One half the \(\gamma\) rays from \(^{99 \mathrm{m}} \mathrm{Tc}\) are absorbed by a 0.170-mm-thick lead shielding. Half of the \(\gamma\) rays that pass through the first layer of lead are absorbed in a second layer of equal thickness. What thickness of lead will absorb all but one in 1000 of these \(\gamma\) rays?

The exponential decay law is incredibly important when dealing with radioactive materials and their interactions, including the absorption of gamma rays, such as the ones from ^99mTc. In essence, it tells us that a property decreases over time at a rate that is proportional to its current value. For gamma rays travelling through a material, the law helps us understand how their intensity diminishes as they penetrate further.

Mathematically, the exponential decay law is given by the equation: \[N = N_0 \times \text{e}^{-\text{lambda} x}\]where \(N\) is the number of unabsorbed gamma rays, \(N_0\) is the initial number of gamma rays, \(x\) is the thickness of the material they are passing through, \(\text{e}\) is the base of the natural logarithm, and \(\lambda\) is the decay constant. This equation implies that for each additional layer of material, a certain fraction of gamma rays will be absorbed, continuing in an exponential fashion.

Understanding the decay constant, represented by \(\lambda\), is crucial to mastering the concept of gamma ray absorption. It quantifies the absorptive properties of a specific material—here, lead when it comes to absorbing gamma rays. In simple terms, the decay constant tells us how likely it is for a gamma ray to be absorbed by a certain thickness of a barrier.

The higher the value of \(\lambda\), the greater the material's ability to absorb gamma rays, and the more effective it is as a shield. By calculating the decay constant, as demonstrated in the exercise solution, we were able to determine the overall shielding effect of lead for gamma rays emitted from ^99mTc. The decay constant is unique to each combination of radioactive source and shielding material and is determined experimentally or found in literature.

Radioactive shielding is the use of materials to protect against the harmful effects of radiation, such as gamma rays. The effectiveness of shielding depends on the material's density, thickness, and the type of radiation. In our problem, lead serves as the shielding material against gamma rays. To protect against gamma rays, a high-density material like lead is typically used because it can absorb more radiation than materials with a lesser density.

The process of selecting and designing effective shielding must consider the type of radiation, its energy levels, and the decay constant we've calculated. By determining how much material is needed to reduce radiation to a safe level—or in the case of this exercise, how thick the lead needs to be to allow only one in 1000 gamma rays to pass through—we can design proper shielding for various applications, like medical imaging or nuclear power plants.