What is the purpose of the ic pin in a nixie tube? For example, the Z573M? Knowing how to drive one, this pin seems to have no purpose.
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IC in nixie tubes is defined as "Internal Connection". The pins are not supposed to be externally connected (equivalent of
NC in other through-hole devices).
The pin(s) provide mechanical stability, and in the case of circular pinouts such as the one listed in the question, prevent indexing ambiguity by ensuring that only one of the pin locations is "missing": 14 pin locations in a circle in this case, of which 13 have pins present.
IIRC (but cannot find a relevant datasheet at the moment) some nixie tubes designed for extremely low temperature use, have an internal heater coil wired from the
IC pin to the Anode(s), and thus at Anode potential. This is to improve operational life of the tube, as operation below 10 degrees centigrade degrades the longevity of the tube. Please take this last paragraph as possibly faulty memory on my part, rather than as fact.
ic = "internal connection"
As described, it is not required to use the tube, but it is used for internal mechanical support and/or interconnect within the device.