X-ray Paradox

The index of refraction n gives the ratio c/v, the speed of light in vacuum to the speed of the electromagnetic wave in the material. Window glass, for example, can have an index of about n = 1.5 for visible light, with a slight variation in the index with the color of the light. A paradox arises with X-rays because they have an index of refraction value less than one in crystals! What does this behavior mean?

Answer

The index of refraction n for a material is normally stated with regard to the phase velocity, unless indicated otherwise. The phase velocity is vph = c/n(k), where the index is a function of the wave number k. If n(k)<1, then the phase velocity is greater than the speed of the light in the crystal. There is no alarm that the energy is being transported faster than c, for the group velocity is still less than c.

Essentially, travelling harmonic waves in all physical examples require wave packets or groups because of the non-infinite extent of space and/or time. There are two velocities associated with these wave packets or groups: the phase velocity and the group velocity. Harmonic waves or harmonic components have a phase velocity vph = ω/k, where ω = 2πf and f is the frequency. This phase velocity is the velocity at which the wave fronts travel. A group of harmonic waves or wave packet has a group velocity vg = dω/dk, the velocity at which the packet shape or envelope travels—that is, the velocity at which information or energy is transported.

On the atomic level, the slowing of light passing through a material may be considered as a continuous process of absorption and emission of photons as they interact with the atoms of the material. One assumes that between each atom, the photons travel at c, as in a vacuum. As they impinge on the atoms, they are absorbed and nearly instantly re-emitted, creating a slight delay at each atom, which (on a large enough scale) seems to be an overall reduction in the speed of the photons. Quantum mechanically, the scattering is a two-step process of absorbing the incident photon and emitting a new photon.

Experiments in other ranges of the electromagnetic spectrum, particularly in the visible, have shown that by storing the phase information of the incident light beam in a gas vapor temporarily, one can even claim that the light pulse can be brought to rest!