New G
N
Geiger-Nuttall
i
N tt ll law
l
off alpha
l h
decay
y and new law of double-beta
decay
Zhongzhou REN (任 中洲)
• Department of Physics, Nanjing
University, Nanjing, China
1
Outline
• Review on the Geiger-Nuttall of alpha decay
• New Geiger-Nuttall (G-N) law
• New law of double beta decay
• Summary
2
Review on decay (alpha,
(alpha cluster)
Proton radioactivity (Z≥51)
Alpha decay (Z≥52)
Cluster radioactivity (Z≥87)
Spontaneous fission (Z ≥90)
α decay: early days of
nuclear physics (1896,
Becquerel; Curies…).
Rutherford: three kinds
of radioactivity, alpha,
beta gamma
beta,
gamma; e
existence
istence
of nucleus by alpha
scattering.
g
3
从这张图里你能发现什么问题？
Page 120-125 Geiger-Nuttall law：Relation between
alpha-decay
l h d
energies
i and
d alpha-decay
l h d
half-lives
h lf li
4
Geiger-Nuttall law for half-lives of α-decay
ZD
log10 T = a
+b
E
•
H. Geiger and J.M. Nuttall "The ranges of the α particles from
various radioactive substances and a relation between range and period
of transformation,"
, Philosophical
p
Magazine,
g
, Series 6,, vol. 22,, no. 130,,
613-621 (1911).
•
H. Geiger and J.M. Nuttall "The
h ranges off α particles
i l from
f
uranium," Philosophical Magazine, Series 6, vol. 23, no. 135, 439-445
(1912).
G. Gamow "Zur Quantentheorie des Atomkernes" (On the quantum
theory of the atomic nucleus)
nucleus), Zeitschrift für Physik,
Physik vol.
vol 51
51, 204
204-212
212
(1928).
1 First: quantum mechanics (Atom) to Nuclear Physics
1.First:
2. beta decay(GT) 3.Big bang 4.Biophysics 5.play? 6.科普
Rext
↓
Ψint →
Internal region
External region
← NΨext
Al h 衰变 著名的盖格 努塔尔定律
Alpha衰变:著名的盖格-努塔尔定律
Z
LogT = c1
+ c2
E
1911年, 由盖格和努塔尔发现 上述推导思想是由
Gamow以及Condon和Gurrey在1928年给出
以及
和
在
年给出
G. Gamow, Z. Phys. 51 (1928) 204
E. U. Condon and R. W. Gurrey, Nature 122
(1928) 439
核中的alpha衰变: 一般 2<E<12 MeV
Vmax ~ 30 MeV
V
E
0
4He
r1
r2
r
There are more than 400 nuclei that exhibit the
alpha decay phenomenon (yellow one)
alpha-decay
one).
9
Review on cluster radioactivity
y
• 1980 Săndulescu, Poenaru, and Greiner (theoretical
prediction) , Sov. J. Part. Nucl. 11 (1980) 528
• 1984 Rose and Jones (experimental observation 14C
from 223Ra), A new kind of natural radioactivity,
Nature 307 (1984) 245
• 1984
1984-2001:
2001 ffrom 221Fr
F to
t 242Cm;
C
C
C, O
O, F
F, N
Ne, M
Mg, Si
radioactivity (14C—34Si)
• 2008: radioactivity of 223Ac by 14C and 15N emissions,
y Conf. Ser. ((2008)) 111012050…
J. Phys.:
10
The first experiment
p
on the cluster emission
223Ra→209Pb+14C: Nature 307 (1984) 245.
Cluster
Parent nucleus
Core
14
C,20O,
C
O 24Ne…
Ne
•
The experimental data of cluster radioactivity
from 14C to 34Si have been accumulated in past
years.
years
•
Cluster emitters:
221Fr, 221,222,223,224,226Ra,
228 230Th,
228,230
232 233 234U,
236 238Pu,
Th 231Pa,
P 232,233,234
U 236,238
P 242Cm…
C
•
Systematic analysis on these data is needed for
cluster
l t radioactivity.
di
ti it
Focus on researches of my group
Formulas of half-lives (My talk today):
1 Half-lives
1.
H lf li
off cluster
l t radioactivity
di
ti it (PRC2004)
2. Unified formula of half-lives for alpha decay and
cluster radioactivity
y ((PRC2008))
3. New Geiger-Nuttall law of alpha-decay half-lives:
effects of quantum numbers (PRC2012) …
Theoretical models (PRC2004-2014…):
1. Density
Density-Dependent
Dependent Cluster Model for spherical nuclei
2. DDCM for deformed nuclei
3. Generalized DDCM
4. Multi-Channel Cluster Model (MCCM) for even-even,
odd-A, and odd-odd nuclei
13
Alpha decay formulas
• The Geiger-Nuttall law
•
log10(T1/2)=aQ-1/2+b
• The Viola-Seaborg formula:
•
log10(T1/2)=(aZ+b)Q-1/2+cZ+d+h
Ren et al
al., PRC 70 (2004) 034304: New formula
and DDCM calculations for cluster radioactivity
15
Comparison of the calculated half-lives using the formula with
the experimental data for emission of various clusters
clusters.
log10 T1/ 2 = aZ c Z d Q
−1/ 2
+ cZ c Z d + d + h
16
Deviations between experimental half-lives and theoretical one for
cluster radioactivity.
radioactivity Calculations are performed within the DDCM.
DDCM
17
Half-lives
Half
lives of cluster radioactivity (PRC, 2004)
Decay
Q/MeV
p
Log10 Texpt
Log10 TFormula
Log10RM3Y
221Fr—207Tl+14C
31.29
14.52
14.43
14.86
221Ra—207Pb+14C
32.40
13.37
13.43
13.79
222Ra—208Pb+14C
33.05
11.10
10.73
11.19
223Ra—209Pb+14C
31.83
15.05
14.60
14.88
224Ra—210Pb+14C
30 54
30.54
15.90
15 90
15.97
15 97
16.02
16 02
226Ra—212Pb+14C
28.20
21.29
21.46
21.16
228Th—
Th 208Pb+20O
44.72
44 72
20.73
20 73
20.98
20 98
21.09
21 09
230Th—206Hg+24Ne
57.76
24.63
24.17
24.38
18
Half lives of cluster radioactivity (PRC
Half-lives
(PRC, 2004)
D
Decay
Q/M
Q/MeV
V
231Pa—207Tl+24Ne
L
Log10 Texpt
L 10 TFormula Log
Log
L 10RM3Y
232U—208Pb+24Ne
60.41
62.31
22.89
20.39
23.44
21.00
23.91
20.34
233U—209Pb+24Ne
60.49
24.84
24.76
24.24
234U—206Hg+28Mg
74.11
25.74
25.12
25.39
236Pu—208Pb+28Mg
79.67
21.65
21.90
21.20
238Pu—206Hg+
g 32Si
91.19
25.30
25.33
26.04
242Cm—208Pb+34Si
96.51
23.11
23.19
23.04
19
PRC 78 ((2008)) 044310: Unified description
p
of alpha
p
decay and cluster radioactivity （大学生1作)
20
Derivation from quantum tunneling
Γ ≡ h ln 2 / T1/ 2 = P0 FP
⎛ 2 RC
⎞
2 μ[V (r ) − Q]dr ⎟
P = exp ⎜ − ∫
⎝ h Rt
⎠
V(R)
Q
log10 T1/ 2 = log10 (h ln 2 / P0 F ) + c1 μ Z c Z d Q −1/ 2
+ c2 μ ( Z c Z d )1/ 2
log10 P0 = −c3 μ ( Z c Z d )1/ 2 + c4
21
Deviation
De
iation of the theoretical res
results
lts from the e
experimental
perimental
data for the alpha decay of nuclei with Z>=84 and N>=128
((Ni,, Ren…,, PRC78,, 2008))
22
Unified description of alpha decay and cluster radioactivity
for even-even nuclei: one set of parameters is used
Phys.
y Rev. C 78 ((2008)) 044310,, Ni,, Ren,, Dong,
g, and Xu
23
PRC 85 (2012) 044608: Effects of the quantum numbers
of quasibound states are included into the formula.
24
量子力学 量子化和薛定谔方程
量子力学：量子化和薛定谔方程
• 量子和经典的差别：物理量量子化
• G-N定律在量子力学之前出现（1911）
• Gamow虽然导出G-N定律，但未引入量子化
G
虽然导出G N定律 但未引入量子化
• 所以
所以，量子数的影响未包含在G-N定律(盖革-努塔
量子数的影响未包含在G N定律(盖革 努塔
尔定律)中
• 由于alpha衰变是纯量子效应（无经典对应），应
引入量子数变化
25
量子数 G=2n+l
G 2 l 对谐振子势
rˆ 2
{ H , L , Lˆ z , P }
G 为 主 量 子 数 ，n 为 径 向 量 子 数 ，
l为 角 动 量 , P为 宇 称 算 子
rˆ 2
2
L Y lm (θ , φ ) = l ( l + 1) h Y lm (θ , φ )
l = 0 ,1,
1 2 , ....
Some basic observables such as quantum numbers can
be absorbed in the formula for a better description of
alpha-decay
alpha
decay data.
⎧⎪ h 2 d 2 ⎡
l(l + 1)h 2 ⎤ ⎫⎪
− ⎢VN (r ) + VC (r ) +
⎨−
⎬ unlj (r ) = E unlj (r )
⎥
2
2
2 μ r ⎦ ⎭⎪
⎩⎪ 2 μ dr ⎣
4
G = 2n + l = ∑ giAc
i =1
偶偶核基态到基态L=0
log10 T1/ 2 = a μ Z c Z d / Q + b μ Z c Z d + c + S + Pl(l + 1)
Effects of G (or n) quantum number on alphay data: S=0 for N>126 and S=1 for N<=126
decay
Effects of angular momentum and parity
of alpha particle
27
Ratios between experiment
p
and theory
y for even-even Po nuclei
with the original law and with the new law: new law also agrees
well with the data for N<=126.
28
Ratios between experimental data and theoretical results
for Rn nuclei with the original law and with the new law
(PRC, 2012)
29
Ratios between experimental data and theoretical results
for odd-A Po nuclei with original law and with new law
(PRC, 2012)
30
31
PRC2014：合成新核素205Ac
PRC2014：合成新核素205Ac
PRC2014：合成新核素205Ac
The calculated half-life (15 ms) with the new GeigerNuttall law [16,17]
[16 17] agrees well with the measured data
(20 +97-9ms).
Systematic
y
of ((a)) Qα-decay
y energies
g
and (b)
( ) α-decay
y
half-lives for favored α transitions of Ac isotopes
Red solid point:
Present
measurement
Blue line:
Bl
li
Calculated results
[ , ]
[16,17]
Black open point:
Literature values
[4,5,12-14]
New law for double-β decay: PRC 89, 064603 (2014)
36
Experimental
p
data for 2νβ
β-β- decay
y of 11 isotopes
p
Nucl. T1/2(expt) (Ey)
log10T1/2(ex Q2β (MeV) [log10T1/2(expt)хQ2β]1/2
pt))
p
48Ca
44(+6 -5)
1.643
4.267
2.649
76Ge
1840(+140 100)
3.265
2.039
2.580
82Se
92(7)
1.964
2.996
2.424
96Zr
23 5(0 21)
23.5(0.21)
1 371
1.371
3 349
3.349
2 143
2.143
100M
7.1(0.4)
0.851
3.034
1.606
116Cd
28(2)
1.447
2.813
2.017
128Te
1.9(0.4)х106
6.279
0.8665
2.333
130Te
700(140)
2.845
2.528
2.682
136Xe
2300(120)
3.362
2.458
2.876
150Nd
9.11(0.68)
0.960
3.371
1.799
238U
2000(600)
3.301
1.144
1.944
o
37
The effect from the Coulomb potential
ρ ( Z , ε ) = 2πη (1 − e−2πη ) , with η = (+ Z /137)(ε / cp )
A correction to double-β--decay half-lives log10T1/2
−2 log10 [ (2π Z ) /137 ]
The leading effect of nuclear structure (shell effect) is
simulated by introducing an addition quantity S:
S=2 when neutron numbers of parent nuclei are magic,
S=0 when neutron numbers are nonmagic.
log10 T1/2 (Ey) = [ a − 2 log10 (2π Z /137) + S ] Q2 β (MeV),
parameter a is obtained as 5.843 by fitting 11 ground-state data.
38
Comparison of the experimental and theoretical doubledouble
β-decay half-lives for ground-state transitions of 11 eveneven nuclei
39
Summary
y
• Review on alpha
p decay
y and cluster radioactivity
y
• Analytical formulas for half-lives of alpha decay and
cluster radioactivity
• New Geiger-Nuttall
Geiger Nuttall (G
(G-N)
N) law:
Effects of the quantum numbers of quasibound states
• New law for half-lives of Doouble beta decay
• Thanks
40
41
Review on models (alpha and cluster)
Alpha-decay models:
Buck et al, Gupta et al: Preformed cluster model
Lovas, Liotta, Delion et al: Phys. Rep. 294 (1998) 265
Ren and C.
C Xu: Density
Density-dependent
dependent cluster model …
Ni and Ren: Multi-channel cluster model …
Denisov and Ikezoe: UMADAC (Cluster model)
Royer et al: Generalized liquid drop model…
Analytical formula for cluster decay half-lives:
Ren and C. Xu, PRC 70 (2004) 034304;
Ni and Ren…,PRC
Ren PRC 78 (2008) 044310
044310…
New G-N law: Ren and Ren, PRC 85 ((2012)) 044608 …
42
新G N定律对Z=85同位素
新G-N定律对Z=85同位素
43
新G N定律对Z=87同位素
新G-N定律对Z=87同位素
44
45
46
47
48
49
50
51
52
53
54
Effect of different hindrance in even-even,, odd-A,, and
odd-odd emitters: values of the parameter c
same c
values
various c
values
Phys. Rev. C 78 (2008) 044310, Ni, Ren, Dong, and Xu
55
该文多处引用了我们的工作，举例如下
GN定律和VS公式的推广见文献[6-8](其中文献[7,8]为我们
工作)。作者特别强调了新GN定律包含了量子数效应 [8]。
56
Comparison
p
of the calculated half-lives with the experimental
p
data for cluster radioactivity (PRC, 2008)
57
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