The_Cambridge_Handbook_of_Physics_Formulas

8.8 Line radiation
173
8.8 Line radiation
Spectral line broadening
Natural
broadening a
Natural
half-width
Collision
broadening
Collision and
pressure
half-width c
Doppler
broadening
Doppler
half-width
I(ω)=
∆ω = 1 2τ
I(ω)=
(2πτ) −1
(2τ) −2 +(ω −ω 0 ) 2 (8.112)
(8.113)
(πτ c ) −1
(τ c ) −2 +(ω −ω 0 ) 2 (8.114)
∆ω = 1 τ c
= pπd 2 ( πmkT
16
( ) mc
2 1/2
I(ω)=
2kTω0 2π exp
[− mc2
2kT
) −1/2
(8.115)
(ω −ω 0 ) 2 ]
ω 2 0
(8.116)
∆ω = ω 0
( 2kT ln2
mc 2 ) 1/2
(8.117)
I(ω) normalised intensity b
τ lifetime of excited state
ω angular frequency (= 2πν)
∆ω
ω 0
τ c
p
d
m
k
T
c
half-width at half-power
centre frequency
mean time between
collisions
pressure
effective atomic diameter
gas particle mass
Boltzmann constant
temperature
speed of light
ω 0
a The transition probability per unit time for the state is = 1/τ. In the classical limit of a damped oscillator, the
e-folding time of the electric field is 2τ. Both the natural and collision profiles described here are Lorentzian.
b The intensity spectra are normalised so that ∫ I(ω)dω = 1, assuming ∆ω/ω 0 ≪ 1.
c The pressure-broadening relation combines Equations (5.78), (5.86) and (5.89) and assumes an otherwise perfect
gas of finite-sized atoms. More accurate expressions are considerably more complicated.
I(ω)
∆ω
Einstein coefficients a
Absorption R 12 = B 12 I ν n 1 (8.118)
Spontaneous
emission
Stimulated
emission
R 21 = A 21 n 2 (8.119)
R ′ 21 = B 21 I ν n 2 (8.120)
R ij transition rate, level i → j (m −3 s −1 )
B ij Einstein B coefficients
I ν specific intensity of radiation field
A 21 Einstein A coefficient
n i number density of atoms in quantum
level i (m −3 )
8
A 21
= 2hν3 g 1
h Planck constant
Coefficient B 12 c 2 (8.121)
g 2 ν frequency
ratios B 21
= g 1
c speed of light
(8.122)
B 12 g g 2 i degeneracy of ith level
a Note that the coefficients can also be defined in terms of spectral energy density, u ν =4πI ν /c rather than I ν .Inthis
case A 21
B = 8πhν3 g 1
12 c 3 g .SeealsoPopulation densities on page 116.
2