No, that's not at all what he means, NOT an actual atomic clock!
This is an atomic "wrist watch." But no, that's not what he's talking about...
I recall a friend bought what was called an "atomic clock" about 12 years ago. I wondered about that (it was only $40!), but then found out it was actually just a plain quartz clock with a small radio receiver that gets powered up once a day and picks up the signal from
WWVB (whose signal is of course controlled by some of the best and most accurate atomic clocks in the world), and resets the clock's time to the received time. The receiver is powered up for only a minute or two a day, and so it's always accurate to about a second, or whatever the quartz clock drifts in 24 hours. And yes, I was appalled that a sync-by-radio clock was called an "atomic clock." They're still called that, and are about $10 at any Target or Walmart.
Joe, thanks for your call and question. The radio in these things doesn't take up a whole lot of power, and as it only operates a minute a day, battery life shouldn't be too much shorter than a regular quartz clock.
A lithium battery should indeed have a substantially longer life than an alkaline battery, perhaps two or three times as long. However, lithium batteries cost many times many times alkalines cost, so it may actually cost more of the same time period to use lithium batteries. If changing the battery every couple of years instead of every six months (or however long you've found alkalines to last in your clock) is worth it to you, then go for the lithium.
I think watch batteries are lithium (or even silver, like the batteries that ran the Moon Buggy in the later Apollo missions) to give them greater power density than alkaline, which is arguably needed in a watch - it would be really inconvenient to change a watch battery every few months, when it's possible without too much expense to make them last a few years.