0198506961.pdf

152 Doppler-free laser spectroscopy

In this section, we consider the Doppler effect on the absorption by a

gas where each atom absorbs radiation at frequencyω 0 in its rest frame,

(^2) Section 8.3 describes what happens i.e. whenω′=ω 0. (^2) Thus atoms moving with velocityvabsorb radiation
when the atoms absorb a range of fre-
quencies, given by the homogeneous
width, in addition to any Doppler
broadening. Absorption is considered
here because of its relevance to laser
spectroscopy, but Doppler broadening
of an emission line arises in the same
way—atoms emit atω 0 in their rest
frame and we see a frequency shift in
the laboratory.
whenδ=ω−ω 0 =kv,orequivalently
δ
ω 0

=

v

c

. (8.2)

In a gas the fraction of atoms with velocity in the rangevtov+dvis

f(v)dv=

√

M

π 2 kBT

exp

(

−

Mv^2

2 kBT

)

dv≡

1

u

√

π

exp

(

−

v^2

u^2

)

dv.(8.3)

Hereu=

√

2 kBT/Mis the most probable speed for atoms of massM

(^3) This can easily be shown by differen- at temperatureT. (^3) Relatingvto the frequency via eqn 8.2, we find that
tiating the Maxwell speed distribution
which is proportional tov^2 times the
velocity distribution, see Table 8.1.
the absorption has the Gaussian line shape function^4
(^4) Note that∫∞
−∞g(ω)dω=1.
gD(ω)=
c
uω 0

√

π

exp

{

−

c^2

u^2

(

ω−ω 0

ω 0

) 2 }

. (8.4)

The maximum value occurs atω=ω 0 and the function falls to half its

maximum value atω−ω 0 =δ 1 / 2 ,where

(

cδ 1 / 2

uω 0

) 2

=ln2. (8.5)

The Doppler-broadened line has a full width at half maximum (FWHM)

(^5) The simple estimate of the FWHM as of ∆ωD=2δ 1 / 2 given by 5
∼ 2 u/cthat leads to eqn 6.38 turns out
to be quite accurate. ∆ωD
ω 0

=2

√

ln 2

u

c

 1. 7

u

c

. (8.6)

Kinetic theory gives the most probable speed in a gas as

u= 2230 m s−^1 ×

√

T

300 K

×

1a.m.u.

M

. (8.7)

Table 8.1The characteristic velocities in a gas with a Maxwellian distribution of

speeds and in an effusive atomic beam;u=

√

2 kBT/M,whereTis the temperature

andMis the mass. The extra factor ofvin the distribution for a beam, as compared

to that of a gas, arises from the way that atoms effuse through a small hole of area

A. Atoms with speedvare incident on a surface of areaAat a rate ofN(v)vA/4,

whereN(v) is the number density of atoms with speeds in the rangevtov+dv—

faster atoms are more likely to pass through the hole. Integration overvleads to

the well-known kinetic theory resultNvA/4 for the flux that arrives at the surface,

whereNis the total number density. The mean speedvhas a value between the

most probable and the root-mean-square velocities.

Gas Beam

Distribution v^2 exp

(

−v^2 /u^2

)

v^3 exp

(

−v^2 /u^2

)

Most probablevu

√

3 / 2 u

Root-mean-square velocity

√

3 / 2 u

√

2 u