II X-rays for Diagnosis                                Medical Physics course notes by jfK © 1995

 

Production of x-rays: Intensity-wavelength distribution

 

X-rays were discovered in 1895 by Wilhelm Röntgen while investigating the glow produced by cathode rays in a discharge tube. X-rays are short wavelength (typically

l » 1Ĺ) electromagnetic radiation produced either when accelerated energetic electrons bombard elemental metallic targets or energy is released from K shell transitions by electrons returning to the ground state.

The principle of the X-ray tube is shown in figure 1, the design by Coolidge in 1912   (now the rotating anode variety is more widely used in medical applications) was made possible by advances in vacuum technology. Less than 1% of the electron energy is converted into photons the rest is dissipated as heat within the target material.

 

 

 

The short wavelength cut-off of an X-ray beam is determined by the tube voltage:

 

                               Energy E  =  eV  =  hn  =  h c                                                       2.1

                                                                          l

                              

     lmin =  h c

                eVmax

 

                                                                  

 

This assumes that an electron looses all its kinetic energy in only one collision whereas in practice an electron will be involved in impacts with many atoms producing lots of low energy photons, this so called Bremsstrahlung - braking radiation produces a continuous spectrum. Ionisation of the K shell by electrons with just the right amount of energy results in a characteristic line spectrum, when electrons in the atom return to the ground state with the emission of an x-ray photon.

The observed intensity-wavelength (often intensity-energy is plotted) profile from a typical tube is shown in figure 2. (Note that this is essentially a current-voltage characteristic as the number of photons is proportional to the number of electrons and wavelength proportional to reciprocal voltage). Figure 2 shows the profile for two different tube currents, there is no difference in the shape of the curve or the short wavelength limit. However the area under the graph does alter, more electrons generate more photons. On the other hand increasing the tube voltage (equation 2.1) produces more penetrating shorter wavelength photons, thereby shifting the curve to shorter wavelengths. The conversion of electron energy into x-ray photons is also more efficient with fast rather than slow electrons.

 

X-ray beam “Quality”

 

The x-ray flux from a Coolidge tube contains photons of all energies up to the short wavelength limit. However the lower energy (longer wavelength) photons are more readily absorbed (particularly in the skin) and are therefore less desirable in diagnosis. The term quality refers to the ratio of shorter to longer wavelengths in an x-ray beam and finds a quantitative expression in the Halve Value Thickness (HVT) see page 5.

Figure 3 illustrates both the role of an ideal filter for x-ray diagnosis and the effect in practice of adding a filter to the output of an x-ray beam. The amount to which an absorber has filtered out the low energy photons depends both on the metal used and on tube voltage and therefore determines the type of filter used (The HVT expresses this). Aluminium is sufficient up to 120 kV, Al and Copper for the shorter wavelengths up to 200 kV, Al Cu and Tin up to 400 kV and above this Lead is used in addition.

 

 

Interaction of x-rays with matter

 

X-rays interact with matter via the electromagnetic force i.e. their oscillating electric and magnetic fields induce resonance in the electrons in atoms. There are four principal attenuation mechanisms to distinguish: two scattering and two absorption processes and each has a different dependence on photon energy.

Scattering processes - simple (Rayleigh) scattering and Compton scattering.

Absorption processes - the photoelectric effect and pair production.

 

( i ) Simple or Rayleigh scattering

       Low energy photons (1 - 20 keV) simply bounce off an atom with no change in

       momentum as shown in figure 4. The scattered and incident photons are coherent.

       This process is largely insignificant in radiography.

 

                                                                  

 

 

( ii ) Compton scattering

       When the energy of a photon is large compared with the binding energy of an  

        electron then the electron may be considered to be free (therefore the Compton

        mass absorption coefficients are independent of the mass number of the

        scattering material). In this case the photon transfers some of its momentum to                  

        the electron suffering an increase in wavelength and change of direction as                

        energy and momentum are conserved in the interaction as shown in figure 5. The

        kinetic energy of the electron is given by:                       

                                                                             hn - hn’

       

        Compton scattering is the dominant absorption mechanism in the energy range

        30 keV - 20 MeV in soft tissues.

 

 

( iii ) The photoelectric effect

        When an x-ray photon has an energy equivalent to the K shell binding energy of

         an atom then the photon will be totally absorbed and an electron ejected from the   

         atom see figure 6. The atom is ionised and in this excited state higher orbital

         electrons return to the K ground state with the emission of a characteristic x-ray

         line. Photoelectric absorption is important in soft tissue in the energy range

         1 - 30 keV.

 

 

 

 

 

( iv ) Pair production

        When the energy of the incoming photon is large (the threshold is 1.022 MeV)              

        then the photon can interact with the nucleus of an atom producing an electron

        and positron (an antielectron, identical properties except electric charge ) pair,

        subsequently the antiparticle annihilates with another electron to produce two       

        511 keV photons see figure 6. This effect dominates at high energies and is only

        important in tissue above 5 MeV.

 

 

 

Absorption Edges

If the mass absorption coefficient of a material is plotted against wavelength as shown in figure 7 for a monochromatic x-ray beam, mm shows sharp discontinuities at particular wavelengths. These correspond to the ionisation energy of a K shell electron and indicate the increased probability of photoelectric absorption, however this drops sharply as the difference between the photon and electron binding energy increases. The variation of mm with photon energy E and atomic number Z for the various scattering and absorption processes is summarised in the following table and shown graphically in figure 10:                            

                                 

                                Summary of Main Attenuation Mechanisms

Mechanism

Variation of mm with E

Variation of mm with Z

Energy range in tissue

Rayleigh

       µ 1 / E

        µ  Z2

1 - 30 keV

photoelectric

       µ 1 / E3

        µ  Z3

1 - 100 keV

Compton

falls gradually with E

     independent

0.5 - 5 MeV

pair production

rises slowly with E

        µ  Z2

> 5 MeV

 

 

 

Attenuation of an x-ray beam

 

The foregoing section dealt with the physical description of the absorption and scattering mechanismsof x-ray beams with matter. This section will deal with a quantitative description of the linear absorption and mass attenuation coefficients of a monochromatic x-ray beam. For a non divergent, homogenous beam of x-rays as shown in figure 8 the decrease in intensity of the beam in passing through a film of material is proportional to the thickness x:

                                    - dI   µ  dx  introducing a constant    dI  =  -  m x

                                        I                                                       I

Integrating                     dI / I   =   - mdx       Ln ( I - I o )  = - m x

                Taking logarithms            I  =  I o exp ( - m x )                                              2.2

 

m (the linear absorption coefficient units m-1 ) is a constant for a particular wavelength passing through a particular material. It is large if Z is high or l is low.

A more useful measure incorporates the density of the material as well:

 

                                                       mm =  m / r                                                             2.3

 

The mass absorption coefficient mm has the units m-1 / kg m-3 = m2 kg-1.

                                                                 

It is usual practice to quote the Half Value Thickness x˝  (HVT) rather than the absorption coefficient of a material. The HVT is defined (rather like radioactive half life) as the thickness of an absorbing material which reduces the intensity of an x-ray beam to one half of the incident intensity. This is shown graphically in figure 9.

Now              I  = ˝  =  exp ( - m x )             ˝  =  exp ( - m x˝  )

                     I o  

                     2  = exp ( m x˝  )       ln 2 =  m x˝  = 0.693

 

 

    x˝  =  0.693                2.4

                m

 

 

 

 

 

Heterogeneous beams and filtration

The x-ray tube output shown in figure 2 contains a spread of wavelengths; if a sheet of metal were placed in the beam more low energy photons would be absorbed than high-energy photons. This filtration of the beam increases the proportion of higher energy photons in the beam. The energy distribution curve alters as shown in figure 3. Although the peak intensity has been reduced the beam has become relatively more penetrating and is said to be “harder”. In diagnostic applications, filtration is necessary to remove the longer wavelengths from being absorbed in a patient’s skin as well as to reduce the overall dose. Suitable filters should have sufficiently high Z to have large photoelectric absorption for low energy photons. In radiography aluminium filters (Z = 13) a few mm thick are commonly employed.

                                                                 

                                                                 

 

Summary Questions

 

Where necessary take:

h = 6.6 x 10 -34 Js, c = 3 x 10 8 m s -1, e = 1.6 x 10 -19 C, mAl = 1.5 mm -1

 

1. An x-ray tube was operating at an accelerating voltage of 30 kV with a beam current of 20 mA. Calculate:

( i )  The power of the tube.

( ii ) The number of electrons reaching the target each second.

( iii ) The maximum energy of the resulting x-rays.

( iv ) The minimum x-ray wavelength.

 

2. What is the effect of varying the following on the spectrum from an x-ray tube:

( i ) tube voltage.

( ii ) target material.

( iii ) tube current.

 

3. What is the difference between “soft” and “hard” x-rays. What is the effect of a filter on the x-ray spectrum ?

 

4. Describe the difference between photoelectric absorption and Compton scatter. Explain which of these effects is most useful in x-ray radiography.

 

5. “Filtration of a heterogeneous beam of x-rays is essential before taking a radiograph of a patient”; explain the significance of this statement.

 

6. If the intensity of an x-ray beam is 6 x 10 8 W m -2 calculate the intensity after passing through 5.7 mm of Aluminium.

 

7. Estimate the ratio of mm for muscle to that for bone observed as contrast differences on a photographic plate by x-rays passing through a patient, when the dominant attenuation mechanism is photoelectric absorption for 30 keV photons.