X-Ray Mirror

X-rays which strike a metal surface at an angle of incidence to the normal greater than a critical angle θ₀ are totally reflected. Assuming that a metal contains n free electrons per unit volume, calculate θ₀ as a function of the frequency ω of the X-rays. The metal occupies the region x > 0. The X-rays are propagating in the x-y plane (the plane of the picture) and their polarization vector is in the z direction, coming out of the picture (see Figure 1.1).

Figure 1.1

SOLUTION

Calculate the dielectric constant of the metal under the assumption that the electrons in metals are free and disregarding any scattering on the atoms, since they are much heavier than electrons (M >> m). Under the influence of the X-rays, the electrons only move a small fraction of the distance between atoms, so the field due to the atoms is nearly uniform over one period of the X-ray. Thus, we may write

Where E = E₀e^{i(k.r – ωt)}, or

(1)

The current density in the field is

(2)

where n is the density of electrons in the metal. Using the Maxwell equation,

(3)

Substituting (2) into (3), we have

(4)

in which we used D = εE. From (4)

(5)

Figure 1.2

For two media with ε₂ < ε₁ (see Figure 1.2),

For θ₂ = π/2, the largest angle of refraction corresponds to the critical angle θ₀ for the incident wave:

So