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Nucleon Effective Mass in the DBHF. 同位旋物理与原子核的相变 CCAST Workshop 2005 年 8 月 19 日- 8 月 21 日 马中玉 中国原子能科学研究院. Introduction. Importance of isospin physics in many aspects: exotic nuclei; astrophysics; heavy ion collision etc. Isospin dependence of many quantities: - PowerPoint PPT Presentation
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Nucleon Effective Mass in the DBHF
同位旋物理与原子核的相变
CCAST Workshop
2005 年 8月 19 日- 8月 21 日
马中玉
中国原子能科学研究院
Introduction
Importance of isospin physics in many aspects:
exotic nuclei; astrophysics; heavy ion collision etc.
Isospin dependence of many quantities:
asymmetric energy as function of density
effective interaction; effective mass etc.
Effective mass characterizes the propagation of a
nucleon in the strongly interacting medium
reaction dynamics of nuclear collisions by unstable nuclei
neutron-proton differential collective flow, isospin equil.
neutron star properties
Introduction
Less knowledge of isospin dependence from experiments
Study from a fundamental theory
NN interaction + SR correlation
Many works in non-relativistic and relativistic approaches
RMF : success in describing g.s. properties
isospin dep. based on stable nuclei
Us, U0 are energy indep.
Our work Relativistic approach DBHF
Definition of effective mass
In non-relativistic approach
effective mass m* describe an independent quasi-particle
moving in the nucl. medium
characterizes the non-locality of the microscopic potential
in space (k-mass) and in time (E-mass)
Jeukenne, Lejeune, Mahuax ‘76
1 2
2 2
0*
( , ) ( , )
( , )2 2
V r r V k
k kV k V
m m
* ** k Em mm
m m m
Effective mas in non-relativistic appr.
1*
( )
1 ( , ( )) 1 ( ( ), )k k
m m d dV k k V k
m k dk d
Effective mass derived by the two equivalent expression
Jaminon & Mahaux’89
It can be determined from analyses of experimental data
in nonrel. Shell model or optical model
Typical value is m*/m ~ 0.70 ± 0.05 at E = 30 MeV
by phenomenological analyses of experimental
scattering data
Dirac mass
Relativistic approach
Effective mass is usually defined as
M* = M – Us Dirac mass (scalar mass)
describe a nucleon in the medium as a quasi nucleon
with effective mass and effective energy, which satisfies
Dirac equation.
M* and m* are different physical quantities
Can not be compared with each other
* *
*
*
( ) ( ) ( )
( )
( )
s
o
p M r E r
M M U r
E E U r
Dirac and Lorentz mass
RMF Us Uo are constant in energy
Us= 375±40 MeV Ring’96
~0.60±0.04 Dirac mass (scalar mass)
Schroedinger equivalent equation
Schroedinger equvalent potential
,
Lorentz mass
*
1 sM U
M M
2 21( ) ( )
2s o s oV U U U UM M
*
1 ( ( ), )m d
V km d
Isospin dep. of effective mass
Isospin dependence of effective mss in RMF
,
Energy dep. of nucleon self-energy is not considered
~0.7 Lorentz mass (vector mass)
Lorentz mass
not related to a non-locality of the rel. potentials,
can be compared with the effective mass in non-relativistic
appr.
Isospin dep. of Lorentz mass, compare with that in non-rel.
2 21( ) ( )
2s o s oV U U U UM M
*
1 oUm
m m
*
1 ( ( ), )m d
V km d
DBHF approach
Relativistic approaches
NN + DBHF
Success in NM saturation
properties
DBHF G Matrix ––-
Nucleon effective int.
Information of isospin dependence
Dirac structure of G Matrix
Bethe-Salpeter equation
3-dimensional reduction: (RBBG)
G=V+VgQG V NN int.(OBEP) g propergator
Q Pauli operator
Self-consistent calculations important
G ? Us, U0 Dirac eq. s.p. wf
G matrix --- do not keep the track of rel. structure
Extract the nucleon self-energy with proper isospin dep.
New decomposition of G
Decomposition of DBHF G matrix
V : OBEP
G a projection method
(1, ) (1, )
Short range
m (g/m)2 finite
E. Shiller, H. Muether,
E Phys. J. A11(2001)15
G V G
Nucleon self-energy
Nucleon self-energy for scattering (k is related with E)
Calculated in DBHF by G = V + G
Direct Exchange
: V OBEP G pseudo meson (1, ) (1, )
: vertex a,b : isospin index
single particle Green’s function
( )kG
Isospin dep. of Nucleon self-energy
Single particle Green’s function
T t : isospin operator
Direct terms isoscalar
isovector
Exchange terms isoscalar
isovector
Nucleon potential
The optical potential of a nucleon
the nucleon self-energy in the nuclear medium
Nucleon self-energy in the nuclear medium with E > 0
0
*0
2 2 20
[ ( )]
[1 ( )] [ ( )] ( )
k
v s
E M k
k k M k k
k – E E incident energy
1 10 1001
2
3
4
10 100
k(p)
k (
fm-1)
E (MeV)
= 0.0 = 0.3 = 0.6
kF = 1.36 fm-1 k(n)
E (MeV)
= 0.0 for free N
Self-Energy of proton and neutron
-350
-300
-250
0 100 200 300 400
200
250
300
100 200 300 400
(a)
Us
p
(MeV
)
(b)
Us
n
(c)U0
p
(MeV
)
Energy (MeV)
(d)
U0
n
Energy (MeV)
= 0, .3, .6, 1
Isospin dep. of effective mass in RMF
Dirac Lorentz
Neutron-rich asymmetric NM (RMF)
, ,
Taking account of isovector scalar meson
, , ,
Us Uo should be of momentum and energy dependence
* * * *
,
,
n p n ps s o o
n p n pM m
U U U
M m
U
* * * *
,
,
n p n ps s o o
n p n pM m
U U U
M m
U
*
1 sM U
M M
*
1 oUm
m m
Nucleon self-energy in DBHF
0 20 40 60 80 100
250
300
350
Nuc
leon
Sel
f-en
ergy
/MeV
Insident Energy /MeV
=0 =.3 Proton =.3 Neutron
Us
U0
0.0 0.2 0.4 0.6 0.8
250
300
350
Us
Nu
cle
on
Se
lf-e
nerg
y /M
eV
Proton E=0 MeV Neutron E=0 MeV Proton E=50 MeV Neutron E=50 MeV
U0
In neutron-rich matter
Us Uo of neutrons stronger than of protons
DBHF
0.0 0.2 0.4 0.6 0.80.60
0.65
0.70
0.75
M*/M
M*/
M
Proton
Neutron m*/m Dirac mass in DBHF :
Lorentz mass:
Isospin dep. of OMP is
consistent with Lane pot.
*0 0 011 1s sU dU U dm U U
M d M Mm M d
* *n pM M
* *n pm m
Ma, Rong, Chen et al., PLB604(04)170B.A.Li nucl-th/0404040
Summary
Isospin dependence of the nucleon effective mass is
studied in DBHF
New decomposition of G matrix is adopted
G=V+G
RMF approach with a constant self-energy can not
account the isospin dep. of m* properly
Isospin dep. of effective mass
* *n pm m
Thanks Thanks
G Us Uo
Single particle energy
R. Brockmann, R. Machleidt PRC 42(90)1965
Momentum dep. of Us & U0 are neglected
works well in SNM, inconsistent results in ASNM
wrong sign of the isospin dependence
2 20( ) [ ( )] ( )i i
i sk k M U k U k
2 20( ) [ ]i i
i sk k M U U
Asymmetric NM
Inconsequential results for asymmetric
nuclear matter
Us U0
isospin dep. with
a wrong sign
S. Ulrych, H. Muether,
Phys. Rev. C56(1997)1788
2 20( ) [ ]i i
i sk k M U U
Projection method
Projection method
F. Boersma, R. Malfliet,
PRC 49(94)233
Ambiguity results are
obtained
for with PS and PV
Shiller,Muether, EPJ. A11(2001)15
5 51, , , ,G 0,sU U
Asymmetry Energy
0.1 0.2 0.3 0.4
20
30
40
50
60
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
60
E/A
(,
) -
E/A
(,0
) [M
eV]
2
1.0 fm-1
1.2 fm-1
1.4 fm-1
1.6 fm-1
1.8 fm-1
DBHF BHF
a asym
()=
E/A
(,1
)-E
/A(
,0)
[MeV
]
[fm-3]
3-body force
Parabolic behavior increase as the density
Ma and Liu PRC66(2002)024321;Liu and Ma CPL 19 (2002)190
asya
Dirac and Lorentz mass
RMF
Us Uo are constant in energy
~0.60 Dirac mass (scalar mass)
Schroedinger equivalent potential
,
~0.70 Lorentz mass (vector mass)
Although not related to a non-locality of the rel. potentials,
comparable with the effective mass in non-relativistic appr.
*
1 sM U
M M
2 21( ) ( )
2s o s oV U U U UM M
*
1 oUm
m m
*
1 ( ( ), )m d
V km d
Nucleon effective mass
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