Here is the full list of all 20 Crystal Crisis achievements.
Perform a quad-crystal combo
Perform a tri-crystal combo
Complete the tutorial twice
Detonate a Cluster with at least 12 Crystals in it
Win a round with your grid at least 90% full
Perform 10 different bursts
Detonate a linked chain of 20 or more Crystals
Detonate a linked chain of 10 or more Crystals
Beat the game with Astro Boy and Black Jack
Beat the game with Aban, Jim and Tina
Beat the game with Astro Boy, Curly, Ninja and Quote
Beat the game with all the female characters
Perform a quin-crystal combo
Win a match without ever rotating a Crystal pair
Detonate a Cluster with at least 18 Crystals in it
Play a ranked online match
Play an online match with a friend
Win two ranked online matches in a row
Win five ranked online matches in a row
Win 10 ranked online matches in a row