(II) A 1.0 -cm-thick lead target reduces a beam of gamma...

(II) A 1.0 -cm-thick lead target reduces a beam of gamma rays to 25$\%$ of its original intensity. What thickness of lead will allow only one $\gamma$ in $10^{6}$ to penetrate (see Problem 20$) ?$

Key Concepts

Logarithmic Methods in Exponential Decay

When dealing with exponential decay, logarithms are essential for solving equations that involve exponential functions. These mathematical techniques are used to isolate variables—such as the thickness of the material—in equations that describe the decrease in radiation intensity. This approach is critical for determining the necessary parameters for effective radiation shielding.

Exponential Attenuation

This concept refers to the phenomenon where the intensity of radiation decreases exponentially as it passes through a material. The decrease in intensity follows the relation I = I0 exp(-?x), where I0 is the initial intensity, ? is the attenuation coefficient, and x is the thickness of the material. This principle is fundamental in understanding how various materials, such as lead, can shield against or attenuate radiation like gamma rays.

Attenuation Coefficient

The attenuation coefficient is a key parameter that quantifies the effectiveness of a material in reducing the intensity of radiation per unit thickness. It is determined by the interaction between the photons and the atoms in the material, including processes like scattering and absorption. Knowing this coefficient allows us to predict the required thickness of any shielding material to achieve a desired reduction in radiation intensity.

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