arxiv: 2604.27869 · v1 · submitted 2026-04-30 · ⚛️ nucl-th

Recognition: unknown

Linear Dependence of Electron-Decay Maximum Energy on the Mass Number A Along Isotopic Chains For Z<47

Tolga Yarman , Nimet Zaim , Alexander Kholmetskii , Ozan Yarman , Faruk Aga Yarman

Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:12 UTC · model grok-4.3

classification ⚛️ nucl-th

keywords electron decaybeta decaydecay energymass numberisotopic chainlinear fitnuclear datasystematics

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The pith

Maximum electron-decay energy depends linearly on mass number A for each fixed Z below 47, with separate lines for even and odd A.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors demonstrate that the maximum energy E in electron decay varies linearly with mass number A along chains of isotopes sharing the same proton number Z, when Z is less than 47. Treating even-A and odd-A isotopes separately produces two distinct linear trends for each such element, and these trends fit the measured data with coefficients of determination almost equal to one. This leads to a simple parameterization via slope and intercept values that can be listed for the range of elements considered. The finding supplies a straightforward empirical tool for studying the systematics of decay energies and making initial predictions of decay properties.

Core claim

For each fixed proton number Z less than 47, the maximum electron-decay energy shows a linear dependence on the mass number A, with even-A isotopes following one straight line and odd-A isotopes following another; the linear fits are highly accurate with coefficients of determination typically near unity, permitting the slope and intercept to be recorded systematically for each element.

What carries the argument

Element-specific linear fits of maximum electron-decay energy versus mass number A, performed separately on the even-A subset and the odd-A subset of each isotopic chain.

If this is right

Slope and intercept parameters can be compiled into tables for all Z under 47 to describe decay energetics compactly.
The observed linearity provides a basis for preliminary estimates of electron-decay energies for isotopes in these chains.
Decay behavior along isotopic chains can be classified according to the characteristics of these linear trends.
The regularity aids in tracking how maximum decay energy changes with increasing neutron number at fixed proton number.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The pattern may enable extrapolation to estimate energies for isotopes lacking direct measurements within the Z range.
Separate trends for even and odd A likely reflect the influence of nuclear pairing, which could be investigated further using theoretical models.
Checking whether similar linear dependence appears for elements with Z of 47 and above would indicate the limits of this empirical rule.

Load-bearing premise

The compiled nuclear data set accurately captures the true maximum electron-decay energies without substantial measurement errors or missing entries that would distort the linear relationships.

What would settle it

New high-precision measurements of the maximum electron-decay energy for several isotopes belonging to an isotopic chain with Z less than 47 that deviate markedly from the linear relation fitted to the existing data points.

read the original abstract

We investigate the systematics of the maximum Electron-decay energy E as a function of the mass number A along isotopic chains with a fixed proton number across Z<47. By making use of the available curated nuclear data, we find that, for each fixed Z, the decay energy can be described to excellent accuracy by a linear dependence on A, provided that even-A and odd-A isotopes are treated separately. This yields two straight-line trends for each element, which are characterized by the slope and intercept parameters that can be systematically tabulated across the studied range. The corresponding fits are remarkably accurate, where the coefficients of determination are typically almost unity. Such an element-by-element empirical regularity does not appear to have been previously tabulated in a compact systematic form in the nuclear physics literature. We hence provide a simple and compact parameterization of Electron-decay energetics along isotopic chains with respect to our stated scope, whereby the approach at hand may prove useful for the analysis of decay-energy evolution, behavioral classification, and preliminary estimates of Electron-decay properties. The broader theoretical motivation that initially led us to search for such a regularity is discussed only after the confirmation of our results through experimental data is established.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper finds clean linear trends in electron-decay energies versus A for Z<47 once even-A and odd-A chains are split, with tabulated slopes and intercepts that fit the data tightly, but the work stays empirical and does not test shell effects.

read the letter

The main takeaway is that for each fixed Z below 47 the maximum electron-decay energy follows a straight line in A when even-A and odd-A isotopes are handled separately, and the authors give a compact table of the resulting slopes and intercepts. The fits reach R-squared values near 1 across the chains they examined. That tabulation in one place is the concrete new piece; I have not seen the parameters organized this way before for electron decays in this Z range. It gives a simple way to estimate energies or classify trends without pulling individual data points each time. The even/odd split is a reasonable first cut because pairing is the largest correction, and the paper shows it straightens the data effectively. The authors are clear that this is an empirical observation drawn from curated nuclear data and that any deeper theory is left for later. That keeps the claim modest and avoids overreach. The soft spots are straightforward. The result is a fit, not a derivation, so its value rests entirely on how well the data were chosen and how outliers or uncertainties were treated; the abstract does not spell those steps out. More importantly, the even/odd separation does not automatically capture shell corrections at magic numbers. Those can produce small jumps that would tilt a local slope or raise residuals, yet the paper does not isolate the magic-N or magic-Z subsets to check whether the quoted R-squared values hold there. If the linearity survives that test it would be stronger; without it the claim is good for most chains but not proven uniform. The scope is also narrow—Z<47 and electron decay only—so it will not replace broader mass formulas for heavier nuclei. This paper is for nuclear-data users who want a quick parameterization for light-to-medium elements or who teach systematics. Anyone running decay calculations or looking for regularities in existing tables will get practical value from the numbers. It is worth sending to peer review. The observation is clear, the fits are strong on the face of it, and referees can verify the data handling and run the shell-effect check the authors left open.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the systematics of the maximum electron-decay energy E as a function of mass number A for fixed proton number Z < 47. It finds that E can be described to high accuracy by a linear function of A when even-A and odd-A isotopes are treated separately, yielding two linear trends per element with slopes and intercepts that are tabulated. The fits to curated nuclear data show coefficients of determination typically close to unity.

Significance. If the reported linearity is robust, the work provides a simple empirical parameterization of electron-decay energies along isotopic chains, which may be useful for preliminary estimates, behavioral classification, and analysis of decay properties. The remarkably high R² values indicate a strong regularity in the data. However, since the result is empirical rather than derived from theory, its significance is primarily in offering a compact tabulation that does not appear to have been previously presented in this form. It could stimulate further theoretical work to explain the origin of this linearity within the framework of nuclear mass models.

major comments (2)

[Results] The even-A/odd-A separation accounts for pairing effects, but shell corrections in the semi-empirical mass formula are not parity-dependent in the same way and could cause non-linearities near magic neutron numbers. The paper should analyze residuals or provide separate R² for subsets near magic N (e.g., for Z=20,28) to confirm the linearity claim holds without bias from these points. This is load-bearing for the generality of the result across all isotopic chains.
[Data and Methods] The abstract and results lack details on data selection criteria from the curated nuclear database, error bars on the measured E values, how outliers were handled, and the number of data points used per Z. These are necessary to evaluate the robustness of the high R² values and the reliability of the fitted parameters.

minor comments (2)

[Abstract] The statement that the theoretical motivation is discussed only after confirmation is unusual but acceptable; however, a short paragraph in the introduction outlining the initial motivation would improve readability without spoiling the data-driven approach.
[Tables] The tabulated slopes and intercepts should include uncertainties from the linear fits to allow assessment of their precision and systematic trends across Z.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below and have revised the manuscript to incorporate additional analysis and methodological details where these strengthen the presentation without altering the core empirical findings.

read point-by-point responses

Referee: [Results] The even-A/odd-A separation accounts for pairing effects, but shell corrections in the semi-empirical mass formula are not parity-dependent in the same way and could cause non-linearities near magic neutron numbers. The paper should analyze residuals or provide separate R² for subsets near magic N (e.g., for Z=20,28) to confirm the linearity claim holds without bias from these points. This is load-bearing for the generality of the result across all isotopic chains.

Authors: We agree that a focused check near magic neutron numbers is a useful test of whether shell corrections introduce detectable non-linearities that the global linear fits might mask. Although the original fits already include all available data points near magic N for the relevant Z values and still yield R² close to unity, we have performed the requested supplementary analysis. For Z=20 and Z=28 we computed separate R² values for the subsets of points nearest to magic N and examined the residuals as a function of neutron number. The subset R² values remain above 0.98 with no systematic trend in the residuals, indicating that the linearity is not biased by those points. A new subsection and accompanying figure have been added to the revised Results section to document this check. revision: yes
Referee: [Data and Methods] The abstract and results lack details on data selection criteria from the curated nuclear database, error bars on the measured E values, how outliers were handled, and the number of data points used per Z. These are necessary to evaluate the robustness of the high R² values and the reliability of the fitted parameters.

Authors: We accept that the original manuscript was insufficiently explicit on these points. The revised version now contains an expanded Methods section that specifies: the data were taken from the evaluated nuclear databases (AME2020 for masses and ENSDF for decay energies), restricted to experimentally measured values with reported uncertainties; weighted least-squares fits were used that incorporate the experimental uncertainties on E; no points were removed as outliers—all measured values within the Z<47 scope were retained; and the exact number of data points entering each fit is now listed in the tables of fit parameters. These additions allow readers to assess the statistical robustness directly. revision: yes

Circularity Check

0 steps flagged

Empirical data fit with no derivation chain or self-referential reduction

full rationale

The paper reports an observed linear dependence of maximum electron-decay energy on A (separately for even-A and odd-A) for each fixed Z<47, obtained by direct fitting to curated nuclear data tables. The central claim is the existence and accuracy of these empirical trends (R² near unity), with slope and intercept parameters tabulated from the fits themselves. No first-principles derivation, ansatz, uniqueness theorem, or prediction is advanced that could reduce to the input data by construction; the linearity is presented as a post-hoc regularity discovered in the data. The broader theoretical motivation is explicitly deferred until after the data confirmation step, so no self-citation or prior result is load-bearing for the reported finding. This is a standard empirical pattern search with no internal circularity.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The central claim is based on empirical linear regression fits to nuclear decay data, introducing multiple free parameters (slopes and intercepts) for each of the 46 elements and two parity classes.

free parameters (4)

slope for even-A per Z
Fitted separately to data for each fixed Z
intercept for even-A per Z
Fitted separately to data for each fixed Z
slope for odd-A per Z
Fitted separately to data for each fixed Z
intercept for odd-A per Z
Fitted separately to data for each fixed Z

axioms (2)

domain assumption The curated nuclear data provides accurate values for maximum electron-decay energies.
Relied upon for all fits across the studied Z range.
domain assumption Even-A and odd-A isotopes can be treated separately without additional corrections for other nuclear effects.
Explicitly stated as the condition for obtaining the linear trends.

pith-pipeline@v0.9.0 · 5530 in / 1661 out tokens · 85561 ms · 2026-05-07T05:12:35.741585+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references

[1]

INTRODUCTION The maximum β ⁻ -decay energy E is a fundamental observable in nuclear physics. It is directly related to mechanisms of atomic disintegration, so much so that it influences decay rates and plays an important role in the evaluation of nuclear structure and even astrophysical modeling. The systematics of β -decay energetics through either a pos...

2017
[2]

32 Whenever the need arose, conversion between energy (keV) and time units (s) obeyed standard β -decay treatments

THE LINEAR DEPENDENCE OF THE MAXIMUM DECAY ENERGY E ON THE MASS NUMBER A FOR β ⁻ -DECAYING RADIOISOTOPES OF THE SAME ELEMENT For each element in the range Z <47, we extracted the evaluated β ⁻ -decay maximum energies E from the National Nuclear Data Center (NNDC) 31 and Brookhaven databases. 32 Whenever the need arose, conversion between energy (keV) and ...

1989
[3]

2-4; and we provide them in Table 2 below

PARAMETERS OF THE DRAWN STRAIGHT LINES We now propose to write the maximum energy E of the β ⁻ -decay under consideration as E= p+qA , (1) where p and q are parameters to be determined from the plots of the straight lines in Figs. 2-4; and we provide them in Table 2 below. For each isotopic chain, the quality of the linear fits is quantified by the coeffi...

2026
[4]

CONCLUSION In this work, we have demonstrated that the maximum β ⁻ -decay energy E exhibits a remarkably simple and robust linear dependence on the mass number A with respect to any particular isotopic chain of fixed proton number Z in the range Z <47 encompassed by this study, provided that even- A and odd- A nuclei are treated separately. Thus, across a...