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
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).