Interaction with External Fields
187
∆ω = 1/τ
(6.74)
The average lifetime τ and therefore the linewidth are related to the transition
probability.
If A is the transition probability, the number of particles (– dN) which undergo
transition in time dt is [see Eq. (1.80)]
dN = – AN (t) dt
(6.75)
which on integration gives
N(t) = N(0) e
–At
(6.76)
Now, the average lifetime of the particles is
τ =
( )
(
)
(0)
−
+∆
∑
i
i
i
i
N t
N t
t
t
N
=
0
∞
−
∫
At
A
te dt
(6.77)
=
1
A
so that the linewidth in Eq. (6.74) is equal to the transition probability A. For
most atomic systems which admit to electric dipole transitions, τ ~ 10
–8
s. This
gives rise to a spread of ∆ω ~ 10
8
s
–1
. For λ ~ 5000 Å, the corresponding spread
in the wavelength is
∆λ ~ 10
–4
Å
(6.78)
For some excited states which are stable against electroid dipole transitions,
e.g., the 2
2
S 1/2 state in the hydrogen atom, known as metastable states, the
lifetime is usually about 10
5
times larger, i.e., τ ~ 10
–3
s. Metastable states play
a very important role in lasers and masers.
There are other effects which also contribute to the observed linewidth.
One of them is due to Doppler effect. Since the atoms are moving around (thermal
motion), the observed radiation is Doppler shifted from the frequency ω 0
expected for atoms at rest. If the velocity of the particle makes an angle of
α with the line of observation, the observed frequency is
ω ~ 0 1
cos


ω
−
α




v
c
(6.79)
For v ~ 6000 m/s (corresponding to atomic hydrogen at about 1400 K), and
λ ~ 5000 Å, the Doppler shift is
∆ω ~ 7.5 × 10
10
rad/s.
∆λ ~ 0.1 Å
(6.80)
Another phenomenon that contributes to the linewidth is atomic collisions
which effectively change the lifetime of the excited states. The observed
linewidth ∆ω is the sum of the linewidths arising from the different effects.