Nuclear Symmetry Energy in Relativistic Mean Field Theory

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Nuclear Symmetry Energy in Relativistic Mean
Field Theory
Asia-Europe link workshop, Beijing, 2005
Shufang Ban
School of Physics, Peking University, Beijing,
China Royal Institute of Technology (KTH),
Stockholm, Sweden
S. F. Ban, J. Meng, W. Satula, and R. Wyss,
submitted
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Outline

• 1 Introduction
• 2 Formalism and numerical details
• 3 Results and discussions
• 4 Summary

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1. Introduction
Neutron-proton exchange interaction
A. Bohr and B. R. Mottelson, Nuclear Structure
Vol. I
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What have been done
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T_z Cranking in Equidistant level model
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What have been done
Skyrme-HF theory have been used to verify this
new concept
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Present work
Understand the origin of the nuclear symmetry
energy in RMF Theory
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2. Formalism and numerical details
Lagrangian density in RMF theory
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2. Formalism and numerical details
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3. Results and Discussions

• 1) The mean level spacing e

• 2) The average effective strength of isovector
  potential ?

• 4) The origin of the linear term in nuclear
  symmetry energy

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1) The mean level spacinge Hint from A48
isobaric chain

• Similar as SHF
• e is nearly a constant at large T.
• Scaled by m/m, all curves
• are within the the empirical limits.
• At small T, there are strong
• variations, which is associated with
• shell structure.
• Different from SHF
• 1. Results are very close to each other
• 2. Before scaled, the value is bigger.

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Wenhui Longs presentation
J. Meng, H. Toki, S. G. Zhou, S. Q. Zhang, W. H.
Long, and L. S. Geng, Prog. Part. Nucl. Phys.
(2005) in press
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2) The average effective strength of isovector
potential ?
Similar In the left panel, ? decreases with
T. Different In the right panel, ? is still
decreasing along T, while for SHF, it is almost
no-T dependence.
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3) Nuclear symmetry energy
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3) Nuclear symmetry energy
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Least squares fitting
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4) Linear term eT
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Without isoverctor potential
With full potential
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Pk1 is switching on the rho meson without
self-consistence
PK1 is 2/3 of PK1 but very similar. The linear
term exists in RMF theory whenever the isovector
potential is switched on. The self-consistensity
of the mean field contribute to another 1/3
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4. Summary
1. The mean level spacing, e,is nearly a constant
at large T, while it can be affected by shell
structure at small T, similar as SHF case.
2. The isovector potential can be characterized
by a single number, ?, along an isobaric chain,
but following T(T1 ? /e) dependence.
3. The symmetry energy is following T(T1) very
well and the coefficient asym has good agreement
with experimental data.
4. The lack of the linear term in mean level
spacing is compensated by a corresponding term of
isovector potential in RMF theory.
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Thank you !
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5) The symmetry energy of nuclear matter
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5) The symmetry energy of nuclear matter
Semi-empirical mass formula
A. Bohr and B. R. Mottelson, Nuclear Structure
Vol. I
Droplet model of nuclear masses referred to as
the Semi-empirical mass formula
In Nuclear matter
In RMF Theory
N. K. Glendenning, Compact Stars, New York, 1997,
p164-166
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5) The symmetry energy of nuclear matter
Two possible ways to compare
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e
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T(T4)
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4 Summary
RMF Theory has been used to study the nuclear
symmetry energy
1. The mean level spacing, e,is nearly a constant
at large T, while it can be affected by shell
structure at small T, similar as SHF
2. The isovector potential can be characterized
by a single number, ?, along an isobaric chain,
but following T(T1k/e)
3. The symmetry energy is following T(T1) very
well and the coefficient a_sym has good
agreement with experiment data.
4. The lack of the linear term in mean level
spacing is compensated by a corresponding term of
isovector potential in RMF theory.
5. The discrepancy of the nuclear symmetry
energies between nuclear matter and finite nuclei
are not so clear to us. But for the heavy nuclei,
it follows T(T4) as well.
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Normal Formula
1. Eigenvalue equations for Angular momentum
2. In isospin space, T
3. Wigner super multiplet model