Investigations of nuclear equation of state in nucleus-Investigations of nuclear equation of state in nucleus-nucleus collisions and fissionnucleus collisions and fission

Martin VeselskyMartin Veselsky

Institute of Physics, Slovak Academy of Sciences, Dubravska Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, Bratislava, Slovakiacesta 9, Bratislava, Slovakia

G. A. Souliotis, P. Fountas, N. VontaG. A. Souliotis, P. Fountas, N. Vonta

Department of Chemistry, University of Athens, Athens, Department of Chemistry, University of Athens, Athens, GreeceGreece

Yu-Gang MaShanghai Institute of Applied Physics of CASc, Shanghai,

China

Nuclear Equation of State

Definition : A relation that determines how nuclear matter (composing of neutrons & protons) behaves when subjected to different pressure, density & temperature

In real gas, this relation is very well known, because we

know exactly the force of interaction (Van der Waals) between the gas molecules

The force of interaction between nucleons (neutrons & protons) in nuclear matter is

not very well known

molecular force of

attraction

The “stiffer” (“softer”) the nuclear interaction the larger (“smaller”) is the predicted neutron star mass

and radius

J. Stone et al, Phys. Rev. C 68 (2003) 034324

The long-sought mass-radius relation of neutron star depends on the strength of interaction

“Stiff”

“Soft”

Nucleon interactions

too “Soft”

Radius (km)

Ruled Out

Credit : S. Lee, CXC, NASA

Mass

(M

Sol)

Neutron star cooling depends on the strength of symmetry energy

“Stiffer” symmetry energy can lead to faster cooling of the

neutron stars

Nucleon interactions

“Stiff”

“Soft”

Neutron stars are way too cooler than originally expected

Esym=Esym(ρ0)(ρ/ρ0)γδ2

Nucleus-nucleus collisions - Introduction Nucleus-nucleus collisions - Introduction

Nucleus-nucleus collisions allow to study properties of the nuclear Nucleus-nucleus collisions allow to study properties of the nuclear matter. matter.

Properties of the nuclear matter are formulated in terms of nuclear Properties of the nuclear matter are formulated in terms of nuclear equation of state. equation of state.

Nuclear matter is a two-component system and the effect of changing Nuclear matter is a two-component system and the effect of changing neutron-to-proton ratio is typically treated in terms of symmetry energy neutron-to-proton ratio is typically treated in terms of symmetry energy and its density dependence. and its density dependence.

Various implementations of Boltzmann equation (Fokker-Planck Various implementations of Boltzmann equation (Fokker-Planck equation, Boltzmann-Uehling-Uhlenbeck equation, Quantum Molecular equation, Boltzmann-Uehling-Uhlenbeck equation, Quantum Molecular Dynamics) are used for modeling of the nucleus-nucleus collisions and Dynamics) are used for modeling of the nucleus-nucleus collisions and testing of various equations of state. testing of various equations of state.

Successful equations of state can be used for modeling of astrophysical Successful equations of state can be used for modeling of astrophysical objects (neutron stars) and processes (supernova explosions)objects (neutron stars) and processes (supernova explosions)

Boltzmann equationBoltzmann equation

Various implementations of the collision term lead to different physicsVarious implementations of the collision term lead to different physics

No collision term – Liouville or Vlasov equation, explains e.g. No collision term – Liouville or Vlasov equation, explains e.g. Boltzmann distribution, behavior of plasma in electromagnetic fieldBoltzmann distribution, behavior of plasma in electromagnetic field

Collision term – BGK form – describes simple transport phenomena – Collision term – BGK form – describes simple transport phenomena – Ohm's law, thermodiffusion, thermo-electric phenomena, Fokker-Ohm's law, thermodiffusion, thermo-electric phenomena, Fokker-Planck equation (deep-inelastic transfer)Planck equation (deep-inelastic transfer)

Collision term with pair collisions – Boltzmann-Uehling-Uhlenbeck, Collision term with pair collisions – Boltzmann-Uehling-Uhlenbeck, Vlasov-Uehling-Uhlenbeck, Landau-Vlasov – suitable for description of Vlasov-Uehling-Uhlenbeck, Landau-Vlasov – suitable for description of dynamical stage of nucleus-nucleus collisions. In-medium nucleon-dynamical stage of nucleus-nucleus collisions. In-medium nucleon-nucleon cross sections ? nucleon cross sections ?

Boltzmann-Uehling-Uhlenbeck equationBoltzmann-Uehling-Uhlenbeck equation

EoS

In-medium nucleon-nucleon cross section in the nuclear matter are In-medium nucleon-nucleon cross section in the nuclear matter are important component in the nuclear implementations of the important component in the nuclear implementations of the Boltzmann equations, nevertheless they are practically unknown.Boltzmann equations, nevertheless they are practically unknown.

Typically, free nucleon-nucleon cross sections are used in Typically, free nucleon-nucleon cross sections are used in simulations of nucleus-nucleus collisions, occasionally scaled simulations of nucleus-nucleus collisions, occasionally scaled down by some factor. down by some factor.

Density-dependence of nucleon-nucleon cross sections is Density-dependence of nucleon-nucleon cross sections is approximated only empirically. approximated only empirically.

True in-medium nucleon-nucleon cross sections must depend on True in-medium nucleon-nucleon cross sections must depend on equation of state of the nuclear matter and thus the dependence of equation of state of the nuclear matter and thus the dependence of collision term on EoS needs to be implemented !!! collision term on EoS needs to be implemented !!!

Calculation of in-medium nucleon-nucleon cross sections in the nuclear Calculation of in-medium nucleon-nucleon cross sections in the nuclear matter presents a considerable challenge to the nuclear theory. matter presents a considerable challenge to the nuclear theory.

G-matrix theory was used to estimate in-medium nucleon-nucleon cross sections by G-matrix theory was used to estimate in-medium nucleon-nucleon cross sections by Cassing et al. (W. Cassing, U. Mosel, Prog. Part. Nucl. Phys. 25, 235 (1990)). Cassing et al. (W. Cassing, U. Mosel, Prog. Part. Nucl. Phys. 25, 235 (1990)).

Density-dependence of in-medium nucleon-nucleon cross section was studied for Density-dependence of in-medium nucleon-nucleon cross section was studied for symmetric nuclear matter (G. Q. Li, and R. Machleidt, Phys. Rev. C 48, 1702 symmetric nuclear matter (G. Q. Li, and R. Machleidt, Phys. Rev. C 48, 1702 (1993); Phys. Rev. C 49, 566 (1994); T. Alm et al., Phys. Rev. C 50, 31 (1994); (1993); Phys. Rev. C 49, 566 (1994); T. Alm et al., Phys. Rev. C 50, 31 (1994); Nucl. Phys. A 587, 815 (1995)), and significant influence of nuclear density on Nucl. Phys. A 587, 815 (1995)), and significant influence of nuclear density on resulting in-medium cross sections was observed in their density, angular and resulting in-medium cross sections was observed in their density, angular and energy dependencies. energy dependencies.

Using momentum-dependent interaction, ratios of in-medium to free nucleon-Using momentum-dependent interaction, ratios of in-medium to free nucleon-nucleon cross sections were evaluated at zero temperature via reduced effective nucleon cross sections were evaluated at zero temperature via reduced effective nucleonic masses (B. A. Li, and L. W. Chen, Phys. Rev. C 72, 064611 (2005)) and nucleonic masses (B. A. Li, and L. W. Chen, Phys. Rev. C 72, 064611 (2005)) and used for transport simulations.used for transport simulations.

Still, transport simulation are mostly performed using parametrizations of Still, transport simulation are mostly performed using parametrizations of the free nucleon-nucleon cross sections, eventually scaling them down the free nucleon-nucleon cross sections, eventually scaling them down empirically or using simple prescriptions for density-dependence of the empirically or using simple prescriptions for density-dependence of the scaling factor (D. Klakow et al., Phys. Rev. C 48, 1982 (1993)). scaling factor (D. Klakow et al., Phys. Rev. C 48, 1982 (1993)).

Van der Waals-like equation of state

By combining a long-range attractive and short-range repulsive interaction it allows to describe by a relatively simple model the main features of the liquid-gas phase transition in isospin-asymmetric nuclear matter, including the possible interplay of 1st and 2nd order phase transition with variation of the isospin asymmetry (M. Veselsky, IJMPE 17 (2008) 1883; arXiv.org:nucl-th/0703077)

It provides a parameter, called “proper” or “excluded volume”, which describes the volume per constituent particle of the non-ideal gas.

Can the proper volume estimate in-medium nucleon-nucleon cross section ?

Method:

Formally transform EoS into Van der Waals form, extract the proper volume.

M. Veselsky and Y.G. Ma, PRC 87 (2013) 034615

Starting from EoS:

Fermionic effects taken into account - Fermionic effects taken into account - Two effects which cancel out mutuallyTwo effects which cancel out mutually

Global 1/ρ2/3 dependence of nucleon-nucleon cross sections from EoS

EoS-dependent

free

It is apparent that while the isospin-dependent nucleon-nucleon It is apparent that while the isospin-dependent nucleon-nucleon cross sections essentially follow the 1/cross sections essentially follow the 1/ρρ2/32/3-dependence, the -dependence, the nucleon-nucleon cross section parametrization of Cugnon et al. (J. nucleon-nucleon cross section parametrization of Cugnon et al. (J. Cugnon, T. Mizutani, and J. Vandermeulen, Nucl. Phys. A 352, Cugnon, T. Mizutani, and J. Vandermeulen, Nucl. Phys. A 352, 505 (1981)) leads to much larger spread, mostly due to its explicit 505 (1981)) leads to much larger spread, mostly due to its explicit energy dependence. Nevertheless, one observes that both energy dependence. Nevertheless, one observes that both parametrization cover essentiallyparametrization cover essentially the same range of values of the the same range of values of the nucleon-nucleon cross sectionsnucleon-nucleon cross sections. . Furthermore, from the comparison of the parametrization of Furthermore, from the comparison of the parametrization of Cugnon et al. to in-medium cross sections at saturation density, Cugnon et al. to in-medium cross sections at saturation density, calculated using the G-matrix theory by Cassing et al. (W. calculated using the G-matrix theory by Cassing et al. (W. Cassing, and U. Mosel, Prog. Part. Nucl. Phys. 25, 235 (1990)), it Cassing, and U. Mosel, Prog. Part. Nucl. Phys. 25, 235 (1990)), it can be judged that the in-medium cross sections, obtained using can be judged that the in-medium cross sections, obtained using the proper volume of the Van der Waals-like equation of state, are the proper volume of the Van der Waals-like equation of state, are in better in better agreement with somewhat higher values of G-matrix in-agreement with somewhat higher values of G-matrix in-medium cross sections of Cassing et al.medium cross sections of Cassing et al., which reflect properly the , which reflect properly the Fermionic nature of nucleons. Fermionic nature of nucleons.

Systematics of proton directed flow in Au+Au, b=5-7fmSystematics of proton directed flow in Au+Au, b=5-7fm

N. Herrmann, J. P. Wessels, and T. Wienold, Ann. Rev. Nucl. Part. Sci. 49 (1999) 581N. Herrmann, J. P. Wessels, and T. Wienold, Ann. Rev. Nucl. Part. Sci. 49 (1999) 581.

Directed flow is a 1st Fourier

coefficient of the angular

distribution in azimuthal

(transverse) plane.

Systematics of proton elliptic flow in Au+Au, b=5-7fmSystematics of proton elliptic flow in Au+Au, b=5-7fm

A. Andronic, J. Lukasik, W. Reisdorf, and W. Trautmann, Eur. Phys. J. 30 (2006) 31A. Andronic, J. Lukasik, W. Reisdorf, and W. Trautmann, Eur. Phys. J. 30 (2006) 31.

Elliptic flow is a 2nd Fourier

coefficient of the angular

distribution in azimuthal

(transverse) plane.

ΔxΔp>2h

Stiff EoS (K0=380 MeV)

Semi-stiff Esym (γ=1.)

Directed flow:

symbols- calc.

line - experiment

Elliptic flow:

Symbols - calc.

line - experiment

EoS-dependent collision term (with in-medium cross sections) leads to correct (positive) directed flow, while free cross sections lead to incorrect (negative) directed flow !!!

Soft EoS (K0=200 MeV)

Semi-stiff Esym (γ=1.)

Directed flow:

symbols- calc.

line - experiment

Elliptic flow:

Symbols - calc.

line - experiment

Starting from potential

After applying standard conditions for saturation we arrive to solution

And get compressibility(linear to )

Semi-stiff EoS (K0=270 MeV)

Semi-stiff Esym (γ=1.)

Directed flow:

symbols- calc.

line - experiment

Elliptic flow:

Symbols - calc.

line - experiment

Simultaneous analysis of directed and elliptic flow in Au+Au semi-peripheral collisions (0.4-10 AGeV)

Production of n-rich nuclei at 15 AMeV

Comparison of the model of deep-inelastic transfer (DIT) vs constrained molecular dynamics (CoMD) vs experimental data

Peripheral Collisions, Deep Inelastic Transfer (DIT)*

DIT : Phenomenological model (Monte Carlo implementation of Fokker-Planck equation)

Formation of a di-nuclear configuration

Exchange of nucleons through a “window” formed by the superimposition of the nuclear

potentials in the neck region

Dissipation of Kinetic energy into internal degrees of freedom *DIT : L. Tassan-Got, C. Stephan, Nucl. Phys. A 524, 121 (1991) DIT(modified): M. Veselsky, G.A. Souliotis, Nucl. Phys. A 765, 252 (2006)

b: impact parameterθ: scattering angle

Approaching phase:

Target (Zt,At)

θ

b

Projectile (Zp,Ap) •Neutrons• Protons

excited projectile-likefragment (PLF) or quasi-projectile

excited target-likefragment (TLF) orquasi-target

Overlapping (interaction) stage

Exchange of nucleons:

Deep Inelastic Transfer(DIT) Model

L. Tassan-Got and C. Stephan,

Nucl. Phys. A 524, 121 (1991)

Modified DIT (Nucl.Phys. A765, 252 (2006)), phenomenological correction, effect of shell structure on nuclear periphery ( assuming validity of the R

n-R

p vs μ

n-μ

p correlation ) and thus on transfer

probability estimated as:

where δS... = S...exp - S...

mac , is a free parameter ( = 0.53 determined as optimal value ), s > 0 fm ( only non-overlapping configurations considered )

Microscopic Calculations: Constrained Molecular Dynamics (CoMD)*

CoMD: Quantum Molecular Dynamics model (Semiclassical)

Nucleons are considered as Gaussian wavepackets

N-N effective interaction (Skyrme-type with K=200 MeV/fm3)

Several forms of N-N symmetry potential Vsym (ρ)

Pauli principle imposed via a phase-space constraint

Fragment recognition algorithm (Rmin = 3.0 fm)

Monte Carlo implementation

*M. Papa, A. Bonasera et al., Phys. Rev. C 64, 024612 (2001)

86Kr+58,64Ni

15 AMeV

Experimental data measured at MARS spectrometer at Texas A&M

CoMD (red) and DITm (blue) reproduce data

86Kr+112,124Sn

15 AMeV

Experimental data measured at MARS spectrometer at Texas A&M

CoMD (red) and DITm (blue) reproduce data

92Kr+64Ni

15 AMeV

Estimate of possible production of n-rich nuclei in reaction with secondary beam

CoMD more optimistic than DITm, symmetry energy ?

Motivated also by results of our work (with GS) new RIB facility is built in Korea including secondary beams with energy within the Fermi energy domain. After 15 years finally someone pays attention !!!

p (30 MeV) + 235U, CoMD simulation of nuclear fission

Comparison between theoretical and experimental results: p (660 MeV) + 238U

Grey dots: Exprimental data: [12] A. R. Balabekyan, et Al Phys. Atom. Nucl. 73, 1814-

1819 (2010)

Red line: standard Vsym ~ ρBlue line: standard Vsym ~ ρstrict selection in Zfiss = 93

Red line: standard Vsym ~ ρBlue line: soft Vsym ~ ρ1/2

Circles: fission of 235USquares: fission of 238U

Triangles: fission of 232Th

Open symbols: experimental data[12] A. R. Balabekyan, et Al Phys. Atom. Nucl. 73, 1814-1819 (2010)

[1] P. Demetriou et al Phys. Rev. C 82, 054606 (2010)

[10] M. C. Duijvestijn, et Al Physical Review. C 64, 014607 (2001)

Total fission cross section and fission cross section/residue cross section

Sensitivity of the ratio fission cross section/residue cross section to the choice of the nucleon-nucleon symmetry potential

Total kinetic energy of the fission fragments

Red line: standard Vsym ~ ρBlue line: soft Vsym ~ ρ1/2

Circles: fission of 235USquares: fission of 238U

Triangles: fission of 232Th

Open sympbols: experimental dataP. Demetriou et al Phys. Rev. C 82, 054606 (2010)

M. C. Duijvestijn, et Al Physical Review. C 64, 014607 (2001)S.I. Mulgin et Al Nuclear Physics A, 824, 1–23, (2009)

Neutron multiplicity

Red line: standard Vsym ~ ρBlue line: soft Vsym ~ ρ1/2

Circles: fission of 235USquares: fission of 238U

Triangles: fission of 232Th

Open symbols: experimental data[1] P. Demetriou et al Phys. Rev. C 82, 054606 (2010)

[10] M. C. Duijvestijn, et Al Physical Review. C 64, 014607 (2001)

Conclusions:Conclusions:

- new version of BUU with N-N cross sections derived from the equation - new version of BUU with N-N cross sections derived from the equation of state and applied in the collision term of state and applied in the collision term

- systematics of directed and elliptic flow reproduced and constraints on - systematics of directed and elliptic flow reproduced and constraints on incompressibility and symmetry energy found incompressibility and symmetry energy found (K(K

00=260-310 MeV and =260-310 MeV and =0.8-1.25)=0.8-1.25)

- consistent reproduction of experimental production cross sections of n-- consistent reproduction of experimental production cross sections of n-rich nuclei using both CoMD and DITm rich nuclei using both CoMD and DITm

- collective dynamics in fission decribed using the CoMD - collective dynamics in fission decribed using the CoMD

- consistent constraints on symmetry energy from fission, low-energy - consistent constraints on symmetry energy from fission, low-energy and relativistic collisions ? and relativistic collisions ?

- hard work ahead- hard work ahead

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