= Emulsion range measurement
{title2=$R$}
An <emulsion range measurement> follows a stopping <charged particle> through the coating to estimate its complete <particle range>. A projected length on one image can underestimate the distance travelled. In an original geometry example, a straight segment with projected length $L_{xy}=240\,\mathrm{\mu m}$ and corrected depth change $\Delta z=180\,\mathrm{\mu m}$ has length
$$
L=\sqrt{L_{xy}^2+(\Delta z)^2}=300\,\mathrm{\mu m}.
$$
Using the projection alone would underestimate this length by $20\%$. A bent track requires a sum over reconstructed segments. These are stipulated coordinates, not measurements of a historical track.
Suppose <photographic processing> contracts the coating only in depth, with shrinkage ratio $s=t_{\rm developed}/t_{\rm exposed}$. Then an observed depth difference $\Delta z_{\rm developed}$ corresponds to $\Delta z=\Delta z_{\rm developed}/s$. For $s=0.60$, an observed $108\,\mathrm{\mu m}$ becomes the $180\,\mathrm{\mu m}$ used above. This correction assumes uniform contraction and unchanged lateral dimensions; it cannot repair a track that leaves the coating before stopping.
A simple continuous-loss model connects <particle range> to <kinetic energy> $T_0$ through <stopping power> $S(T)=-\mathrm dT/\mathrm dx>0$:
$$
R(T_0)=\int_0^{T_0}\frac{\mathrm dT}{S(T)}.
$$
This describes the mean trend for a specified particle and material. Fluctuating energy loss produces a spread of stopping distances, called <range straggling>. An exit point supplies only a lower bound on <particle range>. Consequently, a narrow distribution from fully contained tracks and one assembled by treating exits as stops answer different questions.
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