 Two-target recoil mass constraint

A single recoil endpoint leaves a tradeoff between incident speed and projectile mass. Two targets can remove the unknown incident speed. Assume both runs sample the same maximum projectile speed $v$ and that single elastic collisions reach the endpoint. For initially stationary target masses $M_1<M_2$, let their maximum recoil speeds be $u_1,u_2$. The maximum elastic recoil energy fraction calculation givesIf energies rather than speeds are inferred, use $R=\sqrt{T_{1,\max }{M}_2/(T_{2,\max }{M}_1)}$. The energy ratio itself is not the speed ratio.

For the stipulated $4\mathrm{MeV}$ example, hydrogen and nitrogen giveThe inferred mass then determines $v=u_1(m+M_1)/(2m)$. Changing the assumed mass to improve one recoil endpoint changes the prediction for the other: this is the constraining power of the comparison.

James Chadwick's June paper instead used approximate endpoints $u_p=3.3\times 10^{7}m/s$ and $u_N=4.7\times 10^{6}m/s$, obtaining $m/m_p\simeq 1.15$; it allowed roughly ten percent uncertainty in $u_N$. This was a rough mass comparison.^[ref]

The inverse calculation has useful built-in checks. A positive finite mass requires $1<R<M_2/M_1$. As $R$ approaches $1$, distinguishing large projectile masses becomes difficult. Differentiation givesUncertain recoil endpoints therefore produce an uncertain inferred mass; the collision model cannot turn imprecise range measurements into exact spectroscopy. If different target runs select different incident energy groups, the common-$v$ cancellation also fails.

For comparison, Compton scattering of a photon of energy $E_{\gamma }$ from a stationary free target of mass $M$ has the exact relativistic endpointThis follows by combining momentum conservation with the target's relativistic energy, maximizing transfer when the photon reverses direction. Unlike the massive-projectile calculation, this equation does not approximate the incident photon as a nonrelativistic particle.

As a separate, explicitly hypothetical test, take $m_p{c}^2=938\mathrm{MeV}$ and target rest energies $938\mathrm{MeV}$ and $14\times 938\mathrm{MeV}$. To reproduce the stipulated hydrogen recoil endpoint $T=4\mathrm{MeV}$, the photon would need $E_{\gamma }\simeq 45.36\mathrm{MeV}$. The same photon would give the heavier target at most $0.311\mathrm{MeV}$, whereas the stipulated massive projectile gave $0.996\mathrm{MeV}$. This is the structure of a two-target discrimination, not a transcription of a 1932 measurement. Actual radiation can contain more than one energy group or both photons and neutrons; a useful model must account for the observed endpoints and their origin together.