A student argues: 'A 2 MeV photon has more than enough energy to create an electron-positron pair (which requires only 1.022 MeV), so pair production can occur anywhere in free space.' This is wrong because:
A2 MeV is actually insufficient — pair production requires at least 4 MeV to account for the kinetic energy of the products
BA single photon in free space cannot simultaneously conserve both energy and momentum for the reaction — a nearby nucleus must absorb recoil momentum
CPhotons can only interact with matter at nuclear surfaces, not in open vacuum
DThe photon must have the correct wavelength (not just sufficient energy) to match the electron's de Broglie wavelength
The student's error is checking only energy conservation, not momentum conservation simultaneously. A photon has E = pc; an electron-positron pair at rest has E = 2m_e c² and p = 0. No single photon can supply E = 2m_e c² while also having zero momentum — if E > 0, then p = E/c > 0 as well. This is a kinematic impossibility, not an energy insufficiency. A nearby nucleus absorbs recoil momentum, allowing both conservation laws to be satisfied at once. Even a 100 MeV photon cannot pair-produce in free space for this reason.
Question 2 Multiple Choice
In a PET scanner, two photons are detected in coincidence in exactly opposite directions. This back-to-back emission is directly explained by:
AThe radiotracer emitting photon pairs during its radioactive decay
BMomentum conservation: in the center-of-mass frame of the annihilating electron-positron pair, total momentum is zero, requiring two photons of equal energy in opposite directions
CPET scanner design — detectors are placed opposite each other and only accept anti-coincident signals
DThe two photons carrying kinetic energy and rest-mass energy separately, so they travel in perpendicular directions
When an electron and positron annihilate (approximately at rest), total three-momentum is zero. The final state must also have zero total momentum. Two photons can achieve this only if they have equal energy and travel in exactly opposite directions (their momenta cancel). A single photon is forbidden by this same argument — it cannot have zero momentum if it has nonzero energy. Three or more photons are allowed but extremely rare. The back-to-back geometry is pure momentum conservation, which is what PET imaging exploits for precise three-dimensional localization.
Question 3 True / False
An electron and positron annihilating at rest could produce a single photon carrying 1.022 MeV, since this conserves total energy.
TTrue
FFalse
Answer: False
Energy is conserved in this hypothetical, but momentum is not. In the center-of-mass frame, the electron-positron system has zero total momentum. A single photon must carry momentum p = E/c ≠ 0 (since E = 1.022 MeV > 0). The final state (one photon) has nonzero momentum while the initial state has zero total momentum — a violation. Two photons traveling in opposite directions can each carry 0.511 MeV and have their momenta cancel, satisfying both conservation laws. Single-photon annihilation is kinematically forbidden by the same argument as single-photon pair production.
Question 4 True / False
The positron is the antiparticle of the electron, having the same mass but opposite electric charge.
TTrue
FFalse
Answer: True
The positron has the same mass as the electron (m_e ≈ 9.11 × 10⁻³¹ kg) and the same magnitude of charge (e ≈ 1.6 × 10⁻¹⁹ C) but positive charge rather than negative. This is the defining relationship between a particle and its antiparticle: identical mass, opposite charges (electric and otherwise). When they meet, all rest mass converts entirely to photon energy — 2m_e c² ≈ 1.022 MeV — making annihilation the most complete form of mass-energy conversion possible.
Question 5 Short Answer
Why does pair production require a nearby nucleus, even when the incoming photon has more than sufficient energy to create the electron-positron pair?
Think about your answer, then reveal below.
Model answer: Both energy AND momentum must be conserved simultaneously. A photon has E = pc, so its momentum is p = E/c. An electron-positron pair created at rest has total energy 2m_e c² but zero total momentum. A photon with E = 2m_e c² therefore has momentum p = 2m_e c ≠ 0 — it cannot produce a pair with zero total momentum. This is a kinematic impossibility independent of the photon's energy: even with 100 MeV, a single photon in free space cannot pair-produce. A nearby nucleus resolves this by absorbing the recoil momentum: it moves very slowly (its large mass means it takes little energy), allowing the energy and momentum ledgers to balance simultaneously. The nucleus is a silent participant — it is not consumed, but its presence is required.
The distinction between 'insufficient energy' and 'kinematic impossibility' is the key insight. No amount of energy makes single-photon pair production in free space possible. The nucleus is not a threshold requirement but a structural necessity arising from the simultaneous requirements of two conservation laws.