Statistical Research of Nuclide’s Shell Structure
Chen Dayou
Institute for Condensed Matter Physics & Materials under Northwest University, Northwest University, Xi’an, China
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To cite this article:
Chen Dayou. Statistical Research of Nuclide’s Shell Structure. American Journal of Modern Physics. Vol. 5, No. 5, 2016, pp. 87-134. doi: 10.11648/j.ajmp.20160505.12
Received: August 28, 2015; Accepted: July 19, 2016; Published: September 6, 2016
Abstract: This thesis, after a systematic and in-depth analysis of known nuclides, pro-poses a new model of nuclides’ shell structure and offers a table of the shell structures of 935 nuclides. With this theoretic approach, the thesis studies the shell combination with a bias towards the statistical analysis of nuclide structures. This thesis distinguishes between the basic models of nuclides and gives 7criteria for nuclide binding, the maximal nucleonic number of each shell (Δ), combination of proton and neutron (p/n) and graphs of the nuclide growth. Based on magnetic moment, it also conducts a quantitative analysis of p/n on the shell. The nuclide structure has the characteristic of a shell and on every shell the combination of proton and neutron features clear regularity. Among the 106 elements from H to Sg the serial number of the most outside shell in structure is 7, and nuclides Ha and Sg are respectively even A and odd A 7 shells. It is not a coincidence but a reflection of the nuclide shell structure. The thesis uses the result of a statistical analysis to confirm the existence of "the magic Number" and reveals the fact that the magic number" is a reflection of p/n on nuclide shell, particularly on the outer shells. The statistical analysis reveals that the nuclide stability and its way of decay are dependent on the nucleonic combination on the most outside shell and the matching between full-filled and semi-full filled p/n, thus unveiling the general law governing the stability and decay of nuclides.
Keywords: Nuclide Shell Structure, p/n (Mass Rate of Proton and Neutron), Criteria of Nuclide Binding, Graphs of Nuclide Growth, Table of Nuclide Shell Structure
1. Introduction
In 1940s M.G. Mayer discovered that the number of protons and neutrons is 2, 8, 20, 28, 50, 82, 126 and so on. This kind of nuclides has a special stability. This characteristic is called "the magic number" law. The existence of "the magic number" indicates that the nuclide is characteristic of a shell structure. Afterwards, M.G. Mayer and J.H.D Jensen proposed, with the nuclide independent motion as its theoretic basis, the shell structure of nuclides and as a result explained the "magic number" law. [1]
The Mayer’s shell structure model solved the magic numbers of 2, 8 and 20 first by using potential energy function of nuclear central force field in the model of harmonic oscillator potential well and square potential well. Then, with the analysis of splitting of energy levels, other magic numbers are obtained.
Mayer’s shell structure is good in many ways. For an example, it successfully explains the characters of double magic-number nuclides and their near-by ones in both theory and experiment. But there are many examples showing grent differences between prediction and experiment such as electric quadruple moment of nucleon and magnetic moment of baryon odd-A nuclide. To solve the problem, A. Bohr B. R. Mottelson and L. J. Rainwater proposed the model of collective motion. [2] However neither of the models gave specific from of the nuclide shell structure.
We hold that circumstances inside and outside nuclear are entirely different. Inside it is similar to a free space while outside it has a powerful nuclear force. So there is no electronic orbiting motion and no steady-state distribution inside nuclear. Therefore, the model of nucleon shell structure is most likely an approximate description of the nuclide shell structure. The model we offer in this paper is different from the thought of Mayer, A. Bohr et al and it is based on classification of basic models of nuclides and on statistical analysis of nuclides.
2. Statistical Characteristics of Nuclide Shell Structure
Shell structure of a nuclear is a necessary result of direct proportion between its volume and its nuclide number. The volume V of a nuclear is
V = (1)
In which radius (=1.21×10^{-15}m) is a constant obtained from experiment.[3] It is known that each nucleon has similar mass and identical volume. Suppose the nucleon inside the nuclear takes up an average space of a sphere with a radius of, the nuclide shell structure could be composed with diameter.Suppose the average space between shells is a sphere with diameter", "and the distance between two nearby shells is "", too, the geometric space ΔN_{i} of No."i" shell is as follows, for the volume is directly proportional to nucleonic number:
(2)
In the formula (2), if "i" is 1, 2, 3, 4, 5, 6 or 7, ΔN_{i} must respectively be 1, 7, 19, 37, 61, 91 or 127, indicating geometric space of shell layers in terms of nuclide numbers. To make a distinction, nuclide with "k" shells is called k-shell nuclide. For instance O_{8} could be called 3-shell nuclide and its second shell has 4 nucleons.
In the diagram, the distance between two nearby cells is "r_{o}". Black circles stand for protons and grey ones for neutrons. Full-filled nucleon numbers of the 4^{th}, 5^{th}, 6^{th}, and 7^{th}, shells are 24, 48, 72 and102 respectively.
There is nuclear force between nucleons, so they cannot be indefinitely close to each other. Except for i=1, no shell can be covered with the maximal number of nucleons as given in formula (2). Nucleons do not fully occupy the geometric space of the shell either. The nuclide shell structure is shown in Fig. (1)
Suppose the actual maximal number of nucleons contained on I shell structure is Δ, ΔΔN_{i.} Δ/ΔN_{i} represents the ratio of nucleons occupation of the shell space and reflects the fullness of nucleons. The proton-neutron ratio of each shell, called p/h for short, indicates the nucleonic combination on each shell. Interdigitational distribution of nucleons on shells.
It is found that, if even A nuclides are "hollow" and odd A nuclides are "neutron centered" except for Hydrogen (H) nuclides, the shapes of nuclide shell structures may be obtained according of the pairing characteristics of nucleons and the quantitative relationship of magnetic moment. Through statistical analysis of nuclides, we propose the 7-shell structure of known nuclides. Statistical analysis shows:
(1) There could be only one nucleon on the first shell, and it is always filled by neutron except for element H;
(2) The maximal number ΔA_{2} of nucleon on the second shell is 4, that is, ΔA_{2} = 4, so ΔA_{i }/ΔN_{I} = 0.5714. When the second shell is full filled, the p/n = 2/2;
(3) On the 3^{rd} shell, ΔA_{3} = 12, ΔA_{3}/ΔN_{3} = 0.6316. If the 3^{rd} shell is the most outside and full-filled, its p/n has only two combinations: p/n = 6/6 and p/n = 5/7;
(4) On the 4^{th} shell, ΔA_{4} = 24, ΔA_{4}/ΔN_{4} = 0.6486, the p/n of the full-filled shell has only two combi-nations:12/12 and 10/14;
(5) On the 5^{th} shell, ΔA5 = 48, ΔA_{5}/ΔN_{5} = 0.7868, the p/n of the full-filled shell has only two combinations: 20/28 and 18/30;
(6) On the 6^{th} shell, ΔA_{6} = 72, ΔA_{6}/ΔN_{6} = 0.7912, the p/n of the full-filled shell has only two combinations: 28/44 and 26/46;
(7) On the 7^{th} shell, ΔA_{7} = 102, ΔA_{7}/ΔN_{7} = 0.8031, the p/n of the full-filled shell has only two combinations: 44/58 and 42/60.
Except for the firs and second shells, there are tow kinds of stable combinations when the shell is full-filled. The first kind, represented by I, is full-proton combination and the second kind, represented by II, is full-neutron combination. The objectivity of p/n combination on each shell is a decisive factor for determine the model quality. Further quantitative analysis will be given in the following discussion of nucleonic magnetic moment.
1, ΔN_{i} = i^{3}- (i-1)^{3} is the geometric space of the"i" shell indicated by the nucleonic number.
2, ΔAi is the maximal number of nucleons contained in the "i" shell. The determination of ΔAi and p/n is the basis for compilation of Table of Nuclide Shell Structure.
3, The combinations of "I" type belong to the category of full-filled protons while those of "II" type are of the category of full - filled neutrons.
4, The even - A nuclides are of the hollow type and are indicated with"○".the odd - A nuclides are of the neutron - filled type and mad are indicated with "⊙".
3. Basic Classification of Nuclides of Determination of ΔA_{i}
For the maximal nucleonic number a Ai the shell can actually contain, the statistics of ΔA_{1} =1, ΔA_{2}=4, ΔA_{3}=12, ΔA_{4}=24, ΔA_{5}=48, ΔA_{6}=72, ΔA_{7}=102 provided by us is of important significance for the statistic models of plastic shell structure. In implementation this group of statistics may be obtained from the ratio between ΔA_{i} and ΔN_{i} and the natural abundance of corresponding nuclides.
The nucleonic action inside the nucleus is related to the included angle of nucleonic magnetic moment. The acting force gradually increases as the shell structure enlarges, the nuclear radius becomes longer and the magnetic moment angle becomes smaller between nuclides, making the ΔA_{i}/ΔN_{i} ratio tend to increase. Therefore, we come to the following judgment:
ΔA_{i} /ΔN_{1}<ΔA_{2} /ΔN_{2}<ΔA_{3} /ΔN_{3}<ΔA_{4} /ΔN_{4}<ΔA_{5} /ΔN_{5}<ΔA_{6} /ΔN_{6}<ΔA_{7} /ΔN_{7}<α (3)
Α being an actual number smaller than 1. [4]
The ΔA_{i} may be deduced from the relationship shown in Formula (3) and the stability of corresponding nuclide. Taking even A nucleus as an example, we know that the helium (He) nucleus has stable nuclides of sphere symmetry and is often used as bullet to attack other nuclei. From this we infer that He is the even A full-filled nuclide of the second shell level and ΔA_{2} =4, ΔA_{2} /ΔN_{2} =4/7=0.571.
For even A nuclides of the 3^{rd} shell level,
Because
ΔA_{3 }/ΔN_{3 }= ΔA_{3} /19>0.571 (4)
So ΔA_{3}>10.894. Noticing the characteristic of even integer of even A, ΔA_{3} can only be chosen from among 12, 14, 16 and 18. Since 18/19→1, as a matter of fact ΔA_{3} can only be chosen from 12, 14 and 16. Again, because the nucleus number of full-filled nuclides of Even A of 2^{nd} shell level is 4, the nucleus number of full-filled nuclides of even A of 3^{rd} shell level can only be taken from 16, 18 and 20. Seeing that the nuclides hose nucleus numbers are 18 and 20 lack high abundance stability, the nucleus number of even A full-filled nuclides of the 3^{rd} shell level can be none other than 16 and the corresponding nuclide si O_{8}. ΔA_{3}=12, ΔA_{3}/ΔN_{3}=12/19=0.6316.
For full-filled nuclides of even A of the 4^{th} shell level, because
ΔA_{4}/ΔN_{4} = ΔA_{4 }/ 37>0.6316 (5)
ΔA_{4}>23.369, so ΔA_{4} can only be chosen from among 24, 26, 28 and 30. The corresponding full-filled nucleus numbers of even A are respectively 40, 42, 44 and 46. We notice that none of the nucleus numbers 42, 44 and 46 have nuclides of high abundance stability while A=40 has two nuclides of high abundance of stability: Ar_{22} and Ca_{20}, and their graduations are 99.60 and 96.94. The full-filled nuclides have good stability, and from this we can judge that the full-filled nucleus number of even A of the 4^{th} shell level is 40. ΔA_{4}=24, ΔA_{4 }/ ΔN_{4}=24/37=0.6486.
For full-filled nuclides of even A of the 5^{th} shell,
ΔA_{5 }/ ΔN_{5} = ΔA_{5 }/ 61>0.6486 (6)
ΔA_{5}>39.56, so A5 can only be chosen from among 40, 42, 44, 46, 48 and 50. Because even A nucleus A4=40, the full-filled nucleus numbers of even A of the 5^{th} shell level are respectively 80, 82, 84, 86, 88 and 90. From the analysis of the natural abundance of stable nuclides, we can infer that the full-filled nucleus number of even A of the 5^{th} shell level is 88, the corresponding nuclide is Sr_{50} and the abundance is 82.60. ΔA_{5}=48, ΔA_{5 }/ΔN_{5}=48/61=0.7869.
For full-filled nuclides of even A of the 6^{th} shell level,
ΔA_{6 }/ΔN_{6}=ΔA_{6 }/ 91>0.7869 (7)
ΔA_{6 }>71.608, so ΔA_{6} can only be selected from among 72, 74, 76, 78 and 80 and the corresponding the full-filled nucleus numbers of even A are respectively 160, 162, 164, 166 and 168. From the ratio between ΔA_{i} and ΔN_{i}, we can see that the ΔA_{i}/ΔN_{i} values of the 3^{rd} and 4^{th} shell levels are close to each other and the ΔA_{5}/ΔN_{5} value of the 5^{th} shell level is clearly enlarged. Because ΔA_{i}/ΔN_{i} <α<1, the ΔA_{6}/ΔN_{6} and ΔA_{7}/ΔN_{7} can not possibly maintain the increase rate of ΔA_{5}/ΔN_{5}. From the analysis of the abundance of stable nuclides, it can be determined that the full-filled nucleus number of even A of the 6^{th} shell level is 160 and the corresponding nuclides are respectively Gd_{96} and Dy_{94}, ΔA_{6}=72, ΔA_{6} /ΔN_{6}=72/91=0.7912.
Experiments reveal that the maximal nucleus number of even A nucleus is 262, which conforms to the characteristics of even A full-filled nucleus number of the 7^{th} shell level. Since 262 is the biggest nucleus number of even A, it must be the number of full-filled nuclei. If A7=262, ΔA_{7}=102, ΔA_{7 }/ΔN_{7}=102/127=0.8031, which fully agrees to the relationship shown in Formula 3. From this we can come to the following judgment: 262 is the full-filled nucleus number of even A of the 7^{th} shell level, and the corresponding nuclides are respectively Ha_{157} and Bh_{155}.
After the full-filled nucleus number of even A is determined, that of odd A is at the same time determined. Because the even A nuclei are hollow nuclides and the odd A nuclei are neutron-star nuclides, the addition of one nucleus to even A full-filled nuclei does not alter the nucleus number at various shell levels. From this we know that the full-filled nucleus numbers of odd A are respectively A1=1, A2=5, A3=17, A4=41, A5=89, A6=161 and A7=263. The corresponding nuclides are respectively H_{1}, O_{9}, K_{22}, Y_{50}, Dy_{95} and Sg_{157}. Here, the nucleus numbers 41, 89 and 161 correspond to the nuclides K_{22}, Y_{50} and Dy_{95}. This is an important enlightenment for us to better understand the fundamental categorization of nuclides. Especially, the heaviest nuclide Sg_{157} of the laboratory exactly fills up the position of odd A full-filled nucleus of the 7^{th} shell level. It provides a convincing evidence for the fundamental classification method of nuclides.
4. Magnetic Moments and Combinations of Nuclei
The nucleus number ΔA_{i} of the shell level offers a general description of the nuclei of the level. To conduct an in-depth analysis of the shell-level structure, we have to probe into the combinations of the shell level nuclei, so as to find the specific forms combination between protons and neutrons. The proton-neutron combination ratio p_{i}/n_{i} at the shell level is determined by the pairing characteristics of nuclei and the quantitative relation of nucleus magnetic moments.
Nuclei pairing is the basic condition for the formation of nuclides. The fact that nuclei have magnetic moments means that, apart from nuclear force, there also exists electromagnetic force. Experiments show that the force between nuclei is related to the included angle of the nucleus spin angular momentum [5].
Let’s take even A nuclei as an example. He2 is the nuclide of full-filled 2nd shell level. The nucleus of the first shell level is vacant. The 2nd level has 4 nuclei and the proton-neutron is p2/n2=2/2. O8 is the nuclides of the full-filled 3rd shell level. The proton-neutron ratios of the 2nd and 3rd levels are p2/n2=2/2 and p3/n3=6/6. Ca20 refers to the nuclides of the full-filled 4th shell level and the proton-neutron ratios of the 2nd, 3rd and 4th levels are p2/n2=2/2, p3/n3=6/6 and p4/n4=12/12. The 3 kinds of nuclides are all highly abundant and stable. This shows that p2/n2=2/2, p3/n3=6/6 and p4/n4=12/12 are the stable combinations of proton-neutron ratios of the 2nd, 3rd and 4th shell levels. This type of combinations is characteristic of one-to-one pairing between protons and neutrons. Please refer to Fig. 2 for nuclide pairing.
Nucleons of the same kind expel each other when in the same direction and attract each other when in opposite directions.
Nucleons of different kinds attract each other when in the same direction and expel each other when in opposite directions.
With more shell levels and more neutrons, the nuclei of high levels no longer have conditions for one-to-one pairing between protons and neutrons. But the proton-neutron ratio (pi/ni) can be obtained by quantitative analysis of the ratio of nucleus magnetic moments. The magnetic force of nucleons is shown in Fig. 3 [6].
Magnetic moments of the same kind of nucleons attract each other when in the same direction and expel each other when in opposite directions.
Those of different kinds of nucleons attract each other when in the same direction and expel each other when in opposite directions.
Experiment tests show that the proton magnetic moment μ_{p}= 2.792847386 (63) μ_{N}, the neutron magnetic moment μ_{n} = －1.91304275 (45) μ_{N} and their relative rate is
(8)
This relative rate represent the strength level of the eddy field caused by proton or neutron spin. Nucleons on stable shells are in a state of balance of electromagnetic force, i. e. the eddy fields of protons and neutrons are in mutual balance. Except such nuclides as O_{8} and Ca_{20}, total balance of the proton and neutron eddy fields on all shells should be maintained, Formula (8) indicates that, when the p/n of all shells approaches 1/1.46, the electromagnetic force is in balance on the whole. The nucleon number is a natural one and even-even nucleons tend to be stable. With the above characteristics in mind, we can give a semi-quantitative explanation about the combinations of p/n’ s on all shells.
Statistical analysis reveals that p/n = 5/7, p/n = 10/14 and p/n = 20/28 are respectively a stable combination of the full-filled p/n of the 3, 4and 5 shell. It is no co-incidence that the p/n rate of the 3 shells is 1/1.4. It is a manifestation of the magnetic moment of protons and neutrons and also a manifestation of the strength level of the eddy field caused by a proton or neutron spin. A stable shell is in a balance state of electromagnetic force and is full-filled with neutrons. For any shell, if the quantity of protons is , the quantity of neutrons ≈1.46. Thus, we have the following:
(9)
In the formula, Δ stands for the maximal number of nucleon "i" shell. If p_{i} is figured out, the nucleonic combination on "i" shell can be known.
For the 3^{rd}shell
p_{3}+ 1.46 p_{3}= 12, p_{3}= 4. 88,
After rounding it off to an integer
p_{3}= 5, n_{3}= 7, p_{3}/n_{3 }= 5/7.
For the 4^{th} shell
p_{4} + 1.46p_{4} = 24, p_{4} = 9.76,
After rounding it off to an integer
p_{4 }= 10, n_{4} = 14, p_{4}/n_{4} = 10/14.
For the 5^{th}shell
p_{5} +1.46p_{5} =48, p_{5} = 19.51,
After rounding it off to an integer
p_{5} = 20, n_{5} = 28, p_{5}/n_{5} = 20/28.
For the 6^{th}shell
p_{6} + 1.46 p_{6} = 72, p_{6 }= 29.27,
After rounding it off to an integer
p_{6} = 28, n_{6} = 44, p_{6} / n_{6} = 28/44.
For the 7^{th} shell
p_{7} + 1.46 p_{7} = 102, p_{7} = 41.46,
After rounding it off to an integer
p_{7} = 42, n_{7} = 60, p_{7}/ n_{7} = 42/60.
The calculated results are in agreement with the stable combination of protons and neutrons on full-filled shells and also with the growing graph of nuclides. If these results reflect the overall balance of nucleonic electromagnetic force on full-filled shells, the p/n ’s p_{2}/n_{2} = 2/2, p_{3}/n_{3} = 6/6 and p_{4}/n_{4 }= 12/12 reflect the single balance of the nucleonic electromagnetic force. It is a manifestation of the one-to-one pairing between protons and neutrons. And the other p/n's p_{5}/n_{5} = 18/30, p_{6}/n_{6} = 26/46 and p_{7}/n_{7} = 44/58 are stable combination of dynamic balance of nucleonic electromagnetic force.
Although there is a con-firmed Δ (maximal number of nucleons) and a stable combination of protons and neutrons on each shell, the nucleons do not remain unchanged, for they keep exchanging nucleons against the background of energy exchange with the outside world. In an instant, a quasi-full-filled shell state is formed. In general, if the combination of nucleons on the 3^{rd} shell in a nuclide is a full-filled one (p_{3}/n_{3} =6/6), the combination of nucleons on the 4^{th} shell is mostly a full-neutron one ( p_{4}/n_{4 }= 10/14 ). After a pair of nucleons are exchanged, we have p_{3}/n_{3} = 5/7, p_{4}/n_{4 }= 11/13, of which p_{4}/n_{4 }= 11/13 is a stable combination of inside quasi-filled shell. See Fig. 4 (a) for details. [7]
If the nucleonic combination on the 4^{th} shell is a full-proton one (p_{4}/n_{4} = 12/12), that on the 5^{th} shell is a full-neutron combination (p_{5}/n_{5} = 18/30). The exchange of two pairs of nucleons results in the combination p_{4}/n_{4} = 10/14, p_{5}/n_{5} = 20/28, which is shown in Fig. 4 (b). The newly-formed combination may be restored to the original state after exchange of two pairs of nucleons. Similarly, such exchange may take place between the 5^{th} and 6^{th} shells and between the 6^{th} and 7^{th} shells. To sum up, nucleons on shells fluctuate and exchange between individual balance (e. g. p/n = 6/6, p/n = 12/12) and overall balance (e. g. p/n = 5/7, p/n = 10/14, p/n = 20/28), which is controlled by the relative rate between the magnetic moment of protons and that of neutrons.
(a)
Exchange of a pair of nucleons between the 3^{th} and 4^{th} shells results in quasi-full shell combination.
(b)
Exchange of two pairs of nucleons between the 4^{th} and 5^{th} shells results in switching of combinations.
5. Magic Numbers and Nuclide Stability
In the preparation of the table, no particular attention is attached to the condition of magic numbers which nevertheless do exist as a natural character of the shell structures. Let’ s take He_{2} as an example. Its p/n = 2/2 and it is a 2-shelled nuclide with full-filled structure. O_{8} is a 3-shelled nuclide with full-filled shell structure and its p/n = 8/8. Ca_{20}, its p/n =20/20, is a 4-shelled nuclide with fall-filled structure. Sr_{50}, its p/n =38/50, is a 5-shelled nuclide with full-filled shell structure. Nuclides with full-field structures are stable in character.
Another feature of stable nuclides is that the p/n of most outside shell equals. Nuclides with N = 20 have 5 kinds of stable nuclides, of which 4 have their p/n equal one on the most outside shells. They are, Cl_{20}, Ar_{30}, K_{20}, and Ca_{20}. Their p/n’s of most outside shells are respectively 10/10, 11/11, 11/11 and 12/12.
Of the 5 kinds of stable nuclides with N = 28, 4 have the characteristic of p/n equaling one on the most outside shell. They are Ti_{28}, V_{28}, Cr_{28} and Fe_{28}. Their p/n’s of the most outside shells are respectively 5/5, 5/5, 6/6 and 7/7.
There are 6 kinds of stable nuclides with N =50: Kr_{50}, Rb_{50}, Sr_{50}, Y_{50 }Zr_{50} and MoY_{50}.It is known from Table I that, of all the p/n’s of full-filled shells on the 5^{th} shell have two combinations of p/n =20/28 and p/n = 18/30. The p/n’s of the most outside shells of Kr_{50} and Rb_{50} are18/28 and 18/28, close to the p/n of the 5^{th} shell. For Sr_{50} and Y_{50}, the p/n’s are both 20/28, identical with the combinations on the 5^{th} full-filled shell. Zr_{50}’s and Mo_{50} are 6-shelled nuclides with their p/n’s on the 6^{th} shell being 1/1 and 2/2.
Of the heavy nuclides with N =82, seven are stable ones:, Xe_{82}, Ba_{82}, La_{82}, Pr_{82}, Nd_{82} and Sm_{82}.They are all 6 shelled and their p/n’s of the most outside shells are respectively 18/30, 20/30, 20/30, 22/30, 22/30, 24/30 and 24/32.
Nuclides have shell structures and stable nuclides have stable p/n’s on the most outside shells. But with the increase of nucleonic number A, the filling level of neutrons grows higher and the p/n of the most outside shells is smaller than one. For the 7 stable nuclides with N = 82, the p/n valve of the most outside shells changes around 1/1.5. So it is known that this characteristic is relevant to the magnetic moment of nucleons.
Nuclides with N = 126 are 7-shell structured and there are two stable nuclides: Pb_{126 }andBi_{126}. Their p/n’s of the most outside shells are 18/30 and 20/28, identical with the two stable combinations of the 5^{th} shell when it is full-filled. This shows that a nuclide may become stable when its combination of protons and neutrons of the most outside shells in consistent with the stable nucleonic combination of an inside shell.
Heavy nuclides with Z≥84 are unstable except Th_{142, }U_{143 }and U_{146}.They are 7-shell structured and their p/n’s of the most outside shells are respectively 26/46, 28/46 and 28/50. Except even-even nucleonic combination of the most outside shells, being identical with or close to the p/n’ s of the 6th shell is also a prerequisite for the nuclide stability.
To sum up, it could be presumed that the stability of a nuclide is decided by p/n combination on the most outside shells and on the p/n filling level of each shell. The "magic number" is the reflection of this feature. Except the case that the numbers of protons or neutrons are 2, 8 or 20, other magic numbers reflect nucleonic number of unfull-filled shells. So magic numbers reflect the combinations of protons and neutrons of stable nuclides.
Decay modes of unstable nuclides is dependent on the nucleonic combinations of outside shells. The Table of Nuclide Shell Structure (See the appendix) indicates that a nuclide decays in the (ε) way when its p-n of the most outside shell is 2, 4 or 6 and it decays in the (β^{-}) way when its outside shell n-p is 2, 4 or 6.This characteristic remains true after nearly 1000 unstable nuclides are tested.
Unstable nuclides decaying in the (ε) way are characterized by the nucleonic numbers on outside shells being even numbers of 2, 4, 6, etc. Judging from the condition of forming a nuclide, we know from Fig. 2 (a) that pairing of protons in the abnormal (reverse) direction caused by magnetic moment is a kind of pairing style. The magnetic moment of a proton is l.46 times more powerful than that of a pair of a neutron and the electromagnetic force of a pair of protons is l.46 times more powerful than that of a pair of neutrons. We know that the pairing of protons and neutrons is an important pre-requisite for a nuclide to form. Therefore, a pair of protons is unstable which can distribute on the most outside shell for a short time. By absorbing an electron, a proton turns into a neutron, thus forming a stable nucleonic pair. So the nuclide becomes stable and this is the cause of (ε) way of decay.
The unstable nuclides which decay in the (β^{-}) way are characterized by the even number of nucleons on the outside shells, 2, 4, 6, etc. When protons and neutrons on outside shells fail to strike individual or overall balance, superfluous neutrons pair in reverse direction caused by magnetic moment, as is shown in Fig 2 (b). But the electromagnetic force between pairs of neutrons is l.46 greater than that between proton-neutron pairs. It is less powerful than the combination ability of proton-neutron pairs, so the neutron pair is also unstable and can only be distributed on outside shells. By the force of proton-neutron pairs in the neighboring field, one of the neutrons becomes a proton after discharging an electron. A stable nucleonic pair is formed and the nuclide is made stable. And this is the cause of "β^{‑}" way of decay.
It is surprising that No. 42 element Mo and No. 44 element Ru each have 7 stable isotopes while No. 43 element Tc between them has no stable nuclides at all. The Table of Nuclide Shell Structure tells us that element Tc could not form structure with suitable p/n among the cells and its p/n of the most outside shell is not one.
The (α) way of decay of heavy nuclides is a reflection of the evolution of the p/n combination of most outside shells from unstable to stable. For instance, the outside shell p/n of stable nuclideBi_{126} is 20/28. As for unstable nuclide At_{128}, its outside shell p/n is 22/30 and the product after its (α) way of decay is Bi_{126}, tending to be stable. Let’s cite another example, the stable nuclide U_{146} has its outside shell p/n at 28/50. The unstable nuclide Pu_{148} has it at 30/52, tending to be stable after its decay in the (α) way. Thus, the conclusion is drawn that the decay mode depends on nucleonic combination of the most outside shells. Unstable nuclides which decay in the (f) way result from the imbalance of p/n’s between shells.
The heavy nuclide confirmed by experiment isSg_{157}. It is a neutron-filled nuclide with a 7-shelled full-filled structure. It decays by free fission. If there exists a heavier nuclide with a super-large N number, it must be 8-shell structured. It is presumed from Table I about the specific value (ΔA_{i}/ΔN_{i} ) of the shell space nucleons take up that the maximal number of nucleons ΔA_{8} which can be accommodated by the 8^{th} full-filled shell should be 136 [ (8^{3}-7^{3})×0.81]. If the number of protons on the 8^{th} shell is equal to that of neutrons on the 7^{th} shell, it is a pre-requisite for the stable combination of p/n’ s of the 8^{th}shell. The p/n’s of full-field 8^{th} shell are 60/76 and 58/78. From this we may calculate that the nucleonic number A of an even-A nuclide with 8 full-filled shells is 398.
M. G. Mayer predicted the existence of Z=114 supper-heavy nuclide. At the end of last century, scientists of Joint Instiute for Nuclear Research announced that they had successfully produced Z=114 nuclide, its atomic weight being 289 and its half of decay being 30 seconds which is much longer than other nearby nuclides. [8] Our theory on nuclide shell structure tells us that, if the nuclide whit Z=114 and A=287 tends to be stable, it structure should be as follows:
/l | 2/2 | 6/6 | 10/14 | 18/30 | 26/46 | 42/60 | 10/14 | Σ:114/173 |
6. Binding Energy of a Nuclear and Characteristics of Nuclear Force
The mass average of a nuclear is lighter than the mass sum of free nucleons of the nuclear. The difference between the two is called mass loss. Take Δm (Z, A) for an example,
(10)
In the formula, m (Z, A) is the mass of the nuclide. All nuclear suffer mass loss, i.e. Δm (Z, A)>0.
When the nuclear mass is represented by M (Z, A),
Δm (Z, A) = ΔM (Z, A)
=Z (^{1}H)+ (A－Z)m_{n}－M (Z, A) (11)
In the formula, (^{1} H) stands for the mass of atom hydrogen. According to the relationship between mass and energy in relativity theory, the binding energy of an atom is
B (Z, A)≡ Δm (Z, A)c^{2} (12)
The nucleonic radium calculated by the mass formula is R = 1. 21 (F), but it is R = 0. 8 (F) according to the test conducted by R. Hofstadter in his experiment on electronic scattering. This proves that in the nucleon is a rim with a thickness of t = 0. 4 (F). [9]
The binding energy of a nuclide increases as the nucleonic number grows larger. The binding energy difference between different nuclides is great, but no regularity is discovered. Theoretically, the average bind- ing energy of each nuclide is used to represent the level of tightness of the binding energy. The specific bind- ing energy is
ε (Z, A)= B (Z, A)/A (13)
It represents the average work done on each nucleon when the nuclear with mass number A and electric charge number Z is fragmented into free nucleons. Graph of specific binding energy obtained from experiments is shown in Fig. 5. [10]
(from Evans. R. D.; The Atomic Nucleus, 1995)
Both theory and experiments prove that the binding energy ε value of an even-A nuclide with full-filled shells is relatively high at its peak value. This rule can be confirmed by working out the binding energy of the last nucleon. The significance of the last nucleon’s binding energy refers to the energy released when a free nucleon and other nucleons of the nuclear combine into a nuclide. In other words, it is the energy needed to separate a nucleon from the nuclear.
The binding energy of the last proton is
S_{p} (Z, A) = B (Z, A) - (Z-1, A-1). (14)
The binding energy of the last neutron is
S_{n}_{ }(Z, A) = B (Z, A) -B (Z, A-1). (15)
From the value surplus in Δ ( Z, A) in Table of Nuclide Shell Structure, the definition of Δ (Z, A) and Formula
Δ (Z, A)= [M (Z, A)-A]c^{2}, (16)
can be used to work out the binding energy of a nuclide and that of the last nucleon.
O_{8 }is a 3-shelled full-filled nuclide of "○"category. From the definition of S_{p}, S_{n}, we can work out the following:
S_{p} (O) = 12.12MeV, S_{n }(O) = 15.66MeV
S_{p} (F) = 0.61MeV, S_{n} (O) = 4.15MeV
Ca_{20} is a 4-shelled nuclide of ""category. S_{p}, S_{n}, values of neighboring nuclides are:
S_{p} (Ca) = 8.38MeV, S_{n} (Ca) = 15.68MeV
S_{p} (Sc)=4.27MeV, S_{n} (Ca)= 8.32MeV
Sr_{50} is a 5-shelled full-filled nuclide of the "○" category. S_{p}, S_{n}, values of neighboring nuclides are:
S_{p} (Sr) = 9.94MeV, S_{n} (Sr ) = 10.45MeV
S_{p} (Y) =7.74MeV, S_{n} (S)r= 7.03MeV
Gd and Dy are 6-shelled full-filled nuclides of the "○"category. S_{p}, S_{n }values of neighboring nuclides are:
S_{p} (Gd) =8.51MeV, S_{n} (Gd)= 7.45MeV
S_{p} (Tb)=6.81MeV, S_{n} (Gd)=5.63 MeV
and
S_{p} (Dy)=7.43MeV, S_{n} (Dy) = 8.58MeV
S_{p} (Ho)= 6.22MeV S_{n} (Dy) = 6.45MeV
The above results show that the peak values of S_{p}_{ }and S_{n} appear on full-filled nuclides.
Table 2 shows the binding energy of some isotopes S_{n} (Z, A) of nuclides with full-field shells. We can see that nuclides O, Ca, Sr, Y, andDy show their peak values of S_{n}_{ }(Z, A) when full-filled.
* The nuclide with full-filled shells in high in binding energy.
** Such as nuclides have protons with high filling level and are strong in electromagnetic force and high in binding energy. With the increase of the filling level of protons, the binding energy decreases.
7. Criteria of Nuclide Shell Structure and the Graph of Its Growth
Table 1 shows that the ratio ΔA_{i}/ΔN_{i} is between 0.571 and 0.803 and increases as the number of shells becomes larger, which is consistent with the fact that the distance between nearby nucleons decreases as the radius of curvature becomes larger.
The full-filled shell nuclide refers to the nuclide, each of whose shells has been filled with ΔA_{i}. Only after the inside shells are fully filled, will the outside ones begin to full fill. Therefore, the unfull - filled shells only refer to those outside ones whose proton is smaller than ΔA_{i}.
The statistical analysis shows that, except forAr, in the stable nuclides of the 2^{nd}, 3^{rd} and 4^{th} shells, the p/n is 1 and the protons and neutrons are very likely to pair with each other. If the shells are naturally stable, the p/n of most outside shells is 1.
When the 5^{th}, 6^{th} and 7^{th} shells are full-filled, the p/n' s of the most of their outside shells are 1, showing a big regularity. So we can make out the shell structure table of all the nuclides with the principle of the table, their pairing characteristic and the combining criteria shown in the nuclide structure. The major criteria of nuclide combination are the following 7.
1. The proton cannot occupy the first shell of a nuclide except for element H.
2. Every shell of a nuclide is filled with nucleons of even number except the first shell which is either unfilled or filled with a neutron. The nuclide with an unfilled first shell is called hollow nuclide symbolized by "○". The nuclide with a filled first shell is called neutron-filled nuclide symbolized by "⊙". They are two basic kinds of nuclides. The even A nuclides are "○" kind and the odd A nuclides belong to "⊙" kind.
3. For nucleons show the characteristic of pairing with each, nucleonic number of the most outside shells is even with the exception of element H. The p/n of the most outside shells of stable nuclides is most likely to be 1.
4. For any nuclides with k shells, there are only 2 kinds of combinations of p/n in full-filled shells except for the 2^{nd} shell, as is shown in the following:
(1) 2^{nd} shell: p/n = 2/2, and if k >2, then p_{2} = 2, n_{2} = 2;
(2) 3^{rd} shell: p/n = 6/6, p/n = 5/7; and if k > 3, then p_{3}≤6, n_{3}≤7;
(3) 4^{th} shell: p/n = 12/12, p/n = 10/14; and if k > 4, then p_{4}≤12, n_{4}≤14;
(4) 5^{th} shell: p/n = 20/28, p/n = 18/30; and if k > 5, then p_{5}≤20, n_{5}≤30;
(5) 6^{th} shell: p/n = 28/44, p/n = 26/46; and if k > 6, then p_{6}≤28, n_{6}≤46;
(6) 7^{th} shell: p/n = 44/58, p/n = 42/60; and if k > 7, then p_{7}≤44, n_{7}≤60.
5. If the p/n of the most outside shell is not 1, generally =2. For the nuclide of , the p/n is an even number.
6. The mode of nuclide decay depends on the nucleonic combination of its most outside shell.
7. The stability of a nuclide depends on the p/n relationship between shells. Generally, the filling level of the p/n of each shell of a stable nuclide is invariably I, but its shells are full-filled or unfull-filled alternatively in the II kind of nuclides.
To determine values of ΔA_{i} and the p/n combinations is the principal basis for the preparation of the nuclide shell structure table. The regularity shown in ΔA_{i} and p/n is embedded in the graph of nuclide growth. The developing route of full-filled nuclides is shown in Fig. 6 (a) (b). Fig. (a) is the route of the development of even A full-flied nuclides and Fig. (b) is that of odd A full-filled nuclides. The 1^{st}, 2^{nd}, 3^{rd} and 4^{th}shells are the same as the shell structure proposed by Mayer, but the 5^{th}, 6^{th} and 7^{th} shells are obviously different in the number of nucleons.
Even A nuclide belongs to "○" category and its first shell is vacant. With the second shell full-filled, its stable nuclide is He_{2}; with the third shell full-filled, its stable nuclide is O_{8}; with the fourth shell full-filled, its stable nuclides are Ar_{22} and Ca_{20}; with tile fifth shell full -filled, its stable nuclide is Sr_{50}; with the sixth shell full-fall ed, its stable nuclides are Gd_{96} and Dy_{94}; with the seventh shell full- filled, its stable nuclides are Ha_{157} and Bh_{155}, Ha_{157} and Bh_{155} are exactly the nuclides of the seventh shell. This figure derives from Table I.
Odd A nuclide belongs to "⊙" category. Except for Element H, the first shell is invariably filled with neutrons. With the second shell full- filled, its nuclide He_{3} is unstable; with the third shell full- filled, it is the isotopeO_{9 }of the lowly full - filled O; with the fourth shell full-field, the nuclides include stable nuclide K_{22} and unstable nuclide Ca_{2}; with the fifth shell full -filled, the nuclide is Y_{50} stable; with the sixth shell full- filled, the nuclide Dy_{95} is stable. Sg_{157} is exactly the odd A nuclide of the full -filled seventh shell, The figure derives from Table I.
Any nuclide is first of all categorized according to the nature of nucleon A, i.e. whether it is odd or even in number, and then it is filled with nucleons one shell after another from inside to outside. The p/n of each shell is determined by the afore-mentioned 7 criteria. For an example, Cl_{18} is an odd A nuclide belonging to "⊙" kind. So its first shell is filled with one neutron and its p/n's on the 2^{nd}, 3^{rd}, 4^{th} and the most outside shells are respectively 2/2, 6/6, 9/9 and 1. It is therefore identified as a stable nuclide.
The second example isNi_{32}. It is an even A nuclide belonging to "○" kind. Its first shell is not filled. Its p/n's of the 2^{nd}, 3^{rd}, 4^{th} and 5^{th} shells are respectively 2/2, 6/6, 10/14 and 10/10. It is a stable nuclide because p/n on the most outside shell is 1.The third example is the even A nuclide_{ }Th_{142} belonging to "○" kind. Its first shell is not filled. Its p/n's of the 2^{nd}, 3^{rd}, 4^{th}, 5^{th} and 6^{th}shells are respectively 2/2, 6/6, 10/14, 20/28 and 26/46.The 7^{th} shell is not full-filled, but its p/n (26/46) is the same as that of the 6^{th} shell. So it is presumed to be stable.
Nuclide shell structures may either be directly indicated or shown in a table. For instance, the shell structures of O_{9}, Sr_{50} and U_{146} are illustrated as follows:
Shell structures of stable nuclides are illustrated in Table II which is prepared in accordance with the afore-mentioned criteria. All tile stable nuclides are included and special nuclides are marked with an asterisk "*" The clear regularity shown in the shell structures of stable nuclides is the basis for the preparation of this table.
The Table of Nuclide Shell Structure is completed on the basis of The Table of Shell Structures of Stable Nuclide, giving consideration to the stability and decay modes of nuclides and even to the above-mentioned 7 criteria. Consideration should be given to matching between full-level and unfull level of p/n’ s between shells and to decay modes of unstable nuclides in the combination of nucleons on the most outside shells. The Table of Nuclide Shell Structures prepared in this way can very well explain and predict the stability of nuclides and the decay patterns of unstable nuclides. Please refer to the appendix for the shell structures, with special nuclides marked with asterisks. Tables of Shell Structure of Stable Nuclides are included in Appendix One. Tables of Shell Structure of Nuclides are included in Appendix Two.
8. Conclusion
We’ve arrived at the following conclusions after statistics and analysis of nuclides.
The known highest position of nuclides is a structure of 7 shell levels and the structure is composed with the nucleus ratios r0 as its unit. Nuclides are categorized into two: odd and even nuclides. Except for hydrogen nuclides, even A are hollow and odd A are neutron-star type. The statistic model based on the fundamental categorization of nuclides and the tables of shell structure of nuclides prepared on the basis of the model reveal the general law governing the stability and decay of nuclides. This law is both the effect and a proof of the fundamental categorization method of nuclides.
Appendix
* is two-shelled full-filled nuclide of the "○"category.Its p/n is 2/2.There is only one combination for its full-filled p/n on the 2^{nd} shell.
* * The proton-neutron pairing on the outside shell is an important prerequisite for a stable nuclide.Most of the stable nuclides bear this characteristic.
***O_{8} O_{9}are full-filled nuclide of 3 shells. The 3^{rd} shell p/n is 6/6; the 3^{rd} shell p/n of O_{10} is 5/7.The stable isotope of oxygen shows that p/n =6/6 and p/n =5/7 are the stable combinantions of the 3^{rd} shell.
****O_{10} posses the characteristics of a 4-shelled nuclide.
* For a full-folled proton, the p/n=1 on the most outside shell is not a neceseaary pre-requisite for a stable nuclide.However, the stability of indicates that the nucleons whose p/n’s on all shells are even combinations are more and more stable.
** and show that p/n=10/14, p/n=12/12 are the two stable combinations of the p/n’s on the 4^{th} shell.
*** and bear obvious characteristics of the 5^{th} shell.
****The proton-neutron pairing is the necessary prerequiste for the formation of nuclides. After the proton and neutron of have pared on the outside shell the rematinting 4 neutrons and the inside-shell neutrons become pairs, showing that the nucleons of neighboring shells may also pair. The nuclides whose nucleons are all even-even combinations are more stable.
*The outside-shell p/n = 10/14 of Ni_{36} is a stable combination of the p/n on the 3^{rd} shell. p/n = 5/7, p/n = 10/14 and p/n = 20/28 are respectively the stable combinations of the 3^{rd}, 4^{th} and 5^{th}shell, the ratio being invariably 1/1.4.The nuclides whose outside-shell p/n is 1/1.4 are more likely to be stable.
* Sr_{50} andY_{50 }show that the p/n=20/28 is a stable p/n combination on the 5^{th} shell.The element Y has only one stable nuclideY_{50}. the filling level of the isotopeSr_{50} of element is 82.6, for lower than other isotopes, showing thatSr_{50}and Y_{50} are 5-shelled nuclides of even A and odd A. Their stability depends on the outside-shell p/n=20/28.That is to say, it depends on "the magic numbers"20, 28 instead of "magic number"50.
** Zr_{50 }and Zr_{51 }show the structural characteristics of the 6^{th} shell.
*** Zr_{54 }and Mo_{56 }shows that p/n=18/30 is a stable combination of the 5^{th} shell p/n.
**** Nb_{52} Shows that the nuclides who alternate between "full-filled" and "unfulfiled" in the two kinds of l and Ⅱp/n enjoy a high filling level. Most of the nuclides bear such a feature.
*In the stable nuclides whose p/n"s on outside shells do not equal 1, the nucleons of all shells are mostly even-even combinations, except for the first shell.
*Gd_{96} Shows that p/n =26/46 is a stable combination of the full-filled shell p/n’s on the 6^{th} shell.
** Dy_{94} Shows that p/n =28/44 is a stable combination of the full-filled shell p/n’s on the 6^{th} shell.The neighboring shells exchange nucleons instantly. The p/n’s of the 4^{th}and 5^{th}shells fluctuate between p/n =10/14, p/n = 20/28 and p/n =18/30. But the quasi-full filled combinations on inside shells do not affect the stability of nuclides.
*Dy_{96 }andEr_{94} show the characteristics of nuclides on the 7^{th} shell.
*Bi_{126} is astable nuclide and the p/n of the outside shell is p/n =20/28. The specific value of p/n =5/7, p/n =10/14, p/n =20/28 is 1/1.4 and it a stable combination. The number of neutrons of this nuclide is n=126 and its stability does not depend on "the magic number"126, but on the magching of filling level between shells. So is. Pd_{126}. The outside shell p/n’s are p/n =20/28 and p/n =18/30, which happen to be the 2 stable combinations of the full-filled shell p/n’s on the 5^{th} shell.
*Th_{142} is a stable nuclide and its outside shell p/n is p/n=26/46, identical with the full-filled shelled shell p/n’son the 6^{th} shell.
**U_{146} is the heaviest stable nuclide in the natural world. Its outside shell p/n is p/n=28/50 and its protons and neutrons are all "magic number". However, the nuclide stability do not depend on such "magic numbers" as 28, 50, but on the p/n combinations on outside shells and the matching of filling levelv between shells.
Notes: [11] The fundsmental date of the table comefrom:V. S. Shirley et al, Nuclear Wallet Cards, 1979
K. S. Krane, Introductory Naclear Physics, 1987.
[12] The fundamental data of the table come from:
Nuclear Physics, P390~P405, Xu Side, published by Qinghua university Press, 1992.
* He_{2} is a 2 – shelled fulled full-filled nuclide with its p/n being 2/2. The p/n on 2^{nd} full-filled shell has only one combination.
** It is an important characteristic of a stable nuclide for the p/n on outside shells to be equal to one. The nuclides of n –p =2, 4, 6, …decay in the way of (β^{-}) radiation.
*** Be_{4} is a 3-shelled non-full-filled nuclide with its p/n being 2/2 and decays in (α) style.
**** from is structure, we know that B_{4} decays in (ε) style. In the original table it was listed as a nuclide which decays in (α) style, which should be deemed as the second decay after its (ε) style of decay.
**** The nuclides of p – n =2, 4, 6, …decay in (ε)style. C_{4} decays into the stable nuclide B_{5} after absorbing an electrom.
**** The nuclides of p-n =4, 6, …become stable after more than two decays in (ε)style, e.g. the nuclideC_{3} decays in this procedure:.C_{3} +e→ B_{4}+e_{ }→Be_{5}
* O_{8 }is a 3-shelled nuclide with full-filled shells, with its p/n on the 3^{rd}shell being 6/6.We know from the Table of Development Route of Nuclides that p/n=6/6 is a relatively stable combination of full-filled p/n on the 3^{rd} shell.
** Although there is such a nuclide as one whose p/n is 5/7 when the 3_{rd }shell is the most outside, p/n=5/7 is a combination when the 3^{rd} shell is inside. Such stable nuclide as Ne_{12} Si_{16 }and S_{18 }all indicate that p/n=5/7 is a combination when the 3^{rd} shell is full-filled.
*** Restricted by the proton number p≤6 of the 3^{rd} shell, the two extra protons line on the 4^{th} shell. After two (ε)decays, they form the 3-shelled nuclide O_{9}.
* When the protons are full, the p/n on outside shells of stable nuclides are not always equal to one, but S_{20} shows that those nuclides in which nucleons on all shells are even-even combinations tend to be stable.
** Non-stable nuclides between two isotopes with high filling levels generally have two styles of decay, e.g. Di_{19 }with p/n =9/11 on outside shells. It decays into S_{20} after absorbing on electron and decays into Ar_{18} after discharging on electron, tending to be stable.
*** Ar_{22} shows that p/n =10/14 is a stable p/n combination on the 4^{th}full-filled shell.
**** Ar_{24 }is obviously a 5-shelled nuclide. Only when it is included into 5-shelled nuclides can the nuclide nature be reflected. Likewise, K_{23} also shows the characteristics of a 5-shelled nuclide.
* K_{22} and Ca_{20} show that p/n=12/12 is a stable combination of the 4^{th} full-filled shell.
** Ca_{21} show that the nuclide stability not merely depends on the outside shell structure. Both of Ca_{21} and Ca_{20 }have their outside shell p/n equai to 12/12, but their different structure categories makes different stsbility, Ca_{21}+e→K_{22}, tending to be stable.
*** The proton-neutron pairing is a fundamental pre-requisite for the forming of a stable nuclide. The remaining 4 neutrons, after the protons and neutrons pair on outside shell of Ca_{28,} pair with the superfiuous neutrons on inside shells. This show that nucleons of neighboring shells also may pair. The nuclides whose protons and neutrons of all shells are even numbered have a bigger tendency to be stable.
**** The p/n of outside sheiis ofCa_{30 }p/n=3/7. Such nuclides cannot become stable until after several (β^{—})radiations.Thier decay procedure is:Ca_{30}—eSc_{29}—eTi_{28}. There are many nuclide which decay in the similar way.
* It is an obvious that Cr_{31} is_{ }listed_{ }as (ε) decay in the original table. This nuclide should be in the (β^{-}) decay. The unstable isotopes after nuclides with high filling levels all belong to (β^{-})decay.
* The 3^{rd} and 4^{th} shells of Ni_{27} are all of p/n structure of full protons. After 3 consecutive (ε) decays, it becomes stable. Its decay procedure is Ni_{27}+e→Co_{28}+e→Mn_{30}
* The (β)decay feature of Ni _{39} is not confirmed in the oeiginal table.From its shell structure, we know it should deay in the (β^{-})way.
** Cu_{35} decays intwos. In (ε)decay it absorbs an eiectron and into Ni_{36}. In (β^{-})decay it release an electron and decays into_{ }Zn_{34}_{，.}
*** The nucides with p/n on outside shells not being equal tone and the protons andshellsbring in even number tend to be stable.Similar nucides includeZn_{40}，Ge_{40}, Ge_{42}, Se_{46 }etc
* The p/n^{，}s on all shells except the first shell of Ca_{39} and Ca_{40} are the same, but differ in stability. This shows different characteristics of "○"nuclides and "⊙"ones.
* Rb_{53}, Sr_{53} show the characteristicas of 6-shelled stuctures.
** Sr_{50}is a 5-shelled nuclide, showing that p/n=20/28 is a stable combination of 5^{th} full-filled shell.
*** the number of protons of a stable nuclide is smaller than or equal to 20 (p≤20).When the 5^{th} shell is most outside, the proton number may be bigger than 20.But they are all non-stable nuclides
**** Y_{50} is a 5-shelled of "€" category. It shows that p/n=20/28 is a stable combination of the p/n on the 5^{th} shell. Sr_{50} releases an electron. Zr_{49} attracts an electron and decays into Y_{50}. It shows that p/n=20/28 is a stable combination of p/n on the 5^{th} shell.At the same time it shows that such quasi-full structures as p/n=19/29 and p/n =21/27 can not form stable nuclides
* Zr_{47} and Zr_{48} are the_{ }same_{ }with Y_{45}.The number of protons on the 5^{th} shell is bigger than20 (p>20).They are all non-stable nuclides.
** Zr_{54}, No_{56}, and Mo_{56} show that p/n=18/30 is a stable combination of p/n on 5th^{}full-filled shell.
*** Nb_{48}, restricted by the maximal number of protons of all shells, developed into a 6-shelled nuclide, of which P_{2}=2, P_{3}≤6，P_{4}≤12，P_{5}≤20.
* The p/n’s on the 3^{rd}, 4^{th}, and 5^{th} shells of Ru_{50} all belong to "I" categpry, i, e, the p/n stictire of full protons Through absorotion of two electrons, it decays into Mo_{52} and tend to be stable.It shows that nuclides whose p/n ’s of all all shells alternate between full and mon-full are more likely to be stable.Most of the stable nuclides bear this characteristic
*Xe_{82} show that p/n=18/30 is a stable combination of the 5^{th} full shell. The 5^{th} and 6^{th} shells are the same. At the same time it shows the characteristics of magnetic nucleonic pairimg between neighboring shells.
**Ba_{80} and Ba_{81 }show that p/n=18/30 and p/n=20/28 are stable combinations of the 5^{th} full shell.
* Pr_{80}, through two ()decays and exchange of a nucleon with inside shell, generates La_{82 }and becomes stable. The nucleon exchange between shell is a way of nuclear motion, which however does not affect the stable p/n between shells.
** Nd_{82} and Ce_{84} are similar in nuclide shell strucre, both belonging to"○"category with A=142.But they differ in character because of the p/n difference between shells.
*Nd and Sm, which neighbor Pm, each have 7 stable isotopes. But Pm does not have a single stable isotope, which can be explained with its p/n on on outside shells. It seems that the p/n of outside shell becomes an even-even combination by the exchange of a nucleon between the 6^{th}_{ }and 3^{rd} shells. Even so, this nuclide is not stable. The stable shell structures of Nd and Sm tell us that the p/n^{,}s of such nuclides are all even-even combi nations except for the first shell. The Pm, after the nucleonic exchange, fails to meet the requirement, so it can not become a stable nuclide even after the nuclenonic exchange between the 6^{th} and 3^{rd}shells.
**Pm_{85} should have two ways of decay. It turns into Nd_{86} after (ε) decay and becomes Sm_{84} after (β^{-}) decay.
***Sm_{84} decay in the (α)style and its decay procedure is:Sm_{84}-He_{2}→Nd_{82}. Now it become stable.There are many such examples.Sm_{85} is the same asSm_{84} in the p/n’s between shells the first shell But they differ in stability. this example shows that not all even –even nucleons tend to be stable.TheSm_{84} is the high filling level of outside shells and it is an even –even combination
*The stability of Gd_{96} tells us that the p/n=26/46 is a stable combination of the 6^{th} full shell.
* Dy_{94} and Dy_{95} are respectively even-A 6shelled nuclied, showing that p/n=28/44is a stable combination of the 6^{th} full shell.
** Dy_{96} and Ho_{95} clealy bear the characteristics of a 7-shell structure.
*** Er_{92} is a 6-shell nuclide of the "○"category. All the p/n’s are in agreement with the stable p/n combination, but they are all "I"combination. This shows that the nuclide composed of nothing but full protons is not stable. After two ()decays it generates Dy_{94} and becomes stable.
**** Er_{100} shows that the p/n =26/46 is a stable combination of the 6^{th} full shell. Similar nuclideEr_{102}, Yb_{102}, Yb_{103}, etc.
* ForPb_{121} andPb_{123}, the p/n’s are respectively 17/25 and 17/27.After absorbing an electron and exchanging nucleons with 6^{th} shell, they turn into TI_{122}andTI_{124}, tending to be stable.
** Bi_{126} is a stable nuclide with p/n=20/28.It shows that p/n=5/7, p/n=10/14and p/n=20/28 is a stable combination of interaction of interaction between protons and neutrons.
* Po_{124} becomes stable after (α)decay. Its decay procedure is:Po_{124 }-He_{2}→Po_{122}.
** The decay procedure of Po_{125} is: Po_{125}-He_{2}→Po_{123}+e→Tl_{124}, tending to be stable.
***Rn_{124} becomes stable after (α) decay and (ε) decay. Its decay procedure is:Rn_{124} -He_{2}→Po_{122}+e-Bi_{123} + e-Pi_{123} →Pb_{124}
* Th_{142} is a stable nuclide with p/n=26/46 on outside shells. It is the same as the p/n combination of the 6^{th}full shell. Its hows that p/n=26/46 is a stable combination. U_{144}，after (α) decay, generates Th_{142}, tending to be stable.
** The protons and neutrons on outside shells of U_{143} and U_{146 }are all even-even combinations and are close to the p/n of the 6^{th} full shell. Its numbers of protons and neutrons on outside shells are 28 and 50, dearing the characteristic of "magic number"
*** , after (α) decay, generates , tending to be stable.
****, after (α) decay, generates , tending to be stable.
* Am_{148}, after (α) decay, generates Np_{146}. In the course of nuclear reaction of (α) decay, the proton exchange between neighboring shells may take place at the same time. Np_{146} after (β^{-}) decay, generates Pu_{145} and again after (α) decay, generatesU_{143}. This process indicates that all nuclides have the tendency of becoming stable, only with different modes of decay, finally generating stable nuclides.
** Ha_{157} shows that p/n =44/58 is a p/n combination on the 7^{th} full shell. This nuclide decay in (f) style anf the dilect cause lies in the non-matching between full p/n and unfull p/n on inside shells. Most of the nuclides with (f) decay bear this feature. Ha_{156} decays in (α) way on outside shells because of its good matching between full p/n and unfull p/n.
*** Bh_{155} and Sg_{157} accurately display the characteristics of a 7-shelled nuclide.
****Bh_{155 }shows that p/n=42/60 is a stable p/n combination of the 7^{th} full shell.
Notes: 1) The percentage in the table indicates the filling level of the isotopes. (β^{-}) stands for the (β^{-}) decay; () for the rail electronic capture and (β^{+}) decay; (α) for (α) decay; (f) for spontaneous fission.
2) The basic data of the table are taken from: V. S. Snirley et al, Nuclear Wallet Cards, 1979.
K. S. Krane Intro. ductory Naclear Pnysics, 1987
3) The basic data of the table are taken from: Nuclear Physics (P390~P405), Xu Sida, QingHua University Oress, 1992.
4) The stable lists the shell structures of 935 nuclides of the original table. Lost if new nuclides have been discovered by experiments since the table was made. For all the newly-discovered nuclides, their shell structures can be worked out according to the criteria given in this thesis.For an examole, Rn_{: }, From its structyre we know that it decays in (α)style and the decay product is Po.
5) Although we have managed to work out the table of shell structures of all knoen nuclides, the correctness of the table has yet to be verified by lost of experiments. We are sure there are bound to be exceptions. For an example, B_{ }may be a special proton-core nuclide and its structure:, This may explain why B is entirely different in stability form its mirror-image nuclide _{}Be_{.}
References