A simple way to make a black-body is to put a small hole in an otherwise fully enclosed cavity (Fig. 2.1). Any ray entering the cavity via this hole will have great difficulty escaping, particularly if the cavity walls have moderately good absorption properties. The cavity therefore acts as an excellent absorber of radiation entering the hole. As a result, the area of the hole acts like an almost ideal black-body radiator for the inverse emission process.
From experimental data for such cavities, Max Planck first guessed the correct expression for the spectral variation of black-body radiation in 1900 and then attempted to derive this expression theoretically (Duck 2000; Bailyn 1994). Simply stated, he had to stretch both statistics and physics to achieve his goal! In
Fig. 2.1 : A cavity with a small hole in it acts as an almost perfect absorber of light and therefore emits light from the hole with almost ideal black-body properties.
the latter area, he attempted to understand how his expression arose from the interaction of light with molecules of the black-body, modelled as oscillators. By working backwards to get the correct result, he was forced to hypothesise that these oscillators possessed energy only in multiples of a fundamental quantum.
Albert Einstein first saw the full ramifications of the quantum idea. In 1905, he suggested that light itself must have an intrinsic particle aspect (Duck 2000; Bailyn 1994). By assuming that an electromagnetic wave was not continuous but lumped into particles, he could explain existing experimental results from the photoelectric effect very simply (ejection of electrons from illuminated metals). This also allowed a different approach from Planck’s to calculating the properties of a black-body (Bose 1924). As summarised by Richard Feynman (Feynman et al. 1965), the two approaches are equivalent. From one point of view, the radiation emitted by a black-body can be regarded as being in equilibrium with a large number of oscillators, one for each frequency component of light, with each oscillator in different excited states. From the other, the same property can be analysed in terms of Einstein’s particles. The number of particles having a particular energy correspond to the state of excitation of the corresponding oscillator.
Satyendra Nath Bose was first to derive the black-body formula from the quantum particle point of view in 1924 (Feynman et al. 1965). Bose showed that the quantum particle hypothesis combined with the statistics of these particles were enough in themselves to give Planck’s formula (Bailyn 1994). The particle hypothesis is that light consists of quanta of energy E = hf and momentum k = hf/c, giving the following expression for the spatial components of this momentum:
kX + k2y + k = h 2 f 2/c 2 (2.1)
Using a forerunner of Heisenberg’s Uncertainty Principle of 1926 (AxAkx > h), Bose imagined a 6-dimensional xyzkxkykz phase space with the phase space volume divided into a mosaic of cells each corresponding to a separate state and each of volume:
h = dxdy dz dkx dky dkz (2.2)
At fixed x, y, and z, the three dimensional kx ky kz sub-space is of interest. A spherical cell in the kx ky kz sub-space between radius k and k + dk contains photons with approximately the same frequency, f = ck/h. The total phase volume dQ associated with these photons is:
dil = 4nk 2 dk JJJ dxdydz = 4nV( hf/c )2 d( hf/c) (2.3)
where V is the physical volume of the cavity being considered. The number of states enclosed is:
Pfdf = (g/h3)d Q = g(4nV/c3)f2df (2.4)
where the degeneracy factor, g equals 2. This is because light, considered as a planar transverse wave, needs two parameters to specify how it is orientated (i. e., has two possible polarisation states for each value of f ) or has two possible spins, if considered as a particle.
Most readers, as for the author, probably find it easier to think in 3dimensional rather than 6-dimensional space. Some feeling for the previous mathematics can be gained by thinking in the 3-dimensional xkxky space of Fig. 2.2, as might be important for fixed y, z and kz. The kxky plane is the analogue of the 3-dimensional kxkykz space, in this example. The analogue of the spherical shell is the annular ring shown for three such planes in Fig. 2.2. The volume corresponding to kx and ky values within this ring within the 3-dimensional space is just the area of the ring multiplied by the extent of the region in the x-direction, for the fixed values of y and z. This corresponds to the tubular volume shown. The calculation is simplified since the geometry in the kxky plane is independent of x – value. For the 6-dimensional case, the volume of the shell in the kxkykz sub-space is independent of x, y and z co-ordinates. The volume in the 6-dimensional space is found for an incremental physical volume around any particular x, y, z co-ordinate and integrated over the entire physical volume. The latter converts to a multiplication due to the non-dependence upon spatial co-ordinates.
Bose then calculated the most probable distribution of photons amongst his mosaic of cells within phase space. He was able to show that the average number of particles is given by the function that now bears his name (together with
Einstein’s who was quick to realise the significance of Bose’s work and to develop it). The Bose-Einstein distribution function is given by:
fBE — V(ehf/kT -1) (2.5)
The total photon energy per unit volume in the cavity in the frequency range df is therefore given by: