Reaction Threshold Energy
For a non-relativistic two-body reaction a + A -> products with target A initially at rest and Q < 0, the exact threshold follows invariant energy conservation. In the usual small-|Q| approximation, K_th approximately -Q(1 + m_a/m_A).
Why this shows up in the exam
Finding minimum bombardment energy · Distinguishing Q value from laboratory threshold · Checking whether a reaction channel can open
Learn the idea
An endothermic reaction needs more projectile energy than |Q| because final momentum must also be carried. In the laboratory the target starts at rest, but the final center of mass cannot generally stop. Some incident energy must remain as unavoidable kinetic energy of the products, raising the threshold above the mass-energy deficit.
🧠 Memory hook: Negative Q plus recoil makes threshold larger than |Q|.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- K_th approximately -Q (1 + m_a/m_A) — non-relativistic threshold approximation for a stationary target and Q < 0
- K_th = 0 for a spontaneous exothermic channel — energy threshold only; Coulomb barriers may still suppress the rate
How to approach it
- 1Compute Q first
- 2Identify projectile and stationary target masses
- 3Apply the recoil factor only under the stated non-relativistic assumptions
Common slip-ups that cost marks
- •Setting threshold equal to |Q| for a stationary finite-mass target
- •Using the approximation for relativistic projectiles without checking
- •Confusing an energetic threshold with Coulomb-barrier probability
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
In hydrogen, an electron transitions from n = 2 to n = 1. Using E_n = -13.6/n^2 eV, find the emitted photon energy.
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