca - 40Ca

ΔE C

EN (M

eV)

m*/m

C = 0.88

C = 0.82

C = -0.86

C = -0.94

C = 0.32 C = -0.71

15

16

17

18

19

20

21 40Ca

E CEN  (M

eV)  

ISGQR (T0 E2)

15

16

17

18

19

20

21

E CEN

(MeV

) 48Ca

-1

0

1

2

0.5 0.6 0.7 0.8 0.9 1.0 1.1

48Ca - 40Ca

ΔE C

EN (M

eV)

m*/m

C = -0.96

C = -0.98

C = -0.56

17

18

19

20

21 40Ca

E CEN  (M

eV)  

IVGDR (T1 E1)

17

18

19

20

21

E CEN

(MeV

)

48Ca

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

26 27 28 29 30 31 32 33 34 35 36 37 38

48Ca - 40Ca

ΔE C

EN (M

eV)

J (MeV)

No clear value of the symmetry energy, J can be deduced.

C = -0.27

C = -0.32

C = -0.20

17

18

19

20

21 40Ca

E CEN  (M

eV)  

IVGDR (T1 E1)

17

18

19

20

21

E CEN

(MeV

) 48Ca

-1.5

-1.0

-0.5

0.0

0.5

1.0

30 40 50 60 70 80 90 100 110 120 130

48Ca - 40Ca Δ

E CEN

(MeV

)

L (MeV)

17

18

19

20

21 40Ca

E CEN  (M

eV)  

IVGDR (T1 E1)

17

18

19

20

21

E CEN

(MeV

)

48Ca

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-270 -220 -170 -120 -70 -20 30 80 130 180

48Ca - 40Ca

ΔE C

EN (M

eV)

Ksym(MeV)

C = -0.21

C = 0.24

C = -0.32

C = 0.11

C = -0.24

C = -0.30

18

19

20

21

22

23

24

25

0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

αD  (fm

3 )  

rn-­‐rp  (fm)  

208Pb  CAB=0.55  

Conclusions •  We have developed a new EDFs based on Skyrme type interaction

(KDE0, KDE, KDE0v1,... ) applicable to properties of rare nuclei and neutron stars. •  Fully self-consistent calculations of the compression modes

(ISGMR and ISGDR) within HF-based RPA using Skyrme forces and within relativistic model lead a nuclear matter incompressibility coefficient of K∞ = 240 ± 20 MeV, sensitivity to symmetry energy.

•  Sensetivity to symmetry energy: IVGDR, GR in neutron rich nuclei, Rn – Rp, stll open problems.

•  Possible improvements: –  Account for effect of correlations on B.E. Radii, S.P. energies –  Properly account for the isospin dependency of the

spin-orbit interaction –  Include additional data, such as IVGDR (J) and ISGQR (m*)

References

[1] A. Bohr and B. Mottelson, Nuclear Structure, Vol. II, Benjamin, London (1975).

[2] A. deShalit and H. Feshbach, Theoretical Nuclear Physics, Vol. I: Nuclear Structure,

[3] D. J. Rowe Nuclear Collective Motion Models and Theory, Methuen and Co. Ltd.

[4] P. Ring and P. Schuck, The nuclear many-body problems, Springer, New York-

[5] G. F. Bertsch and R. A. Broglia, Oscilations In Finite Quantum Systems, Cambridge

[6] G. R. Satchler, Direct Nuclear Reactions, Oxford University Press, Oxford (1983).

[7] S. Shlomo and G. F Bertsch, Nucl. Phys. A243, 507 (1975).

John Wiley & Sons, Inc. New York (1974).

(1970).

Heidelerg-Berlin (1980).

University Press (1994).

[8] S. Shlomo and D. H. Youngblood, Phys. Rev. C 47, 529 (1993).

[9] A. Kolomiets, O. Pochivalov and S. Shlomo, Phys. Rev. C 61, 034312 (2000).

[10] S. Shlomo and A. I. Sanzhur, Phys. Rec. C 65, 044310 (2002).

[11] B. K. Agrawal, S. Shlomo and A. I. Sanzhur, Phys. Rev. C 67, 0343314 (2003).

[12] B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 68, 031304(R) (2003).

[13] B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 70, 057302 (2004).

[14] N. K. Glendenning, Phys. Rev. C 37, 2733 (1988).

[15] B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 72, 014310 (2005).

[16] S. Shlomo, V. M. Kolomietz and G. Colo, Eur. Phys. A30, 23 (2006).

Acknowledgments

Work done at:

Work supported by:

Grant number: PHY-0355200

Grant number: DOE-FG03-93ER40773

CONCLUSION We have developed a new EDF based on Skyrme type interaction (KDE0) applicable to properties of rare nuclei and neutron stars.

Fully self-consistent calculations of the ISGMR using Skyrme forces lead a nuclear matter incompressibility coefficient of K∞ = 240 ± 20 MeV with sensitivity to symmetry energy.

It is possible to build bonafide Skyrme forces with K close to the relativistic value.

Further improvement

(i) Account for the effect of correlation on B.E., Rch and single particle energy

(ii) Properly account for the isospin dependency of the spin-orbit interaction

(iii) Include additional data, such as IVGDR (J) and ISGQR (m*)

SUMMARY AND CONCLUSIONS

1)  Fully  self-­‐consistent  calcula2ons  of  the  compression  modes  (ISGMR  and    ISGDR)  

using  modern  energy  density  func2onals  (Skyrme  forces)  lead  to        →  K∞  =  240  ±  

20  MeV,  with  sensi2vity  to  symmetry  energy.    

                 Symmetry  energy  density  (IVGDR,  Rn  –  Rp,  ...)    -­‐-­‐Open  problem  

2  )    Accoun2ng  for  post-­‐emission  decay  allows  one  to  obtain  consistent  values  of  

temperature  of  a  disassembling  source  from  the  “double-­‐ra2o”  method.  

3)      Although  ,  at  low  densi2es,  the  temperature  calculated  from  given  yields  changes  

only  modestly  if  medium  effects  are  taken  into  account,  larger  discrepancies  are  

observed  when  the  nucleon  densi2es  are  determined  from  measured  yields,    

4)      Due  to    clusteriza2on  at  low  density  nuclear  maWer,  the  symmetry  energy  is  much  

larger  than  that  predicted  by  mean  field  approxima2on  

Outline 1.  Introduction Definitions: nuclear matter incompressibility coefficient K∞

Background: isoscalar giant monopole resonance,

isoscalar giant dipole resonance

Hadron excitation of giant resonances

1.  Theoretical approaches for giant resonances Hartree-Fock plus Random Phase Approximation (RPA)

Comments: self-consistency ?

Relativistic mean field (RMF) plus RPA

1.  Discussion ISGMR vs. ISGDR

Non-relativistic viz. Relativistic

Nuclear matter properties from collective modes in nuclei

Shalom Shlomo

Cyclotron Institute

Texas A&M University

New Skyrme effective nucleon-nucleon interaction

Shalom Shlomo

Cyclotron Institute, Texas A&M University

Effects of self-consistence violations in HF-based RPA

calculations for giant resonances

Shalom Shlomo

Texas A&M University

Nuclear matter equation of state and giant resonances in nuclei

Shalom Shlomo

Texas A&M University

The effective Skyrme interaction has been used in mean-field models for several decades and many different parameterizations of the interaction have been realized to better reproduce nuclear masses, radii, and various other data. Today, there is more experimental data on nuclei far from the stability line. It is time to improve the parameters of Skyrme interactions. We fit our mean-field results to an extensive set of experimental data and obtain the parameters of the Skyrme type effective interaction for nuclei at and far from the stability line.

Introduction

The total energy

[ ]∫ ++=Φ++Φ=ΦΦ= rdrHrHrHVVTHE SkyrmeCoulombKineticCoulombtotal )()()(ˆ

12

Where )(2

)(2

)(22

rm

rm

rH nn

pp

Kinetic

ττ +=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

−= ∫∫ '

')',(

'')'()(

2)(

22

rdrrrr

rdrrrrerH chch

chCoulomb

ρρρ

[ ]

( )

( ) [ ]( ) [ ]

[ ] ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ ++−⎟⎠

⎞⎜⎝

⎛ ++∇+∇+∇−

+−+−+∇+∇⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ ++⎟⎠

⎞⎜⎝

⎛ ++

∇⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ +−+⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ +−

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ ++⎟⎠

⎞⎜⎝

⎛ +++⎟⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ +=

+

21)()()(

211)(

121))()()()()()(

21

)(161)()(

161)()()()(

211

2113

161

)()(211

2113

161)()()()(

211

211

41

)()(211

211

41)()(

211

21)(

211

21)(

322

32

30

22211

2221

222211

222112211

221122

002

00

xrrrxrtrJrrJrrJrW

rJxtxtrJrJttrrrrxtxt

rrxtxtrrrrxtxt

rrxtxtrrxtrxtrH

npnnpp

npnnpp

nnpp

npSkyrme

ρρρρρρρ

ρρρρ

ρρτρτρ

τρρρρ

αα

MODERN ENERGY DENSITY FUNCTIONAL FOR NUCLEI AND THE NUCLEAR MATTER

EQUATION OF STATE

Shalom Shlomo

Cyclotron Institute

Texas A&M University