|“||Blown to dust. Scattered to the wind. With luck, he's spread so thin that he'll never be able to form a consciousness again, much less a body. But don't mistake him for dead, Percy.||”|
However, a way exists to defeat them permanently: scattering. The process of scattering is where the being in question receives a blow that would destroy a mortal being utterly but they survive as an immortal.
However, this blow, which destroys their physical form, also scatters their essence to such an extent that it is believed that they won't ever be able to form a consciousness again, let alone a body. Thus they are scattered and harmless though they are not dead.
During the Battle of Manhattan, Kronos battles the heroes before Annabeth Chase convinces Luke Castellan to retake control of his body from the Titan. Luke decides to sacrifice himself to stop Kronos as he is the only one who can strike his Achilles spot without Kronos regaining control. After Percy Jackson gives him Annabeth's Knife, Luke stabs himself in the side, near his armpit, where his Curse of Achilles was, mortally wounding himself and causing an explosion that is apparently the destruction of Kronos.
However, when Percy tries to comfort Hermes with his son's sacrifice to kill Kronos, Hermes tells him Kronos is not dead as Titans can't be killed anymore than gods can and is instead scattered. Hermes explains it as "blown to dust. Scattered to the wind. With luck, he's spread so thin that he'll never be able to form a consciousness again, much less a body." However, Hermes warns Percy not to mistake that this means Kronos is dead.
Percy wonders if little particles of Kronos are floating around laughing at him as he walks through Tartarus. Its shown that the other Titans believe that Kronos is so scattered that he can't reform anymore.
After taking inspiration from the defeat of Ouranos, Leo Valdez, Piper McLean and Jason Grace take on the task of defeating Gaea once and for all. Jason and the mechanical dragon Festus fly Gaea high in the air, cutting her off from the ground, her source of power and weakening her. Piper uses her charmspeak to put her to sleep and then Leo blasts Gaea with an intense supernova blast of fire which, combined with the power of an onager shot from Octavian, incinerates Gaea's physical form and scatters her essence (killing Leo and Octavian in the process). Like her son Kronos, Gaea is described as being scattered so thin that she won't be able to form a consciousness again or a physical body.
List of scattered immortals
It should be noted that there are a few other ways to permanently defeat an immortal. For instance it is shown that an immortal's essence can be absorbed by a more powerful immortal (an example being when Krios and Hyperion's esscences were sucked into Tartarus) or even a ghost using the correct spells.
A way to permanently destroy a god is called a shadow execration, a spell in which a god's sheut or shadow is captured and then destroyed. As the sheut is a vital part of the soul and stands for a being's impact upon the world, destroying it will kill the god beyond any sort of ability to return, something that even scattering can't do as the god is not truly dead with scattering.