Travel to the Nucleus where you'll meet Grand Zealot Richter in front of the entrance. The Children of Atom require a ritual before allowing anyone in their sanctuary. The Grand Zealot instructs you to drink from a small spring to prove yourself worthy in their deity Atom's eyes.
Head to the waypoint on the map, and you will arrive at Atom's Spring. Bottoms up.
A ghostly figure of a woman will appear. Follow the figure through the woods.
Follow the figure to a Holy Site. After you put down a few ghouls in the area, head inside the Children of Atom Shrine.
Investigate the scrawled text on the bookcase near the Periodic Table of Elements chart to reveal the Shine password, Mother.
Use the Shrine password with the terminal to unlock the cage.
Collect the Mother Icon. There's a note on the desk you can collect, a second on the shelf outside of shrine on the shelves and a, perhaps, a Caps Stash under the same shelf.
Return to Richter. Discuss the icon and your vision to learn more about The Children of Atom. Opt to join their ranks and Richter will hand you Robes of Atom's Devoted and the quest will be end.