How Do Einstein A And B Coefficients Relate To Emission Rates?

Spontaneous Emission (A-coefficient)

Transition: 2 → 1

Coefficient:A₂₁

Rate:

R_sp= A21 N2

Excited-state lifetime:

τ21=1A21

Stimulated Emission (B-coefficient)

Transition: 2 → 1

Coefficient:B₂₁

Rate (with spectral energy density ρ(ν)):

R_stim= B21 ⁢ρ(ν)⁢N2

Absorption (B-coefficient)

Transition: 1 → 2

Coefficient:B₁₂

Rate:

R_abs=B12⁢ρ(ν)⁢N1

Relations Between Einstein Coefficients (Thermal Equilibrium)

With level degeneracies g₁ and g₂:

g1 ⁢ B12 = g2 ⁢ B21 A21 B21 = 8⁢π⁢h⁢ ν3 c3

The spontaneous-to-stimulated ratio scales as ν³, hence spontaneous emission dominates at higher frequencies.