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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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