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, 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/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. 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) -> UO 2 , 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 Lead has Outer 50 and 32 Density Decreases in Protons = 82, and 50, Center for Big Nuclei 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 - K 1 Volume - K 2 (#Neutrons + #Protons) in outermost shell /A Surface - K 3 Z(Z-1) A -4/3 Coulomb - K 4 (#paired Neutrons - #paired Protons) 2 /A Asymmetry, Magic - K 5 (#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 Z 2 /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

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