SPONTANEOUS EMISSION

is given by [quantum mechanics], says that,so far as problems involving light are concerned, the electrons behave as though they were held by springs.⁴

In fact, the electron oscillator does not correctly describeallaspects of the interaction of light with atoms, and in particular it does not describe some of the most important features of lasers. With some appropriate modifications, however, the electron oscillator model will allow us to proceed rather quickly and easily to a realistic theory of laser oper- ation, and to do so using mainly physical rather than mathematical aspects of quantum theory. In the Appendix we show that the electron oscillator model can be regarded as a good approximation to the quantum theory of the interaction of an atom with light.

electromagnetic energy is radiated is then dW

dt ¼ 1 4pe₀

e²

3c³[v⁴₀x²₀þv²₀v²₀]¼ 1 4pe₀

2e²v²₀ 3mc³

1

2mv²₀þ1 2mv²₀x²₀

¼ 1 4pe₀

2e²v²₀

3mc³ E, (3:3:5)

where we have recognized the quantity in brackets in the second equality as the oscillator energyW.

This radiation by an electron oscillator that has been “excited,” i.e., given a non- vanishingd²d/dt², corresponds to the spontaneous emission of radiation by an excited atom, which was mentioned briefly in Section 1.5. Since the frequency of the field radiated by the electron oscillator is the same as the oscillator frequencyn0¼v0/2p, we associate the electron oscillator with an atomic transition of frequencyn0(Fig. 3.4).

Thus, for an optical transition of wavelength l0¼500 nm, v0¼2pc/l0 3.8 10¹⁵ s²¹ and the rate of spontaneous emission predicted by the electron oscillator model is

1 4pe0

2e²v²₀ 3mc³

910⁷s¹: (3:3:6)

This is a reasonable estimate for spontaneous emission rates of atomic transitions at optical wavelengths. However, spontaneous emission rates are not a quadratic function of transition frequency as predicted by (3.3.6). The 2p – 1s transition of hydrogen at 121.6 nm, for instance, has a spontaneous emission rate 6.2610⁸s²¹, whereas for the 3s– 2ptransition at 656.3 nm the rate is 6.3110⁶s²¹(see Table 3.2). The ratio of these two rates is (6.2610⁸/6.3110⁶)100, whereas according to (3.3.6) this ratio should be (6563/1216)²30. To bring the classical radiation rate into numeri- cal agreement with the rate at which excited atoms jump spontaneously from an energy levelE2to a lower energy levelE1, withE22E1¼hn0, we multiply (3.3.6) by a factor that, to conform to a notational convention, we write as 3f. Thus, denoting the spon- taneous emission rate for the quantum jump from energy levelE2to energy levelE1

n₀

E₁ E₂

Figure 3.4 An atomic transition of frequency n₀¼v₀/2p¼(E₂2E₁)/h and wavelength l₀¼c/n0.

3.3 SPONTANEOUS EMISSION 75

byA21, we write

A21¼ 1 4pe₀

2e²v²₀

mc³ f: (3:3:7)

It is evident thatfmust have different numerical values for different atomic transitions and provides a measure of the “strength” of the transition.

The factorf, called theoscillator strength, was introduced before the development of quantum theory in order to bring the electron oscillator model into numerical agreement with various spectroscopic data. It remains useful as a measure of the “strength” of a transition; numerical values of oscillator strengths are tabulated in various handbooks.⁵ The quantum theoretical formula for the oscillator strength is derived in the Appendix.

A second modification of the classical oscillator model is required to describe spon- taneous radiative transitions: We must take into account that atoms can only be found in certain states with “allowed” energies. Thus, the spontaneous emission rateA21is the rate at which the numberN2of atoms in the upper state of energy E2decreases and the number N1 of atoms in the lower state of energy E1 correspondingly increases (Fig. 3.4). The changes in the “populations”N2andN1due to spontaneous emission are described by the rate equations

dN2

dt ¼ A21N₂ (3:3:8)

and

dN₁

dt ¼A21N2, (3:3:9) implying thatd(N₁þN₂)/dt¼0, i.e., the total number of atomsN₁þN₂in the upper and lower states of the transition stays the same.

As discussed in Section 3.6, most of the light around us is ultimately the result of spontaneous emission, and the phenomenon appears in many different contexts. The termluminescence, for instance, describes spontaneous emission from atoms or mole- cules excited by some means other than heating. If excitation occurs in an electric dis- charge such as a spark, the termelectroluminescenceis used. If the excited states are produced as a by-product of a chemical reaction, the emission is calledchemilumines- cence, or, if this occurs in a living organism (such as a firefly), bioluminescence.

Fluorescencerefers to spontaneous emission from an excited state produced by the absorption of light.Phosphorescencedescribes the situation in which the emission per- sists long after the exciting light is turned off and is associated with a metastable (long- lived) level, as illustrated in Fig. 3.5. Phosphorescent materials are used, for instance, in toy figurines that magically glow in the dark.

5A useful collection of atomic reference data is provided by the National Institute of Standards and Technology (NIST) and may be found on the Web as well as in a variety of published sources. See, for example, A. N. Cox, ed., Allen’s Astrophysical Quantities, 4th ed., AIP Press, New York, 2000, or W. L. Wiese, M. W. Smith, and B. M. Glennon,Atomic Transition Probabilities, U.S. Government Printing Office, Washington, D.C., 1966. A more recent and readily available compendium of useful data on atomic transitions of interest has been prepared by D. A. Steck at http://steck.us/alkalidata/.

In most situations an excited level has several or many spontaneous decay channels, so that the general case is somewhat more complex than our notationA21implies. For example, the solution of Eq. (3.3.8), the exponential decay law,

N2(t)¼N2(0)e^A21t, (3:3:10) indicates that the population of the upper level decays to zero with the characteristic time constantt_2¼1/A21. However, if level 2 has other decay channels open to it, they will obviously shorten the effective lifetime of level 2 and this expression fort₂ will be incomplete.

According to quantum theory the spontaneous radiative lifetime of levelnis deter- mined by the sum of the rates for all possible radiative channels:

A_n¼X

m

A_nm, (3:3:11)

A_n1 A_n2 A_n3

n

3

2

1

Figure 3.6 An atomic statenmay make spontaneous transitions to lower statesmwith ratesAnm. The total spontaneous decay rate of statenisAn¼SmAnm.

1 2 3 4

Phosphorescence (slow) Pump

Non-radiative decay (fast)

Figure 3.5 Model of phosphorescence. A molecule is pumped to level 4 by absorption of radiation, and then decays to level 3. Level 3 is metastable, i.e., it has a very small spontaneous emission rate. As a result the molecule continues to fluoresce long after the source of radiation has been shut off.

3.3 SPONTANEOUS EMISSION 77

and the correct expression for the upper-state lifetime is t_n¼ 1

A_n¼ 1 P

mA_nm, (3:3:12)

where the summation is over all statesmwith energyEmlower than the energy leveln (see Fig. 3.6). Numerical values of the “Acoefficients”Anmare usually included in tables of oscillator strengths. Radiative lifetimes of excited atomic states are typically on the order of 10 –100 ns.