Physical Chemistry , 1st ed.

and the only varying factor in this ratio is , the frequency of the transition.

What this ratio says is that the greater the frequency of the transition, the larger

the chance that spontaneous emission, determined by A, will occur over stim-

ulated emission, which is measured by B.

Example 15.14

Lasers make use of stimulated emission. Using equation 15.33, suggest a rea-

son why red lasers are easy to engineer but blue lasers are more difficult.

Solution

The ratio in equation 15.33 has the third power of the frequency,, in the

numerator. Therefore, the higher the frequency, the higher the ratio of spon-

taneous emission to stimulated emission. An electronic transition that occurs

in the blue region of the spectrum, which has roughly twice (2) the fre-

quency of the red region of the spectrum, has a (2)^3 8 times greater chance

of decaying by spontaneous emission than by stimulated emission. If lasers

depend on stimulated emission, blue lasers are therefore correspondingly

more difficult to produce.

Example 15.15

Determine the ratio of spontaneous to stimulated emission for a transition

that occurs at the following wavelengths.

a.21.0 cm, a wavelength that has implications in astronomy

b.300.0 nm, which is in the middle UVB part of the spectrum.

c.Comment on the difference in the two ratios.

Solution

a.The frequency in s^1 must be determined first:





c



2.997

2

9

1



.0 c

1

m

08 m/s



10

1

0

m

cm

1.428  109 s^1

Substituting into the formula for the ratio A/B:



A

B



8 

c

h

3

^3



1.800  10 30 kg/(ms)

This is a very small ratio (whose units are a consequence of equation 15.33).

b.Again, we determine frequency:





c



2.997

3

9

00



.0

1

n

0

m

(^8) m/s


109

m

nm

9.993  1014 s^1

and substitute it into the formula for A/B:



A

B



8 

c

h

3

^3



6.168  10 13 kg/(ms)

c.Although still a small number, the second result is 17 orders of magnitude

larger than the first answer. This means that spontaneous emission is almost

8 (6.626  10  34 Js)(9.993  1014 s^1 )^3



(2.9979  108 m/s)^3

8 (6.626  10  34 Js)(1.428  109 s^1 )^3



(2.9979  108 m/s)^3

552 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure