Introduction

Abstract

The present textbook is an introduction to measurement techniques in nuclear and particle physics and to particle accelerators. This subject is a part of the standard curriculum in Physics, Nuclear Engineering and Medical Physics. The course should preferably be taken together with a more theoretically oriented course on nuclear physics or particle physics. References [1–3] are a few of the many good handbooks providing this. However, the book can be used as a stand-alone textbook on the experimental aspects of nuclear and particle physics. The only pre-requisite is the general mathematics and physics background that is required for all students in Physics and Engineering.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1.1 Documentation

The present textbook is an introduction to measurement techniques in nuclear and particle physics and to particle accelerators. This subject is a part of the standard curriculum in Physics, Nuclear Engineering and Medical Physics. The course should preferably be taken together with a more theoretically oriented course on nuclear physics or particle physics. References [1–3] are a few of the many good handbooks providing this. However, the book can be used as a stand-alone textbook on the experimental aspects of nuclear and particle physics. The only pre-requisite is the general mathematics and physics background that is required for all students in Physics and Engineering.

The emphasis in this textbook is on the principles of operation and the basic characteristics of measurement systems and accelerators. For a more detailed and complete description of measurement procedures, the readers should consult, for example reference [4], ‘A Handbook of Radioactivity Measurements Procedures’. Such books, however, assume that the reader already has a basic knowledge of the principles of particle measurement, and this is what the present textbook aims to provide.

For preparing these lectures, and lecture notes, I have used material from many different sources, and no attempt is made to give complete references. I found the following books particularly useful:

• ‘Radiation Detection and Measurement’ by Glen Knoll [5]. This is a reference book about nuclear measurement techniques. It contains extensive references to the original literature and a wealth of useful information.

• ‘Review of Particle Physics’ by Amsler [6]. This document is oriented towards particle physics, but several parts of it are also very useful for researchers active in the field of nuclear science. In particular, the tables with numerical values are very convenient and useful. Moreover, this document is freely downloadable from the Web. Mainly the section ‘Constants, Units, Atomic and Nuclear Properties‘, is of interest to students. Several parts of this last document are reproduced as annexes at the end of these lecture notes.

• ‘Principles of Charged Particle Acceleration’ by Stanley Humphries (Ref. [10] in Chap. 3) and ‘An Introduction to Particle Accelerators’ by Edmund Wilson (Ref. [3] in Chap. 3). Both books contain an excellent introduction to particle accelerators.

• ‘Techniques for Nuclear and Particle Physics Experiments’ by W.R. Leo [7].

Data on properties of isotopes and nuclear reactions can be found the National Nuclear Data Centre tables [8]. Other useful information can be found in the Physical Reference Data from the National Institute of Standards and Technology (NIST) [9] and in Kaye and Laby Tables of Physical and Chemical Constants [10]. These three documents are also freely downloadable from the Web.

At the end of each chapter there is a list of other material used in preparing that chapter, followed by a few references.

1.2 Units and Physical Constants

In nuclear and particle physics it is common to use units that are somewhat different from those that are standard elsewhere. The charge is expressed in number of proton charges, 1 proton charge = 1.602×10⁻¹⁹ C, and the electron has the same charge as the proton, but of opposite sign. Energy is usually expressed in ‘electron-volt’. One electron-volt (eV) is the energy that a proton or electron acquires if it goes through a potential difference of 1 V and hence 1 eV = 1.602×10⁻¹⁹ J. To avoid using large numbers, one often uses

• 1 keV = 10³ eV

• 1 MeV = 10⁶ eV

• 1 GeV = 10⁹ eV

• 1 TeV = 10¹² eV

A mass is expressed in kg, but in nuclear and particle physics, it is common to express the mass as the equivalent energy using the well-known relation E = mc ². The mass of a proton is 1.672×10⁻²⁷ kg, but I will usually write that the mass of a proton is 938.272 MeV/c ². This simply means that m _proton c ² = 938.272 MeV. In this relation, c represents the speed of light in vacuum, c = 299 792 458 m s⁻¹. Similarly, for the momentum P, the quantity cP has the dimension ‘energy’, and I will mention the momentum in units of MeV/c. In the physics literature, it is common practice to omit factors c and ћ in the equations, but in these lecture notes, I will always write these factors explicitly.

The mass of atoms and isotopes is usually expressed in ‘unified atomic mass units’. By definition this unit is 1/12th of the mass of a ¹²C atom. One unified atomic mass unit equals 931.494 MeV/c² or 1.660×10⁻²⁷ kg.

For X-rays and gamma rays, there are a few simple and important relations:

$$E = h\nu = \hbar \omega ,\quad {\rm{ and }}\quad \lambda \nu {\rm{ = c}} $$

In these equations, ν = frequency, λ = wavelength; ω = 2πν is the angular frequency and h is the Planck constant. The reduced Planck constant $\hbar$ is also often used:

$${\rm{ }}\hbar {\rm{ = }}\frac{{\rm{h}}}{{{\rm{2}}\uppi }}{\rm{ }} = 1.054 10^{ - 34} {\rm{Js}}$$

In nuclear and particle physics, the quantity hc or $\hbar$ c often enters into the calculations. In convenient units, these quantities are given by

$$\begin{array}{l} \hbar c = 197.610^{ - 15} {\rm{MeV}}.{\rm{m}} \\ hc = 1.242\,eV.{\rm{\mu m}} \\ \end{array}$$

The fine structure constant, α, and the classical electron radius, often enter into theoretical calculations. These quantities are given by

$${\rm{fine\,structure\,constant:}} \; \alpha = \frac{{e^2 }}{{4\uppi \hbar c \varepsilon _0 }} \approx \frac{1}{{137.035}}$$

$${\rm{classical\,electron\,radius:}}\; r_0 = \frac{{e^2 }}{{m_0 c^2 4\uppi \varepsilon _0 }} = 2.818 10^{ - 15} {\rm{m}},$$

where m ₀ is the electron mass; m ₀ c ² = 511 keV.

The fine structure constant is the squared charge of the electron combined with some other fundamental constants so as to give a dimensionless number. The fact that this number is much smaller than unity somehow says in an absolute, dimension-independent way that the charge of the electron is small. The name ‘classical electron radius’ is misleading, because it has nothing to do with the true dimension of the electron. The true dimension of the electron is not known but it is certainly smaller than 10⁻¹⁸ m.

Annex 1 lists most of the numerical constants that are useful in the contexts of nuclear and particle physics.

1.3 Special Relativity

Whenever objects travel at a speed that is a significant fraction of the speed of light, it is necessary to use relativistic formulas for kinetic variables such as speed, energy and momentum. This is the case if the kinetic energy of a particle is a sizeable fraction of, or larger than, the rest energy mc ². In nuclear physics, alpha particles and nuclear fragments are always slow compared to the speed of light, and non-relativistic equations are often sufficient. However, most of the time electrons have a velocity close to the speed of light, and it is essential to use the correct relativistic equations. In high-energy particle accelerators, all the particles move at a velocity close to the velocity of light, and relativistic equations must be used. I have therefore added a section to remind the reader of the main elements of special relativity.

Consider the situation illustrated in Fig. 1.1, two observers, S and S^′ are moving with a velocity ‘v’ relative to each other. In non-relativistic physics, the relation between the position coordinates x, y, and z and the time coordinate t of an object located at point ‘A’, as seen by the two observers, is given by

$$\left\{ \begin{array}{l} x^{\prime} = \displaystyle x - \nu t \\ y^{\prime} = y \\ z^{\prime} = z \\ t^{\prime} = t \\ \end{array} \right.\\$$

Fig. 1.1

The observer S^′ moves with a velocity ‘v’ relative to the observer S. Each observer has his own reference system, {x, y, z} and {x ^′, y ^′, z ^′}

In non-relativistic physics, the kinetic energy ‘E _kinetic’ and the momentum ‘P’ of a particle of mass ‘m’ with a velocity ‘v’ is given by

$$E_{{\rm{kinetic}}} = \frac{1}{2} m \nu^2 \;\; P = m\nu$$

In special relativity theory, the above transformation relations are replaced by the Lorentz transformation:

$$\left\{ {\begin{array}{l}{x^{\prime} = \displaystyle \frac{{x - \nu.t}}{{\sqrt {1 - (\nu/c)^2 } }}} \\ {y^{\prime} = y } \\ {z^{\prime} = z } \\ {t^{\prime} = \displaystyle \frac{{t - \nu.x/c^2 }}{{\sqrt {1 - (\nu/c)^2 } }}} \\\end{array}} \right.$$

The Lorentz transformation can be written in a more elegant way with the help of the parameters β and γ as

$$\left\{ {\begin{array}{*{20}l} {x^{\prime} = \gamma (x - \beta ct)} \hfill & \hfill & {\beta = \displaystyle \frac{\nu }{c}} \\ {y^{\prime} = y} \hfill & {} \hfill \\ {z^{\prime} = z} \hfill & \hfill & {\gamma = \displaystyle \frac{1}{{\sqrt {1 - (\nu /c)^2 } }}} \hfill \\ {ct^{\prime} = \gamma (ct - \beta x)} \hfill & {} \hfill \\\end{array}} \right.$$

An essential property of the Lorentz transformation is that it guarantees that the speed of light is the same for all observers. This transformation has a number of surprising consequences: one is that the time is no longer an absolute time, but depends on the observer.

The total energy E and the momentum P {E, P _x c, P _y c, P _z c} of any object obey the same Lorentz transformation as the time and position coordinates {ct, x, y, z}. We therefore have

$$\left\{ {\begin{array}{l}{P_x ^\prime c = \gamma (P_x c - \beta E) } \\ {P_y ^\prime c = P_y c } \\ {P_z ^\prime c = P_z c } \\ {E^{\prime} = \gamma (E - \beta P_x c) } \\\end{array}} \right.$$

An important consequence of the Lorentz transformation for the energy and the momentum is the relation

$$m_0^{2 } c^4 = E^2 - \vec P^2 c^2$$

((1.1))

In Eq. 1.1, $\vec P^2$ stands for $P_x^2 + P_y^2 + P_z^2$. This equation shows, among other things, that the energy of a particle at rest is E = m ₀ c ².

To obtain the correct relativistic expressions for the energy and the momentum of a particle as a function of its velocity, let us consider a particle at rest in the system S. The energy and the momentum of this particle are given by

$$\left\{\begin{array}{l} P_xc=0\\\noalign{} P_xc=0\\\noalign{} P_xc=0\\\noalign{} E=m_0c^2\\ \end{array}\right.$$

Consider the frame Sʹ moving relative to S with a velocity v in the opposite direction of the x-axis. The energy and the momentum of this particle in Sʹ are given by

$$\left\{\begin{array}{l} P_xc=\beta\gamma\ m_0c^2\\\noalign{} P_xc=0\\\noalign{} P_xc=0\\\noalign{} E=\gamma\ m_0c^2\\ \end{array}\right.$$

Clearly the particle is moving with velocity v in the system Sʹ. We conclude that the energy and the momentum of a particle with velocity v are given by

$$\begin{array}{l} p=\beta\gamma\ m_0c^2\\\noalign{} E=\gamma\ m_0c^2\\ \end{array}$$

For relativistic particles the dependence of the kinetic energy and the momentum on the velocity is therefore is given by

$$E = E_{{\rm{kinetic}}} + m_0 c^2 = \frac{{m_0 c^2 }}{{\sqrt {1 - (\nu/c)^2 } }} $$

((1.2))

$$P = \frac{\nu m_0c^2}{\sqrt{1 - (\nu/c)^2}}$$

((1.3))

From Eq. (1.2), we immediately derive the relation between the velocity ‘v’ of the particle and its kinetic energy:

$$\frac{\nu}{c} = \sqrt {1 - \left( {\frac{{m_0 c^2 }}{{E_{kinetic} + m_0 c^2 }}} \right)^2 } $$

((1.4))

Using the above equation one finds that the velocity of an electron with a kinetic energy of 1 MeV is 95% of the speed of light, while the velocity of a proton with a kinetic energy of 1 MeV is only 5% of the speed of light.

From Eq. (1.1), we obtain the correct relativistic relations between kinetic energy and momentum:

$$E_{kin} = \sqrt {P^2 c^2 + m_0^2 c^4 } - m_0 c^2$$

((1.5))

$$Pc = \sqrt {2m_0 c^2 E_{kin} + E_{kin}^2 }$$

((1.6))

If the energy of the particle is much larger than its rest mass energy, these equations, to a good approximation, simplify to E _kinetic ≈ E ≈ Pc.

Another important consequence of the Lorentz transformation is that the duration of time intervals is no longer the same for all observers. Consider an object at rest and located at the origin in reference system S. Suppose the object is emitting a light flash at time t ₁ and another light flash at time t ₂. The time difference between these two light flashes as seen in the system S is t ₁ – t ₂. In the system S^′, this time interval is given by

$$t^{\prime}_2 - t^{\prime}_1 = \frac{{t_2 - t_1 }}{{\sqrt {1 - \nu^2 /c^2 } }} = \gamma (t_2 - t_1 )$$

The same time interval is longer in the system S^′. This is completely general. All events happening with an object in motion will be slower than that with the same object at rest. In particular, the decay time of object travelling at a larger velocity will be longer by a factor γ than the decay time for the same object at rest.

1.4 Probability and Statistics

Radioactive decays, or interactions created at particle accelerators, are statistical phenomena. There are a few concepts borrowed from probability and statistics that we will use again and again during these lectures. Therefore, I have added a brief paragraph to remind the students of the most important of these concepts. Many more concepts of probability and statistics are essential tools in the analysis of experimental data in nuclear and particle physics. However, these concepts are not used in the present lecture notes and are therefore not mentioned here. Reference [11] is one of many good textbooks on this subject. The sections on probability and statistics in reference [6] also contain very useful methods for the statistical analysis of data.

Gaussian distribution. Consider a continuous statistical variable x with probability density function f(x). The quantity f(a)dx represents the probability that x takes a value between a and (a + dx).

The Gaussian or normal probability density function is given by

$$f(x) = \frac{1}{{\sqrt {2\uppi }\, \sigma }}\exp \left( { - \frac{{(x - \mu )^2 }}{{2\sigma ^2 }}} \right)$$

((1.7))

The Gaussian distribution, Eq. (1.7), has the following properties:

• (1)
  $$\int\limits_{ - \infty }^{ + \infty } {f(x)dx = 1}$$

• (2) The average value of x, often written as <x> or as $\bar x$, is given by

  $$\bar x =< x >= \int\limits_{ - \infty }^{ + \infty } {xf(x)dx} = \mu $$

• (3) For a statistical variable x, with probability density function f(x), the dispersion or root-mean-square (r.m.s.) deviation, usually written as σ, is given by

  $$\sigma ^2 = \left\langle {(x - \bar x)^2 } \right\rangle = \int {(x - \bar x)^2\, f(x)\, dx}$$

For a Gaussian distribution, the r.m.s. dispersion is the parameter σ occurring in Eq. (1.7). Equation (1.7) therefore represents a Gaussian distribution with average value μ and dispersion σ.

A Gaussian distribution has the well-known bell shape shown in Fig. 1.2. The width of such a distribution is often characterised by its ‘full width at half maximum’, usually abbreviated as FWHM. From the probability density distribution Eq. (1.7), one immediately finds that the FWHM of a Gaussian distribution is related to the dispersion σ by ${\rm{FWHM}} = \sigma \sqrt {8\ln 2} = 2.355 \,\sigma .$

Fig. 1.2

Gaussian or normal distribution. FWHM stands for ‘full width at half maximum’. The probability that the value of x is within one standard deviation $\sigma_.$ of the average value is 68% and the probability that x is within two standard deviations is 95%

Assume that we have a large number of statistical variables x _i, with arbitrary probability density functions f _i(x _i). Consider the new variable y defined as

$$y = \sum\limits_i {x_i }$$

It can be shown that, under very general conditions, and regardless of what the distributions f _i(x _i) are, the probability distribution of y is a Gaussian distribution. This is the central limit theorem, and it explains why Gaussian distributions are so common in many experimental situations. Imagine that you are measuring some physical quantity. The value you measure will not be the true value because the measurement will be affected by a measurement error. Usually, there are a large number of different effects, all adding up to the measurement error. It is, therefore, not surprising that a measurement error often has a Gaussian distribution.

Poisson distribution. Let us now consider a statistical variable n that can only take integer values. P(k) represents the probability to observe the value n = k.

For a Poisson distribution these probabilities P(n) are given by

$$P(n) = \frac{{\lambda ^n }}{{n!}}e^{ - \lambda }$$

It is easy to show that the Poisson distribution has the following properties:

• (1)
  $$P(0) = e^{ - \lambda }$$
• (2)
  $$\sum\limits_{n = 0,\infty } {P(n) = 1}$$
• (3)
  $$< n >= \sum\limits_{n = 0,\infty } n P(n) = \lambda$$

• (4) The dispersion or r.m.s., usually written as σ, is given by

  $$\sigma ^2 =< (n - < n >)^2 > = \sum\limits_{n = 1,\infty } {(n - < n >)^2 P(n) = \lambda }$$

It can also be shown that, for large values of λ, the Poisson distribution approaches a Gaussian distribution. This approximation is very good as soon as λ is larger than a few 10! Therefore, the FWHM of a Poisson distribution as a function of the average value λ is given by${\rm{FWHM}} = 2.355\sqrt \lambda$.

The number of occurrences of a particular event A will have a Poisson distribution if

• a large number of primary events, or primary situations, can give rise to the occurrence of A, and each of these primary events has only a small chance to give rise to the occurrence of A and

• there is no correlation between the primary events.

Assume that we have some amount of radioactive material and are observing the decay of this material. Let us further assume that the half-life of the material is long compared to the duration of the observation. This is clearly a situation where the above conditions are met. The probability that the decay of one particular atom is observed is very small, but there are a large number of atoms, and all have a small, but finite, probability to give rise to an observed decay. In addition, the decays are independent of one another. In this case, the observed number of decays will have a Poisson distribution.

Dispersion of a sum of two statistical variables . Let x and y be two statistical variables. Consider the statistical variable z = x + y. Then, we have the following important relations

$$\begin{array}{l} \bar z = \displaystyle \bar x + \bar y\\\sigma _z^2 = \displaystyle \sigma _x^2 + \sigma _y^2 \\ \end{array}$$

These relations hold for any probability distributions for the variables x and y.

1.5 The Structure of Matter at the Microscopic Scale

All matter is made up of atoms, and atoms have a size of the order of 10⁻¹⁰ m. We have known for about a hundred years that atoms are composed of a nucleus surrounded by a cloud of electrons. To the best of our knowledge, the electrons are truly elementary particles. If they have a dimension at all, that dimension is less than 10⁻¹⁸ m. However, the nucleus is a complex object. The dimension of the nucleus is of the order of 10⁻¹⁵ m. It is composed of protons and neutrons. These protons and neutrons, in turn, are composed of quarks. To the best of our knowledge, the quarks are also truly elementary particles. If they have a dimension at all, that dimension is less than 10⁻¹⁸ m.

In ordinary matter there are two types of quarks, called up-quarks and down-quarks. A proton is made of two up-quarks and one down-quark, and a neutron is made of two down-quarks and one up-quark. The quarks have a charge that is a fraction of the proton charge: up-quarks have a charge of +2/3, down-quarks have a charge of −1/3. Hence all ordinary matter is made of three basic components: electrons, up-quarks and down-quarks. This very simple picture is not sufficient to describe the reality. First, there is a very enigmatic particle called the neutrino. It is electrically neutral and has a very small probability to interact. Furthermore, for all these particles there are corresponding antiparticles. Finally, this basic set of four components (up-quark, down-quark, electron and electron neutrino) is repeated two times with heavier versions of each particle. There is the electron and two heavy electrons, namely the muon and the tau-lepton. The muon has a mass of 105.65 MeV/c ², about 200 times larger than the electron, and the tau-lepton has a mass of 1,777 MeV/c ², about two times the mass of a proton! If I say that a muon is a heavy electron, this means that a muon is in all respects the same as the electron except for its mass. The muon decays with a lifetime of 2.2 μs into an electron and two neutrinos. The fact that muon decays does not make it less fundamental than the electron; the muon decays because this is energetically possible.

The forces between those elementary particles are

• gravitational force

• electromagnetic force

• weak force and

• strong colour force.

The gravitational force and the electromagnetic force are quite familiar from observations in the macroscopic world, while the weak force and the strong colour force only manifest themselves at the subatomic scale. This is because the last two forces are short-range forces.

The gravitational force is the overwhelming force at the macroscopic scale, but at the nuclear scale it is essentially unobservable. The ratio of the gravitational force to the electromagnetic force between an electron and a proton is 10⁺⁴⁰! At the atomic or nuclear scale, one can safely ignore the gravitational force.

The interaction between the fundamental components of matter is well described by the equations of relativistic quantum field theory. Mathematically, these equations are very complex, but one can get an intuitive feeling of what is going on by thinking of an interaction between particles as being due to the exchange of quanta of force. A collision between two charged particles, say two electrons, can be seen as due to the exchange of a quantum of electromagnetic force, as illustrated in Fig. 1.3.

Fig. 1.3

A collision between two electrons is due to the exchange of one or more virtual photons. A photon can be seen as a quantum of electromagnetic force

This quantum of electromagnetic force is a virtual photon. It is not a real particle and can exist only for a very short time. Unlike a real photon, the mass of a virtual photon can be different from zero and can even be negative. However, in quantum field theory there are a number of other possible processes. Figure 1.4(a) shows a process where an electron and an anti-electron (called a positron) meet and annihilate into a virtual photon. This cannot be a real, massless, photon because that would violate energy and momentum conservation laws. This virtual photon can materialise in any charged particle–antiparticle pairs. Figure 1.4(b) shows the annihilation of an electron–positron pair into two gamma rays. In this process, the photons are real particles, but the electron connecting the two gamma mission points is a virtual electron. The charge of a particle describes the strength of its coupling to the electromagnetic field. Photons couple only to charged particles. Photons themselves carry no charge and are massless.

Fig. 1.4

Example of electromagnetic interactions of charged particles. In these diagrams the direction of time is from left to right. An arrow pointing against the direction of time, i.e. to the left, represents an antiparticle. (a) An electron and a positron annihilate each other and materialise again as a quark-anti-quark pair. (b) An electron and a positron annihilate each other into two gamma rays

In these few examples we see that the distinction between a ‘particle’ and a ‘force’ is disappearing at the subatomic level. The electromagnetic force manifests itself as quanta, called photons, while a particle, such as an electron manifests force-like or wave-like behaviour.

The most prominent force at the nuclear scale is the strong colour force. This force is also described by equations that are very similar to those describing the electromagnetic interaction. The force quanta of this strong force are ‘gluons’, and similar to photons these are massless particles. There are several very important differences between the strong colour force and the electromagnetic force. The strong colour charges, the equivalent of the electrical charge for the strong force, appear in three kinds. These three charges were whimsically called ‘red’, ‘green’ and ‘blue’, and the force is therefore called the colour force. Only quarks have a colour charge and therefore only quarks feel the strong force. The antiparticles of quarks, called anti-quarks, have charges ‘anti-red’, ‘anti-green’ and ‘anti-blue’, and these charges are different from ‘red’, ‘green’ and ‘blue’. Unlike the electromagnetic force, where the quanta of electromagnetic force carry no charge, gluons themselves have a colour charge. It can be shown that all these give rise to a force between colour-charged objects that does not decrease with distance. It takes an infinite amount of energy to separate two quarks!

As a result, all particles with colour are permanently locked up in colour-neutral systems. Quarks or gluons cannot exist as free particles. Only colour-neutral combinations of quarks can exist. Just as an electron and a positron together have a total charge zero, a quark and its anti-quark too have a total colour charge zero, and a suitable colour-combination of a quark and its anti-quark can form a colour-neutral system.

It is possible to show that the only possible combinations of quarks that are colour-neutral are a quark and an anti-quark, three quarks or three anti-quarks. Of course, all combinations or multiples of these can also be colour-neutral. The only quark combinations that exist in nature are a combination of a quark and an anti-quark, of three quarks or of three anti-quarks! All particles that are composed of quarks are called hadrons. A very large number of different hadrons exist. However, most of these are extremely short-lived and have a lifetime of only ≈10⁻²³ s. This lifetime is so short that such particles cannot travel a macroscopic distance before they decay. Even at a very high energy, the maximum distance they will travel is less than the size of an atom. Therefore, such particles can never be observed directly. Their existence is inferred from the observation of their decay products. A few hadrons have a much longer lifetime, in the range 10⁻⁸ to 10⁻¹⁶ s. This lifetime is also very short, but at a velocity close to the velocity of light, these particles can travel distances in the range from microns to hundreds of meters. Table 1.1 lists some among the most common hadrons with a long lifetime, together with their main properties.

Table 1.1 Some hadrons and their main properties. In this table the antiparticle of particle x is written as $\bar{x}$

The most familiar quark–anti-quark combination is the π meson. It exists as a π⁺ (up-quark + anti-down-quark) and π⁻ (down-quark + anti-up-quark) or as π⁰ (up-quark + anti-up-quark and down-quark + anti-down-quark). All the particles consisting of a quark and an anti-quark are called mesons. The combinations of three quarks and three anti-quarks are called baryons and anti-baryons, respectively.

The most familiar combinations of three quarks are the protons and neutrons, but many more exist. Protons and neutrons, and more generally all hadrons, are colour-neutral systems; therefore, there is no strong force between these particles except at very short distances. This is similar to the electromagnetic force between neutral atoms or molecules. There is no electromagnetic force between two neutral atoms except at very short distances. However, between two neutral molecules that are touching, there is a force called the van der Waals force. It is an electromagnetic force. Similarly, between two nucleons that are some distance apart there is no force. As they come closer together, first there is a force that is attractive, and as they come even closer the force becomes strongly repulsive. Thus, protons, neutrons and all other hadrons appear to have a size, and that size is about 10⁻¹⁵ m.

Another force that is observed at the subatomic scale is the ‘weak force’. This force causes many decay processes in hadrons and nuclei. Mathematically, it is also described by equations that are similar to the electromagnetic force. The main difference comes from the fact that the quanta of the weak force are massive objects: the W⁺, W^– and the Z bosons. They have a mass that is approximately 90 times the proton mass.

All elementary particles have a weak charge. Figure 1.5(a) and (b) illustrate a few reactions that are mediated by the weak force. Figure 1.5(b) represents the ‘decay’ of a down-quark into an up-quark, an electron and a neutrino. This reaction can never be observed since there are no free quarks, but this is the process that explains the neutron decay (see Fig. 1.5(c)). This process can be understood as follows: the down-quark in the neutron emits a virtual, charged, W⁻ boson. The charge needs to be conserved, and that is possible if the down-quark (charge = −1/3) changes into an up-quark (charge = +2/3). The mass of a W boson is 90 times the proton or neutron mass, so this process seems impossible. Remember though that in quantum mechanics a process that is energetically impossible is nevertheless possible, provided this unphysical state lasts only a short time. This is the well-known ‘tunnelling effect’. A particle can cross an energy barrier that it does not have enough energy to cross, provided the barrier is very thin, so the particle is not long in the unphysical state of negative energy.

Fig. 1.5

Examples of some reactions mediated by the exchange of quanta of the weak force. (a) Annihilation of a down-quark and an anti-up-quark into an electron and an anti-neutrino. (b) ‘Decay’ of a down-quark into an up-quark, an electron and an anti-neutrino. (c) The phenomena depicted in (a) and (b) cannot exist as such because free quarks cannot exist. However, the decay of a neutron into a proton, an electron and a neutrino is due to the process (b)

The uncertainty of the energy and the duration of this state are related by $\hbar \le \Delta E \Delta t$. If the uncertainty of the energy needs to be 90 times the proton mass, then the distance the virtual W boson can travel is given by

$$c \Delta t \approx \frac{{\hbar c}}{{90\, {\rm{GeV}}}} \approx 10^{ - 18} {\rm{m}}$$

Hence, the process illustrated in Fig. 1.5(c) is possible, provided it all happens in a distance of the order of 10⁻¹⁸ m. The ‘weak force’ is not really weak, it has a short range!

The complete list of all fundamental building blocks of nature is given in Table 1.2, and the quanta corresponding to the fundamental forces of nature are listed in Table 1.3. These tables summarise all that is presently known about the fundamental building blocks of nature and about the fundamental forces. There are three ‘up-quark like’ quarks: the up-quark, the c-quark and the t-quark; there are three ‘down-quark like’ quarks: the down-quark, the s-quark and the b-quark; there are three leptons: the electron, the muon and the τ-lepton and there are three neutrinos: the electron neutrino, the muon neutrino and the τ-neutrino. The force quanta are the photon, the W- and Z-boson and the gluons. However, many of these particles cannot be observed as real particles. The quarks and gluons are permanently locked in colour-neutral systems such as protons and neutrons, and the W- and Z-bosons are so short-lived that they can never be observed directly. Only their decay products can be observed. Only the proton, the neutron in a nucleus, the electron and the neutrino are stable. The neutrinos interact so weakly with other particles that they are largely decoupled from the world we observe. Hence, ordinary matter only consists of protons, neutrons and electrons. All the other particles only exist if they have been created in some very high-energy collision. Soon after their creation the particles decay into one of the few stable particles. Such particles have also been created in huge numbers in the beginning of the Universe.

Table 1.2 Fundamental building blocks in nature. All these particles have spin = ½. For each of these particles there is a corresponding antiparticle

Table 1.3 Quanta corresponding to the fundamental forces in nature. All these quanta have a spin = 1

Table 1.4 lists the particles that are stable, or at last can exist long enough to travel truly macroscopic distances. For all these particles, except the photons and some neutral mesons, there are corresponding antiparticles. For the rest of the present lecture we will only consider the particles listed in Table 1.4.

Table 1.4 List of the most common directly observable particles

At this point it should be explained that quantum mechanics implies that exploring small dimensions needs high energies. The study of the structure of matter at the subatomic scale requires the use of high-energy probes, and in reactions between small objects, particles of high energy are produced. Indeed, in quantum mechanics a wave is associated with every particle, and the wavelength λ is given by

$$\lambda = \frac{h}{P} = \frac{{hc}}{{Pc}} = \frac{{1237[{\rm{MeV}}] 10^{ - 15} [{\rm{m}}]}}{{Pc}}$$

((1.8))

If the momentum Pc of a particle is 1237 MeV, the corresponding wavelength is 10⁻¹⁵ m. To probe the structure of objects much smaller than 10⁻¹⁵ m, we need particles with energy much larger than 1237 MeV. If the energy of an object is much larger than its mass, then energy E and momentum Pc become the same.

If a particle is enclosed in a one-dimensional box of dimensions ‘a’, the energy levels of this particle are given by

$$E_n = \frac{{n^2 \uppi ^2 (\hbar c)^2 }}{{2 mc^2 a^2 }}$$

((1.9))

The above result is found by solving the Schrödinger’s equation in the very simple case of an infinitely deep potential well with vertical slopes. We see that the spacing between the energy levels in a small object will be very large. Qualitatively this remains true whatever potential confines objects to a small volume. In atoms, the energy levels of the electrons are in the range 1 eV to a few keV. In nuclei, the energy levels spacing is in the range 10 keV to 10 MeV. The energy level differences in hadrons are of the order of 1 GeV.

If the particles we regard as fundamental today, quarks, electrons etc., are in fact composed of some as yet unknown more fundamental objects, we know that the corresponding size has a scale smaller than 10⁻¹⁸ m. Therefore, the corresponding energy levels and particle masses are expected to be of the order of 1 TeV or larger. If this is the case, the particles we know today are only the lowest energy levels of these objects, and a whole spectrum of particles that are much more massive than the particles we know today, exists.

All these explain why it needs very high-energy accelerators to study very small particles. It is the motivation for the vast worldwide effort to build very high-energy particle accelerators.

1.6 Nuclei and Nuclear Decay

All the familiar matter surrounding us is made up of atoms. The atoms are made up of a nucleus containing most of the mass, and of electrons. The nucleus is a bound state of protons and neutrons held together by the strong nuclear force. Neutrons and protons are collectively referred to as ‘nucleons’. The charge of the nucleus is equal to the number of protons. The mass of the nucleus is about 1% smaller than the sum of the masses of the constituent neutrons and protons. The difference is due to the binding energy of the nucleus. Figure 1.6 shows the binding energy per nucleon for all stable elements. Notice that the binding energy per nucleon increases with the mass of the nucleus, reaches a maximum for iron and nickel and then decreases again. The helium nucleus is unusual: it has an exceptionally large binding energy.

Fig. 1.6

Binding energy per nucleus for all stable isotopes

Like the electron structure of the atom, the nucleus also has a number of discrete energy levels. The energy difference between the nuclear energy levels is typically of the order of 1 MeV, and the transitions between such states give rise to particles with energies in the 1 MeV range.

The charge of the nucleus determines the chemical properties of the atom. The presence of neutrons in the nucleus is essential to make the nucleus stable. A nucleus needs an approximately equal number of protons and neutrons to be stable. High-mass nuclei need more neutrons than protons to be stable. Nuclei of the same charge but with a different number of neutrons are called isotopes. Only a few of all the possible isotopes are stable. For example iron has four stable isotopes: ${}_{26}^{54} Fe, {}_{\;26}^{\;56} Fe, {}_{\;26}^{\;57} Fe, {}_{\;26}^{\;58} Fe$. The notation ${}_Z^N A$ stands for a nucleus A containing Z protons and in total N nucleons. The indication of the number of protons in the nucleus is redundant since the element symbol already gives the charge of the nucleus; this number is therefore often omitted. Besides the stable isotopes, there are many unstable isotopes with a larger or a smaller number of neutrons. These decay in one or more steps until a stable configuration is reached.

1.6.1 The Beta Decay

An unstable nucleus can sometimes reach a lower energy state by changing a proton into a neutron or vice versa. To conserve the charge in this transformation process, an electron or a positron must be emitted. The corresponding reactions on free protons or neutrons are

$$\begin{array}{l} n \to p + e^ - + \bar \nu _e \\ p \to n + e^ + + \nu _e \\ \end{array}$$

The symbols $\nu _e$ and $\bar \nu _e$ stand for the electron neutrino and the electron anti-neutrino, respectively. These reactions are mediated by the exchange of W bosons as illustrated in Fig. 1.3. Because the decay is mediated by the weak force, the corresponding lifetime can be quite long. On free particles, only the decay of neutrons to protons is possible since the mass of the protons is less than the mass of the neutrons. Inside a nucleus, each of the two reactions is possible if the mass of the new nucleus plus the mass of the electron or positron is smaller than the mass of the original nucleus. Written as the decay of a nucleus the process is

$$\begin{array}{l} {}_Z^N A \to {}_{Z + 1}^N A + e^ - + \bar \nu _e\\ {}_Z^N A \to {}_{Z - 1}^N A + e^ + + \nu _e\\ \end{array}$$

In nuclear physics these decays are called β⁻ and β⁺ decays. The neutrino or anti-neutrino produced in the β decay is almost never observed, but it takes away part of the energy liberated in the reaction. The mass difference between the initial nucleus and the sum of the masses of the final-state particles appears as kinetic energy of the particles in the final state. The energy corresponding to this mass difference is usually denoted by Q. The mass of the nucleon in the final state is much larger than the mass of the electron or the neutrino and also much larger than the mass difference. As a result the kinetic energy of the final-state nucleus is very small, usually only a few keV. To first approximation all the energy is shared between the electron and the neutrino only. For a proof of this statement see the solution to Exercise 7. Since there are three particles in the final state the momenta of the final state particles are not determined uniquely. The electron and the neutrino can have a kinetic energy varying between zero and the maximum allowed energy. The mass of the neutrino is extremely small, less than 2 eV; therefore the maximum energy of the electron is equal to the energy Q. In tables with nuclear decays, the corresponding maximum energy of the electron is usually listed as E _max.

1.6.2 The Alpha Decay

A very heavy nucleus has another possibility to reach a more stable configuration. It can just fall apart into two lighter nuclei. This will be energetically favourable since for very heavy nuclei the binding energy per nucleon decreases with increasing mass of the nucleus. For a few very high mass isotopes this can happen by splitting the nucleus into two more or less equal parts. For nearly all other isotopes this always happens through the emission of a helium nucleus, also called alpha particle. The corresponding reaction is

$${}_Z^N A \to {}_{Z - 2}^{N - 4} A + \alpha$$

Because the transition is mediated by the strong force, the long decay time of some alpha emitters is surprising. The explanation is that the decay requires the alpha particle to tunnel through a potential barrier of the nucleus. If the final-state nucleus in the alpha emission is in a well-defined nuclear level, energy and momentum conservation fixes the kinetic energy of the alpha particle. Of course a given isotope can decay by alpha emission to several distinct nuclear levels, and the corresponding alpha particles have different energies.

1.6.3 The Gamma Decay

After a β⁻ or a β⁺ decay, or after alpha particle emission, the nucleus is often not in its ground state, but in some excited nuclear state. Transitions between the excited levels and the ground state can give rise to the emission of gamma rays. Most transitions giving rise to the emission of a gamma ray are extremely fast. However, for some transitions the direct decay mechanism is forbidden, and the corresponding decay is much slower. A nucleus that is trapped in one of these metastable states is called an isomer, and this is denoted by a letter m after the mass number, e.g. ^60m Co.

1.6.4 Electron Capture and Internal Conversion

A nucleus also has a number of other possibilities to decay. There is a finite probability that an electron from the cloud of electrons surrounding the atom is present inside the nucleus. Consider a β⁺ decay. The nucleus can reach the same nuclear final state and conserve the charge by capturing an orbital electron of the atom in the reaction

$$p + e^ - \to n + \nu _e$$

This process is called ‘electron capture’. This reaction occurs mainly in heavy nuclei where the nucleus is larger and the electron orbits are smaller. In most cases the captured electron is the K-shell electron, but L-shell electron capture also occurs. After the electron capture the empty level left in the electronic structure of the atom is filled by an outer electron, and the excess energy is liberated by the emission of an X-ray or by the emission of an Auger electron. The relative importance of β⁺ decay and electron capture also depends on the difference in energy between the two nuclear levels. If this difference is less than 511 keV, the β⁺ decay is impossible and only electron capture occurs.

Another process involving the orbital electrons is the ‘internal conversion’. It is a different decay mechanism for transitions that usually emit gamma rays. In this process the nuclear excitation energy is directly transferred to the atomic electron rather than to a gamma ray. Unlike the electron produced in a β⁻ decay, the electron produced by the internal conversion process always has the same energy. The electron that is most likely to be involved in the internal conversion process is the K-shell electron, but the electrons in other orbitals may also receive the conversion energy. An isotope decaying by the internal conversion process will therefore exhibit a group of electron energies, the differences in energy being equal to the differences in the binding energies between the electronic orbitals.

1.6.5 The Radioactive Decay Law

The probability that a nucleus decays in some small time interval dt is given by

$$P = \lambda\, dt$$

where λ is a constant characteristic of the decay. The important fact is that this probability is independent of how long the nucleus is already waiting to decay. Assume there are N ₀ nuclei at time t = 0, and let us denote the number of nuclei at any time after t = 0 by N(t). We have

$$\frac{{dN(t)}}{{dt}} = - N(t) \; \lambda dt $$

The solution of this differential equation with the boundary condition N(0) = N ₀ is

$$N(t) = N_0 \; e^{ - \lambda t}$$

The normalised probability density function for the observation of a decay after a time t is

$$f(t) = \lambda e^{-\lambda t}dt$$

The average decay time of the isotope is given by

$$\tau = \left\langle t \right\rangle = \int\limits^{\infty}_0 t\,\lambda e^{-\lambda t}\ dt =\frac{1}{\lambda}$$

The decay law can therefore be written as

$$N(t) = N_0 \; e^{ - \frac{t}{\tau }}$$

In nuclear physics it is customary to characterise a decay by its ‘half-life’ T _1/2, rather than by the average decay time τ. The half-life is defined as the time it takes for half of the original nuclei to decay. We therefore have

$$\begin{array}{l} \frac{1}{2} = e^{ - \frac{{T_{1/2} }}{\tau }} \\\noalign{} T_{1/2} = \tau \ln 2 \\ \end{array}$$

Until now we have only considered the decay of one isotope into a stable final state. In many cases the decay process under consideration is part of a chain of decays, and the number of each type of isotope increases because new isotopes are added by the decay of some parent isotope and decreases because of its own decay. In that case one should consider the abundance of all the isotopes involved in the chain and a much more complicated expression is obtained.

1.6.6 The Nuclear Level Diagram

A very useful tool for understanding the nuclear decay mechanism is the nuclear level diagram illustrated in Fig. 1.7. In this diagram the x-direction represents the charge of a nucleus, and the y-direction the energy of the nuclear levels. A given isotope has a fixed number of protons and neutrons. The different energy levels of this isotope are represented as short horizontal lines in the level diagram. The gamma emission process and the electron conversion process correspond to a transition between two levels that are situated one above the other in the level diagram, and this emission is represented by a vertical arrow pointing downwards. A β⁻ or a β⁺ decay is represented by arrows pointing in the down-right or down-left direction.

Fig. 1.7

Nuclear level diagram for the ¹³⁷Cs decay. The numbers to the side of the level (e.g. 3/2⁺) represent the spin parity of the level. The number above the line representing the level is the energy in MeV relative to the ground state. The numbers next to the symbol β⁻ represent the end point energy of the electron and the fractional probability of the transition. There are two competing processes for the transition from the excited ¹³⁷Ba level to the ground state. About 90% of the transitions give rise to a gamma of 662 keV, while about 10% proceed through internal conversion. Therefore only 85% of the ¹³⁷Cs decays give rise to a gamma ray of 662 keV

Annex 6 lists a number of isotopes with decay modes that make them useful as radioactive sources for a variety of applications in nuclear and particle physics. For the nuclei undergoing a β⁺ decay, the positron comes to rest after a short range in matter and annihilates with an electron into two gamma rays of 511 keV emitted back to back.

1.7 Exercises

• 1. Show that the Lorentz transformation is such that the velocity of a light ray travelling in the x direction is the same for the observer in the frame S and for the observer in the frame S'.

• 2. What is the mean free path before decay for a charged pion with a kinetic energy of 1 GeV?

• 3. Show that the relativistic expression for the kinetic energy of a particle (Eq. 1.2) reduces to the non-relativistic expressions if the velocity of the particle is small compared to the velocity of light.

• 4. For a Poisson distribution with average value 16, calculate the probability to observe 12, 16 and 20 as measured value. Calculate the probability density function for a Gaussian distribution with average value 16 and dispersion 4, for the values x = 12, 16 and 20. Compare the results.

• 5. Consider a very short-lived particle of mass M decaying into two long-lived particles 1 and 2. Assume you can measure accurately the energies and momenta of the two long-lived particles. How will you calculate the mass of the short-lived particle from the known energies and momenta of the two long-lived objects?

• 6. Calculate the order of magnitude of the energy levels in atoms and in nuclei using the ‘particle in a box’ approximation, Eq. (1.9). Use for the dimension of the atom 10⁻¹⁰ m and for the dimension of the nucleus 10⁻¹⁵ m.

• 7. Show that in a β⁻ or a β⁺ decay only a very small fraction of the energy derived from the mass difference goes to the kinetic energy of the final-state nucleon. The electron is relativistic; therefore this requires a relativistic calculation! Hint: the 3-body problem can be reduced to a 2-body problem by considering the electron–neutrino system as one object with a mass of a few MeV.