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The Liquid-Drop and The Shell Models

The liquid-drop model assumes that the constituents of the nucleus interact only with their nearest neighbors, and the density is constant inside the nucleus, like the molecules in the liquid. Using this analogy, a semiempirical formula has been developed to describe the binding energy
Liquid Drop Model Nuclear Decay as a function of the mass number A (shown by the solid curve in Figure 14-01). The result is not particularly accurate for the lower value of A. The expression is useful in discussing stability, radioactivity, and the fluctuations from the average behavior due to shell effects. The top diagram in Figure 14-04a shows two vibrational energy levels, which split into finer structures due to rotation. Figure 14-04b shows the deformation of the liquid drop, which eventually separates into two pieces (caused by the electrostatic repulsion of the protons).

Figure 14-04a Liquid Drop Model [view large image]

Figure 14-04b Fission

Nuclear Potential Nuclear Energy Levels There is extensive experimental evidence of the contrary hypothesis that the nucleons move in an effective potential well created by all the other nucleons. Since the nucleons are densely packed into a small region, it is expected that the chance of collision is very high. However, the interaction by collision is minimized by the Pauli exclusion principle, which forbids two fermions to occupy the same quantum state. If there are no nearby, unfilled quantum states that can be reached by the available energy for an interaction, then the interaction will not occur.

Figure 14-05a Nuclear Potential
[view large image]

Figure 14-05b Nuclear Energy Levels [view large image]

In the shell model, the potential well can be in the form of a square well or harmonic oscillator. A more realistic one is shown in Figure 14-05a with a round edge to avoid discontinuity and a Coulomb field for the charged protons. The energy levels obtained by solving the Schrodinger equation is shown on the left in Figure 14-05b. Including the spin-orbit interaction would split the levels by an amount depending on the orbital quantum number as shown in the middle of Figure 14-05b. The multiplicity of states (different possible orientations of angular momentum) is calculated by the formula 2j + 1, where j is the total angular momentum (orbit plus spin) quantum number designated as an subscript in the diagram. The "magic numbers" on the right suggests closed shell configuration, like the shells in atomic structure. They represent one line of reasoning which led to the development of a shell model for the nucleus. Other evidences include: The problem with the shell model is in the region of the rare-earth nuclei. The quadrupole moments predicted from the orbital motion of the individual protons are much smaller than those observed. From the shell model point of view, the rare-earth nuclei lie about midway between the neutron magic numbers 82 and 126. This is just the region for which shell model calculations are the most difficult since there are many particles outside a closed shell.

N-N Potential A more realistic nuclear (nucleon-nucleon) potential is the empirical curve shown in Figure 14-05c. As originally proposed by H. Yukawa, the longest range part of the strong internucleon force can be attributed to exchange of the mesons (pions). At shorter distances, exchanges of heavier mesons become important. However, the origin of the repulsive hard-core below 1 fermi (10-13cm) remains unclear until recently in 2007, when numerical results convincingly demonstrate that it is a consequence of QCD. The numerical computation is actually rather involved because of the virtual gluons and quark-antiquark pairs surrounding the three quarks (the components of the nucleon). The required computational power is only available now to reproduce the empirical potential from first principles. This potential represents the residual force derived from the more fundamental forces (as prescribed in QCD) between the constituent particles. The form of this potential is remarkably similar to the molecular potential curve even though these residual forces originated from different sources - one from quantum chromodynamics, while the other from quantum electrodynamics.

Figure 14-05c Nucleon-Nucleon Potential [view large image]

Figure 14-05c depicts a deuterium nucleus. The proton and neutron are composed of d (down), u (up), u quarks and d, u, d quarks respectively (in colors). The gluons are denoted by the coils with the lighter one representing the residual.

Basically the nucleon (proton/neutron) is known to consist of 3 quarks of different colors. They are bound together by the gluons which also carry color charge and thus also interact among themselves. On top of this complicated structure, there are virtual quark pairs and gluons poping in and out of existence constantly (Figure 14-05d). The mathematics of this many body problem with varying particle number is too complicated to solve even numerically. Therefore, the detailed configuration and dynamics inside the nucleon are actually unknown. The followings list some of the missing information as of 2015.

  • Mass - The mass of quarks accounts for 2% of the nucleon. The rest of the 98% is believed to come from the kinetic energy and binding energy via E = mc2. But it is not clear how do the gluons generate such short range interaction since they are massless and the interacting range is in inverse proportion to the mass of the mediating particles. This puzzle has actually been explained by "gluon shielding".

  • Spin - Experiments show that the quarks contribute about 30% while the gluons generate about 20% of the total spin. The missing 50% is attributed to the orbital motion of the various particles - a conjecture that has yet to be confirmed.

  • QCD Quandaries
  • Color Charge - It is not clear why the hadrons and mesons do not have a net color charge. However, this seems to be a non-issue as we used to think that the strong interaction between nucleons is mediated by mesons. The old concept of "strong interaction" is the manifectation of the residual color charges.

  • Saturated Gluon State - There are hints from particle accelerators that at extreme high speed the gluons inside the hadrons would multiply continuously up to a saturated state - called the color glass condensate. This is something that has to be verified and may yield insight into the dynamics of QCD.
  • Figure 14-05d QCD Quandaries
    [view large image]

    Funding application is in progress to build a quark-gluon femtoscope that would collide electrons with polarized protons and lead nuclei to reveal the structure inside the nucleon. Study of exotic states of matter can also contribute to the understanding of QCD (Figure 14-05d). See "Lattice Theory" and "Unitarity Method" for some progress in QCD studies over the last 40 years.

    Pentaquark There were reports on the detection of pentaquark since 2002. It has finally been confirmed in July 2015 by the LHCb experiment at LHC. Re-analysis of the 2009-12 data reveals that two states of the pentaquark have been detected at 4.38 and 4.45 Gev with less than 10-7 probability due to chance. The measurements were taken from the very rare events of the lambda(b) decay into the K- and the pentaquark with baryon # = 1 and electric charge +1 (see Figure 14-05e). There are two possible configurations : (1) the five quarks are tightly bound, (2) a baryon and a meson are loosely bound (something like a strong interaction molecule). The pentaquark itself decays into a proton and a J/ meson (the so-called charmonium) in about 10-24 sec. Note :
    the u, c, t quark has 2/3 unit of electric charge, while the d, s, b quark carries -1/3, sign of charge reversed for respective anti-quark.

    Figure 14-05e Pentaquark
    [view large image]

    The discovery will provide more information about the behavior of quarks and may help to shed light on the nature of quark stars. It is also suggested that dark matter could be dense agglomerations of many quarks called Macros (See more in "Dark Matter" and even more in "Strangely Familiar").

    The heaviest known naturally occurring element is uranium. However, even heavier elements can be created if enough neutrons can be squeezed into the nucleus to minimize the repulsion between the positive charges of the protons. It is suggested that there is an island of stability (Figure 14-05f) with the number of neutrons and protons close to the magic numbers as shown in Figure 14-05b. In 2008, a nuclear physics lab claims that it has synthesized a monstrous nucleus Ubb, which packs a whop-ping 122 protons and 170 neutrons. This element has a half-life of no less
    Nuclear Island Trans-uranium Elements than 100 million years, which seems to be too long even if it happens to be located right in the middle of the island of stability. Figure 14-05g presents all the trans-uranium elements synthe-sized artificially. It shows the steady decrease in half-life with increasing atomic number (# of protons), then this sudden jump in the disputed claim. The color of the square represents the

    Figure 14-05f Nuclear Island

    Figure 14-05g Trans-uranium Elements
    [view large image]

    chemical property of the element as indicated in the traditional periodic table (see also insert in the figure).

    Periodic Table upgrade The entire periodic table compiling of all the s, p, d and f orbitals (electron configurations) has finally been assembled with the synthesis of element 117 in 2010. If more elements are ever synthesized in the future, they will start with element 119 and 120 of the S-type and then follow by a huge block of the G-type with 50 columns (see Figure 14-05g). As the number of positive charges in the nucleus increase steadily, so does the speed of the electrons in the inner orbitals. The effect causes a contraction in the size of the inner orbitals, but the influence reaches all the way to the valence orbitals (the outermost ones), which define the chemical properties of the elements. For example, silver is shiny because the rather large energy gap is unable to absorb visible light; while gold (the element directly under it in the periodic table) displays rich color as the

    Figure 14-05g Periodic Table Upgrade [view large image]

    separation between energy levels shrink. Such relativistic effect becomes more drastic for elements with atomic number more than 100, they do not have the properties predicted by the periodic table and thus the table loses its explanatory power for the very heavy elements.

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