2 Chapter

1

Blackbody Radiation^ Image Plane Intensity, and Units

1.1 Planck's Law

By convention and definition

blackbody

radiation describes the in-

tensity and spectral distribution of the optical and infrared power emit-

ted by an ideal black or completely absorbing material at a uniform

temperature T. The radiation laws are derived by considering a com-

pletely enclosed container whose walls are uniformly maintained at

temperature T, then calculating the internal energy density and spectral

distribution using thermal statistics. Consideration of the equilibrium

interaction of the radiation with the chamber walls then leads to a gen-

eral expression for the emission from a "gray" or "colored" material with

nonzero reflectance. The treatment yields not only the spectral but the

angular distribution of the emitted radiation.

Although we usually refer to blackbody radiation as "classical", its

mathematical formulation is based on the quantum properties of electro-

magnetic radiation. We call it classical since the form and the general

behavior were well known long before the correct physics was available

to explain the phenomenon. We derive the formulas using Planck's

original hypothesis, and it is in this derivation, known as Planck's law,

that the quantum nature of radiation first became apparent. We start by

considering a large enclosure containing electromagnetic radiation and

calculating the energy density of the contained radiation as a function of

the optical frequency v. To perform this calculation we assume that the

radiation is in equilibrium with the walls of the chamber, that there are a

calculable number of "modes" or standing-wave resonances of the elec-

tromagnetic field, and that the energy per mode is determined by therm-

al statistics, in particular by the Boltzmann relation

p(U)

=

Ae-^^^^ (1.1)

where p(U) is the probability of finding a mode with energy, U; k is

the Boltzmann constant; T, the absolute temperature; and A is a

normalization constant.

Example: The Boltzmann distribution will be used frequently in this

text since it has such universal application in thermal statistics. As

an interesting example, let us consider the variation of atmospheric

pressure with altitude under the assumption of constant temper-

1.1 Planck's Law 3

ature. The pressure, at constant temperature, is proportional to the

density and thus to the probability of finding an air molecule at the

energy U associated with altitude /z, given by U = mgh, with m

the molecular mass and g the acceleration of gravity. Thus the vari-

ation of pressure with altitude may be written

and the atmospheric pressure should drop to lie or 37% at an alti-

tude oi h = kT/mg. Using 28 as the molecular weight of nitrogen,

the principal constituent, yields

mg =

28(1.66

X

ICt^^)

9.8

=

4.5

x

KT^^

newtons

kT =1.38x

10-^^(300) =

4.1

xlO'^^ M^^

h(37%) =

9x10^

meters =

9 km

or 30,000

feet.

This is quite close to the nominal observed value of 8 km, deter-

mined by the more complicated true molecular distribution and a

significant negative temperature gradient. We discuss a simpler

way of calculating energies in section 1.5.

Returning to the chamber, each mode corresponds to a resonant fre-

quency determined by the cavity dimensions. In the original treatments,

each mode was considered to be a "harmonic oscillator" having, as we

shall see, an average thermal energy kT. Before we start counting.the

number of these modes versus optical frequency, let us first verify this

average energy of a single mode according to Boltzmann's formula.

First of all, we know that an ensemble of identical modes, either in time

or over many systems, must have a total probability distribution over all

energies U, which adds to unity, i.e.,

rviWdU = r Ae'^'^^dU

= 1

.'.A

=

? (1.2)

The average energy of the mode is the integral over the product of the

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