New NPC in Gridania Ronan Kognan sheds some light on Deaspected crystals and their meaning.
---------------------------------------------
Muaha, muahahaha! You could read all the literature in Eorzea, friend─every last treatise and tome─and still you would know nothing of deaspected crystals. Nothing!

Ronan Kognan: But even if books could help you, you'd have no need of them. Nooo, not a single page. Ronan Kognan knows all! For the Ashcrown Consortium is his employer, and crystals his employment, muahahaha!

Yeees, my precious little beauties are hard to come by of late. Every clan and creed south of the Bloodbrine is after them, each for their own ends. You see, the fate of Eorzea is tied to the crystals, my friend.

Should the crystals die, so too shall the realm. And so we must study whatever dead crystals we can, as best we can. We must find out why their light has left them.

So bring me any deaspected crystals you come across, adventurer. I shall cut them to pieces and grind them to powder, muaha! Touch them, smell them, taste them, muahahaha! I shall do whatever I must and more to discover the cause of their death! Muahahaha!
------------------------------------------------

I don't have any deaspected crystals on hand, but he is not overtly hard to find go check him out.