I'd like to try to get a concise list of essence requirements for each of the tiers, and I figure the best way to go about it is crowdsourcing the values.
If you could post the amount of essence you sacrificed and how many rewards you got, we might be able to get enough to start a chart. I'm willing to bet that the determining amount is a function vs hard-coded arbitrary amounts, but we won't be able to determine that until we get enough examples.
I'll update this list as I get replies
Tier 1 (100-499)
1:
2:
3: 300
Tier 2 (500-1999)
2: 1000
3: 1500
Tier 3 (2000-5999)
2: 4083, 5016, 5999
3: Does not drop
Tier 4 (6000-8999)
1: 8082
2:
3:
Tier 5 (9000-19999)
2:
3:
Tier 6 (20000-49999)
1: 35401
2:
3:
Tier 7 (50000+)
2:
3: