Third Zap's the Charm
Let's consider the Goa'uld and their needs. They were not known for their patience or tolerance, but their technology was definitely first rate. Only the Ori or the Asgard seemed to have working technology as sophisticated. It is likely the Ancients also had technology as effective, but few working samples remained.
The Goa'uld have mastered the ability to compress energy into small devices such as Zat'nik'tel and Staff weapons. Both weapons have amazing capacity for energy emission and matter disruption.
If we consider the effects we have seen:
- the ability to stun almost any creature (there are exceptions but it's a short list)
- the ability to kill any creature struck more than once in a short period of time, likely due to neural collapse
- the ultimate disintegration of a target with a third shot
- The energy of a Zat can be conducted through both metal and water
- The energy of a Zat is destructive to electronic devices (but does not work on the Replicators)
My initial assessment would call this a form of coherent electron-beam weapon or a lightning-gun. The beam of coherent electrons guided by an invisible laser which polarizes the air and allows the electrons to travel to the target. The guidance effect in this case, must allow for a massive number of charged particles to reach the target.
This meets several of the criteria for our weapon:
- Destroys electronics
- Can be conducted by metal or water
- Can cause a shock to human or humanoid-like creatures likely by overloading their neural systems
But how do we get to the matter-destruction possibility?
Perhaps the Goa'uld are advanced enough to use the anti-particle of the electron, the positron. This anti-matter particle would act pretty much like an electron would, but if it could be caused to stay with a target for more than a few minutes, perhaps by clinging to the matter in some undisclosed fashion, a target could:
- Be effected just like they would if zapped by a heavy charge of electrons
- They would retain enough charge that a second exposure would exacerbate the first, killing a potential target.
- A third exposure might cause enough of an anti-particle reaction to cause annihilation (non-explosively?) reducing matter to dust.
This might also explain how the Zat'nik'tel could possibly have been used to boost hyperdrive engines when amplified by Ancient knowledge.
Granted this would make the Goa'uld weaponry very strange by our standards, but if they could control anti-matter streams in that fashion, it would make most armor obsolete as all any excess matter would do once struck by a Zat is to further hold the positron charge even better.
After reviewing all of the sources I could on Zat weaponry, I have to conclude, and is held up by even the show's designers, that no one considered the Zat and its effects very thoroughly and any speculation here is clearly my own.