Question Video: Comparing the Accelerating Potential Difference of a Coolidge Tube to the Characteristic Spectral Lines Produced Physics

The solid line on the graph shows the relative intensity of X-rays in an X-ray spectrum of different X-ray photon energies produced by an electron beam striking a target. The dotted line on the graph shows the bremsstrahlung radiation that would be produced by an electron beam striking the same target but accelerated across a smaller potential difference. Which of the following correctly shows the characteristic lines that would be observed when the smaller-voltage electron beam was used? [A] Spectrum A [B] Spectrum B [C] Spectrum C [D] Spectrum D [E] Spectrum E

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Video Transcript

The solid line on the graph shows the relative intensity of X-rays in an X-ray spectrum of different X-ray photon energies produced by an electron beam striking a target. The dotted line on the graph shows the bremsstrahlung radiation that would be produced by an electron beam striking the same target but accelerated across the smaller potential difference. Which of the following correctly shows the characteristic lines that would be observed when the smaller-voltage electron beam was used?

Before we look at the graphs that show these characteristic lines, let’s clear some room. Now then, what we’re looking for are the characteristic lines or the sharp peaks that show up in an X-ray spectrum like this for the case where the electrons in the electron beam are accelerated through a smaller potential difference. This means that not only has a lower intensity but also a lower maximum energy than normal.

However, characteristic lines are not produced through the same method which produces these smooth curves, which is bremsstrahlung. Instead, they are produced through something called energy level transitions. They also exist at very specific X-ray energies, regardless of the bremsstrahlung curve. With this in mind, let’s start looking at some of the graphs showing these characteristic lines.

Spectrum (A) shows two characteristic lines that seem to be in the same relative position as on the line that represents the larger potential difference. However, characteristic lines always appear at specific X-ray photon energies independent of the bremsstrahlung curve. They would not be shifted over to the left like this if the entire curve had less energy. They would instead be in the same place, something like this. So spectrum (A) cannot be it.

Looking now at spectrum (B), we see that it has the same problem as spectrum (A), with the characteristic lines being shifted over to the left even though they should be in about the same position as on the line representing the larger potential difference. So (B) is not it either.

Spectrum (C) shows the two characteristic lines at the same energy and same intensity as on the line representing the larger potential difference. These characteristic lines’ positions for energy are correct, but not so for intensity. When we look at X-ray intensity, we are looking at the total number of X-rays at a specific energy. And these X-rays are coming from both the bremsstrahlung and form the smooth curve and from electron energy level transitions and form characteristic lines. Together, they form a complete X-ray spectrum, their intensities being added together.

The characteristic lines by themselves may look something like this and actually will have rather small intensities. But these peaks are added to all of the intensity of the bremsstrahlung radiation, causing them to appear on top. Characteristic lines do not have specific, unchanging intensities, only specific ranges of energies that they can exist at. They depend on the bremsstrahlung radiation too.

So when we look at spectrum (C), we see that the characteristic lines are too intense, since they depend on the bremsstrahlung of the line showing the smaller potential difference, meaning that the peaks should be lower but still distinct. So spectrum (C) cannot be it.

Looking now at the characteristic lines of spectrum (D), we see that the characteristic lines are in the correct positions for energy. And they appear to have an appropriate intensity too. Their position changed by the difference in bremsstrahlung between the larger and smaller potential difference lines. However, these characteristic lines seem to have the same intensity, despite the varying potential difference of the spectra. This should not be the case. And to see why, let’s recall our knowledge about energy level transitions.

In order for an energy level transition to occur, an electron from the electron beam must collide with and eject a low-energy-level electron, which then causes a higher-energy-level electron to transition downwards to fill the gap, releasing an X-ray photon as it does so. In order for all of this to happen though, the initial incident electron from the electron beam has to have enough energy to actually eject the low-energy electron in the atom. If the electron from the electron beam doesn’t have enough energy, it will be repelled away from the atom.

The energy of the electrons in the beam is the product of 𝑞, the charge of an electron, and 𝑉 t, the potential difference across the tube. So a smaller potential difference means that the electrons in the electron beam will have a smaller energy and thus have a smaller chance of being able to eject any of the electrons in the atom. Some electrons will still be able to be ejected, but just not as many.

So because the spectrum of the dotted line has a smaller potential difference, there will be fewer low-energy-level electron ejections as more of the incident electrons from the beam are repelled away, which means there will be a smaller amount of energy level transitions. And since the characteristic lines come from energy level transitions, the characteristic lines will be less intense. Spectrum (D)’s characteristic lines are about the same intensity, so we know it can’t be it.

Spectrum (E)’s characteristic lines, though, are appropriately less intense, given the smaller potential difference. Meaning that, out of all of these spectra, the one that correctly shows the characteristic lines that would be observed when the smaller-voltage electron beam was used is spectrum (E).

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