A quick calculation

I talked previously wrote about trying to photo-ionise a gas using the x-rays from a wire array z-pinch. The cylinder of wires implodes when we pass a high current pulse through it, and the imploding wires form a dense column in the centre of the cylinder, called a pinch. The plasma is extremely hot – over two million ‘C, and will emit light a bit like a hot poker. Instead of glowing red hot, most of the energy will be in the ‘soft’ x-ray range (soft here means not as energetic as ‘hard’ x-rays).

I wanted to get an idea of how much of the energy from the pinch will go into the gas cell above. There are several hurdles for the x-ray light to cross. First of all, we need a small window in the gas cell to let x-rays in whilst preventing the gas from leaking out. This window is made from silicon nitride, and is only 100 nm thick – a thousand times thinner than a human hair! However, it still blocks out quite a lot of x-ray light, around 50%.

Now that some of x-rays have made it through, they need to absorbed by the gas in order to heat it. Some x-rays will pass all the way through without being absorbed, and so their energy is wasted. I’m making a lot of approximations in my calculation, so I assume the fraction of photons absorbed is proportional to the density of the gas.


I put all of these calculations into a little computer code, so I can quickly change parameters like the density of the gas, or how hot the plasma in the pinch is. I use python to program in, but I make extensive use of traits, traitsui and chaco, all by Enthought inc. The first two are programming libraries that make it easy to write graphical (as opposed to text based) programs, and chaco is the graph plotting program that works well with traits. It took me a few hours to write this, but it should save me time in the long run – when my advisor asks me what would happen if we halved the gas pressure, I can find out the answer instantly instead of redoing all the calculations!

It doesn’t look like much energy is getting absorbed by the gas – of the 10 kJ (about half an M&M!) emitted by the pinch, only about 50 J are absorbed, or 0.6%. Most of the loss is because the gas just isn’t that good at absorbing photons – instead of looking like a dense fog, it looks like clear day. 

I’ve made quite a lot of assumptions to get some number out – I’ve approximated the pinch as an instantaneous point source of x-rays, as a perfect black-body, and assumed the radiation is deposited uniformly within the gas, which definitely isn’t true. I’ve assumed the gas doesn’t move (not a bad approximation) so I can use the heat capacity at constant specific volume, but simultaneously that the gas is in thermal equilibrium, which is a bit contradictory. I’ve used the ideal gas equations, which are pretty good for low density noble gases. Data for the photon transmission of the window came from LBL, and for the x-ray absorption by gas from NIST.

These sort of quick calculations are extremely useful in physics, especially when setting up a new experiment takes a lot of time and effort. I think we should see some results from the experiment, but it doesn’t look like we’ll produce a very hot gas, certainly not one which is fully ionised. If we see something radically different from this simple calculation, then it might be time to do some proper computer modelling and work out what’s happening!

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