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Spin polarization phenomena in Spin polarization phenomena in dense nuclear matterdense nuclear matter
Alexander IsayevAlexander Isayev
Kharkov Institute of Kharkov Institute of
Physics and TechnologyPhysics and Technology
UkraineUkraine
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OVERVIEW AND MOTIVATIONOVERVIEW AND MOTIVATION
The spontaneous appearance of spin polarized states in nuclearThe spontaneous appearance of spin polarized states in nuclear matter is the topic of matter is the topic of a great current interesta great current interest. .
Could nuclear matter, formed in heavy ion collisions at high and Could nuclear matter, formed in heavy ion collisions at high and intermediate energies, undergo a phase transition to a spin intermediate energies, undergo a phase transition to a spin polarized state?polarized state?
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OVERVIEW AND MOTIVATIONOVERVIEW AND MOTIVATION
The spontaneous appearance of spin polarized states in nuclearThe spontaneous appearance of spin polarized states in nuclear matter is the topic of matter is the topic of a great current interesta great current interest..
Could nuclear matter, formed in heavy ion collisions at high and Could nuclear matter, formed in heavy ion collisions at high and intermediate energies, undergo a phase transition to a spin intermediate energies, undergo a phase transition to a spin polarized state?polarized state?
SSpin correlations in the medium significantlypin correlations in the medium significantly influence the influence the neutrino cross section andneutrino cross section and neutrino neutrino luminocity. Hence, luminocity. Hence, different different scenarios ofscenarios of supernova explosion and cooling of neutron stars can supernova explosion and cooling of neutron stars can be realizedbe realized, depending on whether nuclear matter is spin , depending on whether nuclear matter is spin polarized or not.polarized or not.
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OVERVIEW AND MOTIVATIONOVERVIEW AND MOTIVATION
Precisely:Precisely: Nucleons possess spin. Hence,Nucleons possess spin. Hence, nuclear matter nuclear matter is is a a paramagnetic medium and under loweringparamagnetic medium and under lowering temperature temperature or changing density nuclear matter can undergo a phaseor changing density nuclear matter can undergo a phase transition to transition to the state with nonequal numbers ofthe state with nonequal numbers of spin-up spin-up and spin-down nucleons: spin polarized state.and spin-down nucleons: spin polarized state. Usually: Usually: majority of neutron spins are aligned in the same majority of neutron spins are aligned in the same direction as majority of proton spins (like ferromagnetic direction as majority of proton spins (like ferromagnetic ordering). ordering).
Another possibility: Another possibility: majoritymajority of of neutron spins neutron spins and and majority of protonmajority of proton spins spins have have the opposite directionthe opposite directions s ((likelike a antiferromagneticntiferromagnetic ordering) ordering)..
Main emphasis:Main emphasis: the structure of athe structure of a ground state of nuclear ground state of nuclear matter, whether it is spin polarized ormatter, whether it is spin polarized or not and what type of not and what type of spin spin ordering ordering can becan be realized. realized.
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OVERVIEW AND MOTIVATIONOVERVIEW AND MOTIVATION
The possibility of ferromagnetic phase transition in nuclear The possibility of ferromagnetic phase transition in nuclear and neutron matter:and neutron matter:
Calculations of magnetic susceptibility with Skyrme Calculations of magnetic susceptibility with Skyrme effective forceseffective forces
A. Viduarre, J. Navarro, and J. Bernabeu, Astron. Astrophys.A. Viduarre, J. Navarro, and J. Bernabeu, Astron. Astrophys.
135135, 361 (1984). Ferromagnetic transition occurs at , 361 (1984). Ferromagnetic transition occurs at
The Fermi liquid criterion for the ferromagnetic instabilityThe Fermi liquid criterion for the ferromagnetic instability in nuclear matter with Skyrme interaction is reached atin nuclear matter with Skyrme interaction is reached at being nuclear matter saturation density.being nuclear matter saturation density.
S. Reddy, M. Prakash, J.M. Lattimer, and J.A. Pons, Phys. Rev. C S. Reddy, M. Prakash, J.M. Lattimer, and J.A. Pons, Phys. Rev. C 5959, 2888 (1999)., 2888 (1999).
3fm 26.018.0
,)42( 0 0
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In the models with realisticIn the models with realistic nucleon-nucleon (NN) nucleon-nucleon (NN) interaction theinteraction the f ferromagnetic phaseerromagnetic phase transition seems to transition seems to be suppressed up tobe suppressed up to densities well abovedensities well above ..0
I. Vidana, A. Polls, and A. Ramos, Phys. Rev. C 65, 035804 (2002).
S. Fantoni, A. Sarsa, and E. Schmidt, Phys. Rev. Lett. 87, 181101 (2001).
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Here: the study of spin polarizability of nuclear matter with the use of effective NN interaction (Skyrme, Gogny effective forces).
Framework for consideration: Fermi liquid (FL) description of nuclear matter, allowing to obtain the self-consistent equations for the order parameters, and, after solving them, to calculate the free energy and to make conclusions concerning thermodynamic stability of different phases.
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Basics of FormalismBasics of Formalism
Normal states of nuclear matter are described by the normal distribution function of nucleons
The matrix self-consistent equation for in the state of thermodynamic equilibrium
The single-particle energy
and
1221Tr aaf ),,( p
is the density matrix of the system
1221
)()(
f
fEf
}1))({exp(}1))({exp( 040
11 fYYfYf
f
TY
TY
TYpnYY
pp
nn
1 , , ),,( 044144 21121
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Structure of the normal distribution function and single particle energy
3333
300303300000
)(
)()()()(
pf
pfpfpfpf
ki , are the Pauli matrices in spin and isospin spaces
3333
300303300000
)(
)()()()(
p
pppp
)()()(4
,)(4
,)(4
,)(4
33
30
03
00
ppnpn
p
ppn
p
pfV
pfV
pfV
pfV
Normalization conditions for the distribution functions
pn - the isospin asymmetry parameter
FM spin order parameter
AFM spin order parameter
In symmetric nuclear matter withFM ordering: 0,0
AFM ordering: 0,0
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The energy functional of the system: )()()( int0 fEfEfE
)},()(~
)()(~)()(~)()(~{2)(
,2
)( ),()(4
)(
3333
030330300000int
0
2
00000
pfp
pfppfppfpfE
m
pppfp
VfE
p
p
q
q
qfkUV
p
qfkUV
pqfkUV
p
qpkqfkU
Vp
),()(2
1)(~
),()(2
1)(~ ),()(
2
1)(~
,2
),()(2
1)(~
33333
0320330130
00000
FL amplitudes )(),(),(),( 3210 kUkUkUkU
describe density, spin, isospin and spin-isospin correlations in nuclear medium.
q
q
q
q
qfkUV
p
qfkUV
p
qfkUV
p
qfkUV
pp
)()(21
)(
,)()(21
)(
),()(21
)(
),()(21
)()(
33333
03203
30130
000000
Self-consistent equations
2
,2
pn
pn
Set of integral equations to be solved self-consistently.
Numerical procedure: iterations on the Gaussian grid in momentum spaceuntil convergence with required accuracy is achieved.
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Amplitude of NN interaction for Skyrme effective force
)1(2
1 ,')1(
)')(1()1()1()',(
212
22112
1336
100
22
2
PppPx
ppPxtPxtPxtppv
t
4
)(
2
1
300
22
23
)
()1(),(
iqp
ii
iii
i
ePPMPH
PBWPxtqpv
Amplitude of NN interaction for Gogny effective force
In numerical calculations: SLy4, SkI3, SkI5 parametrizations
Two Gaussian terms reflect the finite range character of the Gogny interaction. In numerical calculations: D1S force.
,p
ppp
n
nnn
Neutron and proton spin polarization parameters:
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Neutron and proton spin polarization parameters as functions of density at zero temperature
SLy4:
1. Antiparallel ordering only!2. Even small admixture of protons to neutron matter strongly decreases the onset density of spin instability.3. Protons become totally polarized in a narrow density domain at strong isospin asymmetry, unlike to neutrons.
04.2 07.3
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SkI5:
1. Parallel ordering only!2. Small admixture of protons to neutron matter insignificantly change the onset density of spin instability.
Neutron and proton spin polarization parameters as functions of density at zero temperature
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Total energy per nucleon, measured from its value in thenormal state, as a function of density at T=0
The energy gain per nucleon isdecreased with isospin asymmetry for SLy4 force whileit is increased for SkI5 force.
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SkI3: 1. Transient behavior from antiparallel ordering to parallel ordering under increasing density.2. There are no long tails in the density profile of neutrons at strong isospin asymmetry.3. The energy gain is increasing function of isospin asymmetry at the given density.
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AFM spin polarization parameter as a function of density at zero temperature for D1S Gogny force and SkM*, SGII Skyrme forces.
For D1S force only AFM spin ordering is realized and there are no solutions, corresponding to FM spin ordering.
08.3 03.4
Symmetricnuclearmatter
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Total energy per nucleon, measured from its value in the normal state, for the AFM spin state as a function of density at zero temperature for D1S Gogny force and SkM*, SGII Skyrme forces.
CONCLUSIONSCONCLUSIONS
Thus, modern effective nuclear forces being relevant for Thus, modern effective nuclear forces being relevant for calculations at wide range of isospin asymmetries and high calculations at wide range of isospin asymmetries and high densities provide us with different possible scenarios of a densities provide us with different possible scenarios of a phase transition to a spin polarized state in dense nuclear phase transition to a spin polarized state in dense nuclear matter:matter:
a) nuclear matter with SLy4 Skyrme and D1S Gogny a) nuclear matter with SLy4 Skyrme and D1S Gogny interactions undergoes at some critical density a phase interactions undergoes at some critical density a phase transition to a spin polarized state with the oppositely transition to a spin polarized state with the oppositely directed spins of neutrons and protons;directed spins of neutrons and protons;
b) for SkI5 interaction, a spin polarized state with the like-b) for SkI5 interaction, a spin polarized state with the like-directed neutron and proton spins is formed; directed neutron and proton spins is formed;
c) nuclear matter with SkI3 interaction under increasing c) nuclear matter with SkI3 interaction under increasing density, at first, undergoes a phase transition to the state density, at first, undergoes a phase transition to the state with the opposite directions of neutron and proton spins, with the opposite directions of neutron and proton spins, which goes over at larger density to the state with the same which goes over at larger density to the state with the same direction of nucleon spins.direction of nucleon spins.
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The different behavior at high densities of the The different behavior at high densities of the interaction amplitudes, describing spin-spin and spin-interaction amplitudes, describing spin-spin and spin-isospin correlations, lays behind this divergence in isospin correlations, lays behind this divergence in calculations with different effective potentials. calculations with different effective potentials.
These results clearly indicate the necessity to These results clearly indicate the necessity to construct a new generation of the energy functionals construct a new generation of the energy functionals with the properly constrained time-odd part. with the properly constrained time-odd part. Probably, these constraints will be obtained from the Probably, these constraints will be obtained from the data on the time decay of magnetic field of isolated data on the time decay of magnetic field of isolated neutron stars.neutron stars.
S.B. Popov and M.E. Prokhorov, Surveys in High EnergyPhysics 15, (2001) 381.
