Nucleus is an item that's used for the game's weapon ordering (summon) system through "Special Orders". There are three types/colors of Nucleus: Black, Gold, and Red.
Black Nucleus is a somewhat common reward that you can get from exploration, achievements, shop, etc. It lets you summon/order from a regular banner that contains common materials and weapon exp materials, but it can also give R/SR/SSR weapons at a low rate.
From exploration alone, these are some examples that can give a Black Nucleus:
Around the world, there are Dandelions that contain a Black Nucleus. Attacking the Dandelion will spread them out and you'll need to attack the Dandelions that are floating in the air to get Dandelion as a material, as well as the Black Nucleus.
There are also other plants with unique properties that can give rewards. Dandelions and other plants can refresh daily, but won't give Black Nucleus again.
When exploring, there will be an icon that shows up on the map if you're close to a supply pod. These can be opened easily since they don't require any items or even the area to be cleared, and they contain other random rewards when opened.
Occasionally, they can be found simply floating in mid-air, with or without a drone. There will be an icon on the map that shows up when you're close to one, so you won't exactly have to look up all the time to find random Black Nucleus in the air.
Some chapter missions also give them as a reward.
Gold Nucleus is a slightly more premium item that is harder to get. Compared to Black Nucleus, Gold doesn't have common materials as part of its list of rewards, so the odds of getting weapons are higher. There is also a hard pity at 80 summons to guarantee that players will get an SSR, and getting an SSR early will not reset the counter.
Daily bounty is a daily quest where you can hunt unique enemies for some rewards. You can at least get a Gold Nucleus here, but doing these can also give Black Nucleus, exp, and other rewards, so it's recommended to do it every day.
Story Simulacra is an optional story content that features characters with SSR weapons. It serves as a way to introduce players to new characters and stories, as well as a trial for the SSR weapons.
Completing these can give gacha related currencies like Gold Nucleus, Proof of Purchase, or Dark Crystals.
Doing some of the difficult missions like beating bosses or completing explorations will also give some Gold Nucleus. You can look at these from time to time to see what to do if you're feeling lost in the game.
They can be obtained from special items found in the world as well, just not as common as Black Nucleus.
Red Nucleus is only obtainable from the shop, and it's used to summon on a limited time banners only. Newly released weapons are released on a limited time banner with an increased rate to get them.
In CBT, new weapons don't go into the regular pool once their limited-time banner is over. This could change in the final release version of the game.
In the shop, you can use Dark Crystals, which is the game's main premium currency, to buy items in the shop like the ones shown above.
Even though you can only get Red Nucleus from here, you'll have to consider the other items here as well. Special Purchase is an item that works just like Red Nucleus, but for summoning on Matrix/Matrices.
