It is true that the name "Immortals" was never in use by the Achaemenids themselves; Ironically, the designation given to the corps of 10,000 was "Companions" or "Ânûshiyâ".

Errata: I previously mentioned a 2,000 strong elite dé elite unit of spear-bearers, however for historicity, I must correct this figure: The most attested figure appears to be a highly disciplined and decorated unit of 1,000 men, surrounded by 9,000 Companions. However the point on whether this household organization remained intact until the Macedonian invasions is still valid.