Oliver Zimmer

Introduction to nuclear and particle physics November 15, 2005

Contents

1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part I. Physics of the atomic nucleus 2.

A first view on nuclear properties and systematics . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The atomic nucleus and its constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nuclear mass ......................................................... 2.3.1 Measurements of nuclear mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Nuclear mass defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Weizsäcker mass formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Nuclear spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Magnetic dipole moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Magnetic moments due to orbital angular momentum and due to spin . . . 2.6.2 Interaction of a magnetic moment with a magnetic field, Zeeman effect . . . 2.6.3 Hyperfine splitting (hfs) of atomic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Measurements using atomic beam methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Electric quadrupole moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7 9 9 12 13 15 17 17 19 19 23 25

3.

Particle accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Relativistic particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Electrostatic accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Ring accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Betatron: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Cyclotron: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Linear accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 34 34 36 38 41 42

4.

Scattering processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 44 45 47

5.

Geometrical shape of the nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Kinematics of electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rutherford scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Scattering from an extended charge distribution - the form factor . . . . . . . . . . . . . . 5.4 Mott scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 52 53

4

Contents

5.5 Measurements of nuclear form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.6 Nuclear charge distributions and radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.

Nuclear decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The decay law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Natural radioactivity and radioactive dating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Alpha-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Nuclear fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Beta-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Decay types and energetic conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Beta-spectrum and lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Application of beta-spectroscopy: direct measurement of the neutrino mass 6.6 Electromagnetic decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Mössbauer effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 61 63 64 67 67 70 74 74 76

7.

Models of the nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.1 Fermi-gas model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.2 Shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.

Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Basic principle of NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Nuclear spins in condensed matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 NMR imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 91 93

9.

Nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 General features of the reaction cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Application to some specific types of reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Nuclear energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 97 99 101

10. Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 10.1 Primordial nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 10.2 Element burning and the fate of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 11. Nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Charge independence of the nuclear force: the isospin . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Nucleon-nucleon potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 115 117

Part II. Particle physics 12. Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Conserved quantities and symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Conservation of electric charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Particles and anti-particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Baryon number conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Leptons and lepton number conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Hypercharge (strangeness) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Order in the hadron zoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Problem with the Pauli principle - quark colour . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 123 124 127 128 129 132 134 136

Contents

12.10Parity and its non-conservation in weak interactions . . . . . . . . . . . . . . . . . . . . . . . . . 12.10.1The parity operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10.2Parity in particle reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10.3Parity violation in nuclear beta decay - the Wu experiment . . . . . . . . . . . . . 12.10.4Helicity of the neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10.5Parity violation in the decay of the muon and the charged pion . . . . . . . . .

5

136 136 137 139 140 142

13. Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 13.1 Nucleon structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 14. Passage of radiations through matter, particle detection . . . . . . . . . . . . . . . . . . . 14.1 Charged particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Definition of basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Bethe-Bloch formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3 Further loss mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.4 Polarisation effect: Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Absorption coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Processes of energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 149 151 152 153 154 154 155 157

1. Introduction

Subatomic physics has various goals. A first aim of particle physics is to understand, what are the fundamental particles and what are the fundamental forces acting among these particles. Second, most particles found in Nature are composite objects, and theory of particles and their interactions should be capable to explain their properties. Third, many applications arise en passant and often completely unexpected from the quest what we all are finally made of and why matter behaves as it does. The present lecture will shed light on each of these aspects. The first question is certainly a driving force of many projects in particle physics. It is a long standing one and might never find a definitive answer. At different times in the evolution of physics concepts and experimentation, different entities were considered as fundamental particles. It is a remarkable fact that matter seems to be organised hierarchically. Atoms are found to consist of electrons and a nucleus, which itself is found to consist of nucleons, coming in two sorts as protons and neutrons. At the next level of the hierarchy, the nucleons are found to be bound systems of quarks. A particle may be called fundamental when it does not reveal any inner structure, such a for example the electron. Scattering experiments are the most direct method to resolve subatomic structures. They are able to reveal structure of the scattering target only if the de-Broglie wavelength of the probing radiation gets comparable to the size of the probed object. Scattering of electrons with several hundred MeV has revealed the distribution of electric charge within the atomic nucleus, corresponding to a typical nuclear radius of the order of a few 10−15 m. Probing the nucleon via electron scattering at even higher energy has revealed the quark structure of the nucleon. Even at highest energies attainable in laboratory, these particles appear to be point-like, which provides no guarantee that they indeed are fundamental ones. This definition of fundamental particles thus might get superseeded with the next step in the development of accelerator technology. Apart from scattering, an inner structure of subatomic particles may also manifest itself in various other ways. The size of a composite object reflects the strength of the interactions between its constituents. The first few lowest excited quantum states of a smaller composite object have larger energies than those of a larger object bound by weaker forces. Spectroscopy of excited states is a well known method to obtain information about interactions: measurements of atomic and nuclear spectra provide information about electromagnetic, respectively, nuclear interactions. Electromagnetic forces keep electrons bound to atomic nuclei (energy range eV to keV), nuclear forces (as a kind of van-der-Waals force) bind nucleons within the nucleus (MeV), and the strong force binds quarks within the nucleon (hundreds of MeV). Spectroscopy of composite objects may provide information about its constituents and at the same time about the forces acting among them: structure and the interactions, which generate structure can’t be discussed separately. A beautiful example is spectroscopy of systems consisting of heavy quarks, the socalled "quarkonia" discussed in the second part of the lecture. Much information also comes from studies of the properties of particle ground states, such as masses and electromagnetic moments, as well as decay properties of instable particles. It is an interesting exercise to trace back the roots of nearly everyday life applications of physics developments and discoveries. Insight into the heart of matter is a goal in itself, and none of the applications so familiar to us were predictable before the structure of matter was

2

1. Introduction

sufficiently well understood. Development of the fission reactor and understanding stellar fusion and element synthesis resides on the knowledge that there is an atomic nucleus with certain properties and behaving in a certain way. The aim of the first part of this lecture is to clarify what this "certain" means in nuclear physics, and the second part will try to do the same for particle physics.

Part I

Physics of the atomic nucleus

2. A first view on nuclear properties and systematics

Properties of nuclear ground states reveal much about nuclear structure and the interactions of its constituents. This chapter first gives a brief review of the great discoveries of the constituents of atoms made in the last years of the 19th and at the beginning of the 20th century. Next we focus on nuclear mass and what can be learned from systematic mass measurements of nuclei all over the nuclear chart. The last parts are about measurements of nuclear magnetic moments and quadrupole measurements, which will be interpreted in later chapters in the frame of nuclear models.

2.1 The atomic nucleus and its constituents • Discovery of the electron (Joseph John Thomson, 1897) — Experiments: set-up: vacuum tubes including electrodes and slits, such that he could observe the deflection of rays emitted from a cathode (called "cathode rays") by magnetic fields. Measurement of the total charge Q and the total energy W of the rays (measured as a heat production).

Fig. 2.1. J.J. Thomson and his gas discharge tubes [Phil. Magazine 44 (1897) 293].

— Interpretation of observations: Q equals the total number N of charged particles in the ray times their individual charge e: Q = Ne The produced heat W is due to the kinetic energy of the particles:

6

2. A first view on nuclear properties and systematics

W =

N mv 2 2

The magnetic field B forces the particles on a circular orbit with radius rB (Lorentz force = zentripetal force): m

v2 = evB rB

⇒

mv = BrB e

— Conclusions: 2W e = 2 m QB 2 rB The value of e/m thus determined turned out to be independent on the chemical composition of the cathode and the type of residual gas in the tube and therefore should be interpreted as a universal constituent of matter, the electron. J.J. Thomson also was the first to create a vacuum at a quality permitting him to apply large electric fields and observe electrical deflection of the electrons, providing additional evidence. • Discovery of the atomic nucleus (Ernest Rutherford, Hans Geiger, E. Marsden, 1908 − 1913) — Experiment: scattering of α-particles from thin gold foils (a few thousand atomic layers. It was later found that α-particles consist of atomic helium-4 nuclei). — Observation: scattering of some α-particles under much larger angles than expected from a previously existing model of the atom. — Explanation: in an earlier model of the atom by Thomson, its mass was assumed to be distributed continuously over the whole atom, with the electrons sitting in a positively charged medium like the olives in an italian ciabatta bread. However, in order to explain the observation, one has to assume a positively charged nucleus from which the α-particles may bounce back in central collisions. Rutherford, very astonished: "...as if a bullet was reflected back by a sheet of paper". • Discovery of the proton (Ernest Rutherford, 1919) — Experiment: bombaring hydrogen and other light atoms with α-particles. — Observation: creation of positively charged particles with larger range (observed in a Wilson chamber filled with nitrogen) than the incoming α-particles. — Conclusion: the secondary particles are nuclei of hydrogen (protons). Rutherford indeed observed the following reaction: 14

N + 4 He −→

17

O+p

• Discovery of the neutron (James Chadwick, 1932) — Hypothesis (E. Rutherford, since 1920): there should be a neutral particle about as heavy as the proton, resulting from a reaction where an electron has fallen into a proton. — Experiment (Frédéric Joliot & Irène Curie, 1932): α-particles from a polonium source hit a beryllium target, resulting in a very penetrating neutral radiation. This radiation was made to hit atomic nuclei within a paraffin converter (a second target in front of a detector). The detector (an ionisation chamber) measured the energy of the recoiling nuclei out of

2.2 Nuclides

7

Fig. 2.2. Observation of the nuclear reaction 14 N + α → 17 O + p in a Wilson chamber (Rutherford already observed this reaction in 1919 by slightly different means, see [Phil. Magazine 37 (1919) 581]). A radioactive source (212 Pb + 212 Bi + 212 Po) emits α particles with a range of 8.6 cm and 4.8 cm. On the right, an α particles with long range hits a nitrogen nucleus and breaks it up into 17 O (short trace) and a proton (long trace) [P.M.S. Blackett and D. Lea, Proc. Royal Soc. London 136 (1932) 325].

Fig. 2.3. Chadwick and the apparatus with which he discovered the neutron [Proc. Royal Soc. London 136 (1932) 692].

the converter (mainly protons out of hydrogen). They interpreted the radiation from the Be-target as a very energetic γ-radiation due to Compton effect. — Experiment (Chadwick, 1932): same as the experiment by Joliot & Curie, but he used additional converters of various materials → see exercise. — Conclusion: Application of momentum and energy conservation to the detected recoil particles required the radiation from the Be-target to consist of neutral particles about as heavy as the proton (as you have shown in the exercise).

2.2 Nuclides • Notations: — A nuclide is characterised by its proton number Z and its neutron number N . The nuclear charge is given by Ze, with the elementary charge e = 1.062 176 462(63) × 10−19 As. In the neutral atom, Z electrons surround the nuclide, which determine the chemical properties of the element. Alternatively to (Z, N ), a nuclide may be identified by (A, N ), with A = Z + N denoting the mass number.

8

2. A first view on nuclear properties and systematics

— Standard notation for a nuclide of element E: E A ). — Examples:

4 2 He2

= 4 He ("helium-4" or "α"),

— Isotopes are nuclei with same Z (e.g.

3 2 He1

and

A Z EN

(or briefly

235 92 U143

=

235

A

E, sometimes also

U ("uranium-235").

4 2 He2 ),

— Isotones are nuclei with same N (but different Z), — Isobars are nuclei with same A (e.g. 32 He1 and 31 H2 , the latter called tritium), — Mirror nuclei are two nuclei with Z and N reversed (e.g 32 He1 and 31 H2 , or

14 6 C8

and

14 8 O6 ),

— An Isomer is an unstable nucleus with an exceptionally long life time. • The nuclear chart is a pictorial compilation of some nuclear properties, represented by coloured squares in the (N, Z) plane. The colour characterises the dominant nuclear instability (black: stable; blue: β − instable; red: β + instable; yellow: α instable, green: spontaneous fission). Each square contains much additional information, like mean nuclear life times, decay modes, nuclear cross sections, etc.. Some systematic features observable in the nuclear chart will be discussed in an exercise.

Fig. 2.4. The nuclear chart, containing nuclides investigated thus far.

2.3 Nuclear mass

9

Fig. 2.5. The nuclear chart in the region around the lead isotopes.

2.3 Nuclear mass 2.3.1 Measurements of nuclear mass • A mass spectrograph consists of an ion source, an arrangement of electric and magnetic fields, slits for beam collimation and a detector. The beam is travelling within a high vacuum (usually 10−6 mbar or better).

Fig. 2.6. A typical mass spectrometer [E. Segrè, Nuclei and Particles, Benjamin, 1977].

— In the cylindrical condenser, the force Fel exerted by the electric field E on the ion with charge q, mass m and velocity v keeps the ions on a circular trajectory with radius rE : Fel = qE =

mv 2 rE

⇒

Ekin =

mv 2 qErE = 2 2

(2.1)

10

2. A first view on nuclear properties and systematics

⇒

the electric field provides an energy filter.

— In the magnetic field B, the force Fmag keeps the ions on a circular orbit with radius rB : Fmag = qvB = ⇒

mv 2 rB

⇒

p = mv = qBrB

(2.2)

the magnetic field provides a momentum filter.

— Ions are detected if they pass both filters (eliminate v from eq.(2.1) and eq.(2.2)): ⇒ ⇒

2 m B 2 rB = q ErE

the mass spectrometer is sensitive to the ratio of ionic mass and charge.

— The figures 2.7 and 2.8 show mass spectra with well-separated intensity lines of atomic and molecular masses (taken with a photographic plate instead of the electron multiplier shown in the figure). They may be obtained by (a very slight) change of the magnetic field. The experimental resolution is determined by the spatial arrangement and the quality

Fig. 2.7. Mass spectrum of atomic and molecular species with mass number A = 16 [E. Segrè, Nuclei and Particles, Benjamin, 1977].

of the electric and magnetic fields, and by the size of the collimation slits (→ exercise). m/q ratios differing by as little as down to 10−8 may be distinguished by the best available mass spectrometers of this type.Important point: the mass spectrometer may not distinguish between different ionised species if they have the same ratio m/q. In the second spectrum shown, however, the resolution of the apparatus was sufficient to separate singly charged 20 Ne from doubly charged 40 A. The m/q ratios are different due to the different nuclear binding energies (see section 2.3.2). • Mass standard: 12 C (for the experimental reason that carbon is an omnipresent element). The atomic mass unit u is defined as: 1u =

1 m(12 C) = 931.494 013(37) MeV c−2 = 1.660 538 73(13) × 10−27 kg 12

• Energy measurements of radiation emitted in nuclear reactions are needed to determine the mass of the neutron and masses of short-lived nuclei.

2.3 Nuclear mass

11

Fig. 2.8. Mass spectrum of particles with mass number A = 20 [E. Segrè, Nuclei and Particles, Benjamin, 1977].

— Example: Determination of the neutron mass mn . Reaction: radiative capture of thermal neutrons (Ekin = 25 meV) by hydrogen protons (with the γ denoting a quantum of electromagnetic radiation): n + 1 H −→ 2 H + γ The reaction is exothermic with the following energy balance: ¡ ¢ B(2 H) = mn + m(1 H) − m(2 H) c2 = Eγ +

Eγ2 ≈ 2.225 MeV 2m(2 H)c2

where m(1 H) and m(2 H) denote the masses of hydrogen and deuterium atoms (including the bound electron). Eγ is the energy of the emitted γ quantum, and the second term is the recoil energy of the deuterium atom (→ exercise). The mass of the neutron is deduced from accurate measurements of Eγ and the mass difference m(2 H) − m(1 H), ⇒

mn = 1.008 664 915 8(6) u

• A Penning trap consists of a quadrupolar electric field, generated by an arrangement of cylindrically symmetric electrodes as shown schematically in the figure. Superimposed is a homogeneous magnetic field along the axis of symmetry. For positive (negative) ions the end caps are charged positive (negative) with respect to the ring electrode, in order to repel them towards the horizontal plane of symmetry.Motion of charged particles in a Penning trap: — Cyclotron motion, due to the Lorentz force: F = mvω c = qvB

⇒

ωc =

qB m

The cyclotron frequency ω c is the quantity finally measured. For a given magnetic field B, one obtains a value for q/m. Comparing the cyclotron frequencies of two different ions measured in the same trap, a relative accuracy better than 10−10 may be reached. — The axial oscillation is due to the repulsive electric force from the endcap electrodes. — The magnetron motion is caused by the E × B drift.

12

2. A first view on nuclear properties and systematics

Fig. 2.9. Electric quadrupole fields of a Penning trap. The arrangement is cylindrically symmetric [L. Brown and G. Gabrielse, Rev. Mod. Phys. 58 (1986) 233].

Fig. 2.10. Motions of a charged particle in a Penning trap [L. Brown and G. Gabrielse, Rev. Mod. Phys. 58 (1986) 233].

2.3.2 Nuclear mass defect • Mass of a nuclide: ¢ ¡ m A Z EN = N mn + Zmp − B(A, Z)

(2.3)

i.e. the sum of neutron and proton masses, reduced by the nuclear binding energy B(A, Z), also called "mass defect" with typical values around 0.8 − 0.9% of the nuclear mass (note that eq.(2.3) defines binding energy as positive). The following values are taken from the Particle Physics Booklet 2002 of the particle data group: — Proton mass: mp = 938.272 00(4) MeV c−2 , — Neutron mass: mn = 939.565 33(4) MeV c−2 , — Electron mass: me = 0.510 998 902(21) MeV c−2 . — Measurements relative to the absolute measurements:

12

C mass standard are usually much more accurate than

— Proton mass: mp = 1.007 276 466 88(13) u, — Neutron mass: mn = 1.008 664 915 8(6) u. • Nuclear binding energy: The following systematic behaviour of the values of B/A is observed:

2.4 The Weizsäcker mass formula

13

Fig. 2.11. Nuclear binding energy per nucleon as a function of the mass number A [B. Povh et al., Particles and Nuclei, Springer, 2002].

— They are nearly constant for mass numbers larger than A ≈ 20. — Values of B/A are in the range 7.5 − 8.8 MeV c−2 .

— There are strong variations for the light nuclei. — The maximum value of about 8.8 MeV c−2 is reached for the nuclei around iron (A ≈ 60, iron is found to be a rather abundant element in the universe). • Consequences of the maximum of B/A: two ways of energy gain by nuclear reactions (the necessary condition in both cases is that the mass of the daughter particles is smaller than the mass of the mother(s)): — Nuclear fusion of two light nuclei with mass numbers A1 and A2 , B(A1 , Z1 ) + B(A2 , Z2 ) < B(A1 + A2 , Z1 + Z2 ) — Nuclear fission of a heavy nucleus into two (in rare cases more) fragments.

2.4 The Weizsäcker mass formula • The observed systematics of the nuclear binding energies, respectively masses, can be parameterised using the mass formula of Weizsäcker: ∙ ¸ 2 δ Z2 (N − Z)2 3 (2.4) m(A, Z) = (N mn + Zmp + Zme ) − av A − as A − aC 1 − aa − 1 4A A3 A2 {z } | 1 c2

(B0 +B1 +B2 +B3 +B4 )= c12 B(A,Z)

14

2. A first view on nuclear properties and systematics

m (A, Z) denotes the mass of the atom, therefore the mass of the electrons is included. On the other hand the binding energy of the electrons is neglected here, which is much smaller than the terms in the bracket, which are various contributions to the nuclear binding energy. They are shown in fig. 2.12 and interpreted below. A fit to the experimental data shown in Fig. 2.11 results in the set of parameters given in Tab. 2.1. Table 2.1. Parameter

Value [MeV c−2 ]

av

15.7

as

17.2

aC

0.714

aa

93.2 ⎧ ⎪ if Z and N are even: "even-even nuclei" ⎪ ⎨ −11.2 0 if A is odd: "even-odd-" or "odd-even nuclei" ⎪ ⎪ ⎩ +11.2 if Z and N are odd: "odd-odd nuclei"

δ

• Interpretation of the mass formula: — The behaviour of the nucleons within the nucleus shares some similarities with the behaviour of molecules within a liquid drop. The main part of the nuclear binding energy per nucleon (called volume term) is constant, B0 ∝ A. Nuclear forces appear to be saturated. This indicates that they have a short range and act only between nearby nucleons, leading to an approximately constant density of nuclear matter as a function of A. The nuclear radii R 1 measured by electron scattering behave accordingly, R ∝ A 3 (this means that nuclei are rather homogeneously charged and have a rather well-defined radius R). ρN ≈ 0.13 nucleon-mass fm−3 = 2 × 1017 kg m−3 The mean distance of nucleons within the nucleus is ≈ 1.8 fm. 2

— The term B1 ∝ A 3 ∝ R2 is called surface term. It decreases the binding energy in proportion to the nuclear surface, thus accounting for missing "neighboring nucleons" at the nuclear boundary. — B2 accounts for the Coulomb interaction of the protons, which further lowers the binding energy and prefers a low number of protons in a nucleus with given A. — The symmetry term (sometimes also called "asymmetry term") B3 prefers N = Z. It counter-balances the effect of the Coulomb term, leading to a mass minimum (valley of stability) at nearly symmetric (N = Z) light nuclei, and increasingly asymmetric (N > Z) heavier nuclei. — The pairing term: pairs of neutrons or protons within the nucleus increase the binding 1 energy. The factor A− 2 describes a reduced overlap of nucleon wave functions within larger nuclei.

2.5 Nuclear spin

15

Fig. 2.12. The various contributions to the nuclear binding energy per nucleon as given by the Weizsäcker mass formula [K.S. Krane, Introductory Nuclear Physics, Wiley, 1987].

2.5 Nuclear spin There are two different types of angular momentum in subatomic physics - orbital angular momentum and spin. The orbital angular momentum, defined in classical mechanics, can be quantised through the correspondence principle. Quanta of orbital angular momentum turn out to be integer multiples of }. However, in Nature also half odd integer values are found. These are manifestations of spin - an angular momentum intrinsic to a particle, which cannot be attributed to orbital motion of particles. A composite particle like the atomic nucleus may possess an angular momentum consisting of both orbital and spin angular momenta. It is customary to talk about the nuclear spin I independently of its origin - as due to spin or orbital angular momentum, or a coupling of both. • Orbital angular momentum is defined as L = r × p. The quantum mechanical operator is obtained through the correspondence principle µ ¶ ∂ ∂ ∂ p → −i}∇ ⇒ Lz = −i} x −y = −i} , ∂y ∂x ∂ϕ where ϕ is the azimuthal angle in polar coordinates. The states of a particle with given orbital angular momentum may be classified by quantum numbers and m, belonging to the operators L2 and Lz : L2 ψ lm = ( + 1) }2 ψ lm Lz ψ lm = m}ψ lm

( integer)

(− ≤ m ≤ )

p Angular momenta are quantised and restricted to values ( + 1)}. The component of L in a given direction (here taken as the z axis) can have only the values m}. For orbital angular momenta, and m are integer, and m may assume only the 2 + 1 values − ≤ m ≤ . This behaviour can be visualised in a vector diagram (see fig. 2.13). • Spin:

16

2. A first view on nuclear properties and systematics

Fig. 2.13. Quantisation of the angular momentum, shown for Henley, Subatomic Physics, Prentice Hall, 1991].

= 2 and m = 1 [H. Frauenfelder, E.M.

— Observation of dublets in spectra of alkali-atoms cannot be explained with integer angular momentum (belonging to 2 + 1, i.e. an odd number of m-states). — Explanation (Pauli 1924 & Uhlenbeck and Goudsmit): Electrons carry a spin, which is an internal angular momentum with quantum number 12 and correspondingly two possible orientations in a magnetic field. — The components of the orbital angular momentum operator L fulfill the commutation rules [Ji , Jj ] = i}

ijk Jk

(2.5)

(where the components of J are to be taken as those of L). Taking these algebraic identities as a definition of a quantum mechanical angular momentum (rather than the quantisation of the classical orbital angular momentum), also half-integer solutions are found to be possible (using an algebraic method found in many textbooks on quantum mechanics): J2 ψ JmJ = J (J + 1) }2 ψ JmJ Jz ψ JmJ = mJ }ψ JmJ

(J = 0,

1 3 , 1, , 2, ...) 2 2

(2.6)

(−J ≤ mJ ≤ J)

— Half-integer values of J cannot be interpreted as orbital angular momentum (a spinning top with any reasonable radius characterising a sub-atomic paticle leads to contradictions, i.e. the velocity of the moving charges at the surface of the particle should then be larger than the velocity of light). — Even particles with zero rest mass like the photon, or with mass close to zero like the neutrino have a spin. — Particles with integer spin are called bosons, those with half odd-integer spin are called fermions. They behave quite differently: the wave function of identical bosons is symmetric under exchange of two particles, whereas that of fermions is anti-symmetric. Fermions fulfill the Pauli exclusion principle: two fermions cannot sit in the same quantum state. This principle strongly influences the spectra of atomic and subatomic systems.

2.6 Magnetic dipole moments

17

— As evident from fig. 2.13, the spin of a particle determines the number of eigenstates of Jz . If the particle has a magnetic moment, this number is given by the number of Zeeman levels in a magnetic field.

2.6 Magnetic dipole moments The magnetic dipole moment of a particle is proportional to its angular momentum, orbital or spin, or a combination of both. Charged particles and neutral particles with charged constituents may possess a magnetic dipole moment. An example is a neutral atoms. A paramagnetic atom contains a net angular momentum of unpaired electrons. Electronically diamagnetic atoms may still possess a magnetic moment associated with nuclear spin. Particles without any angular momentum do not possess a magnetic moment. 2.6.1 Magnetic moments due to orbital angular momentum and due to spin • Classical magnetic moment: Consider a loop of electrical wire carrying a circular current I around a surface A = πr2 characterised by the normal vector A. As shown in classical electrodynamics, the magnetic moment of this current is given by µ = IA,

(2.7)

with µ being parallel to A for positive current flowing around A in the sense defined by the right hand. Let the current be generated by a single particle with charge q and mass m moving on the circular orbit with frequency v/(2πr), I=

qv , 2πr

(2.8)

then the magnetic moment is found to be proportional to the orbital angular momentum, µ=

q L. 2m

(2.9)

Fig. 2.14. Magnetic moment due to the orbital angular momentum of a charged particle with mass m moving on a circular orbit [H. Frauenfelder, E.M. Henley, Subatomic Physics, Prentice Hall, 1991].

• Quantum mechanical magnetic moment:

18

2. A first view on nuclear properties and systematics

— The angular momentum of a particle defines a direction. Only the projection of the magnetic moment onto this direction may appear in a measurement. Both observables therefore have to be proportional, µ = ge µ

J = γJ. }

(2.10)

The proportionality factor γ is called the gyromagnetic ratio. For a particle with elementary charge e, µ e=

e} 2m

(2.11)

is called the magneton. Its value for the electron is very different from that of the proton due to the large mass difference. The magneton of the electron is called Bohr magneton with value µB =

e} = 5.788 381 749(43) × 10−5 eV T−1 . 2me

(2.12)

The nuclear magneton, µN =

e} = 3.152 451 238(24) × 10−8 eV T−1 , 2mp

(2.13)

thus is a factor µB ≈ 2000 µN

(2.14)

smaller than the Bohr magneton. — The "g-factor" measures the deviation of the real magnetic moment from its value expressed by the magneton. It is a number of order unity and is different for the magnetic moment associated with the spin, respectively, orbital angular momentum. — The g-factor g due to the orbital angular momentum is equal to the charge of the particle in units of the elementary charge, e.g. g

,p

= 1 (proton),

g

,n

= 0 (neutron)

(2.15)

This is due to the correspondence of the classical definition of the magnetic moment of a charged particle with orbital angular momentum L, eq.(2.9) and the quantum-mechanical equation with definition of the magneton in eq.(2.10) with J = L. — The g-factor gs due to the spin is predicted by Dirac’s theory to be 2 for a spin 1/2 particle. For the proton and the neutron, there are large deviations from 2 due to the inner structure of these composite particles (the different components contributing to the spin will be discussed later). Even though the neutron is neutral, gs,n 6= 0 due to its charged constituents. The experimental values of the spin-g-factors are gs,p = 5.585 694 674(58) (proton),

(2.16)

gs,n = −3.826 085 4(10) (neutron).

(2.17)

2.6 Magnetic dipole moments

19

2.6.2 Interaction of a magnetic moment with a magnetic field, Zeeman effect • The Hamiltonian of the interaction of the magnetic moment µ of a particle with a magnetic field B, Hmag = −µ · B = −

ge µ J · B, ~

(2.18)

has the 2M + 1 eigenvalues µM B. WM = −ge

(2.19)

If the levels with different magnetic quantum numbers M were previously degenerate, the magnetic field thus lifts the degeneracy and separates levels with ∆M = 1 by ∆W = ge µB. This phenomenon is called Zeeman splitting and provides experimental access to the g-factor, respectively the value of the magnetic moment (the quantity usually tabulated), µ = ge µJ,

(2.20)

with the quantum number J defined in eq.(2.6).

Fig. 2.15. Zeeman energy splitting of the levels with different quantum numbers M, shown for J = 3/2 [H. Frauenfelder, E.M. Henley, Subatomic Physics, Prentice Hall, 1991].

2.6.3 Hyperfine splitting (hfs) of atomic spectra The nuclear magnetic moment interacts with the inner-atomic magnetic fields produced by one or several unpaired electrons of a paramagnetic atom. This so-called hyperfine interaction gives rise to a shift of the energy levels of the atom, i.e. of the excitation energies of its electron shell, and leads to splitting of optical lines in the absorption spectra of atoms. The determination of the nuclear magnetic moment from the measurement of such hyperfine splittings requires the knowledge of the atomic magnetic field at the nuclear site. This needs to be calculated and limits the accuracy of the method. One may study hyperfine effects without or with application of an additional external magnetic field.

20

2. A first view on nuclear properties and systematics

• Paramagnetic atom without external field: — The nuclear spin I and the total angular momentum J of the paramagnetic electron shell couple to the total angular momentum F = I + J.

(2.21)

F is conserved, and the Hamiltonian H0 of the atom fulfills the following commutation relations, £ ¤ H0 , F2 = 0, [H0 , Fz ] = 0, (2.22)

As shown in quantum mechanics, conservation of total angular momentum is related the invariance of the Hamiltonian under rotations. The commutation relations in eq.(2.22) mean that the eigenvalues of F2 and the eigenvalues of Fz are suited to characterise the states (they are "good" quantum numbers, compare with eq.(2.6)). The eigenvalue equation F2 ψ = F (F + 1) }2 ψ

(2.23)

defines the quantum number F , which due to the possibilities for the coupling of I and J may assume the values F = I + J, I + J − 1, ... |I − J| .

(2.24)

For I > J there are 2J + 1 possible values, and for J > I there are 2I + 1 values. Magnetic substates are defined by Fz ψ = mF }ψ,

(2.25)

defining magnetic quantum numbers mF . For a given eigenvalue F , the mF may assume the 2F + 1 values −F, −F + 1, ...F . — The hyperfine interaction of the nuclear magnetic moment µI with the inner-atomic field BJ (0) at the nuclear site is given by Hhfs = −µI ·BJ (0).

(2.26)

This interaction leads to a hyperfine splitting of the energy levels with different quantum numbers F . With J BJ (0) = BJ (0) , J I µI = µI , (2.27) I we obtain µ BJ (0) I · J =: A I · J, (2.28) Hhfs = − I IJ defining the hyperfine constant A. The eigenvalues of Hhfs are given by (→ exercise)

A [F (F + 1) − I (I + 1) − J (J + 1)] . (2.29) 2 BJ (0) has to be calculated, which is difficult and limits the accuracy of the determination of µI (to typically a few percent, but much better for hydrogen, which can be calculated accurately). For an unpaired s-electron (alkalide in the ground state), WF =

8π 2 (2.30) µ |ψ (0)| , 3 B with ψ (0) denoting the electron wave function at the nuclear site (see Segrè, Nuclei & particles). BJ (0) = −

2.6 Magnetic dipole moments

21

Table 2.2. Magnetic field at the nucleus produced by the atomic electrons. BJ (0)

[T]

Atom

n

2

2

H

1

17.4

Na

3

44

S1/2

P1/2

4.2

2

P3/2

2.5

K

4

63

7.9

4.6

Cs

6

210

28

13

— Interval rule: as a consequence of eq.(2.29), the spacing of levels in a hyperfine multiplet obey WF +1 − WF = A (F + 1) .

(2.31)

Fig. 2.16. Examples to demonstrate the interval rule in the level spacings of hyperfine levels. Left: J = I = 3/2, right: J = 5/2, I = 3/2 [D. Kamke, Einführung in die Kernphysik, Vieweg, 1979].

— The hyperfine splitting of the hydrogen atomic ground state (see fig. 2.17) has since 1951 become important for astronomy. The milky way is a strong emitter of the 21 cm radiation corresponding to the hyperfine splitting. Measurements of Doppler shifts provide information about the motion of gas nebula.

A 4

n =1 l=0 j = 1/ 2

3A 4

F =1

∆E = 6 ×10−6 eV λ = 21 cm ν = 1420 MHz F =0

Fig. 2.17. Hyperfine splitting of the ground state of the hydrogen atom.

• Paramagnetic atom with external field:

22

2. A first view on nuclear properties and systematics

— The total Hamiltonian now contains additional contributions due to the Zeeman interaction of the paramagnetic electron shell and the nucleus with the external magnetic field He , H = H0 + Hhfs − µJ ·He − µI ·He ,

(2.32)

where the nuclear Zeeman term µI ·He gives the smallest contribution to the total energy. We have to consider two limiting cases: 1. Low field (Zeeman interaction of the paramagnetic electron ¿ hyperfine interaction): |µJ ·He | ¿ |µI ·BJ (0)|

(2.33)

the total angular momentum stays conserved (F stays a good quantum number) ⇒

Zeeman effect for the F -spin,

lifting the degeneracy of the hyperfine multiplet. The change of energy of each level due to

Fig. 2.18. The total magnetic moment µF of the coupled angular momenta I and J. Since µI ¿ µJ , the size of µF (projection of µI + µJ onto F) is essentially determined by the projection of µJ onto F. However, the angle between µJ and F is determined by the coupling (compare left and right figure) [D. Kamke, Einführung in die Kernphysik, Vieweg, 1979].

the magnetic field is given by WmF = −µB gF mF He ,

(2.34)

where gF is given by gF = gJ

F (F + 1) + J (J + 1) − I (I + 1) µ F (F + 1) + I (I + 1) − J (J + 1) +gI N . (2.35) 2F (F + 1) µB 2F (F + 1)

The term ∝ gI is much smaller than the first term and can usually be neglected. The size of gF depends crucially on the coupling of J and I, as shown in fig. 2.18. This effect is contained in the fractions. 2. High field: |µJ ·He | À |µI ·BJ (0)| µI ·BJ (0) becomes a perturbation of µJ ·He : the angular momenta I and J decouple (Paschen-Back effect). µJ precesses quickly around He , whereas µI senses the time-averaged

2.6 Magnetic dipole moments

23

component of BJ (0) along He . Total angular momentum is no longer conserved (isotropy of space is broken by the strong magnetic field), F is no longer a good quantum number. The energies of the various states (characterised by mJ and mI ) are given by W = −mJ gJ µB He + AmI mJ − mI gI µN He .

(2.36)

The leading first term is the electronic Zeeman energy. The second term is due to the interaction between I and J, and the last one is the nuclear Zeeman energy.

Fig. 2.19. Splitting of atomic levels for a specific case (J = 1/2, I = 3/2) due to hyperfine interaction (a) without external magnetic field, (b) with a weak field, (c) with a strong field [G. Musiol et al., Kernund Elementarteilchenphysik, Harri Deutsch, 1995].

2.6.4 Measurements using atomic beam methods • Stern-Gerlach effect: — A beam of diamagnetic atoms or molecules passes through a field gradient in z direction perpendicular to the flight path. — The force on the nuclear magnetic moments is Fz = −

∂ ∂He,z (−µeff ·He ) = µeff . ∂z ∂z

(2.37)

The value of the effective magnetic moment µeff is given by µeff = −

∂W , ∂He

(2.38)

with W being given either by eq.(2.34) or eq.(2.36), with J = 0 and mJ = 0 for (electronically) diamagnetic particles. — Difficulty: the beam deviating force is very weak → rather inaccurate method (experimental conditions for first measurement of µI of the proton (Stern, Frisch, Estermann, 1933): ∂He,z /∂z = 800 T m−1 , 1.5 m flight path, beam deviation ≈ 10−2 mm).

24

2. A first view on nuclear properties and systematics

• Rabi method : — Apparatus: atomic beam through two Stern-Gerlach magnets, separated by a homogeneous magnetic field region H0 with a superimposed high frequency field for magnetic resonance.

Fig. 2.20. Rabi’s atomic beam apparatus. The two solid curves in the upper figure represent paths of (neutral) particles having different moments and velocities and whose moments are not changed during passage through the apparatus. The two dashed lines indicate the change of the trajectories when the magnetic moment has changed its component parallel to the external field in the homogeneous field region between the two Stern-Gerlach magnets [J.B.M. Kellog and S. Millman, Rev. Mod. Phys. 18 (1946) 323].

— Observation of the beam intensity in a detector for different values of the homogeneous magnetic field.

Fig. 2.21. Measurement of resonance of 7 Li using Rabi’s atomic beam apparatus [I.I. Rabi et al., Phys. Rev. 53 (1938) 318]

2.7 Electric quadrupole moments

25

— Interpretation: decrease of beam intensity when the resonance condition is fulfilled, ω=

µI H0 , }

(2.39)

i.e. when the energy }ω corresponds to the energy difference of two magnetic substates with ∆m = ±1. Magnetic resonance induces nuclear spin flips, such that the beam deviating forces in both Stern-Gerlach devices act in the same direction. — Typical accuracies are in the order of 10−4 : in the strong field limit, one can get rid of the dependence on the (quite uncertain) intra-atomic field, therefore the situation is much better than in hfs studies of atomic spectra. • Experimental facts: — All even-even nuclei (Z even and N even) have µI = 0. Interpretation: nucleons of the same sort couple to pairs with spin 0. — All even-odd and odd-even nuclei have half odd integer spin. — All odd-odd nuclei have integer spin. — Experimental values of nuclear g-factors (see table 2.3). Table 2.3. Some measured values of nuclear magnetic moments. Nuclide

µ [µN ]

n

−1.913

p 2

+2.792

H

+0.857

17

O

57

Fe

57

Co

+4.733

93

Nb

+6.171

−1.894

+0.0906

2.7 Electric quadrupole moments An electric quadrupole moment Q of the nucleus manifests itself as a violation of the interval rule of the hpyerfine spectra, stated in eq.(2.31). The reason of the additional energy shift of energy levels is the interaction of Q with an electric field gradient at the nuclear site. The electric field due to the electrons of a free atom has J as an axis of cylindrical symmetry, and the nuclear charge distribution is cylindrically symmetric to I. • Classical derivation of the energy due to an electric quadrupole moment of the atomic nucleus: — Electrostatic energy of the nuclear charge distribution ρ (r) situated in the potential ϕ (r) originating from the charges surrounding the nucleus: Z Z WE = e ρ (r) ϕ (r) d3 r with the normalisation ρ (r) d3 r = Z (2.40)

26

2. A first view on nuclear properties and systematics

Fig. 2.22. Deformed atomic nucleus situated in an electric field gradient [D. Kamke, Einführung in die Kernphysik, Vieweg, 1979].

Taylor series expansion of ϕ (r) around the nuclear site at r = 0: ⎡ ⎤ Z 3 2 X 1 ∂ ϕ WE = e ρ (r) ⎣ϕ (0) + r · ∇ϕ (0) + rk rl (0) + ...⎦ d3 r 2 ∂rk ∂rl

(2.41)

k,l=1

— The first term in eq.(2.41), Z eϕ (0) ρ (r) d3 r = eZϕ (0)

(2.42)

is the potential energy of the electric nuclear charge q = eZ in the electrostatic potential ϕ (0). — The second term vanishes, because the nuclear electric dipole moment vanishes, Z d = e rρ (r) d3 r = 0

(2.43)

Reason: nuclear states ψ (r) have well-defined parity (with very small deviations induced by weak interactions, which we neglect here), ρ (r) ∝ ψ ∗ (r) ψ (r) is an even function under inversion of the spatial coordinates (r → −r), such that the integrand is odd. Therefore: Nuclei do not possess static electric dipole moments — The third term describes the interaction of the tensor of the electric field gradient ϕkl (0) :=

∂2ϕ (0) ∂rk ∂rl

with the tensor of the nuclear electric quadrupole moment Z Qkl := rk rl ρ (r) d3 r WQ =

3 e X Qkl ϕkl (0) 2

(2.44)

(2.45)

(2.46)

k,l=1

— The electric field is cylindrically symmetric about J, and we consider a nucleus with cylindrically symmetric deformation about I. Let (x, y, z) be a coordinate system with the z axis along J, and (ξ, η, ζ) a coordinate system with the ζ axis along I. The tensor ϕkl (0) is diagonal in the system (x, y, z), and Qkl is diagonal in the system (ξ, η, ζ). Therefore,

2.7 Electric quadrupole moments

27

¢ e¡ Qxx ϕxx (0) + Qyy ϕyy (0) + Qzz ϕzz (0) 2 ¢ e¡ = Qξξ ϕξξ (0) + Qηη ϕηη (0) + Qζζ ϕζζ (0) 2

WQ =

Since

∇ · E (0) = 0

⇔

∆ϕ (0) = ϕxx (0) + ϕyy (0) + ϕzz (0) = 0,

(which is valid in the good approximation that there is no electron charge at the nuclear site), and the field is axially symmetric, 1 ϕxx (0) = ϕyy (0) = − ϕzz (0) 2 Z ¢ ¡ 2 ϕ (0) ⇒ WQ = zz e 3z − r2 ρ (r) d3 r 4

(2.47)

with r2 = x2 + y 2 + z 2 .

— The nuclear quadrupole moment shall be expressed in the nuclear coordinate system, rather than by the integral in eq.(2.47). One defines the (intrinsic) quadrupole moment of the nucleus as Z ¢ ¡ 2 (2.48) Q := 3ζ − r2 ρ (r) d3 r = 3Qζζ − Qrr and deduces via a coordinate transformation µ ¶ ϕ (0) 1 3 WQ = zz eQ cos2 Θ − , 4 2 2

(2.49)

where Θ is the angle between the direction of the nuclear spin I (the axis of nuclear deformation) and the angular momentum J of the atomic electron shell (see fig. 2.22). • Interpretation of the quadrupole moment as nuclear deformation: — Q > 0 : cigar-shaped (prolate) nuclear deformation, — Q < 0 : lens-shaped (oblate) nuclear deformation, — Q = 0 for spherically symmetric nuclei. • Quantum mechanical treatments: — The spherical harmonic projection factor (term in brackets in eq.(2.49), calculated by Casimir 1936): WQ =

B 32 C (C + 1) − 2I (I + 1) J (J + 1) 4 I (2I − 1) J (2J − 1)

with B = eQϕzz (0)

and C = F (F + 1) − I (I + 1) − J (J + 1)

B is called quadrupole constant.

(2.50)

28

2. A first view on nuclear properties and systematics

— Quadrupole moment Q as expectation value ¯ ® ­ ® ­ ¯ Q (I) = 3z 2 − r2 = ψ ¯3z 2 − r2 ¯ ψ mI =I

(2.51)

where the wave function belonging to mI = I (corresponding to maximal alignment of the nucleus with respect to its spin) has to be chosen. Since this expression enters the equations for the atomic spectra, Q(I) is also called spectroscopic quadrupole moment. One can show that (→ exercise): µ ¶ 1 Q I= =0 2

— One may also consider the expectation values for wave functions with mI < I, describing the various possible projections of the intrinsic quadrupole moment on the axis of nuclear spin. This leads to the more general expression Q (mI ) =

3m2I − I (I + 1) Q (I) I (2I − 1)

Positive and negative values of mI result in the same Q (mI ): the two levels with ±mI have the same energy shift WQ (distinct from the magnetic interaction where the energy shift has opposite sign). • Example (fig.(2.23)): Splitting of the λ = 679.9 nm optical line of 173 Yb. In the 3 S1 states, ϕzz (0) = 0, such that there is no interaction with the nuclear quadrupole moment ⇒ only the hyperfine constant A enters, and the states are split according to the interval rule stated in eq.(2.31). In the less symmetric 3 P1 states, ϕzz (0) 6= 0, and both A and the quadrupole constant B determine the energy levels, the latter breaking the interval rule.

Fig. 2.23. Splitting of the λ = 679.9 nm optical line of 173 Yb due to hyperfine and quadrupole interactions [D. Kamke, Einführung in die Kernphysik, Vieweg, 1979].

Table 2.4. Some measured values of nuclear electric quadrupole moments. Nuclide

QN [10−24 m2 ]

2

+0.002 88

H

17

O

59

Co

−0.025 8 +0.40

133

Cs

161

Dy

176

Lu

+8.0

209

Bi

−0.37

−0.003 +2.4

3. Particle accelerators

• Purposes: — Scattering experiments for investigating subatomic structures (e.g. nuclear radii, structure of the nucleon, etc.), — Generation of new (hitherto unobserved) particles, — Generation of secondary particles ("factories" for pions, muons, spallation neutrons, etc.) for applications in nuclear physics (→ nuclear reactions), solid-state physics (→ ion implantation in surfaces, muon spin rotation, etc.) and medicine (e.g. irradiation of cancers with protons). • Basic principles: — charged particles from a source are accelerated due to the force of an electric field. A difference of electric potential U induces a change of kinetic energy of ∆Ekin = ZeU of a particle with electric charge Ze. — In electrostatic accelerators, U is used once (van-de-Graaf accelerator ) or twice (Tandemvan-de-Graaf accelerator ). Much higher final particle energy may be reached with timevarying electric fields, either in circular accelerators (cyclotron, synchrotron) or linear accelerators (e.g. electron and proton linacs). — Beam focusing and deflection are essential features, in particular of large ring accelerators.

3.1 Relativistic particles Many accelerators boost particles to relativistic energies, meaning Ekin & mc2 . This applies in particular to electron scattering studies to investigate nuclear shapes discussed in chapter 5. Therefore, we first briefly review some basic results and concepts of the theory of special relativity. • Relativistic energy and momentum: p E = m2 c4 + p2 c2

(3.1)

In the non-relativistic limit, E = mc2 +

¡ ¢ p2 + O p4 2m

= rest energy + kinetic energy. For the photon (which has no rest mass) E = |p| c = ~kc

(3.2)

32

3. Particle accelerators

• Lorentz transformations: let a coordinate system (ct0 , x0 ) move with velocity v parallel to the z axis of an inertial system (ct, x). Under the assumption that the speed of light is equal in any inertial system, one can deduce the following transformations of the coordinates (see textbooks on special relativity): ct0 = γ (ct − βz) x0 = x, y 0 = y z 0 = γ (z − βct)

(3.3)

with β=

1 and γ = p 1 − β2

v c

(3.4)

In matrix form (with Einstein’s sum convention on the right-hand-side): x0µ =

3 X

Λµν xν =: Λµν xν

(3.5)

⎞ 0 0 −βγ 10 0 ⎟ ⎟ 01 0 ⎠ 00 γ

(3.6)

ν=0

with

⎛

γ ⎜ 0 µ Λν = ⎜ ⎝ 0 −βγ • Four-vectors:

¡ ¡ ¢ ¢ — Covariant 4-vector: aµ = a0 , a1 , a2 , a3 , contravariant 4-vector: aµ = a0 , −a1 , −a2 , −a3 . Transformation between them with the metric tensor g µν : ⎞ ⎛ 1 0 0 0 ⎜ 0 −1 0 0 ⎟ ⎟ aµ = g µν aν with g µν = ⎜ ⎝ 0 0 −1 0 ⎠ 0 0 0 −1 — Scalar product of two 4-vectors: a · a = aµ aµ :=

3 X

µ=0

¡ ¢2 ¡ ¢2 ¡ ¢2 ¡ ¢2 ¡ ¢2 aµ aµ = a0 − a1 − a2 − a3 = a0 − a2

(3.7)

— time and space, xµ = (ct, x)

¡ ¢ — energy and momentum, pµ = Ec , p ¡∂ ¢ , −∇ — gradient, ∂ µ = ∂x∂ µ = ∂t

— electromagnetic potential, Aµ = (φ, A)

— current, j µ = (ρ, j) • Lorentz invariants: — As a consequence of the invariance of the speed of light on the reference system, a transformation from an inertial coordinate system (ct, x) to a second inertial system (ct0 , x0 ) leaves the scalar product of two 4-vectors invariant:

3.2 Electrostatic accelerators

xν xν = x0ν x0ν

33

(3.8)

E2 − p2 = m2 c2 c2 with the second relation being equivalent to eq.(3.1). pν pν =

(3.9)

• Useful relations: E = γmc2

(3.10)

p = γmv

(3.11)

3.2 Electrostatic accelerators • Van-de-Graaf accelerator:

particle source

accelerating electrodes

transport belt

discharging

charging beam

m otor

target

Fig. 3.1. Sketch of a Van-de-Graaf accelerator [K. Wille, Physik der Teilchenbeschleuniger, Teubner, 1992].

— charging of a rotating belt, transport of the charges to a terminal electrode, mechanical removal of these charges to charge up the terminal to a typical maximum value of 15 MV. — The terminal at high positive potential is electrically coupled to a source for positively charged particles, which are accelerated by a set of electrodes with gradually lower potential, defined by a chain of resistors. The electrode system is situated in vacuum or a quenching gas at high pressure to prevent electrical breakdown, the particle beam is propagating in vacuum. — This scheme produces a particle current up to typically 100 µA (corresponding to 10−4 /(1.602× 10−19 ) ≈ 6 × 1014 singly charged particles s−1 ). • Tandem-van-de-Graaf acelerator : — The accelerating potential difference U is used two times.

34

3. Particle accelerators

Fig. 3.2. Sketch of a Tandem-van-de-Graaf accelerator [B. Povh et al., Particles and Nuclei, Springer, 2002].

— Negative ions from an ion source are first accelerated to the terminal, where they pass a thin stripper foil : this removes electrons from the ions and converts them into positively charged ones, which are subsequently further accelerated as in the single step van-de-Graaf accelerator. For heavy ions, the stripping may lead to highly charged ionic states, such that with U = 15 MV final kinetic energies of several 100 MeV may be attained.

3.3 Ring accelerators In ring accelerators, charged particles are held on a bent orbit by a magnetic field B0 at the trajectory, Lorentz force = centripetal force, q v × B0 = −m v × ω.

(3.12)

The minus sign is definition, meaning that a negatively charged particle is moving about the magnetic field according to the right hand rule. The vector ω of angular velocity is then parallel to B0 , and we define the cyclotron frequency ω c > 0 for q < 0, ⇒

qB0 = −mω c = −m

v |p| =− . r r

(3.13)

The sign of the angular velocity is just a convenient means to express the direction of the cyclotron motion. 3.3.1 Betatron: • Principle (Kerst, 1940): — A time-dependent B-field induces an electrical field, which is used for particle acceleration, ∇×E=−

∂B . ∂t

From Stokes theorem it follows that

(3.14)

3.3 Ring accelerators

2πrE = −

35

¢ d ¡ Bπr2 , dt

where B is the mean magnetic field through the area enclosed by the particle orbit. Keeping the radius r at a constant value r0 , the electric field is given by E=−

r0 dB . 2 dt

(3.15)

The force on the charged particle along its trajectory is qr0 dB d |p| = qE = − dt 2 dt qr0 (3.16) ⇒ ∆ |p| = − ∆B. 2 — On the other hand, r0 is determined by the magnetic field B0 at the orbit. According to eq.(3.13), F =

∆ |p| = −qr0 ∆B0 .

(3.17)

Comparison of eq.(3.16) and eq.(3.17) leads to the following condition, necessary to keep the particle on a constant radius r: ∆B = 2∆B0

(3.18)

Note that the particle can be accelerated only once and to a final energy which is reached

field coils

correction coils

vacuum cham ber

iron yoke

beam

Fig. 3.3. Schematic layout of a betatron. The charged particles are moving in a doughnut shaped vacuum chamber. The main part of the accelerating magnetic flux passes through the central gap in the iron yoke [R. Kollath, Teilchenbeschleuniger, Vieweg, 1955].

when the field is ramped up to its maximum value. • Stability conditions for the particles to stay close to the desired orbit: — Radial stability requires: Lorentz force Q centripetal force

for r Q r0 : Eq.(3.13)

⇒

|B0 (r)| has to decrease weaker than

1 r

36

3. Particle accelerators

Fig. 3.4. Magnetic field gradients force the particle back to the median plane between the pole shoes.

— Axial stability requires restoring forces once the particle leaves the median plane between the pole shoes. This is provided by magnetic field gradients (see fig. 3.4). — Sufficiently strong damping of Betatron-oscillations. • Application: acceleration of electrons up to 300 MeV. • Practical limit to the final energy - energy loss due to synchrotron radiation: electrons traveling in a circular orbit radiate electromagnetic energy. The resulting energy loss per turn is given by µ ¶4 4π e2 3 E ∆E = β . (3.19) 3 r0 mc2 Taking, e.g. r0 = 1 m, E = 100 MeV, one obtains ∆E = 8.9 eV. 3.3.2 Cyclotron: • Principle (Lawrence, 1927): — A constant magnetic field guides the charged particles in a spiral path. Particle acceleration is due to an electric field with correct direction any time that the particle passes through it. — Particle acceleration in the gap between two "D"-shaped electrodes ("Dees"). Electric field only in the gap but not inside the Dees (Faraday cage effect). The two Dees oscillate electrically with ω hf = |ω c |. If the particle gets accelerated in the gap at time 0, it gets further accelerated after each half circle in the magnetic field, since its angular velocity is just the cyclotron frequency, qB0 , (3.20) m independent on the orbital radius which increases with particle energy. For protons, ωc = 15.2 MHz · B[T]. 2π ωc = −

• The maximum energy is not determined by the electric potential difference between the Dees, which can be very much lower than for electrostatic accelerators. The gist of the technique is that the particles traverse the potential difference many times, avoiding the particular difficulties of high voltages (arcs, corona discharge etc.).

3.3 Ring accelerators

pum p

37

HV for beam deflection

ion source vacuum cham ber target rf oscillator

ion beam

Fig. 3.5. Schematic layout of a cyclotron, showing the two Dees (hollow semi-circular electrodes for particle acceleration), driven by a hf power source. The charged particles move on a spiral orbit from their point of creation in an ion source in the centre of the apparatus to a point of extraction [R. Kollath, Teilchenbeschleuniger, Vieweg, 1955].

One of the first Cyclotrons...

... and a little bit later

Fig. 3.6. Left: M.S. Livingston and E.O. Lawrence with their 37 inch cyclotron. It was used to produce the first artificial element, the technetium. Right: a 184 inch cyclotron built after the second world war, used for first production of artificial mesons [Lawrence Berkeley Laboratory].

• For relativistic particles, ωc = −

qB0 , γm

(3.21)

where, due to the factor γ (see eq.(3.4)), ω c gets smaller for larger particle energy. The cyclotron is therefore not suited for acceleration of electrons. A typical final energy for protons is 20 MeV, i.e. well below its rest energy of 938 MeV. Methods to attain higher final proton energies: — Adaptation of ω hf to |ω c | during acceleration ("Synchrocyclotron"), only possible in a pulsed mode for a bunch of particles passing through the whole apparatus. — Adaptation of B0 (r) in order to keep ω c constant ("Isochrone-cyclotron"). Advantage: creates a continuous and high current of particles.

38

3. Particle accelerators

— Example: Isochrone-cyclotron at Paul-Scherrer Institut, Villigen, Switzerland (www.psi.ch). Proton current 1.5 mA at 590 MeV.

Fig. 3.7. The isochrone-cyclotron at PSI [www.psi.ch].

3.3.3 Synchrotron • Principle (Veksler, 1945 / Mc Millan, 1946): — Particles are kept in a given closed path by synchronous adjustments of ω hf (t) and B0 (t) during acceleration. A magnetic field is then only required in the path (no magnetic-pole pieces with diameter of the orbit are needed any more, like in the betatron and the cyclotron). The accelerating electric field is provided in one or more gaps. Preaccelerated particles are injected into the ring using fast switchable kicker magnets.

accelerating gap m agnet for beam deviation

injection m agnet

ejection m agnet focusing m agnet

linac

Fig. 3.8. Schematics of a typical synchrotron [K. Wille, Physik der Teilchenbeschleuniger, Teubner, 1992].

3.3 Ring accelerators

39

• Conditions to be fulfilled simultaneously: 1. The accelerating hf-field ω hf needs to be an integer multiple of the angular particle velocity ω, µ ¶ |p| c2 v v |p| c = ω(t) = using β = = , (3.22) r0 r0 E c E in order to result in repeated acceleration. ω grows with the energy of the particles until they are fully relativistic, where ω=

c . r0

(3.23)

For extremely relativistic electrons, ω hf can be kept constant, but for protons it has to be increased synchronously with the particle frequency in the increasing magnetic field: 2. The magnetic field B0 has to fulfil (eq.(3.13), take for simplicity a circle with field B0 at any point of the particle’s path, ignoring straight beam sections between magnets): B0 (t) = −

|p| . qr0

(3.24)

• Requirement of stability: the particles move on very long trajectories, and deviations from the ideal trajectory should therefore be kept small. — Phase stability: Consider fig. 3.9: a particle arriving too early in the accelerating gap will

electron on ideal orbit phase instable region

phase-stable region

too late electron too early electron

Fig. 3.9. Variation of the accelerating potential with time, showing the origin of phase stability [R. Kollath, Teilchenbeschleuniger, Vieweg, 1955].

experience a larger potential difference U and therefore get stronger accelerated, corresponding to a larger radius. An increase of particle energy corresponds to a decrease of angular velocity: |ω| =

|q| B0 |q| B0 c2 = . γm E

(3.25)

40

3. Particle accelerators

Hence, after the next turn, the particle will cross the gap a little later in phase than it did before. For a particle arriving too late in the gap, the situation is reversed. As a result, an equilibrium situation exists and the particles in the beam form bunches in the orbit. — Strong focusing:Figure 3.10 shows the focusing and defocusing of a charged particle in a

iron yoke

coils

hyperbolic pole-tips

Fig. 3.10. Particle focusing and defocusing in a magnetic quadrupole field. The arrows indicate the Lorentz force experienced by a positively (negatively) charged particle moving into (out of) the plane of the drawing. Particles in the (x, y) plane are focused, whereas in the (z, y) plane there is defocusing [K. Wille, Physik der Teilchenbeschleuniger, Teubner, 1992].

magnetic quadrupole field. A pair of such quadrupole magnets in series, twisted by 90 degrees, has a net focusing effect (like in optics a pair of two lenses with focal lengths f and −f , separated by a distance d, have a total focal length ftotal = f 2 /d). Such pairs of alternatingly focusing and defocusing quadrupole magnets (called quadrupole dublets) can be found in any synchrotron (see, e.g. the accelerator Lear in fig. 3.11. Strong focusing dublets are visible before and after each dipole magnet).

Fig. 3.11. Lear at CERN [www.cern.ch].

3.3 Ring accelerators

41

3.3.4 Collider • This is a special type of synchrotron for high-energy physics studies, where particles and their anti-particles are brougth to interactions in the same ring accelerator. • Advantage: the centre-of-mass of two reacting particles a and b is practically at rest ⇒ the energy W available for production of new particles is the sum of the total energy of both particles, Wcollider = Ea + Eb = 2E. On the other hand, in √ a fixed-target experiment, where a particle a is made to hit a particle b at rest, Wfixed target = 2Ea mb c2 ¿ Wcollider . accelerating structure dipole m agnet focusing m agnet

particle detector injection m agnet

injection m agnet

particle detector

e - injection

e + injection

focusing m agnets

Fig. 3.12. Schematics of a collider. Large detectors are installed at one or several points (two in the figure) to study the particle reactions and possibly find new particles. Note however, that such events are rare with respect to the number of particles within the intersected beams, not leading to a significant reduction of the beam intensities in the ring [K. Wille, Physik der Teilchenbeschleuniger, Teubner, 1992].

• Examples: — LEP II (at CERN): e+ e− collider with 2 × 100 GeV, — Fermi National Accelerator Laboratory (FNAL): pp collider with 2 × 0.9 TeV, — Large Hadron Collider (LHC, project at CERN): a double ring for pp collisions with 2 × 7 TeV. • Difficulty: much smaller luminosity L=f ·n·

N1 N2 A

(3.26)

than in fixed-target experiments. L is proportional to the number of events one may observe in an experiment. f is the orbital frequency, A the beam cross section, N1 and N2 the numbers of particles in the two colliding beams and n is the number of particle packets (bunches) in the beam. FNAL has L = 1032 cm−2 s−1 , and the projected value for LHC is 1034 cm−2 s−1 .

42

3. Particle accelerators

3.4 Linear accelerators • Principle (Wideroe): — employ accerating E fields between a series of cylindrical electrodes, driven by a single hf generator (see fig. 3.13). As in the Dee’s of the cyclotron, there is no electric field inside the tubes, so that charged particles will not sense the change of potential when they are traveling through a tube. Useful for electrons and protons (with some difference in the design of the accelerating structures). — there is no magnetic field, the particle trajectories are straight. Therefore, no emission of synchrotron radiation (see eq.(3.19)), and much higher final electron energies should be attainable than in a ring accelerator.

Fig. 3.13. Schematics of a proton linac. The lengths of the tubes are adjusted to the velocity increase of the particles, such as to provide acceleration within each gap between the tubes [B. Povh et al., Particles and Nuclei, Springer, 2002].

Fig. 3.14. Linacs at CERN [www.cern.ch].

— Applications: Proton linacs with final energy of typically 100 MeV are used as injectors for large ring accelerators. — Project: TESLA, a 30 km long electron linac with final electron energy of 500 GeV.

4. Scattering processes

4.1 General considerations • Scattering processes serve to understand structure of subatomic particles and interactions between them. A projectile is sent onto a target, where it may get scattered or induce a reaction, a + b → c + d + ... where a and b are projectile and target particle and c and d are reaction products. Relevant parameters to describe processes are rate, momentum and mass of the reaction products. Detectors register the angle of secondary particles with respect to the incident direction. • Beams of various particles are nowadays available with energies ranging from 10−3 eV for thermal neutrons up to 1012 eV for protons. Short living particles, like muons, pions or Kmesons are produced as secondary particles in high-energy reactions. • A plethora of targets is available in solid, liquid or gaseous state. Targets with nuclear polarisation are available and further developed for studies of spin-dependent interactions. The target may also be a second beam, as realised in a collider. • In elastic scattering, the particles before and after the process are identical, a + b → a0 + b0 . The deBroglie wavelength λ of a particle is related to its kinetic energy Ekin (using eq.(3.1) with E = mc2 + Ekin ) by ( √ } , Ekin ¿ mc2 λ } }c 2mEkin ≈ . (4.1) = =p 2 }c }c 2π |p| 2mc2 Ekin + Ekin Ekin À mc2 Ekin ≈ E ,

It determines the spatial resolution of the particle radiation incident to a target. Figure 4.1 shows, how λ is related to momentum and kinetic energy of various particle radiations (the deviation for the heavy particles from the photon line vanishes, when these particles are highly relativistic, i.e. for Ekin À mc2 , the particles then get photon-like behaviour ). In order to resolve a structure with linear extension ∆x, the wavelength should fulfill λ . ∆x 2π

⇔

|p| &

} , ∆x

|p| c &

}c 200 MeV fm ≈ . ∆x ∆x

(4.2)

Nuclear radii (a few fm) are thus resolved with electrons with momentum in the order 100 MeV c−1 . • A process, where one or both of the particles get excited is called inelastic scattering:

44

4. Scattering processes

Fig. 4.1. Relation between kinetic energy Ek in , momentum |p| and reduced wavelength particles [B. Povh et al., Particles and Nuclei, Springer, 2002].

a + b → a0 + b∗ ,

λ 2π

for various

b∗ → c + d + ....

For example, a may be an electron exciting a nucleus b into a state b∗ which subsequently decays into a state of lower energy under emission of a γ quantum. Measurements with registration of all products occuring in a reaction are called exclusive. In case that only the scattered particle is detected, the measurement is called inclusive.

4.2 Cross section • The cross section is a measure of the probability of an interaction between two colliding particles. We first consider scattering processes. • Flux Φ of incident particles: consider a monoenergetic beam of particles impinging onto a target of a thin foil of material. Φ is defined as the number of particles Na which cross a surface A perpendicular to the beam per unit time and per unit surface, Φ=

dx dNa 1 dNa = = na va , A dt Adx dt

(4.3)

where na is the number density and va the particle velocity. • Rate (number per unit time) dW of particles scattered into an element of solid angle dΩ: dW ∝ ΦNb dΩ.

(4.4)

Nb denotes the number of independent scattering centres in the target (assuming that the target is so thin that each incident particle scatters at maximum once and each scattering centre acts independently from the others). The proportionality factor is called differential scattering cross section, dσ dW = . dΩ ΦNb dΩ

(4.5)

4.3 Fermi’s golden rule

45

The total number of scattered particles per unit time is given by W = ΦNb σ, with the total scattering cross section Z dσ σ= dΩ. dΩ

(4.6)

(4.7)

4π

The proportionality factor between W and σ is called luminosity: L = ΦNb .

(4.8)

It thus does not depend on the scattering of the system to be investigated, but defines the rate of interactions for a given cross section and therefore the sensitivity of the chosen setup of beam and target to measure an unknown cross section. • Standard unit, in which cross sections are usually quoted: 1 b = 1 barn = 10−24 cm2 = 100 fm2 .

(4.9)

• Interpretation of the cross section. Consider the fraction of the particles scattered by the target (with respect to the incident particles): W dNa dt

=

Nb σ . A

(4.10)

It is equal to the effective part of the surface filled by the scattering centres. σ thus is an effective surface for scattering. It depends on the interaction and is in general not equal to the geometrical surface of the scattering centre. • The scattering cross section can also be expressed in terms of the scattering amplitude f (q), dσ 2 = |f (q)| , dΩ

(4.11)

where q = p − p0

(4.12)

denotes the momentum transfer onto the target particle, p is the momentum of the incident particle before and p0 after the scattering.

4.3 Fermi’s golden rule • Fermi’s golden rule is a theoretical tool to calculate cross sections and transition rates. It expresses the probability per unit time Wi→f for a transition of a quantum mechanical system from an initial state ψ i to a final state ψ f : Wi→f =

2π |hf |Hint | ii|2 ρ (E 0 ) . }

There are two factors:

(4.13)

46

4. Scattering processes

— Hint denotes the Hamiltonian of the interaction causing the transition. The matrix element Z Mfi = hf |Hint | ii = ψ ∗f Hint ψ i d3 r (4.14) accounts for the strength of this interaction. — Second, the transition rate depends on how many final states are accessible to the system (consisting of projectile and target). This is accounted for by the factor ρ (E 0 ), which is the density of final states per energy interval : ρ (E 0 ) =

dn (E 0 ) . dE 0

(4.15)

dn (E 0 ) = ρ (E 0 ) dE 0 thus is the number of final states with energy between E 0 and E 0 +dE 0 . • Total scattering cross section: Wi→f expresses the scattering rate per incident particle and per scattering centre, Wi→f =

W . Na Nb

(4.16)

With the scattering rate W given by eq.(4.6), and using eq.(4.3) and putting na = Na /V , we obtain σ=

2π 2 |Mfi | · ρ (E 0 ) · V. ~va

(4.17)

• Calculation of the densitiy of final states for elastic scattering of a spin-0 particle from a scattering centre fixed in space. The target shall look the same before and after the scattering (that’s the meaning of elastic), and we have to consider only the final states of the scattering particle. Due to the uncertainty principle, in each spatial dimension the position and the momentum of a particle cannot be determined more accurately than ∆x∆px = 2π}.

(4.18)

A particle state therefore can be attributed a volume (2π~)3 in phase space, the six-dimensional space (px , py , pz , x, y, z) of momentum and position. We cannot distinguish between two particles sitting in the same phase space volume element, and therefore can count particle states as the total phase space volume accessible to that particle, divided by the phase space volume (2π~)3 of a single state. In order to do this, one defines a "normalisation box " with volume V around the scattering centre (which will fall out at the end). The number of states with momentum between |p0 | and |p0 | + d |p0 | is then given by 2

dn (|p0 |) =

V · 4π |p0 | (2π})3

d |p0 | .

(4.19)

Energy and momentum are related by dE 0 = v 0 d |p0 | .

(4.20)

This equation is valid also for relativistic particles, as can be shown using eq.(3.1), eq.(3.10) and eq.(3.11). It follows that ρ (E 0 ) =

V · 4π |p0 |

2

v 0 · (2π})3

.

(4.21)

4.4 Feynman diagrams

47

4.4 Feynman diagrams • Feynman graphs like those shown in fig. 4.2 are a pictorial representation of a particle interaction amplitude. An elementary interaction between two fundamental particles (these are fermions: the quarks and the leptons) is represented as an exchange of a boson (particle with integer spin), which is a quantum of the force field mediating the interaction. See for example the diagram (a): an electron interacts with a positron (the antiparticle of the electron) via exchange of the boson of electromagnetic interaction, i.e. the photon. The photon does not appear as a real particle and therefore is called a virtual particle. The points where three or more particles come together are called vertices.

Fig. 4.2. Feynman graphs for the electromagnetic (a, b, c), weak (d, e) and strong interactions (f) [B. Povh et al., Particles and Nuclei, Springer, 2002].

• Initially, such graphs were introduced by Feynman as a shorthand to write down terms which contribute to matrix elements (i.e. amplitudes for a transition from an initial state to a final state). Using specific rules, called Feynman rules, any of the graphs can be translated back to a matrix element. As a convention (which is not used by all authors the same), the time axis points upwards, and the space axis to the left. Straight lines depict initial and final fermions. Lines belonging to antiparticles (e+ , µ+ and ν e in the figure) have an arrow pointing back on time (due to an interpretation of antiparticles as particles propagating backwards in time). Photons (γ) are represented as wavy lines, heavy vector bosons (W and Z0 ) as dashed lines, and gluons (g) as curly lines. — Virtual particles donot have to fulfil the energy-momentum relation E 2 = p2 c2 + m2 c4 . This can be interpreted that they have a mass different from the real, free particle, or that energy conservation is violated during the short time interval of the boson exchange. The

48

4. Scattering processes

contribution of a virtual particle to the matrix element is the so-called propagator term, given by 1 , Q2 + M 2 c2

(4.22)

where M is the real mass of the exchanged boson, and Q2 is the squared four-momentum transfer. Hence, for an electromagnetic process dominated by exchange of a single virtual photon, this term induces a factor 1/Q2 in the matrix element and 1/Q4 in the cross section. — Each vertex contributes a factor proportional to the charge√ to which couples the boson. For an electromagnetic process, each vertex contributes e ∝ α (α being the fine structure √ constant), but for weak and strong processes, there are "weak charges", denoted by g ∝ αw , √ and "strong charges" (also called "colour charges") denoted by αs .

5. Geometrical shape of the nucleus

5.1 Kinematics of electron scattering • The electrons used to determine nuclear radii are highly relativistic, i.e. E À me c2

⇒

E ≈ |p| c.

(5.1)

• Derivation of the angular dependence of the energy of elastically scattered electrons: — Considered system: electron scattered of a nucleus at rest (in the laboratory frame). Four-momentum of the electron:

p = (E/c, p) ,

Four-momentum of the nucleus:

P = (M c, 0) ,

p0 = (E 0 /c, p0 )

(5.2)

0 P 0 = (EN /c, P0 )

(5.3)

The primes denote quantities after scattering. — Energy and momentum conservation (valid in any frame): p + P = p0 + P 0 ⇒

(5.4)

2

2

02

0

0

02

p + 2pP + P = p + 2p P + P .

(5.5)

Further, p2 = p02 = m2e c2 ⇒

0

and P 2 = P 02 = M 2 c2

0

pP = p P .

(5.6) (5.7)

Usually, one detects the scattered electron (but not the recoiling nucleus). Using eq.(5.4) and eq.(5.6), pP = p0 (p + P − p0 ) = p0 p + p0 P − m2e c2 .

(5.8)

Furthermore, using the expressions of the four momenta in the laboratory frame (eq.(5.2) and (5.3)), E M c2 = E 0 E − p · p0 c2 + E 0 M c2 ,

(5.9)

where we have neglected −m2e c2 on the right-hand side, using the condition (5.1). Using E ≈ |p| c and E 0 ≈ |p0 | c, we obtain the seeked dependence (with p · p0 = |p| |p0 | cos θ, with the scattering angle θ): E M c2 = E 0 E (1 − cos θ) + E 0 M c2 ⇒

E0 =

1+

E Mc2

E . (1 − cos θ)

(5.10)

• — The angular dependence of the energy of the scattered electrons is described by the term (1 − cos θ). The difference E − E 0 is the recoil energy of the nucleus. The larger the ratio E/(M c2 ) of total relativistic electron energy and rest energy of the nucleus, the larger the recoil, as one would expect naively.

50

5. Geometrical shape of the nucleus

Fig. 5.1. Angular dependence of the energy of scattered electrons normalised to the energy of the incident beam, shown for two different beam energies and two nuclei (mass A = 1 and A = 50) [B. Povh et al., Particles and Nuclei, Springer, 2002].

5.2 Rutherford scattering cross section • Cross section for scattering of a point-like and spinless projectile with charge e (e.g. an electron with neglect of its spin) off a heavy, spinless scattering centre with electric charge Ze (an atomic nucleus). A classical derivation for non-relativistic energies and momenta results in µ

dσ dΩ

¶

Rutherford

=

¡ 2 ¢2 Ze

(4πε0 )2 (4Ekin )2 sin4

θ 2

.

(5.11)

• Derivation of the Rutherford cross section using Fermi’s golden rule: — Assumptions and approximations to be used for this derivation: 1. Born approximation: the incident and the outgoing electron can be described by plane waves. 2. The target nucleus is so heavy that its recoil energy can be neglected. 3. Projectile and target both have spin 0. 4. Projectile and target both have no inner structure (they are point-like). — Plane waves of incident and scattered electron: µ µ 0 ¶ ¶ ip · r ip ·r 1 1 ψ i = √ exp , ψ f = √ exp . } } V V

(5.12)

V is the volume of the normalisation box used in sec. 4.3. ψ i and ψ f thus are normalised to one electron within V . — Fermi’s golden rule (see the equations given in sec. 4.3): σ=

2π |hf |Hint | ii|2 · ρ (E 0 ) · V }va

2

with ρ (E 0 ) =

V · 4π |p0 | v 0 · (2π})

3

.

(5.13)

5.2 Rutherford scattering cross section

51

This equation describes the total scattering cross section. The differential cross section can be obtained, taking only those final states in ρ (E 0 ) where the electron gets scattered into an element dΩ of solid angle (replace the total solid angle 4π by dΩ): dσ =

2

2π |hf |Hint | ii|2 · dρ (E 0 ) · V }va

with dρ (E 0 ) =

V · |p0 |

3 dΩ.

v 0 · (2π})

(5.14)

For large electron energy, we have va ≈ v 0 ≈ c and |p0 | ≈ E 0 /c, dσ V 2 E 02 2 = 2 4 |hf |Hint | ii| . dΩ (2π) (}c)

⇒

(5.15)

The Hamiltonian of the Coulomb interaction is given by Hint = eφ (r) ,

(5.16)

where φ (r) is the electrostatic potential generated by the nucleus. With the definition of the matrix element and the momentum transfer q = p − p0 we obtain µ ¶ Z e iq · r hf |Hint | ii = (5.17) φ (r) exp d3 r. V } For the potential of a point-like source, Ze 1 4πε0 r

φ (r) =

(5.18)

one obtains (→ exercise): hf |Hint | ii =

Ze2 }2 ε0 V |q|2

,

(5.19)

and therefore ¢2 ¡ µ ¶ 4 Ze2 E 02 dσ 4Z 2 α2 (}c)2 E 02 = = , 2 4 4 dΩ Rutherford (4πε0 ) |qc| |qc|

(5.20)

where the right-hand side (if one prefers this form) contains the fine structure constant, α=

e2 1 ≈ . 4πε0 }c 137

(5.21)

— The classical formula eq.(5.11) is obtained from eq.(5.20) using non-relativistic kinematics: E 0 ≈ mc2 , Ekin = mv 2 /2, p = mv, together with |q| = 2 |p| sin

θ 2

(5.22)

— The proportionality µ

dσ dΩ

¶

Rutherford

2

∝ |Mfi | ∝

Ã

e·

1 2

|q|

· Ze

!2

(5.23)

can be understood as an exchange of a virtual photon (factor |q|−2 in the amplitude) coupling to the charged particles with strength proportional to their charges, i.e. e for the electron and Ze for the nucleus.

52

5. Geometrical shape of the nucleus

5.3 Scattering from an extended charge distribution - the form factor • We return to eq.(5.17) and evaluate it now further for an extended charge distribution. — First note that µ µ ¶ ¶ iq · r iq · r ~2 exp = − 2 ∆ exp . } } |q|

(5.24)

— Using Green’s theorem, which is valid for two scalar fields u and v which vanish sufficiently rapidly for |r| → ∞, and for sufficiently large integration volume, Z (u∆v − v∆u) d3 r = 0, (5.25) the matrix element in eq.(5.17) can be written as µ ¶ Z e}2 iq · r hf |Hint | ii = − exp ∆φ (r) d3 r. } V |q|2

(5.26)

— The Laplacian of the electrostatic potential of a time-independent charge distribution can be replaced by ∆φ (r) = −

ρ (r) , ε0

(5.27)

where ρ (r) is the charge density, which is normalised to µ

¶ iq · r ρ (r) exp hf |Hint | ii = d3 r } ε0 V |q|2 µ ¶ Z Ze2 }2 iq · r = f (r) exp d3 r, } ε0 V |q|2 e}2

with the definition

f (r) = ρ (r) / (Ze) ,

Z

Z

Z

ρ (r) d3 r = eZ. We thus obtain

f (r) d3 r = 1.

(5.28)

(5.29)

• The matrix element for electron scattering from an extended charged object is thus given by its value for a point charge (compare eq.(5.19)) times the Fourier transform of the (normalised) charge distribution function f (r), hf |Hint | ii = hf |Hint | ii|point F (q) , with F (q) :=

Z

f (r) exp

µ

iq · r }

¶

d3 r.

(5.30)

(5.31)

This factor contains all information about the spatial charge distribution attainable by such a scattering experiment. It is called the form factor of the charge distribution. — The form factor reduces the scattering intensity for large momentum transfer: 0 ≤ |F (q)| ≤ 1, F (q) → 1 (q → 0),

(5.32) F (q) → 0 (q → ∞).

(5.33)

5.4 Mott scattering

53

— For a spherically symmetric ¡ ¢ scattering object, it depends on the modulus of q and therefore is often written as F q2 . Using polar coordinates, Z ¡ ¢ sin (|q| r/}) 2 (5.34) F q2 = 4π f (r) r dr, |q| r/} with the normalisation Z Z∞ 3 f (r) d r = 4π f (r) r2 dr = 1.

(5.35)

0

— Relation between charge distributions and form factors for a few spherically symmetric cases in Born approximation (see also fig. 5.2): point:

f (r) =

exponential:

a3 8π

Gaussian: hom. sphere:

δ(r) 4π

exp (−ar) ³ 2 ´ 32 ³ 2 2´ a exp − a 2r 2π ½ 3R3 4π for r ≤ R 0 for r ≥ R

F (q2 ) = 1 ³ ´−2 2 1 + aq2 } 2 ³ ´ 2 exp − 2aq2 } 2 ³ 3~ 3 sin |q|R 3 } − (|q|R)

constant dipole Gaussian |q|R }

cos |q|R }

´

oscillating

Fig. 5.2. Relation between the charge distribution ρ (r) and the corresponding form factor in Born approximation [B. Povh et al., Particles and Nuclei, Springer, 2002].

5.4 Mott scattering • Including the spin 12 of the electron, still neglecting nuclear recoil, the cross section is given by the Mott cross section:

54

5. Geometrical shape of the nucleus

µ

dσ dΩ

¶

M ott

=

µ

dσ dΩ

¶

Rutherford

µ ¶ θ · 1 − β 2 sin2 . 2

For β → 1 (highly relativistic electrons), µ ¶ ¶ µ θ dσ dσ = · cos2 . dΩ M ott dΩ Rutherford 2

(5.36)

(5.37)

The Mott cross section thus decreases even stronger as a function of the scattering angle than the Rutherford cross section. • Interpretation: suppression of backscattering (θ = π) due to conservation of the helicity for relativistic particles. The helicity (projection of the particle spin s onto the axis of momentum), h=

s·p , |s| |p|

(5.38)

is a conserved quantity in the limit β → 1 (this follows from investigating the solutions of the Dirac equation). Relativistic particles have either spin parallel or antiparallel to their momentum, corresponding to h = 1 or −1. For a spinless target, at θ = π, conservation of angluar momentum would require that the helicity changes sign, in contradiction with its conservation. Note that the orbital angular momentum, which is perpendicular to s cannot accomodate a change of s by one unit, as required by helicity conservation. • For a target with spin, backscattering of electrons is possible, because conservation of total angular momentum can be achieved via flipping the nuclear spin by one unit along the axis of electron momentum.

5.5 Measurements of nuclear form factors • The determination of a nuclear form factor requires a measurement of the cross section µ µ ¶ ¶ ¯ ¡ ¢¯2 dσ dσ = · ¯F q2 ¯ . (5.39) dΩ exp. dΩ M ott

Hence, the form factor is detectable as a deviation from the theoretical cross section for a point-like nucleus (here written as Mott cross section for a nucleus without spin. For a nucleus with spin, a different theoretical cross section has to be used in eq.(5.39)).

• Procedure to obtain the charge distribution from the measured cross section: in principle, according to eq.(5.31), f (r) should be obtainable from back-Fourier transforming the measured ¡ ¢ distribution F q2 : µ ¶ Z ¡ ¢ iq · r (5.40) f (r) = (2π)−3 F q2 exp − d3 q. } In practice, however, only a limited q range can be measured, because the electron energy is limited and the cross section decreases very rapidly as a function of the scattering angle. The procedure thereofore is:

1. make a model of the nucleus (parameterise the charge distribution), 2. adapt the model parameters until good agreement is obtained using eq.(5.34).

5.5 Measurements of nuclear form factors

55

Fig. 5.3. Apparatus for the measurement of electron scattering off protons and nuclei at the Mainz electron accelerator. The three magnetic spectrometers shown contain two dipole magnets, wire chambers and scintillation counters for analysing the momentum of scattered electrons. The rotating ring has a diameter of about 12 m [B. Povh et al., Particles and Nuclei, Springer, 2002].

• The form factor of a homogeneously charged sphere with radius R has a first minimum at |q| R ≈ 4.5 }.

(5.41)

For 12 C one finds this minimum at |q| ≈ 1.8 fm−1 } (see fig.5.4). If one interprets the nucleus as a homogeneously charged sphere, this corresponds to a radius of R ≈ 2.5 fm. For the Caisotopes, differential scattering cross sections were determined for a much larger range of |q| (see fig. 5.5). The cross section varied over 7 orders of and three minima could be ¡ magnitude, ¢ observed, leading to a very detailed knowledge of F q2 .

• Mean square charge radius: information about the nuclear radius can be obtained not only from the first minimum of the form factor, but even from its behaviour for q2 → 0. In the limit |q| R ¿ },

(5.42)

the form factor can be expanded in powers of |q|: Z

¡ ¢ F q2 = =

Z

f (r)

∞

0

= 4π

Z

Z

0

"

µ

f (r) r2 dr −

1 q2 4π 6 }2

1

−1

∞

µ ¶n ∞ X 1 i |q| |r| cos ϑ d3 r, n! } n=0

Z

0

2π

1 f (r) 1 − 2

|q| r } Z ∞ 0

¶2

2

ϑ = ^ (r, q) #

cos ϑ + ... dφ d (cos ϑ) r2 dr

f (r) r4 dr + ...

(5.43)

56

5. Geometrical shape of the nucleus

Fig. 5.4. Differential cross section of electron scattering, measured for 7 different angles at fixed beam energy. Dashed line: scattering of plane waves at a homogeneous sphere with diffuse surface, solid line: more realistic theoretic analysis (scattering phase analysis) [B. Povh et al., Particles and Nuclei, Springer, 2002].

Defining the mean square charge radius as Z ∞ ­ 2® r2 · f (r) r2 dr, r := 4π

(5.44)

0

i.e. the first moment of the form factor, we obtain ­ ® ¡ 2¢ 1 q2 r2 F q =1− + ... (5.45) 6 }2 ­ ® ¡ ¢ The experimental determination of r2 requires the measurement of F q2 down to very small values of q2 : ¡ 2 ¢ ¯¯ ­ 2® dF q ¯ r = −6}2 . (5.46) ¯ dq2 ¯ 2 q =0

5.6 Nuclear charge distributions and radii • Many very accurate measurements were performed in the 1950’s, from which radial charge distributions ρ(r) were derived. Some measured charge distributions are shown in fig. 5.6. The results show that: — The charge density of most medium and heavy nuclei is distributed as for a sphere with a diffuse surface (see fig. 5.2). In the nuclear centre, ρ (r) is found to be rather constant. It can be parameterised as ρ (r) =

ρ0 ¡ ¢, 1 + exp r−c a

which is called Fermi function.

(5.47)

5.6 Nuclear charge distributions and radii

57

Fig. 5.5. Differential cross sections for electron scattering off 40 Ca and 48 Ca (for better visibility, they were multiplied with 10, respectively, 10−1 ). The position of the minima tells us that 40 Ca is smaller than 48 Ca [B. Povh et al., Particles and Nuclei, Springer, 2002].

— For medium and heavy nuclei, c = 1.07 A1/3 fm,

a = 0.54 fm.

— The mean charge radius of this charge distribution is p hr2 i = r0 A1/3 , r0 = 0.94 fm.

(5.48)

(5.49)

— Considering (as a rough approximation) the nucleus as a homogeneous sphere with radius R, its radius is connected with the mean square radius as 5 ­ 2® R2 = (5.50) r . 3 From the value stated in eq.(5.49) we obtain R = 1.21 A1/3 fm.

(5.51)

The dependence R ∝ A1/3 was used implicitely in the Weizsäcker mass formula (in the volume term, see interpretation of the mass formula). — The definition and the value of the surface thickness t found for medium and heavy nuclei is t = r (ρ/ρ0 = 0.1) − r (ρ/ρ0 = 0.9) = 2a ln 9 ≈ 2.40 fm.

(5.52)

— The charge density at the nuclear centre shows a slight decrease with increasing mass number. This is due to the presence of the neutrons, which dilute the nuclear charge density. The nucleon density is obtained by multiplying ρ (r) with A/Z, for which one finds almost identical density in the interior of nearly all nuclei: ρN ≈ 0.17 nucleons/fm3 .

(5.53)

58

5. Geometrical shape of the nucleus

— The lightest nuclei have approximately Gaussian charge distributions, i.e. no density plateau in the nuclear interior.

Fig. 5.6. Nuclear charge density as a function of distance (in fm) from the centre of the nucleus, determined by electron scattering. Ordinate unit: 1019 As cm−3 [R. Hofstadter, Ann. Rev. Nuc. Sci. 7 (1957) 231].

6. Nuclear decays

• Instable nuclei are those which may spontaneously decay. Possibilities of nuclear decay are — electromagnetic decay: a nucleus in an excited energy level emits energy in form of γ quanta, i.e. electromagnetic radiation. The transitions are between states corresponding to different nucleon configurations without change of Z and N .

γ

γ

γ

Fig. 6.1. Electromagnetic decay of nuclear levels.

— α decay: a special case of particle emission: emission of an α particle (a 4 He nucleus), whereby the mother nucleus (Z, A) transforms to a daughter nucleus (Z − 2, A − 2). — β decay: a weak interaction process with emission of 1. an electron and an electron-antineutrino, whereby within the nucleus a neutron gets converted into a proton (β − decay). 2. a positron and an electron-neutrino, with conversion of a proton into a neutron (β + decay). Instead of positron emission, an atomic electron may be captured by the nucleus (K-capture).

Fig. 6.2. α and β decay of a nucleus (Z, A).

— nuclear fission: very heavy nuclei may decay into two (or less likely more) fragments of similar size.

60

6. Nuclear decays

• Analysis of emitted radiation and nuclear masses provide nuclear level schemes. An example is shown in fig. 6.3.

Fig. 6.3. Energy levels of

17

O.

6.1 The decay law • Consider an assembly of N instable nuclei in the same state. — The activity A of this assembly is proportional to N , A=−

dN = λN, dt

(6.1)

with the decay constant λ, which is the probability per unit time for the decay of a nucleus. From this follows, via integration, the exponential decay law, N (t) = N0 exp (−λt) ,

(6.2)

6.2 Natural radioactivity and radioactive dating

61

where N0 is the number of nuclei at t = 0. The activity decreases exponentially, A (t) = A0 exp (−λt) ,

A0 = λN0 .

(6.3)

It is measured in units of 1 Bequerel = 1 Bq = 1 s−1 ,

1 Curie = 1 Ci = 3.70 × 1010 s−1 .

(6.4)

— The half-life is the time after which half of the nuclei have decayed. Putting N = N0 /2 in eq.(6.2) gives t1/2 =

ln 2 0.69315 ≈ . λ λ

(6.5)

Mean lifetime τ (also simply called lifetime): this is the average time a nucleus survives until it decays. The number of nuclei which decay between t and t+dt is given by |dN (t) /dt| dt = λN0 exp (−λt) dt, therefore R ∞ ¯¯ dN (t) ¯¯ t ¯ dt ¯ dt 0 1 ¯ τ = R ∞ ¯¯ (6.6) = ≈ 1.443 t1/2 . dN (t) ¯ λ dt ¯ ¯ dt 0

Hence, the mean lifetime is simply the inverse of the decay constant. After t = τ , the activity has decreased to 1/e of its intial value. • Consider an assembly of N nuclei which can decay via two different decay modes a and b with corresponding partial decay constants λa and λb , λa = −

(dN /dt)a , N

λb = −

(dN /dt)b . N

The total decay rate is then given by µ µ ¶ ¶ ¶ µ dN dN dN =− − = N (λa + λb ) = N λtotal , − dt total dt a dt b

(6.7)

(6.8)

where λtotal is called the total decay constant. Hence, the activity decays with a single decay constant. However, the partial decay constants determine the probability for the decay to proceed via mode a or b, which determines the yield of daughter nuclei a and b: Na (t) =

λa N0 (1 − exp (−λtotal t)) , λtotal

(6.9)

Nb (t) =

λb N0 (1 − exp (−λtotal t)) . λtotal

(6.10)

6.2 Natural radioactivity and radioactive dating • A simple decay chain is a sequence of instable nuclei, where each species is populated by the preceeding one, N1 −→λ1 N2 −→λ2 N3 −→λ3 ...Nk , with

(6.11)

62

6. Nuclear decays

dNi = λi−1 Ni−1 − λi Ni . dt

(6.12)

The first term describes feeding from nuclide i − 1, and the second describes the reduction of Ni due to decay to the nuclide i + 1. This system of differential equations can be solved with the ansatz N1 = C11 exp (−λ1 t) N2 = C21 exp (−λ1 t) + C22 exp (−λ2 t) ... Nk = Ck1 exp (−λ1 t) + ... + Ckk exp (−λk t) .

(6.13)

The coefficients are found to be Cij = Ci−1,j

λi−1 , λi − λj

i 6= j,

(6.14)

and the coefficients Cii are found from the boundary conditions for t = 0, Ni (0) = Ci1 + Ci2 + ... + Cii .

(6.15)

Example: k = 2, a radioactive mother nucleus with a radioactive daughter, decaying subsequently into a stable state. Hence, N2 (t) = N1 (0)

λ1 (exp (−λ1 t) − exp (−λ2 t)) . λ2 − λ1

(6.16)

The activity A2 is given by λ2 N2 (hence, proportional to N2 , but it is not given by dN2 /dt, since the activity refers only to decay but not to creation of nuclear species 2). • In case of several decay modes (branching), the solutions get more complicated. • Decay chains of naturally radioactice, heavy elements can be characterised... An example of a decay chain is shown in fig. 6.4. • Radioactive dating: knowing the decay constant λ, the exponential decrease in activity of a sample can be used to measure time. Example: 14 C dating: — The radioactive 14 C is continuously formed in the upper atmosphere resulting from neutron production by cosmic rays and the subsequent reaction of the neutron with nitrogen: ¢ ¡ 14 N (n, p)14 C, t1/2 14 C = 5730 a (6.17)

— until the beginning of the 20th century, the isotopic ratio of 14 C and 12 C was about 1.5 × 10−12 . In the first half of the century it got reduced due to extensive burning of charcoal and oil (setting free 12 C but no 14 C, which has already decayed since long). After 1954 the ratio increased due to tests of nuclear weapons. — Any living organism in gas exchange with the earth’s atmosphere incorporates 14 C with 14 C/12 C concentration ratio of the atmosphere. Without gas exhange, the 14 C decays and the ratio 14 C/12 C decreases exponentially. Measuring the activity of 14 C, knowing the abundance of carbon, one can measure the time elapsed after the death of the organism.

6.3 Alpha-decay

Fig. 6.4. Decay chain

238

63

U, indicating the decay modes and half-lifes of the various nuclei.

6.3 Alpha-decay • Why is α decay possible? — The binding energy of the α particle (a 4 He nucleus) is 7.07 MeV per nucleon. A necessary condition for emission of an α particle is M (A, Z) > M (A − 4, Z − 2) + M (4 He).

(6.18)

This condition may be fulfilled for heavy nuclei, where the binding energy per nucleon is already well below its maximum value of ∼ 8.8 MeV (see fig. 2.11), and the nucleus gains enough binding energy due to emission of the α particle. — Fig. 6.5 shows the potential energy of an α particle as a function of its distance from the nuclear centre. Outside the range of the nuclear potential, the α particle senses the Coulomb potential 2 (Z − 2) α~c/r. within the nucleus, there is the strongly attractive nuclear potential well. We consider an α particle leaving the nucleus with positive kinetic energy Tα (this corresponds to eq.(6.18) being fulfilled). Its escape from the nucleus is not possible classically but only via quantum mechanical tunneling through the Coulomb barrier. • The lifetime of an α instable nucleus can be calculated treating the α particle as a quantum mechanical wave propagating in a one-dimensional potential (after separation in polar coordinates).

64

6. Nuclear decays

Fig. 6.5. Barrier penetration in decay. The half-life for α emission depends on the penetration probability and therefore on the barrier thickness. The measured half-lives thus can be used to determine the radius R where the nuclear force ends and the Coulomb repulsion begins [K.S. Krane, Introductory Nuclear Physics, Wiley, 1987].

— The transmission T by tunneling of an α particle through a thin step potential barrier of hight V and thickness ∆r is given by (→ exercise): 1p T ≈ exp (−2κ∆r) , κ= 2mα (V − Tα ). (6.19) ~

— The Coulomb barrier can be thought of being composed of many of such thin potential steps, with the result that the transmission can then be written as T = exp (−2G) with the Gamow-factor Z 0 1 R p 2mα (V (r) − Tα ) dr G= ~ R

(6.20)

(6.21)

It can be shown that, for the Coulomb potential and with R ¿ R0 , C G≈ √ Tα

— The α decay constant is then given by ³ p ´ λ = λ0 exp −2C/ Tα

(6.22)

(6.23)

where λ0 contains the probability of an α particle to form within the nucleus. This relation is called Geiger Nuttal rule. The extremely strong dependence of the of the lifetime on the energy Tα of the α particle was a great support of the existence of the quantum mechanical tunnel effect.

6.4 Nuclear fission • Fission of heavy nuclei is energetically possible, because there is a maximum of binding energy per nucleon for stable nuclei around 56 Fe (see fig. 2.11).

6.4 Nuclear fission

65

Fig. 6.6. The observation of Geiger and Nuttal.

— ⇒ nuclei with Z > 40 gain energy under fission, i.e. M (Z1 + Z2 , A1 + A2 ) > M (Z1 , A1 ) + M (Z2 , A2 ). — However, there is a large potential barrier, such that spontaneous fission in general gets extremely unlikely. Some uranium isotopes are the lightest nuclei for which spontaneous fission becomes competitive with α decay.

Fig. 6.7. Potential energy during nuclear fission. The solid line corresponds to the situation where energy is required to deform the nucleus. The height of the fission barrier determines the probability of spontaneous fission. For nuclei with Z 2 /A & 48 the fission barrier disappears, and the shape of the potential then corresponds to the dashed line [B. Povh et al., Particles and Nuclei, Springer, 2002].

• Interpretation of the fission barrier : competition between Coulomb energy and nuclear surface energy of the nuclear drop (see eq.(2.4)). With deformation of the spherical nucleus at constant density, the surface energy Es = as A2/3 increases, whereas the Coulomb energy EC = aC Z 2 A−1/3 of the protons decreases. A fission barrier exists, when |∆Es | > |∆EC | .

(6.24)

66

6. Nuclear decays

• Sufficient condition for spontaneous fission: |∆Es | < |∆EC | .

(6.25)

In this case there is a local maximum of potential energy for the undeformed nucleus, and it therefore gains energy with deformation. To find the condition when this occurs: — consider an elliptic deformation of a spherical nucleus as in fig. (6.8), at constant volume 4π 3 4π 2 R = ab , 3 3

(6.26)

requiring a = R (1 + ε) ,

b = R (1 + ε/2) .

(6.27)

Fig. 6.8. Sketch to explain the change of potential energy due to deformation of a heavy nucleus.

— The surface energy as a function of the deformation parameter ε is given by ¶ µ 2 2 2/3 Es = as A 1 + ε + ... , 5 and the Coulomb energy by ¶ µ 1 EC = aC Z 2 A−1/3 1 − ε2 + ... . 5

(6.28)

(6.29)

A deformation ε thus changes the total potential energy by ∆E = E (ε) − E (0) =

´ ε2 ³ 2as A2/3 − aC Z 2 A−1/3 . 5

(6.30)

At negative ∆E, the nucleus gets unstable with respect to deformation. — As a result, the fission barrier vanishes for (using the constants quoted in table 2.1) Z2 2as ≈ 48, ≥ A aC

(6.31)

i.e. for nuclei with Z > 114 and A > 270. — For nuclei with a sufficiently low fission barrier, spontaneous fission may occur due to tunnel effect. This happens only for very heavy nuclei not to far from fulfilling the condition eq.(6.31).

6.5 Beta-decay

67

• Neutron induced nuclear fission: — For heavy nuclei in the region of uranium (Z = 92), the fission barrier is only about 6 MeV. Slow neutrons may provide this energy as binding energy in neutron capture reactions. The nucleus thus may get excited to an energy above the fission barrier and splits up. — Important point: the nuclear pairing energy leads to significant differences in the fissibility of nuclei: — Neutron capture by 238 U: the binding energy of 4.8 MeV is released, whereas the fission barrier of 239 U is 6.3 MeV. For immediate fission, the energy difference has to be supplied by kinetic neutron energy (the process is then called fast fission). However, the reaction cross section is ∝ vn−1 , which therefore is small. — Neutron capture by 235 U: 6.4 MeV release of binding energy, whereas the fission barrier of 236 U is 5.8 MeV. Hence, fission of 235 U is possible with slow neutrons, which have large cross sections for capture. Reason of the higher energy release for this isotope: a paired neutron configuration in 236 U with release of the pairing energy (see again eq.(2.4)). — Practical importance: nuclear reactors. Also

233

Th and

239

Pu are suitable materials.

6.5 Beta-decay 6.5.1 Decay types and energetic conditions • There are two different nuclear β decays: (1) β − decay:

n → p + e− + ν e ,

(6.32)

(2) β + decay:

p → n + e+ + ν e .

(6.33)

The second possibility is possible only within the atomic nucleus, since the neutron mass is larger than the proton mass. A variant of (2) is electron capture: (3) e− capture: p + e− → n + ν e .

(6.34)

• Necessary conditions for the different β decays to take place, (1) M (A, Z) > M (A, Z + 1) ,

(6.35)

meaning that the mass of the mother atom has to be larger than the mass of the daughter atom (taking the atomic masses, the generated electron is already taken into account). (2) M (A, Z) > M (A, Z − 1) + 2me c2 .

(6.36)

The last term is due to the mother atom having one electron more, and one positron is created. (3) M (A, Z) > M (A, Z − 1) + ε.

(6.37)

Here, ε means the excitation energy of the electron shell of the daughter atom. • Typical lifetimes τ β of β-instable nuclei (reasons for this strong variation see next section): 10−3 s . τ β . 1016 a. The free neutron has a lifetime of τ n = 885.8(7) s.

(6.38)

68

6. Nuclear decays

• Electron capture (EC): — Requirement: overlap of nuclear proton and electron wavefunctions ⇒ largest probability for electrons in the K-shell, which have the highest amplitude in the nucleus. The process prefers heavy nuclei due to their larger radii and the smaller electronic orbitals. — Secondary processes: EC causes a hole in the electron shell, which gets filled by a cascade of higher-orbit electrons, accompanied by emission of characteristic X-rays or Auger-electrons. — EC competes with β + decay. Due to the energy requirements eq.(6.36) and eq.(6.37), for an EC there is 2me c2 − ε more energy available than for β + decay. It therefore may happen that only EC is energetically possible. • All nuclear beta decay processes happen at constant nuclear mass number A. We consider therefore isobars in the Weizsäcker model, rewriting the mass formula eq.(2.4) as M (A, Z) = αA − βZ + γZ 2 +

δ , A1/2

(6.39)

where α = mn − av + as A−1/3 + aa /4,

(6.40)

β = aa + mn − mp − me ,

(6.41)

−1

γ = aa A

−1/3

+ aC A

> 0.

(6.42)

M (A, Z) is quadratic in Z and has its minimum at Z = β/ (2γ). Due to the pairing term, we have to distinguish two cases: 1. Beta decay in nuclei with odd A: an even-odd nucleus stays an even-odd nucleus ("even-odd" meaning even neutron and odd proton number, or vice-versa). In both the mother and the daughter nucleus we have zero pairing energy, meaning δ = 0 in eq.(2.4). Hence, M (A, Z) is a single parabola. 2. Beta decay in nuclei with even A: a) an even-even nucleus decays to an odd-odd nucleus: two nucleon pairs are broken which requires an energy of 2δ/A1/2 . b) an odd-odd nucleus decays to an even-even nucleus: two nucleon pairs are created which releases an energy of 2δ/A1/2 . ⇒ the masses of the isobars thus lie on two parabolas, separated by 2δ/A1/2 . Examples for both cases: 1. Isobars with A = 101 (see fig. 6.9). The minimum of the mass parabola is close to

101

Ru.

2. Isobars with A = 106 The minimum is near to 106 Pd. The isobar 106 Cd is also β-stable, because the neighboring odd-odd nuclei have larger mass. As visible in fig. 6.10, all odd-odd nuclei have a stronger bound even-even nucleus in the spectrum of isobars ⇒

odd-odd nuclei are unstable,

with a few exceptions among the very light nuclei (2 H, 6 Li, • Example:

40

K

(6.43) 10

B,

14

N).

6.5 Beta-decay

69

Fig. 6.9. Mass parabola of the A = 101 isobars. The arrows indicate possible β decays. The abscissa is the charge number Z. The zero point of the mass scale was chosen arbitrarily [B. Povh et al., Particles and Nuclei, Springer, 2002].

Fig. 6.10. Mass parabolas of the A = 106 isobars. Abscissa: charge number Z, ordinate: abritrarily chosen zero point of mass scale [B. Povh et al., Particles and Nuclei, Springer, 2002].

—

40

K is a long-living β + and β − emitter.

— All β decay processes compete in one atom. —

40

K has

40

Ar and

— The decay of

40

40

Ca as stable daughters.

K contributes 16 % to the natural radiation exposure of the human body.

• Exotic β decays: the transition from

106

Cd to

106

Pd is possible only via double β decay,

70

6. Nuclear decays Energy

Fig. 6.11. β decay of 40 K. The relative probabilities of the competing processes are quoted in parentheses. The bent arrow in β + decay indicates that 1.022 MeV are required [B. Povh et al., Particles and Nuclei, Springer, 2002]. 106 48 Cd58

→

106 46 Pd60

+ 2e+ + 2ν e .

(6.44)

This second-order process of weak interaction is extremely unlikely. Typical half-lives of such decays (also denoted by "2νββ decay" to distinguish it from a hypothetical neutrinoless double β decay, which one searches for in present experiments) are in the range 1020 − 1024 years (meaning that one needs 1020 − 1024 nuclei to observe about one decay per year!). 6.5.2 Beta-spectrum and lifetime • Fermi’s theory of nuclear β decay provides: 1. form of the spectrum of the β particle (electron in β − decay, positron in β + decay), 2. relation between decay energy and lifetime of β-instable nuclear states, 3. classification of nuclear β decays. • Starting point of the theory is Fermi’s golden rule (see eq.(4.13)): Wi→f =

2π |hf |Hint | ii|2 ρ (E 0 ) . ~

(6.45)

— In electron scattering (see chapters 4 and 5), Mfi = hf |Hint | ii was the matrix element of electromagnetic interaction between electron and nucleus, and ρ (E 0 ) was the final state density per energy interval of the scattered electron. — In β decay, we deal with a final state of three particles, i.e. daughter nucleus, β particle and (anti)neutrino. The nuclear recoil energy can be neglected (as was done in section 4.3), the total decay energy (the Q-value), E0 = Ee + Eν ,

(6.46)

is shared among the β particle and the (anti)neutrino. Equation (6.45) thus generalises to a differential expression, dW (Ee ) =

2π 2 dρ (E0 , Ee ) dEe . |hf |Hint | ii| ~ dEe

(6.47)

6.5 Beta-decay

71

• dW (Ee ) is the differential decay probability for β decay with the β particle emitted with energy between Ee and Ee + dEe , • Mfi = hf |Hint | ii is the matrix element of the beta-decay, • dρ (E0 , Ee ) /dEe is the density of (anti)neutrino states in the energy interval between Eν and Eν + dEν at fixed electron energy Ee . — The beta-spectrum and the nuclear lifetime are determined by Mfi and dρ (E0 , Ee ) /dEe , according to eq.(6.47). • Calculation of the final density of states: — Using eq.(4.19), we write separately for the phase space volumina dnj of the β particle and the (anti)neutrino having momenta between |pj | and |pj | + d |pj |: 2

dne (|pe |) =

V · 4π |pe | (2π~)

3

2

d |pe | ,

dnν (|pν |) =

V · 4π |pν | (2π~)

3

d |pν | ,

(6.48)

where V again is the volume of a normalisation box. — The density of the combined final state of β particle and (anti)neutrino for fixed energy Ee (since we need dW (Ee ) and measure the energy spectrum of the β particle) is given by dρ (E0 , Ee ) =

d2 n (E0 , Ee ) . dE0

(6.49)

This equation generalises the one-dimensional final state density of eq.(4.21) to the present situation, where the total energy is shared among two particles (remember: nuclear recoil neglected), d2 n (E0 , Ee ) = dne dnν =

16π 2 V 2 (2π~)6

2

2

|pe | |pν | d |pe | d |pν | .

(6.50)

With E0 = Ee + Eν and Ee held fixed, we have dE0 = dEν , and therefore d |pν | =

dE0 . c

(6.51)

Using further 2

Ee2 = |pe | c2 + m2e c4

⇒

|pe | d |pe | =

1 Ee dEe , c2

(6.52)

we obtain |pe |2 d |pe | =

1 p 2 Ee Ee − m2e c4 dEe . c3

(6.53)

We thus obtain the two-particle state density: dρ (E0 , Ee ) =

16π 2 V 2 (2π~c)

6 Ee

p Ee2 − m2e c4 (E0 − Ee )2 dEe .

(6.54)

— The preceeding analysis treated the outgoing particles as free. For the β particle, however, the Coulomb interaction with the daughter nucleus leads to a correction of the phase space factor. The effect on the beta-spectrum is shown in fig. 6.12. It is taken into account by multiplying the right-hand side of eq.(6.54) with the so-called Fermi function,

72

6. Nuclear decays

Fig. 6.12. Shape of the beta spectrum as due to the phase space factor. Both ends fall off to zero parabolically. The Coulomb interaction of the beta particle with the daughter nucleus modifies the spectrum for a β + , respectively, β − decay as shown.

F (Z 0 , Ee ) ≈

2πη , 1 − exp (−2πη)

η=∓

Z0α Z 0 e2 =∓ 4πε0 ~ve β

for β ± decay,

(6.55)

where Z 0 e is the charge of the daughter nucleus, ve is the velocity of the outgoing β particle far away from the nucleus, β = ve /c and α is the finestructure constant. • The lifetime τ (or decay constant τ −1 , see eq.(6.6)) of the β-instable nucleus is given by 1 = τ

Z

E0

me c2

dW (Ee ) dEe = dEe

Z

E0

me c2

2π 2 dρ (E0 , Ee ) dEe . |Mfi | ~ dEe

(6.56)

Taking the matrix element as energy-independent, the decay constant is therefore proportional to Z E0 p f (Z 0 , E0 ) = Ee Ee2 − 1 (E0 − Ee )2 F (Z 0 , E0 ) dEe , (6.57) 1

where, to simplify the notation, energies were used in units of me c2 , Ee =

Ee , me c2

E0 =

E0 . me c2

(6.58)

— Sargent rule: for sufficiently large electron energy, Ee À 1 (which is not always justified, but it’s just assumed to get an idea about the decay energy dependence of the lifetime), and F (Z 0 , Ee ) = 1 we obtain 1 ∝ τ

Z

1

E0

2

Ee2 (E0 − Ee ) dEe ≈

E05 . 30

(6.59)

— Without these approximations, one still can define the so-called f t-value: f t := f (Z 0 , E0 ) · t1/2 =

2π~7 ln 2 . m5e c4 V 2 |Mfi |2

(6.60)

Hence, if the matrix element were independent on the details of the nuclear decay, the f tvalue should be the same for any nuclear decay. However, f t-values vary between 103 s and 1022 s (therefore one often finds compiled compiled values of log f t with f t in seconds). We have to search the reason for this in the matrix element.

6.5 Beta-decay

73

• The matrix element contains a transition operator between two nuclear wavefunctions. The discussion of the form of this operator will be postponed to a later chapter on weak interaction in the framework of particle physics. Here, we state a few basic facts. — The neutron (or proton) decays within the nucleus. Important is the overlap of the initial and final nuclear wavefunctions, which, if the configurations are very different, will strongly suppress the transition rate. — Fermi decays are those, where the two leptons (β particle and (anti)neutrino) are emitted with total spin zero ⇒ the nucleon has the same spin before and after the decay. — Gamow-Teller decays are those, where the two leptons are emitted with total spin one ⇒ the nucleon involved in the transition has to flip its spin. In general, both Fermi and Gamow-Teller decays contribute to a nuclear β decay, however, with important exceptions. — Angular momentum constraints may have a strong effect: sometimes a nuclear β transition is energetically possible (due to the conditions stated in sec. 6.5.1) only between states with rather different nuclear spin. The matrix element contains the wavefunctions of the two leptons, which in good (Born)approximation are taken as plane waves (as the electron in eq.(5.12) in the derivation of the Rutherford cross section), Ã ! µ ¶ 2 1 ip · r ip · r (p · r) 1 ψ (r) = √ exp 1+ + ... . (6.61) =√ − } } 2}2 V V Since L = r × p, the right-hand side corresponds to an expansion in terms of orbital angular momentum . Since the linear momenta p have typical magnitudes of MeV/c and typical nuclear radii R are of the order fm, |p| R/} =

|p| cR a few MeV fm ≈ ' a few 10−2 . }c 200 MeV fm

(6.62)

2

Each unit of in |Mfi | therefore leads to a suppression of 10−4 − 10−3 . Nuclear β decays are classified according to the minimum orbital angular momentum required to induce the transition. — There are selection rules for parity P and nuclear spin I. The parity of the leptonic orbital angular momentum state is linked to the parity difference of mother and daughter nucleus, ∆P = (−1) ,

(6.63)

since total parity is conserved. Reason: even with Hint containing pseudoscalar parts (which change sign under the parity transformation), the transition probability ∝ |hf |Hint | ii|2 is a scalar, i.e. it stays the same under parity. • Classification of nuclear β decays: — Allowed decays: = 0 with the following selection rules. For a Fermi decay, no angular momentum change is involved, and the spin of the nucleus therefore cannot change. For a Gamow-Teller decay, the vector difference between initial and final angular momentum must be 1. Hence, ∆P = 0, ∆I = 0 ∆P = 0, ∆I = 0, ±1 (no 0 → 0)

for Fermi decays for Gamow-Teller decays

(6.64)

74

6. Nuclear decays

— Superallowed decays: in this special case the overlap of the initial and the final nuclear wavefunctions is nearly perfect. The f t-values are small and close to that of the free neutron. In general, superallowed decays are β + decays: with the nucleons sitting in the same wavefunctions, smaller nuclear charge is energetically preferable. Exemption: β − decay of tritium: 3

H → 3 He + e− + ν e + 18.6 keV.

(6.65)

Here, the increase of Coulomb energy is slightly overbalanced by the neutron-proton mass difference. The decay energy is the smallest known for a nuclear beta-decay. — Forbidden decays: 6= 0. Example: the decay of 40 K from its level with spin and parity I P = 4− to the 0+ ground states of 40 Ca and 40 Ar requires = 3 (via a Gamow-Teller decay with ∆P = 1), see fig. 6.11. The decay to the first excited level of 40 Ar with J P = 2+ is only singly forbidden ( = 1). However, phase space is very small for this decay branch, due to the very small energy difference of 0.049 MeV. This explains the long half-life of 40 K of t1/2 = 1.27 × 109 a. • The beta-spectrum, i.e. the number of electrons emitted with energy in the intervall between Ee and Ee + dEe is proportional to dρ (E0 , Ee ) /dEe . For allowed transitions, Mfi indeed does not depend on the energy of the β particle, and, for vanishing neutrino mass, the theoretical shape of the spectrum is therefore given by p dW (Ee ) 2 ∝ Ee Ee2 − m2e c4 (E0 − Ee ) F (Z 0 , Ee ) . (6.66) dEe 6.5.3 Application of beta-spectroscopy: direct measurement of the neutrino mass • Motivation: recent observations of neutrino oscillations require a non-vanishing neutrino mass (later chapter). • Method of direct measurement: — employs decay kinematics. Eq.(6.66) was derived with the assumption of a vanishing neutrino mass. Any deviation from this assumption affects the shape of the spectrum, in particular the endpoint energy of the β particle. This can be made visible in the so-called Kurie plot, where s dN (Ee ) /dEe p K (Ee ) = (6.67) 0 F (Z , Ee ) Ee Ee2 − m2e c4 is plotted as a function of Ee , and where dN (Ee ) /dEe is the experimentally determined β spectrum. According to eq.(6.66), one expects a straight line.

— the best-suited beta-decay for the search of a non-vanishing neutrino mass is tritium decay with its extremely small decay kinetic energy of only 18.6 keV. The present limit on the neutrino mass is mν e < 2.5 eV/c2 .

(6.68)

6.6 Electromagnetic decays • The spectrum of excited nuclear states is very rich, see for example fig. 6.3. An exception form the very lightest nuclei, which have only a few states below threshold for particle emission.

6.6 Electromagnetic decays

75

Theory tries to understand the observed spectra and decay modes on the basis of nuclear models and appropriate effective degrees of freedom. • Excited states usually decay via emission of electromagnetic radiation, which can be decomposed into components of different multipolarity. Each multipolarity has a characteristic angular distribution, which can be observed for oriented nuclei. The radiation from an electric dipole, quadrupole, octupole etc. is denoted by E1, E2, E3 etc., and magnetic multipoles are correspondingly denoted by M1, M2, M3, etc.

Fig. 6.13. The angular distribution W (θ) of γ emission from nuclei in condensed matter depends on the ratio of the magnetic energy µB to the thermal energy kT . For kT À µB the nuclei are very weakly polarised at thermal equilibrium, and γ emission is very close to isotropic. In the opposite case, kT ¿ µB, nuclear polarisation may give rise to a strong γ anisotropy [K.S. Krane, Introductory Nuclear Physics, Wiley, 1987].

• The possible multipolarities of gamma emission are defined by selection rules of angular momentum and parity. A gamma quantum with multipolarity E has angular momentum and parity (−1) , an M quantum has angular momentum but parity (−1) +1 . Conservation of angular momentum requires for a transition Ii → If : |Ii − If | ≤ ≤ Ii + If .

(6.69)

• The lifetime of an excited state depends crucially on the multipolarity of the emitted gamma radiation, favouring lower multipolarity and electric transitions. Transitions of type E( + 1) are usually suppressed by several orders of magnitude compared to an E transition, and a transition of type M typically has a probability comparable to a transition of type E( + 1). Lifetime and angular distribution are thus signatures for the multipolarity of the transition and can be used to reveal spins and parities of nuclear levels. Typical lifetimes of excited states are in the range

76

6. Nuclear decays

10−15 s − 10−9 s.

(6.70)

• Isomers are untypically long-lived nuclei. Example: the second excited state of 110 Ag with J P = 6+ and excitation energy 117.7 keV. Energy emission via M4 transition to the first excited state with J P = 2− at 1.3 keV. Half-life: t1/2 = 235 d. Table 6.1. Selection rules for electromagnetic nuclear transitions. Multipolarity

electric E

magnetic |∆I|

dipole

E1

1

quadrupole

E2

2

octupole

E3

3

...

...

∆P

M

−

M1

+ −

|∆I|

∆P

1

+

M2

2

M3

3

−

+

...

• Inner conversion is a process, where the nuclear excitation energy is transferred to an electron of the atomic shell. It appears when the emission of gamma quanta is forbidden (spin 0 → spin 0) or strongly suppressed due to high multipolarity or small decay energy, and it occurs preferentially in heavy nuclei (due to the large amplitude of K electrons within the nucleus).

6.7 Mössbauer effect • Nuclear fluorescence: — Nuclear absorption lines have a natural line width Γ , respectively, δω γ given by the inverse of the mean life, Γ =

~ , τ

δω γ =

1 . τ

(6.71)

— Nuclear absorption lines are extremely narrow. Example: 57 Fe (see fig. 6.14). The lifetime of the level at 14.4 keV is τ = 1.4 × 10−7 s, corresponding to Γ = 4.7 × 10−9 eV. The relative line width is Γ/Eγ = 3 × 10−13 . • Nuclear recoil usually shifts the energy, respectively, frequency of the nuclear γ radiation by much more than the natural line width: — The momentum of the γ quantum is p=

Eγ ~ω γ = . c c

(6.72)

The recoil energy of the nucleus with mass M is ∆E =

Eγ2 p2 . = 2M 2M c2

(6.73)

E.g. for 57 Fe: ∆E = (1.4 × 10−2 MeV)2 / (2 × 57 × 931 MeV) = 2 × 10−3 eV, which is much larger than the natural line width of 4.7 × 10−9 eV.

6.7 Mössbauer effect

77

Fig. 6.14. Decay scheme of 57 Fe. The level with excitation energy 0.136 MeV is populated via K capture in 57 Co. The times shown are the half-lifes [E. Segrè, Nuclei and Particles, Benjamin, 1977].

— A free nucleus at rest emits radiation with energy Eγ = E0 − ∆E,

(6.74)

where E0 is the spacing between the excited and the ground state (14.4 keV for 57 Fe). A second nucleus at rest of the same type in its ground state is therefore out of resonance with the radiation emitted by the first one. However, the missing energy can be provided due to the Doppler effect: atomic nuclei of a gas have a thermal motional energy kT comparable with the recoil energy (kT = 0.025 eV at room temperature). The second nucleus moving with energy E in direction towards the γ radiation may absorb the radiation if E ≈ ∆E. — Using gaseous samples, the line absorbs only a very small fraction of radiation out of a continuum, and it is therefore very difficult to observe nuclear fluorescence. The absorption probability may get strongly enhanced if emission and absorption takes place without recoil: • Recoilless resonance absorption (Mössbauer effect): — Recoilless emission and absorption of γ quanta is possible for nuclei bound in crystals. — Einstein model (see lectures on solid-state physics): the crystal is considered as an ensemble of oscillators with frequency ω. The recoil energy is then a multiple of ~ω. Fig. 6.15 shows that recoilless emission is likely only for ∆E ¿ ~ω (where ∆E is the recoil energy of the free nucleus). The γ ray always transfers a momentum ~ω γ /c to the lattice. In absence of recoil, there is no energy transfer to the lattice and the photons have energy E0 . — The Debye model provides a more realistic representation of the spectrum of lattice vibrations. The relevant quantity is the Debye temperature Θ. At temperatures T > 0 K, the modes of lattice vibrations are excited and the probability of energy transfer by the γ ray to the lattice increases with T . The quantitative result for the probability f of recoilless emission is: Ã ( " µ ¶2 #)! Eγ2 3 2 πT f ≈ exp − , T ¿ Θ. (6.75) 1+ 2 2M c2 kΘ 3 Θ Hence, f is large only if the recoil energy ∆E is much smaller than kΘ (compare eq.(6.73)). Observation of recoil-less resonance absorption thus requires cooled solid samples.

78

6. Nuclear decays

Fig. 6.15. Relative probability for a gamma-ray transition with energy Eγ simultaneous with the excitation of 1, 2, 3 ... n Einstein oscillators in the crystal lattice at T = 0 K. Left: ∆E ¿ ~ω, right: ∆E À ~ω [E. Segrè, Nuclei and Particles, Benjamin, 1977].

Fig. 6.16. Fraction of recoilless transitions in iron and rhenium as a function of temperature [R.L. Mössbauer, Ann. Rev. Nucl. Sci. 12 (1962) 123].

• Doppler spectroscopy: the apparatus shown in fig. 6.17 enables a measurement of the natural width of a γ line emitted from a moving source. Changing the relative velocity between source and absorber, in the example shown in fig. 6.18, a maximum of absorption appears at ∆E = 0. • Gamma lines of a nucleus bound in a lattice may be shifted due to various reasons: — Isomer shift (also called chemical shift). The electrostatic energy of the nucleus in the potential caused by the atomic electrons may be different for the ground state and the γ active excited state, due to slightly different nuclear radii of these two states. These shifts are different for different chemical bonds and can thus be observed if the source and the absorber have different chemical composition, see fig. 6.19. — Magnetic dipole splitting: a magnetic field at the nuclear site lifts the degeneracy of the magnetic nuclear levels, leading to a γ ray spectrum with several lines, as visible in fig. 6.20.

6.7 Mössbauer effect

79

Fig. 6.17. Mössbauer’s original set-up: A: cryostat of absorber, S: rotating cryostat with source, D: scintillation detector, M: region to which D is sensitive [R.L. Mössbauer, Naturwiss. 45 (1958) 538].

Fig. 6.18. Fluorescent absorption in 191 Ir at T = 88 K as a function of the relative velocity between source and absorber. Upper scale: Doppler energy corresponding to the velocity on the lower scale [R.L. Mössbauer, Naturwiss. 45 (1958) 538].

— Electric quadrupole splitting: an electric field gradient interacts with the nuclear electric quadrupole moment (see section 2.7), leading to level splitting for the different nuclear orientations and corresponding splittings in the γ ray spectrum, see fig. 6.21. • Applications: — Determination of the physical or chemical environment of a nucleus, via the aforementioned shifts and splittings. Example from biology: determination of the chemical state of iron in hemoglobin. Example from astronomy: determination of the chemical state of iron in stones on mars. — Fundamental physics: determination of the gravitational red-shift. Photons rising (or falling) in the earth gravitational field change their energy according to ∆Eγ gh = 2 Eγ c

⇒

10−16 per meter change in height.

(6.76)

This effect can be understood as a Doppler effect: following Einstein, the effects of a locally homogeneous gravitational field cannot be distinguished from those of a uniformly acceler-

80

6. Nuclear decays

Fig. 6.19. Isomer shift. Right: ground and excited states may have different shifts with respect to the naked nucleus due to different overlap of electronic wavefunctions with the nucleus. Left: The effect on the resonance is to shift it away from zero relative velocity [K.S. Krane, Introductory Nuclear Physics, Wiley, 1987].

Fig. 6.20. Magnetic dipole splitting of nuclear levels observed with the Mössbauer effect. The ground and excited nuclear states have different magnetic moments, which causes different level splitting. In the example shown, the moments have also opposite sign. For dipole transitions, only ∆mI = 0 or ±1 can occur, so 6 individual components are seen [K.S. Krane, Introductory Nuclear Physics, Wiley, 1987].

ated reference frame. Taking therefore the γ emission and its reabsorption at distance h as happening in a frame accelerated with g = 9.81 ms−1 , at arrival of the photon the absorber will have acquired a velocity g∆t = gh/c, causing a Doppler shift according to eq.(6.76). The experimental set-up is shown in fig. 6.22. The result, an effect of ∆Eγ /Eγ = 4.902(41)×10−15 (for the 45 m round trip) was in full agreement with the theoretical prediction, 4.905×10−15 .

Fig. 6.21. Electric quadrupole hyperfine splitting. In the example shown, the upper and the lower level have different isomer shifts that moves the centre of the Mössbauer spectrum away from zero velocity. In addition, the upper level with spin 3/2 is split due to an electric field gradient at the nuclear site, causing a doublet of γ transitions [K.S. Krane, Introductory Nuclear Physics, Wiley, 1987].

Fig. 6.22. Set-up of the gravitational red shift experiment performed at Harvard. 14.4 keV 57 Fe γ quanta from a 1-Ci source of 57 Co travelled through He gas (which has a small absorption for the γ’s) 22.5 m up a tower [Pound and Rebka, Phys. Rev. Lett. 4 (1960) 337].

7. Models of the nucleus

7.1 Fermi-gas model • The theoretical concept of a Fermi-gas may be applied to systems of weakly interacting fermions, i.e. particles obeying Fermi-Dirac statistics leading to the Pauli exclusion principle. — The free electron gas is an example of solid state physics, where electrons move quasi-freely in a background of positively charged ions. — Protons and neutrons within the nucleus can be considered as moving quasi-freely within the nuclear volume. The binding potential is generated by all nucleons. • Simple picture of the nucleus: — Neutrons and protons are distinguishable fermions and are therefore situated in two separate potential wells. — In a first approximation, these nuclear potential wells are considered as spherical square well potentials with same radius. • Fermi momentum, Fermi energy and its relation to the nuclear binding energy: — Neglecting spin, the number of states of protons, respectively, neutrons within the nuclear volume V is given by dn =

4πp2 dp (2π~)

3

V.

(7.1)

— At temperature T = 0, i.e. for the nucleus in its ground state, the lowest states will be filled up to a maximum momentum, called the Fermi momentum pF . The number of these states follow from integrating eq.(7.1) from 0 to pF : n=

V p3F . 6π 2 ~3

(7.2)

Since each state can be filled with two nucleons of the same type, N=

V p3F,n , 3π2 ~3

Z=

V p3F,p , 3π 2 ~3

(7.3)

where pF,n and pF,p are the Fermi momenta of the neutrons, respectively, the protons. Using our prior knowledge from electron scattering, i.e. (see section 5.6), V =

4 3 4 3 πR = πR0 A, 3 3

R0 = 1.21 fm,

(7.4)

84

7. Models of the nucleus

we obtain for Z = N = A/2 and equal radius for the two separate potential wells of the protons and the neutrons a Fermi momentum of pF = pF,n = pF,p =

~ R0

µ

9π 8

¶1/3

≈ 250 MeV/c.

(7.5)

Hence, the nucleons move in the nucleus with large momentum. — The Fermi energy is the energy of the highest occupied nucleon level, EF =

p2F ≈ 33 MeV. 2mN

(7.6)

mN denotes the nucleon mass (mn ≈ mp in the present context). The difference B 0 between the edge of the potential well and the Fermi level is rather constant for different nuclei and equals the average binding energy per nucleon, B/A ≈ 7 − 8 MeV. Hence, the depth of the potential well, V0 , is approximately independent of A and given by V0 = EF + B 0 ≈ 40 MeV.

(7.7)

Kinetic and potential energies of the nucleons are thus of the same order. In this sense, nucleons are rather weakly bound in the nucleus (similar to the case of electrons in a metal).

Fig. 7.1. Sketch of the proton and neutron potential wells and states in the Fermi gas model [B. Povh et al., Particles and Nuclei, Springer, 2002].

• Coulomb repulsion of the protons leads to an asymmetry in Z and N : — The neutron potential well is deeper than the proton one, since the former have no Coulomb interaction. On the other hand, for a stable nucleus, the Fermi levels of the protons and the neutrons have to be the same, otherwise it would decay to an energetically more favourable state through a β transition. As a result, there are more neutron states than proton states occupied, which explains the fact that N > Z for heavier stable nuclei. — The binding energy as a function of N − Z can be estimated using the Fermi-gas model: the mean kinetic energy per nucleon is R pF Ekin p2 dp 3 p2F hEkin i = 0 R pF 2 ≈ 20 MeV. (7.8) = 5 2mN p dp 0 The total kinetic energy of the nucleus is

7.2 Shell model

Ekin (N, Z) = N hEkin,n i + Z hEkin,p i = Hence, using eq.(7.3) and eq.(7.4), Ekin (N, Z) =

3 ~2 10 mN R02

µ

9π 4

¶2/3

¢ 3 ¡ 2 N pF,n + Zp2F,p . 10 mN

N 5/3 + Z 5/3 , A2/3

85

(7.9)

(7.10)

where the radii of the proton and the neutron potential well have again been taken as equal. Expanding this expression in the difference N − Z leads to ! µ ¶2/3 Ã 3 ~2 9π 5 (N − Z)2 Ekin (N, Z) = A+ + ... . (7.11) 10 mN R02 4 9 A For fixed A the kinetic energy has a minimum for N = Z. The first term contributes to the volume energy in the Weizsäcker mass formula (see section 2.4), whereas the second one contributes to the symmetry (or asymmetry) energy. The asymmetry energy grows with the neutron (or proton) surplus, thereby reducing the binding energy (this consideration, although instructive, neglected the change of the nuclear potential connected to a change of N on cost of Z. This additional correction turns out to be as important as the change in kinetic energy).

7.2 Shell model • Magic numbers: Nuclides with certain proton and/or neutron numbers are found to be exceptionally stable. These so-called magic numbers are 2, 8, 20, 28, 50, 82, 126

(7.12)

— Nuclei with magic proton or neutron number possess an unusually large number of stable or long lived nuclides. — A nucleus with a magic neutron (proton) number requires a lot of energy to separate a neutron (proton) from it. — A nucleus with one more neutron (proton) than a magic number is very easy to separate. — The following doubly magic nuclei are stable: 4 2 He2 ,

16 8 O8 ,

40 20 Ca20 ,

48 20 Ca28 ,

208 82 Pb126

(7.13)

• Eigenstates of the nuclear potential : — The wavefunctions of a spherically symmetric potential can be separated into a product of a radial part Rn (r) and an angular part Y m (θ, ϕ). The quantum number n denotes the number of nodes +1 of Rn (r) at r > 0, and = s, p, d, f, g, h, i, ... is the quantum number of orbital angular momentum. The energy is independent of the "magnetic quantum number" m. The levels belonging to n are 2 (2 + 1) degenerate, the first factor 2 due to the two possible orientations of the nucleon spin. — The nuclear force is very short-ranged ⇒ the form of the potential corresponds to the density distribution of the nucleons within the nucleus. For very light nuclei (A . 7), the nucleon distribution has Gaussian form (corresponding to a harmonic oscillator potential),

86

7. Models of the nucleus

for heavier nuclei it may be parameterised by a Fermi distribution (see section 5.6). The latter corresponds to the Woods-Saxon potential : Vcentral (r) =

−V0 1 + e(r−R)/a

(7.14)

Table 7.1. Hierarchy of the energy eigenstates corresponding to the central Woods-Saxon potential. The first three magic numbers 2, 8 and 20 can be understood as nucleon numbers for filled shells, but the larger magic numbers cannot be obtained that way. The number N = 2 (n − 1) + is characteristic for a harmonic oscillator potential, for which the energy of the quanta of vibration is given by Eh.o. = }ω (N + 3/2). For the Woods-Saxon potential, states with the same N but different n are not degenerate. N = 2 (n − 1) + l

0

1

2

2

3

3

4

4

...

nl

1s

1p

2s

1d

1f

2p

1g

2d

...

degeneracy

2

6

2

10

14

6

18

10

...

States with E ≤ Enl

2

8

10

20

34

40

58

68

...

• Spin-orbit coupling: this is the essential ingredient to the nuclear shell model. In atomic physics, this effect causes only a small effect, the fine structure of atomic levels. In nuclear physics, it leads to considerable changes in the hierarchy of the energy levels (see fig. 7.2). Formally, spin-orbit coupling may be introduced as a term proportional to s: V (r) = Vcentral (r) + V s (r)

h si ~2

(7.15)

The expectation values are 1 h si = (j (j + 1) − ( + 1) − s (s + 1)) = ~2 2

½

/2 − ( + 1) /2

for j = + 1/2 for j = − 1/2

(7.16)

The spin-orbit energy thus leads to an energy splitting rising linearly with orbital angular momentum: ∆E

s

=

2 +1 · hV s (r)i 2

(7.17)

Experimentally, V s < 0, hence states with j = + 1/2 are always below states with j = − 1/2 (in contra-distinction to the effect of electron spin-orbit coupling in atomic physics). The nuclear levels are quoted as n j,

e.g. 1f7/2 and 1f5/2

(7.18)

• Nuclear magnetic moments in the shell model: — Starting point for their calculation is the magnetic moment operator, µnucl = µN

A X i=1

(g

i

+ gs si ) /},

(7.19)

7.2 Shell model

87

Fig. 7.2. Single particle energy levels calculated in the shell model. The spin-orbit coupling lifts the degeneracy of the levels denoted by quantum numbers n . Magic numbers appear when the gaps between levels are particularly large [B. Povh et al., Particles and Nuclei, Springer, 2002].

with (see sec. 2.6.1) ½ 1 for protons g = 0 for neutrons

and

gs =

½

+5.58 for protons −3.83 for neutrons

(7.20)

— The expectation value of the magnetic moment, hµnucl i = µN

A X i=1

hψ nucl |g

i

+ gs si | ψ nucl i /},

(7.21)

can be calculated using the Wigner-Eckart theorem, stating that the expectation value of any vector operator of a system is equal to the projection onto its total angular momentum. Explicitely, µnucl has to be projected onto the the nuclear spin J in the nuclear g factor as hµnucl i = gnucl µN

hJi , }

gnucl =

A X hJMJ |(g i=1

i + gs si ) ·J| JMJ i . hJMJ |J2 | JMJ i

• Special case: a single nucleon in addition to closed shells.

(7.22)

88

7. Models of the nucleus

— Angular momenta of all the nucleons in closed shells couple to zero, such that only the quantum numbers of the nucleon in the outmost shell enter the expression of the nuclear g factor in eq.(7.22), with J being defined by the j of this nucleon. Since 2 · j = j2 +

2

− s2

and

2s · j = j2 + s2 −

2

(7.23)

we obtain gnucl =

g {j (j + 1) + ( + 1) − s (s + 1)} + gs {j (j + 1) + s (s + 1) − ( + 1)} (7.24) 2j (j + 1)

— Definition of the value of the nuclear magnetic moment: it is given by the value measured for the nuclear spin being maximally aligned, i.e. |MJ | = J. Then, hJi = J~, such that µ ¶ gs − g 1 µnucl = gnucl J = g ± J for J = j = ± . (7.25) µN 2 +1 2 Good agreement with this theoretical prediction is found for the nuclei quoted in the table, which are doubly magic up to a single nucleon or a single hole (a nucleon missing to fill a shell). Table 7.2. Comparison of magnetic moments in the shell model with experimental values. The nuclides shown correspond to single proton or neutron states (A = 17) or single holes in a proton or neutron shell (A = 15). The latter are denoted with a superscript −1. Nucleus

state

JP

µnu c l /µN (model)

µnu cl /µN (experiment)

15

N

1/2−

15

O

p-1p−1 1/2 n-1p−1 1/2

−0.264

−0.283

17

O

n-1d5/2

5/2+

p-1d5/2

+

17

F

1/2− 5/2

+0.638

+0.719

−1.913

−1.894

+4.722

+4.793

8. Nuclear Magnetic Resonance

We have already briefly considered nuclear magnetic resonance (NMR) as a method to measure nuclear magnetic moments (see section 2.6). We discussed the interaction energy of a magnetic moment with an external static magnetic field and mentioned that through irradiation of a highfrequency field with frequency corresponding to the Zeeman splitting of magnetic levels one may induce spin flips. In this chapter, we first consider the flip process itself in more detail, and proceed then to applications in solid-state physics and in medicine.

8.1 Basic principle of NMR • The classical equation of motion of a magnetic moment in a magnetic field (including timedependent ones) reads 1 dµ = µ × B. γ dt

(8.1)

(remember that µ/γ is the angular momentum associated with µ, and µ × B is the torque on it). As shown in an exercise, the equation for quantum mechanical expectation values of the magnetic moment is exactly the same: 1 d hµi = hµi × B. γ dt

(8.2)

Hence we may omit the brackets for the averages and perform the following calculations using the classical equations. • The solution describes a precession of µ about B0 with angular velocity ω L = −γB0 , which is the Larmor frequency as derived before (see sign conventions below). • Sign conventions: — Precession frequencies: we encode the direction of precession of µ about a magnetic field B0 by the sign of the precession frequency: positive frequency means precession as one would indicate it using the right hand with the thumb as the precession axis and the other fingers showing the direction of precession. The precession shown in fig. ?? thus proceeds with negative frequency, corresponding to a magnetic moment with positive ratio of gyration, γ. — Magnitudes of magnetic fields (like B0 in case I), are always considered to be positive (otherwise the sign convention for the frequencies would make no sense). • Rotating coordinate system: this is an interesting method to solve the equation of motion eq.(8.2), which can be extended to more complex situations with time-dependent fields, like

90

8. Nuclear Magnetic Resonance B0 µ

Fig. 8.1. Larmor precession of a magnetic moment in a static magnetic field. The direction of the precession is determined by the sign of the gyromagnetic ratio γ. In the case shown, γ is positive and, according to our sign convention, we assign a negative precession frequency to it.

in magnetic resonance. Consider a frame with unit vectors e1 , e2 , e3 , rotating with angular velocity Ω with respect to a laboratory frame, dei = Ω × ei . dt

(8.3)

The magnetic moment, µ (t) =

3 X

µi (t) ei (t) ,

(8.4)

i=1

has components µi with respect to this frame (left side: the referential-free vector, right side: its components in a specific frame). The rotation makes a part of the time dependence of µ (t), µ ¶ dµ X dµi dei = ei + µi dt dt dt ¶ µ X δµ dµi = ei + µi Ω × ei = + Ω × µ. (8.5) dt δt

δµ/δt denotes the time derivative of µ in the rotating frame (dµi /dt are the components of this vector constructed as µ (t) in eq.(8.4)). The equation for δµ/δt can be transformed further, using eq.(8.2), δµ dµ = + µ × Ω = µ × (γB + Ω) = µ × γBeff , δt dt

(8.6)

and introducing an "effective field" Beff . Hence, comparing the original equation of motion with eq.(8.6) describing the motion of the magnetic moment in the rotating frame, we see that it has the same form, but the magnetic field B got replaced by Beff , Beff = B +

Ω . γ

(8.7)

• Case I - a static magnetic field: B = B0 ,

dB0 = 0. dt

(8.8)

— A particularly simple form of the solution is obtained by choosing Ω = −γB0

⇒

δµ =0 δt

(8.9)

8.2 Nuclear spins in condensed matter

91

— Hence, in the rotating frame µ stays static. In the laboratory frame, µ thus precesses about B0 with the Larmor frequency −γB0 . • Case II - a static field B0 + a rotating high-frequency field B1 (t) with constant strength: — We have to solve dµ = µ × γ (B0 + B1 (t)) , dt

(8.10)

with B0 = (0, 0, B0 ) ,

B1 (t) = B1 (cos (ωt + ϕ) , sin (ωt + ϕ) , 0) .

(8.11)

— Defining the rotating frame by Ω = ωe3 ,

(8.12)

the rotating field becomes time-independent in this frame. Choosing ϕ = 0 it can be written as B1 (t) = B1 e1 .

(8.13)

Following eq.(8.6), the motion of the magnetic moment is then described by ∙µ ¸ ¶ ω δµ = µ × [γ (B0 + B1 (t)) + ωe3 ] = µ × γ B0 + e3 + B1 e1 . δt γ Hence, the effective field, µ ¶ ω Beff = B0 + e3 + B1 e1 , γ

(8.14)

(8.15)

is constant, and the resulting motion in the rotating frame is Larmor precession about Beff at frequency −γBeff , as shown in fig. 8.2. — At resonance, ω = −γB0

⇒

Beff ⊥B0 .

(8.16)

Only in this case, a complete inversion of the population of two Zeeman levels is possible ("complete spin flip"). The precession frequency about Beff is then given by −γB1 .

8.2 Nuclear spins in condensed matter In condensed matter (in contrast to a particle beam), one observes a net magnetic effect due to many nuclei. Rather than with individual spins one deals with the • magnetisation: + * N + *N X X µj = ~ γ j Ij . M= j=1

(8.17)

j=1

N is the number of nuclei in the sample. The simple equation of motion (written for a single nuclear species, for simplicity),

92

8. Nuclear Magnetic Resonance z ez(B0+ω/γ)

Beff

µ y

B1

x

Fig. 8.2. Precession of a magnetic moment µ in the effective magnetic field Beff in the rotating frame. Whereas in the laboratory frame the magnetic field B0 + B1 is time-dependent, in the frame rotating with angular velocity ω about B0 , µ senses the static field Beff . The angle between Beff and B1 vanishes at resonance. Far off resonance, Beff is nearly parallel to B0 .

dM = M × γB, dt

(8.18)

does neither take into account the mutual interactions between different spins, nor any effect of energy exchange with the environment. Such effects must be included in order to describe spin relaxation phenomena. Modifications to eq.(8.18) can be easily guessed through the following consideration. • In thermal equilibrium in a static external magnetic field B0 = (0, 0, B0 ), the components of the magnetisation are given by Mz = M0 ,

Mx = My = 0,

(8.19)

with M0 been determined by thermodynamics, see next section. • Deviations from this equilibrium situation will relax back towards equilibrium with two different time constants T1 and T2 as Mx dMx , =− dt T2

dMy My , =− dt T2

(8.20)

and dMz M0 − Mz . = dt T1

(8.21)

The time constants are called — T1 : spin-lattice relaxation time or longitudinal relaxation time, — T2 : spin-spin relaxation time or transversal relaxation time. • Bloch’s phenomenological equations (Felix Bloch 1946) include the aforestated relaxation terms in the equation 8.18 of free motion: Mx dMx , = (M × γB)x − dt T2

(8.22)

8.3 NMR imaging

93

My dMy , (8.23) = (M × γB)y − dt T2 M0 − Mz dMz . (8.24) = (M × γB)z + dt T1 These equations form a starting point of many investigations in solid-state physics. Measured relaxation times can be confronted with theoretical estimates. • The principle of an apparatus to measure relaxation times is shown in fig. 8.3. S ignal generator

Radiofrequency bridge

Rf-detector

Modulation coil

Magnet

S am ple

Oscilloscope

S aw -tooth generator

Fig. 8.3. Layout of an apparatus for detection of nuclear magnetic resonance in samples of condensed matter (liquids, solids). The signal generator feeds a resonance circuit (capacitor + coil), the radiofrequency detector measures a voltage accross the resonance circuit. When the frequency meets the nuclear resonance, the sample absorbs energy out of the rf-field, leading to a change of voltage at the detector [Haken, Wolf, Atomic physics, Springer].

8.3 NMR imaging • In order to see NMR effects, we need oriented nuclei. Denoting the magnetic sub-levels with the quantum number m, we can define the polarisation P of a system of (equal) nuclear spins as P =

m=+I hIz i 1 X mW (m) , = I I

(8.25)

m=−I

where W (m) denotes the relative population of the sub-level m. • In thermal equilibrium, the populations are Boltzmann-distributed, ´ ³ 0 exp − gI µkNBmB T ³ ´. W (m) = P m=+I gI µN mB0 m=−I exp − kB T

(8.26)

94

8. Nuclear Magnetic Resonance

For spin 1/2 the thermal equilibrium polarisation is then simply given by PB = tanh

gI µN B0 . 2kB T

(8.27)

• NMR imaging (MRI) of organic tissue usually employs resonance absorption of protons (hydrogen nuclei). Using suitable magnetic field gradients, the resonance condition is made to be fulfilled locally. Scanning the gradient through the body and measuring the absorbed power as a function of time provides a picture which is proportional to the local proton density and polarisation. • Eq.(8.26) tells us that the available polarisation is determined by the ratio of a potential magnetic energy and a thermal energy. That this will be quite small under usual conditions can be read from the numerical values of the nuclear magneton and the Boltzmann constant: µN = 3.15245166 × 10−8 eV T−1 ,

(8.28)

kB = 8.617385 × 10−5 eV K−1 .

(8.29)

The temperature of the living body is T ≈ 300 K, and a typical value for the magnetic field is 1 Tesla. At these conditions, P ≈ 10−6 .

(8.30)

A sizeable signal is nevertheless obtained due to the high number density of protons in the tissue (a few times 1022 cm−3 ), and pictures of fantastic quality with resolution well below 1 mm3 can be obtained. • A recent application of MRI: inhalation of hyper-polarised 3 He gas (PHe ' 50 %) for diagnostics of lung diseases (see fig 8.4).The 3 He is polarised with an optical pumping technique. Non-smoker

Light smoker

Fig. 8.4. MRI of the lung with hyper-polarised 3 He gas [University of Mainz and Hospital of the University, 1999].

The much lower number density of the polarised gas nuclei (compared to hydrogen MRI) is counterbalanced by the much higher polarisation.

9. Nuclear reactions

9.1 Introduction • Nuclear reactions in the proper sense are those, where the available energy is well below the threshold for pion production (mπ0 = 134.9766 (6) keV c−2 ). Above this threshold, meson and particle phenomena become increasingly important. For energies larger than some GeV, the nuclear composition of a target becomes less important, and we deal with pions, single nucleons and other particles. Low-energy nuclear reactions are interesting for — measurements of masses of unstable nuclei (see section 2.3.1), — attribution of quantum numbers to nuclear states, — production of exotic nuclear states. — Many more nuclear levels are accessible in nuclear reactions than in the decay of radioactive nuclei. • A typical reaction involving a projectile particle a and a target nucleus X with reaction products b and Y , a + X → Y + b.

(9.1)

Alternative way to write the same reaction (already introduced in eq.(14.19)): X (a, b) Y.

(9.2)

Examples: 23

Na (p, α)

20

Ne,

p (n, γ) d.

(9.3)

(see also section 14.3 for more neutron-induced reactions). • Coulomb barrier : Charged particles cannot effectively react unless they have sufficient kinetic energy to overcome the energy barrier due to electrostatic repulsion, which is approximately given by Za ZX in MeV, A1/3

(9.4)

where A is the mass number of the target nucleus (here we have considered a light projectile sent to a heavier nucleus, otherwise the approximation looks a bit different). • The Q value of a reaction is defined as the difference of kinetic energies after and before the reaction:

96

9. Nuclear reactions

Q = Eb + EY − (Ea + EX ) = (ma + mX − mb − mY ) c2 .

(9.5)

Note that Q thus defined is independent of the reference frame due to the difference. — exothermic (also called exoergic) reactions: Q > 0, — endothermic (endoergic) reactions: Q < 0. For endothermic reactions, Q is the minimum energy in the centre-of-mass system, which is required to induce the reaction. • Reactions may be classified according to the mechanism governing the reaction: — Direct reactions: only a few nucleons take part in the reaction, the remaining nucleons of the target are "passive spectators". These reactions, where e.g. a single nucleon is inserted or removed from a shell-modell state. Such reactions are suited for spectroscopy of nuclear states. Many excited states of Y can be reached in direct reactions. Reaction times are of order 10−22 s, which is the time a projectile requires to fly through the nucleus. Important subgroups of these reactions are • knockout reactions: the projectile a reappears in the exit channel, but the reaction causes yet another nucleon to be ejected additionally: X (a, ab) Y.

(9.6)

• transfer reactions: one or two nucleons are transferred between projectile and target. Examples: X (d, p) Y,

X (d, n) Y,

X (t, n) Y.

(9.7)

Simple examples of a transfer reaction: the stripping reactions A

Z (d, p)

A+1

Z,

A

Z (d, n)

A+1

(Z + 1) .

(9.8)

• Collective excitation of the nucleus: the giant resonance.

(a)

(b)

(c)

(d)

Fig. 9.1. (a) - (c): direct reactions. (a): single particle excitation in inelastic scattering, e.g. (p,p’γ), (b): collective excitation, e.g. giant resonance, (c): (p,n) knockout reaction. (d) first step of a compound nucleus reaction [T. Mayer-Kuckuk, Kernphysik, Teubner].

— Compound nucleus reactions: the projectile nucleon or nucleus briefly merges with the target nucleus. The kinetic and binding energies are distributed among many degrees of freedom, until one nucleon by chance acquires enough energy to escape the nucleus. The process is similar to evaporation of a molecule from a hot liquid. Typical reaction time: 10−16 s. — Resonance reactions: the incident particle forms a quasi-bound state in the nuclear potential before the outgoing particle is ejected.

9.2 General features of the reaction cross section

97

Fig. 9.2. Pictorial representation of the projectile’s wavefunction inside and outside the nucleus. Left side: if wavefunction at the nuclear boundary has a horizontal tangent, the wavefunction within the nucleus gets very large and therefore also the probability of a reaction [D. Kamke, Einführung in die Kernphysik, Vieweg 1979].

9.2 General features of the reaction cross section The considerations presented here are based on Fermi’s golden rule. They are valid in cases where no particular nuclear level plays a prominent role. To be specific, we consider a reaction of the type X (a, b) Y . • Transition probability per unit time, E 2π D 2 dn W = |Mfi | , (9.9) ~ dE where dn/dE is the density of accessible states of the reaction products, and Mfi is now the nuclear matrix element causing the reaction considered (compare with the earlier appearances D E 2 of Fermi’s golden rule in sections 4.3 and 6.5.2). |Mfi | denotes the average of the squared 2

matrix element connecting the initial and final states. The basic assumption here is that |Mfi | varies only weakly for the different accessible states.

• The final state density is given by (see eq.(4.19)) dn V · 4π |pb |2 d |pb | = , 3 dE dE (2π~)

(9.10)

where, remember, V is a normalisation volume. • The matrix element, Mfi = hf |Hint | ii =

Z

ψ ∗f Hint ψ i d3 r,

(9.11)

depends on the nuclear structure. For the particles a and X, respectively, b and Y separated well beyond the range of the nuclear forces, considering the reaction in the centre-of-mass system, the wavefunction of the reacting particles (similar for the outgoing particles) is given by 1 1 ψ i = √ ϕa ϕX exp (ikr) = √ ϕi exp (ikr) , V V

(9.12)

where ϕa and ϕX are the internal wavefunctions of the projectile and the target. Since only the very small volume occupied by the nucleus contributes to the integral, it can be written as Z 1 hHint i Vnucleus Mfi ' ϕ∗f Hint ϕi d3 r ' . (9.13) V V hHint i denotes a transition density averaged over the nuclear volume.

98

9. Nuclear reactions

— For charged projectiles, ψ i at the nucleus will be reduced due to the Coulomb repulsion. Hence, in ψ i there will be an additional factor exp (−Ga ) where (see section 6.3), Z 0 1 R p 1 Ga = 2ma (VC (r) − Ea ) dr ∝ √ . (9.14) ~ R Ea

VC denotes the Coulomb interaction between the projectile and the target. Ga is the Gamow factor which we met first in α decay (compare eq.(6.21)), leading to a factor exp (−2Ga ) in D E 2 |Mfi | . A reduction by a factor exp (−Gb ), involving the Coulomb interaction between

the outgoing charged particles will appear in ψ ∗f . Hence, including Coulomb effects we have Mfi '

hHint i Vnucleus exp (−Ga − Gb ) . V

(9.15)

• The reaction cross section is related to the transition probability per unit time by W = na vaX σ,

(9.16)

where na is the number density of projectiles, and vaX is the relative velocity between particles a and X. The product na va,X is the flux Φ of projectiles and we have taken a single target nucleus (compare eq.(4.6)). With the normalisation na = 1/V and dE/d |pb | = vbY , we arrive at E |p |2 V2 D b 2 | . (9.17) |M σ= fi π~4 vaX vbY • The spins of the particles involved can be taken into account by statistical weight factors. For unpolarised particles, these are given by gi = (2Ia + 1) (2IX + 1) ,

gf = (2Ib + 1) (2IY + 1) .

The averaging, taking into account the spin degeneracy, means D E 1 X 2 2 |Mfi | = |Mfi | . gi gf

(9.18)

(9.19)

i,f

In the final state, the particles may come out with the spins randomly oriented. Hence, we have to multiply eq.(9.17) with gf , σ=

E |p |2 V2 D b 2 | (2Ib + 1) (2IY + 1) . |M fi π~4 vaX vbY

(9.20)

• An interesting application of the latter equation is a comparison of a reaction with its inverse: — For Mfi Hermitian, Mfi = M∗if .

(9.21)

meaning that the interaction respects time reversal invariance, the cross section for the reaction forward and backward are related as 2

|pb | (2IY + 1) (2Ib + 1) σ X→Y = σ Y →X |pa |2 (2IX + 1) (2Ia + 1)

(9.22)

The two reactions are considered in the centre-of-mass system, and velocities and momenta are measured in this system. Equation (9.22) is called the principle of detailed balance.

9.3 Application to some specific types of reactions

99

Fig. 9.3. Angular distributions in the centre-of-mass system for the inverse reactions 12 C + α ­ 14 N + d at matched energies: Eα = 41.7 MeV, Ed = 20.0 MeV (lab system) [Bodansky et al., Phys. Rev. Lett 2 (1959) 101].

9.3 Application to some specific types of reactions Here, we classify reactions according to the general behaviour of the cross section given in eq.(9.17). We exclude any possible variations of Hint which is the main effect in resonance phenomena discussed below, Hint ' const

(9.23)

in the small energy range of interest. Only a selection of reactions is discussed here, the dependence of the cross section for the five cases considered are shown in figure 9.4. (a) elastic neutron scattering (a and b uncharged): vn = vaX = vbY ,

|pb |2 = const vaX vbY

(9.24)

At low energy, |Mfi |2 ' const ⇒

σ ' const

(9.25)

(b) exothermic reactions, low-energy uncharged projectile (neutron induced reactions): (n, p) ,

(n, α) ,

(n, γ) ,

(n, f ) ...

(9.26)

(f standing for fission in the last reaction). Q > 0 is typically of the order a few MeV, whereas the energy of the incident neutrons (from a reactor) is of order eV or even less. Therefore,

100

9. Nuclear reactions 2

vbY ' const,

|pb | 1 ∝ vaX vbY vaX

(9.27)

Moreover, Ga = Gn = 0,

exp (−Gb ) = const

1 vn This is the famous "1/v" law. ⇒

σ∝

(9.28)

Fig. 9.4. Behaviour of reaction cross sections at low energy as a function of the projectile velocity v or energy E. (a) neutron elastic scattering, (b) exothermic reactions for neutral projectile, (c) exothermic reaction for charged projectile, (d) endothermic inelastic neutron scattering, (e) endothermic reaction with charged outgoing particle [E. Segrè, Nuclei and Particles, Benjamin, 1977].

(c) exothermic reactions, charged projectile: (p, n) ,

(α, n) ,

(α, γ) ,

(p, γ) ...

(9.29)

2

|pb | 1 ∝ vaX vbY vaX and the Coulomb barrier factor is operative: Ea ¿ Q

⇒

σ∝

⇒

1 exp (−2Ga ) vaX

(9.30)

(9.31)

(d) inelastic neutron scattering: (n,n’) reaction. The nucleus is left in an excited state, the reaction is therefore endothermic. −Q is the excitation energy of the nucleus. For incident neutrons slightly above the threshold energy Ethreshold ,

9.4 Nuclear energy

vaX = vn ' const

101

(9.32)

whereas vbY = vn0 ∝ ⇒

σ∝

p √ excess energy = En − Ethreshold

|pn0 |2 p ∝ En − Ethreshold vn vn0

(9.33)

(e) endothermic reactions, charged outgoing particles: (n, α) ,

(n, p) ...

(9.34)

in addition to the dependence stated in (d), there is the Coulomb barrier factor for the otgoing particle to be included: p ⇒ σ ∝ En − Ethreshold exp (−2Gb ) (9.35)

9.4 Nuclear energy • The maximum of the binding energy at about A = 60 (see section 2.3.2) enables energy gain in nuclear reactions as kinetic energy of the reaction products in 1. fission of heavy nuclei, 2. fusion of light nuclei. • Neutron-induced fission of one 235 U nucleus liberates about 200 MeV, which mostly is available as kinetic energy of the fission fragments. Some part of the energy is created with some delay (radioactivity of the fission fragments), and about 6 % of the energy goes to (anti)neutrinos, and therefore does not contribute to the heating of the reactor fuel (see table 9.1). On average, 2.5 neutrons are ejected per fission process with a spectrum as shown in fig. 9.5. Table 9.1. Energy balance in the fission of 165 MeV

235

U.

kinetic energy of fission fragments

5 MeV

kinetic energy of prompt neutrons

8 MeV

energy of the prompt γ rays

7 MeV

γ rays from radioactive fragments

7 MeV

β decays from radioactive fragments

12 MeV

(anti)neutrinos from radioactive fragments

• As was already explained in section 6.4, the pairing energy in the neutron induced fission leads to a significant reduction of the fission barrier. As a result (see table 9.2), —

235

U is fissile with thermal neutrons (i.e. neutrons with practically no kinetic energy (25 meV)), whereas

— fission of

238

U requires neutrons with a kinetic energy of & 1.5 MeV.

102

9. Nuclear reactions

Fig. 9.5. Energy spectrum of neutrons emitted in the thermal neutron fission of Introductory Nuclear Physics, Wiley, 1987]. Table 9.2. Energies relevant for neutron induced fission of Isotope

fission barrier

neutron binding energy

235

U

5.8 MeV

6.4 MeV

238

U

6.3 MeV

4.8 MeV

235

U and

238

235

U [K.S. Krane,

U.

• A self-sustained chain reaction may take place if on average at least one neutron resulting from a fission process is available to induce a next fission process. A necessary condition is η=

number of fission neutrons σf >1 =ν absorbed neutrons σf + σr

(9.36)

where ν is the average number of neutrons generated per fission, σ f is the fission cross section, and σ r is the cross section for other neutron-induced reactions, by which neutrons get lost for fission. Another source of loss, not included in eq.(9.36), is neutron escape through the surface of the piece of fissile material. As a consequence of the cross sections shown in fig. 9.6, Table 9.3. Some properties of fissile nuclides. Isotope

half-life t1/2

natural abundance

η (at 0.25 eV)

η (at 1 MeV)

235

U

0.72 %

2.07

2.33

238

U

7.04 × 108 a

239

Pu

2.44 × 104 a

−

−

−

4.47 × 109 a

99.28 %

2.12

2.93

— a piece of 235 U with mass exceeding a critical mass is explosive (uncontrolled chain reaction), whereas — natural uranium is not explosive even in very large pieces. • The controlled self-sustained chain reaction can be used for technical production of energy in the nuclear reactor. This consists of

9.4 Nuclear energy

103

envelopes to the resonances

Fig. 9.6. Cross sections for reactions of neutrons with uranium [T. Mayer-Kuckuk, Kernphysik, Teubner 1984].

— fuel (most often uranium compounds, where the

235

U is enriched to 3 %),

— moderator material surrounding the fuel (most often normal water), — movable neutron absorbers (control rods), — shielding around the whole arrangement.

control rods

fuel elem ents

m oderator

shielding and reflector

Fig. 9.7. Principle of a nuclear reactor setup with thermal neutrons [T. Mayer-Kuckuk, Kernphysik, Teubner 1984].

— The moderator slows down the fission neutrons from MeV to thermal energies. A fraction of the moderated neutrons diffuses back into the fissile material, where they may induce fission reactions with a large cross section. The slowing down of the neutrons outside the fissile material considerably reduces losses of neutrons due to reactions other than fission in 238 U.

104

9. Nuclear reactions

At the same time, the moderator acts as a coolant to remove the heat produced mainly due to the kinetic energy of the fission fragments. — An enrichment of 235 U in the natural uranium from 0.72 % to 3 % is necessary in order to operate the reactor with light water (H2 O). If one wants to operate a reactor with uranium of natural isotopic composition, heavy water (D2 O) has to be used as a moderator. This is due to the much lower absorption of the deuteron compared to the proton, σ c,d = 0.0005 barn,

σ c,p = 0.333 barn,

for thermal neutrons.

(9.37)

Nk e ff

235 U (n ,f ) therm al fission

235 U

(n ,γ ) 236 U neutron capture

238 U

(n ,γ ) 239 U → 239 Np → 239 Pu

Nη 238 U (n ,f ) fast fission

N ηε fast neutrons × (1-Ps ) fast neutrons escape

N ηε Ps are m oderated

× (1-p)

N ηε Ps p therm al neutrons are generated

neutrons lost through 238 U (n , γ ) resonances

× (1-Pth ) therm al neutrons escape × (1-f )

N ηε pfPs Pth = Nk eff therm al neutrons are available

therm al neutrons absorbed in the m oderator

Fig. 9.8. Neutron balance in the reactor. ke ff is the multiplication factor, and ε is the fast fission factor. The factors Ps and Pth are geometry-dependent and characterise neutron losses through the surface of the reactor.

— The various possible fates of a neutron created in nuclear fission in a reactor are shown in fig. 9.8. The effective multiplication factor keff has to fulfill keff > 1 keff = 1

to start the reator, for stationary operation.

(9.38) (9.39)

— Control of the reactor : denote with t0 the mean time to pass through a neutron generation. We may write dρ keff ρ − ρ = dt t0

⇒

ρ (t) = ρ0 exp

where the time constant is given by

t τ

(9.40)

9.4 Nuclear energy

τ=

t0 keff − 1

105

(9.41)

Typical values are keff = 1.007,

t0 = 1 ms

⇒

τ ≈ 0.1 s

(9.42)

However, such a rapid control of the reactor is not necessary due to delayed neutrons. These are emitted from highly excited fragments (see fig. 9.9). Their number amounts to about 0.016 neutrons per fission. At small reactivity they considerably slow down the reactor period, making it easier to control.

Energy [MeV]

Fig. 9.9. Chain of successive β − decays of neutron-rich isobars with A = 99. In a few percent the daughters of 99 Sr and 99 Y decay are highly excited and emit neutrons. If the initial isobar was created in a fission process, these neutrons appear delayed to the prompt fission neutrons [B. Povh et al., Particles and Nuclei, Springer, 2002].

• In breeding reactions new fissile material is generated from other nuclides than the rare 235 U. Two possibilities are shown in fig. 9.10. The fast breeder does not use any moderator and has a very high energy density, requiring liquid sodium for heat transport. The thermal thorium breeder is technologically less demanding. It has much smaller breeding rates but could be used in a thermodynamically very efficient graphite-moderated high temperature reactor.

neutrons fast

next fission

fissile nuclide

breeding nuclide

therm al

next fission

fissile nuclide

breeding nuclide

Fig. 9.10. Fission-breeding reaction chains for the fast breeder (upper) and the thermal thorium breeder (lower figure) [T. Mayer-Kuckuck, Kernphysik, Teubner].

10. Nucleosynthesis

According to the standard model of cosmology, nuclides are formed in • primordial nucleosynthesis, also called big bang nucleosynthesis • nucleosynthesis in stars

abundance [Si= 10 -6 ]

Mass num ber A

Fig. 10.1. Abundance of the elements in the solar system as a function of the mass number A, relative to the abundance of Si, which was normalised to 10−6 [B. Povh et al., Particles and Nuclei, Springer, 2002].

10.1 Primordial nucleosynthesis • Primordial nucleosynthesis means formation of light elements during the first few minutes after the big bang of the Universe. The initial state is an infinitely dense and hot plasma of radiation and matter.

108

10. Nucleosynthesis

• Evolution of the Universe: a sequence of phase transitions accompanying the expansion and cooling-down. Here we consider what happened for times ≥ 1 µs after the big bang. — After ' 10−6 s the temperature T of the Universe has fallen to about kB T ≈ 100 MeV (this corresponds to T ≈ 1012 K, remembering that kB = 8.617 × 10−5 eV K−1 ). Previously free quarks and gluons get bound in mesons and baryons (of which the nucleons are the lightest members). The temperature is still much to high for formation of nuclei. The masses of up and down quarks are similar and very small. Therefore, about the same number of neutrons and protons are formed. — ' 10−6 s − ' 1 s: protons and neutrons are in thermal equilibrium through weak processes like ν e + p ­ e+ + n

(10.1)

n + ν e ­ p + e−

(10.2)

or

— After ' 1 s, temperature has fallen to kB T ≈ 1 MeV, i.e. roughly the mass difference of neutron and proton (mn − mp ≈ 1.3 MeV c−2 ). Due to the larger mass of the neutron, the Boltzmann equilibrium gets significantly shifted towards the protons: µ ¶ (mn − mp ) c2 [n] = exp − (10.3) [p] kB T with [n] and [p] denoting the abundances of neutrons and protons. The radiation density gets to small for reactions like in eq.(10.1) to further inter-convert protons and neutrons. The (anti)neutrinos decouple from matter and move in the Universe virtually without any further interactions. — ' 1 s − ' 3 min: neutrons decay freely with a decay time of τ n ' 886 s, which further reduces [n] / [p]. The temperature is still to high for the lightest nucleus to form according to n + p → d + γ + 2.22 MeV

(10.4)

The reason, despite the Q value of 2.22 MeV, is that in the Maxwell spectrum there are enough photons of sufficiently high energy to destroy immediately the newly created deuteron via photo-dissociation, d+γ →n+p

(10.5)

— After ' 3 min, temperature has fallen to kB T ≈ 100 keV. Baryonic matter then consists of 88 % protons and 12 % neutrons. The rate of photo-dissociation gets so low that the deuteron becomes stable. Since the neutron does not decay any more within the nucleus, the ratio [n] / [p] does not change any more. — Most of the deuterons react further in fusion processes with protons and neutrons, n + d → 3 H + γ + 6.26 MeV

(10.6)

p + d → 3 He + γ + 5.49 MeV

(10.7)

Reactions like between 3

H + p,

3

He + n,

3

He + d,

d+d

(10.8)

10.2 Element burning and the fate of stars

109

generate the particularly stable nucleus 4 He. The reaction 4

He +3 He → 7 Li + γ + 2.47 MeV

(10.9)

generates Li nuclei, which, however, mostly get immediately destroyed by the strongly exothermal reaction 7

Li + p → 24 He + 17.35 MeV

(10.10)

The processes which determine the abundance of nuclear species finish about 30 min after the big bang. Subsequently, due to the reduced temperature and the Coulomb barrier prohibits further nuclear fusion reactions. The main constituents of the hadronic matter after this time are H : ' 76 %,

4

He : ' 24 % 3

(10.11) 7

with small admixtures of He and Li. — Due to the absence of stable nuclides with A = 5 and A = 8, simple fusion processes donot produce nuclei heavier than 7 Li. All heavier nuclei are produced much later in the evolution of the Universe in stellar fusion and events like supernova explosions.

10.2 Element burning and the fate of stars • Stars form through contraction of interstellar gas and dust, which consists primordially from hydrogen and helium. Contraction heats up the stellar centre. A star in a stationary state produces as much energy through nuclear reactions as it emits. The equilibrium is determined by the rate of fusion reactions. The gravitational pressure towards the stellar centre is compensated by the radiations from the fusion reactions. Radiation is transported to the stellar surface. The fusion reactions change the chemical composition of a star. • Fusion reactions of charged nuclides: — Basic condition: sufficiently high probability for tunneling through the Coulomb barrier, VC ≈

Z1 Z2 e2 Z1 Z2 1 ≈ 1.44 MeV 4πε0 R1 + R2 (R1 + R2 ) [fm]

(10.12)

The kinetic energy of the nuclei corresponds to the temperature of the gas. A measure of this is Eth = kB T,

kB = 8.617 × 10−5 eV K−1

(10.13)

7

Hence, a temperature as high as 10 K corresponds to a kinetic energy of only ∼ 1 keV! Even in very hot stellar plasms, usually VC À Eth

(10.14)

In the interior of our sun, we have T ≈ 1.5 × 107 K. — Fusion rate in a plasma (per unit volume): dN (10.15) = n1 n2 hσvi dt where n1 and n2 are the number densities of the reaction partners, v is the relative velocity, and σ is the fusion cross section. The brackets denote thermal averaging. hσvi is determined by two effects,

110

10. Nucleosynthesis

1. the Maxwell-Boltzmann velocity distribution is proportional to µ ¶ µ ¶ −mv 2 E n (v) ∝ exp = exp − 2kB T kB T 2. the fusion cross section contains the Gamow factor (see section 9.2), √ σ ∝ exp (−2G) , G ∝ 1/ E

(10.16)

(10.17)

⇒ strong T -dependence of thermonuclear reactions with rather well-defined ignition temperature Tcrit for energy production through fusion in the plasma. • Fusion reactions in stars: — Z-dependence of VC ⇒ the larger Zi , the larger Tcrit . — The first reaction in a star to take place (that with the lowest Tcrit ) is burning of hydrogen: p + p → d + e+ + ν e + 0.42 MeV

(10.18)

This weak process determines the lifetime of the sun. A further chain of reactions leads to the formation of 4 He: p + d → 3 He + γ + 5.49 MeV 3

He + 3 He → p + p + 4 He + 12.86 MeV +

(10.19)

−

e + e → γ + 1.022 MeV In total, the net effect of this "pp-cycle" is 4p → α + 2e+ + 2ν e

(10.20)

An energy of 26.72 MeV is released, of which the neutrinos carry away a fraction of 0.52 MeV. — Another reaction cycle with the same net reaction products is the CNO cycle (BetheWeizsäcker cycle, 1938): 12 C acts as a catalyser. This process gets important only at higher

β+ 13 N

(p, γ) 13 C

14 N

(p, γ)

(p, γ)

β+

(p, α) 12 C

15 N

15 O

Fig. 10.2. The Bethe-Weizsäcker (CNO) cycle.

temperatures (Coulomb barrier!), but it is much faster than the pp- cycle. — After consumption of the hydrogen, gravitation leads to a further collaps of the star, leading to further heating in its interior. Its further fate depends on the mass.

10.2 Element burning and the fate of stars

111

• If it is much smaller than that of the sun, M¯ , no further fusion reaction will ignite and a planet-sized object (a white dwarf ) will form where the collaps is stopped by the Fermi pressure of a degenerate electron gas. • Otherwise, at Tcrit & 108 K and at a density of ρ = 108 kg m−3 , burning of helium will ignite in the centre of the star. The outer region of the star still contains much hydrogen, which gets heated from the interior of the star and blows up under the radiation pressure. The increase of the surface reduces the surface temperature, shifing the emission spectrum to the red. A red giant forms. Burning of helium proceeds through 4

He + 4 He ­ 8 Be

(10.21)

which is possible only at such high temperatures (Coulomb barrier and a negative Q value of 90 keV, see fig. ). At ρ = 108 kg m−3 , in equilibrium there is one 8 Be nucleus per 109 4 He nuclei. This is enough to create a sizeable amount of 12 C via 8

Be + 4 He →

12

C∗

(10.22)

only because there is a 0+ resonance in 12 C which dramatically enhances the reaction cross section (J P = 0+ are just the quantum numbers of the state of reaction partners before the reaction). In total we have 3 4 He →

12

C + 2γ + 7.275 MeV

(10.23)

Almost all nuclei heavier than helium were formed through the 3α process! These are only

Fig. 10.3. Energy levels of the systems 3α, α + 8 Be and 12 C. Just above the ground state of the 3α system the 12 C nucleus has a 0+ resonance which can be populated via fusion of the three α particles. This state decays with a probability of 0.04 % to the ground state of 12 C [B. Povh et al., Particles and Nuclei, Springer, 2002].

about 1 % of all nuclei in the Universe. — After consumption of the helium → further gravitational collaps. If • M < 1.4 M¯ → formation of a white dwarf. • M > 1.4 M¯ → ignition of carbon burning at Tcrit & 6 × 108 K: 12 C +12 C → 23 Na + p + 2.24 MeV → 20 Ne + α + 4.62 MeV → ...

(10.24)

112

10. Nucleosynthesis

Reactions with α particles lead to formation of oxygen, neon, magnesium, silicon. Energy gain is possible up to formation of iron from silicon. This phase takes only a few days. Then no more thermonuclear source of energy is available and the outer shell of matter falls into the centre, leading to a supernova explosion. If the mass of the residual stellar nucleus is smaller than M¯ , a white dwarf forms. If it is between one and two times M¯ , a neutron star forms (a degenerate compressed gas of neutrons). A more heavy supernova residue becomes a black hole. • Heavier elements are generated through neutron capture reactions. — Slow process ("s-process"): during burning of stars, neutrons are formed in reactions like 22

Ne + α →

13

C+α→

25

Mg + n − 0.48 MeV

(10.25)

or 16

O + n − 0.91 MeV

(10.26)

Successive neutron capture processes generate neutron-rich isotopes. Together with β − decays, increasingly heavy isotopes are created along the valley of stability. This process finishes at lead, since more heavy nuclides decay through α decay. — Rapid process (r-process"): happens during a supernova explosion at neutron fluxes of 1032 m−2 s−1 , where many neutrons can be captured subsequently and many nuclides have not enough time to decay. This process leads to production paths far off the valley of stability and generates also elements heavier than lead. The upper limit is given by spontaneous fission.

11. Nuclear forces

The term nuclear forces denotes the interactions between nucleons. Like other hadrons (strongly interacting particles with internal structure), nucleons are composite systems of quarks bound together by the strong force, which is mediated by gluons. Similar to the van-der-Waals interaction, which is a residual electromagnetic interaction between atoms (i.e. composite objects held together by electromagnetic forces), the nucleon-nucleon (NN) interaction is a residual strong interaction between two nucleons. From the weak dependence of the nuclear binding energy per nucleon as a function of mass number A we know already that nuclear interactions have a range much shorter than the extension of medium size nuclei (see section 2.4). In earlier sections we have seen that nuclear properties like magnetic moments and nuclear level schemes may be described rather successfully by nuclear models involving an effective potential, without refering explicitely to the NN potential. Indeed, the exact form of the NN potential can be obtained only in studying the most simple systems and situations available. These are the deuteron and nucleon-nucleon scattering experiments. Symmetry properties of nuclear forces, like conservation of electric charge and conservation of isospin, treated in the present chapter, can also be tested in more complex nuclei.

11.1 The deuteron • The deuteron is the simplest system of bound nucleons. Its ground state is the only known bound state of two nucleons: — There are no bound states between two protons, respectively, two neutrons. — Excited states of the deuteron are unbound. Hence, the single bound state of nucleons offer us only very restricted possibilities to study NN interactions. Nonetheless, the ground state properties of the deuteron already tell us some important features of the binding force. • Properties of the deuteron: — Mass: 1.876 139 MeV, determined by mass spectroscopy using penning trap techniques (sec. 2.3.1). — Binding energy: 2.225 MeV (i.e. ∼ 1.1 MeV per nucleon), determined from measurement of the γ energy in radiative capture, n (p, γ) d. Compared to typical binding energies per nucleon of heavier nuclei (∼ 8 MeV), the deuteron is a weakly bound nuclear system. The binding is so weak that there are no excited bound states. — Spin: J = 1, deduced from observed number of hyperfine components. — Magnetic moment: µd = 0.857 393 µN , measured via nuclear magnetic resonance. The value is close to the sum of the magnetic moments of the free proton and the free neutron,

114

11. Nuclear forces

µp + µn = 2.792 µN − 1.913 µN = 0.879 µN

(11.1)

This implies that the deuteron is essentially a state, where the two spins 1/2 of the nucleons are parallel and add to J = 1 (remember the opposite signs of the proton and neutron magnetic moments, see sec. 2.6.1). This corresponds to an S state (no orbital angular momentum). — Electric quadrupole moment: Qd = 2.86 × 10−27 cm2 , deduced from the magnetic field dependence of hyperfine lines of deuterium. The reason for the deuteron being not spherically symmetric is a non-central part in the NN force, the so-called tensor force. It admixes a state with orbital angular momentum to the ground state, meaning that the deuteron is not a pure S state. Since the strong force conserves parity (remember that the orbital angular momentum of a state influences its parity by a factor (−1) ), the admixture must be a state with = 2 (i.e. a D state), in order to have the same quantum numbers J P . — The sizes of both the magnetic moment and the quadrupole moment can be derived from a wavefunction ¯ ® ¯ ® |ψ d i = 0.98 ¯3 S1 + 0.20 ¯3 D1 (11.2) where the deuteron is found with 4 % probability in the state 3 D1 .

• Simple model of the deuteron (neglecting the non-central force of the NN interaction). The Schrödinger equation of the two nucleons bound by a potential V (r) can be written as (→ exercise) ∙ 2 ¸ ~ ∆ − + V (r) ψ (r) = Eψ (r) (11.3) 2m with r = rp − rn being the relative coordinate of proton and neutron, and m=

mp mn mp + mn

(11.4)

being the reduced mass of the two particles. — We neglect the non-central part of the potential, i.e. take V (r) = V (r)

(11.5)

as direction-independent. As a further simplification, we take V (r) as a square well with depth −V0 : V (r) = −V0 for r < R0 V (r) = 0 for r ≥ R0

(11.6)

— For a central potential, the wavefunction ψ (r) = R (r) Θ (θ) Φ (ϕ)

(11.7)

leads to separation of eq.(11.3) into three independent, one-dimensional equations for R (r), Θ (θ) and Φ (ϕ). One only has to solve the equation for the radial part. The substitution u (r) = rR (r) reduces the corresponding Schrödinger equation to d2 u 2m + 2 [E − V (r)] u = 0 dr2 ~

(11.8)

11.2 Charge independence of the nuclear force: the isospin

115

exp(-κ r) sin(kr)

Fig. 11.1. Potential (left) and wavefunction (right) of a simple model of the deuteron [T. Mayer Kuckuk, Kernphysik, Teubner].

E = −B = −2.225 MeV is the nuclear binding energy. Using the condition u (0) = 0 (avoiding that ψ (r) = u (r) /r diverges at r = 0) and the boundary conditions for u (r) at r = R0 that u and du/dr have to be steady at r = R0 , we obtain the radial wavefunction u (r) of the deuteron as shown in the right-hand side of fig. 11.1 (→ exercise). — The solution of the Schrödinger equation is given by p for r < R0 : uI (r) ∝ sin (kr) with k = ~1 2m √ (V0 − B) for r ≥ R0 : uII (r) ∝ exp (−κr) with κ = ~1 2mB

(11.9)

It does not provide individual values for the potential depth V0 and the radius R0 , but only an expression relating these two quantities: r ¶ µ R0 p B 2m (V0 − B) = − (11.10) cot ~ V0 − B

Putting R0 = 1.4 fm, corresponding to the range of nuclear forces, we obtain V0 ≈ 50 MeV, i.e. a situation as shown in the left-hand side of fig. ??, telling us that

— the potential depth is much larger than the binding energy ⇒ the kinetic energy is comparable with the potential depth. As a result, — the bound state has a large spatial extension: a large fraction of the radial probability density to find the two nucleons separated by r, u2 (r) = r2 |ψ (r)|2 , is outside the range of the potential.

11.2 Charge independence of the nuclear force: the isospin • The concept of isospin was introduced by Heisenberg in 1930 to work out the consequences of the similarity of proton and neutron. — Main idea: proton and neutron are two different states of the nucleon (suggested by the small mass difference of these particles and similarity of their strong interaction as demonstrated below). — Mathematically, one defines a space, called isospin space, in which the particles correspond to specific states. Proton and neutron form a doublet of states, distinguished by the eigenvalue of an Hermitian operator. Defining, e.g. the proton and the neutron isospin states as

116

11. Nuclear forces

|pi =

µ ¶ 1 , 0

|ni =

µ ¶ 0 , 1

(11.11)

together with the isospin operator µ ¶ 1 1 0 1 τ3 = = σ3 , 2 0 −1 2

(11.12)

the proton has isospin +1/2, and the neutron has isospin −1/2. The operator σ 3 is a Pauli matrix, as used in the formalism to describe the internal angular momentum of a spin 1/2 particle. In fact, isospin is defined in full analogy to spin, with operators fulfilling the same commutation rules (as eq.(2.5), without ~). For isospin 1/2, the operators are given by µ ¶ µ ¶ 1 01 1 0 −i 1 1 τ1 = τ2 = (11.13) = σ1, = σ2, 2 10 2 2 i 0 2 and τ 3 as defined before, fulfilling [τ i , τ j ] = τ 1 τ 2 − τ 2 τ 1 = i

ijk τ k .

(11.14)

The ladder operators µ ¶ µ ¶ 01 00 , τ− = τ+ = 00 10

(11.15)

may convert a neutron into a proton and vice versa: τ + |ni = |pi ,

τ + |pi = |0i ,

τ − |ni = |0i ,

τ − |pi = |ni .

(11.16)

— Note that, although there is a formal analogy to spin, isospin is to be understood as an additional property of a strongly interacting particle. The nucleon as an isospin 1/2 particle by chance also has spin 1/2, although spin has nothing to do with isospin (apart from its formal resemblance). — The analogy can be pushed further - it enables us to describe a system of several nucleons without refering to its electric charge (the mathematics is the same as for the coupling of several spins to a total spin). The isospin operator T=

A X

τ (k)

(11.17)

k=1

defines multiplets characterised by the eigenvalue of T2 (these are given by T (T + 1), following the analogy with angular momentum). The number of neutrons and protons of a nucleus with A nucleons are then given by its third component, according to T3 =

A X

k=1

τ 3 (k) =

1 (Z − N ) . 2

(11.18)

• The isospin formalism provides a theoretical frame to test the following conjecture: — The strong force is invariant under rotations in isospin space. In other words: the strong interaction does not distinguish between different states of the same isospin multiplet.

11.3 Nucleon-nucleon potential

117

— Tests of this conjecture are possible with nuclear states (see fig. 11.2), as well as in NN collisions. • So far, all observations support the hypothesis that isospin is a symmetry of the strong interaction, meaning that — isospin stays conserved in reactions governed by the strong force, — for states of particles bound by the strong interaction isospin is good quantum number.

Fig. 11.2. Left: the mirror nuclei 11 B and 11 C. (a) schematics of the nucleon configuration, (b) level schemes of both nuclei: for better comparison, the states of 11 C were shifted by ∆EC such that the ground state energies coincide. Isospin doublets are marked by dashed lines [T. Mayer-Kuckuk, Kernphysik, Teubner]. Right: Isospin triplets in the three isobars 14 C, 14 N and 14 O. The ground state of 14 N was chosen to define the energy scale. Most of the states in 14 N are isospin singlets, i.e. without analog states in the neighbor nuclei [B. Povh et al., Particles and Nuclei, Springer, 2002].

11.3 Nucleon-nucleon potential • The interaction of two nucleons at low energy can be described by a potential of the form £ ¤ (11.19) V (r) = Vcentral (r) + Vss (r) s1 · s2 + VT (r) 3 (s1 · r) (s2 · r) /r2 − s1 s2 +VLS (r) (s1 + s2 ) · L + VLs (r) (s1 · L) (s2 · L) + Vps (r) (s1 · p) (s2 · p) r denotes the relative distance of the nucleons with spins s1 and s2 , p is the relative momentum and L is the total orbital angular momentum. The potential neglects the inner structure of the nucleons and is therefore valid only for bound states and for low-energy NN scattering, from which its form can be derived. Interpretation of the different terms: 1. a central potential. 2. a central spin-spin interaction. 3. the (non-central) tensor potential. It has the same spin dependence as the magnetic interaction of two magnetic moments. This term is the only one which may lead to a mixing of different orbital angular momentum in the physical state. 4. LS coupling, similar to atomic physics, but caused by the strong force.

118

11. Nuclear forces

5. and 6. are terms which respect basic symmetries but may often be neglected due their quadratic dependence on p. • To describe nuclei, additional many-body forces have to be taken into account, leading to a still higher level of sophistication.

Part II

Particle physics

12. Symmetries and conservation laws

• Symmetries play a particular role in particle physics. — They restrict possible forms of an interaction between particles → gauge symmetries. — Symmetries of a physical system lead to conserved quantities and may be used to classify particles and particle states, which appear as members of multiplets of symmetry groups. — Some symmetries seem to be exactly fulfilled in nature, like conservation of the electric charge, others are found to be broken, like the invariance of a system under mirror reflection (parity). • If a physical law is invariant under a particular symmetry operation, there exists a corresponding conservation law. Examples known from classical mechanics: — Invariance of the Hamilton function with respect to translations in time ⇒ energy conservation. — Invariance of the Hamilton function with respect to rotations in space ⇒ conservation of angular momentum. — Invariance of the Hamilton function with respect to translations in space ⇒ momentum conservation.

12.1 Conserved quantities and symmetry Let H be a time-independent Hamilton operator, and ψ a state fulfilling the Schrödinger equation i~

dψ = Hψ. dt

(12.1)

The value of an observable F in this state is given by the expectation value hF i = hψ| F |ψi. We assume F having no explicit time dependence and ask: • Under which condition is hF i time-independent? Z Z Z dψ ∗ dψ d d hF i = d3 r ψ ∗ F ψ = d3 r F ψ + d3 r ψ ∗ F . dt dt dt dt

(12.2)

The complex conjugate of the Schrödinger equation is given by −i~

dψ ∗ ∗ = (Hψ) = ψ ∗ H, dt

and therefore

(12.3)

122

12. Symmetries and conservation laws

i d hF i = dt ~

Z

d3 r ψ ∗ (HF − F H) ψ.

(12.4)

Hence, [H, F ] = HF − F H = 0

⇒

d hF i = 0. dt

(12.5)

— hF i therefore is a constant of motion if the commutator of F with the Hamiltonian vanishes. — The eigenfunctions of H may be selected such that they are simultaneously eigenfunctions of F : Hψ = Eψ,

F ψ = f ψ.

(12.6)

The eigenvalues of F are then called good quantum numbers. • How to find conservation laws? — A usual situation: H is not fully known (and it may be just the aim to get full knowledge of it, the NN interaction was an example of such an incompletely known interaction), and we cannot simply test the commutator with any observable. However, — If H is invariant under a symmetry operation, we may find a conserved quantity. — Symmetry operation: let U be a time-independent operator transforming a physical state ψ according to ψ 0 (r, t) = U ψ (r, t) .

(12.7)

U is taken as unitary in order to maintain the normalisation of the wavefunction, Z Z Z d3 r ψ 0∗ ψ 0 = d3 r (U ψ)∗ U ψ = d3 r ψ ∗ U + U ψ ⇒ U + U = 1.

(12.8)

U is called a symmetry operator if U ψ fulfills the same Schrödinger equation as ψ: i~

d (U ψ) = HU ψ. dt

(12.9)

With U + U = U −1 U = 1, it follows that H = U −1 HU

⇒

[H, U ] = 0.

(12.10)

If U is Hermitian (i.e. U + = U ), it is an observable. If not, but the symmetry operation is a continuous transformation, one may construct an Hermitian operator from it: — Continuous transformations are caracterised by one or several parameters (e.g. an angle α in case of a rotation) which can be varied continuously. They can be made such small that the symmetry operator will approach unity. We consider an infinitesimal operation with parameter , U ψ = (1 + i F ) ψ.

(12.11)

Any continuous transformation can be imagined as resulting from many infinitesimal operations: ³ α ´n exp (iαF ) = lim 1 + i F . (12.12) n→∞ n

12.2 Conservation of electric charge

123

Hence, a continuous operator (for a one-dimensional transformation) can be written as U = exp (iαF ) ,

(12.13)

where F is called the generator of U , and α is real. The operation on ψ is defined by à ! (iαF )2 U ψ = 1 + iαF + + ... ψ. (12.14) 2 Unitarity of U requires ¡ ¢ 1 − i F + (1 + i F ) = 1

⇒

F + = F.

(12.15)

Hence, the generator F is Hermitian, i.e. an observable. It is also a constant of motion, since (compare with eq.(12.5)) [H, U ] = H (1 + i F ) − (1 + i F ) H = 0

⇒

[H, F ] = 0.

(12.16)

— First example: generator of translations. dψ (r) ψ = ψ (r + δr) = ψ (r) + δr = dr 0

µ

d 1 + δr dr

¶

ψ (r) .

(12.17)

Since p=

~ d , i dr

(12.18)

we see that the momentum operator p (divided by ~) is the generator of translations. — Second example: generator of a rotation about the z axis. µ ¶ dψ (ϕ) d 0 = 1 + δϕ ψ (ϕ) . ψ = ψ (ϕ + δϕ) = ψ (ϕ) + δϕ dϕ dϕ

(12.19)

Since ~ Jz = (r × p)z = i

µ ¶ d d ~ d x −y = , dy dx i dϕ

(12.20)

we see that the angular momentum operator Jz (divided by ~) generates a rotation about the z axis.

12.2 Conservation of electric charge • Electric charge appears to be conserved in any process: X X qi = qf .

(12.21)

The sum of electric charge in the initial and final state of a reaction are the same.

• Search for a violation of charge conservation: look for a process which would be allowed by all other known conservation laws.

124

12. Symmetries and conservation laws

— Example: electron decay, e → ν + γ.

(12.22)

Decay of an electron in an atom would leave a hole in the electron shell, giving rise to secondary processes (emission of X-rays or Auger electrons). Experimental lower limit of the electron mean lifetime: τ e > 4.2 × 1024 a.

(12.23)

• Any particle has a well-defined charge, being an integer multiple of an elementary charge e (fractional integer only for quarks, see later), q = N e,

(12.24)

and N is found to fulfill an additive conservation law, e.g. for a reaction a + b → c + d + e,

Na + Nb = Nc + Nd + Ne .

(12.25)

• Following the reasoning in the previous section, conservation of electric charge requires invariance of the Hamiltonian under a symmetry transformation: — Let ψ q denote a state with charge q fulfilling the Schrödinger equation, i~

dψ q = Hψ q , dt

(12.26)

and Q the operator of electric charge. Then, if hQi stays conserved (which is an experimental observation), [H, Q] = 0,

(12.27)

ψ q can be chosen as eigenfunction of Q, Qψ q = qψ q ,

(12.28)

with q being a conserved quantity. The corresponding symmetry transformation is a global phase transformation, ψ 0q = exp (iαQ) ψ q ,

(12.29)

where α is a real parameter (compare eq.(12.13) and the following derivation).

12.3 Gauge symmetry The dynamics of a particle system, i.e. the form of the interactions between fundamental constituents, is intimately related to a symmetry principle, called gauge symmetry. This symmetry is a generalisation of the global phase transformation encountered before to local ones. • Gauge invariance in the Maxwell equations: the classical fields E and B are connected to a vector potential A and a scalar potential V by B = ∇ × A,

(12.30)

12.3 Gauge symmetry

125

∂A . (12.31) ∂t These fields are not uniquely defined: the magnetic field stays the same under the transformation E = −∇V −

A → A0 = A + ∇χ,

(12.32)

where χ is an arbitrary function, since the rotation of a gradient vanishes. Also E stays the same if simultaneously V →V0 =V −

∂χ . ∂t

(12.33)

These transformations are called gauge transformations - they express a certain freedom to define the potentials from which the physical electromagnetic fields are derived. The potentials can be compactely combined to a four vector, Aµ = (V, A) , and the gauge transformations can then be written as µ ¶ ∂ µ 0µ µ µ µ ∂ := A → A = A − ∂ χ, , −∇ , ∂t

(12.34)

(12.35)

with a scalar function χ = χ (r, t) .

(12.36)

• Gauge invariance in quantum mechanics: — The Hamilton function describing the electromagnetic force between charged particles is given by H=

1 2 (p − qA) + qV, 2m

(12.37)

since Hamilton’s equations with this form of H leads to the Lorentz force. From the correspondence principle one obtains the — The Schrödinger equation, ∙ ¸ 1 ∂ 2 (−i}∇ − qA) + qV ψ (r, t) = i} ψ (r, t) , 2m ∂t

(12.38)

describes the state of a particle moving in the potentials Aµ . — What is the influence of a gauge transformation of eq.(12.35) on the state of the particle? How does the solution ψ 0 (r, t) of the modified Schrödinger equation, ∙ ¸ 1 ∂ 0 2 0 (12.39) (−i}∇ − qA ) + qV ψ 0 (r, t) = i} ψ 0 (r, t) , 2m ∂t look like? Answer : ³ q ´ ψ 0 (r, t) = exp i χ (r, t) ψ (r, t) . }

Check :

(12.40)

126

12. Symmetries and conservation laws

³ q ´ (−i}∇ − qA0 ) ψ 0 = (−i}∇ − qA − q∇χ) exp i χ ψ } ³ q ´ ³ q ´ ³ q ´ ³ q ´ = q∇χ exp i χ ψ − exp i χ i}∇ψ − qA exp i χ ψ − q∇χ exp i χ ψ, (12.41) } } } } and hence ³ q ´ (12.42) (−i}∇ − qA0 ) ψ 0 = exp i χ (−i}∇ − qA) ψ. }

Similarly, one finds ¶ µ ¶ ³ q ´µ ∂ ∂ i} − qV 0 ψ 0 = exp i χ i} − qV ψ. ∂t } ∂t

(12.43)

To simplify notation, one defines q D = ∇ − i A, }

D0 =

∂ q + i V, ∂t }

(12.44)

which can be combined to the so-called covariant derivative q Dµ = ∂ µ + i Aµ . } We see that Dµ ψ transforms as ψ, ³ q ´ D0µ ψ 0 = exp i χ Dµ ψ, }

(12.45)

(12.46)

and therefore the same holds also for the higher derivatives. We thus have ³ q ´¡ ³ q ´ ¢ 1 1 2 (−i}D0 ) ψ 0 = exp i χ (−i}D)2 ψ = exp i χ i}D0 ψ = i}D00 ψ 0 , (12.47) 2m 2m } }

which is just the Schrödinger equation after the gauge transformation. The wavefunctions ψ 0 and ψ describe the same physics.

— Summary and comments: the gauge invariance of the Maxwell equations remain an invariance in quantum mechanics provided we simulataneously transform the potentials and the wavefunctions as ³ q ´ Aµ → A0µ = Aµ − ∂ µ χ, ψ → ψ 0 = exp i χ ψ. (12.48) } Any wave equation involving the operator ∂ µ can be made gauge invariant by the prescription q ∂ µ → Dµ = ∂ µ + i Aµ . }

(12.49)

The wave equation of a free particle thus becomes converted into a wave equation of a particle interacting with electromagnetic fields. • The gauge principle inverts the argument: — We require that our theory (the wave equation) shall be invariant under the space-time dependent transformation (local phase transformation) ³ q ´ (12.50) ψ (r, t) → ψ 0 (r, t) = exp i χ (r, t) ψ (r, t) . }

12.4 Particles and anti-particles

127

This is not possible for the Schrödinger equation of a free particle or any other wave equation for particles without interactions. Moreover, this prescription dictates the form of the interactions. The free Schrödinger equation, 1 ∂ 2 (−i}∇) ψ = i} ψ, 2m ∂t must be modified to 1 2 (−i}∇ − qA) ψ = 2m

µ ¶ ∂ i} − iqV ψ, ∂t

(12.51)

(12.52)

with the already stated gauge transformations for the potentials (called gauge field Aµ ) in order to satisfy the local phase transformation. The physical fields are contained in the field strength tensor F µν = ∂ µ Aν − ∂ ν Aµ ,

(12.53)

from which one can construct the Hamiltonian (Lagrangean). With the Euler Lagrange equations one may thus derive the Maxwell equations. — Interpretation: we cannot distinguish between the effect of a local change in the phase convention and the effect of a new vector field. The new vector field appears as a consequence of the local phase invariance prescription, i.e. the invariance of the wavefunction under a unitary transformation belonging to the symmetry group U (1). A vector field corresponds to a spin-1 particle, the photon. • Other interactions than electromagnetic ones are governed by different prescriptions of gauge invariance, built on different symmetry groups. Example: the strong interaction between quarks. This interaction involves three different charges, called colour, wherefore the it is called quantum cromodynamics. The form of the interaction is derived from the prescription that it shall be invariant under the transformation ⎛ 0⎞ ⎛ ⎞ a1 a1 ⎝ a02 ⎠ = exp (iα (r, t) · λ) ⎝ a2 ⎠ , (12.54) a03 a3

which locally rotates a state with three components representing the three different quark colours. The eight 3 × 3 matrices λi are the generators of the group SU (3) (generalising the Pauli matrices of SU (2)), and α = (α1 , α2 , ...α8 ) are the eight rotation parameters. The invariance is obtained with the prescription that the derivatives in the quark’s wave equation be replaced by Dµ = ∂ µ + igλ · Gµ .

(12.55)

introducing eight vector fields (i.e. corresponding to spin-1 particles), called gluons.

12.4 Particles and anti-particles • The relativistic equation for the energy of a particle, 2

E 2 = |p| c2 + m2 c4 ,

(12.56)

has the two solutions q E± = ± |p|2 c2 + m2 c4 .

(12.57)

128

12. Symmetries and conservation laws

• In relativistic quantum mechanics, one cannot simply neglect wavefunctions with negative energy. Any observable has a complete set of eigenfunctions, and it can be shown that the states with positive energy alone donot form a complete set. • Interpretation of states with negative energy (Stueckelberg, Feynman): — Consider a particle moving along the x axis with positive (3-)momentum p and positive energy E+ . Its wavefunction is a plane wave ψ (x, t) = exp (i (px − E+ t) /~) ,

E+ > 0.

(12.58)

This particle moves in positive direction, since the phase stays constant for px − E+ t = const.

(12.59)

— The state of a particle with negative energy is ψ (x, t) = exp (i (px − E− t) /~) ,

E− < 0,

(12.60)

which can be interpreted as a particle with positive energy |E− | moving backwards in time, since the phase stays constant for px + |E− | t = px − |E− | (−t) = const.

(12.61)

— A particle with charge q moving backwards in time fulfills the same equation of motion as a particle with charge −q moving forwards in time. This can be shown exemplary, considering the Lorentz force on a particle with charge −q: m

d2 r dr dr = −q ×B=q × B. 2 dt dt d (−t)

(12.62)

— ⇒ a particle with charge q and negative energy behaves as a particle with charge −q and positive energy. It is called anti-particle. Negative energy wavefunctions can thus be replaced by anti-particles with positive energy. • Particles and anti-particles have opposite additive quantum numbers (which all can be considered as charges).

12.5 Baryon number conservation • Conservation of electric charge is not sufficient to guarantee stability of a particle, e.g. one might expect proton decay according to p → e+ + γ,

(12.63)

which was never observed. The experimental upper limit (dependent of the decay mode) is in the range τ p > 1031 to 1033 years.

(12.64)

One therefore introduces another quantum number, called baryon number A, which, like charge, appears to be a conserved quantity. One attributes

12.6 Leptons and lepton number conservation

129

— Proton and neutron have A = 1, — Antiproton and antineutron have A = −1, — Leptons, photons and mesons have A = 0. Baryon number conservation reads X Ai = const.

(12.65)

• There are other strongly interacting particles than nucleons which carry non-vanishing baryon number. E.g., the hyperons Λ0 , Σ + and Σ − are strange particles (the sense of "strange" will be explained later) which decay (among other possibilities) according to Λ0 → n + π 0 , +

0

−

−

Σ →p+π ,

Σ →n+π .

Λ0 → p + π − , +

0

(12.66) +

Σ → Λ + e + ν,

(12.67) (12.68) 0

From baryon number conservation it follows that Λ , Σ

+

and Σ

−

are baryons with A = 1.

12.6 Leptons and lepton number conservation • Leptons are spin 1/2 particles without strong interactions (thus distinct from quarks and quark composites, the hadrons). There are three leptons with charge −e (e− , µ− , τ − ), and three uncharged ones (ν e , ν µ , ν τ ). For each lepton exists an antiparticle (e+ , µ+ , τ + , ν e , ν µ , ν τ ). They may be put into a scheme of three families of particles with analog properties, apart mass: µ −¶ µ −¶ µ −¶ e µ τ ν ν µ ¶ τ ¶ µ +e ¶ µ + µ ν+ (12.69) e µ τ νe νµ ντ — The lepton masses are me = 0.511 MeV,

m (ν e ) < 2.5 eV,

(12.70)

mµ = 105.7 MeV,

m (ν µ ) < 170 keV,

(12.71)

mτ = 1771.1 MeV,

m (ν τ ) < 18 MeV.

(12.72)

• Lepton number L is defined as — L = 1 for the six known leptons, — L = −1 for the corresponding antiparticles. Similar to the baryon number discussed before, also the lepton number L is a conserved quantity, manifest as absence of reactions which otherwise would be allowed. As a result, like baryons, the leptons can be produced or annihilated only in pairs, like the pair creation processes γ → e− + e+ ,

γ → p + p,

(12.73)

which may happen for high energy photons in the Coulomb force field of an atom (which recoils to fulfill momentum conservation).

130

12. Symmetries and conservation laws

— The charged leptons fulfill separate conservation laws for each family, i.e. Le , Lµ , Lτ are separately conserved. This means that, e.g., the lepton number violating process µ− → e− + γ

(12.74)

was never found. Its branching ratio has a very small experimental limit: Γ (µ− → e− + γ) < 5 × 10−11 . Γ (µ− → all channels)

(12.75)

— Recently, neutrino oscillations were found, transforming one sort of neutrinos into another sort. This means that for these neutral leptons there is a violation of lepton family conservation. However, the total lepton number conservation as defined above is still valid in any process detected so far. • First direct demonstration of the (anti)neutrino (Cowan and Reines 1959): — Setup: a detector of an aquous solution of CdCl2 + scintillator, located close to a nuclear reactor and carefully shielded against ionising radiation. — Antineutrinos from β − decays in the reactor induce the reaction ν e + p → e+ + n.

(12.76)

— The e+ is slowed down in the detector and annihilates with an electron to produce two γ’s with 511 keV. This happens within typically 10−9 seconds. The neutron diffuses for typically a few µs before being captured by the cadmium, which produces a cascade of high energy gamma-rays. Delayed coincidence of these two events provides a clear signature to discriminate the process against background. — From the known flux of the reactor antineutrinos (∼ 1013 cm−2 s−1 ), they deduced from the rate of observed events an energy averaged cross section of σ = (11 ± 2.6) × 10−44 cm2 .

(12.77)

This extremely small value corresponds to a free path in water of 1018 m, i.e. 100 light years! • Neutrino and the antineutrino are different particles, ν e 6= ν e : a crucial experimental test of the validity of the lepton number assignment of the neutral leptons was performed by Davis in 1964: — Neutrinos were first observed indirectly from the recoil kinematics of the electron capture reaction (→ exercise) 37

A + e− →

37

Cl + ν e + 0.8 MeV .

(12.78)

— Davis tested whether an inverse reaction could be induced using the antineutrinos from a reactor: νe +

37

Cl →

37

A + e−

?

(12.79)

— Experiment: a detector of 4 m3 of C2 Cl4 was located close to a reactor. After several weeks, the eventually produced radioactive 37 A was extracted from the detector (washed out with 4 He gas + a cold trap) and put into a well shielded detector ¢ to observe the radiactive decay ¡ eq.(12.78) to count the produced argon atoms (T1/2 37 A = 34 days).

12.6 Leptons and lepton number conservation

131

from reactor liquid scintillator

photom ultiplier

target

gam m a radiation after neutron capture in cadm ium

annihilation quanta

Fig. 12.1. The experiment of Cowan and Reines which first demonstrated directly the neutrino.

— Result: cross section deduced from the experiment: σ < 0.9 × 10−45 cm2 .

(12.80)

Theoretical expectation, if ν e = ν e : σ = 2.6 × 10−45 cm2 .

(12.81)

Interpretation: ⇒

ν e 6= ν e .

(12.82)

• First detection of solar neutrinos (Davis 1968): — Neutrinos with energy larger than 0.8 MeV may induce the reaction (see eq.(12.78)) νe +

37

Cl →

37

A + e− .

(12.83)

Davis used this to detect a flux of neutrinos from the sun, which is much smaller than the antineutrino flux from a reactor. — The detection principle is the same as before. Special care must be taken to shield the setup against the reaction µ+ +

37

Cl →

37

A + νµ,

(12.84)

by installing the detector 1500 m under earth. There, the rate of neutrino events was expected to be 10 times higher than the rate of the reaction induced by muons generated by cosmic radiation in the atmosphere. — Due to the large Q value of the reaction (0.8 MeV), the experiment was sensitive only to the high energy part of the solar neutrino spectrum (neutrinos from decay of 8 B), and hence only to a small fraction of the total neutrino flux.

132

12. Symmetries and conservation laws

• Muon neutrinos are different from electron neutrinos, ν µ 6= ν e : this was demonstrated by Danby et al. in 1962: — A proton beam hitting a beryllium target was used to generate pions, which subsequently decay to muons and µ neutrinos, π − → µ− + ν µ .

π + → µ+ + ν µ ,

(12.85)

— A detector for electrons and muons, able to distinguish these two leptons, was shielded against the muons from pion decay. — Detected processes (34 events collected during 100 hours in the first experiment) ν + n → p + µ− ,

ν + p → n + µ+ .

(12.86)

ν + p → n + e+ .

(12.87)

Not observed : ν + n → p + e− , — Interpretation: ν µ 6= ν e .

p (15 GeV)

(12.88)

Be steel

π±

N

concrete

π + → µ + +ν µ π − → µ − +ν µ

paraffine

detector, able to distinguish e from µ

Fig. 12.2. Experiment of Danby et al. (1962) which demonstrated that muon neutrinos and electron neutrinos are different particles.

12.7 Hypercharge (strangeness) • Observation: in some processes strange particles occur which behave as follows: — they are created with a cross section typical for a strong interaction (mbarn),

12.7 Hypercharge (strangeness)

133

— subsequent decays are slow with decay times typical for weak interaction (order 10−10 s). — they are always created in pairs. — Example: creation of Λ0 and K0 in p + π − → Λ0 + K0 ,

(12.89)

subsequent decay via Λ0 → p + π − ,

K0 → π+ + π − .

(12.90)

• Explanation (Pais, Gell-Mann, Nishijima): introduction of a new, additive quantum number called strangeness S with following properties: — nucleons and pions have strangeness S = 0. — Strong and electromagnetic interactions conserve strangeness: X Si = const.

(12.91)

i

This explains the creation of strange particles in pairs, which then need to have opposite strangeness. In the example, the K0 is attributed strangeness S = 1, the Λ0 then has opposite strangeness, ¡ ¢ ¡ ¢ S K0 = 1, S Λ0 = −1. (12.92)

— The weak interaction does not conserve strangeness (it hence breaks this symmetry of the strong and electromagnetic interactions). The decays of strange particles into "ordinary", non-strange particles therefore have to be weak, wherefore their lifetimes are comparatively long. • Further strangeness assignments. For a few particles, one has to define their strangeness, for all other particles is then can be deduced from particle reactions, respectively tested if additional reactions are considered. — With the definition ¡ ¢ S K+ = 1,

(12.93)

we obtain ¡ ¢ S K− = −1

(12.94)

from the reaction

p + π − → n + K+ + K− . +

−

K and K

(12.95)

form a pair of particle and anti-particle.

— Kaons are well suited to determine the strangeness of many other particles: ¡ ¢ p + π − → Σ − + K+ ⇒ S Σ − = −1, ¡ ¢ p + K− → Σ + + π − ⇒ S Σ + = −1. +

−

(12.96) (12.97)

The baryons Σ and Σ thus have the same strangeness. They donot form a particle antiparticle pair, since not all additive quantum numbers are inverted (despite their opposite electric charge they have same baryon number and strangeness).

134

12. Symmetries and conservation laws

— There are particles with strangeness larger than one, e.g. ¡ ¢ p + K− → Ξ − + K+ ⇒ S Ξ − = −2, d + K+ → Ω + Λ0 + Λ0 + p + π − + π+

⇒

¡ ¢ S Ω = 3.

(12.98) (12.99)

12.8 Order in the hadron zoo There are so many hadrons that already from their sheer number it is not reasonable to assume them to be fundamental. Similar to Mendeleevs periodic system of the elements one may try to order the zoo of hadrons and understand their properties as emerging from their composition of basic constituents. • There are various symmetries observed in the properties of the hadrons. Examples: 1. There are the two nucleons of approximately equal mass, mp = 938, 27200 (4) MeV,

mn = 939.56533 (4) MeV

(12.100)

We have already said that proton and neutron therefore may be considered as two states of the nucleon, a single particle appearing as an isospin dublet, the different mass being due to the electromagnetic interaction. 2. There are three pions with similar masses, π+ , π 0 and π − , ¡ ¢ ¡ ¢ m π0 = 134, 9766 (6) MeV, m π± = 139.57018 (35) MeV

(12.101)

These particles may be considered as an isospin triplet, again the mass difference of its members being due to electromagnetic effects.

3. Including strange particles, the pions can be arranged in a more general multiplet of particles with spin 0 and parity −1, the so-called pseudoscalar mesons.

Fig. 12.3. The multiplet of pseudoscalar mesons, J P = 0− [B. Povh et al., Particles and Nuclei, Springer, 2002].

12.8 Order in the hadron zoo

135

Fig. 12.4. Multiplet of vector mesons, J P = 1− [B. Povh et al., Particles and Nuclei, Springer, 2002].

4. Similarly, there exist particle multiplets for the vector mesons, 5. Also the baryons with spin 1/2 and 3/2 can be arranged in particle multiplets. They are shown in fig.12.5. The regularity of the pattern and the mass differences in vertical direction

-1

0

1

-3/2 -1 -1/2 0 1/2 1 3/2

Iz

Fig. 12.5. The baryon octet and decuplet. The mass values given for the decuplet are mean values for the particles in the corresponding line.

in the diagram of the baryon decuplet led to the prediction of the Ω − and its mass. • The observed patterns of particle states given above can elegantly be explained by introducing basic constituents with the following properties: — three different constituents are required, called u-, d- and s-quark. — They have spin 1/2. — They have fractional electric charges, q (u) = 2e/3,

q (d) = −e/3,

— Strangeness assignments:

q (s) = −e/3.

(12.102)

136

12. Symmetries and conservation laws

S (u) = 0,

S (d) = 0,

S (s) = −1.

(12.103)

• Interpretation of the hadrons in the quark picture: — Mesons are bound states of a quark and an anti-quark, qq, — Baryons are bound states of three quarks, qqq. — The multiplets stated above represent the lowest energy bound states for given spin values. We may thus expect them to be states without orbital angular momentum. The hadron spin is then generated by combination of the quark spins, e.g. anti-parallel spins of the quarkantiquark pair in the pseudoscalar mesons, and all three quark spins parallel in the baryon decuplet.

12.9 Problem with the Pauli principle - quark colour • The resonance ∆++ has spin 3/2 and positive parity. The orbital angular momentum should be = 0. The three u-quarks must then have their spins parallel: ¯ ® (12.104) ∆++ = ¯u↑ u↑ u↑ The wavefunction therefore appears to be symmetric under exchange of two quarks. However, these are fermions, and we should expect the total wavefunction to be anti-symmetric to fulfil the Pauli principle.

• This apparent problem is solved by introducing a new degree of freedom to the quarks - the colour charge, or simply colour. Prescriptions: — Each quark carries one of three different colours, called red, green and blue. — Anti-quark carry anti-colours, called anti-red, anti-green and anti-blue. — For the three quarks bound in a baryon, one may construct a wavefunction which is antisymmetric with respect to colour. The Pauli principle may thus be fulfilled. Also the non-observation of free quarks in nature gets explained that way: • The strong forces are structured such that only colourless objects can be observed as free particles. The simplest colourless objects are — a quark of a certain colour and an anti-quark with the corresponding anti-colour, — a combination of three quarks (or three anti-quarks) of all three colours. One recognizes these possibilities as the mesons and the baryons. • There are other experimental signs of the colour degree of freedom of quarks.

12.10 Parity and its non-conservation in weak interactions 12.10.1 The parity operation • The operation of parity (mirror reflection) changes the sign of an ordinary (polar) vector:

12.10 Parity and its non-conservation in weak interactions

P:

r → −r,

p → −p.

137

(12.105)

It does not change the sign of an axial-vector, like angular momentum, J → J,

(12.106)

as can be seen from the definition of orbital angular momentum, L = r × p, which is constructed from two polar vectors. • A Schrödinger wavefunction is transformed according to Pψ (r) = ψ (−r) .

(12.107)

Hence, P 2 ψ (r) = Pψ (−r) = ψ (r) .

(12.108)

The parity operator thus fulfills P 2 = 1.

(12.109)

and is unitary and Hermitian, i.e. observable. • If parity is a symmetry operation, i.e. if [H, P] = 0,

(12.110)

then Hψ = Eψ

⇒

HPψ = EPψ.

(12.111)

We may distinguish two cases: 1. If ψ is degenerate with respect to E, then the wavefunctions ψ and Pψ may just be two different states with the same energy. 2. If ψ is not degenerate, then Pψ is proportional to ψ, Pψ = P ψ,

(12.112)

and the eigenvalues P of the parity operator P are P = ±1.

(12.113)

12.10.2 Parity in particle reactions • Consider a particle reaction, e.g. a + b → c + d.

(12.114)

— The initial state can be written symbolically as |initial statei = |ai |bi |relative motioni .

(12.115)

Parity acts on each part of the total wavefunction, P |initial statei = P |ai P |bi P |relative motioni .

(12.116)

138

12. Symmetries and conservation laws

— Hence, conservation of parity is an example of a multiplicative conservation law. — P |ai and P |bi refer to inner wavefunctions of the particles a and b. Their parities are well defined, because particle structure is mainly determined by parity conserving interactions. — P |relative motioni = (−1) , where

is the quantum number of orbital angular momentum.

— If parity is conserved in the reaction (i.e. if it is strong or electromagnetic), then 0

P |ai P |bi (−1) = P |ci P |di (−1) .

(12.117)

• Negative parity of the pion may be derived from a suited nuclear reaction. — Deuteron dissociation by pion capture: d + π − → n + n.

(12.118)

Experimentally, a pion is first slowed down in deuterium and replaces the electron in a 2 H atom, thus forming a pionic atom. The pion rapidly rapidly cascades down the levels in the Coulomb potential of the deuteron. Capture of the pion finally takes place for the pion in the lowest state in the Coulomb potential, i.e. an S-state, for which its overlap with the deuteron is maximal. — Parity conservation requires 0

0

Pd Pπ− (−1) = Pn Pn (−1) = (−1) .

(12.119)

— Initial state: for the deuteron we have Pd = Pp Pn ,

(12.120)

since orbital angular momentum of the two nucleons is even (they form S- and D-waves in the deuteron). Further, = 0,

(12.121)

because the π − is captured by the deuteron from the S-shell of a pionic atom. Therefore, 0

Pp Pn Pπ− = (−1) .

(12.122)

— The final state contains two identical fermions, whose wavefunction has to be odd under particle exchange (Pauli principle). Therefore, total spin of n + n is zero ⇒ odd spin wavefunction ⇒ total spin of n + n is one ⇒ even spin wavefunction ⇒

0 0

is even, is odd.

(12.123)

Furthermore, since the pion spin is zero (derived from another reaction) and the deuteron has spin 1 (see section 11.1), the total angular momentum is J =1

⇒

0

= 1.

(12.124)

Hence, in total Pp Pn Pπ− = −1.

(12.125)

12.10 Parity and its non-conservation in weak interactions

139

— This can be solved by Pp = Pn = 1, Pπ− = −1 Pp = Pπ− = 1, Pn = −1

(12.126) (12.127)

which are equivalent experimentally (note that one cannot measure a relative parity between proton and neutron). One defines: Pp = Pn = 1,

(12.128)

from which follows the negative parity of the pion, Pπ− = −1.

(12.129)

12.10.3 Parity violation in nuclear beta decay - the Wu experiment • The hypothesis, that parity might not be conserved in weak interactions, is due to Lee and Yang (1956): — in 1955, two particles with a strange behaviour were discovered: called θ and τ , they appeared to have exactly the same properties apart from their decay. One decays into two pions (parity P = +1) and the other into three pions (P = −1). — Lee an Yang pointed out that there was no experimental prove that parity is conserved in weak interactions. They suggest that θ and τ are identical, and they propose additional experiments to test their hypothesis of parity non-conservation. • Experimental demonstration of parity non-conservation by Wu and co-workers (1957): — Measurement of the β asymmetry in the decay of polarised 60

Co →

60

Ni + e− + ν e .

60

Co nuclei: (12.130)

The term "β asymmetry" means that β particles emitted in a beta-decay (e− in β − or e+ in β + decay, see section 6.5.1) appear at different rates when measured parallel, respectively, anti-parallel to the axis of polarisation of the parent nucleus (here 60 Co). An experimental asymmetry can generally be defined as N+ − N− , N+ + N−

(12.131)

where N+ and N− are particle rates parallel, respectively, anti-parallel to a particular axis, which here is the axis of nuclear polarisation. — Observation of a β asymmetry is a manifestation of parity violation, because the observable is a pseudo-scalar quantity, i.e. changing its sign under the parity operation: P hJ · pi = − hJ · pi .

(12.132)

If parity were conserved in the decay, the expectation value hJ · pi has to vanish. The experiment by Wu et al. showed that this is not true (see fig. 12.6), the electrons are preferentially emitted anti-parallel to the nuclear polarisation of 60 Co.

140

12. Symmetries and conservation laws

— In the experiment, the nuclei are polarised by a strong magnetic field at low temperature (Boltzmann equilibrium, see section 8.3). The radioactive 60 Co atoms are introduced into a single crystal of alaun. The orientation of the electronic shell of the cobalt creates a strong magnetic field at the nuclear site already in a rather modest external field. The nuclear orientation is detected through, with respect to the nuclear orientation axis, anisotropic (but not asymmetric) intensity of the γ rays emitted after the β decay from the 60 Ni nuclei. In fig. 12.6 only a single of two NaI scintillation detectors is shown, the second one is located at the position of the reader’s eye, i.e. perpendicular to the magnetic field. After cooling down the sample by adiabatic demagnetisation, the γ anisotropy was 30%. It was observed to be independent from the magnetic field direction. The β intensity, on the other hand was dependent on the direction of nuclear orientation. As shown on the left side of fig. 12.6, immediately after cooling down the sample, the intensity of β particles was different from that for unoriented nuclei, which one obtains after a few minutes of warming up the sample. In contrast to the γ emission, the intensity changes its sign, when the experiment is done with oppositely oriented nuclei (inverted magnetid field): an excess becomes a lack of intensity, which demonstrates parity violation. — Reason of the observed β asymmetry: The 60 Co nucleus with spin 5~ decays to 60 Ni with spin 4~. Since the decay of 60 Co is allowed, no orbital angular momentum is required, and the difference ∆I = 1 is provided by the spins of the ν e and the electron, which couple to I = 1 for a Gamow-Teller transition. The observed β asymmetry can be understood if one assumes that the e− is preferentially emitted with negative helicity (see left side of fig. 12.6). In section 12.10.4 an experiment is described, which shows that the ν e has positive helicity.

Tim e [m in]

Fig. 12.6. The β asymmetry in the decay of polarised 60 Co nuclei. Left: β particle emission rates antiparallel, respectively, parallel to an external magnetic field, normalised to the rate for unpolarised nuclei: with time, the initially polarised sample warms up, and the nuclear polarisation thereby gets reduced. After 8 minutes there is no polarisation and hence no β asymmetry any more. Right: origin of the β asymmetry (see text).

12.10.4 Helicity of the neutrino • The helicity of a particle was already introduced in section 5.4 in eq.(5.38). It is defined as s·p h= . (12.133) |s| |p| Helicity is a pseudoscalar quantity. Therefore, like the β asymmetry, a non-vanishing expectation value of the neutrino (which has only weak interactions) is another demonstration that the weak interaction violates parity.

12.10 Parity and its non-conservation in weak interactions

141

• Experimental evidence of the left-handedness of the neutrino (Goldhaber 1958): 152 Eu

152 Eu

– source

0-

(T 1/2 = 9.3 h )

Analysing m agnet

EC (950 keV) 1-

3×10 -14 s

152 Sm *

γ (961 keV) 0+

Lead

152 Sm

p( 152 Sm * )

p(ν)

Fe + Pb shielding

Sm 2 O 3 scatterer

(I) Mu – m etal shielding

(II) s( 152 Sm * )

s(ν)

Photom ultiplier

Fig. 12.7. The Goldhaber experiment which demonstrated the left-handedness of the neutrino. Upper left : level scheme of the process used. Lower left: the two kinematic possibilities to combine the spins of the excited 152 Sm nucleus and the neutrino to the spin 1/2 of the electron captured by 152 Eu. (I) corresponds to positive, (II) to negative neutrino helicity, to be experimentally distinguished. Right: set-up of the Goldhaber experiment.

— Goldhaber found a nuclear reaction which allows one to define 1. the neutrino momentum p, 2. the neutrino spin s. The measurement employs the K-capture reaction 152

Eu + e− →

152

Sm + ν e .

(12.134)

1. The definition of neutrino momentum employs nuclear resonance fluorescence: Resonant absorption an re-emission ("resonance scattering") of the γ quanta emitted by 152 Sm∗ nuclei by 152 Sm nuclei in the ground state is only possible if the γ quanta have an excess energy corresponding to the recoil of the emitting and the scattering nuclei (note that the recoil-less emission and absorption by two different nuclei, i.e. the Mößbauer effect is not dominant in this case). This requires the emitting nucleus moving towards the scatterer. The required momentum is due to the neutrino emitted prior to γ emission: • The recoil energy of the 152 Sm∗ nucleus (the ∗ standing for "excited state") due to the neutrino is 3.12 eV. • The recoil energy due to the γ emission is 3.2 eV with width ~/τ = 0.023 eV. The detector is placed such that (see fig. 12.7) any γ quantum reaching the detector was resonantly scattered by a Sm2 O3 scatterer. As a result, the highest intensity of γ quanta is observed if the 152 Sm∗ nuclei emits the neutrino in opposite direction to the γ ray. That way the neutrino momentum is defined.

142

12. Symmetries and conservation laws

2. The direction of the neutrino spin s is measured indirectly: through the two kinematic possibilities shown in fig. 12.7 it is connected to the γ polarisation. This can be measured using the spin-dependence of the γ transmission through magnetised iron, which is due to the spin-dependence of the Compton scattering. — Observation: On inverting the magnetic field applied to the iron the γ count rate changed by a few percent. The change corresponds to the efficiency to measure the γ polarisation, and one observes that only case (II) is realised, ⇒

neutrinos are left-handed.

(12.135)

12.10.5 Parity violation in the decay of the muon and the charged pion • Muon decay is a weak process involving only leptons, µ− → e− + ν µ + ν e .

(12.136)

— In the rest frame of the muon the electron has the largest momentum, respectively, energy, when the ν µ and the ν e are parallel to each other and anti-parallel to the electron momentum (see fig. ). The spins of the ν µ and the ν e are anti-parallel ⇒ the spins of µ− and e− are parallel. — Observation: electron emission with positive helicity is suppressed.

suppressed:

νe νµ

prefered:

e-

e-

µ-

µ-

νe νµ

Fig. 12.8. Muon decay: experimentally, one may focus on events, where the electron is emitted with the largest energy. In this case, the two neutrinos have to be emitted anti-parallel to the electron. Since neutrino and anti-neutrino have opposite helicity, the situation shown on the left-hand side requires the electron to be right handed, which is observed to be suppressed compared to the situation shown on the right-hand side.

• It is a general observation that: In any β-decay, emission of electrons with negative helicity is prefered.

(12.137)

All experimental observations can be incorporated in theory requiring that: The charged W-bosons of weak interaction couple only to left-handed leptons.

(12.138)

• Only for massless particles helicity is invariant under Lorentz transformations ⇒ for particles with non-zero rest mass (e.g. an electron) and for a given reference frame one may always find a second frame moving faster than the electron in the first frame. The helicity of the electron has different sign in both frames. As a result, leptons with finite mass emitted in a β-decay

12.10 Parity and its non-conservation in weak interactions

143

have a right-handed component. Compared to the dominant left-handed component, the size of the right-handed component is proportional to v 1− . c

(12.139)

This has strong implications, e.g. in the decay of charged pions: • The negative pion decay proceeds dominantly via π − → µ− + ν µ .

(12.140)

The process π − → e− + ν e

(12.141)

is strongly suppressed with respect to the first one (by a factor 8000), even though the phase space is larger by a factor 3.5 due to the electron mass being much smaller than that of the muon.

ν

π-

e-, µ-

Fig. 12.9. Pion decay: The anti-neutrino is right-handed. Since the pion has zero spin, the charged lepton (electron or muon) has to be emitted with spin parallel to its momentum. For the electron, this is much stronger suppressed due to the factor 1 − v/c than for the much heavier muon.

• Reason: — The pion has spin zero, and the decay is a two-body decay. In the centre-of-mass system, the lepton and the anti-neutrino are emitted in opposite directions. Conservation of angular momentum requires the spin of the negative lepton to be parallel to its momentum, because the anti-neutrino has positive helicity. But the positive helicity is strongly suppressed by the weak interaction. — The size of the right-handed component of the e− is by a factor 2.6 × 10−5 times smaller than of the µ− , due to the electron velocity being close to c, whereas the µ− are still nonrelativistic due to their much larger mass.

13. Quarks

13.1 Nucleon structure • The nucleon structure provides evidence of more basic constituents of matter. • Nucleon structure can be studied via elastic and inelastic electron scattering: — Elastic electron scattering provides electromagnetic form factors of the nucleon (similar to the electric form factor of nuclei, ¡but¢now at higher momentum transfer Q). One can 2 distinguish an electric ¡ 2 ¢ form factor GE Q related to the charge distribution, and a magnetic form factor GM Q related to the distribution of electric currents. The cross section for elastic electron scattering is given by the Rosenbluth formula, " # ¡ ¢ ¡ ¢ µ ¶ ¡ 2¢ G2E Q2 + τ G2M Q2 dσ dσ Q2 2 2 θ (13.1) = + 2τ GM Q tan , τ= dΩ dΩ Mott 1+τ 2 4M 2 c2 The form factors can be measured separately, using their different dependence on the scattering angle. — In the limit Q2 → 0, the form factors for the proton and the neutron are given by GE,p (0) = 1, GM,p (0) = 2.79,

GE,n (0) = 0 GM,n (0) = −1.91

(13.2) (13.3)

GE,p (0) and GE,n (0) correspond to the charges in units of e, and GM,p (0) and GM,n (0) correspond to the anomalous g-factors (2 would be the expected value for a Dirac particle, i.e. a particle with spin 1/2 which can be described by the Dirac equation). — The measured electric form factor of the proton corresponds to an exponentially decreasing charge distribution with ρ (r) = ρ (0) exp (−ar) ,

a = 4.27 fm−1

The electric charge radius (as defined in section 5.5) of the proton is p hr2 i = 0.81 fm

(13.4)

(13.5)

For the correspondingly defined radius of the distribution of the magnetic moments of proton and neutron one finds the same value.

• At sufficiently high energy, electrons may be scattered inelastically from nucleons. — The nucleon breaks up into two or more particles (most often hadrons) in the final state. This process is called hadronisation.

146

13. Quarks

Fig. 13.1. Electric and magnetic form factor of proton and neutron, as a function of Q2 . The data points are scaled with the indicated factors [B. Povh et al., Particles and Nuclei, Springer, 2002]. hadrons

proton electron

Fig. 13.2. Inelastic electron scattering leads to formation of hadrons

— The spectrum of scattered electrons shows a series of resonances. They correspond to nucleon excitations. The first resonance, appearing at E 0 = 4.2 GeV in fig. 13.3, has a mass of 1232 MeV/c2 . It is called the "delta-resonance", ∆ (1232). This resonance exists in 4 charge states, ∆++ ,

∆+ ,

∆0 ,

∆−

(13.6) +

In electron scattering, only the ∆ can be excited, since there is no charge transfer. It has a width of Γ = 120 MeV, corresponding to a lifetime of ~ 6.6 × 10−22 MeV s = = 5.5 × 10−24 s Γ 120 MeV It decays strongly through τ=

∆+ → p + π 0 ,

∆+ → n + π +

(13.7)

(13.8)

— Scattering data at higher energies reveal that the nucleon contains point-like constituents with spin 1/2 (point-like, because there are form factors, here called structure functions, which in first order do not depend on the momentum transfer).

Elastic scattering (divided by 15)

Fig. 13.3. Spectrum of electrons scattered inelastically by protons. Data were taken at beam energy 4.879 GeV and for a scattering angle of 100 [B. Povh et al., Particles and Nuclei, Springer, 2002].

14. Passage of radiations through matter, particle detection

14.1 Charged particles 14.1.1 Definition of basic terms • Heavy charged particles (heavy compared to electrons, hence any other charged particles) are the only ones which have a range: on passage through matter, a monoenergetic beam of these particles looses energy without significant change of particle number. After having traversed a rather well defined thickness, they will all be stopped. The minimum amount of material needed to stop the particles is called the range. This definition takes into account that the range is (nearly) independent on the density of the material.

Fig. 14.1. Range curve, showing the number N of particles of a monoenergetic beam penetrating into a depth x of a (homogeneous) absorber. The mean range R0 is the thickness, for which the particle number is reduced to half of its initial value. For x ¿ R0 , there is practically no loss of particles out of the beam, and they will only have lost part of its energy, mainly due to collisions with the electrons of the absorber. R1 is called the extrapolated range [E. Segrè, Nuclei and Particles, Benjamin, 1977].

• Specific energy loss: energy lost per unit path length, −dE/dx. The average value is called the stopping power of the absorber. • Specific ionisation: number of electron-ion pairs created per unit path length. • Straggling fluctuations in the range, caused by fluctuations in energy loss. • Bragg curve: specific ionisation as a function of the range or the residual range (to be distinguished for individual particles and as averaged over the particles within a beam, see fig. 14.2).

150

14. Passage of radiations through matter, particle detection

Fig. 14.2. Bragg curves. Left: for an individual α particle [Holloway and Livingston, Phys. Rev. 54 (1938) 29]. Right: Bragg curve for a beam of protons, performed at an air density of 1.166 mg cm−3 [R. Wilson, Cornell University].

• Mass specific energy loss: mean that the thickness of the absorber is measured in g cm−2 . The thickness x (in cm) is related to the effective thickness t (in g cm−2 ) by ρx = t ⇒

dE 1 dE = dt ρ dx

(14.1) (14.2)

• Rutherford scattering: elastic nuclear collisions. They give rise to large changes in the direction of the impinging particle, but not to significant changes of particle energy. Figure 14.3 shows on the right-hand side nuclear collisions of α particles in a cloud chamber. The greatest part of

Fig. 14.3. Cloud chamber tracks of α particles, showing delta rays (collisions with atomic electrons, which are sufficiently violent to create secondary ionisation). In the right picture, large changes of direction due to nuclear collisions are visible [T. Alper, Z. Physik 67 (1932) 172].

14.1 Charged particles

151

energy loss of a charged particle is energy loss due to ionisation, described by the Bethe-Bloch formula (see next section). 14.1.2 Bethe-Bloch formula • describes the specific energy loss of heavy (M À me ) charged particles, caused by collisions of the impinging particle with atomic electrons. Sometimes, an atomic electron receives enough energy to become free and causes secondary ionisation (delta rays, see fig. 14.3), sometimes the atom gets excited but not ionised. In any case, the energy required for these processes reduces the kinetic energy of the incident particle. Using simple approximations, the specific energy loss can be calculated classically, taking the electrons as free and being at rest. The trajectory of the heavy particle is not considerably affected, and the electron acquires an impulse perpendicular to the trajectory. The maximum collision parameter, at which electrons still can acquire energy is limited by the electrons being bound in atoms: a jump from one state to another requires a perturbation, which is sufficiently short to overcome the adiabatic limit (the electrons are bound in atomic states, which have a certain frequency. If the effective frequency of "switching" the external force field, i.e. the inverse of the time, the Coulomb field of the bypassing charged particle gets significantly changed is to low, the electron will stay in its initial atomic state. In this adiabatic limit the energy of the atomic state gets changed during the interaction, without causing any transition (and therefore no ionisation)). The mimimum collision parameter is determined by the maximum energy transfer possible. It is essentially given by µ ¶ dE 2me v 2 z 2 e4 ne 2 − ln (14.3) −β = dx 4πε20 me v 2 I(1 − β 2 ) where ne = natom Z is the number density of electrons in the absorber, z is the charge number of the impinging particle, I is an effective ionisation potential of the absorber atoms.

Fig. 14.4. Specific energy loss due to ionisation as a function of βγ.

• Properties of the Bethe-Bloch formula: — Particles are minimum ionising at βγ ≈ 3, i.e. at E ≈ 3M c2 . There, the mass specific energy loss is approximately equal for all materials (except hydrogen, for which it is about 2 times

152

14. Passage of radiations through matter, particle detection

larger, since it contains about two times more electrons for a given density) ¯ MeV dE ¯¯ (protons) ≈ 2 − ¯ d(ρx) minimum g cm−2

(14.4)

— For electrons, −dE/dx is larger due to bremsstrahlung.

— For small energies, −dE/dx is rising rapidly as −dE/dx ∝ 1/E ∝ v −2 . — The relativistic rise is due to the relativistic increase of the transverse electric field as E⊥ ∝ γ due to Lorentz contraction. The larger range of this field increases the collision parameter at which electrons still may acquire energy. — Part of the energy is carried away as light (Cherenkov effect, see below). 14.1.3 Further loss mechanisms • Bremsstrahlung: — Radiation of photons due to interaction of the charged particle with the Coulomb field of an absorber nucleus. — Specific energy loss: −

¯ µ ¶2 dE ¯¯ Z 2 2 e2 ∝ E z dx ¯brems A mc2

(14.5)

Since −dE/dx ∝ m−2 , energy loss due to bremsstrahlung is particularly important for electrons. • Production of electron-positron pairs: — Mechanism: electron-positron pairs are produced in the Coulomb field of the absorber nuclei, e.g. µ + N → µ + e− + e+ + N — Specific energy loss: ¯ dE ¯¯ ∝E − dx ¯pair

(14.6)

• Photonuclear interactions: — Mechanism: inelastic interactions of the impinging charged particle with an absorber nucleus involving virtual photons (these are photons which are exchanged between charged particles without occuring as real, detectable particles, see sec. 4.4) resulting in nuclear excitation or even nuclear breakup. — Specific energy loss: ¯ dE ¯¯ − ∝E dx ¯photonucl

(14.7)

14.1 Charged particles

• Total energy loss of charged particles: ¯ ¯ ¯ ¯ ¯ dE ¯¯ dE ¯¯ dE ¯¯ dE ¯¯ dE ¯¯ − =− − − − dx ¯tot dx ¯ionisation dx ¯brems dx ¯pair dx ¯photonucl

153

(14.8)

In fig. 14.5 this is shown for muons in iron.

Fig. 14.5. Mass specific energy loss of muons in iron. The individual loss components are shown as well as their sum [C. Grupen, Teilchendetektoren, B.I. Wissenschaftsverlag, 1993].

14.1.4 Polarisation effect: Cherenkov radiation • Cherenkov radiation is an electromagnetic radiation emitted from particles, which traverse a medium with refractive index n with a velocity larger than the velocity of light (in the medium), v > c/n. • Origin: an asymmetric polarisation of the atoms close to the track of the particle. A timedependent dipole field radiates electromagnetic energy. For v < c/n, the dipoles are symmetrically arranged around the particle position, and there is no emission of radiation (see fig. 14.6). • Contribution to the total specific energy loss dE/dx is small (maximum a few % ), even for minimum ionising particles (the minimum in fig.14.4). • The radiation is emitted at a specific angle θc to the path of the charged particle (similar to the supersonic cone in acoustics). From the figure, the condition for emission of Cherenkov radiation follows as 1 cos θc = (14.9) nβ consistent with the Cherenkov condition 1 β≥ n

(14.10)

154

14. Passage of radiations through matter, particle detection

Fig. 14.6. Left: the atomic dipoles are symmetric with respect to the position of the charged particle, there is no radiation of e.m. energy. Right: the particles move too fast for the formation of dipoles on their future track. This asymmetric configuration of dipoles radiates [C. Grupen, Teilchendetektoren, B.I. Wissenschaftsverlag, 1993].

• Limiting cases: β & 1/n : β'1:

emission in forward direction emission under the angle θc = arccos

1 n

• Number dN of radiated photons per path length dx in a wavelength range [λ1 , λ2 ] (derived by Tamm and Frank): dN = 2παz 2 dx

Zλ2 λ1

⇒

dλ λ2

µ 1−

1 β 2 n2

¶

(14.11)

more blue photons than red ones

In the range of visible light: dN £ −1 ¤ ≈ 490 sin2 θc cm dx

(for z = 1)

(14.12)

A larger yield of photons can be attained if also ultraviolet radiation can be detected (gain factor 2 − 3). But there is a limit: for X-rays, n ≈ 1, such that the Cherenkov condition eq.(14.10) cannot be fulfilled any more. • The Cherenkov effect can be used for particle identification in so-called RICHes (Ring Imaging Cherenkov detectors) using the Cherenkov condition eq.(14.10) for particles with same momentum but different mass, such that their β’s are different (→ exercise).

14.2 Photons 14.2.1 Absorption coefficient • Interactions of photons in the absorber lead to either

14.2 Photons

155

Table 14.1. Properties of various materials as Cherenkov radiators. The values for the gases are stated at normal conditions. Material

n

β min

γ min

solid sodium

3.22

0.24

1.029

diamond

2.91

0.26

1.034

flint glas

0.92

0.52

1.17

water

0.33

0.75

1.52

aerogel

0.025 − 0.075

0.93 − 0.976

4.5 − 2.7

2.93 × 10−4

0.9997

41.1

0.99997

123

1.7 × 10−3

pentane air

−5

helium

3.3 × 10

0.9983

17.2

1. complete absorption 2. scattering under large angles (compared to the energy loss of heavy charged particles due to ionisation) • As a result, the absorption by a layer of material is exponential, −

dI = µ dx I

(14.13)

where I is the intensity of the radiation, µ the absorption coefficient, and dx the thickness traversed. µ is the inverse of the mean free path. The mass absorption coefficient µ0 follows from µx =

µt = µ0 t ρ

(14.14)

using the definition given in eq.(14.1). • Detection via conversion into charged particles and subsequent detection of the ionisation. 14.2.2 Processes of energy loss • Photo-effect: — Absorption of a photon by an atomic electron: γ + Atom → e− + Ion The atomic nucleus recoils, such that conservation of energy and momentum can be fulfilled simultaneously. — Particularly large absorption of γ quanta by electrons in the K shell, which are very close to the nucleus. — Absorption edges: sudden change of the absorption cross section, when Eγ gets sufficiently large to bring M-, L-, K-electrons of the atom into the ionised state. — Cross section (probability per second of the process to happen, divided by the flux of incident particles, see next chapter):

156

14. Passage of radiations through matter, particle detection

σ photo ∝ Z 5 Eγ−7/2 ∝Z

4.5

Eγ−1

(at low energy and far away from absorption edges) (0.1 MeV . Eγ . 5 MeV)

The strong dependence on Z indicates that the Coulomb field of the nucleus is strongly involved in the interaction. — Secondary effects after photo-effect: emission of characteristic X-rays, Auger electrons. • Compton effect: — Elastic scattering of the photon at a quasi-free atomic electron: γ + Atom → γ + e− + Ion — Energy Eγ0 of the scattered photon: Eγ0 = Eγ 1+

1 Eγ me c2

(1 − cos θγ )

(14.15)

where θγ denotes the angle, by which the photon gets deflected. — The total cross section (given per atomic electron exactly by the Klein-Nishina formula of quantum electrodynamics) has the following dependence (per atom, and for Eγ À e− − binding energy): σ Compton ∝ Z

ln E E

(14.16)

i.e. proportional to the number of atomic electrons. • Pair production: — In the Coulomb field of the nucleus, the photon generates an electron-positron pair: γ + Atom → e− + e+ + Atom The atomic nucleus recoils to fulfill conservation of energy and momentum simultaneously. — Kinematic requirement: Eγ ≥ 2me c2 +

2m2e c2 ≈ 1.022 MeV M

(14.17)

The second term in this expression is the recoil energy of the atom with mass M . — Cross section: σ pair ∝ Z 2

(14.18)

with the Z 2 -dependence indicating that a coherent scattering off the nucleus as a whole is involved in this process. • The total photoabsorption coefficient for lead is shown in fig. 14.7.

14.3 Neutrons

157

Fig. 14.7. The photoabsorption coefficient of lead [C. Grupen, Teilchendetektoren, B.I. Wissenschaftsverlag, 1993].

14.3 Neutrons • As neutral particles, neutrons have no Coulomb interaction. They interact with matter via strong nuclear forces. Since this interaction is a very local one (with range of the order 10−15 m), even slow neutrons pass through rather easily. • Nuclear processes are often denoted as target nucleus (projectile, light product particle(s)) daugther nucleus

(14.19)

For neutron induced processes we may distinguish: 1. Elastic scattering: A(n, n)A

(14.20)

In an elastic process, the neutron looses energy due to recoil of the nucleus. This is most efficient, when the nucleus has the same mass as the neutron, i.e. for hydrogen. The slowing down of fast neutrons by a few collisions is called moderation. It is an important ingredient for the functioning of a nuclear reactor (see section 9.4). 2. Inelastic scattering:

158

14. Passage of radiations through matter, particle detection

A(n, n0 )A∗ ,

A(n, 2n0 )B,

etc.

(14.21)

In the first process, a neutron with kinetic energy larger than typically En & 1 MeV transfers energy to the nucleus (the prime indicates that an inelastic process has taken place). The nucleus is left in an excited state (denoted by the *). 3. Neutron capture: A(n, γ)(A + 1)

(14.22)

At low neutron energy (En . 1 eV) the cross section for (n, γ) reactions is often inversely proportional to the neutron velocity (when there are no nuclear resonances in the region of thermal energies), σ capture ∝

1 v

(14.23)

4. Neutron induced emission of light nuclear particles: (n, p),

(n, d),

(n, t),

(n, α),

(n, αp),

etc.

(14.24)

d stands for deuteron (2 H), t for triton (3 H). Examples of such reactions: 10 3 6

B(n, α)7 Li∗ ,

He(n, p)3 H

Li∗ → 7 Li + γ

(14.25)

( = 3 He(n, p)t )

(14.26)

7

3

Li(n, α) H

(14.27)

5. Neutron induced nuclear fission (discussed in detail in sec. 6.4): (n, f )

(14.28)

6. Hadron production. Hadrons are any strongly interacting particles. Examples are the neutron and the proton but there are many more others, the lightest of these being the pions, which come in three charge states and have masses of .... Hence, processes like n + n → n + n + π+ + π−

(14.29)

appear only at high energy En À 100 MeV. Table 14.2. Energy ranges of neutrons and their detection. Energy range

name

detection via

En À 100 MeV

high energy neutrons

particle reactions

100 keV . En . 100 MeV

fast neutrons

nuclear recoil

0.1 eV . En . 100 keV

epithermal neutrons

En ≈ 25 meV

thermal neutrons

nuclear reactions

En < 25 meV

cold neutrons

nuclear reactions

En . 250 neV

ultra-cold neutrons

nuclear reactions

moderation → nuclear reactions