The graph shows the X-ray spectrum produced by a Coolidge tube. Which of the wavelengths marked on the graph is produced from the transition of an electron in a higher energy level in the target atom to a lower energy level?
Taking a look at our graph, we see it shows us the radiation intensity of X-rays produced by a Coolidge tube versus the wavelength of those X-rays. And on this wave length access, we see four particular wave lengths marked out. They are labeled 𝑚, 𝑜, 𝑛, and 𝑝. Each one of these wavelengths corresponds to a certain point on this curve. And we want to answer which of these wavelengths has to do with the transition of an electron from a higher to a lower energy level in the target atom. Since the radiation shown in this graph was produced by a Coolidge tube, let’s recall the construction of a Coolidge tube.
These tubes consist primarily of an evacuated glass chamber which has a cathode and an anode within it. Thanks to a potential difference that’s setup between cathode and anode, electrons from the cathode are accelerated toward the anode, the target material. By the time they reach that material, they’re moving at incredibly high speeds and smash in to the target atoms. It’s this collision which leads to the creation of X-rays which are emitted by the tube. And the particular spectral profile of those X-rays which we have shown in our graph has to do with the material that our target is made up of. More specifically, it has to do with the energy levels of the atoms of our target material.
Let’s say that the particular target atom that we’ve picked has three distinct energy levels to it. That is, there are three different levels of the electrons all those atoms can occupy. There is the ground state 𝐸 sub zero, the first excited state 𝐸 sub one, and the second excited state 𝐸 sub two. These energy levels are important for understanding the X-rays that are produced by our Coolidge tube. That’s because it’s in the transition of an electron from a higher to a lower energy level that X-rays are produced.
Not only that, but these X-rays bear the mark of the particular energy structure of the atom from which they were emitted. That’s because the energy of our X-ray is equal to 𝐸 two, the higher energy level, minus 𝐸 one. That is, it’s equal to the energy difference that the electron moved through during its transition. And notice something more, the energy of this X-ray relates to its wavelength. This is because these two quantities are connected through relationship for photon energy which says that the energy of a photon is equal to Planck’s constant times its speed 𝑐, the speed of light, divided by its wave length.
All that to say, the particular energy of our emitted X-ray has to do with the wavelength of that X-ray as well. The wavelength of our emitted X-rays connects back to the energy levels of our particular target atom. We’ve said that this X-ray will have an energy equal to the energy difference between the levels traversed by our moving electron. The question then comes up. If we can get X-rays with energy 𝐸, can we also get X-rays with energy near to, but not equal to, the energy 𝐸? The answer has to do with the separation between energy levels in our target atom.
Here’s a question. If our electron starts out an 𝐸 two, could it transition down to this point we’ve drawn here, that is, somewhere short of 𝐸 one? The answer is no it can’t because there is no energy level there to transition two. And it’s the same if the electron would’ve tried to transition here or here or here or here or anywhere in between 𝐸 two and 𝐸 one. And if we go from 𝐸 two past 𝐸 one, it’s the same story. The electron can’t transition here because there’s no level to go to nor can it transition here nor here. If it starts at 𝐸 two, it either has to go to 𝐸 one or 𝐸 zero. These are the only options.
Here’s what that means. From this particular target atom, we will get X-rays with an energy we’re calling 𝐸. But because transitions across similar energy gaps are disallowed, that means we won’t see energy levels near to 𝐸 from other emitted X-rays. All this means that thanks to the energy structure of our target atom, we would expect an X-ray radiation peak corresponding to the energy level differences in our atom but that, around those radiation peaks, we would expect to see very few emitted X-rays. And again, that’s because of the discrete energy level structure of the target atom. This has implications for our radiation intensity graph.
It means we could expect to see a peak on the graph which corresponds to an energy level difference in our target atom but that, around that peak on either side, there shouldn’t be very much radiation at all. On our graph, we see two such peaks. One is unlabeled and the other is labelled as wavelength 𝑜. The other three wavelength options, 𝑚, 𝑛, and 𝑝, are not located at radiation peaks. But wavelength 𝑜 is. And as we said, this wave length shows the important hallmark of a peak being surrounded by a rapid fall off on either side. So we choose wave length 𝑜 then as a wave length which is produced from the transition of an electron in our target atom from a higher to a lower energy level.