= Bubble-chamber nucleation model
{title2=$r_*=2\sigma/\Delta p$}
Why can hot <liquid> wait for a <charged particle> rather than immediately turn into <vapor>? A small cavity costs <surface energy>. Consider an original macroscopic capillarity model at fixed <temperature> with <surface tension> $\sigma>0$, imposed <liquid> <pressure> $p_l$, and driving difference $\Delta p=p_{\rm sat}(T)-p_l>0$, where $p_{\rm sat}$ is the planar-interface <vapor pressure>. Neglect dissolved <gas>, interface thickness, <heat transfer> and curvature dependence of $\sigma$. The reversible formation-work approximation is
$$
\Delta G(r)=4\pi\sigma r^2-\frac{4\pi}{3}\Delta p\,r^3.
$$
The first term charges for surface area; the second rewards the new <vapor> <volume (thermodynamics)>. This is a macroscopic <free energy> model, not the <energy> that an actual detector must receive as localized <heat>. It shares the capillarity balance used in the <cloud-chamber expansion model>, but the new phase is now a bubble.
Differentiating gives
$$
\frac{d\Delta G}{dr}=4\pi r(2\sigma-\Delta p\,r),\qquad
r_* =\frac{2\sigma}{\Delta p},\qquad
\Delta G_* =\frac{16\pi\sigma^3}{3\Delta p^2}.
$$
At the <critical nucleus>, $d^2\Delta G/dr^2=-8\pi\sigma<0$. The barrier is a maximum: cavities smaller than $r_*$ tend to shrink, while larger ones can grow in this model. Doubling $\Delta p$ halves $r_*$ and quarters $\Delta G_*$ if $\sigma$ stays fixed. This explains the strong sensitivity to operating conditions without claiming a universal detector threshold.
Choose $\sigma=0.010\,\mathrm{J\,m^{-2}}$ and $\Delta p=2.0\times10^5\,\mathrm{Pa}$ solely for a teaching example. Then
$$
r_*=100\,\mathrm{nm},\qquad
\Delta G_*=4.189\times10^{-16}\,\mathrm J\simeq2.614\,\mathrm{keV}.
$$
For the same chosen $\sigma$ and twice the driving <pressure>, $r_*=50\,\mathrm{nm}$ and $\Delta G_*\simeq0.654\,\mathrm{keV}$. These are calculated formation-work scales, not measured bubble thresholds for <diethyl ether> or any other substance.
\Image[/bubble-chamber-critical-radius-model.png]
{title=<Surface energy> and bulk driving compete to set a bubble's <critical nucleus>}
{description=Original calculations for stipulated surface tension and pressure differences. The plotted formation work is neither measured data nor a complete radiation-response model.}
A simplified way to discuss localization is to stipulate deposited <energy> $E_{\rm loc}$ within a segment of <length> $\ell$ and a trial conversion fraction $0<\eta\leq1$. A toy criterion $\eta E_{\rm loc}\gtrsim\Delta G_*$ gives an <energy> per unit <length> scale $\Delta G_*/(\eta\ell)$. It is only a bookkeeping estimate: a real bubble also needs <latent heat>, must exchange <heat>, and is affected by the spatial pattern and time scale of <energy> deposition. Consequently neither a single <stopping power> value nor this barrier alone predicts <detection efficiency> or the number of bubbles. Two tracks with the same mean <stopping power> can distribute their <energy> differently on the scale of a <critical nucleus>.
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