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Speak with Gerolt with your Zenith Relic Weapon equipped who will then direct you to Jalzahn standing opposite him. Jalzahn needs you to bring him 12 crystalline objects known as Atmas. Atmas are dropped randomly from FATEs in certain zones around Earzoa when equipped with a Zenith Relic Weapon. After speaking to Jalzahn, the quest will move to your "completed quest log". Collect all 12 Atmas, bring them to Jalzahn along with your unequiped Zenith Relic Weapon and you will be rewarded your Atma Zodiac Weapon.

Atma pieces drop randomly from any FATE in their respective zone. The drop rate is suggested to be somewhere around 2%. However it has been theorized that certain hours of the day will increase the drop rate of specific Atma pieces. See Atma table below.

You must be equipped with your Zenith Relic Weapon in order to get Atma from FATEs; however this can be any Zenith Relic Weapon / job.
Your rating in the FATE doesn't affect your drop rate; however only successfully completed FATEs will drop Atmas. You also do no need to stay for the entire length of the FATE.
Atmas do no go into your Key Items but into your normal inventory slots. Make sure you have free space in your inventory.

The following are the Atma Weapons introduced in patch 2.2:

Curtana Atma

Lv50 | Gladiator's Arm

Holy Shield Atma

Lv50 | Shield

Thyrus Atma

Lv50 | 2H Conjurer's Arm

Stardust Rod Atma

Lv50 | 2H Thaumaturge's Arm

Bravura Atma

Lv50 | Marauder's Arm

Omnilex Atma

Lv50 | Arcanist's Grimoire

Yoshimitsu Atma

Lv50 | Rogue's Arms

Gae Bolg Atma

Lv50 | Lancer's Arm

Artemis Bow Atma

Lv50 | Archer's Arm

The Veil of Wiyu Atma

Lv50 | Arcanist's Grimoire

Sphairai Atma

Lv50 | Pugilist's Arm

AM/PM is irrelevant. The time listed are in GMT+8 and shows when the drop rates for the respective Atma are at its highest.

Atma of the Scorpion

1:00 | Southern Thanalan

Atma of the Water-bearer

2:00 | Upper La Noscea

Atma of the Goat

3:00 | East Shroud

Atma of the Bull

4:00 | Eastern Thanalan

Atma of the Ram

5:00 | Middle La Noscea

Atma of the Twins

6:00 | Western Thanalan

Atma of the Lion

7:00 | Outer La Noscea

Atma of the Fish

8:00 | Lower La Noscea

Atma of the Archer

9:00 | North Shroud

Atma of the Scales

10:00 | Central Thanalan

Atma of the Crab

11:00 | Western La Noscea

Atma of the Maiden

12:00 | Central Shroud