C.A. Bertulani

Department of Physics and Astronomy Texas A&M University-Commerce

Collaborator: Paolo Avogadro

Nuclear Matter Incompressibility and Giant Monopole Resonances

1 27th Texas Symposium on Relativistic Astrophysics - Dallas, TX - Dec 8-13, 2013

56 Fe→134 He+ 4n p+ e− → n + νe

Iron core in core-collapse supernovae

The iron core grows and becomes dynamically unstable as it approaches the critical mass. The main reactions are

Needs energy (124 MeV) Neutrino escapes

Pressure decreases and the core collapses

2

The Equation of State (EoS) has to cover densities of 109 – 1015 g/cm3

Nuclei are bound into nuclei until they merge (ρ0 ~ 1014 g/cm3) Pressure is essentially dominated by e- (γ = ∂lnP/∂lnρ = 4/3) For ρ > ρ0, nuclear matter becomes very hard (γ ~ 2)

Below nuclear density: - Matter is described by nuclei surrounded by a “gas” of nucleons and alpha particles -  Electrons are uniformly distributed in space, inside and outside nuclei.

-  In equilibrium

-  At high densities nuclear matter fills the space uniformly

-  Techniques for computing EOS: liquid drop, Thomas-Fermi, Hartree-Fock

-  EOS is obtained by minimizing the free energy of the system for each entropy and Ye

µe = µn −µ p

EOS

3

!

0.16 fm-3

-16 MeV

Above nuclear density: - Maximum density occurs when the inner central core reaches ρ ~ 5ρ0 -  the compressibility has to be enough to allow the existence of neutron stars

with M ~ 1.4 M¤ or larger.

EOS

4

E ρ"# $%= E ρ0"# $%+118K

∞

ρ−ρ0ρ0

(

)**

+

,--

2

K∞= 9ρ2

d2 E(ρ) /ρ#$ %&dρ2

ρ0

- Analysis of Giant Isoscalar Resonances

-  Relativistic heavy ion collisions

-  Neutron masses Giant Resonance: Coherent vibration of

nucleons in a nucleus -  Resonances related to incompressibility:

ISGMR, ISGDR, ISGQR

EOS – Experimental Observations

5

EISGMR ≈KA

m r2

KA = K∞1+ cA−1/3( )+K τ

N −ZA

$

%&

'

()

2

+KCoulZ2A−4/3

c ≈1

ISGMR (T=0, L=0)

ISGDR (T=0, L=1)

ISGQR (T=0, L=2)

-  KCoul is basically model independent -  Measurements over several isotopes should give Kτ -  Kτ critical to understand neutron stars

EOS – Theoretical Methods

6

-  Build an energy functional E[ρ] using an mean field calculation Each such a functional characterizes a K∞

-  Get excitations such as the ISGMR from a self-consistent QRPA calculation For the nucleon-nucleon interaction

V(ri ,rj ) =VijNN +Vij

CoulVijCoul = −

e2

4

τ ij2 + τ ijri − rji, j=1

A

∑ ,jiij τττ +=

Vij

NN = t0 (1+ x0Pijσ )δ(ri − rj )+

12t1(1+ x1Pij

σ )[!k ij2δ(ri − rj )+δ(ri − rj )

!k ij2 ]+

t2 (1+ x2Pijσ )!k ijδ(ri − rjj )

!k ij +

16t3(1+ x3Pij

σ )ραri + rj2

&

'((

)

*++δ(ri − rj )+

iW0

!k ijδ(ri − rj )(

!σ i +!σ j )!k ij,

0,,, Wxt ii α are 10 Skyrme parameters

E[ρ]= Φ T+VijCoul +Vij

NN Φ

+ pairing

vNNeff = Skyrme + pairing force

HF + BCS

€

Δ i =12

GijΔ j

ε j − λ( )2 + Δ j2j

∑

HFB

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

+−Δ−

Δ−

k

kk

k

k

HF

HF

vu

Evu

hh

λ

λ

€

V =V0 1−ηρ r( )ρ0

%

& '

(

) *

α,

- . .

/

0 1 1 δ r1 − r2( ), ρ0 = 0.16 fm, α =1

η =

0, "volume"pairing1, "surface"pairing1/2, "mixed"pairing

3

4 5

6 5

7

EOS + Pairing

8

v r,r '( ) = v0 1−ηρρ0

$

%&&

'

())

γ+

,

--

.

/

00δ r − r '( )

Skyrme K∞ SLy5 230 SkM* 216 Skxs20 202

Protons and neutrons tend to pair up, much like Cooper pairs of electrons in superconductors. Pairing is important in nuclei and neutron stars, but a clear understanding of the microscopic foundation of the pairing functional is still lacking.

Volume pairing

Mixed pairing

Surface pairing

Blaizot Phys. Rep. 64, 171 (1980) Shlomo,Youngblood PRC 47, 529 (1993)

From experiment: •  Δ(3) larger for (-1)N = +1 •  Δ(3) smaller for (-1)N = -1 •  Δ(4) reflects average of Δ(3) (N) and Δ(3) (N-1) (Δ(4) no additional information)

Δ(3) =12−1( )

NB(N −1)+B(N+1)− 2B(N)#$ %&

Δ(4) =14−1( )

N3B(N −1)−3B(N)−B(N − 2)+B(N+1)#$ %&four-point

or higher ?

Pairing Measure

three-point

9

experiment

€

Δ o(3) =

12A1/ 3

MeV

rms for binding energies

rms for separation energies

Can Microscopic Models do better than LDM?

rms = 0.3 MeV

Blocking Procedure for Odd Nucleons

EF

v2 → ½, u → 0 for one orbit and its time-conjugate partner

)(kVpairing

EF Λ

Λ ~ 50 MeV

11

12

Pairing Improves Nuclear Properties

Separation energies staggering

Sn

12

Pairing vs Level Properties

( ) [ ])(2)1()1(121)3( NBNBNBN −++−−=Δ ( )

( )λλ

gNNBN 112 2

2)3( =

∂

∂=

∂

∂≅Δ−

Fermi energy (λ= ∂B/ ∂N) s.p. level density (g(e)= dN/ ∂e)

Jahn-Teller mechanism: spherical symmetry spontaneously broken (2j+1) ! double-degenerate orbits Δ(3) alternates for (-1)N = + and -

λ

degenerate shell N

λ does not vary with N ! Δ(3) = 0

λ

valence shell full N=2n en

en+1

(-1)N = +1, dλ/dN ≈ en+1 – en ! Δ(3) = (en+1 – en )/2

measure of gap in single-particle spectrum

13

14

Pairing vs Level Properties ∑=

=+−=A

kkspmacrospsp eEEEEB

1

,~Macroscopic-microscopic model:

( ) MeV23,2

=−

+= IImacro aAZNaE ! MeV23)3(

A=Δ

€

˜ E sp contribution : g λ( ) =3aπ 2 , a ≅ A

8MeV

€

Δ(3) ≅ −1

g λ( )≅ −

25AMeV

€

Δ(3) N( ) ≅ − 12δe

+ LDM corrections

€

2 Δ(3) even( ) −Δ(3) odd( )[ ]≅ en+1 − en

Satula, Dobaczewski, Nazarewicz, PRL 81, 3599 (1998).

14

The uNclear Nuclear Pairing – UNEDF Collaboration •  Mass tables for 2,400 nuclei have

been analyzed using different forms and methodologies for the pairing mechanism

• New functional forms for the pairing interaction have been proposed

15

experiment UNEDF theory

( ) [ ])(2)1()1(121)3( NBNBNBN −++−−=Δ

Bertsch, Bertulani, Nazarewicz, Schunck, Stoitsov, PRC 79, 0343306 (2009)

QRPA: The Role of the Rearrangement Term

•   Fully self consistent EWSR = 99.2%

•  Without Rearrangement in EWSR =116%

16

h =δEkinδρ

+δEskyrmeδρ

+δEpairδρ

+δECoulδρ

δhrearrδρ

=δδρ

δEpairδρ

#

$%%

&

'((

Calculations without rearrangements tend to return higher centroids respect to the fully self-consistent case.

Avogadro, Bertulani, PRC 88, 044319 (2013)

≠ 0 if Epair depends on density

Dependence on Functional

17

Skxs20 very well

SLy5 bad results

mk = Ek S(E) dE0

∞

∫

Skm* well!

S(E) = 0 F0 j2δ E−E0( )

j∑

F0 = ri2

i=1

A

∑

Isovector pairing

18

vpairMSH (r,r ') = v0 1− 1−δ( )ηs

ρρ0

%

&''

(

)**

αs

−δηnρρ0

%

&''

(

)**

αn,

-

.

.

/

0

11δ(r,r ')

vpairMSH (r,r ') = v0 1− η+η1τ3δ( ) ρ

ρ0−η2 δ

ρρ0

&

'((

)

*++

2,

-

.

.

/

0

11δ(r,r ')

ρ =ρn +ρp

δ =ρn −ρpρ

Margueron, Sagawa, Hagino, PRC 76, 064316 (2007)

Yamagami, Shimizu, Nakatsukasa, PRC 80, 064301 (2009)

v r,r '( ) = v0 1−ηρρ0

$

%&&

'

())

γ+

,

--

.

/

00δ r − r '( )

Isovector pairing – Good globabl fits to pairing gaps

19 Bertulani, Liu,, Sagawa, PRC 85, 014321 (2012)

Isovector pairing – Reasonable Nuclear Radii

20 Bertulani, Liu,, Sagawa, PRC 85, 014321 (2012)

MSH IS+IV pairing

12C

Hofstadter, 1953

PWBA

Radii and Skins

dσdΩ

=σM θ( )Z2

Fcharge q( )2

21

Δrnp

Measuring Rn in 208Pb constrains the pressure of neutron matter at ~ 2/3ρ0 = 0.1 fm-3. C. Horowitz

Neutron stars

Isovector pairing – Reasonable Nuclear Skins

22 Bertulani, Liu,, Sagawa, PRC 85, 014321 (2012)

MSH IS+IV pairing

Radii from spin-dipole resonances Krasznahorkay et al., PRL 82, 3216 (1999) & Antiprotonic atoms Trzcinskaet al., PRL 87, 082501 (2001)

Isovector pairing – Improves Centroids of ISGMR

23

Isovector pairing – ISGMR – Comparison to Recent Data

24 Avogadro, Bertulani, PRC 88, 044319 (2013)

Conclusions

25

EOS & Pairing Conundrum

•  Nuclear pairing evidently improves masses, separation energies and staggering effects in microscopic approaches: HF + BCS or HFB.

•  Inclusion of pairing complicates determination of best Skyrme models. E.g., 20% of the nuclei well explained with the SLy5 interaction, and 10% with the SkM* interaction.

•  Isovector pairing improves nuclear properties ß at the cost of additional parameters

•  Detailed studies using HFB + QRPA + isovector pairing with comparison with newest data on ISGMR (RCNP and TAMU) à ISGMR is better reproduced with the soft interaction Skxs20 (K∞ ≈ 202 MeV), in contrast with the generally accepted value for K∞ ≈ 230 MeV.