The Double Tetrahedron Structure of the Nucleus

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1. Introduction

The nucleus has been identified by the scattering experiments of Hans Geiger and Ernest Marsden, carried out under the supervision of Rutherford [1]. Ever since then this tiny center of the atom has been targeted by better and better penetrating probes to reveal its physics. Despite the great amount of knowledge has been gathered about the properties of the atomic nuclei even the phase of the nucleons remains an enigma. The developed models, shell [2] , liquid drop [3] and cluster [4], assume a gas, liquid and semi-solid phase for the nucleus respectively. The observed saturation properties of the nuclear forces, the low compressibility of the nucleus and the well defined nuclear surface are consistent with a liquid phase, the independent quantum characteristics of the nucleons can be explained by assuming a gas phase for the nuclei while the clustering of alpha particles is best explained with a semi-solid phase. Each of these models is able to describe very successfully certain selected properties of the nuclei; however, none of them able to give a comprehensive description. Most of the characteristics of the different phases are mutually exclusive. Like the well defined surface could never be explained by a model assuming a gas phase. The comprehensive description of the nucleus; therefore, most likely should come from a model representing a different phase. The remaining phase left is solid. This phase has not been considered for many decades as a viable option for the nucleus because of the uncertainty principle and the lack of diffraction.

2. Solid phase nucleus models

In the 1960s the discovery of quarks and neutron star researches satisfactorily answered the objections which have been hold against solid nuclear structures. The common features of the developed solid phase models are: the protons and the neutrons have same size and they alternately arranged in a closest packing crystal structure. These assumptions are reasonable. The radii of protons and the neutrons differ only slightly [5]. The same proton and neutron magic numbers indicates same structural development for both protons and neutrons resulting in an alternating proton and neutron array. The equal spheres most likely will be utilizing the available space in the most efficient way which is a closest packing arrangement. Among the highly symmetrical packing arrangements [6] the best agreement between nuclear properties and lattice parameters has been found for face-centered cubic (fcc) structure. Fcc lattice models can easily reproduce the various shell, liquid-drop, and cluster properties [7]. Phenomena like asymmetric fission, heavy-ion multi fragmentation, that the traditional models of nuclear structure theory cannot explain, can be reproduced by fcc lattice models [8]. The only remaining challenge for solid nuclear structure models are the individual properties of the nucleus. Solid structure could explain the individual properties of the nucleus if the nucleon arrangement in the lattice corresponds to quantum numbers. Significant effort has been made to find correlation between lattice positions and quantum numbers with partial success for fcc structure [9]. No correspondence between lattice positions and quantum states has been found for hexagonal closest packing, body centered cubic and simple cubic packing [10]. All previous investigations looked for symmetry elements by expanding the fcc structure spherically. The structure of fcc consist only tetrahedron and octahedron units. The expansion of these units should be investigated first if someone looks for structural symmetry pattern in the fcc lattice. In this study the expansion of tetrahedron and octahedron in an fcc lattice will be considered in details.

3. Expansion of the fcc crystalline structure to a double tetrahedron

The first stage in the nuclear structural development is marked by the end of the first period of the periodic system. Assuming closed packing arrangement for the nucleons, the two protons and two neutrons in the structure of the nucleus should form a tetrahedron. Calculations of potential model, constrained by the hadron spectrum for the confinement of the relativistic quark [11] and colored quark exchange model [12] are also consistent with a tetrahedron formation of the nucleus. Tetrahedron formation of equal spheres arranged in fcc packing can be expanded by adding layers of equilateral triangles packed in two dimensional closest packing. Starting with one sphere and increasing the length of the side of a triangle by one sphere the number of nucleons in each triangle will be 1, 3, 6, 10, 15, 21, 28, and 36 (Fig. 1/a). Placing one layer on to the other the number of spheres in two consecutive layers will be 4, 16, 36, and 54, which gives the number of protons in these double layers to 2, 8, 18, and 32. These numbers are identical with the number of possible states of the principle quantum numbers. If the layers are added to the outer sides of the tetrahedron then a shell like structures can be formed (Fig. 1/d). Assuming that the layers of the double tetrahedron correspond to the radius [R(r)] then the angular momentum quantum number (l) corresponds to , while the magnetic quantum number corresponds to .

In this physical sense the number of protons in one layer of the outer planes should be the same as the number of states determined by the angular momentum quantum numbers and the number of different proton positions in one layer should be the same as the number of magnetic quantum numbers. The correlation is identical for both cases (Fig. 2).

 

3.1. Structural symmetry and the periodic system

In order to cover the two sides of every vertexes of the tetrahedron a double tetrahedron has to be formed around a core tetrahedron. The core tetrahedron contains four nucleons. The arrangements of the nucleons in the fully developed double tetrahedra are shown in Fig. 3. The number of protons in the same layers of the vertexes agrees identically with the multiplicities (Fig. 4.). This agreement further supports the double tetrahedron nuclear structure. The proton numbers in the outer layers of the double tetrahedra are 2-8-8-18-18-32-32. These numbers are identical with the periodicity of the periodic system which exhibits a K-L-L-M-M-N-N development of the shells (Fig. 5/a). The existence of double periodicity in the L, M, and N shells is supported by all the characteristic features of the elements (Fig. 5/b). Quantum mechanics explains the double periodicity by the completion of shells and subshells; however, there is no explanation why certain shell and subshell numbers are outstanding among the many numbers corresponding to the completion of the shells and subshells. If the completion of shell marks the end of the period as in the case of K and L shells then why the completion of M and N shells does not have the same affect. The same argument can be raised against subshells.

3.2. Nuclear moments

All known even-even nuclei have ground state angular momentum of zero. If symmetrical positions are available in the structure then protons and neutrons predictable will favor those positions contrarily to non-symmetrical arrangements. Employing this assumption a symmetrical arrangement with zero magnetic momentum can be produced for every even-even nuclei. The predicted charge distribution of the symmetrical nuclei, up to Ar, has been shown in Fig. 3. The three dimensional images can be seen in Fig. 6., where the fully developed nuclear structures of the noble gases are also plotted [13].

3.3. Clustering properties of the nucleus

If a structure contains sub-structure then in the case of fragmentation the original structure most likely will break down into its substructure. The fcc structure contains tetrahedron and octahedron formations of the nucleons. The charge and structural stability of a tetrahedron is favorable in comparison to an octahedral nucleus arrangement indicated by the end of the first period in the periodic system and magic number two. Based on the tetrahedron sub-structure of the nucleus it can be predicted that the nuclear structure will most likely disintegrate to tetrahedron particles. This prediction is consistent with the observed alpha particle decays and clustering properties of the nucleus.

 

3.4. Nuclear magic numbers

Magic numbers correlate to the completion of a shell or sub-shell [2]. The double tetrahedron nucleus model offers a physical explanation for this correlation. When a structural unit fully developed on the surface of the fcc structure then this unit has higher resistance against forces in comparison to a surface consisting easily removable individual particles or not fully developed nuclear units. The well constrained structure of and , corresponding magic numbers 2 and 8 respectively can be recognized by simple visual inspection (Fig. 6). In an effort to quantify the compactness of an fcc structure the average minimized distance of the nucleons from the mass center have been calculated for structures with given numbers of nucleons. The nucleon positions in the fcc structure have been perturbed and for each arrangement the average distance have been calculated. The smallest value of the calculated average distances were taken as the average minimized distance of the arrangement. The determined minimum values for the second period of the periodic system have been plotted on Fig. 7. The minimum value observed at nucleon number 16 is consistent with magic number 8.

4. Conclusions

Expanding a four nucleons core tetrahedron to a double tetrahedron formation it has been found that the number of protons in the outer two layers of the double tetrahedron corresponds to the principle quantum numbers, the number of protons in a layer of these shells corresponds to the angular momentum quantum number, while the number of different proton positions in one layer corresponds to the magnetic quantum number. These correlations are consistent with the physical meaning of the quantum description. By explaining the individual properties of the nucleus the proposed double tetrahedron fcc crystalline model offers the first comprehensive description of the nucleus.

 

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