Topic Start | Study Unit | Previous Page | Next Page

Presentation

Except for certain background radiation in inter-galactic space, every radiation we know of is to some degree affected by interaction with materials. In particular, the thermal radiation emitted from a surface will show certain characteristics dependent on the material of which the surface is made. There are many other forms of radiation--such as that emitted in radioactive decay, or the emissions from an ionized gas-which are almost wholly dependent for their characteristics on the material sources from which they originate. Such forms of radiation carry information on their sources which can be used to construct models of these sources.

If we want to look at thermal radiation as a system in itself, then we have to eliminate the influence of the actual material surfaces involved. We simply imagine a cavity in which radiation exists. It does not matter whether we think of this radiation in terms of streams of energy passing to and fro from side to side of the cavity, or as something "standing still". We simply take it that in time the system will attain thermal equilibrium. In other words, it will reach its state of maximum entropy, and any information on the materials lining the cavity will then have disappeared.

In Study Unit 3 we discussed the relation between maximum entropy and equilibrium. The point can be simply illustrated. Imagine a room in which on one side there is a line of people holding conversations together. Microphones transmit the words to the opposite wall where they are re-broadcast. The mixture of live conversation and relayed conversation is then re-transmitted, and so on. The eventual result is noise in which it will be impossible to pick out any single word. But this noise will have certain characteristics that depend on the range of possible frequencies and the intensity of the conversations.

The analogy is very exact for thermal radiation. The radiation coming off a surface can tell us what the surface is like-this corresponds to a direct transmission from the speakers-but the radiation inside a cavity scrambles information from the surface and can be characterized purely in terms of the temperarure.

The characterization of radiation takes the form Df the energy distribution with respect to frequency (or wavelength). In fig. 7.1.

we have the characterization of radiation inside a cavity-which is known as . The adjective "black-body" was introduced more than a hundred years ago because it was then simpler to think of a featureless surface, which would absorb and reflect all radiation equally without bias, than to think in terms of entropy, which was then only just emerging as an important concept.

It is true that any actual surface will discriminate between different frequencies of radiation; that is why we have coloured surfaces in the domain of the visible spectrum of radiation. For any actual surface, we can measure the , which is the fraction of the incident radiation of frequency v which the surface will absorb (thus a fraction 1 = a will be reflected). The , , of a surface is the fraction of the radiation emitted which is of frequency . For a black body we have:

a V = V, = 1
For all other surfaces: aV = V, < 1

When a coloured surface is heated, it will often begin to emit its colour. The diagram (fig. 7.2) shows the situation for a blue object at room temperatures. At high temperatures the direction of the arrows can be reversed, for the object is then giving out more energy than it is taking in. So what was initially a blue object then appears orange.

We have talked of frequency and made reference to the visible spectrum. This is because thermal radiation is , and all frequency ranges show the same properties of reflection and refraction that we associate with visible light. We can therefore talk either of wavelength or frequency , since

v = c, the velocity of light.

Equations can be derived from electromagnetic theory and thermodynamics which relate together the pressure, temperature, volume and internal energy of a system of radiation. The following results are important:

(i) It was well known in the late nineteenth century that electromagnetic energy exerted a pressure on the surfaces on which it fell. Maxwell worked out the equation for radiation pressure:

p = 1/3 u, where u is the energy density

(ii) By using the fundamental equation of thermodynamics:

TdS = dU + PdV

and the expression for radiation pressure, it is possible to derive the expression:

( du / dT ) = 4 ( u / T ), where u is the energy density,
which gives u = constant x T4    [7.1]

This equation corresponds to , in which the energy radiated from unit area of a black-body surface in all directions in unit time is said to be directly proportional to the fourth power of its absolute temperature. The law is usually called the Stefan-Boltzmann Law, since Boltzmann first gave a theoretical derivation of it.

(iii) It can also be worked out theoretically that, if is the wavelength corresponding to the maximum energy for a given temperature T, then

T. max = constant    [7.2]
If you look at fig. 7.1, you will see that the peak of the energy distribution curve is "displaced" according to the temperature.

The energy of the system is associated with electromagnetic vibrations, and we have to find out how the total internal energy is distributed amongst the various frequencies. Now, in equilibrium, the energy of the system remains constant, so that there is no overall transference of energy either from one part of the system to another or outside the system. We can imagine an array of different vibrations, equivalent to the collection of particles in a gas. Then the energy is "located" in these vibrations, and we can consider these as standing waves-since there is no transference of energy.

Fig. 7.3 shows two standing waves, and the energy is indicated as residing in the antinodes of these waves. We treat each antinode as having an equal share in the energy associated with a particular frequency. For a given small frequency band dv, we have first to determine the number of antinodes in that band. The technique used starts from this notion of standing waves. If we consider a standard length of one unit, the profiles of the first four standing waves are shown in fig. 7.4.

The only possible wavelengths which could give rise to standing waves in one dimension are

tells us how many antinodes there are in a given wavelength, and when is large (so that we can differentiate) we can work out how many antinodes there will be corresponding to wavelengths between and + :

When this technique is applied to three dimensions instead of one, we find that:

If each antinode has an energy that is, on average, independent of frequency, we can easily see that the energy involved per frequency interval will increase with the frequency (see equation [7.3]). Thus the total energy will be infinite. This unworkable result was obtained by Maxwell in 1884 and became known as the "ultraviolet catastrophe".

Maxwell attributed an equilibrium energy T to every equilibrium vibration. is a constant, Boltzmann's constant, which is of fundamental importance in thermal physics. The inclusion of the absolute temperature T should remind us of the ideal gas correlation between the average kinetic energy of molecules and absolute temperature. Whatever the constant in front of T, the basic idea is simple: there is in classical mechanics why any one "particle of energy" should have an energy different from any other, . This is the notion of the - the total energy is equally partitioned amongst all the independent "particles".

However, the energy distribution curve of black-body radiation tells us categorically that the equilibrium vibrations do differ in their capacity to take on energy! Thus the principle of equipartition of energy does not hold. The task of Planck was to find out just what one had to suppose to derive the right equation for the energy distribution.

In classical mechanics, it was generally assumed that energy changes were always continuous - that is, they could consist of any quantity of energy, no matter how small. Planck saw that it was just this assumption that caused the difficulty. In order to obtain the right equation for the energy distribution, it had to be recognized that energy was taken in and given out in which of the vibration.

The energy of an antinode is thus given by
= n' 0     [7.4]

where ' is an integer. The indivisible unit of energy, or quantum, is given by

0 = hv     [7.5]

where h is Planck's constant, now one of the universal constants of physics. Thus . The higher the frequency, the greater the amount of energy that has to become available before that vibrational mode can gain a quantum of energy; and hence we have the tail-off of energy towards the higher frequencies that is apparent in fig. 7.1 (to the left of the maxima).

We can briefly indicate the results, different from those of Maxwell, which are obtained using Planck's ideas. We will have to assume an important notion for the statistics of energy: the of an energy state is directly proportional to . Here, T is just an average energy quantity for the system as a whole, and e is a for the energy .

Now, in Maxwell's classical scheme of a continuum of energy changes, the average energy is given by:


This is the same for all antinodes of all frequencies. Now in Planck's scheme, = ', and we have to use instead of integration. Thus:


To find the , J, resulting from all the vibrations with frequencies between and + , we use equation 7.5. Thus:

J is then a function of (or _). A graph of J against _ gives the same distribution as that in fig.7.1.

The basic concept is simple, though it requires careful reflection to catch it clearly. In classical mechanics, one had to treat each independent "locus" of energy in a system - whether it was a molecule, an electron, or a vibration - as of equivalent status. They were equivalent because each one could change in energy by any amount whatsoever. So what they were, their structure and so on, was of no importance from the point of view of the . With Planck and his , the "loci" of energy - in this case, the equilibrium electromagnetic vibrations - were . Planck introduced the two new concepts:

(i) = ', energy comes and goes in quanta.

(ii) = , the quantum of energy is associated with the frequency of its locus.