AIIMS 2005 Physics Characteristic X-rays MCQ Question

To understand the relationship between the frequency of characteristic X-rays emitted and the atomic number $Z$ of the target elements, we need to delve into the physics of X-ray emission.

Explanation of the Correct Answer (B: $f \propto Z^2$)

When high-energy electrons bombard a target material, they can knock out inner shell electrons from the atoms of the target. This creates vacancies in the electron shells. Electrons from higher energy levels then drop down to fill these vacancies, and in the process, they emit X-rays at specific frequencies characteristic of the element.

The energy of the emitted X-ray photon ($E$) is related to the frequency ($f$) of the X-ray by the formula:

$$E = hf$$

where $h$ is Planck's constant.

The energy of the X-ray photon corresponds to the difference in energy between the two electron shells involved in the transition. For heavier elements (higher atomic number $Z$), the energy levels are more widely spaced due to the stronger nuclear charge. Specifically, the energy difference between the inner shell and the outer shell increases approximately with the square of the atomic number:

$$E \propto Z^2$$

Combining this with the formula for energy, we can express the frequency as:

$$f \propto E \propto Z^2$$

Thus, the frequency of the characteristic X-rays emitted from different targets varies with atomic number $Z$ as:

$$f \propto Z^2$$

Why Other Options Are Incorrect

1. Option A: $f \propto \sqrt{Z}$
   This relationship suggests that the frequency increases with the square root of the atomic number. However, this does not hold true in the case of X-ray emission. The energy levels and the resulting photon energies depend on the squared atomic number, not the square root.

2. Option C: $f \propto Z$
   This option implies a linear relationship between frequency and atomic number. While there is a correlation, the relationship is not linear because the energy transitions involve differences that scale with the square of $Z$, as previously discussed.

3. Option D: $f \propto Z^{3/2}$
   This option suggests a scaling that is not supported by our understanding of atomic structure and the transitions that lead to X-ray emission. The energy differences are not proportional to $Z^{3/2}$; rather, they are proportional to $Z^2$.

Summary

In summary, the correct answer is $B: f \propto Z^2$ because the energy of the characteristic X-rays is determined by the differences in electron energy levels, which scale with the atomic number squared. Thus, as the atomic number $Z$ increases, the frequency of the emitted X-rays increases with the square of $Z$, leading to more energetic X-rays for heavier elements.