Abstract
“Half-life” is not one mechanism. Alpha decay, beta decay, and gamma transition are called by one word yet depend on the released energy in different forms. This paper asks how large that difference is: the answer is a factor of $10^{17}$. No new mathematical theorem and no new law is claimed.
First, we actually fit the Geiger--Nuttall law to four nuclides. $\log_{10}t_{1/2}=1.470203 (Z_d/\sqrt{Q})-46.749203$, with $R^2=\mathbf{0.99983298}$ (Section 2).
Second, those four points span $24$ orders of magnitude. From $2.99\times10^{-7}$ s for $^{212}$Po to $1.41\times10^{17}$ s for $^{238}$U, with a largest residual of $0.164$ decades (Section 2).
Third, a mere factor of $2.097$ in $Q$ gives a factor of $4.7161\times10^{23}$ in lifetime (Section 3).
Fourth, this is the core of the paper. The same “double the $Q$” gives $\mathbf{5.69\times10^{18}}$ for alpha, $\mathbf{32}$ for beta, and $\mathbf{8}$ for gamma (Section 4).
Fifth, the span the one word covers is $10^{52.84}$. From $10^{-21}$ s for resonances to $6.94\times10^{31}$ s for $^{128}$Te (Section 5).
Sixth, the separator is what has to pass a barrier. Only alpha sends a charged particle under a Coulomb barrier (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: the Geiger--Nuttall law, Sargent’s rule ($\propto Q^5$), the $\propto E^3$ of E1 transitions, and the measured values of the four nuclides are all standard. We do not build a nuclear model: all we use is one straight-line fit and three exponents. We do not derive Gamow’s theory: we merely quote the form $1/\sqrt{Q}$ and do not enter the WKB approximation. We do not discuss nuclear structure: shell model, deformation, and preformation factors are not treated at all. We claim no generality for the fit: the four nuclides of Section 2 are a sample restricted to ground-state transitions of even-even nuclei, and odd nuclei or hindered transitions depart greatly. The $R^2=0.99983$ is a value for these four points. We do not say $Z_d/\sqrt{Q}$ is the only variable: preformation probability, angular-momentum barrier, and nuclear deformation all act. This paper pushes them into the intercept $b$. We claim no precision for the exponents: both Sargent’s $Q^5$ and E1’s $E^3$ are asymptotic forms, and in actual transitions matrix elements move by orders. $32$ and $8$ are numbers showing whether the form is exponential or a power, not predicted values. We do not assert the two ends of $10^{52.84}$: the $10^{-21}$ s is an estimate converted from resonance widths, and the $^{128}$Te value also has measurement spread. They are numbers for showing the structure of the orders. Relation to earlier papers: Paper 272 showed that one and the same “twice as strong” opens by a factor of $6.70$ on the stimulus side. This paper is the case where the same doubling opens into $5.69\times10^{18}$, $32$, and $8$, the same figure at a different order. Paper 240 counted seven things called “mass”. This paper turns “half-life” pointing at three mechanisms into a number, as a difference of sensitivity. Paper 190 measured “rare” on a logarithmic scale: both the $24$ decades and the $52.84$ decades here must be read logarithmically. Paper 262 showed there are three ways to break and that strength does not decide which. This paper likewise sees three mechanisms under one word. What is added is actually fitting the Geiger--Nuttall law to four nuclides and obtaining $R^2=0.99983298$, computing that a factor of $2.097$ in $Q$ corresponds to $4.7161\times10^{23}$ in lifetime, lining up the three mechanisms’ responses to the same “doubling” as $5.69\times10^{18}$ / $32$ / $8$, and measuring the span the one word covers as $10^{52.84}$.
1. Notation and Standing
The half-life $t_{1/2}$ is defined through the decay constant $\lambda$ by $t_{1/2}=\ln2/\lambda$. This definition applies in the same way to every decay.
Question: Even if the usage is the same, is what sets $\lambda$ the same?
Three mechanisms:
| Mechanism | What leaves | Dependence on released energy |
|---|---|---|
| Alpha decay | $^4$He nucleus (charge $+2$) | $\log_{10}t\propto Z_d/\sqrt{Q}$ (exponential) |
| Beta decay | Electron and antineutrino | $\lambda\propto Q^5$ (power) |
| Gamma transition (E1) | Photon | $\lambda\propto E^3$ (power) |
Observation 1.1 Only the first row has a different form. $Q$ sits inside the exponent, and moreover as an inverse square root. This is where the difference in orders is born.
2. Fitting the Geiger--Nuttall Law to Four Nuclides
Setting: For alpha decay we take $x=Z_d/\sqrt{Q}$ ($Z_d$ the daughter’s atomic number, $Q$ in MeV) as abscissa and $\log_{10}t_{1/2}$ as ordinate.
Check 2.1: We line up the values for four nuclides.
| Nuclide | $t_{1/2}$ [s] | $Q$ [MeV] | $Z_d$ | $x=Z_d/\sqrt{Q}$ |
|---|---|---|---|---|
| $^{238}$U | $1.4101\times10^{17}$ | $4.270$ | $90$ | $43.5541$ |
| $^{226}$Ra | $5.0496\times10^{10}$ | $4.871$ | $86$ | $38.9663$ |
| $^{222}$Rn | $3.3035\times10^{5}$ | $5.590$ | $84$ | $35.5282$ |
| $^{212}$Po | $2.9900\times10^{-7}$ | $8.954$ | $82$ | $27.4035$ |
Check 2.2: We fit by least squares.
| Quantity | Value |
|---|---|
| Slope $a$ | $\mathbf{1.470203}$ |
| Intercept $b$ | $\mathbf{-46.749203}$ |
| Coefficient of determination $R^2$ | $\mathbf{0.99983298}$ |
Check 2.3: We look at the residuals.
| Nuclide | Measured $\log_{10}t$ | Predicted | Difference |
|---|---|---|---|
| $^{238}$U | $+17.1492$ | $+17.2841$ | $-0.1348$ |
| $^{226}$Ra | $+10.7033$ | $+10.5392$ | $\mathbf{+0.1641}$ |
| $^{222}$Rn | $+5.5190$ | $+5.4845$ | $+0.0345$ |
| $^{212}$Po | $-6.5243$ | $-6.4606$ | $-0.0638$ |
Observation 2.4 The four points span $\mathbf{23.67}$ decades, yet the largest residual is $0.164$ decades. In relative terms, linearity to better than $\mathbf{0.7\%}$: a single straight line runs through twenty-four orders of magnitude.
Observation 2.5 A point to write honestly. This is a sample chosen to be ground-state transitions of even-even nuclei only. Odd nuclei and transitions that change angular momentum depart by several orders: $R^2=0.99983$ is a value for these four points, not for alpha decay in general.
3. A Factor of $2.097$ in $Q$, a Factor of $4.72\times10^{23}$ in Lifetime
Check 3.1: We compare the two ends of the table.
| $^{238}$U | $^{212}$Po | Ratio | |
|---|---|---|---|
| $Q$ [MeV] | $4.270$ | $8.954$ | $\mathbf{2.0970}$ |
| $t_{1/2}$ [s] | $1.4101\times10^{17}$ | $2.9900\times10^{-7}$ | $\mathbf{4.7161\times10^{23}}$ |
| Spread in $\log_{10}$ | $\mathbf{23.6736}$ decades |
Observation 3.2 A mere factor of $2.1$ in $Q$ gives $\mathbf{4.7\times10^{23}}$ in lifetime. $^{238}$U lives about the age of the Earth; $^{212}$Po is gone in a third of a microsecond, and both are “alpha decay”.
Observation 3.3 This extreme sensitivity follows from $Q$ sitting inside an exponential. $Z_d$ moves slightly too ($90\to82$), but what dominates is the $\sqrt{Q}$ side.
4. The Core: The Same “Doubling” Gives $5.69\times10^{18}$, $32$, and $8$
Setting: We apply exactly the same operation to all three mechanisms. We double the released energy exactly.
Check 4.1:
| Mechanism | Form of dependence | Factor on the decay rate |
|---|---|---|
| Alpha ($Z_d=90$, $Q:4.270\to8.540$) | $\log_{10}t\propto1/\sqrt{Q}$ | $\mathbf{5.6874\times10^{18}}$ |
| Beta (Sargent’s rule) | $\lambda\propto Q^5$ | $2^5=\mathbf{32}$ |
| Gamma (E1 transition) | $\lambda\propto E^3$ | $2^3=\mathbf{8}$ |
Observation 4.2 This is the core of the paper. The input is “exactly double” in every case, yet the outputs are $5.69\times10^{18}$, $32$, and $8$: top and bottom differ by $\mathbf{7.1\times10^{17}}$, that is a sensitivity gap of about $10^{17}$.
Observation 4.3 $32$ and $8$ are close (a factor of $4$). Beta and gamma are both powers, differing only in exponent $5$ versus $3$: the one standing apart is alpha.
Check 4.4: We look at beta’s $Q^5$ numerically.
| $Q$ [MeV] | Relative decay rate $Q^5$ |
|---|---|
| $0.5$ | $0.0312$ |
| $1.0$ | $1.0000$ |
| $2.0$ | $32.0000$ |
| $4.0$ | $1024.0000$ |
Observation 4.5 An eightfold $Q$ gives only $32768$ for beta. For alpha a twofold $Q$ gives $5.69\times10^{18}$: the gap between power and exponential widens the further the comparison is extended (Paper 190).
Observation 4.6 A point to write honestly: both $32$ and $8$ are numbers from asymptotic forms, and in actual transitions matrix elements move by orders. They are numbers showing whether the form is exponential or a power, not predictions for individual nuclides.
5. The Span the One Word Covers Is $10^{52.84}$
Check 5.1: We take the two ends of what is called a “half-life.”
| $t_{1/2}$ | |
|---|---|
| Resonance state (converted from width) | $\sim10^{-21}$ s |
| $^{128}$Te double beta decay | $2.2\times10^{24}$ yr $=6.9427\times10^{31}$ s |
| Ratio | $\mathbf{6.9427\times10^{52}}=10^{\mathbf{52.84}}$ |
Observation 5.2 One word covers a factor of $\mathbf{10^{52.84}}$. The age of the universe is about $4.35\times10^{17}$ s, so the upper end is $1.6\times10^{14}$ times the age of the universe: a region where “wait until half remains” cannot in principle be carried out.
Observation 5.3 The lower end cannot be carried out either. The $10^{-21}$ s was not measured directly but converted from a resonance width via $\Gamma\tau\sim\hbar$: at both ends, nothing was measured as the definition literally says.
Observation 5.4 So “half-life” has one definition but three or more ways of being measured. Wait; fit a decay curve; convert from a resonance width: the same figure by which Paper 240 split “mass” into seven here lies on the side of the measurement method.
6. Separator: What Has to Pass a Barrier
Observation 6.1 Separator. What must be moved for the sensitivity to change from exponential to power? Whether what leaves carries charge.
Observation 6.2 An alpha particle carries charge $+2$ and must tunnel through the daughter’s Coulomb barrier. The transmission depends exponentially on the barrier’s height and width, and that width is set by $Q$. Here the exponential is born.
Observation 6.3 An electron carries charge, but is created at the instant of decay and so does not tunnel. A photon carries no charge: for both there is no barrier problem, only the volume of phase space acts, and so the form is a power.
Observation 6.4 Hence what separates the three is neither the strength of the nuclear force nor the size of the nucleus. It is whether what leaves has a barrier to pass: a conclusion of the same shape as Paper 262’s, that “breaking” cannot be separated by strength.
Check 6.5: Comparison with Paper 272.
| The same “doubling” as input | Spread of outputs | |
|---|---|---|
| Paper 272 (sensation and stimulus) | Sensation doubles | $\mathbf{6.70}$ |
| This paper (three decay mechanisms) | $Q$ doubles | $\mathbf{7.1\times10^{17}}$ |
Observation 6.6 The figure is the same and the order differs by $17$. In Paper 272 only the power exponents differed, whereas here one of them is an exponential: even within “several mechanisms under one word,” there are classes of how widely they open.
7. Roll Call and Closing
- $a=1.470203$, $b=-46.749203$ (Check 2.2): Fit to four nuclides.
- $R^2=0.99983298$ (Check 2.2): One line through $23.67$ decades.
- Largest residual $0.164$ decades (Check 2.3): Under $0.7\%$ relative.
- $Q$ by $2.0970$ (Check 3.1): Lifetime by $4.7161\times10^{23}$.
- Alpha: doubling gives $5.6874\times10^{18}$ (Check 4.1): Because it is exponential.
- Beta: doubling gives $32$ (Check 4.1): $Q^5$.
- Gamma: doubling gives $8$ (Check 4.1): $E^3$.
- Sensitivity gap $7.1\times10^{17}$ (Observation 4.2): Identical input.
- One word covers $10^{52.84}$ (Check 5.1): Neither end measured directly.
- The separator is charge (Observation 6.4): Only alpha passes a barrier.
Closing: “Half-life” has one definition and three mechanisms. For alpha decay, the Geiger--Nuttall law runs through four nuclides and $23.67$ decades at $R^2=\mathbf{0.99983298}$, and a mere factor of $2.0970$ in $Q$ gives $\mathbf{4.7161\times10^{23}}$ in lifetime. We then apply the same operation: doubling the released energy exactly gives $\mathbf{5.6874\times10^{18}}$ for alpha, $\mathbf{32}$ for beta, and $\mathbf{8}$ for gamma. The input is identical, and the outputs open by $7.1\times10^{17}$. And the span the one word covers is $\mathbf{10^{52.84}}$, with neither end measured as the definition “wait until half remains” literally says. One thing separates them: whether what leaves has a barrier to pass. If it has, the form is exponential; if not, a power. Not the strength of the nuclear force, not the size of the nucleus, but the presence or absence of charge makes a sensitivity gap of $10^{17}$.
Revision Record
Second edition (October 2026): the dashes used as separators in the text were replaced by punctuation; section headings containing mathematics, which had been cut off in display, were repaired; no claim, number or reference was changed.
References
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- Yamagishi, Y. (2026). “Rare” Is Measured on a Logarithmic Scale [Paper 190]. Zenodo. 10.5281/zenodo.22106182
- Yamagishi, Y. (2026). “Mass” Is Seven Things, One of Which Goes Negative and One of Which Depends on the Direction of the Push [Paper 240]. Zenodo. 10.5281/zenodo.22129765
- Yamagishi, Y. (2026). There Are Three Ways to Break, and Strength Does Not Decide Which [Paper 262]. Zenodo. 10.5281/zenodo.22169671
- Yamagishi, Y. (2026). One and the Same “Twice as Strong” Opens by a Factor of 6.70 on the Stimulus Side [Paper 272]. Zenodo. 10.5281/zenodo.22169965
Reference
Yamagishi, Y. (2026). One and the Same “Double the $Q$” Becomes $5.69\times10^{18}$, $32$, and $8$ [Paper 283]. YOI: https://www.yamagishiyuuki.com/yoi/283
DOI: 10.5281/zenodo.22171127
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 283. This paper is a preprint and has not been peer reviewed.