99.81.4 Magic Numbers (Iuliano) 

From: Jerryiuliano@aol.com
To: MetPhys@aol.com
Subject: Magic Numbers
Date: 06/07/03

MP

In physics one of the great mysteries of the Independent Particle Model are the so-called "magic" numbers of electronic restrictive shell occupations of the atomic elements...126 ; 82 ; 50 ; 28 ; 20 ; 8 and 2.

magic number

NOUN: 1. Any of the numbers, 2, 8, 20, 28, 50, 82, or 126, that represent the number of neutrons or protons in strongly bound and exceptionally stable atomic nuclei. 2. Sports The number of wins by a leading team and losses by its closest competitor required in order for the leading team to capture a championship or secure a playoff spot.

The American Heritage¨ Dictionary of the English Language, Fourth Edition. Copyright © 2000 by Houghton Mifflin Company. Published by the Houghton Mifflin Company. All rights reserved.

The universal harmonic number, 82944 ,Ê solves this "code"Ê in two actions....

1...the chaos to order form of the Feigenbaum constant...4.669201609...
2...the fine-structure constant, the amplitude for energy (electron) and light(photons) transitions...such that:

82944 = ( 100^Pi ) / ( tan F ) .....F = 4.66920431323..

...and to derive fine-structure by inverting Pi: a(em)= 1/137.036000986..

cos 1/a(em) = 1 / (( tan F ) ^ ( 1/Pi )) = ( 82944 ^ ( 1/Pi )) / 100

...to derive the electronic number of the final shell (126) through the dividing into the whole the previous shell numbers as follows:

(((( 82944 ^ ( 1/Pi )) / 100 ) ^ - Pi ) * ( 10^8 ) / 82 / 50 / 28 / 20 / 8 = 125.9906919

....since it can be demonstrated:

( 82944 ^ ( 1/Pi )) / 100 = cos 1/a(em)

...and

1 / (( tan 4.69920431323 ) ^ ( 1/Pi )) = cos 1/a(em)

...then the "magic" numbers used in totality can demonstrate cosine inverse fine-structure:

(125.9906919 * 82 * 50 * 28 * 20 * 8 / ( 10^8 )) ^ - ( 1/Pi ) = cos 1/a(em)

...for Feigenbaum: tan in radians

( tan^-1 ( 125.9906919 * 82 * 50 * 28 * 20 * 8 / ( 10^8 ))) + Pi = 4.66920431323

...and the universal harmonic 82944:

( 10 ^ (( 2*Pi) + 8 )) / 125.99063919 / 82 / 50 / 28 / 20 / 8 = 82944

A few days later I worked out the non-cosine form for fine-structure using the collective unconscious form....( 10 ^ ( 82944 / 32 / 666 )) / 666...to arrive at:

( 666/(10^((10^((2*pi)+8))/125.9907024/82/50/28/20/8/666/32)))^2 = a(em)

A few days later still I searched Google for references to the magic numbers for electronic shell limitations and ran across this:...........

PressÊReleases

2000/06/12 New Magic Number "16" Where Nuclei Exist Stably Discovered

Researcher Akira Ozawa, Senior Researcher Isao Tanihata and others of RIKEN (the Institute of Physical and Chemical Research) have become the first in the world to discover a new magic number "16" in neutron-rich isotopes, where atomic nuclei can exist stably. The research results appeared in the June 12 issue of "Physical Review Letters," the proceedings of the American Physical Society.

An atomic nucleus consisting of protons and neutrons becomes stable when the number of protons and number of neutrons satisfy given numbers called magic numbers. The magic numbers known so far were "2", "8", "20", "28", "50", "82" and "126." ............................

This was just the missing shells from the Leahy equations!!!!! ...which can now be shown as :

(666/(10^((10^((2*Pi)+8))/125.990702409/82/50/28/20/16/8/2/666)))^2 = a(em)

The proof is the missing magic number ..16.. creates Plato's mysterious cyclic number ...2592.. a collective unconscious number:

(10^((2*Pi)+8))/125.990702409/82/50/28/20/16/8/2 = 2591.999784

....of which:

2591.999784 / 666 = 3.891891567 = the collective unconscious constant.

J.Iuliano

PressÊReleases

2000/06/12 New Magic Number "16" Where Nuclei Exist Stably Discovered

Researcher Akira Ozawa, Senior Researcher Isao Tanihata and others of RIKEN (the Institute of Physical and Chemical Research) have become the first in the world to discover a new magic number "16" in neutron-rich isotopes, where atomic nuclei can exist stably. The research results appeared in the June 12 issue of "Physical Review Letters," the proceedings of the American Physical Society.

An atomic nucleus consisting of protons and neutrons becomes stable when the number of protons and number of neutrons satisfy given numbers called magic numbers. The magic numbers known so far were "2", "8", "20", "28", "50", "82" and "126."

It had been believed that magic numbers reflect the fact that nuclei have a "shell structure" and that magic numbers do not vary even if the mixture ratio of protons to neutrons changes. But recent developments in unstable nuclear research have clarified that rather than protons and neutrons being homogeneously mixed, only neutrons are layered near the surface of nuclei in neutron-rich isotopes.

The research team measured the nuclear radii of light and unstable isotopes' such as nitrogen and oxygen, precisely using the synchrotron at the GSI in Germany, and discovered that the nuclear radii with the neutron numbers "15" and "16" increased abnormally. They also verified that with excess neutrons nuclei with neutron number "16" are especially stable.

(For further information, please contact the Public Relations Office, RIKEN; phone: 048-467-9272) (Source:STA TODAY July 2000)

The Shell Model

One such model is the Shell Model, which accounts for many features of the nuclear energy levels. According to this model, the motion of each nucleon is governed by the average attractive force of all the other nucleons. The resulting orbits form "shells," just as the orbits of electrons in atoms do. As nucleons are added to the nucleus, they drop into the lowest-energy shells permitted by the Pauli Principle which requires that each nucleon have a unique set of quantum numbers to describe its motion.

When a shell is full (that is, when the nucleons have used up all of the possible sets of quantum number assignments), a nucleus of unusual stability forms. This concept is similar to that found in an atom where a filled set of electron quantum numbers results in an atom with unusual stabilityÐan inert gas. When all the protons or neutrons in a nucleus are in filled shells, the number of protons or neutrons is called a "magic number." Some of the magic numbers are 2, 8, 20, 28, 50, 82, and 126. For example, 116Sn has a magic number of protons (50) and 54Fe has a magic number of neutrons (28). Some nuclei, for example 40Ca and 208Pb, have magic numbers of both protons and neutrons; these nuclei have exceptional stability and are called "doubly magic." Magic numbers are indicated on the chart of the nuclides.

Filled shells have a total angular momentum, J, equal to zero. The next added nucleon (a valence nucleon) determines the J of the new ground state. When nucleons (singly or in pairs) are excited out of the ground state they change the angular momentum of the nucleus as well as its parity and isospin projection quantum numbers. The shell model describes how much energy is required to move nucleons from one orbit to another and how the quantum numbers change. The figure above shows an energy diagram of the two filled shells of the ground state of 12C. Promotion of a nucleon or a pair of nucleons to an unfilled shell puts the nucleus into one of the excited states.

Excited nuclear states decay to more stable states, i.e., more stable nucleon orbitals. Measuring transition rates between nuclear energy levels requires specialized alpha, beta, and gamma detectors and associated electronic circuitry to precisely determine the energy and half-life of the decay. Quantum mechanics and shell-model theory permit nuclear scientists to compute the transition probability (rate of decay) between nuclear states. For nuclei whose structure can be described by a small number of valence nucleons outside filled shells, the Shell Model calculations agree very well with measured values of spin and parity assignments and transition probabilities.

An example of transitions between a nucleusÕs energy levels is shown above. The ground states of 12B (5 protons, 7 neutrons) and 12N (7 protons, 5 neutrons) are related to each other and to the 15.1 MeV state in 12C. Each has a nucleon in the third energy level, each has the quantum number 1+, 1, and each decays to the ground state of carbon. The boron and nitrogen beta decay emit an e- and e+ respectively. The 15.1 MeV state in carbon decays by gamma emission. The shell model calculations of these transition probabilities agree quite well with the measured rates even though different decay mechanisms are at work in each case. Nuclear theorists keep refining the shell model to understand the details of nuclear structure and to make that knowledge available for applications in nuclear technology.

magic number

NOUN: 1. Any of the numbers, 2, 8, 20, 28, 50, 82, or 126, that represent the number of neutrons or protons in strongly bound and exceptionally stable atomic nuclei. 2. Sports The number of wins by a leading team and losses by its closest competitor required in order for the leading team to capture a championship or secure a playoff spot.

The American Heritage¨ Dictionary of the English Language, Fourth Edition. Copyright © 2000 by Houghton Mifflin Company. Published by the Houghton Mifflin Company. All rights reserved.

[PDF]New Magic Number, N = 16, near the Neutron Drip-Line Akira Ozawa File Format: PDF/Adobe Acrobat - View as HTML Your browser may not have a PDF reader available. Google recommends visiting our text version of this document.

... show a closed shell structure for the following neutron ( N ) and proton ( Z ) numbers: 2, 8, 20, 28, 50, 82 and 126. They are called nuclear magic numbers. ... wwwndc.tokai.jaeri.go.jp/nds/proceedings/2000/o03.pdf - Similar pages

Magic Numbers.

n the atomic shell model, the shells are filled with electrons in order of increasing energy until they completely fill a closed shell, producing the inert core of a noble gas. These elements have highly stable properties, such as low ionic radius and high ionization energy. As further electrons are added to shells outside the core, the atomic properties are primarily determined by these valence electrons. The shell model arises from observation of similar phenomena in nuclei, with certain numbers of nucleons being particularly stable, these numbers are called `magic numbers'.

Nuclei with either numbers of protons or neutrons equal to Z, N =2, 8, 20, 28, 50, 82, or 126 exhibit certain properties which are analogous to closed shell properties in atoms, including anomalously low masses, high natural abundances and high energy first excited states. The effect can be seen in a plot of separation energy versus increasing N or Z (figure 2.3) [52]. This is similar to the ionization energy of an atom with increasing mass, ie. there is a gradual increase with a definite sharp drop off at each of the magic numbers, corresponding to the filling of major nuclear shells.

Figure 2.3: Two proton (top) and two neutron (bottom) separation energies as a function of N and Z respectively. Taken from [18]

In order to reproduce the observed magic numbers, it was proposed in 1948 by Mayer and Jensen [50,53] that in addition to the static potential, there is also a nuclear spin-orbit potential of the form:

(3.9)

This potential, as in the equivalent atomic case, perturbs the nuclear energy levels causing each one to split into two, characterized by the total angular momentum quantum number , as j is the vector coupled sum of the orbital angular momentum l, and the spin s. Unlike the atomic form, this effect is not due to an electromagnetic interaction, but a nucleon-nucleon interaction, and consequently has a much more dramatic effect.

The total effect of the potentials is shown in figure 2.4, and it can be seen that there are pronounced gaps at the points where the magic numbers of nucleons are filling the orbitals.

Figure 2.4: Theoretical nuclear level structure, labeled nlj, including spin-orbit perturbations and showing energy gaps at the magic numbers.

It is the assumption of the shell model that in nuclei with, for example, one additional proton or neutron outside the closed shell, the total angular momentum and parity of such a nucleus is simply that of the additional valence nucleon due to the Pauli blocking effect. eg. 178O has an additional neutron in the 1d state for which and = +1, thus we predict a ground state of, in agreement with experiment [54].