BioPHYSICAL chemistry

be a photon present in the initial state. The presence of this photon inter-

acts with an electron and induces the transition of the electron down to

the lower state. By conservation of energy,there are two photons of equal

energy in the final state.

The rate of stimulated emission,w′, can be written in a form very similar

to that of stimulated absorption:

w′=B′ρ (14.20)

where B′is thecoefficient for stimulated emission. In contrast, the rate of

stimulated emission,w′, does not depend upon the presence of the initial

photon so is given by:

w′=A (14.21)

where Ais a constant called theEinstein coefficient of spontaneous emission.

The o 9 erall rate of emission, W′, is given by the sum of the two rates

multiplied by the number of electrons in the upper state,N′:

W′=N′(A+B′ρ) (14.22)

298 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Derivation box 14.1

Relationship between the Einstein coefficient and electronic states

It is possible to relate the Einstein coefficients to the wavefunctions of electrons. The intensity

of the transition is determined by the coefficient B, which can be expressed as:

(db14.1)

where μfiis called thetransition dipole moment. The transition dipole moment can be calculated

directly from the wavefunctions determined by Schrödinger’s equation:

(db14.2)

where ψfandψiare thefinal andinitial wa 9 efunctions, respectively. The operator μis electric

dipole moment operator and is given by the product of the charge and the distance between

charges.

Sometimes, a related quantity called the dipole strength,Dfi, is used:

D 3 =⎣⎡∫ψμψ τfid⎦⎤ (db14.3)

2

μψμψτ 3 =∫ *fid

B=

μ

ε

3

Z

2

0

6 2