The Fraction of Black Body Photons in (k,dk)

G.R.Dixon, 12/28/2006

Assume that a unit volume cavity contains n_{total} blackbody photons in thermal equilibrium with the cavity walls. n_{total} is constant. Let a vector __k__ be associated with each photon. The direction of __k__ is the same as the photon’s direction of propagation, and its magnitude equals 2p/l = 2pe/hc where e is the photon’s energy. Each vector __k__ has a unique direction.

Define g(k)dk to be the fraction of the photons with |__k__| in (k,dk). Then n_{total}g(k)dk is the number of photons with |__k__| in (k,dk). And of course

. (1)

** As previously
derived**, the mean photon energy is <e> = 2.7kT. Thus <k> = 2p(2.7)kT/hc, and

. (2)

In postulating a form for g(k)dk we expect no photon’s energy to be either zero or infinite. And (see Eq. 1), g(k)dk must integrate to unity.

Integrals of the form have the potential of satisfying both of these requirements, as well as Eq. 2, provided n>0. Thus we postulate that

, (3)

where the constants a, n and b are to be determined. In this photons case we shall use
** a method first worked out by
Maxwell**, and used on this site to derive the Maxwell-Boltzmann law for the distribution of molecular speeds in a gas.

Let the fraction of __k__’s with x-components in (k_{x},dk_{x}) be f(k_{x})dk_{x}, and similarly for (k_{y},dk_{y}) and (k_{z},dk_{z}). Then the fraction with components in (k_{x},dk_{x}) and (k_{y},dk_{y}) and (k_{z},dk_{z}) is f(k_{x})f(k_{y})f(k_{z})dk_{x}dk_{y}dk_{z}. And the number of __k__’s with components in this range is Nf(k_{x})f(k_{y})f(k_{z})dk_{x}dk_{y}dk_{z}.

In __k__-space each __k__ can be represented as a point (i.e., as the tip of a "displacement" vector from the origin). Since dk_{x}dk_{y}dk_{z} is a volume element in this space, the __density__ of the displacement vector tips at point (k_{x},k_{y},k_{z}) is Nf(k_{x})f(k_{y})f (k_{z}). This density can depend only on |__k__|, the distance from the origin.

The number of __k__ tips in the spherical shell of volume 4pk^{2}dk is Ng(k)dk, and the density in this shell is Ng(k)dk/(4pk^{2}dk). Thus

. (4)

In particular, for k_{x}=k and k_{y}=k_{z}=0,

, (5)

or

, (6)

where the constant C=4pf^{2}(0).

Now Eq. 6 can be satisfied for all k only if g(k) contains a k^{2} term. Thus the correct choice for n in Eq. 3 is n=2, and we postulate that

. (7)

From Eq. 7,

. (8)

Or, since

, (9)

we find that

, (10)

and

. (11)

From Eq. 2 we have

. (12)

Or, since

, (13)

we find that

(14)

and thus

. (15)

Substituting in Eq. 10:

, (16)

and thus

. (17)

As found previously, the average value of k is

. (18)

Fig. 1 plots g(k) vs. k over the range 1E-7<k<5E6 and at a temperature of 1500 Kelvin.

Figure 1

g(k) vs. k

Not surprisingly, the curve in Fig. 1 resembles that for the Maxwell-Boltzmann distribution. A rather interesting feature is its resemblance also to the Planck spectral distribution law for black body radiation energy density. The curves are admittedly not the same. Still, there is food for thought. k is of course proportional to e, the photon’s energy (k = 2pe/hc), and Planck exhibits the same sort of cut-off in energy (or number of photons present at temperature T) as the curve approaches the shorter wavelengths from the right.