Finding Magic Numbers for Heavy and Superheavy

Nuclei

ByRoger 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, Semi-Empirical 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/MuonPhysics, 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 SuperheavyNuclei

� 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. Robley Evan’s 1950s Nuclear Physics for Engineers

2. Magic Numbers and the Semi-Empirical Binding Energy Formula

3. Lucas’ Electromagnetic Model of the Nucleus

4. Superheavy Nuclei

5. New Magic Proton and Neutron Numbers

6. 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 NucleusDensity Decreases in Center for Big Nuclei

Lead has Outer 50 and 32Protons = 82, and 50, 32, 18, 18, and 8 = 126 Neutrons

Lucas’ “Rule” Assignments for Doubly Magic Isotopes

AT A Z N N1 P1 N2 P2 N3 P3 N4 P4 N5 P5 N6 P6

He 4 2 2 2 2

O 16 8 8 8 8

Ca 40 20 20 6 6 14 14

Ca 48 20 28 2 8 6 18 14

Ni 48 28 20 2 6 8 14 18

Sn 100 50 50 18 18 32 32

Sn 132 50 82 6 8 18 18 18 32 32

Pb 208 82 126 8 18 18 32 32 50 50

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 RedPeninsula 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 RedArea Near Z = 108 and N = 182; IslandLies 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 ConsideredPeninsula

� 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 SuperheavyIsotopes

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 HeaviestCerium 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 AlsoStable

� 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 Os-166, 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 86and Large Lower Bound of 132, and 18Width 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 !