Two-body nuclear reaction kinematics

Assume nonrelativistic motion and a target initially at rest. The projectile kinetic energy splits into centre-of-mass translation plus the kinetic energy of relative motion. The target-to-total-mass ratio multiplies the laboratory beam energy to give the initial relative energy. Adding the channel Q-value gives the final relative energy, which must be nonnegative. Treat reaction mass differences in the rest-energy balance while neglecting their small effect on inertial mass sums only when that low-energy approximation is adequate.

In the final centre-of-mass frame, the two momenta are equal and opposite. Consequently the lighter product has greater speed and a larger share of kinetic energy. The reduced mass conveniently packages this partition into a single relative momentum. If a residual nucleus is left excited, lower the Q-value before calculating that momentum. A discrete set of final excitations can therefore produce distinct energy groups even with the same incident beam and nominal reaction channel.

For elastic scattering, the centre-of-mass velocity is constant and the projectile’s relative speed has the same magnitude before and after collision. Its laboratory exit velocity is the vector sum of the centre-of-mass velocity and the rotated relative velocity. Squaring that sum produces an angle-dependent laboratory energy. The cross term can add or subtract depending on direction, so a single centre-of-mass speed corresponds to a range of laboratory energies.

For unequal masses, the laboratory angle requires the direction of that vector sum, preferably using a two-argument arctangent so the correct quadrant is retained. Endothermic reactions near threshold can even have two allowed laboratory energies at one laboratory angle. Conservation laws determine such branches but not their relative intensities. Cross sections require the transition amplitude and, when transforming angular densities between frames, the appropriate solid-angle Jacobian rather than only a relabeling of the angle.

For a head-on elastic collision, solve the one-dimensional momentum and kinetic-energy equations together. A light projectile on a heavier stationary target can reverse direction while retaining part of its initial energy. For equal masses, a head-on collision transfers all kinetic energy to the target. In the infinitely heavy-target limit, the projectile speed is unchanged while its momentum direction can change, because a vanishing target speed can still carry finite recoil momentum.

These limits distinguish energy transfer from momentum transfer. A large scattering angle does not necessarily imply a large energy loss for a very heavy target. Conversely, the equal-mass case permits complete transfer even when no internal excitation occurs. In practice, reaction identification also needs isotope masses, excitation energies, energy loss, angular acceptance, and calibration. An ideal kinematic curve is an allowed locus for specified states, not a full detector response or proof that the scattering was elastic.