One thing I should clear up right at the beginning, this clock is "atomic" not as in "uses special properties of atoms" but instead as in "atomic age" - a radioactive renaissance era in which it one day seemed that plutonium would be available at every corner drugstore. The two are about as unrelated as you can get besides having atoms involved (which describes most of everything, come to think of it).

The kind of ultra-precise, Einstein-confirming atomic clock you are probably thinking of exploits the fact that when some elements (like cesium-133) are hit with a microwave radio signal of a certain frequency, the electron configuration will change just enough to affect the magnetic properties of the atom. You can then use a magnet to determine which atoms were hit with the correct frequency, and really dial it in to a staggering level of precision.

I've seen no better explanation of this than the one CuriousMarc did when he repaired his HP 5061A, which he has incidentally also entered into this very same 1 Hz challenge.

Normal atomic clocks are not radioactive in any way, except for the very latest research into "nuclear clocks" which use the properties of the nucleus itself rather than the electron cloud, and use radioactive materials only by coincidence.

This also isn't an atomic powered clock. I know one of those "nuclear batteries that last 50 years" make the rounds every few years, but even if you could make one yourself the power output would be too low to drive even an LCD clock. We also aren't building an RTG. For what I'm doing, I'm not even capturing any radiation energy at all.