Doesn't clarify this to me either. And I'm gratefull that I'm not so confused about the actual physics here as I read Watsisname's responses.
In fact there are important points still to be adressed. I suspect the amount of energy you need for the gamma ray laser is way beyond any other energetic output in this system. Still, why mercury? why not xenon? Where are the new electrons really coming from?
these ions are called cations which means that they strip electrons off of atoms, it needs to do this so it can add electrons back into it and repel the cations, as the cations fly they push on an atom which propels it forward
This part is confusing to me. If I understanded correctly this is a many-steps process: (1) Input energy to separate negative charge from positive (electrons and nucleai), in your case ionizing mercury atoms. (2) Then you need more input energy to make the electrons turn back and collide with the nucleai (in reality you don't need energy they would tend to return since they atract, that's exactly the reason you needed energy in the first place to separate them... anyway), but why? why separating and joining them which is, as far as I see, reversing the process to return to the initial conditions would give the cations more kinetic energy than the electrons? 3) You want the electrons to kick the cations away to gain thrust? wouldn't the electrons repel in the opposite direction with the same force? Assuming again that both would repel (which is absurd) you seem to want to break Newton's third law in this step. Or maybe you want to push the cations by pushing the electrons with the gamma ray laser (a second use of it)? In that way they would be pushed indeed in one direction, but then why pushing the electrons to make them push the cations? why don't use the gamma ray to directly push the cations without the electrons intervening? And why you even need the first and second step? why you need to separate charges and then make them collide?
I see a lot of steps here, and remember that in each process you transform energy from one object to other and in one form to another so you are inevitably losing a lot of energy by dissipation (imposible to overcome in the end because of the second law of thermodynamics).
Even if electrons and cations attracted each other (which doesn't happen at all) I see this billiard as an ineficient complex path to dissipate more and more energy while you seem to see not only a perfectly efficient path but even a chain that is able to aplify the amount of energy in some magical way by distracting the attention of the inquirer of how conservation laws work by increasing the complexity of the chain of interactions you seem to need until non-physics leak into the explanation silently and we get deceived.
So, as far as your explanation goes I can only see violations of either Newton's 3rd Law, the 2nd Law of Thermodynamics, Coulomb's Law, or a combination of those. Nature is a hard coded restrictor some times.
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By the way I see that both Watsisname and me have talked about Ion engines (like the one used by the Dawn spacecraft). This is because the closest to your invention plausible idea that one can think of is exactly that, an ion engine. It works by accelerating ions with an electric field (no by an energy-expensive gamma ray pushing-electron-pushing-cation laser). Is a clever idea. And it is currently been used in space. But is has some (many) dissadvantages and dosen't fit to interstellar probe needs.
An ion engine can't propel to large speeds (remember, you want to reach proxima centauri, which is far away) and above all it can't give you large accelerations (remember you want to gain huge speeds in a reasonable time to get there in a reasonable time). An ion thruster needs two days of continuous acceleration to gain 0.025 km/sec (a car driving through a highway). You would need 9 years to gain sufficient speed as to escape the Solar System gravitational influence. 10 years just to gain that speed! but you would need a lot more time to actually physically get out of the Solar System since at that speed you would still need 16 years to reach the Voyagers (that's in principle interesting since the Voyagers have taken 40 years to reach that distance while our immaginary ion spacecraft needs 25 years). Been able to get to the escape velocity of the Solar System from Earth means a staggering 42 km/s speed, but with that speed you would still need 30.000 years to reach Proxima Centauri!! The advantage of these thrusters is that you can accelerate constantly even if the acceleration is puny. But it needs fuel (the matter were the ions come from), which is a Xenon gas in in the case of Dawn (not mercury as in your scenario). The need of fuel means that the accelleration is not eternal, it would finish when the Xenon has been depleted (or when your mercury atoms have been fully ionized and ejected). How can one extend the life of the engine? By adding more fuel. But adding more fuel implies more mass for the spacecraft and that means that the same amount of thrust is going to generate less accelleration (inertia is not our friend here), so there's an equilibrium point between the amount of time you want the engines to operate and the amount of acceleration you could possibly have, all controled by the kind of gas you are using and the total amount of it. So this technology has its limits. Also to generate the electric field to expell the ions you need an energy source (in the case of Dawn it was provided by solar panels), analogous to your gamma ray laser using some energy coming from "no one knows yet" to first ionize your mercury disk. If you plan an ion engine to go outside of the solar system first consider if there is sufficient time for it to reach those speeds with that amount of acceleration, second consider if there's sufficient fuel to have that acceleration and third consider if there is too much fuel to been able to push the spacecraft to that speed with your energy limitations. In this case you would need some nuclear battery onboard to power the electric field (Dawn is close to the sun so the solar panels are able to collect some energy but farther from Jupiter this is currently impossible, as you can see by the fact thet Cassini, Voyager, New Horizons etc... are all solar-panel-less spacecrafts). So you would need a nuclear reactor as the one used by the Voyagers to make your extremely slow ion drive work. So assuming that the Xenon is infinite and lightweight you still need to acount for the nuclear fuel to make your engine work. That secondary fuel can endure maybe 50 years? Making your spacecraft reach maybe 100 km/s at the final moment when the engine stops? At that speed you would still need 12.600 years to reach Proxima Centauri. And that is why no ion thrusters have been proposed for missions like that (yours is a special ion thruster that violates physics).
That's why ion engines are not designed for missions that require huge speeds to arrive to far away destinations but rather for nearby missions that requiere progressive changes in orbital parameters. As I said before getting to Proxima Centauri is a very difficult task. No ion drive and no chemical rocket could reach it in our lifetimes or even our civilization lifespan. We need a breakthrough in our understanding of propulsion, and that's where the idea of a Solar sail pushed by lasers comes. You see, chemical rockets, ion engines, even fantastic disk propulsion futuristic device that shots ions from a mercury disk and electrons from a battery, all of those are just the same schema. All of those need onboard fuel to propel and even onboard fuel to make the engine start using the fuel to propel. That's energy stored in matter, and matter has mass, mass that has to push itself (and the entire spacecraft). But what if we made a spacecraft propeled by energy that is not embeded in a fuel onboard the spacecraft? That's the idea behind solar sails; we consume a lot of energy here on earth (nuclear energy is the proposal) to power a laser that shots to a sheet of reflective material in space, no need for fuel, engines or batteries onboard the spacecraft, all of that can be placed and controled here on Earth where we have plenty of material while the spacecraft only need to have the instruments onboard (and a little battery to power them), that means little mass been pushed from home and not by itself. Also the idea is to make instrumets very lightweight (something that you can't do with some fuel because it would mean you are cutting it's mass and it's energy output) as to arrive to the 1 gramm limit for the entire spacecraft. That is trully efficient. The escape velocity of the Solar System from Earth's distance is 0.014% the speed of light, the expected velocity for a sail like this is 20% the speed of light. That is the first realistic proposal to travel to the closest star.
Your device, even if it worked, would never be even close to that, just by the limitations of the rocket equation, by which your disk propulsion is limited and the solar sail is not.