Radiation attenuation, laser gain and cavity modes
Introduced| English |
|---|
| half-value layer/hɑːf ˈvæljuː ˈleɪə/ |
| population inversion/ˌpɒpjʊˈleɪʃn ɪnˈvɜːʃn/ |
A decision before an answer
- A photon beam has no definite range, yet a heavy charged particle does; shielding calculations fail if the two are swapped.
- Your goal: Distinguish charged-particle stopping and range from exponential photon attenuation.
Choose the interaction model
- 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 is ln2/μ.
A narrow photon beam passes through three half-value layers. The transmitted fraction is:
Each half-value layer halves the intensity: (1/2)³=1/8.
Attenuate and count
- 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.
Continuous laser action requires population inversion because in thermal equilibrium:
Boltzmann populations give N2<N1, so absorption wins without pumping.
Invert the population
- 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.
For μ=0.2 /cm the half-value layer is ln2/0.2=3.47 cm, and three such layers pass 1/8. Under a constant 2 MeV/cm loss model a 5 MeV particle stops after 2.5 cm; a photon beam instead keeps an exponential tail. A 25 cm cavity has mode spacing c/(2L)=600 MHz.
A linear laser cavity of length 25 cm has adjacent longitudinal-mode spacing ____ MHz.
c/(2L)=3×10⁸/0.5=6×10⁸ Hz=600 MHz.
Resonate in the cavity
- 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.
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.
Which answer fits this case?
Distinguish charged-particle stopping and range from exponential photon attenuation
Broad-beam shielding transmits exactly the narrow-beam exponential intensity.
Scattered build-up adds transmission; the exponential assumes narrow-beam geometry.
Keep the distinctions
- half-value layer 半值层 — The material thickness that halves an exponentially attenuated beam, equal to ln2/μ.
- population inversion 粒子数反转 — A non-equilibrium state with more population in the upper laser level than the lower, enabling net stimulated emission.
- Distinguish charged-particle stopping and range from exponential photon attenuation.
- Compute attenuated intensity and detected counts with stated coefficients.
- Explain stimulated emission, population inversion and optical cavity resonance conditions.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.