Finding Magic Numbers for
Heavy and Superheavy
Nuclei
By
Roger A. Rydin
Associate Professor Emeritus of
Nuclear Engineering
Foreword
I am a Nuclear Engineer, Specializing in
Reactor Physics
Nuclear Physics = Physics of Nucleus
Theory Taught by Robley Evans,
Experiments by Norm Rasmussen
Fascinated by Magic Numbers, SemiEmpirical Binding Energy Formula
Disturbed by Fast Moving Nucleons in
Nucleus, Coulomb Barrier Penetration
Foreword
Met Dr. Charles Lucas at U. Tulsa Meeting
PhD Theoretical Physics W&M, Newport
News Accelerator, Expert in Pion/Muon
Physics, Now Owns Company
Models of Nucleons as Charge Carrying
Ring Magnets
Nucleus Model in Fixed Static Shells Under
Force Balance
Explained Magic Numbers
New Semi-Empirical BE Formula
Foreword
Joint Letter to NSE, Published 2009
Sent Copy to Professor Hans Weber
He Suggested Application to Superheavy
Nuclei
Summary T/E by Dr. Mohini Gupta
Gupta Suggests Annals of Nuclear Energy
Paper
Published December 2010
Follow On Paper Published August 2011
Order of Presentation
1.
2.
3.
4.
5.
6.
Robley Evan’s 1950s Nuclear Physics for
Engineers
Magic Numbers and the Semi-Empirical
Binding Energy Formula
Lucas’ Electromagnetic Model of the
Nucleus
Superheavy Nuclei
New Magic Proton and Neutron Numbers
Consequences for Selected Isotopes
The Atomic Nucleus - 1955
Heavy on Experimental Data
Analysis of What the Data Implied
Theory of the Time – Not Cut in
Stone
Orderly Treatment: Charge; Size;
Mass; Moments; Isotopes; Nuclear
Systematics; Forces; Nuclear Models
Nuclear Magic Numbers
2, 8, 28, 50, 82, 126 Are Closed Shells of
Some Kind -> Extra Stable Isotopes
Helium-4 (2p, 2n) = Alpha Decay
Oxygen-16 (8p, 8n) -> UO2 , etc.
Double Hump Fission Yields
Light (28, 50) +, and Heavy (50,82) +
Delayed Neutrons, Poisons, i.e. Xe-135
Lead-208 (82, 126) Last Stable Isotope
Semi-Empirical Binding, B/A
1955
Stable Isotope Data
Contribution Terms
Mass Parabolas
Odd A Decay
Even A Decay
Questions
What is the Nature of the Closed
Shells ?
What Produces Liquid Drop Property ?
Why Doesn’t the Semi-Empirical
Binding Energy Formula Match the
Low A Peaks ?
What is the Physical Decay
Mechanism ?
Lucas’ Electromagnetic Nucleus
Protons and Neutrons Occupy Fixed
Positions in Symmetric 3D Space
Under Static Force Balance
They are Distributed in 6 Double
Cycles – Occupying 2, 8, 18, 18, 32
and 50
Inner Neutron Shells Can Expand to
Next Number Like Electron Shells
Lucas’ Electromagnetic Nucleus
Density Decreases in
Center for Big Nuclei
Lead has Outer 50 and 32
Protons = 82, and 50,
32, 18, 18, and 8 = 126
Neutrons
Lucas’ “Rule” Assignments
for Doubly Magic Isotopes
AT
A
Z
N
N1 P1 N2
He
4
2
2
O
16
8
8
8
8
Ca
40
20
20
6
Ca
48
20
28
Ni
48
28
20
Sn
100
50
50
Sn
132
50
82
6
8
Pb
208
82
126
8
18
2
P2
N3
P3
6
14
14
8
6
18
14
6
8
14
18
N4
P4
N5
P5
N6
P6
18
18
32
32
18
18
18
32
32
18
32
32
50
50
2
2
2
Lucas’ Electromagnetic Nucleus
Magic Numbers are Composites of 6 Shells
Proton Shells Fill from Outside
The Neutron Shell Between Outer Proton
Shells Acts Like a Decoupler by Polarizing
Sideways => Liquid Drop Properties
Interior Neutrons Polarize with Plus Ends
Toward Center and Fill Inwards
Decay is a Vibration Process !
Complicated Vibrations
Force Laws Nonlinear
Nucleons Vibrate About Positions
Internal “Bumped” Nucleon Vibrations
-> Beta Decay?
Non-Spherical Rotational Vibrations
Linear Model Analog of Schrödinger
Equation !
Semi-Empirical Binding Energy
B/A - K1
Volume
- K2 (#Neutrons + #Protons) in
outermost shell /A
Surface
- K3 Z(Z-1) A-4/3
Coulomb
- K4 (#paired Neutrons #paired Protons)2 /A
Asymmetry, Magic
- K5 (#unpaired Protons +
#unpaired Neutrons) /A
Pairing
Lucas’ New Semi-Empirical
Binding Energy for 3000 Nuclei
Electromagnetic Nucleus
Computational Confirmation
Superheavy Nuclei
Produced by Bombarding Heavy
Elements, i.e., Uranium, Plutonium,
Curium, Californium, and Berkelium
by Heavy Ions Like Doubly Magic
Ca-48 (20, 28)
Work Done at GSI Darmstadt, JINR
Dubna, ORNL, RIKEN Japan, LLNL
Longest Half Lives are 12 Minutes,
and 22 Seconds
Superheavy Nuclei Sea Extent
Observations
Lower End of the Red Peninsula is Near Z
= 90 and N = 140; Upper End of the Red
Peninsula is Near Z = 100 and N = 158
Low End of the Green Peninsula Area is
Near Z = 82; Upper End Around Z = 108
Shoal is Near Z = 108, and it Lies Between
N = 158 and 164
Island of Stability is Centered with a Red
Area Near Z = 108 and N = 182; Island
Lies Between Z = 102 and 118, and
Between N = 172 and N= 184.
Theoretical Superheavy Nuclei
Magic Numbers
Spherical and Deformed Nuclei,
Multiple Theories, Liquid Drop Plus
Shells
Magic Z at 108, 110, 114, 120 ?
Magic N at 152, 164, 172, 184 ?
Why Not Others, Close Together ?
Z Extension of Lucas’ Shells
Z = 50 + 32 + 8 = 90
Z = 50 + 32 + 8 + 2 = 92
Z = 50 + 32 + 18 = 100
Z = 50 + 32 + 18 + 2 = 102
Z = 50 + 32 + 18 + 8 = 108
Z = 50 + 32 + 18 + 8 + 2 = 110
Z = 50 + 32 + 18 + 18 = 118
Z = 50 + 32 + 18 + 18 + 2 = 120
N Extension of Lucas’ Shells
N = 50 + 32 + 32 + 18 + 8 = 140
N = 50 + 32 + 32 + 18 + 8 + 2 = 142
N = 50 + 50 + 32 + 18 + 8 = 158
N = 50 + 50 + 32 + 32 = 164
N = 50 + 50 + 32 + 32 + 8 = 172
N = 50 + 50 + 32 + 32 +18 = 182
N = 50 + 50 + 32 + 32 +18 + 2 =
184
Consequences
New N and Z Numbers Cover Other
Theoretical Values
Agree with Peninsula, Shoal and
Island Boundaries
Suggestion of Lower A Single and
Double Magic Nuclei in Continent Yet
Unexplored
Requires a Careful Look at Isotope
Data In the Table of Isotopes
N Extension of Lucas’ Shells ?
N = 50 + 32 + 8 = 90
N = 50 + 32 + 8 + 2 = 92
N = 50 + 32 + 18 = 100
N = 50 + 32 + 18 + 2 = 102
N = 50 + 32 + 18 + 8 = 108
N = 50 + 32 + 18 + 8 + 2 = 110
N = 50 + 32 + 18 + 18 = 118
N = 50 + 32 + 18 + 18 + 2 = 120
N = 50 + 32 + 18 + 18 + 8 + 2 = 128
Further Downward Z Extension
of Lucas’ Shells ?
Z = 32 + 18 + 8 = 58
i.e. Cerium, N = 58 First
Suggested in 1981 by Linus Pauling
Z = 32 + 18 + 18 = 68
i.e. Erbium, Near N = 70 by
Pauling
Z = 32 + 18 + 18 + 8 = 76
i.e. Osmium
Linus Pauling Data
Isotopes Considered
Peninsula
Thorium Z = 90
Uranium Z = 92
Fermium Z = 100
Nobelium Z = 102
Continent
Cerium Z = 58
Dysprosium Z = 66
Osmium Z = 76
Lead Z = 82
Thorium, Z = 90 = Holy Grail ?
19 Isotopes
Doubly Magic Th-230 (90, 140) @ 75000 y
Doubly Magic Th-232 (90, 142)@ 1.4E10 y
Th-229, One Short of Double @ 7300 y
N/Z ~ 1.54 for Most Stable
Lighter Isotopes, ns to days
Heavier Isotopes, days to minutes
Uranium, Z = 92
20 Isotopes
Doubly Magic U- 232 (92, 140) @ 68 y
Odd U-233 @ 1.6E5 y
Doubly Magic U- 234 (92, 142) @ 2.4E5 y
U-236 @ 2.4E7 y
U-238 @ 4.5E9 y and N/Z = 1.52
Lighter Isotopes, µs to days
Heavier Isotopes, days to minutes
Neptunium and Plutonium
Long Lived U, Np and Pu Isotopes All Lie
at a Ratio of N/Z Near 1.54
Seaborg Criterion for Even A Spontaneous
Fission Parameter Z2/A > ~ 44
Odd-A Nuclei More Stable to Spontaneous
Fission than Even-A Nuclei
Fission Preferred Mode of Decay for the
Proton Rich Heavy and Superheavy
Isotopes
Thorium, Uranium, Neptunium
and Plutonium Conclusions
N/Z ~ 1.54 Is Important
Magic and Near Magic Gives Longer Half
Lives
Magic Gives More Isotopes
Worse to Have Too Many Protons vs. Too
Many Neutrons
Fermium, Z = 100
19 Isotopes
Odd Fm-257 @ 100 days Longest Lived
Doubly Magic Fm-258 (100, 158), Short
Spontaneous Fission
Lighter Isotopes, ms to days
Heavier Isotopes, days to ms
Nobelium, Z = 102
12 Isotopes
Odd No-259 @ 58 minutes Longest Lived
Doubly Magic Fm-260 (102, 158), Short
Spontaneous Fission
Lighter Isotopes, ms to minutes
Heavier Isotopes, ms
Fermium and Nobelium
Conclusions
Magic Effects Not As Clear
Longest Lived Odd, One Short of Doubly
Magic
Spontaneous Fission More Important, at
Doubly Magic
Magic Gives More Isotopes
Worse to Have Too Many Protons vs. Too
Many Neutrons
Cerium, Z = 58
20 Isotopes
Ce-140, Doubly Magic at N = 82, Almost
90% of Natural Cerium
Ce148, Doubly Magic at N = 90, @ 56
seconds, and Ce-150 Doubly Magic at
N = 92 @ 4 seconds Are Among Heaviest
Cerium Isotopes Known
Dysprosium, Z = 66
30 Isotopes
Dy-160 to Dy-164, Comprise Most of the
Naturally Stable Isotopes
Lighter Dy-158, with a Magic N = 92, and
Dy-156, with a Magic N = 90, Are Also
Stable
Light Dy-148, with a Magic N = 82 @ 3.1
minutes
Heavy Dy-166, with a Magic N = 100 @
81.6 hours, and Dy-168, with a Magic N =
102 @ 8.7 minutes
Osmium, Z = 76
30 Isotopes
Naturally Occurring Osmium Isotopes Lie
Between Os-192 and Os-187
Among Lightest Osmium Isotopes are Os166, with a Doubly Magic N = 90 @ 7.1
seconds, and Os-168, with a Doubly Magic
N= 92 @ 2.2 seconds
Among Heaviest, Os-194, with a Doubly
Magic N = 118 @ 6 years, and Os-196,
with a Doubly Magic N = 120 @ 35
minutes
Lead Z = 82
33 Isotopes
Naturally Occurring and Long-lived Lead
Isotopes Lie Between Pb-204 and Pb-208
Among Lightest Doubly Magic Lead
Isotopes are Pb-202, with a Magic N= 120
@ 5.2E4 years , and Pb-200, with a Magic
N = 118 @ 21.5 hours
Among Heaviest Doubly Magic, Pb-210,
with a Magic N = 128 @ 22.3 years
Better Fission Yield Distribution
With Magic 58
Double Hump Fission Yields , U-235 + n
Light (28, 50), and Heavy (50, 82)
+ 14 + 12 = 26 to Divide ?
Or Better Yet
Light (28, 58), and Heavy (50, 82) = 18 to
Divide ? Matches Small Lower Bound of 86
and Large Lower Bound of 132, and 18
Width Containing 95% of Fission Products
Fission Yield Distributions
Conclusions
Magic Affects Number of Stable
Isotopes
Magic Accounts For Longer Half Lives
and Number of Lighter and Heavier
Isotopes
Large Number of Isotopes Related to
New Magic Numbers
N/Z ~ 1.54 Are Most Stable
Superheavy Half Lives Won’t Be Long
Conclusions
Table of Isotopes Is Now 4 ½ Inches
Thick !
Data on 3000+ Isotopes:
Level Schemes, Half Lives, Reactions,
Abundance, etc.
Probably Not Examined for
Systematic Behavior
Fertile Area for Research !