When carbonate is precipitated or comes out of solution in the form of calcite the crystal structure is too small for Pb (lead) to reside in as the ionic radius of Pb²⁺ is large.
The ionic radius of uranium U⁴⁺ is small enough to fit inside the crystal structure.
The decay products for uranium ²⁰⁶Pb and ²⁰⁷Pb reside inside the crystal, if there is non-radiogenic ²⁰⁴Pb present it means lead has found its way into the crystallization process but can be factored out.
That's pretty neat. That eliminates the issue of pre-existing lead isotopes.
The uranium comes from uranium ions dissolved in water.
When the carbonate comes out of solution uranium ions are incorporated into the calcite.
Umm...hard egg shells are calcite. This implies ingested uranium. Of course, it's not like eating yellow cake but it's somewhat unsettling, even in trace amounts. Radium I knew about, and it does show up naturally in some aquifers. But we get radium from thorium (I'm assuming a shorter-lived isotope) and thorium from uranium, but haven't heard anything about this showing up in well water. As a quick check, looked up the EWG of one municipality that had problems with it, and it lists radium (well below the maximum allowable amount), but not uranium and thorium.
Off-Topic Sidebar Ahead:
Why my interest? Something called the Apalachicola Embayment. This cuts across part of the Florida Panhandle and across part of Georgia, and was the old channel of the Gulfstream. Have the report in PDF, but without looking at it, conditions were such that it allowed radium to be precipitated out and trapped in the channel. This was all shaking out when I had college chemistry, mainly about it showing up in high amounts in some municipalities, and was discussed in class. Remediation consisted of drilling new wells and filtration.
Note: A search of Apalachicola Embayment radium turned up a report with the date of 1990, but this was discussed in class way before then. Skimming over it, there's the implication of focusing on radium because it's a high alpha emitter. I don't know if the tests aren't looking for uranium and thorium, or if it's not present. The implication is that they are.
Note II: Since this is off-topic, I haven't posted a link. The title is
Hydrology of the Gulf Trough - Apalachicola Embayment Area, Georgia.
If it hadn't come up in a chemistry class, probably wouldn't even think about it.