Finding New Magic Numbers for Light, Heavy and Superheavy Nuclei

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1 Finding New Magic Numbers for Light, Heavy and Superheavy Nuclei By Roger A. Rydin Emeritus, University of Virginia Abstract Historically, nuclear modeling came down to two complimentary ideas, the Nuclear Shell Model and the Liquid Drop Model. Certain combinations of neutrons and protons in closed shells led to exceptionally stable isotopes, and these were referred to as the magic numbers: 2, 8, 20, 28, 50, 82, and 126. There was no apparent pattern to this number sequence, only an experimental observation that there is a discontinuity of about 1 MeV in binding energy as each of these boundaries is crossed, i.e., an extra nucleon is loosely bound. A new phenomenological Electro-dynamic Model of the nucleus has been constructed by separately modeling a proton and a neutron using a classical electromagnetic approach first suggested by Arthur Compton. Using these two nucleon representations, different nuclei are arranged by placing neutrons and protons in static geometric positions, where the force balance has minimum energy. These positions have been verified numerically using a variational minimization technique. The intrinsic mass of a nucleon is ~ 99%. Some mass equivalent is stored in EM Fields between nucleons, and some mass equivalent is stored in EM fields inside a nucleon. The masses of all individual isotopes that were calculated are in agreement with the measured values within less than a tenth of a percent, thus mimicking the experimentally measured binding energy per nucleon curve. This suggests that mechanical vibration is a valid physical mechanism for decay and de-excitation, and Wave Mechanics is not needed. For the first time, the pattern of all the magic number closed nucleon-shells is accurately predicted based on the shell pattern 2, 8, 18, 18, 32 and 50, and many new magic numbers have been found. These magic numbers correspond to stable or long lived isotopes, and to the range of existing isotopes. The new magic number 58 explains the double hump fission product distribution almost exactly. 1. Introduction The current model of all particles is called the Standard Model. The model predicts by a tabular inference all of the common and exotic particles. The basic particles come in families of three, such as the electron, muon and tauon, etc. The neutron and proton are said to be composed of fractionally charged quarks, two up and one down for the proton and two down and one up for the neutron. Inside the nucleus, the protons repel each other by the fairly long range 1/r-squared Coulomb Force. Neutrons and protons attract each other by a short range Strong Force, attributed to gluon exchange. Neutrons and protons undergo decay using the Weak Force by emitting beta particles and neutrinos, mediated by the exchange of heavy W+ and W- particles. The neutrons and protons are said to occupy shells of some kind, with extra stable nuclei containing full shells of 2, 8, 20, 28, 50, 82 or 126 nucleons, called the magic numbers. The nucleons are thought to be in rapid motion, and decay takes place by Coulomb Barrier penetration using Wave Mechanics. It is said that quarks, gluons and W particles have been observed in high

2 energy machines such as at CERN, but the nature of the detectors is such that such observations are more inferred than real. 2. New Electro-dynamic Model of the Nucleus In a series of papers, Lucas [1] has developed a new Electro-dynamic Model of particles, based on using +1/3e and -1/3e wrapped charged fibers in a three level scheme. Using these models, decaying particles essentially split into their decay products needing at most some neutral fiber pairs. A model of the nucleus has been developed by arranging these nucleons in double shells containing 2, 8, 18, 18, 32 and 50 neutrons or protons. The major new feature is that these nucleons are in static positions under electromagnetic force balance, and at most vibrate about their equilibrium positions. The magic numbers have been explained as complete fillings of these six basic shells, so that the doubly magic isotope Pb-208 has 32 and 50 = 82 protons in the two outer shells, and 8, 18, 18, 32 and 50 = 126 neutrons in the five outer shells. Hence, the old magic numbers are composites of shell fillings. The Strong Force has been replaced by a 1/r-fourth Electromagnetic attraction, and the Weak Force has been replaced by a repulsive compression resistance to nucleon distortion. Using this model, and the composite shell idea, a new Semi-empirical Binding Energy formula has been developed that is accurate for more than 3000 elements, and gives the low A peaks accurately rather than missing them as the old formula did [2]. In addition, all the spins are correctly predicted, rather than getting a reasonable fraction wrong. Fig. 1. New Semi-empirical Binding Energy Fit from the Electro-dynamic Model On a dare by a member of the UVA Physics Faculty, the methodology was applied to a completely new problem, that of explaining the Superheavy nuclide distribution. Experiments done in the USA, Russia, Germany and Japan using particle accelerators to bombard heavy elements such as Californium with heavy doubly magic projectiles such as Ca-48, allow production of exotic new elements up to Z = 118 and examination of their subsequent decay products. New magic numbers had been predicted theoretically by a variety of methods, and the one thing in common was that there were numerous new numbers fairly close to one another. A variant of the six shell model was used, where an 18 shell could refill to 32, and a 32 shell could refill to 50, in a manner similar to the refilling of electron shells in the Lanthanide rare earths and the Actinide elements. A number of magic numbers larger than 82 were predicted. For the most part, these corresponded to the boundaries of the Peninsula, Shoal and Island of Superheavy Elements [3]. In the Peninsula, it was predicted that the isotope Th-230 was doubly magic!

nucleon and certain nuclear properties such as spontaneous nuclear fission, that quantum nuclear shell models have been unable to adequately describe.

The shell model and the liquid drop model are incompatible models in that the surface of the nucleus in a shell model cannot act like a liquid surface. Fig.

3 nucleon and certain nuclear properties such as spontaneous nuclear fission, that quantum nuclear shell models have been unable to adequately describe. However, these things can be satisfactorily described by the liquid drop model of the nucleus. The shell model and the liquid drop model are incompatible models in that the surface of the nucleus in a shell model cannot act like a liquid surface. Fig. 2 The Heavy and Experimentally Discovered Superheavy Elements (Science Daily, 2008) Since the idea of new magic numbers worked well in the high mass range, it was postulated that it ought to work well in the in the Continent of Isotopes. Indeed, here it explains a number of the stable isotopes, and explains the breadth of the fairly long lived isotopes of a given element in terms of heavy and light magic combinations [4]. The new magic number 58 used in the light fission fragment explains the double hump fission product distribution almost exactly. Fig. 3 Fission Product Yield vs. Mass Number All of the results have been summarized in a review paper [5]. There are some nuclear properties, such as the binding energy per In the geometrical packing model, there is a physical basis for the liquid drop model. This can be seen from Fig. 4. For these figures, the structure of the spherical shells has been symbolically represented by a slice cross section through the center of the nucleus such that each spherical shell shows up as a circle or ring. Each proton shell is shown explicitly. Each neutron shell is depicted as an electron shell coupled to a proton shell, i.e., the neutrons polarize in such a way that the neutron shell appears to be an electron shell plus a proton shell. Note in each of the figures that in the innermost part of the nucleus, polarized neutron and proton shells alternate as one proceeds from the center of the nucleus outward. This alternating sandwich effect keeps them tightly bound together. However, at three shells in from the outermost shell, there are always two proton shells in a row separated by a sidewise-polarized or net neutral neutron shell for the larger nuclides. This causes the last three alternating sandwiches of bound shells to be repulsed by the inner nucleus. Thus, they are only weakly bound to the inner nucleus. This weak binding allows the outermost triplet of shells to have liquid-like properties and forms the proper justification for a liquid drop model of the nucleus. Such an effect does not exist in quantum shell models of the nucleus, because they are based on a central potential instead of allowing a dynamic rearrangement of shells to minimize the binding energy of the nucleus.

is correlated with the quantum mechanical model derived from nuclear wave functions, appears to have some 65 80% over all reliability in reproducing nuclear spins of all isotopes. Fig. 4.

4 It has been our experience that the ring shell model scheme always comes closest to the final shell occupancies. Perhaps this is owing to the fact that this scheme accurately reproduces reported nuclear spins of all isotopes for which values are listed; whereas, the conventional magic numbers shell scheme which is correlated with the quantum mechanical model derived from nuclear wave functions, appears to have some 65 80% over all reliability in reproducing nuclear spins of all isotopes. Fig. 4. (Shell Structure of Lead-208 An exact expression for all electromagnetic static interactions between charged-ringmagnet-like neutrons and protons (except for the self energies), which constitute the nucleon components of an atomic nucleus, has been derived for this study. Coulomb forces are 1-over r-squared, while magnetic forces are taken to be 1-over-r-fourth. Integrations are carried out over all angular orientations within the boundaries of each current loop. The required numbers of nucleons are initially distributed within shells, using either the old magic numbers scheme or the new ring model scheme. In the case of 40 K, for example, there are 19 protons for the atomic number plus 21 polarized neutrons. The assignments for the ring model shell scheme are 1, 0, 18 protons and 2, 1, 18 neutrons. While the methodology of the calculations is independent of initial particle assignments (energy minimization via the variational approach will attain the correct final shell assignments), if the initial assignments are reasonably close to the final, there is a great saving on time constraints for minimization. Fortunately, accurate Nuclear Binding Energies (NBE) are not essential for obtaining reasonably accurate decay energies, which depend on differences in NBE and not their accurate values. Whatever discrepancies in NBE are present in one parent isotope are also present to the same extent in its daughter isotope, for which the differences cancel upon evaluating the decay energy. Note that the decay energies presented are accurate within 90 99% of the accepted values. Table 1. Computed Nuclear Binding Energies The basic mechanism of decay is probably a complicated nonlinear vibration, composed of motion between nucleons, motion within nucleons, and rotary motion of groups of nucleons. The free neutron decays to a proton, an electron and an antineutrino, with a half life of about 13min and with a Q value

5 of MeV. An alternate explanation for the beta decay of a neutron rather than boson exchange is that an accidental bump starts the neutron to begin vibrating internally. The analog of the Weak force acts as a compression resistance to distortion of the neutron, so we have a spring reaction. When the amplitude of vibration takes the neutron s geometric configuration to a point where another configuration at a lower energy state is more stable, it separates fibers into a proton, an electron and an antineutrino, giving up the mass difference between the neutron and proton as excitation. From the standpoint of the Lucas model, the fractionally charged magnetic fibers of the neutron essentially rearrange themselves into the fiber configurations of the three new particles. Whenever a new more stable configuration in an excited nucleus is obtained by vibration amplitude, the parts can separate into the new configuration, releasing the difference in energy and angular momentum in the process. A vibration model is also capable of explaining all the other nuclear transitions from one excited state to another, including delayed neutron emission. The numerical method used above cannot converge on such higher states without using a numerical procedure that excludes the ground state, such as a Wielandt fractional iteration scheme. Nonetheless, a beta transition from one state to another excited state can occur, and this will appear to be quantized because of the fixed configuration energy between the states. The excess energy can then be emitted subsequently as discrete gamma rays from one allowed state to another as the ground state is approached. A simplified nucleus model, which treats the nuclear forces between nucleons as linear springs, produces a classical multi-body vibration problem. Setting F = ma, in 3D space, gives an eigenvalue structure of masses and spring constants with many allowable oscillatory eigenstates. This equation mimics the Schrodinger equation, which is the basis of Wave Mechanics. 3. Summary A simple physical geometrical packing model has been found capable of describing the packing of atomic electrons about the nucleus in layers or shells as well as the packing of neutrons and protons in the nucleus itself. This leads to an assignment in shells for nuclides in the Chart of the Nuclides, and leads to a much improved new Semi-empirical binding energy formula. This formula is much more accurate than the original Semi-empirical binding energy mass formula, and includes more of the detailed features of the experimental measurements, including spin, meaning that it gives a better phenomenological representation of the nucleus. From the limited number of examples presented, it appears, from the consistency of the results, that the simple Lucas classical electro-dynamic ring model of the neutron and proton gives new insights into the organization of the nucleus and provides a simple but crude basis for calculating nuclear structure including binding energy and radioactive decay energies. At this point in time, accuracy in calculating specific NBEs is lacking, since self energies and other higher order refinements have not been incorporated into the model. 4. Extensions As the result of a recent contact with a Russian scientist, we now have additional experimentally derived evidence about the magic number 58. The group at JINR is interested in what they call nontraditional magic numbers, and they ask about the number 6 in C-14 and O-14. For C-14, the 6 protons are in the 8 shell, but are the 8 neutrons in the 8 shell, or are there 6 in the 8

6 shell and 2 in the 2 shell? For O-14, the 8 protons are in the 8 shell, but are the 6 neutrons in the 8 shell or are there 4 in the 8 shell and 2 in the 2 shell? This can be sorted out in the Electromagnetic nucleus model by Baxter's computer code, which iteratively converges on the minimum energy configuration after being given a guessed nucleon distribution. The primary JINR paper is by Angeli, et al [6]. The group uses experimental measurements of nuclear charge radius over a wide span of nuclides, to look for bends, kinks, and peaks in systematic behavior. For this work, they derive a kink strength S. They define, Changes in the nuclear structure as shell effects and deformation give rise to changes, kinks, in the mass number dependence of nuclear charge radii R. Along isotopic sequences, these small variations in the slope of R Z (N) series can be observed easily by investigating the second difference d 2 R/dN 2. Therefore, in what follows, the slope change, kink, will be characterized by the quantity S Z (N) = d 2 R Z (N) / d (2N) 2 A 2/3. A similar effect is shown for N = 82, which has been known to be magic for over 50 years, but again is obvious in Figure 6 for Nd, Ce, Ba, Cs, and Xe. Fig. 5. Kink Strength S versus Neutron Number N (Angeli, et al) The kink for Ce-140 has long been explained by having N = 82, but using Z = 58, Ce-140 is actually doubly magic, with an abundance of about 90%, so this data confirms Z = 58 as well. In fact, there is an old plot in The Atomic Nucleus (Evans, 1958) that shows similar behavior in the B/A curve, shown in Figure 7. We contend here that the strong slope change is due to double magic. The kinks in the isotonic sequences can be defined in an analogous way by simply exchanging N and Z in the equation. In the main text the short form S will be used, where the subscript and argument are not necessary. The following graphs are taken from their paper, which show the kinks for N = 58 and N = 82. The behavior for Zr, Y, Sr and Rb are shown in Figure 5, where the peaks in S are obvious at N = 58. Fig. 6. Kink Strength S versus Neutron Number N (Angeli, et al)

7 with the rest tied up in internal electromagnetic fields. Fig. 7. Binding Energy, B/A versus A (Evans, 1955) Finally, we come to two new ideas, ether and intrinsic mass. It would be very convenient if the ether were made up of a cubic lattice of pairs of unwrapped +1/3e and 1/3e charged fibers. Such fibers would be essentially nonreactive with respect to any other particles, and they would be essentially without intrinsic energy. But they would act as a medium to transmit EM energy, and they would be available to complete the fiber balances for various decay reactions. Other candidates for the ether seem to be too massive to not be detected experimentally. The second idea is about mass, or rather where does the binding energy arise for nuclear reactions? The simplest nucleons are the free proton and free neutron. The free proton is stable. The free neutron decays to a proton, an electron and an antineutrino, with a half life of about 13 minutes and with an end point energy of MeV. Each nucleon has a mass of approximately 1 AMU, each with an energy equivalent of about 931 MeV. We immediately see that if each nucleon has an intrinsic mass, it is at least 99% of the total mass of the particle, For the neutron, the mainstream explanation of beta decay is that it is fostered by massive bosons called W +, W-, and Z 0 that act through the Weak Force. This is patently ridiculous. The charged W bosons cannot contribute because there is no experimentally observed extra charged particle that can complete the balance. It is also ridiculous to think that the massive 90 GeV Z 0 would exist only to foster such a small energy change and then disappear back into the continuum. A far better explanation for the beta decay of a neutron is that an accidental bump starts the neutron to begin vibrating internally. The Weak Force acts as a compression resistance to distortion of the neutron, so we have a spring reaction. When the amplitude of vibration takes the neutron s geometric configuration to a point where another configuration at a lower energy state is more stable, it separates into a proton, an electron and an antineutrino, giving up the mass difference between the neutron and proton as kinetic energy. From the standpoint of the Lucas model, the fractionally charged magnetic fibers of the neutron essentially rearrange themselves into the fiber configurations of the three new particles. The same process occurs for positron decay, but the required extra energy has to be borrowed from the nucleus. When we begin to put neutrons and protons together into nuclei, we find that the resulting nucleus is less massive than the sum of its constituent free nucleons. This mass defect is called the Binding Energy, which has been emitted when the nucleus

8 was formed, and must be supplied back in order to take it apart again. We obtain a pattern (Figure 1) of Binding Energy per nucleon, or B/A, which follows a generally rising curve with some exceptional peaks, with a broad peak beginning at about A = 60 and a B/A of about 9 MeV per nucleon, and then a gradual falloff as the mass number A increases further. The maximum Binding Energy equivalent per nucleon is always less than 1% of the mass of a nucleon. When A is low, heavier nuclei can be formed by fusion of lighter nuclei and release energy in the process. When A is high and greater than A = 208, heavier nuclei can be split up into smaller pieces by fission or alpha decay, and also release energy. When A is near the peak B/A, no more energy can be obtained by nuclear reactions, which for a star means that it can no longer produce enough energy to avoid gravitational collapse. If the star is massive enough, this collapse leads to a supernova. If smaller, it becomes a dead star. motions are isolated from each other and ignore the relative difference in velocities for all behavior except for Doppler shifts based on c being constant in an absolute reference frame. 4. Conclusions A new Electromagnetic model of the nucleus, where the nucleons occupy static positions and decay is caused by vibration to a more stable configuration, seems to answer a large number of open questions about the behavior and energetics of nuclear decay. The model quantitatively explains a vast amount of experimental data to a high degree of accuracy. Mass exists to our senses, as does gravitation. Current theories, such as the Standard Model, do not explain either. With the use of an Electro-dynamic Model of the nucleus, an explanation has been obtained for the Binding Energy properties of nuclei, and for the energetic behavior of nuclear reactions that appear to be quantized. A portion of the mass equivalent of a nucleon is tied up in the internal electromagnetic fields, and this amount can be released as the field configuration changes. The basic physical mechanism of change is vibration, which allows transition from one state to a more stable state, releasing a fixed amount of energy and spin in the process. Mass itself does not increase with velocity, but only appears to do so. Galaxies are isolated from one another, so their internal References 1) Lucas, C.W., Jr, Galilean Electrodynamics 7, 3 (1996).

9 2) Lucas, C.W., Jr. and Rydin, R. A., Electrodynamic Model of the Nucleus, Nuclear Science and Engineering 161, , ) Rydin, R.A., A New Approach to Finding Magic Numbers for Heavy and Superheavy Elements, Annals of Nuclear Energy 38 (2011), ) Rydin, R.A., New Magic Numbers in the Continent of Isotopes, Annals of Nuclear Energy 38 (2011), ) Lucas, C. W., Jr., Baxter, E. C., Boudreaux, E. A., and Rydin, R. A., A Classical Electro-dynamic Theory of the Nucleus, Physics Essays 26, 3, ) Angeli, I., Gangrsky, Yu. P., Marinova, K. P., Boboshin, I. N., Komarov, S. Yu., Ishkhanov, B. S. and Varlamov, V. V. N and Z Dependence of Nuclear Charge Radii, J. Phys. G: Nucl. Part. Phys. 36 (2009) (26pp)

Finding Magic Numbers for Heavy and Superheavy Nuclei. By Roger A. Rydin Associate Professor Emeritus of Nuclear Engineering

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