Physical Chemistry , 1st ed.

right frequency. It is dependent on the concentration of species in the ex-

cited state but independentof the photon density (). The rate of sponta-

neous emission is characterized by A, the Einstein coefficient of spontaneous

emission:

rate of spontaneous emission Achigher (15.29)

The total emission is the sum of the spontaneous and stimulated emission:

rate of total emissionAchigher B() chigher

(15.30)

[A B()]chigher

Under normal circumstances, the rate of emission is always greater than the

rate of absorption, so that most atomic and molecular systems are in their

lowest-energy states. The three interactions are illustrated in Figure 15.26.

By thinking of the electronic transition as a “vibration” of an electron,

Einstein derived the mathematical forms of the coefficients Aand Bas

A

(4

8





2

0

e

)

2

m

^2

ec

 3 (15.31)

B

(4



0 )

e

h

2

me

 (15.32)

where eis the charge on the electron in coulombs,is the frequency of the

transition in s^1 ,meis the mass of the electron in kg,his Planck’s constant,

and cis the speed of light in m/s. The constant 

0 is necessary to relate the units

of charge, coulombs, to SI units. The ratio A/Bis



A

B



8 

c

h

3

^3

 (15.33)

15.12 Lasers 551

© CORBIS
Figure 15.25 Albert Einstein laid down the
basic theory of lasers in 1917, over 40 years be-
fore they were developed.

h Eground

(c)

Eground  2 h

Eexcited Eexcited

Eground

(b)

Eground h

Eexcited Eexcited

h Eground

(a)

Eground

Eexcited Eexcited

Figure 15.26 (a) Stimulated absorption,

which defines Einstein’s coefficient B. (b) Spon-

taneous emission, which defines Einstein’s coeffi-

cient A. (c) Stimulated emission, which defines

Einstein’s coefficient B. In stimulated emission,

the two photons have the same wavelength and

phase, as indicated.