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Isospin effect in asymmetric nuclear matter
(with QHD II model)
Kie sang JEONG
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Effective mass splitting• from nucleon dirac eq. here energy-
momentum relation
• Scalar self energy• Vector self energy (0th )
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Effective mass splitting• Schrodinger and dirac effective mass
(symmetric case)
• Now asymmetric case visit• Only rho meson coupling
• + => proton, - => neutron
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Effective mass splitting• Rho + delta meson coupling
• In this case, scalar-isovector effect appear
• Transparent result for asymmetric case
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Semi empirical mass for-mula
• Formulated in 1935 by German physicist Carl Friedrich von Weizsäcker
• 4th term gives asymmetric effect
• This term has relation with isospin density
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QHD model• Quantum hadrodynamics• Relativistic nuclear manybody theory• Detailed dynamics can be described
by choosing a particular lagrangian density
• Lorentz, Isospin symmetry• Parity conservation *• Spontaneous broken chiral symmetry
*
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QHD model• QHD-I (only contain isoscalar
mesons)
• Equation of motion follows
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QHD model• We can expect coupling constant to
be large, so perturbative method is not valid
• Consider rest frame of nuclear sys-tem (baryon flux = 0 )
• As baryon density increases, source term becomes strong, so we take MF approximation
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QHD model• Mean field lagrangian density
• Equation of motion
• We can see mass shift and energy shift
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QHD model• QHD-II (QHD-I + isovector couple)
• Here, lagrangian density contains isovector – scalar, vector couple
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Delta meson• Delta meson channel considered in
study
• Isovector scalar meson
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Delta meson• Quark contents
• This channel has not been consid-ered priori but appears automatically in HF approximation
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RMF <–> HF • If there are many particle, we can as-
sume one particle – external field(mean field) interaction
• In mean field approximation, there is not fluctuation of meson field. Every meson field has classical expectation value.
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RMF <–> HF • Basic hamiltonian
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RMF <–> HF • Expectation value
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Hartree Fock approximation
Classical interaction be-tween one particle - sysytem
Exchange contribution
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H-F approximation• Each nucleon are assumed to be in a
single particle potential which comes from average interaction
• Basic approximation => neglect all meson fields containing derivatives with mass term
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H-F approximation• Eq. of motion
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Wigner transformation• Now we control meson couple with
baryon field• To manage this quantum operator as
statistical object, we perform wigner transformation
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Transport equation with fock terms
• Eq. of motion
• Fock term appears as
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Transport equation with fock terms
• Following [PRC v64, 045203] we get kinetic equation
• Isovector – scalar density• Isovector baryon current
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Transport equation with fock terms
• kinetic momenta and effective mass
• Effective coupling function
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Nuclear equation of state• below corresponds hartree approximation• Energy momentum tensor
• Energy density
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Symmetry energy• We expand energy of antisymmetric
nuclear matter with parameter
• In general
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Symmetry energy• Following [PHYS.LETT.B 399, 191]
we get Symmetry energy
nuclear effective mass in symmetric case
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Symmetry energy• vanish at low densities, and still
very small up to baryon density• reaches the value 0.045 in this
interested range
• Here, transparent delta meson effect
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Symmetry energy• Parameter set of QHD models
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Symmetry energy• Empirical value a4 is symmetry energy
term at saturation density, T=0
When delta meson contribution is not zero, rho meson cou-pling have to increase
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Symmetry energy
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Symmetry energy• Now symmetry energy at saturation
density is formed with balance of scalar(attractive) and vector(repulsive) contribution
• Isovector counterpart of saturation mechanism occurs in isoscalar chan-nel
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Symmetry energy• Below figure show total symmetry energy
for the different models
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Symmetry energy• When fock term considered, new effective
couple acquires density dependence
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Symmetry energy• For pure neutron matter (I=1)
• Delta meson coupling leads to larger re-pulsion effect
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Futher issue• Symmetry pressure, incompressibility• Finite temperature effects• Mechanical, chemical instabilities• Relativistic heavy ion collision• Low, intermediate energy RI beam
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reference• Physics report 410, 335-466• PRC V65 045201• PRC V64 045203• PRC V36 number1• Physics letters B 191-195• Arxiv:nucl-th/9701058v1