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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
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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
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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 ...
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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.
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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
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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
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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
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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
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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
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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)
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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)
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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
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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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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.
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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
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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
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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