Natural and Man-Made Sources of Radiation

Tavernier, Stefaan

doi:10.1007/978-3-642-00829-0_3

Stefaan Tavernier²

7082 Accesses

Abstract

In 1895, Henri Becquerel discovered that some uranium salts emit penetrating radiation that can be made visible with ordinary photographic emulsions. This discovery was the beginning of nuclear science. Today we know that most natural radioactivity is due to a few very long-lived unstable isotopes that were formed 4.5 × 10⁹ years ago. At that time the Earth was formed, and presumably, a very large number of different radioactive isotopes were produced. Those with a shorter half-life have all decayed and only those with a very long half-life still exist today. The isotopes contributing most to the natural radioactivity are uranium-235, uranium-238 and thorium-237. The half-life of each of these isotopes is listed in Table 3.1. After a long decay chain with a succession of alpha and beta emissions, these isotopes eventually decay to stable lead (²⁰⁶Pb). There are many other isotopes with a very long lifetime occurring naturally; the most important of these is potassium-40. Several short-lived isotopes also occur naturally; these are either decay products of long-lived isotopes or are produced by cosmic rays. It is worth mentioning radon-222, which is an alpha emitter with a half-life of 3.8 days. This isotope is produced in the decay of uranium. Although in general it is very rare, it can occasionally be found in high concentrations inside buildings or in thermal springs and can represent a health hazard.

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3.1 Natural Sources of Radiation

In 1895, Henri Becquerel discovered that some uranium salts emit penetrating radiation that can be made visible with ordinary photographic emulsions. This discovery was the beginning of nuclear science. Today we know that most natural radioactivity is due to a few very long-lived unstable isotopes that were formed 4.5 × 10⁹ years ago. At that time the Earth was formed, and presumably, a very large number of different radioactive isotopes were produced. Those with a shorter half-life have all decayed and only those with a very long half-life still exist today. The isotopes contributing most to the natural radioactivity are uranium-235, uranium-238 and thorium-237. The half-life of each of these isotopes is listed in Table 3.1. After a long decay chain with a succession of alpha and beta emissions, these isotopes eventually decay to one of the stable isotopes of lead. There are many other isotopes with a very long lifetime occurring naturally; the most important of these is potassium-40. Several short-lived isotopes also occur naturally; these are either decay products of long-lived isotopes or are produced by cosmic rays. It is worth mentioning radon (²²²Rn), which is an alpha emitter with a half-life of 3.8 days. This isotope is produced in the decay of uranium. Although in general it is very rare, it can occasionally be found in high concentrations inside buildings or in thermal springs and can represent a health hazard. Some degree of radioactivity is present in all materials. The amount of activity present in natural materials varies by orders of magnitude. A value of up to 74,000 Bq/kg is usually considered ``not radioactive’.

Table 3.1 Some naturally occurring radioactive isotopes

Full size table

The best known example of the cosmogenesis is carbon-14. This isotope is constantly produced in the atmosphere by cosmic rays and is therefore present in approximately constant concentration in air. Any living organism takes its carbon from the air, either directly or indirectly. Any living biological material therefore contains the same fraction of ¹⁴C. After the organism has died, it no longer absorbs carbon from the air and the fraction of ¹⁴C starts to decrease. Measuring the ¹⁴C concentration is, therefore, a very powerful method for dating archaeological samples.

Besides these naturally occurring radioactive isotopes, there are many artificial radioactive isotopes. These are produced in nuclear reactors or with particle accelerators. Often the production of such isotopes is an undesirable side effect, but some of these radioactive isotopes have useful applications and are made on purpose. Annexe 6 lists some commonly used radioactive isotopes.

Another important natural source of radiation are the ‘cosmic rays’. In 1912, Victor Hess carried electrometers (see Sect. 4.1) to an altitude of 5300 m in a balloon flight. He found that the ionisation rate increased approximately four-fold over the rate at ground level. He concluded that this was caused by radiation from outer space. Today we know that this radiation primarily consists of positively charged nuclei. Of the primary charged particles in cosmic rays, ≈90% are protons, ≈9% are helium nuclei and ≈1% are electrons. This radiation spans an enormous energy range, from ≈1 GeV [10⁹ eV] up to ≈10²⁰ eV. The flux of primary cosmic ray particles decreases approximately like E^−2.7. The flux for particles with energy exceeding 10¹⁷ eV is ≈1/(km² h).

The Sun is an intense source of energetic particles, but the energy of these particles rarely exceeds ≈1 GeV. These particles are accelerated in plasma waves in the corona of the Sun and the intensity fluctuates considerably depending on the solar activity. Occasionally, there are bursts of activity, and the intensity of the radiation increases by several orders of magnitude during the bursts. The Earth’s atmosphere will completely stop any radiation with energy less than 1 GeV and this radiation is therefore harmless for people living on the surface of the Earth, as most of us do. This stream of particles is deflected by the Earth’s magnetic field towards the poles, where it causes eerie phenomena such as the Northern Lights.

The term ‘cosmic rays’ usually only refers to particles with a primary energy above 1 GeV. The Earth’s atmosphere corresponds to about 10 times the hadronic interaction length and 30 times the radiation length. A high-energy particle coming from outer space will always interact somewhere at high altitude in the atmosphere. In the collision, a large number of secondary particles will be produced, mainly protons, neutrons and pions. This is illustrated in Fig. 3.1. The secondary protons and neutrons will again interact, producing new secondary particles of lower energy and so on. Eventually, the energy is so low that particles are stopped by ionisation of the air molecules. The result is that protons and neutrons very rarely reach the Earth’s surface. However, charged pions have a lifetime of 2.6 × 10⁻⁸ s. The average distance travelled by a high-energy pion before it decays is given by

$$\frac{{E_\pi }}{{\rm{m}_\pi }}\,c\,2.6\,10^{ - 8} \,{\rm s} = \frac{{E_\pi }}{{\rm{m}_\pi }}\,7.8\,\rm{m}\,.$$

The symbols E _π, m _π and c stand for the energy and mass of the pion and the velocity of light. The pions are sensitive to the strong colour force, but high in the atmosphere the mean free path before a nuclear interaction for a pion is several kilometers; therefore, most of the pions will have decayed before they can interact. A pion decays into a muon and a neutrino as indicated below:

$$\begin{array}{l} \pi ^ + \to \mu ^ + + v_\mu \\ \pi ^ - \to \mu ^ - + \bar v_\mu \\ \end{array}$$

The neutrino is almost unobservable. A muon has an electric charge and is therefore easily observable. However, a muon is insensitive to the strong colour force; a muon will almost never undergo a nuclear interaction. The muon will lose energy according to the Bethe–Bloch equation discussed in Sect. 2.2. In travelling from the upper atmosphere down to the surface of the Earth, it will lose about 2 GeV. Many of the muons have more than 2 GeV of energy and can, therefore, reach the surface of the Earth. The muon has a lifetime of 2.2 × 10⁻⁶ s and decays into an electron and two neutrinos. However, the average energy of muons at sea level is about 4 GeV and the mean free path before decay of a muon with this energy is about 25 km. Many of the muons produced high in the atmosphere will therefore reach the surface of the Earth.

In the interaction of primary cosmic rays, neutral pions are also produced. Such neutral pions will decay in 8.4 × 10⁻¹⁷ s into two gamma rays. This lifetime is so short that a neutral pion will only travel a microscopic distance before decaying. At high energy, a gamma ray will initiate an avalanche consisting of a large number of electrons, positrons and secondary gamma rays. On average in about one radiation length the original gamma ray gives rise to an electron–positron pair. This electron and positron will create a large number of secondary gamma rays by bremsstrahlung. In one radiation length, an electron or a positron will radiate about half of its energy in this way. Many of these secondary gamma rays will again create electron–positron pairs and these will again undergo bremsstrahlung and so on. If the energy of the initial gamma ray is large enough, the number of particles in the shower will grow exponentially. However, at each step the average energy of the particles in the shower decreases, and fewer of the secondary gamma rays have sufficient energy to produce electron–positron pairs. After a few radiation lengths, the number of the particles in the shower reaches a maximum and thereafter starts to decrease. Eventually, all electrons, positrons and gamma rays are absorbed or stopped.

Because of these cosmic rays, everywhere on the Earth’s surface there is a constant flux of muons. The intensity of this flux is of the order of 1/(cm²·min). The energy spectrum of cosmic ray muons is shown in Fig. 3.2. These muons typically have a few GeV of energy, but the spectrum extends beyond 100,000 GeV. In addition to the muons, we will also see the end of the electromagnetic shower caused by gamma rays from neutral pion decay or by primary electrons. This will give electrons and gamma rays with energy rarely exceeding a few 10 MeV.

The origin of cosmic rays is not known with any certainty. It is widely believed that most cosmic rays have been accelerated in the plasma shock waves caused by supernova explosions in our galaxy. There are indications that the cosmic rays with energy above 10¹⁴ eV are of extragalactic origin, possibly accelerated in the extremely intense electromagnetic fields that are known to exist near massive black holes.

The radiation dose from cosmic rays is small for people living on the surface of the Earth. At an altitude of 10 km, cosmic radiation is much more intense and reaches an average of 5 μSv/h. This is negligible for an occasional traveller, but is of some concern for airline crews. Astronauts in low orbits are at moderate risk because the magnetic field of the Earth shields out most cosmic rays. Outside low Earth orbit, this radiation is much more intense, it is an important concern for astronauts and it represents a major obstacle to future long-term human exploration of the Moon or Mars.

3.2 Units of Radiation and Radiation Protection

Much of this section is reproduced from the ‘Review of Particle Physics’, Ref. [6] in Chap. 1. The International Commission on Radiation Units and Measurements (ICRU) recommends the use of SI units. We also mention CGS units, and some other non-SI units, because these units are still widely used.

Unit of activity: The amount of radioactivity present in a sample can be characterised by the number of radioactive decays per second. The corresponding unit is the becquerel (Bq); the corresponding non-SI unit is the curie (Ci):
$$1\ {\rm Bq} = 1\ \hbox{disintegration s}^{-1} = 1/(3.7 \times 10^{10}) {\rm Ci}$$
Unit of absorbed dose: The amount of radiation absorbed in a sample can be characterised by the amount of energy deposited by the radiation in the sample. The corresponding unit is the gray (Gy); the corresponding non-SI unit is the ‘rad’:
$$1\ {\rm Gy} = 1\ \hbox{joule kg}^{-1} = 6.24 \times 10^{12}\,\hbox{MeV kg}^{-1} = 100\ {\rm rad}$$
Unit of exposure: This unit is somewhat obsolete, but it continues to appear on many measuring instruments. It is a measure of the photon fluence at a certain point in space integrated over time, in terms of ion charge pairs produced by secondary electrons in a small volume of air around the point. The name of the unit simply is ‘unit of exposure’; the corresponding non-SI unit is roentgen (R). One ‘unit of exposure’ creates 1 Coulomb of ionisation charges in one kilogram of air.
$$1\ {\rm R} = 1\ \hbox{esu cm}^{-3} \hbox{in air} = 2.58\times10^{-4}\hbox{`unit of exposure'}$$
Implicit in the definition is the assumption that the small test volume is embedded in a sufficiently large and uniformly irradiated volume, and that the number of secondary electrons entering the volume equals the number of secondary electrons leaving the volume, i.e. there is charged particle equilibrium.
Unit of equivalent dose: The amount of biological damage caused by ionising radiation in a sample depends on the type of radiation and on the amount of energy deposited by the radiation in the sample. The corresponding unit is the sievert (Sv), the corresponding non-SI unit is ‘rem’ (roentgen equivalent for man). The conversion factor is 1 Sv = 100 rem. The equivalent dose H _T in an organ T is equal to the absorbed dose in the organ in gray, times the radiation weighting factor ω _R, formerly called the quality factor Q. The equivalent dose expresses the long-term risks, primarily cancer and leukaemia, from low-level chronic exposure. It depends on the type and energy of the radiation as indicated in Table 3.2 [1]:
Effective dose: The amount of biological damage an irradiated person suffers is called the ‘effective dose’ E. This is the sum of the equivalent doses in each tissue H _T, weighted by the tissue weighting factors ω _T of the organs and tissues in the body that are considered to be the most sensitive [1]:
$$E=\sum_T\omega_T\times H_T$$
The tissue weighting factors are listed in Table 3.3.
Radiation levels [2]: The natural annual background radiation dose, summed over all sources, in most world areas, amounts to a whole-body equivalent dose rate in the range 0.4–4 mSv/year. The world average is 2.5 mSv/year. It can reach up to 50 mSv/year in certain areas. The most important component, ≈ 2 mSv, comes from the inhaled natural radioactivity, mostly radon and radon daughters. The average quoted is for a typical house but varies considerably. It can be more than two orders of magnitude higher in poorly ventilated mines. It is only 0.1–0.2 mSv/year in open areas. The US average is ≈3.6 mSv/year. In Europe it varies from 2 mSv/year in the UK to 7.5 mSv/year in Finland.

Table 3.4 gives some typical average values for the different contributions to the annual background doses received by an average person. The average contribution from medical interventions has increased in recent years and is probably underestimated in this table.
Cosmic ray background in counters: At sea level a detector for charged tracks will count: < 1 min⁻ ¹ cm⁻ ² counts due to cosmic rays penetrating in the detector. Most of the counts are due to muons, the rest is due to low-energy electron or positron tracks caused by gamma interactions.
Dose from external gamma emitting sources: The dose rate in air from a gamma point source of ‘C’ Curies emitting one photon of energy E MeV, with energy 0.07<E<4 MeV, per disintegration, at a distance of 30 cm is about 6×C×E rem/h, or 60×C×E mSv/h. The uncertainty on this number is ≈20%. The dose rate from a point source decreases approximately proportional to 1/r ² as a function of the distance r. The dose rate in air from a semi-infinite uniform photon source of specific activity C (in μCi/g) and gamma energy E (in MeV) is about 1.07×C×E rem/h or 10.7×C×E mSv/h.
Recommended limits to exposure of radiation workers (whole-body dose):
- EU&Switzerland: 20 mSv year⁻ ¹
- US: 50 mSv year⁻ ¹
Lethal dose: The whole-body dose from penetrating ionising radiation resulting in 50% mortality in 30 days (assuming no medical treatment) is 2.5–4.5 Gy. For this number it is assumed that the dose is measured internally on body longitudinal centre-line. The surface dose varies due to variable body attenuation and may be a strong function of energy.
Cancer induction by low-LET radiation: The probability to induce cancer, on average, is about 5% per Sv. [1]

Table 3.2 Radiation weighting factors

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Table 3.3 Tissue weighting factors ω _T

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3.3 Electrostatic Accelerators

Most of the radiation we use is not natural but made by artificial means. Nuclear reactors produce a huge amount of radioactive material, but this subject is not discussed in these lecture notes. We will only discuss the production of high-energy particles with accelerators. The book ‘An Introduction to Particle Accelerators’ by Edmund J. N. Wilson [3] contains an excellent introduction to the physics of particle accelerators. A more advanced discussion of accelerator technology can be found in [4–9].

The most straightforward way to accelerate charged particles is by using an electrostatic potential difference. A particle with a charge Z = 1 travelling a potential difference of X volts receives a kinetic energy of X eV. All that is needed is a high-voltage power supply and a source of charged particles.

The only charged particles that are easily available are electrons and nuclei including protons. Any material heated to a high temperature under vacuum will copiously emit electrons. The hot filament is covered with a suitable substance with low work function (usually alkali oxides) to increase the electron yield. There are many different designs of ion sources, but essentially they are all based on causing a glow discharge in a low-pressure gas. The pressure is typically 0.01 mbar. The glow discharge is a plasma, i.e. a state of matter where most atoms are ionised. There is usually a hot filament producing electrons and a magnetic field. The electrons spiral in the magnetic field and ionise the gas, helping the plasma formation. The beam of positive ions radiates from the plasma chamber through one or more small holes.

If a particle only needs to be accelerated to a few 100 keV, standard commercial high-voltage power supplies can be used. To obtain a much higher energy, special devices are necessary. There are basically two methods to generate the very high potential difference that is needed to accelerate a particle to high energy. This gives rise to two different types of accelerator; the Cockcroft–Walton and the Van de Graaff. These are discussed below. The first successful particle accelerator was built by Cockcroft and Walton and was used in 1932 to create the first example of transmutation of elements using an accelerator.

The Cockcroft–Walton accelerator. The layout of a Cockcroft–Walton accelerator is illustrated in Fig. 3.3. The high voltage is generated with the circuit shown on the right-hand side of the figure. This type of circuit is commonly used in many applications that need a high DC voltage source. It uses a voltage multiplier ladder network of capacitors and diodes to generate a high voltage. The principle is as follows: A moderately high-voltage transformer creates an alternate voltage at one end of the secondary winding, the other end being connected to ground. The capacitor transmits this voltage to point A by capacitive coupling. At first the voltage at point A will also oscillate between +V and −V, but every time this point is at a voltage below zero, a current will flow through the diode, charging the point A to a positive voltage oscillating between 0 V and +2 V. The diode between point A and point B will then cause the charging of point B to the potential +2 V. The same scheme can then be repeated many times to reach higher and higher voltages, but eventually the problems associated with very large electrostatic potentials will also limit this type of accelerator to ≈10 MV.

Until recently a Cockcroft–Walton accelerating structure was often used as the first acceleration step in the high-energy accelerators that will be discussed in the next sections. Today this method is abandoned in favour of RF quadrupole acceleration structures. Only the first ≈100,000 V of acceleration is usually still obtained with an electrostatic voltage difference.

The Van de Graaff accelerator. A completely different approach to reach a high voltage is used in the Van de Graaff accelerator. This device looks like a 19th century electrostatic instrument, but it is still used today for accelerating ions. The working principle of a Van de Graaff accelerator is schematically shown in Fig. 3.4. The high-voltage electrode is a hollow sphere. A circular rubber band runs continuously between the high-voltage electrode and the low-voltage side of the accelerator. A high-voltage power supply of a few 10 kV at the low-voltage side of the accelerator provides the positive charge. The electric charges are generated by field emission at the tip of fine needles and sprayed on the rubber belt. The belt transports the charges mechanically to the high-voltage electrode. Inside this high-voltage electrode, the charges are collected by reverse field emission. The charge collection is done inside the hollow electrode, where the potential is constant, regardless of how large the potential of this electrode is. The ion source is inside the high-voltage electrode. The ions are accelerated in high vacuum inside a straight tube connecting the high-voltage electrode to the target area outside the accelerator. In the tube there are field-shaping electrodes ensuring that the electric field lines guide the ions towards the target. In modern Van de Graaff accelerators, to avoid sparking between the high-voltage electrode and any other metal object at ground potential, the complete accelerator is in a pressure vessel filled with a suitable gas such as freon or SF₆. However, also with such precautions, high static potentials in excess of 1 million volts are an enormous technical challenge. A voltage of 10 million volts will cause sparking over a distance of the order of 10 m in air at atmospheric pressure; the exact value depends on the shape of the electrodes. These technical problems with very high voltages limit the maximum energy that can be reached with electrostatic accelerators to about 25 MeV.

An interesting variant on this instrument is the ‘Tandem Van de Graaff’ accelerator. This instrument takes advantage of the tendency of protons to form negative ions by capturing two electrons. The high-voltage electrode is brought to a large positive potential. A negative hydrogen ion source is outside the detector at zero potential, and the negative ion is accelerated towards the centre of the Van de Graaff. There the negative ion is stripped of its two electrons by letting it pass through a thin metal foil, and the resulting positive ion is accelerated a second time by the same potential difference. This machine allows doubling of the energy of the protons, and it also has the advantage that the delicate proton source is easily accessible outside the accelerator structure.

3.4 Cyclotrons

The difficulties with very high voltages led Rolf Wideröe in 1928 to propose accelerating particles by using a lower voltage difference several times. The principle is illustrated in Fig. 3.5. A beam of particles passes through a succession of metallic tubes. The voltage difference between the tubes is changed while the particles are inside the tube, in such a way that the particles always see an accelerating field on passing from one tube to the next. Several accelerators using this principle were actually built, but the highest radio frequency (RF) that could be made in the 1930s was ≈10 Mz. With this frequency, the linear accelerator becomes impractically long. Today, frequencies in the GHz range are possible and thus make this a practical proposal. Such linear accelerators will be discussed later.

To obtain a more compact accelerator, in 1930 Lawrence proposed bending the particles into a circular path with a magnetic field. In this way the same electrodes can accelerate the particles several times. The idea is illustrated in Fig. 3.6. The essential components of a cyclotron are a homogeneous and parallel magnetic field that forces the particles to travel in circles and an accelerating cavity in the shape of a pillbox cut into two halves. The two electrodes are called ‘Dee’s’ because of their shape.

A large and alternating electric potential difference is applied on the two D-shaped electrodes. The particles emanating from the ion source in the centre are accelerated by the field in the gap between the two D-shaped electrodes. The magnetic field bends the charged particles back towards the gap. If the frequency is right, the field will have reversed when the particles pass the gap a second time and the particles receive a second acceleration. This is repeated again and again. The energy of the particles increases and so does the radius of the particle trajectory. Eventually, the radius becomes too large and the particles leave the cyclotron with a high energy.

When a charged particle is travelling in a plane perpendicular to a magnetic field, the particle will travel in a circular orbit. The radius of the orbit is found by requiring the centrifugal force and the Lorentz force to be equal. Using MKSA units we have

$$M\frac{{v^2 }}{r} = Ze\,\vec v \times \vec B = Ze\,v\,B$$

The notations used are

M: mass of the particle
r: radius of the trajectory of the particle
v: velocity of the particle
e: charge of the proton
Ze: charge of the particle
B: magnetic induction

The equation above is only correct in the non-relativistic limit. To make it relativistic we need to make the substitution

$$ M \Rightarrow M\gamma = \frac{M}{{\sqrt {1 - (v/c)^2 } }} = M\frac{{E_{\rm{kinetic}} + Mc^2 }}{{Mc^2 }} $$

We therefore obtain a very simple relation between the momentum P of the particle, the magnetic induction B and the radius of curvature of the trajectory r.

$$ P = M {\nu} \gamma = Ze\,B\,r $$

In convenient units this relation is written as

$$ \{Pc\}\;[\rm{GeV}] = 0.29979 \cdot Z \cdot B[\rm{tesla}] \cdot r[\rm{m}] $$

((3.1))

The rotation frequency of the charged particles is therefore given by

$$f = \frac{v}{{2\pi r}} = \frac{{ZeB}}{{2\pi M\gamma }} $$

In convenient units this relation is written as

$$f[\rm{MHz}] = 14.3 \frac{{Z\,B[\rm{tesla}]}}{{\gamma \,\{ Mc^2 \} [\rm{GeV}]}}$$

((3.2))

It follows from Eq. (3.2) that the frequency is independent of the energy of the particle as long as the particle remains non-relativistic and therefore γ ≈ 1. With a field of about 1 tesla, the frequency needed is of the order of 10 MHz for protons. Note that in the 1930s a large RF field with this frequency was feasible.

In the simplest version of the cyclotron, there is a dipole magnet with a soft iron core. The pole faces are flat and parallel and create a constant and parallel magnetic field. In between the pole faces there are two accelerating electrodes, and the RF field has a constant frequency given by Eq. (3.2).

The cyclotron just described will certainly accelerate particles, but we would expect that only very few particles will be accelerated. Only particles that start off travelling exactly in the mid-plane between the two magnet poles will eventually reach the exit port. If the particle has a small momentum component parallel to the magnetic field, it will soon hit one of the D-shaped electrodes. The more turns the particles make, the more severe this limitation becomes. What is required is a mechanism to force the particles back towards the mid-plane of the accelerator.

We need to study more closely the geometry of the magnetic field to understand how this is achieved. The field in the dipole will not be exactly constant and parallel. In the centre, the field will have the maximum value and it will decrease slowly from the centre towards the edge of the poles. As a result, the magnetic field lines cannot be exactly parallel, but must have a shape as shown in Fig. 3.7. To prove this let us consider the Maxwell equations in the integral form. The loop integral over the loop shown in Fig. 3.7 must be zero, since there is no current inside this loop. We therefore can write

$$\oint {\vec H\,d\vec l\, = \,} \int\limits_1 {\vec H\,d\vec l\, + \,} \int\limits_2 {\vec H\,d\vec l\, + \,} \int\limits_3 {\vec H\,d\vec l\, + \,} \int\limits_4 {\vec H\,d\vec l\,} = \,0$$

The four parts of the integral represent the line integrals over the four sides of the loop. Because of the overall symmetry, the line integral over part four is equal to zero and we have

$$\int {\left| {H_1 } \right| l_1 } + \int {\vec H_2 \vec l_2 } - \int {\left| {H_3 } \right| l_3 } = 0$$

If the field in the centre is larger than at the edge we have |H ₁|>|H ₃|. If follows that

$${\rm{ }} \int {\vec H_2 \vec l_2 } < 0$$

This shows that the field lines must be bending outward as shown in Fig. 3.7.

This shape of the field will have a focusing effect. This is made clear on the left-hand side of Fig. 3.7. The Lorentz force is always perpendicular to the magnetic field lines, while the centrifugal force always points radially outwards. In the mid-plane of the magnet, both forces exactly compensate, but away from the mid-plane, a small component towards mid-plane remains. To a good approximation, this restoring force is proportional to the distance of the particle from the mid-plane. The particles will have a harmonic oscillation in the vertical direction around the mid-plane.

Also the radial trajectory of the particles in the plane should be stable around the nominal trajectory. It can be shown that the condition for radial stability is

$$\frac{{dB_z }}{{dr}} \le 0$$

For the proof of this statement, I refer the reader to [10]. In the simple geometry we are discussing, this condition is also satisfied. It is remarkable that a simple dipole with flat pole pieces has exactly the magnetic field that is needed to have stable particle acceleration conditions.

However, there remains one problem. If the field is radially decreasing, the rotation frequency is no longer constant. At a small radius, we should have a slightly larger RF frequency than at a large radius. If the particles are only making a small number of turns, we can get away with taking an average value for the RF frequency. Assume a particle starts its journey exactly in phase with the RF field as shown in Fig. 3.8. In the beginning, the particle will travel too fast and it will gradually be more and more early relative to the maximum of RF phase. However, at the same time, its trajectory will have a larger radius and the mismatch between the rotation frequency and the RF frequency will decrease. If the number of turns is not very large, the particle will reach the point where the rotation frequency and RF frequency are equal before it is completely out of phase. From this point on, the particle will be too slow and will start lagging behind relative to the maximum of the RF phase. If the number of turns is not very large, the particle will reach the exit before it is too much out of phase. If the cyclotron should only accelerate particles to a modest energy, this method is possible, and early cyclotrons worked in exactly this way.

As the energy increases, more turns are necessary and the method described above can no longer be used. Moreover, as the energy increases, the relativistic correction to the frequency in Eq. (3.2) can no longer be neglected. It is no longer possible to have a constant RF frequency.

The most straightforward solution is to follow one particular bunch of particles from the source to the ejection and adjust the frequency throughout as the particles are accelerated. A cyclotron using this principle is called a synchro-cyclotron.

Many years ago high-energy cyclotrons worked in the way just described. The drawback is that only a small fraction of the ions produced at the source is accelerated. The higher the energy, the more severe this effect becomes.

For this reason, most cyclotrons today have a very different focusing mechanism using a much more complicated magnetic field shape, namely, focusing with azimuthally varying fields.

In this design, the magnet is subdivided into azimuthal sectors with alternatively larger and smaller values for the magnetic field, as shown in Fig. 3.9. In this figure, darker sectors represent large values of the field and lighter sectors smaller values of the field. In this field geometry, the particles no longer travel in circles, but according to a trajectory as shown in the figure. The particle, therefore, acquires a periodically varying radial component of the velocity. The magnetic field lines acquire a variable azimuthal component, as shown in Fig. 3.9(b). At the edge of the high-field section, the combination of the radial component of the velocity and the azimuthal component of the field together cause a force that pushes the particles back to the mid-plane of the magnet. This force is focusing both when the particles enter and leave a high-field sector.

In addition, the sectors are also given a spiralling shape as shown in Fig. 3.10. It is possible to show that this will further enhance the focusing effect on the beam. The focusing effect obtained with this azimuthal variation of the field is strong enough to compensate the defocusing effect due to a magnetic field that slightly increases with the radius. In this way, it is possible to make an isochronous cyclotron, i.e. a cyclotron where the rotation frequency of the particles remains constant during the acceleration cycle. With an isochronous cyclotron, a much larger beam current can be achieved. Figures 3.11 and 3.12 show examples of cyclotrons.

Table 3.4 Typical average values for the contribution of different sources to the radiation dose for an average person

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Table 3.5 Main parameters of the cyclotron at PSI Switzerland

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For the extraction of the beam, one can use electrostatic fields, but often one prefers to accelerate H^– ions and remove the electrons with a thin metal foil to convert these ions into protons. The magnetic field deflects the positive proton in the opposite direction from the negative H^– ion and therefore immediately ejects it from the magnet.

3.5 The Quest for the Highest Energy, Synchrotrons and Colliders

The maximum energy that can be reached with a cyclotron is limited by Eqs. (3.1) and (1.1). For a given magnet, there is a maximum radius the trajectory of the particles can have and therefore a maximum momentum and a maximum energy. To reach higher energies, one must either increase the magnetic field or increase the diameter of the magnet. Therefore, we will first discuss what magnetic fields can be achieved.

Most large magnets used in accelerators are electromagnets. The magnetic induction B in a solenoid, in MKSA units, is given by

$$B[\rm{tesla}] = \mu _0 \,\mu _r \,I[\rm{A}]\frac{N}{{L[\rm{m}]}}$$

The notations are

N: total number of windings of the solenoid
μ ₀: magnetic permeability of vacuum (μ ₀ = 4π × 10⁻⁷)
μ _r: relative magnetic permeability of the core
I: current in the windings
L: length of the solenoid

The shape of a magnet for a cyclotron is not a simple solenoid, but the maximum field that can be reached in the gap between two pole pieces is given by a similar equation. Note that throughout this book, the term ‘magnetic field’, will usually mean the magnetic induction B. If no ferromagnetic core is used, the maximum field that can be reached is limited by the amount of heat produced in the coils. Even with forced water-cooling, it is difficult to reach a magnetic field larger than ≈ 0.1 tesla. All magnets in accelerators therefore use a ferromagnetic yoke such that, for the same current, the field is increased by the relative magnetic permeability of the yoke material. The values of μ _r for some commonly used ferromagnetic materials are shown in Fig. 3.13. Soft iron has μ _r > 1000 and allows much larger fields to be reached. However, in this case, the maximum field that can be reached is limited by the saturation of the ferromagnetic material. All ferromagnetic materials saturate at ≈2 tesla, setting an upper limit of about 2 tesla to the field that can be reached with conventional (i.e. non-superconducting) magnets.

Superconducting magnets can reach higher fields, because almost no heat is generated in the coils; therefore, there is also no need for a ferromagnetic yoke. In this case the maximum field is limited by the properties of the superconducting material. Indeed, the magnetic field destroys the superconductive property of the superconducting wires. With presently used niobium-based superconductors ≈20 tesla is the upper limit for the magnetic induction that can be reached. Practical considerations limit the field to values well below this number.

When accelerating protons in a cyclotron with a conventional magnet with a diameter of 2 m, the maximum energy that can be achieved is ≈100 MeV (see Exercise 1). To achieve higher energies, the diameter of the magnet needs to be increased and the cost of the magnet will increase faster than the energy! A different approach is necessary. The solution is the synchrotron and we now discuss this type of accelerator.

Figure 3.14 shows the layout of a synchrotron. In a synchrotron, the particles have a fixed trajectory. The beam pipe has the shape of a torus and all around this ring there are bending magnets to keep the particles on the circular track. The particles must already have a minimum energy before they can be accelerated in a synchrotron. If a bunch of particles is injected into the synchrotron, these particles will circulate inside the torus. The magnetic field, the curvature of the track and the particle energy must obey equations (3.1) and (1.1). If the bending magnets are designed such as to have a magnetic field that slightly decreases with increasing radius, the trajectory of the particles will be stable, as we have shown when discussing the cyclotron. The bunch of particles can remain stored in this stable orbit for a very long time. To be used as an accelerator, the synchrotron also needs an RF cavity. An RF cavity is a large enclosure, usually made of copper, with a precise shape. In the box a standing electromagnetic wave is generated. The geometry of the RF cavity is such that this standing electromagnetic wave will have an oscillating electric field pointing either parallel or anti-parallel to the direction of the beam. When the beam passes through the RF cavity, the particle will either be accelerated or be decelerated, depending on its phase relative to the RF field. RF cavities will be discussed further in Sect. 3.6 on linear accelerators.

For the sake of argument, I will now assume that the stored particles are electrons of several 100 MeV. In this case, the energy and the momentum of the electrons are, to a very good approximation, related by E = Pc, and the velocity is very close to the velocity of light. The revolution frequency is determined by the length of the trajectory divided by the speed of the particle. The frequency of the RF cavity should be equal to, or a multiple of, the revolution frequency of the particles.

Figure 3.15 shows the field experienced by an electron passing through the RF cavity. Consider an electron passing exactly through the centre of the beam pipe.

This electron has the nominal trajectory and it will have the same phase relative to the RF field every time it passes the RF cavity. Consider now an electron with the nominal trajectory passing through the cavity with a phase as indicated by the arrow ‘stable point’. This electron is neither accelerated not decelerated and it can continue to turn for a long time. Consider now an electron with the nominal trajectory passing through the cavity with the phase indicated by the arrow ‘particle is accelerated’. This electron will acquire energy each time it passes through the RF cavity. The radius of curvature will become larger and the trajectory will become longer. It will need longer to make one full turn and each time it passes the RF cavity somewhat later. It will move in the direction of the ‘stable point’ and continue in that direction beyond the ‘stable point’. There the electron will experience a decelerating electric field. It will lose energy and as a result the trajectory will become shorter and it will need less time to make a complete revolution. The electron will again move in the direction of the ‘stable point’.

The result is that the energy of the electron will be making oscillations around the ‘stable point’. All electrons will converge to this point and will therefore become grouped in bunches. The number of bunches is equal to the ratio of the RF frequency over the particle revolution frequency.

To turn the machine just described into an accelerator, all that is necessary is to increase the magnetic field in the bending magnets very slowly. If the magnetic field is slightly increased, the trajectory of the particles is shorter and the particles come early relative to the ‘stable point’ and will again experience an accelerating field. The electrons acquire more energy and a new equilibrium is reached, but this time with a slightly higher energy for the electrons. We can again slightly increase the field, the electrons will again acquire a higher energy, and so on. This can be continued until the maximum magnetic field in the bending magnets is reached.

To use a synchrotron as a particle accelerator, one must first turn the field in the bending magnets to some low value and inject a bunch of particles at the energy corresponding to the value of the field. It is not possible to make this field much lower that ≈0.1 tesla, because then it becomes impossible to accurately control the field parameters. After injection, the field is increased slowly to the maximum value possible. The beam is then ejected to the target area. The acceleration cycle can be quite long, e.g. it is 12 s for the CERN SPS synchrotron with a maximum energy of 450 GeV.

So far, we have been considering the acceleration of high-energy electrons. These electrons move at a speed very close to the speed of light, such that the revolution time of a particle with the nominal trajectory will always be the same, regardless of the energy. The frequency of the RF cavity is constant. However, protons of the same energy will have a speed that is lower and the speed will change with energy. As a result, for protons it is necessary to slightly change the frequency of the RF cavity during the acceleration cycle.

The synchrotron we have just described is a ‘weak focusing’ synchrotron, and such machines have been used in the past. Modern synchrotrons, however, use a different focusing system called ‘strong focusing’. Strong focusing is based on the use of quadrupole magnets such as shown in Fig. 3.16. A quadrupole magnet has four poles that are alternatively of north and south magnetic type. The beam passes through the magnet perpendicularly to the plane of the drawing and the centre of the beam passes through the centre of the magnet. From the geometry it is clear that the magnetic field in the centre is zero. A few magnetic field lines are drawn in Fig. 3.16. Let us assume that a beam of protons is passing through the quadrupole magnet shown in Fig. 3.16 from the front to the back. From simple inspection of the direction of the field lines, we see that the Lorentz force on the beam particles is focusing in the horizontal plane and defocusing in the vertical plane. With a correct shape of the pole pieces of the magnet, the magnetic field will increase linearly with distance from the centre. A quadrupole magnet system will therefore behave like an optical lens for the particle beam, except that it will be focusing in one plane and defocusing in the other plane. We now have to recall a well-known property of optical lenses. Two optical lenses, one focusing and one defocusing, will behave like a focusing lens, regardless of the order of the two lenses. More generally, if we have two lenses with focal length f ₁ and f ₂, separated by a distance d, this system of lenses behaves like a single lens with focal length F given by

$$\frac{1}{F} = \frac{1}{{f_1 }} + \frac{1}{{f_2 }} - \frac{d}{{f_1 f_2 }}$$

A positive value for f means a focusing lens and a negative value a defocusing lens. A doublet of two lenses with equal focal length, one focusing and one defocusing, will behave like a focusing lens with focal length given by

$$F = \frac{{f^2 }}{d}$$

To obtain a system that will focus the beam both in the horizontal and vertical direction, we only need to use two identical quadrupoles with reversed magnetic fields. The focusing effect of a doublet in both planes is not symmetric, because the two effective lenses will seem to be displaced by a distance 2f relative to one another. Therefore, it is often preferable to use triplets of quadrupoles. In such a triplet, we have lenses with focal lengths (2f, −f, 2f) in one plane and (−2f, f, −2f) in the other plane. It is straightforward to show that such a triplet will also behave like a focusing lens in both planes. For triplets the optical properties of the two planes are much more similar.

All very high-energy accelerators in use today are synchrotrons. The layout of these machines is similar to what is shown in Fig. 3.14, except that there are many more bending magnets and sets of quadrupole lenses are added between the bending magnets. The advantage of using quadrupoles is that a much stronger focusing effect on the beam is obtained. This will result in a larger particle flux and/or a smaller beam pipe diameter and therefore smaller magnets. This type of synchrotron is therefore called a ‘strong focusing synchrotron’. As an example of a synchrotron, the properties of the Super Proton Synchrotron (SPS) of the international research centre CERN are listed in Table 3.6. This accelerator is entirely installed in an underground tunnel near the city of Geneva, Switzerland. This accelerator was completed in 1976 and will certainly still be used for many years. All the magnets are conventional magnets. Figure 3.17 shows a view of the inside of the tunnel housing the accelerator.

Table 3.6 Main properties of the CERN Super Proton Synchrotron

Full size table

Proton synchrotrons are the highest energy accelerators available in the world. The energy is limited by the maximum magnetic field obtainable in the magnets and the diameter.

Such high-energy accelerators are mainly used for fundamental research. The aim is to study interactions between elementary particles such as protons on protons or electrons on positrons at the highest possible energy. The energy that really matters is the energy in the centre of mass system.

Consider a beam of particles with mass m _b and energy E _b colliding with particles of mass m _t at rest. For any particle, the quantity $E^2 - \vec P^2 c^2 $ is an invariant. If we consider a system of two particles, the quantity

$$(E_1 + E_2 )^2 - (\vec P_1 + \vec P_2 )^2 c^2 $$

is also an invariant. Evaluating the quantity in the centre of mass frame we see that this invariant is simply the square of the centre of mass energy. We therefore have

$$\begin{array}{l} (E_b + m_t c^2 )^2 - P_b^2 c^2 = E_{\rm cm}^2 \\ \\ E_{\rm cm} = \sqrt {(E_b - m_t c^2 )^2 - P_b^2 c^2 } \\\end{array}$$

For a very high energy accelerator the beam energy is much larger than the mass of the beam and target particles, and the centre of mass energy can be written as

$$E_{cm} \approx E_b \sqrt {\frac{{2m_t c^2 }}{{E_b }}} $$

We see that the centre of mass energy is smaller than the beam energy by a large factor. The importance of the effect increases with the energy. It is much better to use two beams of particles colliding head on. In that case, the laboratory system is also the centre of mass system and all the energy is useful.

A particle accelerator arranged in such a way as to allow studying head-on collisions between two beams of particles is called a ‘collider’. One could build two accelerators and let them send two particle beams against one another. A better way is as follows: We have already seen that a synchrotron accelerator can store particle beams. In addition, particles and their anti-particles have the same mass but opposite charge and these can circulate in the opposite direction in the same machine. If we have two stored beams, the particle bunches will meet each other at a number of points along the circular orbit. How many events will we observe when two beams meet each other? Let us assume that the bunch of particles has the shape of a cylinder with length l and section s. Let us further assume that the particle density all over this volume is constant and that the two bunches meet head-on. The total number of particles in the bunches are n ₁ and n ₂. Using the definition of the cross section (Eq. 2.1) and assuming there is only one particle in bunch 2, we have

$$dW\, = \frac{{n_1 }}{{s\,l}}\sigma \,l = \frac{{n_1 }}{s}\sigma $$

If there are n ₂ particles in bunch 2, the number of interactions that will be observed is

$$\frac{{{{n}}_{1\,} \,{{n}}_2 }}{{{s}}}\sigma $$

To have a large number of events there must be as many particles as possible in each beam and the transverse dimensions of the beam should be as small as possible.

If the beams meet with a frequency f, the number of interactions per second will be given by

$$f\frac{{n_1 n_2 }}{s}\sigma = L\sigma $$

The quantity $L = f\frac{{n_1 n_2 }}{s}$ is called the luminosity.

Of course, the density of particles in the beam is not a constant over the volume of the beam, but rather has a Gaussian shape with standard deviation σ _h and σ _v in the horizontal and the vertical directions. It is straightforward to show that the expression for the luminosity in this case becomes

$$ L = f\frac{{n_1 n_2 }}{{4\pi \sigma _h \sigma _v }}$$

As an illustration we give the main parameters of the new Large Hadron Collider (LHC) now being finalised in CERN. It will be the largest energy accelerator in the world and is due to start operation at the end of 2009. This machine is designed to study proton–proton collisions. Collisions of Pb on Pb ions will also be possible. LHC has two interleaved accelerators that cross the beams under a small angle in eight points along the rings. The accelerators are synchrotrons that will first accelerate the beam to 7 TeV and then store the beams to let the protons interact in the crossing points. The parameters of the LHC accelerator and collider are summarised in Table 3.7.

Table 3.7 Main parameters of the LHC accelerator and collider

Full size table

3.6 Linear Accelerators

Circular machines such as cyclotrons and synchrotrons have been very successful, but linear accelerators have not been abandoned. There are many reasons for this. In 1928 when Wideröe first proposed linear accelerators, the technology did not allow the production of very high frequency, high power electromagnetic fields. This technology was developed in the 1940s and 1950s, mainly for the use in radar systems. Today linear accelerators can be much shorter than what was possible in 1930. Moreover, cyclotrons are not well suited for the acceleration of electrons. The reason is the γ factor in Eq. (3.2). For electrons, this effect is very important and the only possible solution is using the cyclotron as a synchrocyclotron, but this results in a strong reduction of the beam intensity. As a result, linear accelerators are the preferred solution for accelerating electrons up to a few 10 MeV. The synchrotron is well suited for accelerating electrons up to a few GeV. However, at much higher energy the ‘synchrotron radiation’ makes it more and more difficult to use circular accelerators for the acceleration of electrons or positrons. The phenomenon ‘synchrotron radiation’ is briefly discussed below.

Any charged particle undergoing acceleration will emit electromagnetic radiation. If a charged particle is travelling in a circular orbit in a magnetic field, it is being accelerated in the direction perpendicular to its direction of motion and the particle will emit electromagnetic radiation. The energy radiated per turn by a particle of mass m, unit charge, energy E, and velocity v is given by

$$\frac{{\Delta E}}{{turn}} = \frac{{4\pi }}{3} \frac{{\alpha\,\hbar c}}{r} \left( {v/c} \right)^3 \left( {\frac{E}{{mc^2 }}} \right)^4 $$

Notice the effect is proportional to (E/mc ²)⁴, therefore synchrotron radiation is usually negligible for all particles except electrons and positrons.

For electrons this equation can be written in convenient units as

$$ \frac{{\Delta E}}{{turn}}[\rm{MeV}] = 0.0885 \frac{{E^4 [\rm{GeV}]}}{{r[\rm{m}]}}$$

((3.3))

The electromagnetic radiation is emitted with a broad spectrum with a maximum around

$${\rm h} \omega [\rm{keV}] \approx \frac{{E^3 [\rm{GeV}]}}{{r[\rm{m}]}}$$

((3.4))

We can see that the amount of energy radiated becomes quite large if the energy of the electron exceeds 1 GeV. This equation makes it clear that it is impossible to accelerate an electron to very high energy in a circular accelerator!

The accelerating structures in synchrotrons or linear accelerators consist of one or more resonant radio frequency cavities. In a synchrotron, the magnetic field bends the beam such that the particles pass many times through the same accelerating element. In a linear accelerator, the beam passes the same structure only once and one usually needs a large number of RF cavities.

We will now briefly discuss the main properties of such resonant RF cavities. Electromagnetic waves can be induced inside any conducting box. We are all familiar with this phenomenon, because a microwave oven is nothing else that a conducting box in which electromagnetic waves are induced. For the purpose of particle acceleration, we are mainly interested in resonant waves in cylindrical cavities, and we will discuss this case in more detail. Consider a box with walls made of conductive material. Any electromagnetic wave in this cavity should satisfy the Maxwell equations in vacuum with the boundary condition that, at the surface of the cavity, the electric field component parallel to the surface must vanish. The general solution of the Maxwell equations in a cavity is rather lengthy and we will only consider the most important solution for the purpose of particle acceleration. For this application, we naturally consider a resonating cavity with the shape of a cylinder with length ‘L’ and radius ‘R ₀’. Because of the cylindrical geometry we will use cylindrical coordinates, ‘z’ denoting the direction along the axis of the cylinder, ‘r’ the distance to the axis of the cylinder and ‘θ’ the angle in the plane perpendicular to the axis of the cylinder. The Maxwell equations are

$$\begin{array}{l} \vec \nabla \bullet \vec D = \rho \\ \vec \nabla \times \vec H - \dfrac{{\partial \vec D}}{{\partial t}} = \vec J \\ \vec \nabla \bullet \vec B = 0 \\ \vec \nabla \times \vec E + \dfrac{{\partial \vec B}}{{\partial t}} = 0 \\ \end{array}$$

Using the well-known vector relation

$$\vec \nabla \times (\vec \nabla \times \vec A) = \vec \nabla (\vec \nabla \vec A) - \vec \nabla ^2 \vec A $$

it is straightforward to show that the electric field should satisfy the following differential equation

$$\vec \nabla ^2 \vec E - \frac{1}{{c^2 }}\frac{{\partial ^2 \vec E}}{{\partial t^2 }} = 0\; {\rm{with}}\; c = \frac{1}{{\sqrt {\varepsilon _0 \mu _0 } }} $$

((3.5))

We are looking for solutions of Eq. (3.5) satisfying the following conditions:

(1)
Stationary solutions (also called standing wave solutions), i.e. solutions where the electric field can be written as a product of a spatial and a temporal function
$$\vec E(z,\ r,\ \theta\ ,t) = \vec f(z,\ r,\ \theta ) \times g(t)$$
(2)
Solutions where the electric field has no longitudinal or azimuthal variation
(3)
Solutions satisfying the boundary condition that the component of the electric field parallel to the surface of the cavity vanishes

From these conditions it immediately follows that only the component of the field in the z-direction is different from zero and that the function f(z, r, θ) is a function of ‘r’ only. Using the first condition, Eq. (3.5) can be written as the sum of two terms, one term depending only on ‘r’ and one term depending only on ‘t’.

$$\frac{{\nabla ^2 f(r)}}{{f(r)}} = \frac{1}{{c^2 }}\dfrac{{\dfrac{{\partial ^2\,g}}{{\partial t^2 }}}}{{g(t)}}$$

This equation can only be satisfied if both terms are equal to the same constant. As it will turn out, this constant must be negative in order to obtain a stationary solution and therefore the constant is written as ‘–k ²’. Writing the Laplace operator in cylindrical coordinates we obtain the following two equations

$$\nabla ^2 f(r,\ \theta,\ z) = \frac{1}{r}\frac{{\partial f}}{{\partial r}} + \frac{{\partial ^2 f}}{{\partial r^2 }} + \frac{1}{{r^2 }}\frac{{\partial ^2 f}}{{\partial \theta ^2 }} + \frac{{\partial ^2 f}}{{\partial z^2 }} $$

$$\left\{ {\begin{array}{*{20}c}{\dfrac{{\partial ^2g}}{{\partial t^2 }} + c^2\;k^2\;g(t) = 0 } \\ {\dfrac{{\partial ^2 f}}{{\partial r^2 }} + \dfrac{1}{r}\dfrac{{\partial f}}{{\partial r}} + k^2 \;f(r) = 0} \\\end{array} } \right.$$

The solution for the first equation is given by: g(t) = A exp(iωt) with ω = kc. The second equation is an equation of the Bessel type with α = 0 and the solution is a linear combination of the zero-order Bessel functions J ₀(kr) and Y ₀(kr). The Y ₀ term is eliminated by the requirement that the electric field has to have a finite value on the axis. A solution for the electric field satisfying Eq. (3.5) and satisfying the three conditions therefore exists and is given by

$$\left\{ \begin{array}{l} E_{zn} (r,\;t)\, = \,E_{0n} \;J_0 \,(k_n r)\,\exp (i\omega t) \\E_r \, = \,0 \\E_\theta\, = \,0 \\\end{array} \right.$$

The constant k is not a free parameter, but is constrained by the requirement that the electric field must vanish on the surface of the cylinder mantel, i.e. J ₀(k R ₀) = 0. There is one possible value of k for each zero of the Bessel function, and this is indicated by the index n. If the first zero of the Bessel function coincides with the wall of the cylinder, we obtain the condition k ₁× R ₀ = 2.405. The other values of k _n are given by

$$\begin{array}{l} k_2 \, \times \,R_0 \, = \,5.520. \\ k_3 \, \times \,R_0 \, = \,8.654. \\ k_4 \, \times \,R_0 \, = \,11.792. \\ {\rm{etc}}. \\ \end{array} $$

The magnetic induction B obeys a similar equation as the electric field, and the solution to this equation can be derived from the fourth Maxwell equation. This gives

$$\left\{ \begin{array}{l} B_z = 0 \\ B_r = 0 \\ B_{\theta n} = ( - i/c)E_{0n}\; J_1 (k_n r)\,\,\exp (i\omega t) \\\end{array} \right.$$

Together with the conditions k ₁ R ₀ = 2.405 and ω = kc this completes the solution of the problem. The shape of the corresponding electric and magnetic fields is illustrated in Fig. 3.18. This mode of oscillation is referred to in the literature as the TM₀₁₀ mode. Here TM stands for ‘transverse magnetic’. Clearly it should be understood that the true fields are the real parts of the expressions above. For the TM₀₁₀ mode, the frequency of the oscillation is given by

$$f = 2.405\frac{c}{{2\pi \,R_0 }}$$

The symbol R ₀ stands for the radius of the cylinder and c is the velocity of light. The length of the cylinder does not enter in the equations. However, if this RF cavity is to be used for accelerating particles, the length is constrained by the condition that the particle should only see an accelerating field while passing through the cavity. The time spent inside the cavity must be less than, or equal to, ½ period of the oscillation. For a particle of velocity ‘v’, we therefore have

$$L \le \frac{v}{{2f}} = \frac{{\pi R_0 }}{{2.405}}\frac{v}{c}$$

A numerical example is instructive. If the frequency of the RF cavity is 200 GHz, the diameter of the cylinder is 115 cm. For accelerating particles that travel at almost the speed of light, the length of the cavity should be less than or equal to 75 cm. These dimensions scale inversely proportional to the frequency.

In its simplest geometry a linear accelerator hence consists of a series of aligned RF cavities as shown in Fig. 3.19. If each cavity has the maximum allowable length, the phase difference between two successive cavities must be equal to π. In the accelerating structure we must create a stationary wave with a phase difference of π between any two successive cavities. Sometimes one prefers to use shorter cavities and in that case the phase difference between two successive cavities is less than π. This corresponds to an RF wave travelling along the structure.

A very important consideration in accelerating structures is the heat dissipation. Let us again consider a cylindrical cavity. The energy content and the heat dissipation of a stationary wave inside this cavity can be calculated as follows: The energy present in the electromagnetic wave is switching back and forth between the electric and the magnetic fields. When the electric field reaches its maximum, the magnetic field is zero and vice versa. The calculation of the energy content of the electromagnetic field ‘U’ is therefore reduced to a volume integral over the electric field at its maximum value

$$\begin{array}{*{20}c}{U = \mathop {\int \int \int }\limits_{\rm volume} \dfrac{{\varepsilon _0 \,E_{\max }^2 }}{2}dv} \\ {U = L\int\limits_0^{R_0 } {(\varepsilon _0 [E_0 \;J_0 (kr)]^2 /2)\,\,2\pi r\,dr} } \\ {U = (\pi R_0^2 L)\,(\varepsilon E_0^2 /2)\;J_1^2 (2.405)} \\\end{array}$$

The heat dissipation is due to Ohmic heating caused by the currents in the walls of the cavity. These currents can be found by considering the second Maxwell equation in the integral form over the loop shown as a dotted line in Fig. 3.20.

$$\oint {\vec{\rm{H}}}{{\rm d}{\vec{\rm{l}}} = \int\int{{\left({\vec{\rm{j}}} + \frac{{\partial {\vec{{\rm D}}}}}{{\partial \rm{t}}}\right)\,{\rm{ds}}} } } $$

The line integral of the magnetic field for the part of the loop inside the metal is zero, because the magnetic field has no time to penetrate in the metal. Also the surface integral of the derivative of the dielectric displacement D over the whole surface of the loop is zero, because the tangential component of the electric field is zero. Therefore, the magnitude of the current in the metal is equal to the magnetic field component parallel to the surface and the orientation of the current is perpendicular to this magnetic field. Consider an infinitesimal surface element dxdy as shown on Fig. 3.20(B). The current flowing through the surface element dxdy is given by J = Hdx and is flowing in the y direction. This current will be restricted to a thin layer of thickness δ by the skin effect. The skin depth δ depends on the frequency f and on the resistivity of material. If ρ is the resistivity of the wall of the cavity, the resistance ‘R’ of the surface element dxdy is given by

$$R\,{\rm{ = }}\,\,\rho \frac{{{{dy}}}}{{\delta \,{{dx}}}}$$

The power dissipation in this surface element is given by

$${{dW}}\,{\rm{ = }}\,{{RJ}}^2 = \rho \frac{{dy}}{{\delta \,dx}}(H\,dx)^2 = \frac{\rho }{\delta }H^2 \,dxdy$$

The total power dissipation in the walls of the cylinder is therefore given by

$$W\, = \,\frac{\rho }{\delta }\,\int {\int {H^2 } \,ds} $$

The magnetic field is proportional to sin(ωt) and only the time averaged power dissipation matters. Since we have

$$\frac{1}{{2{\rm{\pi }}}}\int\limits_0^{2{\rm{\pi }}}{\sin ^2 \,({\rm{\omega t}})\,dt = \frac{1}{2}}$$

This time averaged power dissipation is given by

$$W = \frac{\rho }{{2\delta \mu _0^2 }}\int\int {B_{\max }^2 \,ds}$$

The integral is to be taken over the total surface of the cylinder.

For a numerical example, consider again the cylindrical cavity oscillating at 200 MHz and with a length of 75 cm; the total power dissipation for a maximum electric field of 1 MV/m is ≈ 20 kW! To find this result we used that the resistivity of copper is ρ = 1.7 10⁻⁸ Ωm and that the skin depth of copper is given by

$$\delta = \frac{{66\,\rm{mm}}}{{\sqrt f }}$$

Notice that the power dissipation is increasing proportionally to the square of the electric field. Notice also that the power dissipation per unit length of the accelerator is decreasing inversely proportional to the square root of the frequency.

The Q value of an oscillator is the energy content of the system divided by the energy dissipated in one half oscillation. This quantity is a measure of how accurately the oscillation frequency is determined. In our example the energy content of the field is ≈ 1 J; therefore Q = 1.6 × 10⁻⁴.

The above calculation underlines the importance of the power dissipation in accelerating cavities. The maximum field that can be maintained in a cavity is ultimately limited by the extraction of electrons from the metal walls. This phenomenon depends on the surface smoothness and the practical limit is about 100 MV/m. At such fields the power dissipation is enormous, of the order of 300 MW per meter! It is clear that it is impossible to operate cavities continuously at this value of the accelerating field.

As an example we will describe the structure of SLAC 2-mile linear accelerator. This accelerator is located near San Francisco, California, and it is the largest linear electron accelerator in the world today. The accelerating structures in this accelerator are of the travelling wave type and consist of a series of coupled oscillating cavities that support travelling waves. This structure is illustrated in Fig. 3.21. Inside each of the cavities there is an oscillating electromagnetic field with a geometry similar to what we discussed before. The oscillations in each cavity are coupled and the distance between the discs and the diameter of the iris adjusts the degree of coupling. This coupling determines the phase velocity along the structure.

The accelerating structure is made of pure copper and each element is about 3 m long. A high frequency wave from a high power klystron is injected at one end. The phase velocity equals the speed of the electrons, so that all the electrons stay in phase with the electric field in the cavities as they travel along the structure. The wave is attenuated because of resistive losses in the copper. After 3 m, the amplitude is reduced so much that it is not useful to make the structure any longer. The remaining energy is absorbed in a dump. The average accelerating electric field in the structure is ≈15 MeV/m.

The main properties of SLAC accelerator are listed in Table 3.8. The heat dissipation in the structure is of the order of 1 MW/m. It is clear that the amount of heat produced is so large that it is impossible to use the accelerator continuously. The accelerator therefore produces beam pulses lasting 2.5 μs with a repetition rate of 360 Hz.

Table 3.8 Main properties of the SLAC linear accelerator

Full size table

Iris-loaded waveguides cannot be used for accelerating particles with a velocity much below the speed of light. If the particles travel at a few percent of the speed of light, each cavity becomes much shorter than its diameter, and it becomes difficult to avoid exciting other modes than the T₀₁₀ mode in the structure. A possible way to accelerate slow ions is the structure shown in Fig. 3.22(a). In this geometry, there are a number of cavities, and around the beam there are drift tubes shielding the beam from the electric field when it has the wrong orientation. In such an accelerator the length of the drift tubes must vary to stay in step with the changing velocity of the particle. The shape of each cavity must therefore change in such a way that all the cavities oscillate at the same frequency. If the oscillations in the cavities are all in phase, the wall separating two successive cavities carries zero net current. This wall can therefore be omitted, leaving only bars to support the drift tubes. This is called an ‘Alvarez structure’, and it is shown in Fig. 3.22(b). In a linear accelerator with an Alvarez structure, a standing electromagnetic wave in a large conductive tube is created, with drift tubes in the centre containing the beam. Figure 3.23 shows the inside of an accelerating structure of the Alvarez type.

This type of linear accelerator is often used as an injector for large proton synchrotron accelerators. The linear accelerator will typically receive a beam of ≈1 MeV from an RF quadrupole accelerator, and it will accelerate the beam to a few 100 MeV. After this the energy is sufficient for the beam to be injected into a synchrotron. The RF cavities in Super Proton Synchrotrons also have a similar geometry.

The most important recent advance in linear accelerator technology is the development of superconducting accelerating cavities. Figure 3.24 shows such a superconductive cavity working at 1.3 GHz. It consists of nine sub-cavities and in each sub-cavity a standing wave with a field geometry similar to what is shown in Fig. 3.18 is generated. There is a phase difference of 180° between any two successive cavities. If the centre-to-centre distance between two successive sub-cavities equals the distance travelled by a particle in one half period of the oscillation, a particle experiencing a maximum accelerating field in sub-cavity one, will again be in phase with the field in the next sub-cavity two, and so on. Accelerating fields of 35 MeV/m have been achieved in such structures.

3.7 Secondary Beams

So far we have only considered accelerators producing beams of electrons, protons or nuclei. These are the only charged particles that are easily available. However, often one is interested in other particles.

Positrons or gamma rays are produced in large numbers whenever an electron beam penetrates any piece of target material. If the electrons have more than 10 MeV of energy, they will produce a large number of gamma rays by bremsstrahlung. These gamma rays have a 1/E spectrum, but the photons with a very low energy are absorbed in the target. Most of the gamma rays are going in the direction of the initial electron beam. The higher the energy, the better the collimation in the forward direction. The gamma rays, in turn, will produce electron–positron pairs by the pair creation process. If the target has a thickness of a few tenths of a radiation length, the beam spot in the target will be a copious source of both gamma rays and positrons.

A magnet placed behind the target will remove the electrons and positrons, leaving only the gamma beam (see Fig. 3.25). This is the standard method for the production of gamma beams. This gamma beam will have a broad energy spectrum. By using suitable absorbers one can somewhat reduce the bandwidth of this spectrum.

This setup can also serve as a source of positrons. Positrons can be accelerated by any of the methods that can be used for accelerating electrons. However, there is a problem. Any accelerator has a very limited acceptance in energy and direction of the particles it will accelerate. The positrons produced in the target have a broad energy spectrum. In electron accelerators, the electron source produces large amounts of electrons coming from the same point and all with about the same energy, i.e. almost zero energy. The result is that the number of positrons available for acceleration is many orders of magnitude smaller than the number of electrons and can only produce fairly low-intensity positron beams. To increase the intensity of the positrons, the particles are often kept in a storage ring. This is basically a synchrotron, but optimised for storing particles rather than for accelerating particles. Immediately after injection, the positrons fill the complete phase space of position and momentum that the storage ring will accept. The positron will oscillate around their equilibrium trajectory; the amplitude of the oscillations being determined by the aperture of the magnets and the diameter of the beam pipe. However, positrons emit significant amounts of synchrotron radiation and this emission is equivalent to some sort of friction; therefore, with time the amplitude of these oscillations is damped, and the positrons all converge to the equilibrium trajectory. Because of the stochastic nature of the synchrotron radiation, the particles will not all end up in the equilibrium trajectory; there will remain some spread in position and momentum. If a second bunch of positrons is injected in the storage ring, they will at first have a different trajectory from those already stored, but with time they will converge to the same equilibrium trajectory. In this way one can accumulate a large number of positrons in one bunch and eventually have a positron bunch with the same intensity as that possible for electron bunches. Such a storage ring for positrons is often called a damping ring.

It is also possible to produce anti-protons by a method very similar to the one just described. If a proton beam of well above 10 GeV interacts with a target, a very large number of secondary particles will be produced. About 90% of these particles will be pions, the rest will be a collection of other hadrons; heavier hadrons being less abundant than lighter ones. Among these there will also be some anti-protons. With the help of magnets we can now select those anti-protons with an energy and direction interval that the next accelerator will accept. However, the number of anti-protons produced is very small and their energy is spread over a large range. Very few of these anti-protons will be accepted by the storage ring or the accelerator. Because of the absence of synchrotron radiation, there is no natural damping mechanism for anti-protons. There are methods to reduce the oscillations of such stored anti-protons, but discussion of these is beyond the scope of the present lecture notes.

It is also possible to produce beams with unstable particles such as pions, but because of their short lifetime it is impossible to store these particles in a storage ring. The intensity of such beams will therefore be many orders of magnitude lower than the intensity of proton beams. The principle of the production of secondary pion beams is illustrated in Fig. 3.26. To produce a beam of pions, a primary beam of protons is allowed to hit a solid target. A large number of secondary particles, mostly pions, are produced. With the help of bending magnets, collimators and quadrupoles, pions in a certain energy range are selected. To obtain a beam of muons, one starts from a beam of pions and keeps the pions with the help of quadrupoles in a sufficiently long decay tunnel. A pion decays in a muon and a muon–neutrino with an average decay path equal to 7.8 m multiplied by the relativistic gamma factor of the pion, γ = E/m. Depending on the energy of the primary proton, it can require several 100 m before most of the pions have decayed. At the end of the decay tunnel, the beam is sent into a dump to remove the remaining pions. After about 10 hadronic interaction lengths, all of the pions have undergone interactions and the secondary hadrons have been absorbed, thus a beam containing only muons emerges from the dump.

To make a neutrino beam, the same set-up is used but with a much more massive dump. A sufficiently massive dump will not only stop all the pions but also all the muons. After such a massive dump only neutrinos are left.

All these secondary beams are available in the CERN accelerator complex shown in Fig. 3.27. In this accelerator complex, an RF-quadrupole accelerating section delivers protons of 750 keV. These protons are injected into a linear accelerator of the Alvarez type that brings the energy to 50 MeV. The protons are subsequently injected into a first synchrotron, the PS-booster that accelerates them to 1.4 GeV. A second synchrotron, called the PS, increases the energy further to 13.1 GeV. The Super Proton Synchrotron (SPS) accelerates the protons to 450 GeV. This last accelerator injects the protons into the (LHC) Large Hadron Collider, where the protons are finally accelerated to 7 TeV.

The PS and SPS each provide a wide variety of secondary beams for particle physics experiments. One neutrino beam is sent in the direction of the Gran Sasso laboratory at 730 km distance, where these neutrinos are detected.

3.8 Applications of Accelerators

The initial motivation for developing particle accelerators was to probe the structure of matter at the subatomic scale. However, these machines have found numerous applications in many other fields.

Accelerators are widely used for the implantation of ions in the semiconductor industry or for hardening steel objects such as ball bearings or cutting tools. Ions of tungsten, chromium, tantalum, nitrogen and boron, among others, are used for this purpose. This application usually only requires accelerating particles to a few 100 keV and electrostatic acceleration is the most economic approach.

When particles with a larger energy are required, it is more appropriate to use one of the methods for particle acceleration described in the previous sections.

Accelerators for medical applications. One important use of accelerators is as sources of X-rays and gamma rays. This is achieved by accelerating electrons and letting them interact with a target. The X-rays and gamma rays are produced in the target by bremsstrahlung and by photoelectric effect followed by the emission of X-rays. The energy of the electrons, the target thickness and the target material together determine the spectrum of the gamma rays. Worldwide, many thousands of electron accelerators are in use in hospitals for gamma radiation therapy. The energy of these accelerators varies typically from 4 to 25 MeV. In radiation therapy, one needs an arrangement where the beam can come from all directions on a circle around the patient. By rotating the beam one ensures that a maximum of radiation dose is delivered at the position of the tumour and a lower dose is delivered to the surrounding tissue.

Figure 3.28(a) shows the general layout of an electron accelerator for this application. If the beam energy is not very large, the accelerator has a modest size and it is possible to rotate it around the patient with a rotating gantry as shown on Fig. 3.28(b). If the energy of the electrons is larger, the accelerator is very bulky. It is better to have the accelerator in a fixed position on the rotation axis of the gantry and steer the beam with magnets fixed on the gantry as shown in Fig. 3.28(c). Figure 3.29 shows an example of an accelerating cavity used in this type of electron accelerators. Very similar accelerators are used for the purpose of sterilisation of food or other materials.

Another approach to radiation therapy is the use of protons or light ions such as carbon. The motivation of this method comes from the property of protons or heavy ions to have an increase of the energy loss by ionisation towards the end to the trajectory. In this case there is also no radiation damage behind the irradiated area. One therefore obtains a distribution of the radiation dose as shown on Fig. 3.30. This is particularly important if the area to be irradiated is close to a vital organ. The range of the hadrons in tissue should be up to about 30 cm, and this requires 220 MeV and 5280 MeV for protons and carbon ions, respectively. This energy can be reached either with a cyclotron or with a synchrotron, and systems of either type are in use in several places. Figure 3.11 shows a cyclotron for used proton therapy. The limited integrated beam flux obtainable with a synchrotron is not a problem in this application because only a small beam flux is needed. Figure 3.31 shows the layout of a typical synchrotron for radiation therapy. Typically a synchrotron for hadron therapy consists of an ion source and an electrostatic potential that bring the protons to 80 keV. Next there is an RF-quadrupole accelerating structure and a drift tube linac that brings the protons to a few 10 MeV. After this point, the protons can be injected in the synchrotron to be accelerated to their final energy. One acceleration cycle takes 2 s and is followed by a slow extraction of the beam in a time that can be up to 10 s. The number of protons accelerated in one cycle is ≈10¹⁰, the number of carbon ions is ≈10⁹.

One of the most expensive components in such a treatment system is the massive gantry needed to bend the beam such that the patient can be irradiated from all directions. This requires heavy magnets that are rotating around the patient.

Another important medical application of accelerators is the production of isotopes for Positron Emission Tomography (PET). This medical imaging modality will be discussed in Chap. 6. To apply this imaging method, one needs to produce short-lived positron-emitting isotopes such as ¹⁸F or ¹¹C with decay times of 109.7 min and 20 min, respectively. Because of the short lifetime of these isotopes the cyclotron has to be near to the hospital. The energy needed for the production of isotopes is typically in the range 8–40 MeV and a cyclotron is perfectly suited for this application. Worldwide several 100 cyclotrons are being used for the production of isotopes for nuclear medicine.

Accelerators for nuclear power stations. A potentially very important application of proton accelerators is as ‘drivers’ for nuclear fission reactors in ADS (Accelerator Driven Systems). In a nuclear power station using this approach, the reactor core is sub-critical, i.e. not enough neutrons are produced by the fission process to keep a steady chain reaction going. A spallation source provides the additional neutrons to sustain the nuclear chain reaction. Figure 3.32 shows that the optimal proton energy for this application is about 1 GeV. At this energy about 25 neutrons are produced for each proton incident on a heavy target such as lead. The total power in the beam is therefore 10 MW or more. The advantages of this approach are the following

The reactor is subcritical; as soon as the accelerator is switched of it will stop immediately. This will give an additional safety margin to the reactor.
Different fuel compositions can be used compared to a conventional reactor. In particular spent fuel from reactors could be used, or thorium. Thorium is about three times more abundant in the world than uranium and consists of 100% of ²³²Thorium. This isotope itself cannot sustain a nuclear chain reaction, but it can absorb neutrons and form the fissile isotope ²³³U.
The most attractive aspect of ADS is that the energy spectrum of the neutrons in such a reactor is different from the neutron spectrum in a conventional reactor, and as a result a large fraction of the very long-lived isotopes in the reactor will be converted to short lived ones. ADS systems built for the sole purpose of ‘nuclear waste transmutation’ are being considered.

For this kind of application, the average beam intensity needed is 10 mA or more, which excludes the use of synchrotrons and probably also the use of cyclotrons. The only proven accelerator technology allowing to reach the energy and beam intensity needed for ADS is the linear accelerator. The advantage of linear accelerators over cyclotrons is due to the fact that in a linear accelerator it is possible to obtain a strong focussing effect with the help of quadrupole magnets and this in turn allows obtaining much higher beam intensities. The much weaker focussing effect in a cyclotron limits the beam intensity that can be reached. However, the accelerating gradient that can be achieved in room temperature linear acceleration cavities is limited to about 1 MeV/m if we want to keep the power dissipation in the cavity walls at an acceptable level. Such a linear accelerator therefore will be more than 1 km long and will be too expensive. A possible solution is the use of superconducting linear accelerator structures as shown in Fig. 3.24. Such accelerating structures are used successfully in high-energy electron synchrotrons, such as synchrotron radiation sources. For particles travelling at a velocity close to the speed of light, average accelerating gradients of 25 MeV/m and conversion efficiencies of electrical power to beam energy as high as 50% have been routinely achieved. The theoretical limits for the accelerating gradient that can be reached with niobium cavities are 55 MeV/m, 44 MeV/m and 37 MeV/m for particles with β = v/c, of 1, 0.65, and 0.5, respectively. A value of β = 0.5 corresponds to a proton energy of 145 MeV. At lower energy it is necessary to have a different geometry for the accelerating cells, and it is more difficult to obtain large accelerating gradients. This fact, and the need to have a very reliable accelerator, probably limits the accelerating gradients that can be used to values well below 25 MeV/m. A considerable research effort is under way to design superconductive accelerators for ADS.

Circular accelerators such as fixed field alternating gradient (FFAG) accelerators are also being considered for ADS. This type of accelerator would be more compact and therefore possibly less expensive than linear accelerators. Demonstrating the feasibility of this approach to ADS is also an active field in accelerator research today. This type of accelerator will probably need a proton source followed by a RF-quadrupole accelerating structure and linear accelerator of the Alvarez type accelerating the protons to about 50 MeV. The protons will then be injected in two (or more) circular fixed field machines in succession.

Synchrotron radiation sources: Synchrotron radiation is a major problem when trying to accelerate electrons to the highest possible energy, but the phenomenon is very interesting in its own right. It allows making very intense beams of electromagnetic radiation. The radiation is tuneable in frequency from the visible region to the X-ray region, by choosing the energy of the beam and the strength of the bending magnet. The emission is concentrated in a narrow cone with an opening angle given by ≈m _e/E in the direction of motion of the electron. The radiation is also polarised.

The first synchrotron radiation sources were initially built as electron synchrotrons for research in high-energy physics, later converted to synchrotron radiation sources and these used the synchrotron radiation emitted from the bending magnets. The synchrotron is used as an accelerator and as a storage device. The beam is accelerated to the desired energy and kept in orbit for a long time. The lifetime of the electron beam in such a storage ring is several 10 h. The energy radiated by synchrotron radiation is compensated by the RF cavities.

Current third-generation synchrotron radiation sources typically have several insertion devices called wigglers or undulators. In an undulator the straight sections in the storage ring are used for inserting periodic magnetic structures composed of many magnets that form a repeating row of magnetic fields with alternating direction. Instead of a single bend, many tens or hundreds of ‘wiggles’ with a fixed period add up and multiply the total intensity that is seen at the end of the straight section. In such a structure stimulated emission can occur. It can therefore work as a free electron laser.

There are more than 50 electron accelerators in the world dedicated for the use as synchrotron radiation sources.

3.9 Outlook

There are two parameters determining what future accelerators will be possible: the maximum magnetic field that can be achieved and the maximum accelerating gradient that can be achieved.

Regarding magnetic fields, only modest progress is to be expected in the coming decades. With the currently used type of superconductors, 20 tesla is the upper limit for the field that can be achieved. New high temperature superconductors could possibly one day reach a field of 30 tesla, but experience shows that progress in superconductive magnets is very slow.

Regarding accelerating gradients, the situation is very different. It is well known that the electric field in lasers is many orders of magnitude larger than the maximum field that can now be reached in cavities. However, nobody has been able to come up with a realistic proposal on how to use these fields for the purpose of accelerating particles. The electric fields in plasma waves are also very large and this seems a realistic approach to particle acceleration. In a recent experiment, electrons were accelerated to 1 GeV in a plasma wave over a length of only 4 cm. The idea is illustrated in Fig. 3.33. The plasma, and the plasma wave, are caused by a powerful laser pulse in a low-pressure gas.

It is possible that electron accelerators based on this principle will replace conventional linear accelerators as the source of electrons reaching a few 10 MeV of energy. Such machines could be operational by the year 2020.

A somewhat similar idea is the plasma wake field accelerator. In this approach, a high intensity low-energy electron beam excites the plasma. This beam causes a strong wake field that can be used for accelerating particles. A high-intensity low-energy electron beam could, in this way, give rise to a low-intensity high-energy beam of electrons or other charged particles.

For the future of very high-energy accelerators for basic research, there are two approaches: proton colliders, such as the present LHC machine and electron linear accelerators used for studying electron–positron collisions. It is very unlikely that a proton–proton collider of higher energy than the present LHC accelerator will be built in the near future, if ever.

The international scientific community agrees that the next high-energy accelerator to be built should be the International Linear Collider (ILC). This machine will be based on the use of superconductive accelerator cavities similar to the one shown in Fig. 3.24. The accelerating field will be 31.5 MeV/m. The main parameters of the ILC are summarised in Table 3.9. Such a machine could optimistically be operational by 2019. ILC will allow the study of electron–positron collisions with a centre of mass energy of 500 GeV. This seems modest compared to the LHC, but a proton is composed of quarks, it is not an elementary object. What matters is the energy in the quark–quark collision and this energy is only about 1.2 TeV at LHC. The interpretation of events observed in proton–proton collisions is also much more difficult than the interpretation of electron–positron collisions.

Table 3.9 Main parameters of the International Linear Collider

Full size table

A much more ambitious proposal is the CLIC project [18], see Table 3.10. It aims at building an electron–positron collider that could reach a total centre of mass energy of 3 TeV. The two accelerators and associated equipment could fit in a site 38 km long. Figure 3.34 shows the layout of the CLIC collider and Table 3.9 summarises its main properties.

Table 3.10 Main parameters of CLIC (Compact Linear Collider)

Full size table

The size of the budgets involved makes it clear that this will, in the end, be a political decision. Many different options are possible. It is difficult to be certain what the outcome will be, but it remains very likely that, if a new very high-energy accelerator is built one day in next 30 years, it will be a linear electron–positron collider.

3.10 Exercises

1.
Assume a linear accelerator as shown in Fig. 3.5 and an alternating voltage source of 10 MHz. Assume that we want to use it to accelerate electrons. After a few steps, the electrons will have a velocity close to the velocity of light. How long should each of the tubes be to accelerate each electron further?
2.
Assume that you have a cyclotron with a magnet of 1.5 tesla field. The useful diameter of the magnet is 2 m. What is the maximum energy you can reach for protons with this machine?
3.
Show that in a cyclotron the distance between the successive orbits becomes smaller as the energy of the particles becomes larger.
4.
Show that the equation for the radius of curvature of the track of a charged particle in a magnetic field: P = Ze B r, can be rewritten as Eq. (3.1).
5.
Assume that to drive a nuclear reactor one needs a beam of protons with an energy of 1 GeV with a beam current of 20 mA. Assume that the accelerator has an efficiency for converting electrical energy to beam energy of 33%. How much electrical power will this accelerator use?
6.
What is the speed of the train that has the same kinetic energy as the energy stored in one of the proton beams of LHC. A typical train weighs 400 metric tons.
7.
Assume that we accelerate protons and make them collide with protons at rest. What should be the energy of the proton beam to produce the same centre of mass energy as the LHC collider.
8.
In the SPS proton synchrotron, the frequency of the RF cavities at the maximum energy of 450 GeV is 200.2 MHz. How much should the frequency be at the injection energy of 10 GeV?
9.
Assume a synchrotron for electrons with a beam energy of 1 GeV. What is the power dissipated by synchrotron radiation? Assume that the bending magnets have a field of 2 tesla that the number of particles stored is 10¹² and that 33% of the circumference is occupied by the bending magnets. The rest of the circumference has quadrupoles and straight sections. Neglect the power dissipated in the quadrupoles.

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Fak. Wetenschappen, Dept. Natuurkunde (DNTK), Vrije Universiteit Brussel, Pleinlaan 2, 1050, Bruxelles, Belgium
Prof. Stefaan Tavernier

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Tavernier, S. (2009). Natural and Man-Made Sources of Radiation. In: Experimental Techniques in Nuclear and Particle Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00829-0_3

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DOI: https://doi.org/10.1007/978-3-642-00829-0_3
Published: 14 September 2009
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Natural and Man-Made Sources of Radiation

Abstract

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3.1 Natural Sources of Radiation

3.2 Units of Radiation and Radiation Protection

3.3 Electrostatic Accelerators

3.4 Cyclotrons

3.5 The Quest for the Highest Energy, Synchrotrons and Colliders

3.6 Linear Accelerators

3.7 Secondary Beams

3.8 Applications of Accelerators

3.9 Outlook

3.10 Exercises

References

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