Oliver Thewalt

    Oliver Thewalt

    Theoretical Physics | Quantum Biology | Dark Matter Research Cluster

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    Michael Balmer's notes: Nuclear Configuration and the nuclear shell model

    Author: Michael Balmer, Montclair

    This is a  long and difficult subject to discuss, this reason is in the details that must be put forth to ensure a reasonable amount of understanding to those not familiar with the nuclear structure of atoms,most everyone knows what the principles of the nucleus are,what the functions are and why, unlike the electronic configuration is indeed more widely known to chemist mostly and physicist, generally it is known how molecules are formed even just water and air, without really knowing the images of the structure, like most of what we know about the atom on a whole is relatively new, starting at the "bomb" and progressing to energy to sustain our daily undertakings and on to the possibility of off world travel and residence, new to us also the hazards and dangers of this step in our advancement as a species.

        In this short history, which began to flourish ... be it with limited knowledge ... even before the famous or infamous, depending on your prospectus, work on atomic use, the proposal of a shell model for the nuclear structure was put forth in 1932 by Dmitry Ivanenko and Igor Gapon, later in 1949 the model was developed in separate research by Eugene Paul Wigner,J. Hans D. Jensen and Maria Goeppert Mayer, during and after the development of the first atomic weapon, which received contribution from the researchers.

      

    this is to show partial shell structure and magic numbers...courtesy of Wikipedia:"

    1st shell: 2 states (n = 0, j = 1⁄2).

    2nd shell: 6 states (n = 1, j = 1⁄2 or 3⁄2).

    3rd shell: 12 states (n = 2, j = 1⁄2, 3⁄2 or 5⁄2).

    4th shell: 8 states (n = 3, j = 7⁄2).

    5th shell: 22 states (n = 3, j = 1⁄2, 3⁄2 or 5⁄2; n = 4, j = 9⁄2).

    6th shell: 32 states (n = 4, j = 1⁄2, 3⁄2, 5⁄2 or 7⁄2; n = 5, j = 11⁄2).

    7th shell: 44 states (n = 5, j = 1⁄2, 3⁄2, 5⁄2, 7⁄2 or 9⁄2; n = 6, j = 13⁄2).

    8th shell: 58 states (n = 6, j = 1⁄2, 3⁄2, 5⁄2, 7⁄2, 9⁄2 or 11⁄2; n = 7, j = 15⁄2)."

    and so on.



    The magic numbers are then

    2

    8=2+6

    20=2+6+12

    28=2+6+12+8

    50=2+6+12+8+22

    82=2+6+12+8+22+32

    126=2+6+12+8+22+32+44

    184=2+6+12+8+22+32+44+58 

    This short exert courtesy of Wikipedia:


     "The shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a filled shell results in greater stability. When adding nucleons(protons or neutrons) to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. This observation, that there are certain magic numbers of nucleons: 2, 8, 20, 28, 50, 82, 126 which are more tightly bound than the next higher number, is the origin of the shell model."

     

     this excerpt courtesy of Wikipedia:                                                                                                                                                                                                                                                                        "The rules for the ordering of the nucleus shells are similar to Hund's Rules of the atomic shells, however, unlike its use in atomic physics the completion of a shell is not signified by reaching the next n, as such the shell model cannot accurately predict the order of excited nuclei states, though it is very successful in predicting the ground states."


    The order of the first few terms are listed as follows: 1s, 1p3⁄2, 1p1⁄2, 1d5⁄2, 2s, 1d3⁄2... For further clarification on the notation refer to the article on the Russell-Saunders term symbol.

    For nuclei farther from the magic numbers one must add the assumption that due to the relation between the strong nuclear force and angular momentum, protons or neutrons with the same n tend to form pairs of opposite angular momenta.

    Therefore a nucleus with an even number of protons and an even number of neutrons has 0 spin and positive parity. A nucleus with an even number of protons and an odd number of neutrons (or vice versa) has the parity of the last neutron (or proton), and the spin equal to the total angular momentum of this neutron (or proton). By "last" we mean the properties coming from the highest energy level.

    In the case of a nucleus with an odd number of protons and an odd number of neutrons, one must consider the total angular momentum and parity of both the last neutron and the last proton. The nucleus parity will be a product of theirs, while the nucleus spin will be one of the possible results of the sum of their angular momenta (with other possible results being excited states of the nucleus).

    The ordering of angular momentum levels within each shell is according to the principles described above - due to spin-orbit interaction, with high angular momentum states having their energies shifted downwards due to the deformation of the potential (i.e. moving from a harmonic oscillator potential to a more realistic one).

    For nucleon pairs, however, it is often energetically favorable to be at high angular momentum, even if its energy level for a single nucleon would be higher. This is due to the relation between angular momentum and the strong nuclear force.


    Nuclear magnetic moment is partly predicted by this simple version of the shell model. The magnetic moment is calculated through j, l and s of the "last" nucleon, but nuclei are not in states of well defined l and s. Furthermore, for odd-odd nuclei, one has to consider the two "last" nucleons, as in deuterium.

    Therefore one gets several possible answers for the nuclear magnetic moment, one for each possible combined l and s state, and the real state of the nucleus is a superposition of them. Thus the real (measured) nuclear magnetic moment is somewhere in between the possible answers.

    The electric dipole of a nucleus is always zero, because its ground state has a definite parity, so its matter density (where is the wavefunction) is always invariant under parity. This is usually the situations with the atomic electric dipole as well.


    Higher electric and magnetic multipole moments cannot be predicted by this simple version of the shell model, for the reasons similar to those in the case of the deuterium

    image

     

      "Illustration of the many-body wave-function in the nuclear shell model. The total wave-function ∣ Ψ ⟩ {\displaystyle \mid \Psi \rangle } {\displaystyle \mid \Psi \rangle } is obtained as a linear superposition of Slater determinants ∣ Φ k ⟩ {\displaystyle \mid \Phi _{k}\rangle } {\displaystyle \mid \Phi _{k}\rangle } which differ by which single-particle states are occupied in the valence space. The inert core is made of all single-particle states which are fully occupied in all Slater determinants."



    Image Courtesy: By Schunck (Own work) [CC BY 3.0 (http://creativecommons.org/licenses/by/3.0)], via Wikimedia Commons

     

    this is one of many illustrations available on the model.

     

     

            As i said earlier,most,maybe all of the research of this model is derived from the work of the Atomic Bomb,but we all are aware of this,it is still taught,studied,researched and applied in all aspects of nuclear technology and medical technologies with outstanding results, it has given so much to the development of even more forms of the atoms, but what it has also given us is uncontrolled radioactive events, it does not offer a solution to itself since the purpose is not neutralize or eliminate,but create or to instruct in the creation and to teach why, the model is excellent in these purposes and will and should be used for these purposes for some time to come,however to make it more efficient and to be able to understand and stabilize the results of the research, development and applications, it must be revised or expanded with a more precise model for this purpose,since this is a long and difficult topic of discussion it will be taken in segments....

     

    as always you may present your ideas or challenge these assertions....sharing must have prior approval...just ask