Are they inversely proportional to each other?
This is the case for the atoms, I think.
The problem is that, for those isotopes like uranium 238, the half-time is as long as 4.4 billion years, and this would imply an extremely narrow line-width. But the interaction inside the nucleus is very strong.
The intrinsic width of a line due to a decay process is related to the lifetime $\tau$ (not the half-life) by the energy-time version of Heisenberg's Uncertainty Principle $$ \sigma_E \, \tau = \frac{\hbar}{2} \,.$$
Recall that the standard deviation of an exponential decay is the lifetime.
Be careful that you do not confuse the experimental resolution with the intrinsic width (in other words, you can't use a NaI crystal for this measurement).
