Experimental Correlation of Mercury (Hg) Isotopes with the Alikhan Symmetry-Based Nuclear Stability Model ()
1. Introduction
Classical nuclear models such as the shell model, the liquid-drop model, and the “magic number” concept have historically been used to describe nuclear stability [1]-[4].
While these approaches successfully explain many isotopic behaviors, they fail to account for transitional regions, particularly in heavy nuclei such as Mercury (Hg) [3] [5]-[7].
The Alikhan Symmetry Model proposes a complementary framework in which the nucleus is interpreted as a layered proton-neutron structure [5].
In this model, stability emerges from the spatial symmetry between polar and equatorial regions, even-odd neutron parity, and the balance between nuclear attraction and Coulomb repulsion [5].
It should be noted at the outset that the present model is qualitative in nature and is intended to reveal systematic structural correlations rather than provide exact quantitative predictions.
The primary objective of this study is to test the explanatory power of the Alikhan Symmetry Model against experimental half-life data of Hg isotopes across a wide mass range.
2. Methodology
The neutron framework is constructed using symmetrical layers distributed around the equator.
Each layer follows a numeric sequence (1 + 3 + 6 + 9 + 12 + …), representing spatial shells [5], arising from the progressive filling of concentric equatorial layers with increasing circumference, reflecting a simplified geometric growth of nuclear cross-sections.
When the equatorial balance (“Ekvator balansı”) equals zero, the nucleus is perfectly symmetric, ensuring stability [5] [8] [9].
In this framework, a value of 0 corresponds to an equal number of neutrons on both sides of the equatorial plane, whereas deviations of ±1 arise when one additional neutron occupies either the upper or lower equatorial-adjacent layer, producing mechanical and energetic asymmetry.
This geometric interpretation does not replace quantum mechanical descriptions but provides a complementary macroscopic symmetry criterion.
A deviation of ±1 introduces mechanical and energetic imbalance, leading to reduced stability [9].
The “stability interval” refers to the range of isotopic compositions where nuclear attraction and Coulomb repulsion reach equilibrium [6].
In practical terms, this interval corresponds to a limited N-Z range in which the equatorial balance remains equal to 0 or fluctuates minimally (±1) without destroying global symmetry.
Deficit (“def”) and surplus (“artıq”) zones in the N-Z balance define regions of instability [6] [10].
3. Results and Analysis
3.1. Low-Mass Region (A = 171 - 193)
Half-lives range from microseconds to several hours (80 μs - 3.8 h) [6] [10].
This region is dominated by neutron deficit (N-Z = −25 to −3), leading to reduced symmetry and short lifetimes.
The presence of an odd neutron in the equatorial layer introduces a ±1 equatorial balance, creating mechanical asymmetry and further destabilizing the nucleus [9].
As the neutron number increases, the gradual reduction of neutron deficit leads to a steady increase in half-life, indicating partial symmetry restoration.
3.2. Transition Region (A = 193 - 196)
A sharp transition in stability occurs: Hg-193 (3.8 hours) → Hg-194 (444 years) → Hg-196 (stable) [6] [10].
Symmetry restoration (Ekv. Balans = 0) corresponds to equilibrium of nuclear forces [5].
Hg-194 lies near the lower boundary of the stability interval, where nuclear attraction and Coulomb repulsion approach equilibrium.
In contrast, Hg-195 reintroduces an odd equatorial neutron (±1), resulting in a sharp decrease in half-life.
Hg-196 is located at the center of the stability interval, representing a perfectly balanced configuration [5] [9].
3.3. Stability Plateau (A = 198 - 202)
Hg-198, Hg-199, Hg-200, Hg-201, and Hg-202 remain stable despite neutron parity variation [6].
This plateau indicates that force equilibrium dominates over parity deviations [9].
Even in isotopes containing odd neutrons (Hg-199 and Hg-201), global symmetry and force balance are preserved because these nuclei lie well within the stability interval.
Nuclear-Coulomb forces are harmonized, ensuring stability even under minor symmetry imperfections [5].
3.4. Upper Stability Edge (A = 203 - 204)
Transition from Hg-203 (odd N) to Hg-204 (even N) results in a lifetime increase (46.6 days → stable) [6].
Near the upper boundary of the stability interval, small changes in neutron parity produce pronounced inertial and symmetry-driven effects.
This reflects an inertial disbalance consistent with predictions of the Alikhan model [5].
3.5. Neutron-Rich Region (A = 205 - 216)
Beyond A = 204, neutron surplus (N-Z = +9, …, +20) drives instability [6].
Half-lives decrease rapidly (minutes → nanoseconds).
These isotopes lie outside the stability interval, where excessive neutron density disrupts equatorial symmetry and force equilibrium, leading to strong instability [9] [10].
4. Discussion
These results suggest that spatial symmetry considerations can serve as a unifying framework linking empirical decay data with structural nuclear models.
A clear correlation between spatial symmetry, neutron parity, and experimental half-lives is established [6] [9] [10].
The Alikhan Symmetry Model complements the magic number concept (N = 82, 126) by interpreting magic closures as the completion of symmetric structural layers [1] [2] [5]. In particular, the neutron magic number N = 126 corresponds in the model to the completion of a fully symmetric neutron framework layer, after which additional neutrons necessarily occupy asymmetric positions, leading to instability.
Hg (Z = 80), being close to the magic proton number 82, exhibits quasi-magic behavior aligned with harmonic shell closure in the model [1] [8].
Stability is thus understood not only as quantum shell filling but also as a dynamic equilibrium between symmetry and force interactions [5] [9].
5. Conclusions
The Alikhan symmetry-based framework provides a coherent explanation for the stability of Mercury isotopes consistent with experimental observations [6].
By uniting structural symmetry, proton-neutron parity, and nuclear-Coulomb balance, the model refines the understanding of isotopic stability [5] [9].
Although qualitative, the approach uncovers systematic relationships that align with established nuclear principles, demonstrating that spatial symmetry considerations can enhance modern nuclear theories [1]-[4]. Future studies may extend this approach to other heavy nuclei to further evaluate its general applicability.
Appendix
Table of Mercury (Hg) Isotopes
A |
Neutron Framework |
N |
N-Z |
Deficit/Excess |
Equatorial Balance |
T1/2 |
Hg |
1 + 3 + 6 + 9 + 12 + 18 + 12 + 9 + 6 + 3 + 1 = 80 (Z = 80, Proton framework); fully symmetric, harmonic, ideally stable |
|
|
|
|
|
171 |
1 + 3 + 7 + 9 + 12 + 27 + 12 + 9 + 7 + 3 + 1 = 91 |
91 |
11 |
−25 |
1 |
80 μs |
172 |
1 + 3 + 7 + 9 + 12 + 28 + 12 + 9 + 7 + 3 + 1 = 92 |
92 |
12 |
−24 |
0 |
420 μs |
173 |
1 + 3 + 7 + 10 + 13 + 25 + 13 + 10 + 7 + 3 + 1 = 93 |
93 |
13 |
−23 |
1 |
1.1 ms |
174 |
1 + 3 + 7 + 12 + 12 + 24 + 12 + 12 + 7 + 3 + 1 = 94 |
94 |
14 |
−22 |
0 |
2 ms |
175 |
1 + 3 + 7 + 12 + 12 + 25 + 12 + 12 + 7 + 3 + 1 = 95 |
95 |
15 |
−21 |
1 |
10.8 ms |
176 |
1 + 3 + 7 + 12 + 13 + 24 + 13 + 12 + 7 + 3 + 1 = 96 |
96 |
16 |
−20 |
0 |
20.4 ms |
177 |
1 + 3 + 7 + 12 + 13 + 25 + 13 + 12 + 7 + 3 + 1 = 97 |
97 |
17 |
−19 |
1 |
127.3 ms |
178 |
1 + 3 + 7 + 12 + 13 + 26 + 13 + 12 + 7 + 3 + 1 = 98 |
98 |
18 |
−18 |
0 |
269 ms |
179 |
1 + 3 + 7 + 12 + 13 + 27 + 13 + 12 + 7 + 3 + 1 = 99 |
99 |
19 |
−17 |
1 |
1.09 s |
180 |
1 + 3 + 7 + 12 + 13 + 28 + 13 + 12 + 7 + 3 + 1 = 100 |
100 |
20 |
−16 |
0 |
2.58 s |
181 |
1 + 3 + 7 + 12 + 13 + 29 + 13 + 12 + 7 + 3 + 1 = 101 |
101 |
21 |
−15 |
1 |
3.6 s |
182 |
1 + 3 + 7 + 12 + 13 + 30 + 13 + 12 + 7 + 3 + 1 = 102 |
102 |
22 |
−14 |
0 |
10.83 s |
183 |
1 + 3 + 7 + 12 + 13 + 31 + 13 + 12 + 7 + 3 + 1 = 103 |
103 |
23 |
−13 |
1 |
9.4 s |
184 |
1 + 3 + 7 + 12 + 12 + 34 + 12 + 12 + 7 + 3 + 1 = 104 |
104 |
24 |
−12 |
0 |
30.6 s |
185 |
1 + 3 + 7 + 12 + 12 + 35 + 12 + 12 + 7 + 3 + 1 = 105 |
105 |
25 |
−11 |
1 |
49.1 s |
186 |
1 + 3 + 7 + 12 + 12 + 36 + 12 + 12 + 7 + 3 + 1 = 106 |
106 |
26 |
−10 |
0 |
1.38 min |
187 |
1 + 3 + 7 + 12 + 12 + 37 + 12 + 12 + 7 + 3 + 1 = 107 |
107 |
27 |
−9 |
1 |
1.9 min |
188 |
1 + 3 + 7 + 12 + 16 + 30 + 16 + 12 + 7 + 3 + 1 = 108 |
108 |
28 |
−8 |
0 |
3.25 min |
189 |
1 + 3 + 7 + 12 + 16 + 31 + 16 + 12 + 7 + 3 + 1 = 109 |
109 |
29 |
−7 |
1 |
7.6 min |
190 |
1 + 3 + 7 + 12 + 17 + 30 + 17 + 12 + 7 + 3 + 1 = 110 |
110 |
30 |
−6 |
0 |
20 min |
191 |
1 + 3 + 7 + 12 + 17 + 31 + 17 + 12 + 7 + 3 + 1 = 111 |
111 |
31 |
−5 |
1 |
49 min |
192 |
1 + 3 + 7 + 12 + 18 + 30 + 18 + 12 + 7 + 3 + 1 = 112 |
112 |
32 |
−4 |
0 |
4.85 h |
193 |
1 + 3 + 7 + 12 + 18 + 31 + 18 + 12 + 7 + 3 + 1 = 113 |
113 |
33 |
−3 |
1 |
3.8 h |
194 |
1 + 3 + 7 + 12 + 19 + 30 + 19 + 12 + 7 + 3 + 1 = 114 |
114 |
34 |
−2 |
0 |
444 y |
195 |
1 + 3 + 7 + 12 + 19 + 31 + 19 + 12 + 7 + 3 + 1 = 115 |
115 |
35 |
−1 |
1 |
10.53 h |
196 |
1 + 3 + 7 + 12 + 18 + 34 + 18 + 12 + 7 + 3 + 1 = 116 |
116 |
36 |
0 |
0 |
Stable |
197 |
1 + 3 + 7 + 12 + 18 + 35 + 18 + 12 + 7 + 3 + 1 = 117 |
117 |
37 |
1 |
1 |
64.14 h |
198 |
1 + 3 + 7 + 12 + 18 + 36 + 18 + 12 + 7 + 3 + 1 = 118 |
118 |
38 |
2 |
0 |
Stable |
199 |
1 + 3 + 7 + 12 + 18 + 37 + 18 + 12 + 7 + 3 + 1 = 119 |
119 |
39 |
3 |
1 |
Stable |
200 |
1 + 3 + 7 + 12 + 19 + 36 + 19 + 12 + 7 + 3 + 1 = 120 |
120 |
40 |
4 |
0 |
Stable |
201 |
1 + 3 + 7 + 12 + 19 + 37 + 19 + 12 + 7 + 3 + 1 = 121 |
121 |
41 |
5 |
1 |
Stable |
202 |
1 + 6 + 6 + 12 + 18 + 36 + 18 + 12 + 6 + 6 + 1 = 122 |
122 |
42 |
6 |
0 |
Stable |
203 |
1 + 6 + 6 + 12 + 18 + 37 + 18 + 12 + 6 + 6 + 1 = 123 |
123 |
43 |
7 |
1 |
46.595 d |
204 |
1 + 7 + 6 + 12 + 18 + 36 + 18 + 12 + 6 + 7 + 1 = 124 |
124 |
44 |
8 |
0 |
Stable |
205 |
1 + 7 + 6 + 12 + 18 + 37 + 18 + 12 + 6 + 7 + 1 = 125 |
125 |
45 |
9 |
1 |
5.14 min |
206 |
1 + 7 + 7 + 12 + 18 + 36 + 18 + 12 + 7 + 7 + 1 = 126 |
126 |
46 |
10 |
0 |
8.15 min |
207 |
1 + 7 + 7 + 12 + 19 + 37 + 19 + 12 + 7 + 7 + 1 = 127 |
127 |
47 |
11 |
1 |
2.9 min |
208 |
1 + 7 + 7 + 12 + 19 + 36 + 19 + 12 + 7 + 7 + 1 = 128 |
128 |
48 |
12 |
0 |
42 min |
209 |
1 + 7 + 7 + 12 + 19 + 36 + 19 + 12 + 7 + 7 + 1 = 129 |
129 |
49 |
13 |
1 |
37 s |
210 |
1 + 4 + 6 + 12 + 18 + 48 + 18 + 12 + 6 + 4 + 1 = 130 |
130 |
50 |
14 |
0 |
10 min |
211 |
1 + 4 + 6 + 12 + 18 + 49 + 18 + 12 + 6 + 4 + 1 = 131 |
131 |
51 |
15 |
1 |
>300 ns |
212 |
1 + 4 + 7 + 12 + 18 + 48 + 18 + 12 + 7 + 4 + 1 = 132 |
132 |
52 |
16 |
0 |
>300 ns |
213 |
1 + 4 + 7 + 12 + 18 + 49 + 18 + 12 + 7 + 4 + 1 = 133 |
133 |
53 |
17 |
1 |
>300 ns |
214 |
1 + 4 + 7 + 12 + 19 + 48 + 19 + 12 + 7 + 4 + 1 = 134 |
134 |
54 |
18 |
0 |
>300 ns |
215 |
1 + 4 + 7 + 12 + 19 + 49 + 19 + 12 + 7 + 4 + 1 = 135 |
135 |
55 |
19 |
1 |
>300 ns |
216 |
1 + 4 + 7 + 10 + 19 + 54 + 19 + 10 + 7 + 4 + 1 = 136 |
136 |
56 |
20 |
0 |
>300 ns |
Analysis
Isotopes Hg-171 to Hg-193 are radioactive, with half-lives ranging from 80 μs to 3.8 hours.
In this region, the presence of an odd neutron in the equatorial layer creates a small imbalance (1n), leading to a moderate decrease in stability. The neutron deficit (from −25 to −3) dominates, resulting in a gradual increase of half-life with rising neutron number.
The transition from Hg-193 to Hg-194 represents a sharp leap: from 3.8 hours to 444 years, because Hg-194 lies near the lower boundary of the stability interval, and its equatorial balance equals zero.
From Hg-194 to Hg-195, the half-life drops dramatically from 444 years to 10.53 hours due to the reappearance of an odd neutron (1n) in the equatorial layer, creating a disbalance.
From Hg-195 to Hg-196, the half-life jumps from 10.53 hours to complete stability, since the equatorial layer now contains an even number of neutrons, restoring balance and priority stability.
In the interval Hg-198 to Hg-202, Coulomb and nuclear forces reach an equilibrium limit. Even though Hg-199 and Hg-201 have odd neutrons, they remain stable because they lie within this equilibrium zone.
Transitions Hg-202 → Hg-203 and Hg-203 → Hg-204 are also accompanied by abrupt changes. Near the upper boundary of the stability interval, the neutron excess and inertial imbalance (odd 203 vs. even 204) become pronounced.
In the region Hg-205 to Hg-210, the half-life is in the minute range, where neutron excess predominates.
From Hg-211 to Hg-216, increasing neutron excess leads to very short half-lives (~300 ns), indicating strong instability in this neutron-rich domain.