The basic idea hasn’t changed since Maxwell’s time. Take a cavity at temperature t.
We use a cavity to trap radiation, thus causing it to act as a perfect absorber. The radiation that the cavity emits will be in thermal equilibrium with the walls.
When the cavity is in thermal equilibrium, the radiation in the cavity must form standing waves in the cavity. Thus we can ask what are the allowed modes for the radiation. From our knowledge of standing waves, we know that the allowed modes are those with integer and half integer values of the frequency. From quantum mechanics, we know that the frequency of a wave is related to the energy of the wave by Planck’s constant, and so the allowed modes for the radiation are those with an energy
But these are just the allowed energy values of the harmonic oscillator! Thus we see that each mode can be replaced by a simple harmonic oscillator with energy .
The main difference is that before we talked about s as the quantum number for the harmonic oscillator, now we treat s as the number of photons in a particular mode in the cavity.
Solution to Maxwell’s Equations
To see that the allowed values are those of a harmonic oscillator, consider the Maxwell’s equations for an electromagnetic field. These are
If the waves are confined to a cube of length L on each side, the solutions are of the form
This states that the field vectors must be perpendicular to the vector n. Thus the electromagnetic field in the cavity is a transversely polarized field. The polarization direction is defined as the direction of E0. Similarly, (9.1) yields
where n2 = nx2 + ny2 + nz2. Thus the frequency wis determined in terms of the integers nx, ny, and nz. Finally, note that the time dependent part of the solutions satisfies the equation
where Ei0(t) = Ei0 sin (wnt) and wn2is given by (9.3). But this is just the equation of a simple harmonic oscillator, so we see that the solutions are those of simple harmonic oscillators.
What is the thermal average number of photons in the cavity? That is, what is the expectation value of the parameter s? Recall that the partition function is given by
Then the expectation value of s is
This is the Planck distribution function for the thermal average number of photons in a single frequency mode. Equivalently, it is the average number of phonons in that mode. As we will see, a phonon is the quantum of energy of an elastic wave moving through a solid.
Using (9.5), we can determine the total energy contained in the cavity. By definition, the total energy is the sum of the energy in each mode, so
where the sum is over nx, ny and nz. Assume that the temperature is large compared to the change in wnso that we may replace the summation with an integral. Then the integral becomes
Here dn = dnxdnydnz and the factor of 1/8 arises from the fact that we are only integrating over the positive octant of the parameter space. Now, one result of is that there are two independent polarization directions. Thus, we must multiply (9.6) by two. Using (9.3) to replace wn, we finally get
Let . Then the integral becomes
This integral can be looked up in a table. It is found to be p4/15. This leads to the final result
where V = L3. This result is called the Stefan-Boltzmann law of radiation. It shows that the total energy density of a black body is proportional to the fourth power of the temperature. This law is of immense use in astrophysics, as we are able to measure or determine the energy density of a star, and thus we can determine its equivalent black body temperature.
Most of the time we are interested in the energy per unit volume per unit frequency range. This is called the spectral density of the radiation, and is denoted uw. Starting from
We can derive a relation for uw. Recall that wn= npc/L, so in terms of wnthis becomes
This is known as the Planck radiation law. It gives the frequency distribution of thermal radiation. The entropy of thermal radiation can be determined from tds= dU. Using (9.7), we get
or, upon integrating,
Finally, we define the energy flux density JU as the rate of energy emission per unit area. In terms of the energy, it can be written
or, upon substituting in U(t),
is called the Stefan-Boltzmann constant. In MKS units it has a value of 5.670 x 10‑8 W m-2 K-4. Any object that radiates at this rate is said to radiate as a black body.
Another way of looking at blackbody radiation is as a photon gas. This introduces periodic boundary conditions and running waves. Look first at one dimension. Suppose we have a box of length l. We can represent waves by complex notation with the condition
This implies l = 2np, where . Let k = 2np/l. We know that v = w/k, which implies x = 2npv/l. The usefulness of this approach is that we can neglect edge (or surface) effects. This holds if the surface to volume ratio is small. Extending this to three dimensions, we now require that
where we have gone to a period of 2pto eliminate negative integers, and thus, the total energy becomes
as before, with
Suppose we have a cavity and a body enclosed in it. Let a be the fraction of radiation absorbed by the body. This is called the absorptivity of the body. If the amount of radiation incident on the body is JU, then if the body is in thermal equilibrium it must emit an amount of radiation equal to eJU, where e £1 is called the emissivity of the body. Since the body is in thermal equilibrium, the amount of average thermal energy going into the body must be the same as that which is leaving the body, so aJU = eJU, or
a = e (9.11)
This is known as the Kirchhoff law. For the special case of a perfect reflector, a = 0 and so e = 0, which implies that a perfect reflector does not radiate. This can be applied to the spectral density, with the result that, for all frequencies, a(w) = e(w).
We can apply our laws of radiation to determine the thermal fluctuation of current in a circuit, or the electrical noise. We can treat a resistor as an ideal one dimension absorber. Consider a resistor at the end of a transmission line of impedance R.
Then the transmission wave is totally absorbed. Now put another resistor, R‘, a length l down the transmission line and suppose we have a uniform temperature t.
We can treat the loop between the two resistors as a closed loop,
so the current in the loop is
i= V / (R + R‘)
where V is the emf in the circuit. So the average power to R‘ is
If R‘ = R, this becomes
<P> = V2 / 4R
The line will have modes of propagation with k = 2pn/l and wn= vk = 2pvn/l. Let fn = vn/l. Then wn= 2pfn. For Dn= 1, Df = v/l and the density of modes is l/v. Classically the average energy per mode is t. So the power in Dfis
where v is the wave velocity, (t/l) is the energy density and (l/v) is the mode density. So the total power in the resistor is
<V2> = 4R tDf (9.12)
This is known as the Nyquist theorem, and in words it states that the average of the square voltage across a resistor of resistance R is proportional to the product of the temperature of the resistor and the frequency bandwidth within which the voltage fluctuations are measured. Here frequency is in cycles per unit time. To account for quantum mechanics, this result becomes
Transmission of Heat Through a Solid
Another application of this is to the transmission of heat through a solid. This is known as the Debye theory of specific heat. Recall that we said that the phonon is the quantum of energy associated with an elastic wave in a solid. The average number of phonons at a specific wavelength was given by the Planck distribution function
Is there a limit to this number? This is the same thing as asking if there is a limit to the number of possible modes. For an electromagnetic wave, there was no limit, but since the elastic wave is dependent on the material in the solid, it is limited by the solid. If there are N atoms in the solid, each with 3 degrees of freedom, then there are a total of 3N modes possible. In addition, unlike an electromagnetic wave, there is no condition on the polarity of the elastic wave, so it has three polarizations; two transverse and one longitudinal. Let D(w) dwbe the number of modes in the range (w, w+dw). Then
is the number of degrees of freedom, which we saw was equal to 3N. We can treat the atoms in a solid as being connected by springs. Therefore they must have wavelengths on the order of the interatomic spacing. We will assume all of the atoms are the same. Since each wave has three degrees of freedom, we must have
But w= 2pnv/L, so this becomes
which implies D(w) = 3Vw2/2v3p2. Integrating this, we get
Finally, recall that the energy associated with the spring is , and that energy is related to temperature by e µ t. This allows us to define the Debye temperature
The thermal energy of the phonons can be calculated in the usual way
or, setting ,
where . In the limit that t<< q, we see that xD becomes very large, and we can replace it with infinity. Then the solution is similar to that for photons and we get
Recall that the heat capacity was defined to be . Then
This is known as the Debye T3 law. For t>> q, we get the Dulong Petit law, which states that CV = 3NkB.