1

Nuclear charge radii of exotic nuclei

and superheavy nuclei from

experimental decay data

Zhongzhou Ren

1Department of Physics, Nanjing University, Nanjing, China

2Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou,

China

2

Outline

• The history of determining nuclear radii

• Charge radii of heavy and superheavy

nuclei from alpha-decay data

• Charge radii of exotic nuclei from the data

of proton emission and cluster emission

• Summary

3

Research background

• 1911 Rutherford : existence of nucleus in an atom by alpha scattering experiment.

• In 1950s, electron scattering on nuclei has been used to probe nuclear density distributions and radii.

• Other methods (p, μ …) were also used for researches of nuclear radii.

• Since1950s , charge radii of many stable nuclei were obtained by electron scattering ...

4

Nuclei:

S, L, P, T

Lifetimes T½ & BR

Energy B, Sp, Sn, Qα… Radius R, Rn, Rc

Using nuclear decay models, extract nuclear radii

of superheavy nuclei and exotic nuclei from the

experimental decay data.

5

Idea: α decay half-life is sensitive to

charge density distribution

Proton emission (Z≥51)

Alpha decay (Z≥52)

Cluster emission (Z≥87)

Spontaneous fission (Z ≥90)

α decay: early days of

nuclear physics.

α decay half-life is

sensitive to the Coulomb

potential and charge

density of daughter nuclei

6

First result on charge radii of superheavy nuclei by decay data

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GDDCM for alpha decay

2 2 2

2 2

( 1)( ) ( ) ( ) ( )

2 2N C n j n j

dV r V r u r E u r

dr r

In the cluster representation, we solve the stationary

S-eq describing the relative motion of the cluster

with respect to the core nucleus

The nuclear and Coulomb potentials between cluster and daughter are numerically constructed in the double-folding model.

1 2 1 1 2 1 2 2( ) ( ) ( | |) ( )NorCV r drdr r s r r r r

8

The density distribution of spherical alpha-particle (e-A scattering) is

The density distributions of spherical core has the Fermi form

is fixed by integrating the density distribution equivalent to mass number of nuclei.

2

1 1 1( ) 0.4299exp( 0.7024 )r r

1

1/3

2 2 0 0( , ) 1 exp ,r R

r R r Aa

0

9

Correlation between radii and decay data

2point

0 2

4( ) ( ) ( ) ( ) ( )N C C n jP F kr V r V r V r u r dr

k

1/24

1/2 22

2

2

( )

( )

r r drR r

r r dr

Alpha-decay half-life

RMS charge radius

Density distribution of daughter nuclei

radius r0 and diffuseness a

1/2 ln 2T

10

1

2 0

1/3

0 2 20 4 40

( )( , ) 1 exp

( ) [1 ( ) ( )]

r Rr

a

R r A Y Y

1/24

2

2

2

( , ) sin

( , ) sin

r r drdR

r r drd

/2

0( )sin d

Attempts to include nuclear deformation

The effect of nuclear deformation on half-lives can be

evaluated by integrating the partial width along the direction

The rms charge radius is calculated as

Axially deformed

density distribution

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Dependence of the theoretical results on the radius

parameter of charge density distribution: (a) alpha-decay

half-life of 212Po, (b) rms charge radius of 208Pb

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Dependence of the theoretical results on the diffuseness

parameter of charge density distribution: (a) alpha-decay

half-life of 212Po, (b) rms charge radius of 208Pb

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The diffuseness a is fixed at its standard value of 0.54 fm because the results show weak sensitivity to it.

The parameter r0 can be considered as the connection between decay half-lives and radii.

Key points of our calculations

The r0 value is exactly extracted to reproduce the

available experimental data of alpha-decay half-lives.

Next, the rms charge radius of the daughter nucleus is

evaluated from the density distribution with the

resulting r0 value.

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Comparison of the extracted rms charge radii with the

experimental data versus the mass number A for even-

even nuclei with Z=58-96

1/2

812

expt calc

1

81 0.1284 fmi i

i

R R

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List of the extracted rms charge radii together with the

error bars for even-even nuclei with Z=98-116. Note that

the experimental data for these nuclei are not available.

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Simple formula for nuclear charge radii

Rt RC0

V(r)

V0

Q

r

1/2 0

2

ln 2 /

2exp 2 [ ( ) ]

,

C

t

R

c t

R

C d

T P FP

P V r Q dr

R Z Z e Q R cR

1/2

10 0 10 1/2 1 2 1

2

1 2

4 ln10 log ln 2 log

, 2c d

cR P F T Q

Z Z e

1/2

1 2 10 1/2 1 2 3 1logR X X T X Q

X1, X2, X3 are the parameters to be determined

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Through a least-square fit to the available charge radii

for even-even nuclei with Z≥82, N≥126, the three

parameters are determined as follows:

1

2

3

15.8767(942)

0.6213(30)

0.7975(26)

X

X

X

The standard deviation of the calculations is

1/2

292

expt calc

1

29 0.0557 fmi i

i

R R

The formula is not only simple in form but also easy

to see the physical meanings.

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from nuclei Z≥82, N≥126

from actinide nuclei 89<Z<97

Extracted charge radii for even-even nuclei with Z=98-116

within the different models

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Isotopic trend of the extracted rms charge radii for

even-even Cf isotopes, which is correlated with the

deformed N=152 subshell effect on alpha decay.

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PRC 89 (2014) 024318: Nuclear charge radii of superheavy

odd-mass and odd-odd nuclei from α-decay data

Element A Rexpt (fm) Rcalc (fm) Rform (fm)

Hg 187 5.40 5.35 5.44

Tl 191 5.42 5.39 5.38

Pb 189 5.42 5.27 5.31

Pb 191 5.42 5.32 5.37

Pb 193 5.43 5.33 5.40

Pb 195 5.44 5.39 5.47

Pb 197 5.44 5.37 5.48

Pb 201 5.46 5.38 5.51

Pb 203 5.47 5.29 5.44

Pb 209 5.51 5.48 5.47

Pb 211 5.53 5.65 5.66

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Experimental and extracted rms charge radii

for odd-A and odd nuclei (I)

Element A Rexpt (fm) Rcalc (fm) Rform (fm)

Bi 203 5.49 5.38 5.51

Bi 205 5.50 5.26 5.42

Bi 209 5.52 5.55 5.48

Fr 213 5.60 5.73 5.64

Tb 147 4.92 5.07 4.95

Tb 149 4.94 4.95 4.90

Ho 151 5.04 5.25 5.16

Tm 153 5.06 5.10 5.02

Tb 148 4.93 5.00

Tb 150 4.95 4.84

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Experimental and extracted rms charge radii

for odd-A and odd nuclei (II)

Nucleus Rcalc (fm) ∆Rcalc (fm) Rform (fm) ∆Rform (fm)257No 6.21 0.16 6.19 0.14

255Lr 5.60 0.13 5.64 0.11

267Rf 5.73 0.27 5.87 0.23

267Db 5.74 0.43 5.84 0.36

259Sg 5.68 0.16 5.67 0.13

261Sg 6.13 0.03 6.07 0.03

269Sg 5.85 0.27 5.93 0.23

271Bh 5.79 0.21 5.85 0.19

263Hs 6.20 0.28 6.07 0.22

265Hs 5.74 0.13 5.74 0.11

269Hs 5.86 0.16 5.86 0.13

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Extracted rms charge radii for odd-A superheavy

nuclei with Z=102-115 (I)

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Nucleus Rcalc (fm) ∆Rcalc (fm) Rform (fm) ∆Rform (fm)275Hs 5.78 0.12 5.91 0.10

275Mt 5.73 0.15 5.64 0.34

277Ds 5.76 0.26 5.88 0.22

279Ds 6.18 0.13 6.27 0.11

281Ds 6.19 0.15 6.31 0.13

279Rg 6.19 0.34 6.26 0.29

281Rg 5.89 0.50 6.03 0.43

285Cn 6.28 0.15 6.38 0.12

283113 6.20 0.22 6.28 0.18

285113 6.01 0.27 6.15 0.23

289115 6.24 0.37 6.33 0.32

Extracted rms charge radii for odd-A superheavy

nuclei with Z=102-115 (II)

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Extracted charge radii for odd-A superheavy nuclei with

Z=102-115 within the different models

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List of the extracted rms charge radii together with the

error bars for odd-odd superheavy nuclei with Z=105-115.

2/3 4/3 1/3

0 1 2( )R c c A c A A

I. Angeli, At Data Nucl. Data Tables 87, 185 (2004)

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PRC 87 (2013) 054323: Nuclear charge radii from

decay data of cluster and proton emissions

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Extracted rms charge radii of light neutron-rich nuclei

2/3 4/3 1/3

0 1 2( )R c c A c A A

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Summary

• A new way to investigate nuclear size:

Heavy and superheavy nuclei with Z=98-116,

proton-rich nuclei with Z=68-82,

light neutron-rich nuclei

• Other common methods such as electron scattering

are not available for these nuclei.

• Their charge radii are respectively extracted through

alpha decay, proton emission, cluster emission.

• This is the first result on nuclear charge radii of

superheavy nuclei and some exotic nuclei based on

nuclear decay data.

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Thanks for discussion with Prof.

Oganessian on nuclear radii.

Thanks for your attention!

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Modified two-approach (MTPA) for decay

The cluster (particle)-daughter potential is divided

into two regions by the separation radius R, one

introduces two auxiliary potentials:

S. A. Gurwitz, P.B. Semmes, W. Nazarewicz and T.Vertse, PRA 69 (2004) 042705

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Once the bound state wave function is solved

in the potential U(r), the decay width is obtained as

( )n j r

r r

The value of is chosen well

in such a way that the

potential V(r) can be well

approximated by the repulsive

part (i.e. the attractive part

disregards) for

22 ( )

( )

n j rk

G kr

r

S. A. Gurwitz, P.B. Semmes, W. Nazarewicz and T.Vertse, PRA 69 (2004) 042705

35

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Extracted rms charge radii of 290116 with various P0

values. Based on the experimental rms radii, the P0

factor is taken as 0.1265 for all even-even nuclei.

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40

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Current researches about electron scattering

With the development of radioactive ion beam facilities,

it is possible to produce short-lived exotic nuclei and

investigate their properties in laboratories

Facilities under

construction:

RIKEN in Japan

(133Cs)

GSI in Germany

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Electron scattering on unstable nuclei

Configuration of the

SCRIT-based radioisotope-

electron scattering system

in RIKEN

Facilities for short-lived

unstable nuclei in Tohoku

University

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Available methods to measure nuclear charge radii

• (1) Transition energies in muonic nuclei

• (2) Elastic electron scattering experiments

providing information on charge radii R

• (3) Kαx-ray isotope shifts (KIS)

• (4) Optical isotope shifts (OIS)

providing information on isotopic changes δR

• The (1-3) methods have been performed only on stable nuclei (several tens of milligrams of a target material are required)

• The (4) method can be performed for radioactive atoms with lifetimes down to 1 ms.

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High energy electron scattering

Measuring nuclear charge radii

e-A electron scattering apparatus(Hofstadter 1954 )

Cross Section of scattering electron

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Proton emission from drip-line nuclei

The residual daughter nuclei are

very proton-rich with short

lifetimes. So the known methods

to measure their radii are not

available at present.

The interaction potentials between proton and daughter are numerically constructed in the single-folding model.

1 1 1 1

1/301 1 0

1

( ) ( ) ( | |)

( ) ,1 exp ( ) /

NorCV r dr r s r r

r R r Ar R a

＋

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Charge radii from proton emission

Proton emssion half-life

RMS charge radius

1/2 ln 2 ppT S

The decay width Γp is calculated using the modified two-

potential approach with the single-folding potential.

The spectroscopic factor Sp is calculated using the

relativistic mean-field theory.

The r0 value is exactly determined to reproduce the available experimental half-lives of proton emission.

Next, the rms charge radius of the daughter nucleus is evaluated from the density distribution with the resulting r0 value.

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2/3 4/3 1/3

0 1 2( )R c c A c A A

I. Angeli, At Data Nucl. Data Tables 87, 185 (2004)

Comparison of the extracted rms charge radii from the

proton-emission data with the results of the formula for

the proton-rich nuclei with Z=68-82.

49

Extracted rms charge radii of proton-rich nuclei with

Z=68-82. The available data for 184Pb are also shown.

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Cluster emission in the trans-lead region

The residual daughter nuclei are

near 208Pb with long lifetimes.

Their radii are already known. We

pay attention to the emitted

clusters that are neutron-rich.

The interaction potentials between cluster and daughter are numerically constructed in the double-folding model.

＋

1 2 1 1 2 1 2 2( ) ( ) ( | |) ( )NorCV r drdr r s r r r r

1,21/30

1,2 1,2 0

1,2

( ) ,1 exp ( )

r R r Ar R a

51

Charge radii from cluster emission

Cluster emssion half-life

RMS charge radius

1/2 ln 2 cT P

The decay width Γ is calculated using the modified two-

potential approach with the double-folding potential.

The cluster preformation factor is given by

D. Ni, Z. Ren, T. Dong, and C. Xu, PRC 78 (2008) 044310

The density distribution of daughter nuclei are specified

by their experimental charge radii.

The r0 parameter of the cluster density distribution is

exactly determined to reproduce the experimental half-

lives of cluster emission.

1/2

10log ( )c c dP b Z Z c

52

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Referee’s report on the manuscript

… In the present paper, a completely new method to

determine the nuclear radius is presented … the

present method can be a powerful tool to determine

the nuclear radius in nuclei …

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PRC论文引用举例 (一)：

In the introduction, our works [19-21] are emphasized.

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PRC论文引用举例 (二)：

Their results for 286114 and 290116 are comparable with

our results [37].

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PRC论文引用举例 (三)：

In the introduction, Our works [6,7] are cited.

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Comparison of the extracted rms charge radii with the

experimental data versus the mass number A for odd-A

nuclei with Z=65-87