The packing model of the nucleons in an atomic nucleus always arouses curiosity of many scientists and amateurs. The model determines the shape of a nucleus as well as its various properties, such as the stability, the magnetic moment, the spin, the binding energy and radioactivity. Sophisticated theories of nuclear model were established and many facts were well explained [1] [2] . However, there is obviously still a mysterious veil on some characteristics of nuclei to be revealed, especially the packing structure of nucleons inside a nucleus. Thanks to X-ray diffraction, chemists know plenty of details about how atoms pack to build a molecule. Molecular structure establishes the basis of modern chemistry. However, since nuclei are too small and are usually embedded in electron “cloud”, it is very hard to “see” them.

In nature, the equilibrium shape of a centered direction-independent object is a sphere. These objects vary vastly in composition and size, including planets, liquid droplet, bubble, most nucleoli in cell, crystalline spherulite, latex micelles, isolated atoms (at least those which can be packed in cubic lattices) and possibly some elementary particles. A plant leaf is far from a sphere because the petiole to the stem and the surface toward the sun are very direction-related. The non-sphere shape of a molecule comes from the direction-related chemical bonds. Many atomic nuclei are generally not sphere and may possess remarkable non-zero quadrupole moment [2] . That means tha the nucleons, protons (P) and neutrons (N) are bound in specific direction and are packed in a particular structure. In this paper, the authors try to propose a model that depicts the packing of nucleons in a nucleus based on the conception of covalent molecule configuration.

From the reported data [3] -[5] , there are 7 isotopes of hydrogen, or single-P nuclides: ¹H, ²H, ³H, ⁴H, ⁵H, ⁶H and⁷H. In addition, reported single-N nuclei include ²H, ³He, ⁴Li, ⁵Be and ⁶B. This gives a clue that N, as well as P, can be a 6-coordinate center bound with 6 “ligands” at maximum as shown in Figure 1(a), where the proper octahedral structure seems to be a right way to ensure the ligands to be indistinguishable. Although these structures need to be further verified, especially the 90˚ triangular ³H and ³He, it is obvious that, in the case of single center, when the “coordination number” is higher than 2, the isotope will be unstable and tends to kick out the excess P or N.

Based on these structures, we assume that P can only bind with N and vice versa (considering deuterium, PN, is stable, but there is no conclusive evidence about the existence of NN and PP). The “bond distance” between bound P and N, D_NP, is in the range of 1 to 2 fm, and the “bond angle” for 3 continuous nucleons is 90˚ or 180˚.

The single-center nuclei are characterized by extremely low binding energy E_B (and therefore a low E_B/A), even for the stable nuclei as ²H and ³He, noted in Figure 1(a) comparing with other nuclei. This is probably because of the existence of suspending open P or open N. Therefore, we assume that, for a nucleus with ratio of N:P = 1, open nucleon should be avoided and all P, as well as all N, should be equivalent. Thus, P and N should be bound alternatively to form a ring. It is obviously that the ring is not shaped like a pearl necklace on neck; instead, the ring must fold somehow, like a necklace in casket. The most possible way of folding a ring is taking a “bond angle” of 90˚ for every 3 continuous nucleons, to ensure the nuclide to adopt the densest packing and the most nearly sphere symmetrical shape. However, for most nuclei, the ratio of N/P is more or less than unit. Among hundreds of stable nuclei, only 12 are composed purely of a ring: ⁴He, ⁶Li, ¹⁰B, ¹²C , ¹⁴N, ¹⁶O, ²⁰Ne, ²⁴Mg, ²⁸Si, ³²S, ³⁶Ar and⁴⁰Ca. Thus, extra N(‘s) or extra P(‘s) will be necessary to connect or insert the ring somehow in most nuclei.

Based on the “ring plus extra nucleon” hypothesis, the shape of ⁴He should be a square, which is the smallest ring. Figure 1(b) demonstrates the shapes and E_B of nuclei of the P₂N₂ family, where the extra open N generally contributes very little to E_B, while extra open P generally gives minus contribution. Thus, in the P₂N₂ family, all the nuclei which possess open nucleon are unstable. In addition, the E_B and life time of P₂N_2+X are always in larger number than P_2+XN₂. This corresponds to the fact that stable nuclei present a ratio N/P ≥ 1(except ³He).

In Figure 1(b), some nuclei may have 2 or more isomers, for example, ⁵He may have 2 isomers, ⁶He 3, ⁷He 6 and so on. It is reasonable to assume that favored structure should be the highest symmetry and the most compact. It should be noted that, for saturating the 6 coordinates, ¹²He will be the possibly heaviest isotope of helium (¹⁰He is the maximum found). For the same reason, for elements of Z = 7 to 10, ⁹N, ¹⁰O, ¹¹F and ¹²Ne will be the possibly lightest isotopes respectively, which have not been found yet.

Figure 2 gives the possible shapes of some light rings and their stable isotopes. For odd-Z rings, although

there are a few possibilities and more when Z becomes large, a common feature is that the gravity centers of P and N are separated and are eccentric to the gravity center of nuclide. This can be easily proved mathematically. In addition, one can always find one or more arrangement with antisymmetric center, through which each P can find an N by space reflection and vice versa. We assume that the most possible odd-Z ring structure is antisymmetric and the most densely packed. For ⁷Li and other nuclei, the extra N may not likely bind to a ring P as an open N like unstable nuclides of P₂N₂ family. Instead, when 2 or more ring P’s are geometrically available to share an extra N with 90 “bond angles” and same D_PN, the extra N tends to be stable. It seems this is the most likely way the extra N combines with the ring considering the geometric arrangement and indistinguishableness of N’s. Actually, among all stable nuclei, the ratio of N/P is in the range of 1.00 to 1.54 (²⁰⁸Pb), that means ring P is able to stably bind a little more than 0.5 extra N (or 2 ring P’s share one extra N) at maximum, while no extra P can stably bind to ring N. This is the reason that, in the P₂N₂ family, all the nuclides with extra P or extra N are unstable because of the valence limitation of the square shaped ring. From the shape of the ring structure, one can predict the possibly heaviest and the lightest isotopes of the related elements. As an example, based on P₃N₃ ring, the possibly heaviest isotope of lithium is ¹⁶Li and the possibly lightest isotope of aluminium is ¹⁶Al. For other elements, the possibly heaviest and lightest isotopes can also be predicted if the structure of related ring is known.

The possible packing structures of some light even-Z rings and their stable isotopes are also shown in Figure 2. By contrast to odd-Z rings, all the even-Z rings possess a superimposed gravity center of P and N, which was tested by hundreds of examples without any exception and is destined to be proved mathematically. In addition,

some of the structures are center-symmetrical. We assume that the most possible ring structure of even-Z nuclei is the most densely packed one with center-symmetry and is indistinguishable when all P and N interchange. In Figure 2, the framed ¹²C is marked “impossible” because it mutates to another structure when P and N interchange. It is interesting to find that ⁸Be is composed of two P₂N₂ squares and is ready to yield into two α-particles. Only ⁹Be, with an extra N that bridges two squares by the manner mentioned above, is stable. Therefore, as another restriction, the most possible ring structure should avoid the arrangement that can split into two normal rings.

Because of the symmetrical and geometrical restrictions of ring structure, empty open space in a large nucleus (Z > 20) is unavoidable. An important function of extra N is to fill the open space, making the nuclide packed more densely and establishing adequate non-bond interaction. Figure 3 shows the nuclide size, characterized by radius of gyration, [R²]^0.5 (as defined in the following equation) versus the nucleon number A. The points of stable nuclides locate around the line of slope = 1/3, or the nuclide volume is roughly proportional to A. In addition, one can find a general trend that, for rings, the radius expands related to the line with Z increases due to the open space (¹⁶O is special possibly because of double magic numbers, which will be revealed in future).

From the possible structures shown in Figure 2, one can figure out that if nuclei possess too many N’s to be bind properly by 2 or more ring P’s, some open N will be unavoidable as a neutron halo and the nucleus will be unusually large, such as ¹¹Li in Figure 2. This is qualitatively accordant with the scattering cross-section observations for extremely N-rich nuclides [6] .

The E_B/A for all the P_xN_x rings, in Figure 4, display a zigzag feature, where even-Z rings are at the vertex and odd-Z rings at the nadir. As mentioned earlier, the fundamental difference for even-Z and odd-Z ring is that the former has superimposed gravity centers of P and N, while the later is eccentric. It is obvious that eccentricity results in lower E_B, which is possibly because the eccentric nucleus inside an atom requires more energy to spin faster to avoid the surrounding electrons to “feel” uneven.

It is well known that E_B/A of the isotopes of an element displays a specific zigzag feature, where the vertex always relates odd A for odd-Z elements, and relates even A for even-Z elements, as shown in the examples (₂₆Fe and ₂₉Cu) in Figure 4. Weizasäcker’s semi-empirical mass formula treated this feature by a selective function, f₅ in an ad hoc way [2] .

In this work, the degree of eccentricity, S_g, is defined as the separation of the gravity centers of P and N, D_g(PN), over 2 times the radius of gyration, [R²]^0.5,

where D_i is the length from i’th nucleon to the gravity center of the nucleus. For any even-Z ring, S_g is always zero because D_g(PN) = 0; while for odd-Z rings, S_g is higher than 0. From the definition, deuterium, ²H, has the highest S_g of 1. For other light odd-Z rings, such as ⁶Li and ¹⁰B, S_g is also quite high, and therefore leads up to a very low E_B. The depression of E_B/A by eccentricity, ΔE_ec, is defined in Figure 4 for some simple odd-Z rings and even-odd nuclides. The relationship between S_g and ΔE_ec is demonstrated in Figure 5. It should be noted that this relationship is not very quantitative because ΔE_ec is also affected by non-bond interactions of a particular nuclide as will be mentioned later.

Cohesive interactions between metallic atoms are mostly governed by their electron configuration. Since the high S_g means the high nuclear “polarity”, which provides an extra interaction between isolated atoms, like polar molecules have higher intermolecular interactions than nonpolar ones. That can explain Li, Be and B respectively demonstrate the unusually highest melting point and boiling point in their group (similar electron configuration) of periodic table.

To understand the zigzag feature, Figure 6 depicts the D_g(PN) and eccentricity S_g shift caused by extra nucleon in different cases, that results in fluctuation of E_B/A exactly as the way as the rings or isotopes of any element do (vertex for even Z at even A while odd Z at odd A).

The envelope line of E_B/A for even-Z rings in Figure 4 demonstrates that, without the energy depression by eccentricity, E_B/A increases from very low for light Z ring to maximum at P₂₈N₂₈, and then decreases slowly. This should be due to the variation of non-bond interaction.

For most elements, E_B/A of a ring, as basic structure, is not the highest among the isotopes, as the examples shown in Figure 4. It is because extra N, which fills the open spaces in large ring and builds adequate non-bond interaction, may lead to higher E_B/A and stability. Nevertheless, special stability of ring structure can be detected

in many facts. Weizasäcker formula includes a symmetry term f₄ emphasizing P_ZN_Z is more favored in energy [2] . This can also be shown in Figure 4 that, except P-rich nuclides, P_ZN_Z always possesses the highest one-N separation energy, S_n (for most odd-Z nuclei the highest is P_ZN_Z+1 because of lower eccentricity). That is to say, N in a ring structure is generally more stable than extra N. Since extra P is much more unstable, the normal extra nucleons can only be N or P, excluding N and P.

Basically, E_B/A of a nucleus is governed by many factors. Strong force between binding P and N in a ring plays the most important role, up to about 7 MeV (as in ⁴He), which should be equal for all rings. The weak force acts between non-bond nucleons (P..N, N..N and P..P) in a ring that provides higher E_B/A up to 8.5 MeV (as in ⁴⁰Ca). The open spaces hinders non-bond interaction, and makes nuclide very unstable even E_B/A is not so low. The extra N shared by 2 or more ring P’s, which contributes similar strong force, provides more weak force and, therefore, can increase E_B/A up to 8.8 MeV (as in ⁶²Ni). When ring is too large, the open spaces may not be completely filled by extra N, and E_B/A lowers gradually. Eccentricity is an important factor that decreases E_B especially for light nuclei as depicted in Figure 5. In addition, excess extra N or P can only connect to the ring as open nucleons, which have very little or minus contribution to strong force and thus result in many very unstable nuclei with very low E_B/A. Because of the complicated effects, the nuclide shapes shown in Figure 2 are only the most possible arrangements and need to be confirmed or revised by new theory considering more details of nuclear forces and new experimental observations.

Mirror nuclei are the pair of nuclides P_X+nN_X and P_XN_X+n. Some correlations of the pair have been reported since long ago [7] -[10] . The most remarkable fact is the identical spin and parity, J^π, of mirror nuclei in ground state as summarized in Table 1, with very few exceptions or uncertainties. Since the ring is indistinguishable when P and N interchange, any nuclide with n extra N(¢s) will have a counterpart with n extra P(¢s) in similar way of binding to the ring. Thus, the shapes of the pair nuclides are similar but P and N interchange, just like a black-white photo and the negative film. This should be the basic reason for the following equation.

It is also reasonable to predict the identity of degree of eccentricity for the pair.

⁷Be is present in the atmosphere in trace amount [11] as the only natural occurring radioisotope with the ratio of N/P < 1. As the mirror nuclide of ⁷Li, ⁷Be has an extra P and can also obtain a large energy bonus from the highest eccentricity in ⁶Li although, as expectation, ⁷Be is still not stable enough as ⁷Li.

Figure 7 demonstrates the E_B/A of mirror nuclei. It is clear that the E_B/A of P_XN_X+n is always larger than that of P_X+nN_X. Compared with the zigzag feature of ring P_XN_X mentioned earlier, when n is odd, the curves are smooth because odd extra P or N diminishes the depression energy by eccentricity for odd-X nuclides. Of course, when n is even, the curves zigzag again. In addition, it is notable that the difference of E_B/A for any pair is nearly a constant: averagely 0.179, 0.368, 0.559, 0.763, 0.926 and 1.084 MeV for n = 1, 2, 3, 4, 5 and 6 respectively (the curves for n = 5 and 6 a re omitted in Figure 7). This can be described with following equation.

In summary, the ring plus extra nucleon model seems properly meet the structure requirement to the correlations of mirror nuclei.

A nucleus P_XN_Y (X, Y ≥ 2) is composed of a ring P_XN_X with P and N alternative binding plus Y − X extra N(¢s) if Y > X, or a ring P_YN_Y plus X − Y extra P(¢s) if X > Y. The ring folds with a “bond angle” of 90˚ for every 3 continuous nucleons. Extra N can bound to ring-P with the same “bond angle” and “bond distance”. Extra P can bound to ring N in a similar way but the binding is weaker than that of extra P. When 2 or more ring P’s are geometrically available, the extra N tends to be stable. Excess extra N results in open N with low E_B/A and neutron halo with large radius. Even-Z rings, as well as normal even-even nuclei, always have superimposed gravity centers of P and N; while for odd-Z rings, as well as odd-A nuclides, both centers of P and N must be eccentric. The eccentricity results in a depression of E_B. Therefore, the zigzag features of E_B/A of an element differing for odd and even Z can be simply explained by the shift of eccentricity by extra nucleons. Symmetry center may present in any even-even nuclei. While for odd-Z rings, only antisymmetry center is possible. In both cases of even and odd-Z, the rings are indistinguishable when P and N interchange. Based on the ring plus extra nucleon model, the mirror nuclei are equivalent in structure and demonstrate identical spin and parity. As the eccentricity shift, the E_B/A curves of mirror nuclei display smooth when the number of extra nucleon n is odd, and zigzag

*Not include the mirror nuclei n > 5, which satisfy the identity as the cases in the table.

when n is even. The E_B/A difference between the pair is nearly a constant of 0.184n MeV.

The authors thank Prof. Jubo Zhu and Prof. Jianshu Luo for their effort in mathematical proof as well as many valuable discussions with Qinghua Wang, Tianjiao Hu and Zhenghua Jiang.

1. The nucleon coordinates (x, y, z) for the nuclei in Figure 2

(P: bold, N: normal, extra N: in parenthesis)

⁶Li 0,0,0 & 1,1,0 & 0,1,1 & 1,0,0 & 1,1,1 & 0,0,1

⁷Li ibid & (0,1,0)

¹¹Li ibid & (2,1,0) & (0,1,2) & (0,-1,0) & (1,2,0)

¹⁰B 0,0,0 & 1,0,1 & 2,1,1 & 1,2,1 & 1,1,0 & 1,0,0 & 1,1,1 & 2,2,1 & 1,2,0 & 0,1,0

¹¹B ibid & (0,0,1)

¹⁴N 0,0,0 & 0,1,1 & 1,0,1 & 2,1,1 & 3,1,0 & 2,0,0 & 1,1,0 & 0,0,1 & 1,1,1 & 2,0,1 & 3,1,1 & 3,0,0 & 2,1,0 & 1,0,0

¹⁵N ibid & (0,1,0)

¹⁸ F 1,0,0 & 2,0,1 & 3,1,1 & 2,1,0 & 2,2,1 & 1,3,1 & 0,3,0 & 1,2,0 & 1,1,1 & 2,0,0 & 3,0,1 & 2,1,1 & 2,2,0 & 2,3,1 & 1,3,0 & 0,2,0 & 1,2,1 & 1,1,0

¹⁹ F ibid & (1,0,1)

⁸Be 0,0,0 & 1,1,0 & 2,1,1 & 1,0,1 & 1,0,0 & 2,1,0 & 1,1,1 & 0,0,1

⁹Be ibid & (2,0,1)

¹² C 1,0,0 & 2,1,0 & 3,1,1 & 2,2,1 & 1,1,1 & 0,1,0 & 2,0,0 & 3,1,0 & 2,1,1 & 1,2,1 & 0,1,1 & 1,1,0

¹³ C ibid & (1,0,1)

¹⁶O 1,0,0 & 2,0,1 & 2,1,0 & 3,1,1 & 2,2,1 & 1,2,0 & 1,1,1 & 0,1,0 & 1,0,1 & 2,0,0 & 3,1,0 & 2,1,1 & 2,2,0 & 1,2,1 & 0,1,1 & 1,1,0

¹⁷O ibid & (0,0,0)

¹⁸O ibid & (3,2,1)

The coordinates are determined according to the rules:

1) Even Z ring nuclei are center-symmetrical, indistinguishable when P and N interchange.

2) Odd-Z ring nuclei are antisymmetrical.

3) Avoid the structures that may split into two normal rings if possible.

4) Be of smallest radius of gyration [R²]^0.5.

5) Extra N is bound by 2 or more ring P’s, high abundance isotopes are generally bound by 3 P’s.

2. The proof: “odd-Z rings have separated gravity centers of P and N”

A ring with ZP’s and ZN’s is arranged as P-N alternating connections in Cartesian coordinate, where the distance between two bound nucleons is 1 unit, and the adjacent connection lines perpendicular to each other. Let any of a P be A₁ at (0,0,0), while A₂ (an N) is at one of 6 possible coordinates: (±1, 0, 0) or (0, ±1, 0) or (0, 0, ±1).

For any A_i, the Equations (1) and (2) must be satisfied.

when Z is an odd number, if the gravity center of P (i = 1, 3, 5, ・・・, 2Z − 1) were superimposed with that of N (i = 2, 4, 6, ・・・, 2Z), the following equations must be satisfied.

Then, the necessary condition for the superimposition is

Adding Equations (3), (4) and (5), the resulted Equation (6) is the necessary condition for the superimposition.

However, the left side of Equation (6) must be even, while the right side must be odd. Obviously, the Equation (6) is impossible. Therefore, for odd Z ring, the gravity centers of P and N must be separated.

3. Calculation of gravity center of n nucleons, x_g, y_g and z_g

4. Calculation of radius of gyration of a nucleus with A nucleons

x_g, y_g and z_g are the coordinate of gravity center of the nuclide.

5. Calculation of the separation of gravity centers of P and N

x_g, y_g and z_g with subscript (P) and (N) are the coordinates of gravity center of all P’s and all N’s in a nucleus respectively.