X-ray Generation in Theory and Practice

Marius M. Strauß, summer semester 2018

X-rays were first discovered in 1895 by Wilhelm Conrad Röntgen who thereby was awarded the first Nobel Prize in Physics, in 1901. In their nature, X-rays belong to the electromagnetic waves and as such contribute to the electromagnetic spectrum. (cf. Figure 1)

One easily observes that X-rays do not possess wavelengths in the visible region but rather connect to the ultraviolet bandwidth, resulting in smaller wavelengths, i.e. higher frequencies and therefore higher energy of radiation. (Note that every electromagnetic wave is characterized by the relation $//<![CDATA[ \begin{array}{l}c=\lambda\cdot f\end{array} //]]>$, with $//<![CDATA[ \begin{array}{l}c: speed\:of\:light\:[\frac{m}{s}]\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}f: wave\:frequency\:[\frac{1}{s}]\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\lambda: wavelength\:[m]\end{array} //]]>$

X-ray generation: theory

The simplest way to generate X-rays is the set-up of a classical X-ray tube depicted in Figure 2.

Note that this is only a schematic drawing to demonstrate the basic physical effects of X-ray generation. In practice, one finds a huge variety of different tubes. The appliance consists of an evacuated glass tube with a thermionic cathode on the one side and opposite an anode. Both are connected through an electrical conductor and subjected to an electric potential. Electrons are removed from the cathode and subsequently accelerated towards the anode. Consequently, the potential between cathode and anode is often denoted as acceleration voltage $//<![CDATA[ \begin{array}{l}U_{acc}\end{array} //]]>$. When the electrons reach the anode (which is therefore also commonly referred to as target), the deceleration of the electrons together with several atomistic effects (see below) leads to the emission of radiation.

Thermionic cathode

In order to be able to excavate electrons from the heating coil of the cathode, high temperatures must be provided such that the thermal energy of the electrons inside the coil is sufficient to increase the kinetic energy of the same. Therefore another voltage, the so called heating voltage, is being created on the cathode. If the kinetic energy becomes larger than the binding energy, i.e. the energy that holds the electrons inside the metal, then the electrons can be set free. Hence, it must hold that that $//<![CDATA[ \begin{array}{l}E_{kin}(e^-)\gt E_{binding}(e^-)\end{array} //]]>$ which already implies a requirement to the filaments’ material, meaning a high melting point to be able to increase the kinetic energy without failure.^[1]

The binding energy is described as the work that has to be done by an electron to withstand the bond to the filaments’ metal. The two main contributions to this work stem from the formation of a dipole layer as well as work done by a so called mirror-image force. Hence, $//<![CDATA[ \begin{array}{l}E_{binding}=W_{dipole}+W_{mirror}\end{array} //]]>$. The appearance of the dipole layer is attributed to the fact that there is a charge difference between electrons that have left the metal surface and the (positive) ionized metal atoms. Consequently, following electrons are slowed down by the electric field resulting from this layer, i.e. their energy must be high enough to overcome this field. Furthermore, the work contribution $//<![CDATA[ \begin{array}{l}W_{mirror}\end{array} //]]>$ can be explained as a result of electrostatic induction between electrons that have left the metal and the remaining charge inside. The consequence is again the formation of an electric field which can be described as a symmetric field with respect to the metal surface (mirror) having the electric charges $//<![CDATA[ \begin{array}{l}-e\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}+e\end{array} //]]>$ above and below the surface, respectively.^[1]

Note that the work $//<![CDATA[ \begin{array}{l}W_{mirror}\end{array} //]]>$ necessary in order to increase the distance d from the metal surface to an infinite value is given by:^[1]

$//<![CDATA[ \begin{array}{l}W_{mirror}=\frac{e^2}{4\pi\epsilon_0} \int_{d}^{\infty} \frac{dx}{(2x)^2}=\frac{e^2}{16\pi\epsilon_0 d}\end{array} //]]>$^[1]

Wehnelt cylinder

The Wehnelt cylinder (not depicted in Figure 2) is a hollow-cylindrical metal electrode which has a negative potential. It is located between cathode and target which means that it is a component of the acceleration setup of the electrons. It must be highlighted that due to this same (negative) electric potential the electron beam is filtered in such a way that only electrons that can overcome this potential, i.e. have a certain kinetic energy, will be further accelerated. In addition to that, the negative potential forces the remaining electrons onto the central axis of the cylinder. Hence, it is also a way of focussing the electron beam onto a certain spot.

Interaction with Target Matter: X-ray spectrum

Just before reaching the target, the electrons have a kinetic energy equal to the potential energy of the acceleration field which at this point is basically the energy the electrons have when having passed through the potential difference between cathode and anode, i.e. the acceleration voltage $//<![CDATA[ \begin{array}{l}U_{acc}\end{array} //]]>$. Note that due to energy conservation it follows that:

$//<![CDATA[ \begin{array}{l}E_{pot}=E_{kin} \rightarrow e\cdot U_{acc}=\frac{1}{2}m_ev^2\end{array} //]]>$

which implies the impact velocity to be $//<![CDATA[ \begin{array}{l}v=\sqrt{\frac{2 \cdot e \cdot U_{acc}}{m_e}}\end{array} //]]>$. Obviously the impact velocity does only depend on the acceleration voltage $//<![CDATA[ \begin{array}{l}U_{acc}\end{array} //]]>$. The conversion of the electron beam into photons happens when the fast electrons interact with the anode material as they are suddenly being decelerated by the latter. When this happens, two main physical effects can be observed, the bremsstrahlung and the characteristic radiation.^[1]

Bremsstrahlung

The word bremsstrahlung is actually a German word and does literally translated mean braking radiation. At this point in the workflow of generating X-rays it is the first time that actual radiation in the sense of electromagnetic waves is being emitted. Simply speaking, one can explain the occurrence of radiation as follows: if an electron is being slowed down, then its overall energy reduces due to the reduction in velocity. The difference of the energies before and after the deceleration process gives rise to the energy that has been released during this process (cf. Figure 3). This release happens by emitting a light quantum, i.e. a photon. Hence, the energy of the photon is given as: $//<![CDATA[ \begin{array}{l}E_{photon}=E_{before}-E_{after}=h\cdot f=h\cdot \frac{c}{\lambda}\end{array} //]]>$ with $//<![CDATA[ \begin{array}{l}h:\:Planck\: constant\end{array} //]]>$.

Note that the energy of a photon and thus the energy of X-rays is fully determined by its frequency or wavelength. Accordingly, one can plot the wavelength against the intensity of radiation as shown in Figure 4.

Figure 4 depicts the principle drawing of an X-ray spectrum. All the parts of this spectrum, i.e. of the red line, without the two peaks result from the effect of bremsstrahlung. It is important to understand that this is an uninterrupted curve since one electron might be decelerated in several steps and therefore emits several photons with different energies, i.e. wavelengths. The special case when an electron reaching the target is fully slowed down in just one step which means that all of its energy is transformed into radiation energy, states precisely the maximum radiation energy $//<![CDATA[ \begin{array}{l}E_{max}\end{array} //]]>$. The maximum radiation energy is thus equal to the energy an electron possesses after the acceleration phase, i.e.: $//<![CDATA[ \begin{array}{l}E_{max}=e\cdot U_{acc}=h\cdot f_{max}=h\cdot \frac{c}{\lambda_{min}}\end{array} //]]>$

From this relation one can easily obtain the minimum wavelength which corresponds to the maximum radiation energy $//<![CDATA[ \begin{array}{l}\lambda_{min}=\frac{h\cdot c}{e\cdot U_{acc}}\end{array} //]]>$. This relation is also known as Duane-Hunt law. The minimum wavelength can be found in Figure 4. For the sake of completeness one has to mention that this is a quite rare phenomenon which can be observed by looking at the intensities of such a threshold when seeing different X-ray spectra.^[1] (Please note that equivalent statements can be made concerning $//<![CDATA[ \begin{array}{l}f_{max}\end{array} //]]>$ by using $//<![CDATA[ \begin{array}{l}c=\lambda_{min}\cdot f_{max}\end{array} //]]>$)

The continuous intensity of the braking radiation spectrum follows the Kramers’ rule:^[2]

$//<![CDATA[ \begin{array}{l}J(\lambda)=K\cdot I\cdot Z\cdot (\frac{\lambda}{\lambda_{min}}-1)\cdot \frac{1}{\lambda^2}\end{array} //]]>$^[2] with $//<![CDATA[ \begin{array}{l}K:\: Kramers' \: constant\: [V^{-1}], \: I: \: electron\: current\: [A], \: Z: \: atomic \: number\end{array} //]]>$

Hence, the intensity of braking radiation $//<![CDATA[ \begin{array}{l}J\end{array} //]]>$ is direct proportional to the atomic number of the target atoms.

Characteristic spectrum

The peaks in Figure 4 correspond to the characteristic spectrum. Note that in reality there might be significant more than just two peaks and one speaks of the characteristic spectrum as it depends on the anodes’ material. This is because this spectrum stems from the interaction of the inner electrons of the target with the fast electrons coming from the cathode.

In Figure 5 there is a Bohr atom depicted with the respective shells in order to model these effects. For the sake of clarity, only the K-, L- and M-shell are shown as well as there are only three electrons whereas in reality all shells would be filled since the anodes’ metal atoms are not a priori ionized. However, on the left side there is an electron heading with a certain velocity (black arrow) towards an inner electron (K-shell) of an atom of the target. Now if the energy of the impacting electron is high enough to bounce out the K-electron, both electrons can leave the atom, as shown on the right side. What happens now is that the electron from the energetic higher L-shell changes its state since now there is a free spot on the K-shell which is energetically more stable. In this process electromagnetic radiation is emitted and the energy of this radiation calculates again as $//<![CDATA[ \begin{array}{l}E_{photon}=h\cdot f=h\cdot \frac{c}{\lambda}\end{array} //]]>$. From this we can follow that the transition of an electron between two of these shells is again related to a certain characteristic wavelength, namely for our example $//<![CDATA[ \begin{array}{l}\lambda_{c,K_{\alpha}}=\frac{h\cdot c}{E_{photon}}=\frac{h\cdot c}{E_L-E_K}\end{array} //]]>$, where $//<![CDATA[ \begin{array}{l}E_L\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}E_K\end{array} //]]>$ is the energy an electron has when being on the L- and K-shell, respectively. In Figure 4 the respective $//<![CDATA[ \begin{array}{l}K_{\alpha}\end{array} //]]>$ peak is shown. Nevertheless, it is also possible that the collision happens at other shells: for example, a collision occurring at the L-shell could lead to the $//<![CDATA[ \begin{array}{l}L_{\alpha}\end{array} //]]>$ peak. (Please note that the peak resulting from a transition L- to K-shell is denoted as $//<![CDATA[ \begin{array}{l}K_{\alpha}\end{array} //]]>$, from M- to K-shell as $//<![CDATA[ \begin{array}{l}K_{\beta}\end{array} //]]>$ and accordingly the one from M- to L-shell is denoted as $//<![CDATA[ \begin{array}{l}L_{\alpha}\end{array} //]]>$ and so on.)^[1]

Influence factors on X-ray spectrum

The main variables that can be changed during the generation of X-rays are the acceleration voltage $//<![CDATA[ \begin{array}{l}U_{acc}\end{array} //]]>$ and the heating current $//<![CDATA[ \begin{array}{l}I\end{array} //]]>$.

We have already seen that the kinetic energy of the incident electrons is direct proportional to $//<![CDATA[ \begin{array}{l}U_{acc}\end{array} //]]>$. Thus, increasing $//<![CDATA[ \begin{array}{l}U_{acc}\end{array} //]]>$ leads to an increase in energy of the impacting electrons and therefore also the radiation energy coming from the effect of bremsstrahlung increases, which will basically lead to an expansion of the wavelength range keeping in mind that also the minimum wavelength becomes smaller.

It is important to understand that an increase in acceleration voltage does not change the characteristic wavelengths as these do only depend on the characteristic energies of the anodes’ material atoms (discrete energies of the respective shells in the Bohr atom). Hence, the choice of target material is actually also an influence factor on the X-ray spectrum as it changes the position of the characteristic wavelengths and has furthermore also an effect on the deceleration process.

An increase in heating current leads however to an increase in intensity of the electron beam and thus increases the intensity of radiation.

Consequences for tube design

As we have already seen it is very important that both, the filament of the cathode as well as the target, are made of a material that can in particular stand high temperatures without melting. Tungsten is a chemical element that fulfills this property and has in fact a good thermal conductivity, too.^[1] The fact that only 1 % of the kinetic energy of the electrons is transformed into radiation, whereas 99 % of this energy is transformed into heat suggests to additionally cool down the anode in operation to prevent failure. ^[3]

Furthermore, incorporating the glass tube into a metal case serves the purpose that the generated X-rays can only leave the equipment through a certain exit hole. This leads to directed radiation which is crucial for all applications.

X-ray generation: practice

In practice the principles of X-ray generation theory have led to an ongoing optimization process concerning the X-ray equipment. It has to be highlighted that one distinguishes between installations where for example the filament transformer as well as the (water) cooling gadget for the target are performed in several smaller devices or whether everything is unified in one package.^[3] Note that the choice of when to take which X-ray device depends highly on the desired application. Hence it is also clear that the respective devices might vary also in size.

Furthermore, one mainly distinguishes X-ray tubes that work in reflection or transmission. The former case is the one depicted in Figure 2 where the target has a certain inclination as well as a comparatively high thickness such that the main radiation is being reflected towards the exit hole of the tube. In the latter case one uses a thin target such that the emitted radiation can hit the anode and leave the tube through an exit hole behind the target. The main difference between these two tube types is the maximum applicable acceleration voltage since the targets in a transmission tube are significantly harder to cool so that the acceleration voltage has to be decreased in order to prevent damage to the target. As a consequence, higher photon energies can be obtained with tubes operating in reflection.^[4]

Moreover, regarding the high amount of heat released at the target, in practice one often finds the application of a rotating anode which has the effect of constantly changing the position where the electron beam penetrates the anode in order to obtain a uniform distribution of heat along the target. Note that this leads to a rotationally symmetric anode. Another possibility of constantly changing the focal spot is the utilization of a liquid metal jet such that the target material is continuously reconditioned. Note that the liquid metal jet is an example of emitting radiation in transmission.^[4]

For the sake of completeness it must be mentioned that in practice X-rays are not only generated using X-ray tubes but rather different particle accelerators (Betatron, Synchrotron, Magnetron, Linear particle accelerator (Linac), etc.) fulfill this task, too.

Although, correctly speaking $//<![CDATA[ \begin{array}{l}\gamma\end{array} //]]>$-radiation is not the same as X-rays in the sense of the origin of radiation (radioactive decay vs. excitation via fast electrons) both are electromagnetic waves. Since $//<![CDATA[ \begin{array}{l}\gamma\end{array} //]]>$-radiation does usually have higher radiation energy (cf. Figure 1) it is reasonable to also name some common $//<![CDATA[ \begin{array}{l}\gamma\end{array} //]]>$-radiators used for non-destructive testing, as for applications they might sometimes already be sufficient if not even necessary, concerning for example the penetrability of thick materials. In practice one therefore often uses the isotopes $//<![CDATA[ \begin{array}{l}^{60}Co\end{array} //]]>$ as well as $//<![CDATA[ \begin{array}{l}^{75}Se\end{array} //]]>$ in combination with silver bromide sheets for the industrial radiography, and in particular for non-destructive testing in civil engineering. This means that the aforementioned $//<![CDATA[ \begin{array}{l}\gamma\end{array} //]]>$-radiators radiograph the test specimen and thus material failures inside the sample become visible on the silver bromide sheets located alternate to the radiation source.^[3]

Literature

1. Buzug, Thorsten M. Computed tomography: from photon statistics to modern cone-beam CT. Springer Science & Business Media, 2008.
2. Kiehlmann, Sebastian: Bremsstrahlung. www.lp.uni-goettingen.de, 2010. [Date: 17.08.2018]
3. Große, C. U.: Durchstrahlungsprüfung (Radiographie). In: Grundlagen der Zerstörungsfreien Prüfung. Arbeitsblätter im Rahmen der Vorlesung, S. 112 ff. Lehrstuhl für Zerstörungsfreie Prüfung der TU München. München, 2018.
4. Carmignato, Simone; Dewulf, Wim; Leach, Richard (Hg.). Industrial X-ray computed tomography. Springer International Publishing, 2018.