To establish the relationship between the stopping distance \( d \) of an alpha particle and its initial kinetic energy \( K \), principles of energy conservation and Coulomb force are applied.
1. Initial Kinetic Energy: The alpha particle begins with a kinetic energy denoted by \( K \).
2. Potential Energy at Distance \( d \): Upon halting, the alpha particle's kinetic energy is fully converted into electrostatic potential energy. The potential energy \( U \) between the alpha particle (charge \( +2e \)) and the gold nucleus (charge \( +79e \)) at distance \( d \) is calculated as:
\[
U = \frac{1}{4\pi\epsilon_0} \cdot \frac{(2e)(79e)}{d}
\]
Here, \( \epsilon_0 \) represents the permittivity of free space.
3. Energy Conservation: At the point of cessation, the alpha particle's kinetic energy is zero, and its potential energy is equivalent to the initial kinetic energy:
\[
K = \frac{1}{4\pi\epsilon_0} \cdot \frac{158e^2}{d}
\]
4. Solving for \( d \): The equation is rearranged to isolate \( d \):
\[
d = \frac{1}{4\pi\epsilon_0} \cdot \frac{158e^2}{K}
\]
This equation demonstrates an inverse proportionality between \( d \) and \( K \):
\[
d \propto \frac{1}{K}
\]
Consequently, the correct answer is:
\[
\boxed{C}
\]