Download 国家杰出青年科学基金 申请书

Time reversal symmetry broken triaxial relativistic
mean field approach for magnetic moment and
nuclear current in odd mass nuclei
孟

杰
北京大学物理学院
兰州重离子加速器国家实验室
 中国科学院理论物理研究所

Jie Meng
School of Physics/Peking University
HIRFL/Lanzhou
Institute for Theor.Phys./AS
Contents




Introduction
Triaxial RMF with time-odd component
Numerical details
Results and discussion
1. Single particle energy
2. density distribution
3. Magnetic moment
4. Nuclear current

Summary
Introduction
 Magnetic moments are measured with high precision.
Traditionally it provided a sensitive test for nuclear models.
Blin-Stoyle R J 1957 Theories of Nuclear Moments (Oxford: Oxford University Press).
Wilkinson D H and Rho M (Eds.) 1979 Mesons in Nuclei vol I1 (Amsterdam: North-Holland ).
Arima A 1984 Prog. Part. Nucl. Phys. 11 53
 Because the single particle state can couple to more
complicated 2p-1h configurations and there are mesons
exchange corrections caused by the nuclear medium effect,
the configuration mixing provide a better foundation to
describe the observed values . The mean field may not be
expected to describe the magnetic moment well.
Arima A Horie H 1954 Prog. Theor. Phys. 11 509
Arima A, Shimizu K, Bentz W and Hyuga H 1988 Adu. Nucl. Phys. 18 1.
Introduction
 However, it should be appropriate for the isao-scalar magnetic moment
in LS closed shell nuclei plus or minus one nucleon, as
1.
LS-closure, no spin-orbit partners on both sides of the Fermi surface,
therefore the magnetic moment operator can not couple to magnetic
resonance.
2.
Pion-exchange current contribution turned to be very small to iso-scalar
current, as well as others processes.
 Although relativistic mean field approach has achieved great success
during the last two decades:
Serot & Walecka, Adv. Nucl. Phys. 16 (86) 1
Reinhard, Rep. Prog. Phys. 52 (89) 439
Ring, Prog. Part. Nucl. Phys. 37 (96) 193
Meng, Toki, Zhou, Zhang, Long & Geng, Prog. Part. Nucl. Phys. 2006, in press
 Straightforward application of the single-particle relativistic model
does not agree with the experimental magnetic moments
Introduction
 The Sigma and the time-component vector mesons of Omega
fails to reproduce the corresponding Schmidt values:
H.Ohtsubo, et. al., Prog. Theor. Phys. 49(1973 ) 877
Miller L D, Ann. Phys., NY 91 (1975) 40.
Bawin M, Hughes C A and Strobel G L Phys. Reu. C 28 (1983) 456.
Bouyssy A, Marcos S and Mathiot J F Nucl. Phys. A 415 (1984) 497.
Kurasawa H., et. al., Phys.Lett.B165 (1985) 234
 Taking into account the contribution of the back-flow to the
current operator can solve this problem. This back-flow is
caused by the polarization of the core by the external
particle. H. Kurasawa, et. al., Phys.Lett.B165(1985)234
J. A. McNeil, et. Al., Phys. Rev. C34(1986)746
S. Ichii, W. Bentz and A. Arima, Phys. Lett. B 192(1987)11.
J. R. Shepard, et al., Phys.Rev.C37(1988)1130
P. G. Blunden, Nucl. Phys. A 464 (1987)525
Introduction
 In these the widely investigated mean field theories there are only
the time-even fields which are most sensitive to physical observables.
 The time-odd fields, which appear only in the nuclear systems with
time-reversal symmetry broken, are very important for the description
of the magnetic moments, rotating nuclei, N=Z nuclei, and pairing
correlations.
U. Hofmann and P. Ring, Phys. Lett. B 214, 307(1988) .
J. Koenig, and P. Ring, Phys. Rev. Lett. 71, 3079 (1993).
W. Satuła, in Nuclear Structure 98, edited by C. Baktash, AIP Conf. Proc. No. 481
~AIP, Woodbury, NY, 1999!, p. 114.
K. Rutz, M. Bender, P.-G. Reinhard, and J. A. Maruhn, Phys.Lett. B 468, 1 (1999)
 The broken time reversal symmetry  a non-vanishing vector part of
the ω-field  a magnetic potential and changes the nuclear wave
function and the resulting magnetic moments.
 The magnetic field created by magnetic potential will influence the
magnetic moment, single-particle spin and angular momentum.
Introduction
 The core polarization is always neglected in Spherical cases,
 For the axial deformed case, the RMF with time-odd components are
developed and the isoscalar magnetic moment are well reproduced:
U. Hofmann and P. Ring, Phys. Lett. B 214, 307(1988) .
R. J. Furnstahl, C. E. Price, Phys. Rev. C40 (1989) 1398.
 Time-even triaxial RMF have been developed to investigate the triaxial
deformation and MD
D. Hirata, et al.., Nucl. Phys. A609, 131 (1996).
J. Meng, et al., Phys. Rev. C 2006
 Purpose: developing the time reversal symmetry broken triaxial RMF
approach, investigating the non-vanishing vector part of the ω-field,
nuclear current,, magnetic potential and magnetic moments
Starting point of RMF theory
Nucleons are coupled by exchange of mesons via an effective Lagrangian
s
(Jp T)=(0+0)
Sigma-meson:
attractive scalar field
S(r)  gss(r)
r
w
(Jp T)=(1-0)
(Jp T)=(1-1)
Omega-meson:
Short-range repulsive
Rho-meson:
Isovector field
1 - 3 0
V( r )  gww ( r ) + gr 3r ( r ) + e
A (r)
2
Serot & Walecka, Adv. Nucl. Phys. 16 (86) 1
Ring, Prog. Part. Nucl. Phys. 37 (96) 193
0
0
Reinhard, Rep. Prog. Phys. 52 (89) 439
Meng, Toki, Zhou, Zhang, Long & Geng,
Prog. Part. Nucl. Phys. 2005, in press
Lagrangian of RMF theory
1 - 3




L  i  i   - M - gs s - gw   w - gr   r  - e
 A i
2


1
1

+   s s - U(s) -  + U(w)
2
4
1   
1

1
1
1
- RR + U(r) - FF 
U (s   ms2s 2 + g 2s 3 + g 3s 4
2
3
4
4
4
2
1
1
U (w   mw2w w + c3 (w w 
p
Jp
0-
T
1
s
0+
0
w
r
11-
0
1
meson
2
4
2
1
1
U ( r   mr2 r   r  + d 3 ( r   r  
2
4
    w -  w



 R   r -  r - gr (r  r 
 F  A - A
 
 
 
Equations of Motion
    

(

  p - V ( r )  + V( r ) +  M + S( r ) i  ii
Magnetic field
V( r)  g w0 ( r) + g  r0 ( r) + e 1 - 3 A0 ( r) Same footing for
w
r 3

2
 Deformation



1 - 3 

 Rotation
A( r)
V( r)  gw ω( r) + gr 3 ρ( r) + e
2
 Pairing (RHB,BCS,SLAP)

Magnetic potential
S(r)  gs s( r)
…

-space-like components of vector mesons
Nuclear magnetism
-behaves in Dirac equation like a magnetic field
(    + ms2 )s  - gsr s - g2 s2 - g3 s3

μ


(    + mw2 ) w  gw jv - c3 w w w

μ

2

(


+
m
)

g
j
 
r r
r R
  
   A ( r)  ejP
r s ( r)  A i ( r)i ( r)
i1

+

j ( r)  A 
(
r
)

i ( r)
v
i1 i

A +
 

j
(
r
)


(
r
)

i ( r)

R

i1 i
 
 1- 
A
+
3
j
(
r
)


(
r
)
i ( r)
c


i1 i

2
Numerical techniques for time reversal invariance violation
Nucleon
 1 
 (r , s)  nx ( x)ny ( y )nz (Tz ) (r , s) nx +-
1
(r , s)
2 ( -1)


T in y(r , s) nx + ny +1 (r , s1) 
 (r , s)  n x ( x)n y ( y )n z ( z )
( -1)


nx 
2
(
1)
T  -is K 
i
Expanded on 3D HO Basis
 nf

j
  f   ( r , s ) 

  (t )
 j ( r , s, t )  
n
 g
 t
j
  ig  ( r , s ) 




 

 nf
 f

 j ( r , s, t )  
 ng
  ig

 
y
T j (r , s, t )  - j (r , s, t )

(
r
,
s
)


  (t )
 t
j
  ( r , s ) 


j
ny
T j (r , s, t )   j ( r , s, t )

Dirac equation
meson
 ( x)    n ( x)n ( y )n ( z )

x
y
z
where   s , w , r
Coulomb field: the standard Green function method
Nuclear Magnetic potential: vector part of the
ω-field
Nuclear Magnetic Fields due to the vector part
of the ω-field
Magnetic field B = w at y=z=1.29 fm
Single nucleon levels with time reversal
invariance violation
Single nucleon levels with time reversal
invariance violation
Density distribution of the last odd nucleon
The other degree of
freedom was
integrated.
Density distribution for proton, neutron and
matter
Magnetic Moment in Relativistic approach
2Mc 2 1
m
( r   ) +  
c 2
Magnetic moment

 
A

i 1

 f (r , s) 
 ( r , s, t )  
 t (t )
 ig (r , s) 


nucleon wave function
i



i
Relativistic effect
 2Mc2  1 +

+
   d r
 i (r)r  i (r) +  i (r)i (r) 


 hc  2
3
A
A
 2Mc2 1

D
   d r
r   ji (r) +   jiA (r)
i 1
i 1
 hc 2

3
D
A
Dirac current j (r )   + (r ) (r ) Anomalous current j (r )   + (r )   (r )
Magnetic Moment
magnetic moments of light nuclei near closed shells (N)

15O
17O
39Ca
41Ca
15N
17F
39K
41S
Exp.
0.72
-1.89
1.02
-1.60
-0.28
4.72
0.39
5.43
Schmidt
0.64
-1.91
1.15
-1.91
-0.26
4.79
0.12
5.79
Spher.
0.66
-1.91
1.17
-1.91
-0.03
5.05
0.72
6.32
Axial
0.65
-2.03
0.96
-2.13
-0.29
4.99
0.33
6.07
Triaxial
0.57
-2.00
0.98
2.13
0.19
4.89
0.37
6.04
Spherical and axial RMF results with NL1 taken from Hofmann 1989
Triaxial RMF with PK1
Magnetic Moment
Iso-scalar magnetic moment (N)
  (Z, N + (Z + 1, N - 1/ 2
s
A
Schmidt
Landau
Spher.
Axial
RHA
Triaxial
Exp.
15
0.19
0.19
0.32
0.18
0.20
0.19
0.22
17
1.44
1.41
1.57
1.48
1.44
1.45
1.41
39
0.64
0.64
0.94
0.64
0.66
0.67
0.71
41
1.94
1.91
2.21
1.97
1.95
1.96
1.92
Landau: taking into account the current by linear response theory
Magnetic Moment
Iso-vector magnetic moment (N)
  (Z, N - (Z + 1, N - 1/ 2
V
A
Schmidt
s-w+config. Spher
mixing
Triaxial
Exp.
15
0.451
0.357
0.345
0.376
0.501
17
-3.353
-3.487
-3.480
-3.446
-3.303
39
0.512
0.217
0.225
0.305
0.312
41
-3.853
-4.141
-4.115
-4.086
-3.513
s-w including config. mixing: Y Nedjadi and J R Rook, J. Phys. G: Nucl. Part.
Phys. 15 (1989) 589
U. Hofmann, P. Ring, Phys. Lett. B214 (1988) 307
Dirac and Anomalous parts of Magnetic Moment (N)
D
15O
17O
39Ca
Schmidt
-
-
-
Spher.
-
-
41Ca
15N
17F
39K
41S
-
0.33
3.00
1.20
4.00
-
-
0.59
3.26
1.81
4.54
Axial
-0.13
-0.13
-0.30
-0.22
0.44
3.21
1.50
4.29
Triaxial
-0.11
-0.13
-0.16
-0.28
0.46
3.15
0.64
4.31
A
15O
17O
39Ca
41Ca
15N
17F
39K
41S
Schmidt
0.64
-1.91
1.15
-1.91
-0.60
1.79
-1.08
1.79
Spher.
0.66
-1.91
1.17
-1.91
-0.62
1.79
-1.09
1.79
Axial
0.78
-1.90
1.26
-1.87
-0.73
1.78
-1.17
1.79
Triaxial
0.68
-1.86
1.13
-1.85
-0.64
1.75
-1.00
1.73
Nuclear current in 17F and 17O in y-z plane
Dirac current
Anomalous nuclear current in 17F and 17O in y-z plane
Dirac and anomalous current in 17F
Dirac current
Anomalous current
Summary and perspective
 Triaxial RMF without time reversal symmetry is developed
 Ground-state properties of light odd mass nuclei near
double-closed shells, i.e., E/A, single-particle energy,
density distribution, etc., are calculated self-consistently
 The broken time reversal symmetry leads to a non-vanishing
vector part of the ω-field, which creates a magnetic
potential and changes the nuclear wave function and the
resulting magnetic moments.
 The first calculated nuclear magnetic moments of light LS-
closed shells nuclei plus or minus one nucleon agree well with
the Schmidt values and the data.
奇核子系统问题-时间反演对称性破缺
正确确定激态及价核子组态-绝热与非绝热约束计算
constraints
MD
奇核子系统问题-时间反演对称性破缺
s.p. levels in 106Rh
奇核子系统问题-时间反演对称性破缺
和 40Ca 的中子（左）和
质子（右）的单粒子能级
41Ca
考虑磁势后41Ca 中互为时
间反演态的能级劈裂
How to define the single-particle property
in dense, strongly interacting many-body system?
 Laudau and Migdal answer:
The responds of the system as a whole when a quasi-particle is removed.
Thus, the single quasi-particle current is defined as the difference in
the total baryon current when the particle is removed.
ji 
 JB
,
 ni
J B   n j j  j  
j
j
nj
Ej
( k j - gw J B )
Relativistic extension of Landau’s Fermi-liquid theory
based on sigma-omega model
J. A. McNeil, et. Al., Phys. Rev. C34(1986)746
s -w
Nucleon:
model
Meson fields:
Total
current
w  g 2w / m 2w
Self-consistent
Dirac equation:
m*
Landau quasiparticle current:
Enhancement is reduced
Especially
T=0 K:
j i |T 0
0.6m
Backflow effect
Renormalization of current
k 2i -1
k i [m + w r0 - s r s +
]
r0   + , rs   
2m
Remark: The cancellation of the scalar enhancement due to the vector meson
Spin ½ particle
One-body matrix
element of current
-ig
Vector
fields
Vertex correction
g
g
Renormalized current
Transfer momentum
Electric
form factor
magnetic
form factor
q  p '- p  0
Remark: Dirac current is related to the electric form factor!
Relativistic extension of Landau’s Fermi-liquid theory
J. A. McNeil, et. Al., Phys. Rev. C34(1986)746
The relativistic wave functions are obtained from a relativistic
Woods-Saxon well with parameters adjusted to give the
separation energy and elastic electron scattering form factor.
The interaction vertex is renormalized by consideration of
backflow effect in nuclear medium, namely,
3
2
 R   {1 + w r0 [( p 2 r0 )3/ 2 + (m*) 2 ]-1/ 2 }-1
Effective (renormalized) Dirac current
j
D
R
(r )   (r ) R (r )
Remark1: the wave function and the interaction vertex are not consistent!
Remark2: The anomalous current is not renormalized in this paper.
Remark3:
The renormalization are considered without the consideration of
iso-vector meson fields, i.e., rho and pi, thus the iso-vector current and
magnetic moment are still enhanced. Even if rho meson is considered, the
enhancement of iso-vector current still can not be reduced significantly
because of the small rho-N coupling constant.
P. G. Blunden, Nucl. Phys. A 464 (1987)525
Comment:
mw
mr
r / w  1/ 4 - 1/ 9
Remark4:
In additional, the anomalous iso-vector moment, which is
much larger that the Dirac moment, does not get affected
by the scalar field, so that the total iso-vector spin moment
will not be enhanced much.
Magnetic Moment in Relativistic approach
Dirac current
D
j (r )   + (r ) (r )
can be decomposed into an orbital current and a spin current
1

 (r )  (r )  *  (r )[( p '+ p)  - is  q ] (r )
2m
spin
convection
Anomalous current
A
j (r )   + (r )   (r )
Iso-scalar and iso-vector magnetic moment
1 Mc 2
s 
(r   ) +  (0)  
2 c
 3 Mc 2
v  - [
(r   ) +  (1)  ]
2
c
 (0)  -0.12 N
 (1)  3.706 N
 3  +1, n
 3  -1, p
Introduction
 Extensive shell model calculations within the full Ohw shell-model
space show good agreement between theoretical and observed values.
The remaining deviations arising from higher order corrections, i.e.
meson exchange currents, isobar currents and higher-order
configuration mixing, are removed through the use of effective
operators to be determined empirically: Arima A, Shimizu K, Bentz W and
Hyuga H 1988 Adu. Nucl. Phys. 18 1. Brown B A and Wildenthal B H 1983 Phys. Reu. C 28
2397.
 Relativistic s-w model + the configuration mixing within one major shell for the
mirror pairs, 150-15N, 170-17F, 39K-39Ca and 41Ca-41S, removes most of the
discrepancies for isovector moments while leaving the isoscalar moments unaltered,
i.e. also in agreement with experiment when vertex corrections are included. For
isovector moments, this agreement is better than in similar non-relativistic
calculations: Y Nedjadi and J R Rook, J. Phys. G: Nucl. Part. Phys. 15 (1989) 589-600.
奇核子系统问题-时间反演对称性破缺
 奇核子系统: 未配对核子破坏时间反演对称性, 从而导致矢量介
子场的空间部分不为零, Dirac 方程中出现磁势
 球对称: 奇A核处理成偶偶核额外加入一个核子, 体系核子波函
数仍具有球对称性. 无法考虑时间反演对称性破缺对整个原子核
的影响
 轴对称: Hofmann等人(88) 和 Furnstahl等人(89)自洽地考虑了
磁势, 研究了核芯极化效应对整个原子核性质的影响. 这种核芯
极化效应能抵消标量场引起的相对论效应对同位旋标量磁矩的
增强, 给出与 Scnmidt 值一致的原子核磁矩.
 本工作: 三轴形变框架下研究时间反演对称性破缺