0198506961.pdf

126 The interaction of atoms with radiation

7.2 The EinsteinBcoefficients

In the previous section we found the effect of an electric fieldE 0 cos (ωt)

on the atom. To relate this to Einstein’s treatment of the interaction

with broadband radiation we consider what happens with radiation of

energy densityρ(ω) in the frequency intervalωtoω+dω.This produces

an electric field of amplitudeE 0 (ω)givenbyρ(ω)dω= 0 E 02 (ω)/2.

For this narrow (almost monochromatic) slice of the broad distribution

eqn 7.12 gives

|Ω|^2 =

∣

∣

∣

∣

eX 12 E 0 (ω)

∣

∣

∣

∣

2

=

e^2 |X 12 |^2


^2

2 ρ(ω)dω

 0

. (7.17)

Integration of eqn 7.15 over frequency gives the excitation probability

for the broadband radiation as

|c 2 (t)|^2 =

2 e^2 |X 12 |^2

 0 
^2

∫ω 0 +∆/ 2

ω 0 −∆/ 2

ρ(ω)

sin^2 {(ω 0 −ω)t/ 2 }

(ω 0 −ω)^2

dω. (7.18)

We integrate the squares of the amplitudes, rather than taking the square

of the total amplitude, since contributions at different frequencies do

(^6) As in optics experiments with broad- not interfere. (^6) The range of integration ∆ must be large compared to
band light, it is intensities that are
summed, e.g. the formation of white-
light fringes in the Michelson interfer-
ometer.
the extent of the sinc function, but this is easily fulfilled since as time
increases this function becomes sharply peaked atω 0. (In the limit
t→∞it becomes a Dirac delta function—see Loudon (2000).) Over
the small range aboutω 0 where the sinc function has an appreciable
value, a smooth function likeρ(ω) varies little, so we takeρ(ω 0 ) outside
the integral. A change of variable tox=(ω−ω 0 )t/2, as in eqn 7.16,
leads to
|c 2 (t)|^2 
2 e^2 |X 12 |^2
 0
^2
ρ(ω 0 )×
t
2
∫+φ
−φ
sin^2 x
x^2
dx. (7.19)
The integration has limits ofx=±φ=±∆t/ 4 πand the integral ap-
proximates closely to

∫∞

−∞x

− (^2) sin (^2) xdx=π. We make this assumption
of a long interaction to find the steady-state excitation rate for broad-
band radiation. The probability of transition from level 1 to 2 increases
linearly with time corresponding to a transition rate of

R 12 =

|c 2 (t)|^2

t

=

πe^2 |X 12 |^2

 0 
^2

ρ(ω 0 ). (7.20)

Comparison with the upward rateB 12 ρ(ω) in Einstein’s treatment of

radiation (eqn 1.25) shows that

B 12 =

πe^2 |D 12 |^2

3  0 
^2

, (7.21)

where|X 12 |^2 →|D 12 |^2 /3, andD 12 is the magnitude of the vector

D 12 =〈 1 |r| 2 〉≡

∫

ψ∗ 1 rψ 2 d^3 r. (7.22)