LM.3 · Radiation attenuation, laser gain and cavity modes

A photon beam has no definite range, yet a heavy charged particle does; shielding calculations fail if the two are swapped.

Prerequisites: 7, 17, 21.

• Distinguish charged-particle stopping and range from exponential photon attenuation
• Compute attenuated intensity and detected counts with stated coefficients
• Explain stimulated emission, population inversion 粒子数反转 and optical cavity resonance conditions

Heavy charged particles lose energy continuously through many small collisions, so they have an approximate range: under a constant-loss model a particle with initial energy E0 and loss rate dE/dx stops after E0/(dE/dx). Electrons straggle more and radiate. Photons instead interact in single events, so a narrow beam attenuates exponentially as I=I0 e^(−μx) with no definite maximum depth; the half-value layer 半值层 is ln2/μ.

Apply the exponential only to the stated narrow-beam geometry: μ=0.2 /cm gives a half-value layer of 3.47 cm and I/I0=e^(−2)≈0.135 after 10 cm. Multiply by detector efficiency for recorded counts: with μx=ln4 and ε=0.25 the recorded fraction is 0.25×1/4=1/16. Build-up from scattered photons makes broad-beam shielding transmit more than the narrow-beam exponential; state the geometry.

Stimulated emission produces a photon matching the stimulating photon in frequency, direction and phase, giving coherent amplification. It competes with absorption; net gain needs population inversion N2/g2>N1/g1 (N2>N1 for equal degeneracies). Positive-temperature equilibrium has N2/g2<N1/g1 by the Boltzmann factor, so a two-level system in equilibrium cannot lase continuously; practical lasers pump a third level or a metastable state.

A linear cavity of length L supports standing modes with L=mλ/2, i.e. frequencies ν_m=mc/(2L) with spacing c/(2L): for L=30 cm the spacing is 500 MHz. Gain must exceed the round-trip losses for oscillation. Interferometers such as Michelson use the same coherence: fringe counts track optical path changes, as in the gas-cell measurement.

A narrow beam has attenuation $I=I_0e^{-\mu x}$ and half-value layer $x_{1/2}=\ln2/\mu$.

Three half-value layers transmit 1/8. This model excludes scattered build-up into the detector. For a linear vacuum laser cavity, adjacent longitudinal modes are separated by $c/(2L)$. Net stimulated gain requires $N_2/g_2>N_1/g_1$; the simpler $N_2>N_1$ assumes equal degeneracies.

Assigning photons a definite range or charged particles a single exponential law, and forgetting that equilibrium two-level populations cannot invert. Check which interaction model the beam species requires.

half-value layer: The material thickness that halves an exponentially attenuated beam, equal to ln2/μ.

population inversion: A non-equilibrium state with more population in the upper laser level than the lower, enabling net stimulated emission.