Lesson Explainer: X-ray Tubes Physics

In this explainer, we will learn how to describe the production of X-rays using an X-ray tube and how the spectrum of X-rays produced can vary.

X-rays are produced by decreasing the energy of charged particles.

An electrically charged particle that is moving has an associated kinetic energy.

When the energy of such a particle decreases, one way that the energy can be transferred from the particle is as the energy of electromagnetic waves, which the particle emits. X-rays are electromagnetic waves, specifically with frequencies, , given by the range

For electrons within atoms, an electron decreases in energy by emitting electromagnetic radiation.

Electrons in atoms have specific energies, which are termed energy levels. When an electron transitions from an energy level to a lower energy level, it emits electromagnetic waves with an energy equal to the difference between the energies of the higher level and the lower level.

This can be expressed as where is the energy of the waves and is the difference in energy between the higher energy level and the lower energy level.

The energy of the wave can be modeled as the energy of a photon, in which case this relationship is expressed as where is the Planck constant and is the frequency of the photon.

To produce an X-ray photon with a frequency of Hz, the energy level difference is given by

The energy needed to remove an electron from a hydrogen atom is only J. This means that to produce even the lowest energy of an X-ray photon, a larger energy change is needed than is possible for a hydrogen atom.

We see from this that an atom of an element with many electrons is required so that some of the electrons have enough energy to produce an X-ray photon when they transition to a low energy.

Elements such as tungsten, rhodium, and molybdenum are used to produce X-ray photons. The following figure shows the basic structure of the electron energy levels of a tungsten atom.

The nucleus is at the center of the atom. Electrons at different distances from the nucleus have different energies. The greater the distance from the nucleus, the greater the electron energy. The electrons themselves are not shown.

The number of electrons that can have a given energy increases with distance from the nucleus. The following diagram represents the approximate density of electrons at different distances from the nucleus. Where the blue color is densest, the density of electrons is greatest. This representation simplifies the issue of showing the positions of individual electrons.

The diagram does not represent the full complexity of the electron structure of a tungsten atom. There are in fact 20 different possible energies that an electron in a tungsten atom could have. These energies cannot easily be represented in a simple diagram, and it is not necessary to understand the details of the electron energy level structure to understand the basic principles of X-ray production.

Let us consider how a high-energy electron in a tungsten atom can transition to a low energy.

Let us suppose that an electron very close to the nucleus is ejected from the atom. The density of electrons close to the nucleus is then decreased, as shown in the following figure.

Electrons with greater energy than the ejected electron can now transition to the energy level of the ejected electron.

A transition of a high-energy electron to a low energy level is shown in the following figure.

We have seen that X-rays can be produced by ejecting electrons at low energy levels from atoms. We will now consider how this can be achieved.

A Coolidge tube is a device for producing X-rays. The following figure shows a Coolidge tube.

A Coolidge tube contains an anode and a cathode that are within an evacuated vessel. The basic principle of a Coolidge tube is, therefore, to produce a potential difference across a cathode and an anode within a vacuum. This potential difference is called the acceleration potential difference.

The Cathode of the Coolidge tube is a coil. The coil is also connected to a second potential difference, called the thermionic potential difference.

The thermionic potential difference provides energy to the coil that is dissipated within the coil, increasing the temperature of the coil. In a coil that is at a sufficiently great temperature, some of the free electrons in the coil have sufficiently great velocities that they are ejected from the coil. This is called thermionic emission, as it is the process of ionization of atoms by the thermal motion of free electrons.

It has been stated that X-ray production can be induced by the ejection of electrons from atoms. The ejection of free electrons from the coil is not, however, what produces X-rays. This is because of the following:

• X-ray production due to electron ejection requires electrons to be ejected from low energy levels of atoms. Free electrons in the coil are emitted from high energy levels of atoms.
• Thermionically emitted electrons are no longer bound to atoms after they are emitted, so they do not transition to an energy level of any atom.
• Atoms in the coil do not have unoccupied low energy levels for emitted electrons to transition to.

Producing X-rays requires that thermionically emitted electrons be accelerated from the cathode to the anode. The acceleration potential difference accelerates such electrons. The accelerated electrons travel in a vacuum, so they do not lose energy by collisions with atoms of a gas as they travel to the anode target.

The target in a Coolidge tube is made from an element such as tungsten, rhodium, or molybdenum. We recall that electron energy transitions in atoms of these elements can produce X-ray photons if a low energy level electron is ejected.

Electrons from the cathode that strike the target must be moving at very great velocity in order to eject low energy level electrons. The following figure compares the interaction of a slower electron with an atom of the target and the interaction of a faster electron with the atom.

Electrons striking an atom are repelled by the electrons in the atom.

For a slower electron, the repulsive force deflects the path of the electron away from the atom without the electron getting close enough to any low energy level electrons to interact with them.

A faster electron has sufficient initial momentum that the repulsive force from the electrons in the atom only deflects the electron away from the atom after the electron has passed far enough inside the atom to interact with a low energy level electron. The faster electron can transfer energy to the low energy level electron and so eject the low energy level electron from the atom.

Let us now look at an example involving production of X-rays by electron ejection.

Example 1: Comparing the Energy of X-ray Photons Produced by Electron Energy Level Transitions

The diagram shows an atom in the target material used in a Coolidge tube that generates X-rays. An electron in the electron beam used in the tube could eject either an electron in the K-shell or an electron in the L-shell of the atom. Which of the following would result in the greatest energy X-ray photon being emitted from the atom?

1. An electron in the K-shell of the atom is ejected.
2. An electron in the L-shell of the atom is ejected.
3. The energy of the X-ray photon emitted from the atom will be the same whichever electron is ejected.
4. The energy of the X-ray photon emitted from the atom will depend on the initial energy of the electron in the beam as well as which electron in the atom is ejected.

Answer

The energy, , of the emitted X-ray photon is given by where and are the energy levels of the electron in the atom that transitions from a higher energy level to a lower energy level.

An electron in the atom is ejected by an electron in an electron beam that strikes the target of the Coolidge tube.

The energy of the electron in the beam must be sufficient to eject an electron in the atom. This is the only consideration of the energy of the electron in the beam that is necessary. The energy of the X-ray photon is not transferred from the energy of the electron in the beam. It is useful to note that the equation does not contain a term related to the energy of the electron in the beam. We can then eliminate the option that the energy of the X-ray photon emitted from the atom will depend on the initial energy of the electron in the beam as well as which electron in the atom is ejected.

The equation shows that the energy of the X-ray photon depends on the difference in energy of the initial and final energy levels of an electron in the atom that decreases in energy.

The greatest decrease in energy that can occur due to an electron changing energy level is for the high energy level electron shown transitioning to the energy level of an ejected electron. We see from the question that the two possible ejected electrons have different energy levels. The L-shell electron has greater energy than the K-shell electron.

We see then that the equation could have the values or where

We see from this that

This tells us that the photon energy is greater for the high energy level electron transitioning to the K-shell than for that transitioning to the L-shell.

The correct answer is then that an electron in the K-shell of the atom is ejected.

We have seen how X-rays can be produced by electrons transitioning between energy levels in atoms. It is, however, also possible for X-rays to be produced by free electrons decreasing in energy. It is possible that a fast electron does not eject an electron from an atom with which it interacts. In such a case, a fast electron is still acted on by repulsive forces from the atom and so can still decrease in energy. This is shown in the following figure.

It is possible for the decrease in the energy of a free electron to be emitted as a single photon. In such a case, the maximum energy of the photon emitted is the initial energy of the free electron. This corresponds to the decrease in the energy of the free electron being equal to its initial energy. X-rays emitted that are not due to ejection of electrons from atoms are called bremsstrahlung X-rays. The word “bremsstrahlung” is a German word that means “braking.”

It is convenient to use the unit electron volt (eV) when considering the energy of accelerated electrons. An electron volt equals the work done, , on an electron that is accelerated by a potential difference of 1 volt. This is given in joules by where is approximately .

We see then that

Let us now look at an example involving the energy of X-ray photons produced by a Coolidge tube.

Example 2: Determining the Maximum X-ray Energy for a Coolidge Tube

The diagram shows a Coolidge tube used for the production of X-rays. The potential difference and the potential difference . What is the maximum energy of X-rays that the tube can produce?

Answer

The maximum energy of X-rays that the Coolidge tube can produce is equal to the energy of the electrons accelerated by the tube when they reach the target.

The energy of electrons accelerated by the tube when they reach the target equals the energy transferred to them by the potential difference that accelerates them.

The diagram shows two potential differences, and .

The value of is 60 kV. This is 60 kilovolts, or 60‎ ‎000 volts.

The potential difference that is applied across the cathode and anode of the Coolidge tube, and hence the potential difference that accelerates the electrons, is . The maximum energy of the electrons is, therefore, given by

This can also be expressed as 60‎ ‎000 electron volts (eV), or 60 kilo-electron volts (keV).

We have seen that an X-ray photon can correspond to a free electron decreasing to zero energy from its initial energy. As well as this, however, two other possibilities must be considered:

• A free electron decreases in energy, but not to zero energy.
• A free electron emits more than one photon while decreasing in energy.

These possibilities mean that the energies of bremsstrahlung X-ray photons emitted by a Coolidge tube can vary, depending on how much the energy of free electrons decreases and how many photons free electrons emit while decreasing in energy.

The following figure shows a spectrum of bremsstrahlung X-ray photons produced by a Coolidge tube.

We can see from the spectrum that the high-energy limit of the spectrum is the maximum photon energy that the tube can produce, which corresponds to the energy of the accelerated electrons.

This spectrum does not include the X-ray photons due to electron ejections. The following figure shows the spectrum of a Coolidge tube. The spectrum contains a “spike” that shows a very narrow range of photon energies within which many more photons are produced than those with similar energies that are outside of the narrow range.

The spike appears in the spectrum due to electron energy transitions due to electron ejection. The energies of the spike correspond to transitions between electron energy levels in an atom. The percent of photons produced by these transitions is added to the percent of bremsstrahlung photons that have this energy.

Such a spike is called a characteristic line of the spectrum. It is important to note that the spike has some width and is not literally a line. This might seem inconsistent with the idea that there is a specific set energy levels within an atom, but it is consistent with an object made of many such atoms. For such an object, an energy level associated with an atom splits into a range of very close but different energy levels.

Let us now look at an example involving the production of characteristic lines.

Example 3: Identifying an Energy Change of an Electron Energy That Produces a Characteristic X-ray Spectrum Line

The diagram shows an atom in the target material of a Coolidge tube that generates X-rays. An electron in the electron beam used in the tube ejects an electron from the K-shell of the atom and is scattered. Either the high energy level electron or the L-shell electron can transition to the K-shell. Which electron would produce a photon that would be part of a characteristic line of the spectrum with an energy closer to the maximum energy value of the spectrum?

1. The high energy level electron
2. The ejected electron
3. The scattered electron
4. The electron in the L-shell
5. All of the electrons

Answer

A characteristic line of an X-ray spectrum is a very narrow range of photon energies within which many more photons are produced than those with similar energies that are outside of the narrow range.

More photons are produced with the narrow range of energies because photons with these energies can be produced both by bremsstrahlung electrons and by the ejection of electrons from atoms.

A characteristic line of a spectrum is important because it produces photons in a narrow frequency range that exceeds the bremsstrahlung production for those frequencies. The bremsstrahlung production process is not then considered part of the characteristic line.

This means that it cannot be the case that the scattered electron contributes to the characteristic line and, therefore, nor can it be the case that all the electrons in the diagram contribute to the line.

The ejected electron may produce bremsstrahlung X-rays, as its energy may decrease after it is ejected. Such X-rays are not though part of a characteristic line.

This leaves the options of the electron in the high energy level or the electron in the L-shell.

The difference between these electrons transitioning to the K-shell is that the high energy level electron decreases in energy more than the L-shell electron. The X-ray photon produced by the transition of the high energy level electron, therefore, has greater energy. Two possible positions of characteristic lines are shown in the following figure.

A characteristic line of an X-ray spectrum cannot have an energy value greater than the maximum energy value of the spectrum. This means that the high-energy electron line must be closer to the maximum energy value of the spectrum.

The correct answer is that the high energy level electron would produce a photon that would be part of a characteristic line with an energy closer to the maximum energy value of the spectrum.

Two factors affecting the X-ray spectrum produced by a Coolidge tube have been identified:

• The target substance used in the tube
• The accelerating potential difference

We have seen that the target substance used determines which characteristic lines can be present in the spectrum.

We have seen that the accelerating potential difference determines the maximum energy value for the spectrum.

To explain more completely, the accelerating potential difference also determines the energy at which the greatest percent of photons are generated.

The following figure shows the bremsstrahlung photon spectra for two Coolidge tubes with accelerating potential differences and , where

We see that these spectra are shaped a bit like hills. The peak of the hill for the spectrum that has the greater maximum energy value corresponds to a greater energy value than the peak of the hill for the spectrum with the lesser maximum energy value.

It is important to note that the graph used to compare these spectra does not show the intensities of X-ray radiation emitted from a Coolidge tube at different energies, only the distributions of the energies of the photons that comprise the radiation emitted.

If the intensity of X-ray radiation is compared for and , we see that the shapes of the spectra are affected, as in the following figure.

We can understand the difference between the shapes of these spectra by considering the energy of the electrons producing them. Bremsstrahlung X-rays are emitted by electrons when these electrons decrease in energy. Increasing the initial energy of these electrons means that these electrons can change how they emit photons in two different ways:

• An electron could emit a photon with greater energy.
• More photons of a given energy can be emitted.

The intensity of X-ray radiation at a given energy from a Coolidge tube depends on both the number of photons produced at a given energy and the energy of the photons.

It is also important to note that an electron with greater initial energy can more easily approach close enough to the nucleus of an atom to eject an electron from a low energy level than an electron with less initial energy.

Let us now look at an example that compares the accelerating potential difference of a Coolidge tube to the X-ray spectrum for the tube.

Example 4: Comparing the Accelerating Potential Difference of a Coolidge Tube to the Characteristic Spectral Lines Produced

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?

Answer

The characteristic lines of an X-ray spectrum have energy values fixed by the target substance used by a Coolidge tube. Changing the potential difference that accelerates the electrons cannot change this energy.

This means that the position of the characteristic lines on the energy axis of the graph cannot change. The graphs shown with the pink and green borders can, therefore, be eliminated.

The remaining graphs all show characteristic lines with correct energies. The intensities of the lines are different for each graph, however.

We can see the following:

• For the graph with the blue border, the peaks of the characteristic lines have decreased in intensity by approximately the same amount as the bremsstrahlung intensities for those energies.
• For the graph with the yellow border, the peaks of the characteristic lines have not decreased in intensity.
• For the graph with the red border, the peaks of the characteristic lines have decreased in intensity by considerably more than the decrease in the bremsstrahlung intensities for those energies.

To decide which of these three graphs is correct, we consider the following points.

The characteristic lines have energy values less than the maximum photon energy produced by electrons after the reduction of the accelerating potential difference. This means that after the potential difference decrease, electrons still have sufficient energy to eject electrons from atoms and produce characteristic spectral lines. We see then that electrons in the beam that would have ejected electrons from target atoms would still do so after the reduction of the accelerating potential difference. We assume that electrons in the beam all have exactly the same energy.

Reducing the potential difference that accelerates the electrons does not reduce the number of electrons in the electron beam, only the energy of the electrons. The number of electrons in the beam that originally ejected electrons from atoms is not decreased by reducing the energy of the electrons, provided that the electrons in the beam still have sufficient energy to eject electrons. We assume that electrons ejected from atoms are ejected from the same energy levels before and after the potential difference is decreased.

Taking these points into consideration, we would expect the intensity of the spectrum at the energies of the characteristic lines to only decrease due to the decrease in the bremsstrahlung intensity caused by reducing the accelerating potential difference for the same energy values as the characteristic lines.

The graph with the yellow border shows no reduction in the intensity of the characteristic lines, so it can be eliminated.

The graph with the blue border shows an intensity reduction approximately equal to the decrease in the bremsstrahlung intensity. This might seem to be the correct graph, as we have seen that it seems sensible to conclude that only the decrease in bremsstrahlung intensity causes a decrease in the intensities for the energy values of the characteristic lines.

We must consider though that atoms in a target repel electrons that approach them. An electron must come closer to the nucleus of an atom to eject an electron than to just be scattered by the atom. This means that electrons with less energy would be less likely to eject an electron from an atom, despite having enough energy to do so. Electrons with less energy would then produce less X-ray photons of the energy value of a characteristic line.

The graph with the red border shows an intensity reduction at the energies of the characteristic lines considerably greater than the decrease in the bremsstrahlung intensity.

The graph with the red border most correctly represents the change in the X-ray spectrum.

We have seen that the intensity of X-ray radiation of a given energy produced by a Coolidge tube depends on the number of X-ray photons produced at that energy. Each X-ray photon is produced by a decrease in energy of an electron. The number of X-ray photons of a given energy produced, therefore, is affected by the number of electrons emitted by the cathode of the Coolidge tube.

The rate at which electrons are emitted by the cathode of a Coolidge tube is the magnitude of the current for the beam of electrons from the cathode. This is called the beam current.

The beam current can be varied by varying the thermionic potential difference. The greater the thermionic potential difference, the more electrons are thermionically emitted and the greater the beam current.

The maximum thermionic potential difference is very small compared to the accelerating potential difference, so increasing the thermionic potential difference has negligible effect on the energies of the accelerated electrons that strike the target.

Let us now look at an example involving variation of beam current in a Coolidge tube.

Let us now summarize what has been learned in this explainer.