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First, I know there is a question similar to this already, but I am asking it in much more detail and I have done much more research than the original asker. Say I wanted to ionize a 3 metre long path of air using a laser. The ionization energy of diatomic nitrogen is about 1503 kj/mol, which translates to 0.00000000000000000249668 J, or 15.583051547 eV.

Using single photon ionization, I would need each photon to be about 0.0000000795634687903593572263966547575179839 metres in wavelength, aka roughly 80 nm, given the formula E = hc/y, where E is the energy of one photon, h is Plancks constant, and y is the wavelength of the photon.

Since the density of air is 0.01225 g/m3, and the mols per gram of nitrogen is 1/28.013, the number of nitrogen molecules in one metre cubed of air is about n = 2.6x10^21 m^-3. V = (pi)(r^2)(L). The radius of the beam will be important later. The number of molecules to be ionized is nV. Now I have no idea what the recombination rate of diatomic nitrogen is, but that will be vital in determining power requirements, since P = E/t. Im assuming radiant energy density is also pertinent.

Formula for total energy needed is E = nhv, where n is total number of molecules, h is Plancks constant, and v is the frequency. Now before I ask about the recombination rate, there's another matter that id like to touch upon. One might look at the formula for volume and say, "just make the radius super small mate! Easy solution." Sure, you can decrease the waist beam radius or whatever you wanna call it. However, a laser is not uniformly thick, as Gaussian optics tells us. The function for the width of a laser along its length, denoted by z, is w(z) = w0 sqrt(1+(yz/(pi)(w0^2))^2). My apologies, I dont know how to format the formulas yet, im new to posting. y is the wavelength, w0 is the waist beam radius at its least value. Basically what this function tells us is that the further the laser beam propagates and the smaller your w0 value, the faster your laser beam will diverge, or widen. Obviously this is annoying and a major hindrance.

So now here come the questions: 1. What is the recombination rate of diatomic nitrogen? 2. What is the limit to how small you can make the waist beam radius of a laser? 3. Is there any way to overcome the divergence thing using a lens or something of the sort? 4. Do I have to ionize ALL of the molecules along the length of the beam, or will some cascading avalanche effect take over, or is it sufficient to only ionize a certain percentage of the path so it becomes conductive? 5. Would some other method of ionization, such as multiphoton ionization, tunneling ionization, or collisional ionization be more feasible? 6. Is there any other gaseous molecule, compound, etc that would be more easily ionized?

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