130_notes.dvi

As Γ→0 this line shape approaches a delta function,δ(ω−ω 0 ).

For the 2p to 1s transition in hydrogen, we’ve calculated a decay rate of 0. 6 × 109 per second. We
can compute the FWHM of the width of the photon line.

∆E= ̄hΓ =

(1. 05 × 10 −^27 erg sec)(0. 6 × 109 sec−^1 )

1. 602 × 10 −^12 erg/eV

≈ 0. 4 × 10 −^6 eV

Since the energy of the photon is about 10 eV, the width is about 10−^7 of the photon energy. Its
narrow but not enough for example make an atomic clock. Weaker transitions, like those from E2
or M1 will be relatively narrower, allowing use in precision systems.

29.11.1Other Phenomena Influencing Line Width

We have calculated the line shape due to the finite lifetime of a state. If we attempt to measure
line widths, other phenomena, both of a quantum and non-quantumnature, can play a role in the
observed line width. These are:

• Collision broadening,


• Doppler broadening, and


• Recoil.

Collision broadening occurs when excited atoms or molecules have a large probability to change state
when they collide with other atoms or molecules. If this is true, and it usually is, the mean time
to collision is an important consideration when we are assessing the lifetime of a state. If the mean
time between collisions is less than the lifetime, then the line-width will bedominated by collision
broadening.

An atom or molecule moving through a gas sweeps through a volume per second proportional to its
cross sectionσand velocity. The number of collisions it will have per second is then

Γc=Ncollision/sec=nvσ

wherenis the number density of molecules to collide with per unit volume. We canestimate the
velocity from the temperature.

1

2

mv^2 =

3

2

kT

vRMS=

√

3 kT

m

Γc=n

√

3 kT

m

σ

The width due to collision broadening increases with he pressure of the gas. It also depends on
temperature. This is basically a quantum mechanical effect broadening a state because the state
only exists for a short period of time.