Modeling of Clusterized Nuclear Matter under

Stellar ConditionsIgor Mishustin

with T. Buervenich, C. Ebel, U Heinzmann, S. Schramm, W. Greiner

FIAS, J. W. Goethe University, Frankfurt am Main, Germanyand

National Research Center “Kurchatov Institute”, Moscow, Russia

NUFRA2015, Kemer, October 4-11, 2015NUFRA2015, Kemer, October 4-11, 2015

ContentsContents

Introduction: Macro- and Micro-Superovae Uncertainties in Nuclear Statistical Ensemble Nuclear structure calculations in stellar

environments Pasta phases with realistic electrostatic

potentials Conclusions

A.S. Botvina, I.N. Mishustin, Phys. Lett. B584, 233 (2004); Phys. Rev. C72, 048801 (2005}; Nucl. Phys. A 843, 98-132 (2010);

A. Botvina and I. Mishustin, Phys. Rev. C72 (2005) 048801;

Th. Buervenich, I.N. Mishustin, W. Greiner, Phys. Rev. C 76, 034310 (2007); + Ebel, paper in preparation

3} N. Buyukcizmecl, A.S. Botvina, I.N. Mishustin, R.Ogul, M. Hempel, J. Schaffner-Bielich, F.K. Thielemann, S.Furusawa, K.Sumiyoshi, S. Yamada, H. Suzuki, Nucl. Phys. A (2013)

N. Buyukcizmecl, A.S. Botvina, I.N. Mishustin, Astrophys.J. 789 (2014) 33

Claudio Ebel, Igor Mishustin, Walter Greiner, J.Phys. G42 (2015) 10, 105201

U. Heinzmann, I. Mishustin, S. Schramm, paper in preparation

Relevant publicationsRelevant publications

Macro-Supernovae: Numerical Macro-Supernovae: Numerical simulations simulations

H.-T. Janka, K. Kifonidis, M. Rampp Lect.Notes Phys.578:333-363,2001

t~230 ms

~70 km~300km

~150km

Sketch of the post-collapse stellar core during the neutrino heating and shock revival phase

hot bubble

Creation of Micro-Supernovae in laboratory

THeating : T~10 MeV <0, P~0slow expansion

t>100 fm/c

PeripheralAA collision(ALADIN)

Multifragmentation – creation of micro-supernovae in laboratory,Power-law mass distributions: (liquid-gas phase tr.) Can be well understood within the equilibrium statistical approachdeveloped in 80s (following ideas of N. Bohr and E. Fermi):

Randrup&Koonin, D.H.E. Gross et al, Bondorf-Mishustin-Botvina, Hahn&Stoecker,...

Expanding equilibrated source made of hot nuclei

( ) , 2Y A A

P or T spectatorP or T spectatort=0 fm/c

P

Similarity of conditions in nuclear reactions and supernova explosions I. Thermal multifragmentation of

nuclei: • Production of hot fragments at T≈ 3-8

MeV ≈ (0.1 -0.3)0

• A chance to investigate properties of hot fragments in dense environment, which may differ from their ground state properties.

II. Collapse of massive stars leading to Supernova II explosions:

• Production of hot nuclei in stellar matter at T≈ 1-10 MeV 0.3 0

• Characteristic times (milliseconds) are very long for nuclear statistical equilibrium to be established.

• Properties of hot clustered nuclear matter may influence essentially the dynamics of collapse and explosions.

nucl

ear

liqui

dnucleon gas

PHASE DIAGRAMof nuclear matter

S/B=1

246

PB

CB

BB

0

10-4 10-3 10-2 10-1 100

T(M

eV)

1

10

coexistence lineisentropic trajectories (SMSM)coexistence line (TM1)BB, CB and PB

multifragmentation

Multifragmentation reactions and properties of hot stellar matter at sub-nuclear densities Botvina&Mishustin, PRC72 (2005) 048801

Statistical description of stellar Statistical description of stellar mattermatter

i i B i Q i LB Q L

AZFor nuclear species ( , ) : B QA Z A Z

-eelectrons : Q L e

e

neutrinos : L

( , )

( , )

Baryon number conservation :

fixed

Electric neutrality

0

B AZA Z

Q AZ eA Z

BA n

V

QZ n n

V

B

e

Lepton number conservation

(trapped ν) (no or = )eL e

B B

L n n nY Y

B

Q

( )

Grand-canonical Ensemble

with fixed , ,B L eT Y

calculations are done in a box containing 1000 baryons,density of individual fragments is fixed at

-30ρ =0.16 fm

Nuclear statitical Nuclear statitical ensemblesensembles

Grand Canonical version of SMM: Botvina, Mishustin et al., Sov. J. Nucl. Phys. 42 (1985) 712)

3/2

fAZ 3

Number density of nuclear species ( , ) :

V A 1=g exp -

V TAZ AZ AZT

A Z

n F

nuc Coul

( , )

Pressure = electrons

nuc AZA Z

P P P

P T n

5/ 42 2 2 2B 2 / 3AZ 0 0 2 2

0

Internal free energy of species ( , ) for 4

( 2 )F

liquid drop parametrizat

( ) , ,

Reduced Coulomb energy

ion

d

B S sym CAZ AZ AZ AZ AZ

S symcAZ AZ

c

A Z A

F F F F F

T T T A ZT w A F A F

T T A

1/ 32

1/ 30 0 0

ue to the electron screening

3 ( ) 3 1( ) ( ) , ( ) 1

5 2 2C e eAZ e e e

p p

eZ n nF n c n c n

r A n n

B

3Q LConsider box with 10 nucleons, μ , μ , μ are found iteratively

<1

Mass distributions : SMSM, FYSS and HS (/0=10-1)(from talk of Nihal Buyukcizmeci)

Various versions of the statistical model using “reasonable” assumptions on nuclear properties give very different results!

T=0.5 MeVYe=0.2

X Data

50 60 70 80 90 100

iso

top

e y

ield

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

T=0.5 MeVYe=0.4

X Data

50 60 70 80 90 100

SMSMHSFYSS

T=1 MeVYe=0.2

X Data

50 60 70 80 90 100

iso

top

e y

ield

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

T=1 MeVYe=0.4

X Data

50 60 70 80 90 100

T=2 MeVYe=0.2

A50 60 70 80 90 100

iso

top

e y

ield

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

T=2 MeVYe=0.4

A50 60 70 80 90 100

10-3

Z=26 T=1 MeVYe=0.2

X Data

50 60 70 80 90 100

iso

top

e y

ield

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

T=1 MeVYe=0.4

X Data

50 60 70 80 90 100

SMSMHSFYSS

T=2 MeVYe=0.2

X Data

50 60 70 80 90 100

iso

top

e y

ield

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

T=2 MeVYe=0.4

X Data

50 60 70 80 90 100

T=3 MeVYe=0.2

A50 60 70 80 90 100

iso

top

e y

ield

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

T=3 MeVYe=0.4

A50 60 70 80 90 100

10-2

Z=26

Isotopic distributions : 26Fe

Evaluation of nuclear properties in stellar environments, i.e. at finite temperature (T), baryon density (ρ) and electron or lepton fraction (Y): 0<T<10 MeV, 10-5<ρ/ρ0<0.5, 0.1<Y<0.6.

One should evaluate a) binding energies, b) excited states, c) decay probabilities of huge numbers of nuclei (1<A<1000, 0<Z<A), i. e. construct a “Nuclear Chart” for each set (T, ρ, Y).

Existing nuclei represent only a very small subset at (0,0,0). Altogether up to 108 new “data” points should be calculated using any “reasonable” approach (LDM, HFB, RMF, CEFT, …)

Such data are needed as inputs for a Statistical Model to find the most probable Nuclear Statistical Ensemble (NSE) in a specific stellar environment.

Challenging task for nuclear community

This slide is borrowed from Valery Zagrebaev

Nuclear Chart is not fully explored even at laboratory

conditions

Wigner-Seitz approximation

neutrons+electrons

spherical nucleus

deformed nucleus

spherical cell deformed cell

Requirements on the cells: 1) electroneutrality, 2) fixed average barion density and Z/A

The whole system is subdivided into individual cells each containing one nucleus, free neutrons and electron cloud

Nuclear Coulomb energy is reduced due to the electron screening:

1/32( )3 3 1( ) ( ), ( ) 1 15 2 2p pA

eC ee e eAZ

nneZF n c n c nR nn

neutrons+protons

Possible tool: RMF model + electrons

First step: constant electron density

Second step: self-consistent calculation

parameter set: NL3

T. Buervenich, I. Mishustin and W. Greiner, Phys. Rev. C76 T. Buervenich, I. Mishustin and W. Greiner, Phys. Rev. C76 (2007) 034310;(2007) 034310;C. Ebel, U. Heinzmann, I. Mishustin, S. Schramm, work in C. Ebel, U. Heinzmann, I. Mishustin, S. Schramm, work in progressprogress

deformed ground state

behind barrier

charge density

240240PuPu

kF = 0.5 fm-1=100 MeV

2 0.28 0.60

Nuclear structure calculations in uniform

electron background

Nuclear chart in SN environments

with increasing kF theβ-stability line movestowards the neutron drip line,they overlap already at kF=0.1 fm-1=20 MeVfree neutrons appearat higher kF (“neutronization”)

protondripline

neutron dripline

Energy of ellipsoidal deformation

1) Deformation becomes less favourable because of reduced Coulomb energy2) Energy of isomeric state (or saddle point) goes up with ne

Suppression of spontaneous fission

Fissility parameter

2 2 48

( )S

C F

aZ

A a c k

increases with kF due to reduced Coulomb energy

At kF=0.25 fm-1 =50 MeV

280Z

A

Decreasing Q-values disfavor fission mode

Adding neutrons into the WS cell 1

Sn, N=82 Z=108, N=3492

Sn, N=650

Sn, N=1650

Th. Buervenich, I. Mishustin, C. Ebel et al., work in progressTh. Buervenich, I. Mishustin, C. Ebel et al., work in progress

Adding neutrons into the WS cell 2

1) Dripping neutrons distribute rather uniformly outside the nucleus2) Protons are distributed rather uniformly inside the nucleus 3) With increasing A the surface tension decreases (smaller density gradients)

Adding neutrons into the WS cell 3

1) Neutrons as well as protons develop a hole at the center 2) Central proton density drops gradually with increasing nucleus size

Single-particle levels in β-equilibrium

1) All protons are shifted down due to the attractive potential generated by electrons.2) Neutrons have attractive mean field inside and outside the nucleus. 3) Neutron level density in the continuum is very high.

n p e

Electrostatic properties of clustered matter

Proton density parameterized by Woods-Saxon distribution, neutrons neglected, electrons treated “exactly”

Claudio Ebel, Igor Mishustin, Walter Greiner, J.Phys. G42 (2015) 10, 105201

Modeling pasta phases in WZ approximation

Spherical cell Cylindrical cell Cartesian cell

2) all cells have equal average baryon density and proton fraction

3) each cell has zero net charge, i. e. the number of protons (dark) is equal to the number of electrons (bright), i. e. Ye=Yp, and vice versa for inverse sapes

4) any interaction between the cells is neglected

/B B cellN V /p p BY N N

1) The whole system is divided into cells of different geometry

Realistic electron distributions

• Poisson equation for electrostatic potential

• Thomas-Fermi approximation

• Chemical potential from the normalization

• Boundary conditions

( ) ( , )p ee r r

3/ 22 22

1( , ) ( ( ))

3e r e r m

( )e r dV Z

( 0) 0r ( ) 0r R

Results for droplet (left) and tube (right)

Over the cell electron density changes up to 30!

Results of numerical calculationsEnergy density – bulk contr. Optimum cell radius

Standard sequence of shapes: droplets (d), rods (r), plates (p), tubes (t), bubbles ( b, uniform (u) Using realistic electron densities leads to lowering energies and shifting boundaries between different shapes

dr p t

b u

Ye=0.3 Ye=0.3

Electron chemical potential

Using realistic electron densities leads to lowering electron chemical potential up to 5 MeV!

Conclusions Microscopic (HFB, RMF, CEFT,...) calculations are needed to

obtain information about nuclear properties (binding energies, level densities etc) in dense and hot stellar environments.

Partly such information can be obtained also from experiments studying multifragmentation reactions.

This information is crucial for calculating realistic NEOS and nuclear composition of supernova matter within the Statistical equilibrium approach.

Survival of (hot) nuclei may significantly influence the explosion dynamics through both the energy balance and modified weak reaction rates.

Using “exact” electron distributions may lead to a significant change of electron fraction (δμ~5 MeV)and shift of the inner crust boundary to deeper layers.