THERMAL RADIATION

In thermal equilibrium the processes of absorption and emission balance each other in such a way that the spectral density of radiation is completely characterized by the temp- eratureT. The spectral energy density of thermal radiation is thePlanck spectrum:

r(n)¼ 8phn³=c³

e^hn=kBT1, (3:6:1)

wherek_B (¼1.38010²²³J/K¼0.0861410²³ eV/K) is Boltzmann’s constant.

The historical significance of the Planck spectrum in the development of quantum

n n+Dn

Du(n)=r(n)Dn r(n)

Figure 3.9 Spectral energy densityr(n) is defined such thatDu(n)¼r(n)Dnis the electromagnetic energy per unit volume in the narrow frequency interval fromntonþDn.

3.6 THERMAL RADIATION 85

theory is discussed in many textbooks. Our aim here is to see what the classical electron oscillator model and the quantum theory of emission and absorption, as formulated thus far, imply about thermal equilibrium radiation. Because the thermal spectral densityr(n) is certainly “broadband,” we will use the absorption formulas appropriate to broadband radiation.

Consider first the classical electron oscillator model. The rates of emission and absorption are given by Eqs. (3.3.5) and (3.5.6), respectively. The equilibrium condition that emission and absorption are equal in the electron oscillator model is then

1 4pe₀

2e²v²₀

3mc³E¼ 1 4pe₀

pe²

m r(n), (3:6:2) or, sincev0¼2pn0,

r(n0)¼8pn²₀

3c³ E: (3:6:3)

It is a well-known theorem of classical physics that, in thermal equilibrium at temperatureT, the average energyEof an oscillator free to oscillate in three dimensions is 3kBT. Using this result in (3.6.3), we have

r(n)¼8pn²

c³ k_BT (3:6:4)

for the spectral density of thermal radiation predicted by the classical electron oscillator model. We have written this result in terms of an arbitrary frequencyn rather than a single resonance frequencyn0in order to model an idealized blackbody that, as dis- cussed below, absorbs radiation atallfrequencies.

The spectrum (3.6.4) is theRayleigh – Jeans spectrum, and it is an inexorable conse- quence of classical physics. It is an approximation to the Planck spectrum (3.6.1) when the quantum of energyhn is small compared tokBT, and so can be regarded as the

“classical limit” of the Planck spectrum.

Let us now use the formulas (3.3.8), (3.3.9), and (3.5.5) of quantum theory for the emission and absorption. FordN1/dt, for instance, we have

dN₁

dt ¼A21N2A₂₁ 8ph

c³

n³₀N1r(n0), (3:6:5) the first term being due to spontaneous emission and the second to absorption. Setting dN₁/dt (ordN₂/dt) equal to zero, since the populations of the atomic states must be constant when the atoms and the radiation are in equilibrium, we obtain the equilibrium radiation spectrum:

r(n₀)¼8phn³₀ c³

N₂

N₁: (3:6:6)

Now in thermal equilibrium at temperatureTthe ratio of the populationsN2andN1must satisfy a general result of quantum statistical mechanics:⁶

N2

N₁¼e^(E2E1)=kBT ¼e^hn0=kBT: (3:6:7) Therefore, the spectrum of thermal radiation predicted by this (not quite correct) argu- ment is

r(n)¼8phn³

c³ e^hn=kBT, (3:6:8) where, for the reason noted following Eq. (3.6.4), we have written the spectrum in terms of an arbitrary frequency nrather than a specific frequencyn₀. The spectrum (3.6.8), which is called theWien spectrum, is an approximation to the Planck spectrum when the quantum of energyhnis large compared tokBT.

To see why we have not obtained the correct spectrum of thermal equilibrium radi- ation using the (correct) results of the quantum theory of spontaneous emission and absorption, let us use (3.6.7) to write the Planck spectrum as

r(n₀)¼8phn³₀ c³

1

N1=N21¼8phn³₀ c³

N₂ N1N2

, (3:6:9)

or

c³

8phn³₀r(n0)(N1N2)¼N2, (3:6:10) and therefore

A₂₁ 8ph

c³

n³₀N1r(n0)¼A21N2þ A₂₁ 8ph

c³

n³₀N2r(n0), (3:6:11) where we have multiplied through by the spontaneous emission rateA₂₁. This equation forr(n₀), together with the Boltzmann condition (3.6.7), yields the Planck spectrum.

Without the second term on the right-hand side of (3.6.11) we would have Eq. (3.6.6), which was obtained by equating the rates of absorption [the left-hand side of (3.6.11)]

and spontaneous emission (the first term on the right). To obtain the correct thermal radi- ation spectrum, therefore, we require another effect in addition to absorption and spon- taneous emission. This “new” effect is described by the second term on the right-hand side of (3.6.11). Like spontaneous emission, the rate for this effect is proportional to the upper-state populationN2, so that it too is associated with the emission of radiation.

Unlike spontaneous emission, however, the rate for this emission process is proportional

6We are ignoring any degeneracy of the energy levelsE₂andE₁, which does not affect the thermal radiation spectrum (Problem 3.8).

3.6 THERMAL RADIATION 87

to the spectral densityr(n0) of radiation already present. That is, the process described by the second term on the right-hand side of (3.6.11) isstimulated emission.

According to (3.6.11) the rate coefficient for stimulated emission, which we write as A₂₁

8ph c³

n³₀r(n₀)¼B₂₁r(n₀), (3:6:12) is identical to the rate coefficient for absorption, which we write asB₁₂r(n₀). That is,⁷

B12¼B21: (3:6:13)

With this notation we can write Eq. (3.6.11) as

N1B12r(n₀)¼N2A21þN2B21r(n₀), (3:6:14) the left-hand side being the rate of absorption and the right-hand side the rate of spon- taneous and stimulated emission.

† Equation (3.6.14) was first presented by Albert Einstein in 1916, more than a decade before what are now called the “EinsteinAandBcoefficients” could be derived from quantum theory. To obtain the Planck spectrum using discrete energy states and other aspects of the Bohr theory, Einstein postulated the processes of spontaneous emission, absorption, and stimulated emission, and that these processes could be characterized by rate coefficients as in (3.6.14). Arguing that (3.6.14) must be true for all temperatures and therefore for arbitrarily large spectral densities r(n), Einstein concluded thatB12¼B21.

Equations (3.6.14) and (3.6.7) imply

r(n₀)¼ A21=B21

e^hn0=kBT1, (3:6:15) which in the “classical limit”hn₀=k_BT1 reduces to

r(n0)¼A21

B21

kBT

hn0: (3:6:16)

Reasoning that this limit should yield the Rayleigh – Jeans spectrum (3.6.4), Einstein deduced that A21

B21

¼8phn³₀

c³ , (3:6:17)

a result implied by our Eq. (3.5.5) when it is written as dN1

dt ¼ N1B12r(n0), (3:6:18)

7Equation (3.6.13) and a few others are generalized in Section 3.7 when degenerate energy levels are considered.

where, using (3.3.7), we identify

B12¼ 1 4pe₀

pe²f

mhn0: (3:6:19)

Einstein was thus able to derive essentially all the results for emission and absorption in broad- band fields that we have obtained starting from the electron oscillator model and modifying it to incorporate results of quantum theory. A truly new insight achieved by Einstein was that the observed Planck spectrum implied that there must be stimulated emission in addition to spon- taneous emission. The practical utilization of this new concept—the laser—was to occur more

than 40 years later. †

It is instructive to write the energy per unit volume of thermal radiation in the small frequency intervalDnaboutn,Du(n)¼r(n)Dn, in the form

r(n)Dn¼hn 8pn² c³ Dn

1

e^hn=kBT1: (3:6:20) The first factor on the right is the energy of a photon of frequencyn. The quantity in parentheses is the number of electromagnetic field modes per unit volume in the small frequency interval [n,nþDn], assuming that any cavity containing the thermal radiation is large compared to the wavelength c/l (Section 3.12). The last factor, 1=(e^hn=kBT1), can therefore be identified as the average number of photons of fre- quencynin thermal equilibrium at temperatureT.

This last quantity has a further significance that can be inferred by considering the ratio of the rate of stimulated emission in thermal equilibrium,N2B21r(n0), to the rate of spontaneous emission,N2A21, for a transition of frequencyn0:

N₂B₂₁r(n₀) N2A21

¼B₂₁ A21

r(n₀)¼ 1

e^hn0=kBT 1, (3:6:21) where we have used Eq. (3.6.15). In other words, the stimulated emission rate is equal to the spontaneous emission rate times the average number of photons at the transition fre- quency. Although this result has been inferred for the case of thermal radiation, it is more generally valid and may be stated as follows:The rate of stimulated emission into any mode of the field is equal to the spontaneous emission rate into the mode, times the average number of photons already occupying that mode.(See also Section 3.7.)

The Planck spectrum is independent of the atomic or molecular properties of the material in thermal equilibrium with radiation. According to (3.6.7), there will be more atoms in the lower level than the upper level of any transition. This means that any radiation incident on the material will lose energy. It is convenient to define an ideal blackbodyas an object that absorbsall the radiation, ofanyfrequency, incident upon it. In such a blackbody the absorption and emission of radiation are exactly balanced in a steady state of thermal equilibrium, and any radiation incident upon its surface would be completely absorbed.

Although no perfect blackbody is known to exist, it is possible to construct an excel- lent approximation to an ideal blackbody surface. Consider a cavity inside a metal block, with a small hole drilled through to provide an opening to the outside, as illustrated in Fig. 3.10. Any radiation incident on the hole from the outside is repeatedly reflected

3.6 THERMAL RADIATION 89

within the cavity, and eventually absorbed, so that the amount of incident radiation escaping back through the hole to the outside is negligibly small. The hole itself thus acts as the “surface” of a blackbody. Furthermore a small amount of equilibrium thermal radiation inside the cavity, produced by spontaneous and stimulated emission from the cavity walls, can escape through the hole to the outside. This radiation escaping through the hole is a sampling of the thermal radiation inside the cavity, and therefore the spectral density of this “cavity radiation” should satisfy the Planck formula. If the block is placed inside an oven, it can be kept in thermal equilibrium at some fixed temperatureT. The earliest accurate measurements of such cavity radiation in the far-infrared spectral region from 12 to 18mm, where the Wien law fails to agree with the data, were carried out by O. Lummer and E. Pringsheim in 1900. These and other measurements, particu- larly those of H. Rubens and F. Kurlbaum, motivated Planck to reconsider the existing theory of thermal radiation and led to his announcement of formula (3.6.1) at the October 19, 1900, session of the Prussian Academy of Science.

Many sources of radiation have spectral characteristics approximating those of an ideal blackbody. Stars, for instance, are certainly not perfect blackbodies, but they come sufficiently close to the ideal that we can estimate their surface temperatures by fitting their spectra to Planck’s law (Fig. 3.11). In particular, the peak emission wave- lengthl_maxof a blackbody at temperatureT(K) is given by (Problem 3.1)

l_max¼2:89810⁶

T nm: (3:6:22)

Thus, the sun, which has a spectrum approaching that of a blackbody at 5800K, has a peak emission wavelengthl_max500 nm. Its total intensity at Earth’s surface is about 0.14 W/cm²¼1.4 kW/m². Equation (3.6.22) is consistent with the observation that the color of hot bodies shifts to shorter wavelengths with increasing temperatureT. Thus,

“white hot” is hotter than “red hot”; the filament of an incandescent lightbulb glows white, whereas a (cooler) toaster glows red. This shift of peak wavelength with tempera- ture is evident in Fig. 3.12.

For wavelengths in the visible, and for temperatures less than several tens of thou- sands of kelvins, the ratio (3.6.21) is much less than unity. For the solar temperature T¼5800K, for instance, and l_¼500 nm, the ratio is about 1/142. Thus, we can infer that more than 99% of the light from the sun is due to spontaneous rather than stimulated radiation processes.

Figure 3.10 A cavity inside a metal block kept at constant temperature. A small hole allows radiation to enter the cavity, and the radiation is diffused by repeated internal scattering. The hole itself acts as the surface of a blackbody.

The total electromagnetic energy density of a blackbody is found by integrating (3.6.1), which leads to

ð₁

0

r(n)dn¼ 8p⁵k_B⁴

15c³h³T⁴: (3:6:23)

The total intensity, or power radiated per unit area, is then I_total¼c

4 ð₁

0

r(n)dn¼ 2p⁵k⁴_B

15c²h³T⁴¼sT⁴, (3:6:24)

Relative brightness

0 500 nm

8000K

5800K

1000 nm Optical ir

uv

Figure 3.11 Comparison of blackbody emission (smooth curves) and stellar emission spectra for two temperatures, 8000 and 5800K. (After W. M. Protheroe, E. R. Capriotti, and G. H. Newsom,Exploring the Universe, 3rd ed., Merrill, Columbus, 1984.)

20,000K

10,000K 1000K

300K

100 1

Wavelength, l(μm) Log10r(l)

0.01 –6

0 6

Figure 3.12 Log10r(l) vs.lfor an ideal blackbody radiator at four temperatures.

3.6 THERMAL RADIATION 91

where s_¼5.6710²⁸ J-m²²-s²¹-K²⁴¼5.6710²¹²W/cm²/K⁴ is the Stefan – Boltzmann constant. Note that the intensity at each frequency in this formula is not simply the speed of light c times the spectral energy density at that frequency. The extra factor ¹₄arises for two reasons. First, a factor ¹₂arises because, along any axis through the blackbody, there are equal intensities of radiation propagating in opposite directions, but in (3.6.24) we are only interested in radiation propagating outward through the surface. A second factor of ¹₂ arises because the average component of light velocity normal to the surface is

(cosu)_avg¼ 1 2p

ð_2p

0

df ð_p=2

0

cosusinudu_¼¹

2: (3:6:25) Except at very high pressures, an atomic gas radiates only at certain discrete wave- lengths corresponding to the spectral lines of the atoms. In solids, however, the inter- actions of the closely packed atoms cause the emitted radiation to have a continuous spectrum, which is approximately the Planck spectrum for the temperatureTof the solid. The ratio of the power radiated by a given solid to the power radiated by a black- body of equal area and temperature is called thetotal emissivity,1, of the solid. Thespec- tral emissivity,1_l, is defined similarly in terms of the power radiated within a narrow wavelength interval betweenlandlþdl. The spectral emissivity and the reflectivity r_lsatisfy1_lþr_l¼1, so that the spectral emissivity can be determined by measuring the reflectivity. For passive surfaces (not part of some laser device) we haver_l,1, so emissivities are always less than 1. Typically 0.2,1,0.9, but emissivities can be very small for highly reflecting surfaces.

The radiation of a perfect blackbody is isotropic, that is, the Planck spectrum does not depend on any direction of propagation. For real bodies, however, the spectral emissivity can depend not only on wavelength but also on direction and temperature. Emissivities of different materials are tabulated in handbooks.⁸

Figure 3.13 shows the radiation spectrum of tungsten, the filament material in house- hold incandescent lightbulbs, compared with that of a blackbody for the temperature T¼3000K. In order for the filament to produce significant visible radiation, it must be heated to high temperatures; tungsten is used because, among other things, it has a high melting temperature (3655K) and proper working gives it the strength and ductility necessary for it to be formed in fine wire filaments.⁹As can be seen from Fig. 3.13, how- ever, only a small part of the radiation from the heated filament lies in the visible.

† The limited lifetime (1000 h) of tungsten filament lightbulbs is due to the evaporation of tungsten, which causes the blackening of a bulb over time. An inert fill gas (usually argon) is used to reduce the evaporation of tungsten particles, which are deposited on the upper part of the bulb as a result of convection currents that carry them upward. Thus, a bulb used “base up” on a ceiling will blacken near the stem of the bulb, whereas “base down” use results in black- ening over the dome. The tungsten halogen lamps marketed in recent years employ a tungsten – halogen regenerative cycle in which the evaporated tungsten combines with a halogen (e.g., iodine) to form a compound, thus preventing the tungsten from being deposited on the glass

8See, for instance, R. C. Weast, ed.,CRC Handbook of Physics and Chemistry, CRC, Boca Raton, FL, 1988, pp. E-390 – E-392.

9This was one of the first successes of American industrially organized science. It came from the discovery in 1908 of how to make tungsten ductile by W. D. Coolidge in the General Electric Company laboratory in Schenectady, NY.

bulb. When this compound comes in contact with the filament, however, the heat is sufficient to dissociate the compound into tungsten, which is redeposited on the filament, and the halogen, which is then available to continue the regenerative cycle.

The more efficient fluorescent tubes are, of course, not thermal light sources, but like thermal and all other nonlaser sources the light they generate derives from spontaneous emission. At each end of the tube is an electrode, one of which is a tungsten coil coated with a material that increases the efficiency with which electrons are ejected when the coil temperature exceeds about 1300K.

The low-pressure (0.008 Torr mercury and 1–3 Torr of other gases, 1 Torr¼1/760 atm of pressure) electric discharge along the axis of the tube causes the emission of 253.7 nm radiation from mercury atoms excited by collisions with electrons. The inner walls of the tube are coated with a “phosphor” that absorbs in the ultraviolet and emits in the visible. The operating lifetime in this case is determined primarily by the erosion of the electron emissive coating each time the lamp is turned on, so that the rated average life of fluorescent tubes is based on the number of starts, assuming 3 h of operation per start.

Lighting technology remains an active area of research and development, with particular focus in recent years on light-emitting diodes (LEDs). The reader wishing to pursue some of the many interesting aspects of the subject is referred not only to books and journals devoted to it but also to lighting and optical company catalogs or websites that sometimes contain tutorial information

about lighting products. †

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