= Mass and range of a nuclear exchange field
{title2=$\lambda=\hbar/(mc)$}
A free scalar <Klein-Gordon equation> with <rest mass> $m$ is
$$
\left(\frac1{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2+\frac{m^2c^2}{\hbar^2}\right)\phi=0.
$$
Here $c$ is the <speed of light> and $\hbar$ the reduced <Planck constant>. In the static limit, comparison with the <static Yukawa Green function> identifies
$$
\kappa=\frac{mc}{\hbar},\qquad \lambda=\kappa^{-1}=\frac{\hbar}{mc}=\frac{\hbar c}{mc^2}.
$$
This $\lambda$ is the reduced <Compton wavelength>, rather than $h/(mc)$; the two differ by $2\pi$. It is an exponential decay length of this exchange field, not the geometric size of a <nucleon> or an exact radius of a <bound state>. This article uses $\lambda$ for a length; the 1935 paper used that letter for its inverse.
For a deliberately rounded worked example, stipulate $\hbar c=197.3\,\mathrm{MeV\,fm}$ and $mc^2=140\,\mathrm{MeV}$. Then
$$
\lambda=\frac{197.3}{140}\,\mathrm{fm}=1.4092857\ldots\,\mathrm{fm}\approx1.41\,\mathrm{fm},\qquad \kappa\approx0.70958\,\mathrm{fm}^{-1}.
$$
A <femtometre> is $10^{-15}\,\mathrm m$. The chosen <rest energy> is a convenient <pion>-scale input; neither it nor the resulting <Yukawa potential> is asserted to be an exact measured nuclear interaction. Holding $\hbar c$ fixed, chosen <rest energies> $70$, $140$ and $280\,\mathrm{MeV}$ give lengths $2.819$, $1.409$ and $0.7046\,\mathrm{fm}$. Doubling the exchange <rest mass> halves the decay length.
\Image[/yukawa-exchange-mass-and-range-comparison.png]
{title=Heavier exchange fields have shorter Yukawa decay lengths}
{description=Original comparisons using stipulated $\hbar c=197.3\,\mathrm{MeV\,fm}$ and illustrative <rest energies> $70$, $140$ and $280\,\mathrm{MeV}$. Left: inverse relation between <rest energy> and $\lambda$. Right: the relative scalar <potential energy> magnitude $e^{-r/\lambda}$ after dividing out the common $1/r$ factor. No interaction strength or experimental <nuclear force> curve is fitted.}
Range alone does not guarantee binding: the <potential energy> must overcome the <kinetic energy> cost of localizing a <wavefunction>. The existing <exponential variational bound for the Yukawa potential> makes this precise. For the same toy $V$, with <reduced mass> $M$, choosing a normalized exponential <wavefunction> proportional to $e^{-\alpha r}$ with $\alpha=\kappa/2$ yields
$$
\langle H\rangle=\frac{\hbar^2\kappa^2}{8M}-\frac{A\kappa}{8}.
$$
For additional stipulated inputs $Mc^2=470\,\mathrm{MeV}$ and $A=100\,\mathrm{MeV\,fm}$, this gives $\langle H\rangle\approx-3.66\,\mathrm{MeV}$. A negative value proves a <bound state> in this scalar model by the <quantum variational principle>; it is an upper bound on its <ground-state energy>, not a prediction for the <deuteron>. A nonnegative trial value would not prove the absence of binding.
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