Question

The atomic number of the anode material in an X-ray tube is \( Z \). If the X-ray tube has an anode-to-cathode voltage of \( V \), the efficiency of X-ray production is proportional to:

• A. \( \frac{Z}{V} \)
• B. \( ZV^2 \)
• C. \( \frac{V^2}{Z} \)
• D. \( ZV \)

Hint:

X-ray production is highly inefficient. At typical diagnostic voltages (e.g., \( 100 \text{ kV} \)) with a tungsten target (\( Z = 74 \)), the efficiency is roughly \( \eta \approx 10^{-9} \times 74 \times 10^5 \approx 0.0074 \), meaning less than 1% of the energy turns into X-rays while over 99% is wasted as heat!

Answer 1

The Correct Option is D

Concept: The production of X-rays occurs when high-velocity electrons accelerated through a potential difference strike a metallic target (anode). Most of the kinetic energy of the electrons is converted into thermal energy (heat), while only a small fraction is converted into electromagnetic radiation (X-rays). The efficiency (\( \eta \)) of this conversion process depends on both the material characteristics of the target and the accelerating electrical energy.

Step 1: Establishing the theoretical formula for efficiency.
Empirically, the efficiency of X-ray production in a standard thick-target X-ray tube is described by the relation: \[ \eta = k \cdot Z \cdot V \] Where:
• \( \eta \) is the efficiency of X-ray production (the ratio of emitted X-ray energy to the total kinetic energy of incident electrons).
• \( k \) is an empirical proportionality constant, typically valued around \( 1 \times 10^{-9} \text{ V}^{-1} \) to \( 1.4 \times 10^{-9} \text{ V}^{-1} \).
• \( Z \) is the atomic number of the target (anode) material.
• \( V \) is the operating tube potential (anode-to-cathode voltage) expressed in volts.

Step 2: Analyzing the proportionalities.
From the linear expression, we can isolate the direct relationships:
• \( \eta \propto Z \): A higher atomic number material provides a larger positive nuclear charge, increasing the Coulombic interactions that cause Bremsstrahlung radiation.
• \( \eta \propto V \): A higher accelerating voltage increases the kinetic energy of the incident electrons, yielding more energetic and efficient radiative interactions. Combining these individual dependencies leads directly to: \[ \eta \propto ZV \] This mathematically matches option (D).