%% Scientific Reports submission
%% Title: A long-range nuclear potential as the origin of the 1/v: ...
%% Author: Kazuo Ooyama
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\begin{document}

%% ===== Title block =====
\title{A long-range $r^{-4}$ nuclear potential as the origin of
the $1/v$ law and energy-independent elastic scattering
in light nuclides}

\author{Kazuo Ooyama\\[6pt]
\small Ooyama Power Inc., Shinjuku-ku, Tokyo 160-0033, Japan\\
\small \texttt{k2@ooyama-power.com}}

\date{\today}

\maketitle

%% ===== Abstract =====
\begin{abstract}
Advances in evaluated nuclear data, as represented by the
JENDL-5 library, have clarified that the reaction cross sections
of $^3$He, $^6$Li, $^{10}$B, and $^7$Be follow the $1/v$ law
over nine decades of neutron energy, with effective interaction
radii of 173--1344~fm inferred from thermal data, while the
elastic scattering cross sections of $^1$H, $^2$H, and $^3$H
remain essentially constant from $10^{-5}$~eV to $10^4$~eV.
Both features are difficult to account for by short-range nuclear
forces alone, whose range (${\lesssim}2$~fm) falls six orders of
magnitude below the thermal neutron de~Broglie wavelength.
We propose a long-range nuclear potential $V(r)=-N_A/r^4$ as a
common origin for both observations.
The $r^{-4}$ form is derived from the $1/v$ law by a contrapositive
argument; $N_A$ is determined from a single cross-section measurement
with no free parameters.
Two-dimensional time-dependent Schr\"odinger equation simulations
reproduce both the $1/v$ reaction law and the energy-independent
elastic scattering cross section, in agreement with JENDL-5 data
within 4.8\% for elastic scattering and 3.8\% for absorption.
The $-N_A/r^4$ potential provides the non-resonant physical basis
lacking in Breit-Wigner and Lane-Lynn frameworks, which require
resonance parameters absent from JENDL-5 for these nuclides, and
complements Yukawa theory at short range.
\end{abstract}

\noindent\textbf{Keywords:} long-range nuclear force; few-body nuclear systems;
neutron-nucleus scattering; $1/v$ law; neutron cross section; JENDL-5; effective
interaction radius; inverse-power potential; Schr\"odinger equation simulation

\newpage

%% -------------------------------------------------------------------
\section{Introduction}

The JENDL-5 evaluated nuclear data library~\cite{JENDL5} provides
highly accurate neutron cross sections for light nuclides over a wide
energy range.
Two features of these data call for attention.
First, the reaction cross sections of $^3$He, $^6$Li, $^{10}$B, and
$^7$Be follow the $1/v$ law over nine decades of energy, and their
effective interaction radii inferred from thermal cross sections ---
413~fm for $^3$He, 173~fm for $^6$Li, 349~fm for $^{10}$B, and 1344~fm for $^7$Be ---
exceed the nuclear radius $R\sim A^{1/3}$~fm (where $A$ is the mass
number) by two to three orders of magnitude.
Second, the elastic scattering cross sections of $^1$H (20.4~b),
$^2$H (3.39~b), and $^3$H (1.66~b) remain essentially constant from
$10^{-5}$~eV to $10^4$~eV.
At thermal energies the neutron de~Broglie wavelength reaches
${\sim}1.81\times10^5$~fm, six orders of magnitude larger than the
range of any Yukawa-type nuclear potential ($r\lesssim 2$~fm).
The spin-dependent scattering-length treatment reproduces the $^1$H
cross section by back-calculating $a_s$ from the measured value and
therefore provides no independent theoretical prediction of this
energy independence.

These observations suggest that short-range nuclear forces alone,
effective only within ${\sim}2$~fm, are insufficient to account for
the data at such large length scales.
We are thus led to postulate the existence of a long-range nuclear
potential and to ask whether a single such potential can account for
both the $1/v$ reaction law and the energy-independent elastic
scattering cross section simultaneously.

In this paper we derive the form $V(r)=-N_A/r^4$ from the $1/v$ law
by a classical mechanical argument and obtain a provisional value of
the coefficient $N_A$ from a single cross-section measurement
(Sec.~3).
We then validate $N_A$ through two-dimensional time-dependent
Schr\"odinger equation simulations, which simultaneously reproduce
the $1/v$ energy dependence of the reaction cross section to within
3.9\% and the energy independence of the elastic scattering cross
section to within 4.3\%; the residual discrepancy is attributed to
the two-dimensional geometry of the simulation (Sec.~4).
The experimental evidence from JENDL-5 is presented in detail in
Sec.~2, and the limitations of existing frameworks ---
resonance-based theories (Breit-Wigner and Lane-Lynn) and the
spin-dependent scattering-length treatment --- are examined in
Sec.~5 in the light of the proposed potential.

%% -------------------------------------------------------------------
\section{Experimental evidence from JENDL-5}
%% -------------------------------------------------------------------

JENDL-5 (Japan Evaluated Nuclear Data Library, version~5) is a
comprehensive evaluated nuclear data library published by the Japan
Atomic Energy Agency~\cite{JENDL5}, providing neutron cross sections
for a wide range of nuclides over an energy range from $10^{-5}$~eV
to 20~MeV.
We examine the cross-section data for hydrogen isotopes and several
light nuclides that exhibit notably large cross sections at low
energies.

\subsection{Energy-independent elastic scattering cross sections}

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.48\textwidth]{fig1_H1.png}
  \includegraphics[width=0.48\textwidth]{fig2_H2.png}
  \caption{Neutron cross sections of (a)~$^1$H and (b)~$^2$H from
    JENDL-5~\protect\cite{JENDL5}.
    Thin black line: elastic scattering cross section.
    The cross sections are essentially constant at 20.4~b and 3.39~b
    respectively from $10^{-5}$~eV to $10^4$~eV.}
  \label{fig:Hiso12}
\end{figure}

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.48\textwidth]{fig3_H3.png}
  \includegraphics[width=0.48\textwidth]{fig4_He3.png}
  \caption{Neutron cross sections of (a)~$^3$H and (b)~$^3$He from
    JENDL-5~\protect\cite{JENDL5}.
    Thin black line: elastic scattering cross section.
    For $^3$H the elastic scattering cross section is constant at
    1.66~b; no reaction cross section exists.
    For $^3$He, black solid line: $(n,p)$ reaction cross section
    following the $1/v$ law (846.4~b at 1~eV);
    thin black line: elastic scattering cross section
    (${\approx}3.1$~b, essentially constant).}
  \label{fig:H3He3}
\end{figure}

Figure~\ref{fig:Hiso12} shows the neutron cross sections of $^1$H
and $^2$H from JENDL-5.
The elastic scattering cross section of $^1$H is approximately 20.4~b,
remaining essentially constant from $10^{-5}$~eV to $10^4$~eV.
That of $^2$H is similarly constant at approximately 3.39~b over the
same range.

Figure~\ref{fig:H3He3} shows the corresponding data for $^3$H and
$^3$He.
The elastic scattering cross section of $^3$H is approximately 1.66~b,
essentially constant over the same energy range; no nucleon-emission
reaction cross section exists for $^3$H.
For $^3$He, both the elastic scattering cross section
(${\approx}3.1$~b, essentially constant) and the $(n,p)$ reaction
cross section are shown; the reaction cross section is discussed in
Sec.~2.2.

At thermal energies ($E_0 = 0.025$~eV, where $E_0$ is the neutron
kinetic energy in the laboratory frame and $v$ is the corresponding
neutron speed), the neutron de~Broglie wavelength reaches
${\sim}1.81\times10^5$~fm, six orders of magnitude larger than the
range of Yukawa-type nuclear forces ($r\lesssim 2$~fm).

\subsection{The $1/v$ law}

The neutron reaction cross sections of $^3$He, $^6$Li, and $^{10}$B
follow the $1/v$ law --- that is, $\sigma_R \propto 1/v \propto
E_0^{-1/2}$ --- over the range $10^{-5}$--$10^4$~eV, as shown in
Figs.~\ref{fig:H3He3} and~\ref{fig:LiB}.
At 1~eV the reaction cross sections are 846.4~b for $^3$He, 149.6~b
for $^6$Li, and 607.3~b for $^{10}$B.
The elastic scattering cross sections of these same nuclides at 1~eV
are 3.135~b, 0.739~b, and 2.144~b respectively, giving
$\sigma_R/\sigma_E$ ratios of 270, 202, and 283.

In the JENDL-5 file format, resonance parameters are stored in MF=2;
MT=151 contains scattering radii.
For $^3$He, $^6$Li, $^{10}$B, and $^7$Be, JENDL-5 lists no resonance
parameters in MF=2; only scattering radii (MT=151) are provided.
For $^6$Li, JENDL-5 states explicitly that the capture cross section
below 100~keV is obtained by normalizing a $1/v$ curve to thermal
data~\cite{JENDL5}.

Figure~\ref{fig:CdBe} shows $^{112}$Cd and $^7$Be.
For $^{112}$Cd, resonance parameters are provided in MF=2, yet
JENDL-5 itself states in MF=2/MT=151: ``A negative resonance was
added so as to reproduce the thermal capture and scattering cross
sections given by Mughabghab et al.''~\cite{JENDL5}, confirming
that a negative-energy resonance must be artificially introduced
to reproduce the thermal cross sections.
For $^7$Be (half-life 53~days), the $(n,p)$ reaction cross section
reaches 8722.8~b at 1.062~eV and the elastic scattering cross section
is 384~b at thermal energies.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.48\textwidth]{fig5_Li6.png}
  \includegraphics[width=0.48\textwidth]{fig6_B10.png}
  \caption{Neutron cross sections of (a)~$^6$Li and (b)~$^{10}$B
    from JENDL-5~\protect\cite{JENDL5}.
    Black solid line: reaction cross section following the $1/v$ law
    (149.6~b and 607.3~b at 1~eV respectively);
    thin black line: elastic scattering cross section.}
  \label{fig:LiB}
\end{figure}

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.48\textwidth]{fig7_Cd112.png}
  \includegraphics[width=0.48\textwidth]{fig8_Be7.png}
  \caption{Neutron cross sections of (a)~$^{112}$Cd and (b)~$^7$Be
    from JENDL-5~\protect\cite{JENDL5}.
    For $^{112}$Cd: thin black line, elastic scattering cross section;
    red solid line, $(n,\gamma)$ capture cross section.
    According to the JENDL-5 MF=2/MT=151 documentation, a
    negative-energy resonance was explicitly added to reproduce
    the thermal capture and scattering cross sections~\cite{JENDL5}.
    For $^7$Be: black solid line, $(n,p)$ reaction cross section
    (8722.8~b at 1.062~eV); thin black line, elastic scattering cross
    section (384~b at thermal energies).}
  \label{fig:CdBe}
\end{figure}

\subsection{Anomalously large effective interaction radii}

Interpreting the thermal cross section as a geometric area
$\sigma = \pi r_{\rm eff}^2$, one obtains:
$r_{\rm eff} = 413$~fm for $^3$He ($\sigma_R = 5363$~b at
$E_0 = 0.025$~eV),
$r_{\rm eff} = 173$~fm for $^6$Li,
$r_{\rm eff} = 349$~fm for $^{10}$B, and
$r_{\rm eff} = 1344$~fm for $^7$Be ($\sigma_R = 56777$~b at
$E_0 = 0.025$~eV).
These values exceed the nuclear radius $R\sim A^{1/3}$~fm
(where $A$ is the mass number) by two to three orders of magnitude.
The elastic scattering cross section of $^7$Be (384~b) corresponds
to $r_{\rm eff}\approx 550$~fm.

%% -------------------------------------------------------------------
\section{Derivation of the $-N_A/r^4$ potential}
\label{sec:derivation}
%% -------------------------------------------------------------------

\subsection{Contrapositive approach from the $1/v$ law}

Rather than starting from a theoretical model, we derive the form
of the nuclear potential directly from the experimentally observed
$1/v$ law.

The observation that $\sigma(E_0)$ forms a straight line with slope
$-1/2$ on a log-log plot over many decades of energy indicates that
the cross section is a monomial in $E_0$.
This in turn requires that the nuclear force $F$ acting between
neutron and nucleus be a monomial in their separation $r$: any
superposition of power laws would produce curvature on the log-log
plot, contrary to what is observed.
Furthermore, since $F$ must vanish as $r\to\infty$, it must
decrease with increasing $r$.
These two constraints --- monomial form and decay at infinity ---
restrict $F$ to the family
\begin{equation}
  F = -Nr^{-x},\quad x>0,
  \label{eq:force}
\end{equation}
where $N$ is a positive coefficient and $x>0$ is the exponent to
be determined by matching the observed energy dependence of the
cross section.
The corresponding potential, with the boundary condition
$V\to 0$ as $r\to\infty$, is
\begin{equation}
  V(r) = -\frac{N}{x-1}\,r^{-(x-1)},\quad x\neq 1.
  \label{eq:Vgeneral}
\end{equation}

A classical two-body analysis in the center-of-mass frame,
balancing the centrifugal force against Eq.~(\ref{eq:force}) and
invoking conservation of angular momentum and energy, yields the
reaction cross section as a function of $E_0$
(see Appendix~\ref{app:classical} for the full derivation):
\begin{equation}
  \sigma(E_0)\propto E_0^{-2/(x-1)}.
  \label{eq:sigmaE}
\end{equation}

The experimentally observed $1/v$ law requires
$\sigma\propto E_0^{-1/2}$, which fixes the exponent uniquely:
\begin{equation}
  \frac{2}{x-1} = \frac{1}{2} \implies x=5.
  \label{eq:x5}
\end{equation}

By contrapositive: if $x\neq 5$, then
$\sigma\propto E_0^{-2/(x-1)}\neq E_0^{-1/2}$,
contradicting the experimental fact.
Therefore $x=5$ is the unique solution consistent with the
$1/v$ law, giving
\begin{equation}
  V(r) = -\frac{N}{4}\,r^{-4} \equiv -\frac{N_A}{r^4},
  \label{eq:V}
\end{equation}
where $N_A\equiv N/4$ is the long-range nuclear potential
coefficient.
The reaction cross section takes the explicit form
\begin{equation}
  \sigma = \pi\sqrt{\frac{4M}{M-m}\cdot\frac{N_A}{E_0}},
  \label{eq:sigmaNA}
\end{equation}
where $M$ is the total mass of the neutron-nucleus system and
$m$ is the neutron mass.
Equation~(\ref{eq:sigmaNA}) reproduces the $1/v$ law exactly
and allows a provisional value of $N_A$ to be obtained from a
single measured cross section at a known energy.

This two-step procedure --- classical mechanics yields a provisional
Eq.~(\ref{eq:sigmaNA}), which is then verified by quantum mechanics
in Sec.~4 --- parallels the treatment of the hydrogen atom, where
the classical Bohr radius provides the length scale and the
Schr\"odinger equation yields the most probable radius.

\subsection{Determination of $N_A$ for each nuclide}

\begin{table}[htbp]
  \caption{Long-range nuclear potential coefficient $N_A$ determined from JENDL-5
    reaction cross sections~\protect\cite{JENDL5} for four nuclides exhibiting
    the $1/v$ law.
    Target mass and neutron mass are given in atomic mass units (u).}
  \label{tab:NA}
  \centering
  \begin{tabular}{lccccc}
    \toprule
    Nuclide & $E$ [eV] & $\sigma_R$ [b] & Target mass [u] & Neutron mass [u] &
      $N_A$ [J\,m$^4$] \\
    \midrule
    $^3$He   & 1.000 & 846.43  & 3.016029  & 1.008665 & $2.178\times10^{-71}$ \\
    $^6$Li   & 1.000 & 149.55  & 6.015122  & 1.008665 & $7.773\times10^{-73}$ \\
    $^7$Be   & 1.062 & 8722.8  & 7.016929  & 1.008665 & $2.867\times10^{-69}$ \\
    $^{10}$B & 1.006 & 607.32  & 10.012937 & 1.008665 & $1.368
    \times10^{-71}$ \\
    \bottomrule
  \end{tabular}
\end{table}

Solving Eq.~(\ref{eq:sigmaNA}) for $N_A$ and inserting the JENDL-5 cross section
data~\cite{JENDL5} at neutron energies near 1~eV, where the $1/v$ law is well
established, gives the values listed in Table~\ref{tab:NA}.
Atomic mass units are used throughout; all masses are taken from standard nuclear
data tables.

The values of $N_A$ span three orders of magnitude across the four nuclides,
reflecting the different nuclear structures of the targets.
The anomalously large value for $^7$Be is consistent with its exceptionally large
effective interaction radius of 1344~fm noted in Sec.~2.3; a stronger
long-range potential naturally produces a larger capture cross section.
The verification of these provisional values through quantum
mechanical simulation is described in Sec.~4, with $^3$He
serving as the primary test case.

%% -------------------------------------------------------------------
\section{Quantum mechanical verification}
%% -------------------------------------------------------------------
 
\subsection{Simulation setup}
 
To verify that the $-N_A/r^4$ potential derived in Sec.~3 is consistent with
quantum mechanics, we solved the time-dependent Schr\"odinger equation
numerically for a neutron wave packet incident on a $^3$He target nucleus.
The governing equation is
\begin{equation}
  i\hbar\frac{\partial\psi}{\partial t}
    = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(r)\right]\psi,
  \label{eq:TDSE}
\end{equation}
where the potential $V(r)$ is taken to be complex,
\begin{equation}
  V(r) = V_{\rm real}(r) + i\,V_{\rm imag}(r).
  \label{eq:Vcomplex}
\end{equation}
 
For $^3$He, where the reaction cross section exceeds the elastic
scattering cross section by a factor of 270 at 1~eV (Sec.~2.2),
absorption dominates the interaction: a neutron captured by the
nucleus triggers the $(n,p)$ reaction.
We therefore assign the $-N_A/r^4$ form to the imaginary (absorptive)
part as a leading-order approximation and set
\begin{equation}
  V_{\rm real}(r) = 0,\quad
  V_{\rm imag}(r) = -\frac{N_A}{r^4},
\end{equation}
with $N_A=2.178\times10^{-71}$~J\,m$^4$ determined in Sec.~3.2.
Setting $V_{\rm real}=0$ is justified a posteriori by the simulation
results in Sec.~4.2: the elastic scattering cross section, which
arises entirely from diffraction by the imaginary potential, is
reproduced within 4.3\% of the JENDL-5 value, with the residual
overestimate attributable to the less efficient wave diffraction
around the target in two-dimensional geometry, as discussed in
Sec.~4.2.
 
The equation was integrated using the split-operator fast Fourier transform (FFT)
method on a two-dimensional $(x,z)$ Cartesian grid.
The $^3$He nucleus was placed at the origin; the neutron wave packet was
initialized as a Gaussian envelope modulating a plane wave propagating in the
$+z$ direction, with the packet center at $z_0<0$.
Absorbing boundary layers were applied at all grid edges to suppress spurious
reflections.
 
Calculations were performed at three neutron energies: 250~eV, 500~eV, and
1~keV.
The computational domain and grid were scaled with wavelength so that each
calculation samples the same physical geometry relative to the neutron de~Broglie
wavelength.
At 1~keV the wavelength is half that at 250~eV, so the domain edge length was
halved and the time step reduced by a factor of four, keeping the number of
wavelengths across the domain constant.
The grid density near the target was adjusted across energies to maintain a
consistent sampling of the $-N_A/r^4$ potential at the nearest grid point, which
otherwise would vary as $r^{-4}$ and introduce energy-dependent truncation errors.
Table~\ref{tab:sim} lists the computational parameters for each energy.
 
The neutron wave packet is normalized so that the total domain contains exactly
one incident neutron.
The scattering and absorption probabilities in Table~\ref{tab:sim} are therefore
dimensionless ratios; multiplying each by the $x$-extent of the domain gives the
corresponding cross-section length in fm, which is the two-dimensional analogue
of the reaction cross section in three dimensions.
 
Non-uniform grids and cylindrical coordinates were considered but found to
introduce spurious internal reflections at grid boundaries; cylindrical
coordinates in particular produced unphysical absorption at the $x=0$ singularity
with compensating emission at the outer boundary.
A uniform Cartesian grid was therefore adopted throughout, at the cost of higher
memory requirements for low-energy calculations.
 
\begin{table}[htbp]
  \caption{Computational parameters and results for the two-dimensional
    time-dependent Schr\"odinger equation simulation of a neutron incident on
    $^3$He.
    The nuclear potential coefficient $N_A=2.178\times10^{-71}$~J\,m$^4$ is the same
    for all three energies.
    Cross-section lengths are the 2D analogue of cross sections; multiplying by
    the $x$-extent of the domain gives the effective cross-section length in fm.
    zs: zeptosecond ($10^{-21}$~s).}
  \label{tab:sim}
  \centering
  \begin{tabular}{lccc}
    \toprule
    Parameter & 250~eV & 500~eV & 1~keV \\
    \midrule
    Wavelength $\lambda$ [fm]            & 1810         & 1280          & 905        \\
    Wave packet width $\sigma_0$ [fm]    & 2000         & 1414         & 1000          \\
    Domain $[x\times z,\;\text{nm}]$     & $40\times20$ & $28.28\times14.14$ & $20\times10$  \\
    Grid $N_x\times N_z$                 & $2293\times1380$ & $1929\times1160$ & $1621\times976$ \\
    Initial position $z_0$ [nm]          & $-5$         & $-2.5$       & $-2.5$      \\
    Total time [zs]                      & 48000        & 24000        & 12000         \\
    Time step $dt$ [zs]                  & 2            & 1          & 0.5        \\
    Scattering probability               & $1.351\times10^{-3}$ & $1.993\times10^{-3}$ & $2.766\times10^{-3}$ \\
    Scattering cross-section length [fm] & 54.04        & 56.36        & 55.32        \\
    Absorption probability               & $4.179\times10^{-3}$ & $4.386\times10^{-3}$ & $4.343\times10^{-3}$ \\
    Absorption cross-section length [fm] & 167.16       & 123.95        & 86.86        \\
    \bottomrule
  \end{tabular}
\end{table}

\subsection{Results and Discussion}
 
At lower neutron energies the de~Broglie wavelength increases,
requiring a proportionally larger computational domain; 250~eV
was therefore chosen as the lower energy limit of the present
calculations.

Figure~\ref{fig:wavepacket} shows the probability density $|\psi|^2$ and the
scattered wave intensity $|\psi_{\rm sc}|^2$ at the final time step for the 500~eV
calculation.
The neutron wave propagates upward through the frame, and the distortion of the
probability density around the origin reflects the action of the $-N_A/r^4$
potential.
The 250~eV and 1~keV cases yield geometrically similar patterns when scaled to
the respective wavelengths and are not shown separately.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.85\textwidth]{fig9_n-He3.png}
  \caption{Neutron probability density $|\psi|^2$ (top) and scattered wave
    intensity $|\psi_{\rm sc}|^2$ (bottom) at the final time
    $t_{\rm max}=24000$~zs for a 500~eV neutron incident on $^3$He.
    The $^3$He nucleus is located at the origin.
    The neutron wave propagates in the $+z$ direction (bottom to top).}
  \label{fig:wavepacket}
\end{figure}

\begin{table}[htbp]
  \caption{Comparison of simulation cross-section lengths with JENDL-5
    data~\protect\cite{JENDL5}
     (values in parentheses are interpolated from nearby data). Ratios are normalized to 1 at 500 eV.}
  \label{tab:compare}
  \centering
  \begin{tabular}{lcccccc}
    \toprule
     & 100~eV & 200~eV & 250~eV & 275~eV & 500~eV & 1~keV \\
    \midrule
    Elastic [b], JENDL-5       & 3.107 & - & (3.096)  & - & (3.079) & 3.045 \\
    Scattering cross-section length [fm]  & - & - & 54.04  & - & 56.36 & 55.32 \\
    Ratio (sim/JENDL-5)        & - & - & 1.048 & - & 1.000 & 1.007 \\
    $^3$He$(n,p)$ [b], JENDL-5 & - & 61.59 & (55.45) & 52.75 & 39.60 & 28.62 \\
    Absorption cross-section length [fm]  & - & - & 167.16 & - & 123.95 & 86.86 \\
    Ratio (sim/JENDL-5)        & - & - & 1.038 & - & 1.000 & 1.031 \\
    \bottomrule
  \end{tabular}
\end{table}

The key results from Table~\ref{tab:sim} are as follows.
The scattering cross-section length is approximately constant across the three
energies (54.04, 56.36, and 55.32~fm at 250~eV, 500~eV, and 1~keV respectively),
reproducing the energy-independent elastic scattering observed in the JENDL-5
data.
The absorption cross-section length decreases with increasing energy (167.16, 123.95,
and 86.86~fm), consistent with the $1/v$ law: a factor of~4 increase in energy
yields a factor of~2 decrease in cross-section length, as required by
$\sigma\propto E_0^{-1/2}$.

Table~\ref{tab:compare} compares the simulation results with
JENDL-5 values~\cite{JENDL5}, normalized to unity at 500~eV.
The simulation reproduces the elastic scattering cross section
within 4.8\% and the reaction cross section within 3.8\%.

At 1~keV the simulation scattering and absorption cross-section
lengths approach each other (55.32~fm and 86.86~fm respectively),
whereas the JENDL-5 data maintain a ratio of approximately 7
at the same energy.
This convergence is a consequence of the two-dimensional geometry:
in 2D, wave diffraction around the target is less efficient than
in 3D, so the scattering probability is overestimated relative
to the absorption probability.
As a result, the present simulation overestimates the scattering
cross section even though $V_{\rm real}(r)=0$, because the
deflection of the neutron wave by the imaginary potential alone
is amplified by the 2D geometry.
Accurately reproducing the $1/v$ linearity and the energy
independence of elastic scattering over a wider energy range
would require three-dimensional simulations demanding several
orders of magnitude greater computational resources.

Two conclusions follow from these results.
First, the $-N_A/r^4$ potential simultaneously reproduces both
cross sections within a single quantum mechanical framework, with
no additional free parameters beyond $N_A$ determined in Sec.~3.2.
Second, the classical derivation of Sec.~3.1 is supported
quantum mechanically: the potential inferred from the $1/v$ law
through a classical argument produces consistent results when
inserted into the Schr\"odinger equation.

%% -------------------------------------------------------------------
\section{Limitations of existing theoretical frameworks}
%% -------------------------------------------------------------------

The derivation and quantum mechanical validation of the
$-N_A/r^4$ potential in Secs.~3 and~4 allow us to examine
the existing theoretical frameworks in a new light.
We note that the foundational components of nuclear theory
are not derived from first principles but adopted as working
hypotheses whose validity is established post hoc by their
numerical success.
Yukawa's meson-exchange theory~\cite{Yukawa1935} postulates
the $\pi$-meson and its coupling to the nucleon field; the
resulting potential is an effective description, not derived
from QCD.
Chiral effective field theory~\cite{Machleidt2011} introduces
low-energy constants fitted to nucleon-nucleon scattering data
rather than calculated from the QCD Lagrangian.
The Breit-Wigner formula~\cite{BreitWigner1936} parameterizes
the cross section near an isolated resonance but provides no
account of the interactions that generate the resonance.
The spin-dependent scattering lengths $a_t$ (triplet,
spin-parallel, $S=1$) and $a_s$ (singlet, spin-antiparallel,
$S=0$) are back-calculated from measured cross sections,
constituting a fit rather than a prediction.
Lane-Lynn theory~\cite{Lane1959,LaneLynn1960} leaves the
non-resonant background term explicitly unspecified.
We examine below how each framework fares when confronted
with the JENDL-5 data presented in Sec.~2.

\subsection{Breit-Wigner theory and the implicit nature of
the $1/v$ law}

The Breit-Wigner single-level resonance
formula~\cite{BreitWigner1936} places the description of
neutron cross sections on a formal mathematical footing.
Near an isolated resonance at energy $E_r$, the reaction
cross section takes the form
\begin{equation}
  \sigma_R(E) = \pi\bar{\lambda}^2\,g\,
    \frac{\Gamma_n\Gamma_r}{(E-E_r)^2+(\Gamma/2)^2},
  \label{eq:BW}
\end{equation}
where $\bar{\lambda}$ is the reduced de~Broglie wavelength;
$\Gamma_n\propto\sqrt{E}$ is the neutron partial width,
representing the decay width for the compound
nucleus to decay back into the neutron channel;
$\Gamma_r$ is the reaction partial width, representing
decay into the reaction channel (capture, emission, etc.);
$\Gamma = \Gamma_n + \Gamma_r + \cdots$ is the total width,
equal to the sum over all open decay channels; and $g$ is
the statistical spin factor.

In the limit $E\ll E_r$ (i.e., at energies far below the
resonance, including the thermal region $E\approx 0$),
Eq.~(\ref{eq:BW}) reduces to
$\sigma_R \propto \bar{\lambda}^2\,\Gamma_n/E_r^2
  \propto (1/E)(\sqrt{E}/E_r^2)
  = 1/(\sqrt{E}\,E_r^2)
  \propto 1/\sqrt{E}$,
reproducing the $1/v$ law.
The $1/v$ behavior thus appears as the low-energy tail of a
resonance, entering through the neutron partial width
$\Gamma_n\propto k\propto\sqrt{E}$, which is equivalent to
the penetration factor $P_c\propto k$ in the $s$-wave channel.

This derivation, however, presupposes the existence of a
resonance at some energy $E_r > 0$: the quantities $\Gamma_n$,
$\Gamma_r$, and $\Gamma$ are properties of the compound nucleus
formed at the resonance energy and have no meaning in the
absence of a resonance.
The $1/v$ law appears in Breit-Wigner theory only as the tail
of such a resonance at $E \approx 0$, not as an independent
phenomenon with its own physical origin.
As shown in Sec.~2.2, JENDL-5 lists no resonance parameters
in MF=2 for $^3$He (Fig.~\ref{fig:H3He3}(b)), $^6$Li
(Fig.~\ref{fig:LiB}(a)), $^{10}$B (Fig.~\ref{fig:LiB}(b)),
or $^7$Be (Fig.~\ref{fig:CdBe}(b)) --- all of which exhibit
clear $1/v$ reaction cross sections --- so the theory
provides no basis for the observed behavior in these nuclides.

\subsection{Lane-Lynn theory and the absence of resonance
parameters in JENDL-5}

Lane and Lynn extended the Breit-Wigner framework into a
comprehensive theory of direct and resonance contributions
to neutron capture~\cite{Lane1959,LaneLynn1960}.
The $1/v$ law enters Lane-Lynn theory through the same
penetration factor $P_c\propto k$; no $r^{-4}$ term appears
in the external region potential, and the $1/v$ law is not
explicitly derived in either of the original papers.

The most direct evidence comes from JENDL-5 itself.
For $^3$He, $^6$Li, $^{10}$B, and $^7$Be, MF=2 contains
no resonance parameters; only scattering radii (MT=151)
are listed.
JENDL-5 states explicitly for $^6$Li that the capture cross
section below 100~keV is obtained by normalizing a $1/v$
curve to thermal data~\cite{JENDL5}, confirming that no
resonance-based description was found adequate for these
nuclides.
The contrast with $^{112}$Cd is instructive
(Fig.~\ref{fig:CdBe}(a)): even for a nuclide where
resonance parameters do exist in MF=2, JENDL-5 reports
that ``a negative resonance was added so as to reproduce
the thermal capture and scattering cross sections given
by Mughabghab et al.''~\cite{JENDL5}, meaning that a
physically unobservable resonance at negative energy must
be artificially introduced to recover the $1/v$ behavior
in the thermal region.
This demonstrates that the resonance framework cannot
account for the $1/v$ law even in principle at thermal
energies.
The $-N_A/r^4$ potential derived in Sec.~3 provides the
non-resonant physical basis that these frameworks lack.

\subsection{Conventional explanation of elastic scattering
and its limitations}

The standard treatment of low-energy neutron-proton elastic
scattering invokes the spin dependence of the nuclear force.
The deuteron ($^2$H), the only bound state of the
neutron-proton system, exists in the spin-triplet
configuration ($S=1$).
In neutron-proton scattering, both the spin-triplet ($S=1$,
statistical weight $2S+1=3$) and spin-singlet ($S=0$,
statistical weight~1) channels contribute, giving
\begin{equation}
  \sigma_E = 3\pi a_t^2 + \pi a_s^2,
  \label{eq:scatlen}
\end{equation}
where $a_t$ is the triplet scattering length (spin-parallel,
$S=1$) and $a_s$ is the singlet scattering length
(spin-antiparallel, $S=0$).
This treatment faces three difficulties.

First, $a_s \approx \pm 23.7$~fm is not independently
determined but back-calculated from the measured cross
section using Eq.~(\ref{eq:scatlen}); agreement with
experiment is guaranteed by construction rather than
achieved as a theoretical prediction.

Second, at thermal energies the neutron de~Broglie
wavelength (${\sim}1.81\times10^5$~fm) exceeds the nuclear
force range (${\sim}2$~fm) by five orders of magnitude.
Classical intuition would suggest that the cross-sectional
area swept by the neutron wave
(${\sim}\lambda^2\sim 10^{10}$~fm$^2$) vastly exceeds the
area within which the short-range force acts
(${\sim}10$~fm$^2$), so the cross section would be expected
to decrease with increasing wavelength.
Quantum-mechanical $s$-wave theory predicts
$\sigma = 4\pi a^2$, an energy-independent cross section
that formally matches the observations of
Figs.~\ref{fig:Hiso12} and~\ref{fig:H3He3}(a), but this
agreement is achieved by fitting $a$ to the measured cross
section rather than deriving it from the underlying nuclear
force.
It provides no explanation for why the singlet scattering
length $a_s \approx -23.7$~fm is approximately ten times 
the force range (${\sim}2.5$~fm) in magnitude, nor for why
the elastic scattering cross section of $^7$Be reaches
384~b, corresponding to an effective interaction radius of
${\sim}110$~fm --- far beyond any scale set by the
short-range potential.

Third, the treatment requires separate calibration for
each nuclide and provides no unified predictive framework.
Even within the hydrogen isotopes, the elastic scattering
cross sections decrease from 20.4~b ($^1$H) to 3.39~b
($^2$H) to 1.66~b ($^3$H) as the number of nucleons
increases (Figs.~\ref{fig:Hiso12}
and~\ref{fig:H3He3}(a)), an inverse correlation with
nuclear size that cannot be explained by simple geometric
neutron-nucleus elastic scattering.
This framework has persisted for nearly a century without
a predictive first-principles derivation of these values
from the short-range nuclear force.
The energy independence of elastic scattering cross sections
in heavier nuclides has been assumed to follow by analogy,
without independent theoretical justification.

The $-N_A/r^4$ potential accounts for all three of these
observations within a single framework, as demonstrated
in Secs.~3 and~4.

%% -------------------------------------------------------------------
\section{Discussion}
%% -------------------------------------------------------------------

\subsection{Complementary relationship with Yukawa theory}

The $-N_A/r^4$ potential does not contradict Yukawa's meson exchange
theory~\cite{Yukawa1935} but occupies a distinct and complementary role.
Yukawa-type potentials of the form $V\propto\exp(-r/r_0)/r$ describe
the strong interaction through $\pi$-meson exchange and dominate at
short distances ($r\lesssim 2$~fm), accounting for nucleon binding and
nuclear structure.
At longer distances ($r\gtrsim 2$~fm), the Yukawa potential decays
exponentially and becomes negligible, while the $-N_A/r^4$ potential
decays only as a power law and continues to influence low-energy
neutrons.
The two potentials together provide a more complete description of
neutron-nucleus interactions than either alone: Yukawa theory governs
the short-range binding regime, and the $-N_A/r^4$ potential governs
the long-range scattering regime accessed by thermal and epithermal
neutrons.
In this sense the present work extends, rather than replaces, the
Lane-Lynn framework~\cite{Lane1959,LaneLynn1960} by identifying the
physical origin of the non-resonant background term that Lane-Lynn
theory leaves unspecified.

The quantum mechanical validity of the $-N_A/r^4$ potential in this
long-range regime is established directly by the two-dimensional
time-dependent Schr\"odinger equation simulations of Sec.~4, which
reproduce both the $1/v$ reaction law and the energy-independent
elastic scattering cross section without invoking any short-range
tunneling mechanism.
A full quantum mechanical treatment of the matching between the
short-range Yukawa-dominated core and the long-range $-N_A/r^4$ tail
--- for example, by matching interior and exterior wavefunctions in
the manner of effective-range theory --- is left for future work.

\subsection{Correspondence with the optical potential model}

The standard optical potential~\cite{Feshbach1954}
$V_{\rm opt}=V_{\rm real}(r)+iW(r)$ is widely used in nuclear
reaction theory precisely because a complex potential can
simultaneously describe both elastic scattering (governed by the
real part) and absorption into reaction channels (governed by the
imaginary part) within a single Schr\"odinger equation.
The present work fits naturally into this framework, with both parts
sharing the $r^{-4}$ radial dependence:
\begin{equation}
  V(r) = -\frac{N_A^{\rm real}}{r^4} - i\,\frac{N_A^{\rm imag}}{r^4}.
  \label{eq:optical}
\end{equation}

The relative magnitudes of $N_A^{\rm real}$ and $N_A^{\rm imag}$ are
expected to vary with nuclear species, reflecting the different balance
between elastic scattering and reaction for each nuclide.
For $^3$He in the two-dimensional simulation of Sec.~4, we set
$V_{\rm real}=0$ because the 2D geometry overestimates scattering;
a full three-dimensional calculation is expected to require
$N_A^{\rm real}\neq 0$ for $^3$He as well.
For $^7$Be, whose elastic scattering cross section of 384~b is
anomalously large, $N_A^{\rm real}$ is expected to be substantial.
At the other extreme, $^1$H and $^2$H undergo only radiative capture
with neutrons ($^1$H$+n\to{}^2$H$+\gamma$, $^2$H$+n\to{}^3$H$+\gamma$),
so $N_A^{\rm imag}$ is very small and the potential is predominantly
real.
For $^3$H, which undergoes neither nucleon emission nor radiative
capture with neutrons, $V_{\rm imag}\approx 0$; the elastic scattering
cross section suggests a repulsive real component, possibly arising
from the positive potential associated with the mass increase that
would result from neutron capture forming $^4$H.
The identity of form between real and imaginary parts implied by
Eq.~(\ref{eq:optical}) suggests that elastic scattering and absorption
share the same underlying long-range mechanism, with the proportion
between the two channels determined by the nuclear structure of the
target.

\subsection{Physical origin of the $r^{-4}$ form}

The $r^{-4}$ dependence is reminiscent of the charge-induced dipole interaction,
in which a point charge polarizes a neutral particle and attracts it with a potential
$V \propto -\alpha_d/r^4$, where $\alpha_d$ is the electric polarizability.
A nucleus of charge $Ze$ (where $Z$ is the atomic number and $e$ the elementary charge)
could in principle induce an electric dipole moment in an approaching neutron,
producing an attractive $r^{-4}$ potential.
 
To quantify this analogy, we compare the observed coefficient $N_A$ with the prediction
of the electrostatic charge-induced dipole mechanism.
The charge-induced dipole potential takes the form
\begin{equation}
  V_{\rm dipole}(r)
  = -\frac{\alpha_n (Ze)^2}{2(4\pi\varepsilon_0)\,r^4}
  \equiv -\frac{C_{\rm EM}}{r^4},
  \label{eq:dipole}
\end{equation}
where $\alpha_n$ is the electric polarizability of the neutron (in SI units, m$^3$).
Using the experimentally measured value
$\alpha_n \approx 1.16 \times 10^{-3}$~fm$^3 = 1.16 \times 10^{-48}$~m$^3$~\cite{Schmiedmayer1988}
and $Z = 2$ for $^3$He, the electrostatic coefficient evaluates to
\begin{equation}
  C_{\rm EM}
  = \frac{\alpha_n (Ze)^2}{2(4\pi\varepsilon_0)}
  \approx 5.3 \times 10^{-76}\ \text{J\,m}^4.
  \label{eq:CEM}
\end{equation}
This is approximately $4.1 \times 10^4$ times smaller than the observed value
$N_A = 2.178 \times 10^{-71}$~J\,m$^4$ for $^3$He,
indicating that a purely electromagnetic interpretation of the $-N_A/r^4$ potential
is not viable.
 
Further evidence against a simple electrostatic origin comes from the hydrogen isotope series.
If the potential arose purely from the proton charge, the coefficient $N_A$ would be the same
for $^1$H, $^2$H, and $^3$H since all three contain a single proton.
Instead, the elastic scattering cross sections of these isotopes decrease with increasing
neutron number, suggesting that $N_A$ depends on the neutron-to-proton ratio of the target
and not on the proton charge alone.
 
We propose that the physical origin of the $-N_A/r^4$ potential lies in
\textbf{np-binding-induced attraction}, analogous to nuclear polarization.
When an incident neutron approaches a nucleus containing protons with unoccupied
np-bonding degrees of freedom, the neutron--proton $T=0$ channel interaction induces
an effective long-range attractive potential of the $r^{-4}$ form, in analogy with
the polarization potential generated by charge-induced dipoles.
The energy scale of this interaction is set by the nuclear fusion Q-value $Q(n,\gamma)$,
which represents the large energy gain upon np bond formation in compound nucleus
creation --- a force substantially stronger than any electromagnetic polarization effect.
 
This interpretation is consistent with both observations above.
The decrease of the elastic cross section along
$^1$H $\to$ $^2$H $\to$ $^3$H reflects the progressive saturation of np-bonding
degrees of freedom with increasing neutron number:
in $^1$H the single proton is entirely free to bond with the incident neutron,
whereas in $^2$H the np bond is already occupied ($n_p^{\rm free} \approx 0$),
and in $^3$H the proton is increasingly screened by excess neutrons.
This trend persists despite the larger neutron capture Q-value of $^2$H (6.257~MeV)
compared to $^1$H (2.224~MeV), demonstrating that $n_p^{\rm free}$ is the dominant
factor when np-bonding degrees of freedom are saturated.
Similarly, the anomalously large $N_A$ of $^7$Be is consistent with the absence of
an established $\alpha$+$^3$He cluster structure, leaving all four protons available
for np bonding with the incident neutron.
 
We conjecture that $N_A$ scales as
\begin{equation}
  N_A \propto Q(n,\gamma)\times n_p^{\rm free},
  \label{eq:NA_conjecture}
\end{equation}
where $Q(n,\gamma)$ is the neutron capture Q-value representing the energy gain upon
compound nucleus formation, and $n_p^{\rm free}$ is the number of protons available
to form new np bonds with the incident neutron.
This conjecture accounts for the two-orders-of-magnitude difference between $N_A$
of $^7$Be and $^6$Li, as well as the near-equality of $N_A$ for $^3$He and $^{10}$B
despite their differing Q-values.
A rigorous microscopic derivation of this relation from isospin-dependent
nucleon--nucleon interactions remains an open question for future work.

\subsection{Physical picture of the $1/v$ law and energy-independent
scattering}

The classical two-body analysis of Sec.~3 provides an intuitive
picture of both phenomena.
Under a $-1/r^4$ potential, a neutron approaching with impact
parameter $r_0$ --- defined as the perpendicular distance between
the initial neutron trajectory and the target nucleus --- is captured
if $r_0$ falls within a critical distance determined by the balance
of the long-range attractive potential and the centrifugal barrier.
This critical distance scales as $r_0\propto(N_A/E_0)^{1/4}$, so
the reaction cross section $\sigma_R\propto r_0^2\propto E_0^{-1/2}$,
reproducing the $1/v$ law.

Elastic scattering arises from the deflection of neutrons by the same
$-N_A/r^4$ potential; the deflection is largest for neutrons whose
impact parameter is close to the capture radius $r_0$.
An intuitive picture of the energy independence of $\sigma_E$ is
provided by an analogy with wave reflection: if the elastic scattering
cross section is proportional to $\sigma_R/\lambda$, where $\lambda$
is the de~Broglie wavelength, then since both $\sigma_R\propto
E_0^{-1/2}$ and $\lambda\propto E_0^{-1/2}$, their ratio is
energy-independent.
This picture is consistent with the simulation results of Sec.~4,
which confirm the energy independence of $\sigma_E$ directly;
a quantitative derivation is left for future three-dimensional work.
This cancellation is a special property of the $r^{-4}$ potential
and does not hold for other power laws, providing an additional
self-consistency check on the uniqueness of $x=5$ established in
Sec.~3.1.

\subsection{Deviation from the $1/v$ law at higher energies}

The JENDL-5 data show that both the reaction cross section and the
elastic scattering cross section begin to deviate from their
respective low-energy behaviors above approximately 1~keV for $^3$He.
Both cross sections fall below their low-energy extrapolations,
the reaction cross section dropping below the $1/v$ trend and the
elastic cross section decreasing slightly from its constant value.
This is consistent with the onset of a competitive effect between
the two channels as the neutron energy increases, and with the
growing influence of the short-range nuclear force at higher energies.
Quantitative reproduction of this deviation requires a full
three-dimensional simulation that correctly captures the wave
propagation geometry, and is left for future work.

%% -------------------------------------------------------------------
\section{Conclusion}
%% -------------------------------------------------------------------

The JENDL-5 evaluated nuclear data library provides neutron cross
sections that display two notable features: the reaction cross
sections of $^3$He, $^6$Li, $^{10}$B, and $^7$Be follow the $1/v$
law over nine decades of energy with effective interaction radii of
173--1344~fm, and the elastic scattering cross sections of $^1$H,
$^2$H, and $^3$H remain essentially constant over the same energy
range.
This paper has proposed and validated a long-range nuclear potential
$V(r)=-N_A/r^4$ as a common origin for both observations.

The $r^{-4}$ form is derived from the $1/v$ law by a
contrapositive argument in classical mechanics:
requiring $\sigma\propto E_0^{-1/2}$ constrains the force
law to $F=-Nr^{-5}$, giving $V(r)=-N_A/r^4$.
A provisional value of the coefficient $N_A$ is obtained from a
single cross-section measurement, taking values between
$7.6\times10^{-73}$ and $2.8\times10^{-69}$~J\,m$^4$ across the
four nuclides studied.
Two-dimensional time-dependent Schr\"odinger equation simulations
with this potential simultaneously reproduce the $1/v$ energy
dependence of the reaction cross section to within 3.8\% and the
energy independence of the elastic scattering cross section to
within 4.8\%, validating the provisional $N_A$.
The residual discrepancy is attributable to the two-dimensional
geometry; three-dimensional calculations are expected to reduce it.

The examination of existing frameworks in Sec.~5 shows that
Breit-Wigner and Lane-Lynn theories can account for the $1/v$ law
only as the tail of a resonance, yet JENDL-5 lists no resonance
parameters for $^3$He, $^6$Li, $^{10}$B, or $^7$Be.
For $^{112}$Cd, where resonance parameters do exist, a
negative-energy resonance must be artificially introduced to
reproduce the thermal cross section, demonstrating that the
resonance framework cannot account for the $1/v$ law at thermal
energies even in principle.
The $-N_A/r^4$ potential provides the non-resonant physical basis
that these frameworks lack, extending rather than replacing the
Lane-Lynn framework by identifying the origin of its unspecified
background term.

The positive correlation between $\sigma_R$ and $\sigma_E$ across
$^3$He, $^6$Li, and $^{10}$B ($\sigma_R/\sigma_E = 202$--$283$)
emerges naturally from their shared $-N_A/r^4$ origin.
Within the optical potential framework~\cite{Feshbach1954}, the
real and imaginary parts share the same $r^{-4}$ dependence, with
their relative magnitudes determined by the nuclear structure of
the target.

The physical origin of the $r^{-4}$ form remains an open question,
but the empirical values of $N_A$ suggest a systematic trend.
A comparison with the charge-induced dipole mechanism shows that
the electromagnetic coefficient $C_{\rm EM}\approx
5.3\times10^{-76}$~J\,m$^4$ (Sec.~6.3) falls nearly about four and a half orders of magnitude below the observed $N_A=2.178\times10^{-71}$~J\,m$^4$
for $^3$He, ruling out a simple electrostatic origin.
We conjecture that $N_A$ arises from np-binding-induced attraction,
scaling as $N_A\propto Q(n,\gamma)\times n_p^{\rm free}$, where
$Q(n,\gamma)$ is the neutron capture Q-value and $n_p^{\rm free}$
is the number of protons available to form new np bonds with the
incident neutron (Sec.~6.3).
A rigorous microscopic derivation of this relation and
three-dimensional simulations across a wider range of nuclides
are the principal directions for future work.
Three-dimensional simulations are needed to establish quantitative
agreement with JENDL-5 data across a wider range of nuclides and
energies.

%% -------------------------------------------------------------------
\section*{Acknowledgements}
%% -------------------------------------------------------------------
The author thanks the Japan Atomic Energy Agency for providing the JENDL-5
database and associated online plotting tools, which formed the observational
foundation of this work.

%% -------------------------------------------------------------------
\section*{Funding}
%% -------------------------------------------------------------------
This research received no external funding.
The author conducted this work independently at Ooyama Power Inc.

%% -------------------------------------------------------------------
\appendix
%% -------------------------------------------------------------------

\section{Classical mechanics derivation of the $-N_A/r^4$ potential}
\label{app:classical}

We consider the two-body problem in the center-of-mass frame.
Let $M$ be the total mass of the neutron-nucleus system, $m$ the neutron mass, and
$(M-m)$ the target nucleus mass.
The separation between neutron and nucleus is $r$; the neutron's distance from the
center of mass is
\begin{equation}
  r_{\rm MS} = \frac{M-m}{M}\,r.
  \label{eq:rMS}
\end{equation}

Assuming the nuclear force takes the power-law form $F=-Nr^{-x}$ ($x>0$, $N>0$),
the corresponding potential vanishing at infinity is
\begin{equation}
  V(r) = -\frac{N}{x-1}\,r^{-(x-1)},\quad x\neq 1.
  \label{eq:Vapp}
\end{equation}

\paragraph{Circular orbit condition.}
For a neutron moving in a circular orbit at radius $r$ with speed $v_n$ in the
center-of-mass frame, the centrifugal acceleration equals the nuclear force per
unit mass:
\begin{equation}
  \frac{mv_n^2}{r_{\rm MS}} = Nr^{-x}.
  \label{eq:circ}
\end{equation}
Substituting Eq.~(\ref{eq:rMS}):
\begin{equation}
  v_n^2 = \frac{M-m}{Mm}\,Nr^{-(x-1)}.
  \label{eq:vn2}
\end{equation}

\paragraph{Energy and angular momentum conservation.}
The neutron kinetic energy in the laboratory frame is
$E_0=\tfrac{1}{2}m(v_{n0}^{\rm lab})^2$.
Transforming to the center-of-mass frame, the center-of-mass velocity is
$v_G^{\rm lab}=(m/M)v_{n0}^{\rm lab}$, so the neutron speed in the center-of-mass
frame is
\begin{equation}
  v_{n0} = \frac{M-m}{M}\,v_{n0}^{\rm lab},
  \label{eq:vcm}
\end{equation}
giving
\begin{equation}
  E_0 = \frac{mM^2}{2(M-m)^2}\,v_{n0}^2.
  \label{eq:E0}
\end{equation}

In a two-body central-force problem, the change in potential energy is shared
between the two bodies in inverse proportion to their masses.
The neutron's share is $(M-m)/M$ times the total potential change, so energy
conservation in the center-of-mass frame gives
\begin{equation}
  \tfrac{1}{2}mv_n^2 = \tfrac{1}{2}mv_{n0}^2 - \frac{M-m}{M}\,V(r).
  \label{eq:Econs}
\end{equation}
Substituting Eqs.~(\ref{eq:Vapp}) and (\ref{eq:vn2}) into Eq.~(\ref{eq:Econs})
and collecting terms in $r^{-(x-1)}$:
\begin{align}
\frac{1}{2}mv_{n0}^2
  &= \frac{M-m}{M}\,Nr^{-(x-1)}
     \left(\frac{1}{2}-\frac{1}{x-1}\right) \notag \\
  &= \frac{M-m}{M}\,Nr^{-(x-1)}
     \frac{x-3}{2(x-1)},
\label{eq:vn02a}
\end{align}
which yields ($x\neq 3$)
\begin{equation}
  v_{n0}^2 = \frac{x-3}{x-1}\cdot\frac{M-m}{Mm}\,Nr^{-(x-1)}.
  \label{eq:vn02}
\end{equation}
The case $x=3$ gives $v_{n0}=0$ regardless of $r$, which is physically
meaningless, and is therefore excluded.

\paragraph{Impact parameter and cross section.}
Defining $r_0$ as the impact parameter --- the perpendicular distance between the
initial neutron trajectory and the target nucleus --- angular momentum conservation
gives $mrv_n=mr_0v_{n0}$.
Squaring both sides and substituting Eqs.~(\ref{eq:vn2}) and~(\ref{eq:vn02}):
\begin{equation}
  r_0^2 = \frac{x-1}{x-3}\,r^2.
  \label{eq:r02}
\end{equation}
Substituting Eq.~(\ref{eq:vn02}) into Eq.~(\ref{eq:E0}) to eliminate $v_{n0}^2$
and solving for $r$ gives the critical radius:
\begin{equation}
  r = \left[\frac{2E_0(x-1)}{M(x-3)}\cdot\frac{Mm}{(M-m)N}\right]^{-1/(x-1)}.
  \label{eq:rc}
\end{equation}
Any neutron whose perpendicular distance from the target nucleus is less than
$r_0$ approaches the nucleus more closely than $r$ and is captured.
The reaction cross section is therefore $\sigma=\pi r_0^2$.
Substituting Eqs.~(\ref{eq:r02}) and~(\ref{eq:rc}):
\begin{equation}
  \sigma = \pi
    \left(\frac{x-1}{x-3}\right)^{(x-3)/(x-1)}
    \times
    \left(\frac{1}{2}\cdot\frac{M}{M-m}\cdot\frac{N}{E_0}\right)^{2/(x-1)}.
  \label{eq:sigmaFull}
\end{equation}

\paragraph{Determination of $x$ by contrapositive.}
Equation~(\ref{eq:sigmaFull}) shows that $\sigma\propto E_0^{-2/(x-1)}$.
The experimentally observed $1/v$ law requires $\sigma\propto E_0^{-1/2}$.
If $x\neq 5$, then $\sigma\propto E_0^{-2/(x-1)}\neq E_0^{-1/2}$, contradicting
the experimental fact.
Therefore, by contrapositive, $x=5$ is uniquely determined.
Eq.~(\ref{eq:sigmaFull}) reduces to
\begin{equation}
  \sigma = \pi\sqrt{\frac{4M}{M-m}\cdot\frac{N_A}{E_0}},
  \label{eq:sigmaFinal}
\end{equation}
where $N_A\equiv N/4$ is the long-range nuclear potential coefficient, and the
potential is $V(r)=-N_A/r^4$.\hfill$\square$

\section{Python code for 2D Schr\"odinger equation simulation}
\label{app:code}

The Python code used for the two-dimensional time-dependent Schr\"odinger equation
simulations described in Sec.~4 is provided as Supplemental Material.
The code implements the split-operator FFT method on a uniform Cartesian grid and
was executed on Google Colaboratory.

%% -------------------------------------------------------------------
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\end{document}
