Briefly explain why the magnitude of the absorption coefficient \((\beta \text { in Equation } 21.18)\) depends on the radiation wavelength.

The absorption coefficient (β) is a parameter that characterizes how easily a material can absorb radiation. A higher absorption coefficient indicates that the material absorbs more radiation at a given wavelength, whereas a lower absorption coefficient indicates that the material is more transparent to radiation at that wavelength.

The absorption coefficient depends on several factors, including: 1. Material properties: Different materials have different electron structures, which determine how efficiently they can absorb photons (particles of electromagnetic radiation) with different energy levels, and therefore, different wavelengths. 2. Temperature: The temperature of the material can cause changes in its electron structure, which can in turn affect the absorption coefficient. 3. Pressure: Variations in pressure can alter the density of the material, modifying its absorption properties.

For a given material, the absorption coefficient generally varies with the radiation wavelength. The reason for this variation lies in the electron structure of the material: when the energy of an incoming photon (which is proportional to the inverse of its wavelength) matches the energy gap between different electron energy levels, the photon is absorbed, and an electron is excited to a higher energy level. Since different materials have different electron structures, the energy gaps and therefore the wavelengths at which absorption occurs will be different. Since the absorption coefficient depends on the probability of photons being absorbed by the material, and this probability depends on the wavelength of the radiation, it follows that the absorption coefficient will also depend on the radiation wavelength. In conclusion, the magnitude of the absorption coefficient depends on the radiation wavelength because different wavelengths correspond to different photon energies. This, in turn, affects the probability of a photon being absorbed by a material, which is the basis for the absorption coefficient.