= Two-target recoil mass constraint
{title2=$m=(M_2-RM_1)/(R-1)$}
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 gives
$$
u_i=\frac{2m}{m+M_i}v,\qquad
R=\frac{u_1}{u_2}=\frac{m+M_2}{m+M_1},\qquad
m=\frac{M_2-RM_1}{R-1}.
$$
If <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> give
$$
\frac{M_1}{m_p}=1,\qquad \frac{M_2}{m_p}=14,\qquad
R=\sqrt{\frac{4\times14}{224/225}}=\frac{15}{2},\qquad
\frac{m}{m_p}=\frac{14-15/2}{15/2-1}=1.
$$
The 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 $u_p=3.3\times10^7\,\mathrm{m/s}$ and $u_N=4.7\times10^6\,\mathrm{m/s}$, obtaining $m/m_p\simeq1.15$; it allowed roughly ten percent uncertainty in $u_N$. This was a rough <mass> comparison.<The Existence of a Neutron by James Chadwick>[Section 3, pages 698–699: endpoint equations and uncertainty]{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 gives
$$
\frac{dm}{dR}=\frac{M_1-M_2}{(R-1)^2}.
$$
Uncertain 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 endpoint
$$
T_{\gamma,\max}=\frac{2E_\gamma^2}{Mc^2+2E_\gamma},\qquad
E_\gamma=\frac{T+\sqrt{T^2+2TMc^2}}{2}.
$$
This 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 $m_pc^2=938\,\mathrm{MeV}$ and target rest <energies> $938\,\mathrm{MeV}$ and $14\times938\,\mathrm{MeV}$. To reproduce the stipulated <hydrogen> recoil endpoint $T=4\,\mathrm{MeV}$, the <photon> would need $E_\gamma\simeq45.36\,\mathrm{MeV}$. 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.
Back to article page
