Hidden symmetries in atomic nuclei: a brief introduction

Jie Meng 孟 杰

北京大学物理学院

School of Physics, Peking Universitymengj@pku.edu.cn

iTHES workshop on "Exploration of hidden symmetries in atomic nuclei“Saturday, 27 July 2013 from 9:00 to 18:00at RIKEN Wako Campus

two-nucleon separation energy

ΔS2p

much smaller after 8, 20, 28, 50, 82, 126

ΔS2n

Nuclear magic number

Nuclear properties change periodicallywith proton or neutron number原子核性质随中子数或质子数周期性地变化

ARn + A-4Po TSudden rise at N = 126

Neutron capture cross section

Very small at N = 28, 50, 82, 126

Abrupt change in nuclear radius at N = 20, 28, 50, 82, 126R

Ravg

RRavg

Nuclear magic number

Only smallest magic numbers are reproduced. Why ?

Spin-orbit term is necessary

Mayer and Jensen et al., 1949

srrV

rVV sso .)(1

M. G. Mayer, Phys. Rev. 75(1949)1969; 78(1950).O. Haxel, J. H. D. Jensen, and H. E. Suess, Phys. Rev. 75(1949)1766;Z. Phys. 128(1950)295;

原子核壳结构 Nuclear Shell Model

Great for:magic numbersground state propertiessome low lying excited states

强自旋-轨道壳模型是原子核结构的所有微观理论的出发点

S. G. Nilsson的轴对称变形核的强自旋-轨道壳模型能级系

S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No.16(1955).S. G. Nilssion, et al., Nucl. Phys. A131(1969) 1.

原子核壳结构 Nuclear Shell Model

2013-07-27 6

s1

p1

d1s2

f1p2g1

s3

h1

2/1

2/3

2/12/5

2/32/72/32/1

2/1

2/52/9

2/72/52/32/12/11

2/9

2

6

1614

20283240

8

3850

5864687082

92

d2

3/21/2,p~

5/23/2,d~

3/21/2,p~

7/25/2,f~

2

321

,2,,,1

ljlnljln

0

1

2

3

4

1/2s~

2/1~:spinpseudo1~:orbitpseudo

sll

Hecht & AdlerNPA137(1969)129

sl

Arima, Harvey & ShimizuPLB30(1969)517

Spin and pseudospin symmetry in atomic nuclei

Woods-Saxon

2013-07-27 7A. Bohr, I. Hamamoto, and B. R. Mottelson, Phys. Scr. 26,267 1982

Earlier attempt to understand PSS

Earlier attempt to understand PSS

Since PSS was suggested, lots of nuclearphenomena have been interpreted inconnection with the PSS, for example …

Nuclear superdeformed configurations

Interpretation of the identical bands

Observation of the PSS partner bands…Meng and Zhang, JPG37, 064025 (2010)

Even for Magnetic moments, transitions and γ-vibrational states

On the validity of the pseudo-spin concept for axially symmetric deformed nucleiD. Troltenier, W. Nazarewicz, Z. Szymański, J.P. Draayer, Nucl. Phys. A 567, 591 (1994).

l-forbidden M1 transitions and pseudospin symmetryP. von Neumann-Cosel and J. N. Ginocchio, Phys. Rev. C 62, 014308 (2000)

Neutron number dependence of the energies of the γ-vibrational states in nuclei with Z∼100 and the manifestation of pseudospin symmetryR. V. Jolos, N. Yi. Shirikova, and A. V. Sushkov, Phys. Rev. C 86, 044320 (2012)

Nucleon Dirac Equations

15 2013-07-27

Relativistic mean field / Covariant density functional theory

( ) ( ) i i iV M S p r r

33

1( ) ( ) ( ) ( )2

( ) ( )

V g g e A

S r g

r r r r

r

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Dirac Spinor quantum numbers

2/1

2/1

)(~)2/1()(

2/1

lj

lj

ll

jlj

)()(

)()(1)( ~~

ljmn

ljmnN

mnYrF

YrGi

r

r

Pseudo quantum numbers are nothing but the quantum numbers of the lower component.

21

21

2/1 ~,1~,2~,0,2

s2ljln

ljln

21

21

2/3 ~,1~,2~,2,1

d1ljln

ljln

3/21/2,2/32/1 p~d1,s2

1 number node n

2/3,2/1p~2~ n

Ginocchio PRL78(97)436

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Origin of the symmetry - Nucleons

FFVMdr

dVrMr

lldrd

drdV

Mdrd

M N

~

41)1~(~

21

21

222

2

For nucleons, d[V(r)S(r)]/dr=0 spin symmetry d[V(r)+S(r)]/dr=0 pseudo-spin symmetry

GGVMdr

dVrMr

lldrd

drdV

Mdrd

M N

222

2

41)1(

21

21

)()()( rSrVrV

)(),( rVMrM NN

)()(

)()(1)( ~~

ljmn

ljmnN

mnYrF

YrGi

r

r

Meng, Sugawara-Tanabe, Yamaji, Ring, Arima, PRC58 (1998) 628R Meng, Sugawara-Tanabe, Yamaji, Arima, PRC59 (1999) 154

2013-07-27 18

Origin of the symmetry - Anti-nucleons

GGVMdr

dVrMr

lldrd

drdV

Mdrd

M

~

41)1~(~

21

21

222

2

FFVMdr

dVrMr

lldrd

drdV

Mdrd

M

222

2

41)1(

21

21

)()()( rSrVrV

)(),( rVMrM AA

For anti-nucleons, d[V(r)S(r)]/dr=0 pseudo-spin symmetry d[V(r)+S(r)]/dr=0 spin symmetry

)()(

)()(1)( ~~

ljmn

ljmnA

mnYrGi

YrF

r

r

Zhou, Meng & Ring PRL92(03)262501

The factor is ~400 times smaller for anti nucleons!

241

M

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Origin of the symmetry

FFMlrSrVdrd

rMrSrV

M A

1)()(14

1)()(2 2*

2

FFMlrSrVdrd

rMrSrV

M N

~~1)()(1

41)()(

2 2*

2

For nucleons, the smaller component F

For anti-nucleons, the larger component F

Zhou,Meng&RingPRL92(03)262501

2013-07-27 20

Origin of the symmetry - Anti-nucleons

Zhou, Meng & Ring, PRL92 (03) 262501

0 1 2 3 4 5r [fm]

0

200

400

600

800

1000M

−V +

(r) [

MeV

]

0 1 2 3 4 5

880

900

920

940

sp

df

g

h

ij

kl

s

pd

16O neutron

p1/2 p3/2

2013-07-27 21

Experimental verification: Anti- in 16O

Song, Yao, Meng, CHIN. PHYS. LETT. Vol. 26, No. 12 (2009) 122102

0 1 2 3 4 5

-500

-400

-300

-200

-100

0

0 1 2 3 4 5

-50

-40

-30

-20

-10

0

k

j

i

h

g

f

d

ps

r (fm )

S+V (MeV) anti-L in

16O

d

p

sL in

16O

Experiment: exploration of the relativistic symmetry by hyperon

2013-07-27

Summary / Introduction

Thank you for your attention!22

Origins of nuclear shell structure Spin symmetry in Dirac negative energy spectrum Perturbative interpretation of relativistic symmetries possible Pseudospin symmetry in resonances Pseudospin symmetry in supersymmetric quantum mechanics Pseudospin for deformed nucleus …

Liang, Shen, Zhao, Meng, Phys. Rev. C 87, 014334 (2013) [13 pages] B.-N. Lu, E.-G. Zhao, and S.-G. Zhou, Phys. Rev. Lett. 109, 072501 (2012). Bing-Nan Lu, En-Guang Zhao, Shan-Gui Zhou arXiv:1305.1524

Kenichiro Arita (Nagoya Institute of Technology) Haozhao Liang (RIKEN) Shan-Gui Zhou (Institute of Theoretical Physics, CAS, China)…