Thermal Radiation

 

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.

 
 

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

 
 

(9.1)

 
 

(9.2)

If the waves are confined to a cube of length L on each side, the solutions are of the form

 
 

implies

 
 

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

 
 

or

 
 

(9.3)

where n² = n_x² + n_y² + n_z².Thus the frequency w is determined in terms of the integers n_x, n_y, and n_z.Finally, note that the time dependent part of the solutions satisfies the equation

 
 

where E_i0(t) = E_i0 sin (w_nt) and w_n² is 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.

Thermal Average

 

(9.4)

Then the expectation value of s is

 
 

(9.5)

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.

Thermal Energy

 
 

where the sum is over n_x, n_y and n_z.Assume that the temperature is large compared to the change in w_n so that we may replace the summation with an integral.Then the integral becomes

 
 

(9.6)

Here dn = dn_x dn_y dn_z 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 w_n, 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

 
 

(9.7)

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.

 
 

We can derive a relation for u_w.Recall that w_n = npc/L, so in terms of w_n this becomes

 
 

so

 
 

(9.8)

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,

 
 

(9.9)

Finally, we define the energy flux density J_U as the rate of energy emission per unit area.In terms of the energy, it can be written

 
 

or, upon substituting in U(t),

 
 

(9.10)

where

 

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.

Photon Gas

e^ik(x+1) = e^ikx

or

 

e^ikl = 1

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

 

which implies

 

where we have gone to a period of 2p to eliminate negative integers, and thus, the total energy becomes

 

as before, with

 

Kirchhoff's Law

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).

 

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 w_n = vk = 2pvn/l.Let f_n = vn/l.Then w_n = 2pf_n.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 Df is

 

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

 

which implies

 

 

<V2> = 4RtDf

(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

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) dw be 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

 

 

(9.13)

Butw = 2pnv/L, so this becomes

 

which implies D(w) = 3Vw²/2v³p².Integrating this, we get

 

or

 

 

(9.14)

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

 

 

(9.15)

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 x_D becomes very large, and we can replace it with infinity.Then the solution is similar to that for photons and we get

 

 

(9.16)

Recall that the heat capacity was defined to be .Then

 

 

(9.17)

This is known as the Debye T³ law.For t >> q, we get the Dulong Petit law, which states that C_V = 3Nk_B.