Einstein's A & B coefficients

Einstein's A & B coefficients

1. Three Processes Between Two Levels $E_1$ (lower) and $E_2$ (upper), $hν=E_2-E_1$

• Absorption: atom in $E_1$ + photon $hν$ → $E_2$; rate $∝$ energy density $ρ(ν)$:

• Spontaneous emission: atom in $E_2$ → $E_1$ + photon $hν$ randomly; independent of $ρ$:

• Stimulated emission: atom in $E_2$ + photon $hν$ → $E_1$ + clone photon (same $ν, phase, direction$); rate $∝ ρ$:

$N_1,N_2$ populations; $A_{21}, B_{12}, B_{21}$ are Einstein coefficients (intrinsic atomic properties, not dependent on $ρ$).

2. Einstein's Derivation (equilibrium with blackbody)

At thermal equilibrium, upward = downward:

with Boltzmann

and Planck

Comparing forms gives:

and

(If counting energy density per unit frequency $ρ_ν$; for per unit angular $ω$, factor differs by $2π$). This is the famous Einstein relation.

3. Meaning

• $A/B ∝ ν³$ ⇒ at optical frequencies $A≫Bρ$ (spontaneous dominates) → need population inversion and cavity to make stimulated dominate.
• At microwave (low $ν$), $A/B$ small ⇒ stimulated dominates more easily (maser first).
• $B_{12}=B_{21}$ (for $g_1=g_2$) means absorption and stimulated emission equally likely per atom — net amplification needs $N_2>N_1$.

4. Worked Implications

Equilibrium at high T? N2≈N1 then spontaneous >> stimulated for ν large.

Ratio stimulated/spontaneous = Bρ/A = 1/(e^{hν/kT}-1) = average photon number n̄ in mode.

Laser needs n̄ ≫1 (large ρ) and N2>N1.

5. Exam Points

• Define three processes with $A,B$ — 5 marks, very frequent.
• Derive $B_{12}=B_{21}$ and $A/B=8πhν³/c³$ — 7-10 marks, very frequent (the core derivation).
• Explain why $A∝ν³$ makes optical lasers harder than microwave — 3-5 marks, frequent.
• Give units: $A$ (s⁻¹), $B$ (m³/J·s²) depending on $ρ$ definition — 3 marks.