University of Cape Town Department of Physics PHY3022S Nuclear and Particle Physics Nuclear Physics Part 1 Basics

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1 University of Cape Town Department of Physics PHY3022S Nuclear and Particle Physics Nuclear Physics Part 1 Basics Andy Buffler UCT Physics andy.buffler@uct.ac.za Room 5.01, RW James Building 1

2 Atomic Physics (David Aschman) 20 lectures PHY3022S Course Outline Nuclear and Particle Physics (Andy Buffler and Andrew Hamilton) 20 lectures Solid State Physics (Mark Blumenthal) 20 lectures For nuclear and particle physics Prescribed book is BR Martin, Nuclear and Particle Physics: An Introduction (Wiley, 2006) Also used: RJ Blin-Stoyle, Nuclear and Particle Physics (Chapman and Hall, 1991) Tutor for Nuclear Physics: Mawande Lushozi 2

3 Conservation of energy and charge in e + e - production 3

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6 Periodic Table of the Elements 6

7 The nuclear atom Chapter 1: RJ Blin-Stoyle, Nuclear and Particle Physics (Chapman and Hall, 1991) 7

8 Radioactivity - history 1885 JJ Thomson discovers the electron 1896 Bequerel β-radiation from uranium salts 1896 Roentgen X-rays 1898 Pierre and Marie Curie α-radiation 1900 Villard γ-radiation

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10 Radioactivity α-radiation: Emission of an α-particle ( 4 He nucleus) from a nucleus. Energy around 5 MeV (short range in matter). β -radiation: Electron emission in nuclear decay β + -radiation: Positron emission in nuclear decay γ-radiation: De-excitation of a nucleus from excited state to lower state (kev to MeV) 10

11 Development of atomic theory JJ Thomson (Nobel Prize in Physics, 1906) Plum pudding model. Poor agreement with experiment. 11

12 The Rutherford model Ernest Rutherford (Nobel Prize in Chemistry, 1908) 12

13 Scattering of a 4 He nucleus off (a) Thomson s atom (b) Rutherford s atom Simulation of Rutherford scattering off a gold nucleus 13

14 Rutherford: It was the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15 inch shell at a piece of tissue paper and it came back and hit you. Rutherford model (1911): The atom has a small hard central core (nucleus) where all the positive charge is concentrated. The negative charge inhabitants the nearly empty space around the nucleus. Thus most of the alpha particles migrate through the gold foil with some or no (Coulomb) interaction, but some will experience a head-on collision with a nucleus and return in a backwards direction. 14

15 But there were still unanswered questions why does the nucleus (all positive charge) not fly apart due to Coulomb repulsion? why do the negative charges not radiate energy, spiral inwards and collapse into the nucleus due to Coulomb attraction? the model did also not explain existing experimental observations. 15

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17 Atomic size In a solid, assume all atoms are spherical and packed in a tight grid. For radius R, separation is 2R and volume of each atom (2R) 3. One mole contains N A atoms and occupies volume 8N A R 3. Volume also given by Thus R 3 M 10 ρ where ρ is density and M molar mass (in g) M 10 = 2 ρn A Gold has M 197 and ρ = kg m 10. thus R m 3-3 All atomic radii are about the same: range ( ) m

18 Nuclear phenomenology nucleus positive core of the atom discovered by Rutherford in α-scattering experiments almost all the mass of the atom concentrated in the nucleus the nucleus consists of A nucleons = Z protons + N neutrons held together by strong nuclear forces positively charged no charge 18

19 nuclide: a particular combination of protons and neutrons nuclides are characterized and represented by: ( AZ) A A A, X X X X A ( ) atomic number mass number Z Z N X: chemical symbol e.g. 60,27 Co Co Co Co 60 A= Z + N neutron number isotopes: Nuclides with the same Z isotones: Nuclides with the same N isobars: Nuclides with the same A e.g. 17Cl 17Cl e.g. 20Ca 19K e.g. 9F 10 Ne 19

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21 Nuclear and atomic masses exactly Mass of 1 neutral atom of 12 6 C = u unified mass unit One mole of a substance contains Avogadro s number, N A = atoms (or molecules) One mole of a substance of molecular weight M u is M 10-3 kg Then kg of 12 C contains N A atoms of mass 12 u giving 1 u = kg Then Eu ( 27 )( kg 3 10 m s ) ( J ev ) = = = 2 u c MeV 2 So we can write 1 u = MeV/c 2 21

22 Energy equivalent masses: M p = u = 938 MeV M n = u = 940 MeV M e = u = MeV 22

23 Nuclear sizes and shapes Size and shape of a nucleus can be found from scattering experiments use electrons to learn about charge distribution and hadrons (often neutrons) to learn about matter distribution Charge distribution In principle the charge distribution of a nucleus may be obtained from the measurements of the differential cross section of electron scattering, but more reliable determinations are obtained from numerical solutions of the Dirac equation fitted to experimental data 23

24 Radial charge distributions ρ () r of various nuclei, in units of e fm 3 ch ; the thickness of the curves near r = 0 is a measure of the uncertainty in ρ () r ch. 0 () ρch ρch r = r a b 1+ e Saxon Woods For medium and heavy nuclei a A fm b 0.54 fm 0 ρch 0.06 to

25 Mass distribution Almost identical nuclear density in the nuclear interior of all nuclei. ρ ch 0 the decrease in with increasing A is compensated by the increase in A/Z with increasing A Interior nuclear density then ρnuclear 0.17 nucleons / fm 3 and for medium and heavy nuclei R 13 nuclear 1.2 A fm 25

26 Elastic differential cross-sections for 52 MeV deuterons on 54 Fe 26

27 Differential cross-sections (normalized to the Rutherford cross-section) for the elastic scattering of 30.3 MeV protons, for a range of nuclei compared with optical model calculations; the solid and dashed lines represent the results using two different potentials. 27

28 Periodic Table of the Elements 28

29 Nuclear binding energy Use atomic masses to define the mass deficit / defect / excess M( Z, A) M( Z, A) Z( M + m ) NM Define the nuclear binding energy to be: p e n B = M( Z, Ac ) 2 Measure nuclear masses with a mass spectrometer 29

30 B/A is the binding energy per nucleon for a particular nuclide. Plotting B/A versus A gives the binding energy curve... Average binding energy per nucleon (MeV) Mass number A Note that the region of greatest stability is at a maximum of 8.7 MeV per nucleon, occurring at A =

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32 Example: Binding energy of the deuteron. What is the binding energy of the deuteron if B D = M c where D 2 M D = u M p = u M n = u ( ) M = M + M M = u D p n D MeV B = ( u) c = uc 2 D MeV 2 n p capture: n + p d + γ MeV 32

33 Stability of nuclides Not all nuclides are stable. Unstable nuclei decay spontaneously to become stable. i.e. emit particles e.g. Gold has 30 isotopes ( 175 Au 204 Au). Only 197 Au is stable. The other 29 isotopes are radioactive nuclides (radionuclides) Organising the nuclides The periodic table shows the most common or most stable isotope of each element. Usefulness limited since different isotopes have very different nuclear properties. The nuclidic chart shows all known nuclides is a plot of Z vs. N 33

34 Stable nuclides shaded dark Radioactive nuclides shaded light For a stable nuclides with N 20, N Z This increases to N 1.5 Z, for N

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36 Valley of stability 36

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38 α decay Nuclear decay 4 α = helium nucleus 2He2 the unstable parent nucleus emits an α-particle. A Z X Y + He A 4 4 Z 2 2 β decay β = an electron occurs when n p inside a neutron-rich nuclide X Y + e + ν A A Z Z+ 1 N 1 e Possible when M( Z, A) > M( Z + 1, A) 38

39 β + decay β + = X a positron occurs when p n inside a proton-rich nuclide Y + e + ν A A + Z Z 1 N+ 1 e Possible when M( Z, A) > M( Z + 1, A) + 2m e Electron capture A nucleus captures an atomic electron which has ventured too close to the nucleus X + e Y + ν A A Z N Z 1 N+ 1 e Possible when M( Z, A) > M( Z + 1, A) + ε where ε is the excitation energy of the atomic shell of the daughter nucleus 39

40 Energy released in radioactive decay In nuclear decay, binding energy is released, becoming kinetic energy of the decay products Q = energy released in the decay {( mass) ( mass) } c 2 = before Q > 0 is a condition for the decay to occur. A A 4 4 For example, consider the α decay: X Y + He Z after Z 2 2 The Q value or disintegration energy is Q= MXc ( MYc + Mαc ) This can be shown to be the same as: Q= B B B X Y α B B Bα A ( A 4) 4 A A 4 4 X Y = 40

41 Ground and excited states of nuclei 41

42 Gamma decay Decay is a step towards stability. Nuclei, like atoms have discrete energy states. Excited nuclei decay by transitions in which high energy photons (γ-rays) are emitted. A * A ZX ZX + γ excited state e.g. the decay of Co: β γ γ 2.50 MeV (1.17 MeV) 1.33 MeV (1.33 MeV) 0 60 Co is therefore a source of both γ s and 27 β s, both radiation exhibiting a half life of 5.3 years. 42

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44 Natural radioactivity 44

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46 Angular momentum of a nucleon 1 Nucleons are fermions, therefore spin quantum number s = + 2 Eigenvalue of Eigenvalue of s is ( ) s is ± ( ) z 1 2 Nucleon has orbital angular momentum quantum number = 0,1,2,... Eigenvalue of Eigenvalue of Add spin and orbital angular momentum vectorially to get total angular momentum j= l+ s Eigenvalue of Eigenvalue of is 2 l ( ) 2 z is is +1 m j j+ 1 2 j ( ) 2 jz is m j m = l,...,0,..., 1, + m = j,...,0,..., j 1, + j j 46

47 Angular momentum of a nucleus Add all spins of nucleons vectorially to get total nuclear spin S. Add all orbital angular momenta vectorially to get total orbital angular momentum L. Add total spin and total orbital angular momentum vectorially to get total angular momentum (or nuclear spin) J = L+ S Write nucleus wavefunction as ψ JM Therefore ( ) 2 2 J ψ JM = J J + 1 ψ JM and Jzψ JM = M ψ JM where M = J, J + 1,...,0,..., J 1, J 47

48 Parity Parity is the transformation r r ( x ) ( x ) Pˆ Ψ, t = PΨ, t The intrinsic parity of a single proton and a single neutron is defined as +1 Under the parity transformation ˆ PY ( θφ, ) = ( 1 ) Y ( θφ, ) Therefore for a single particle nuclear state P = ( 1) Total parity of a multiparticle state is the product of parities of individual particles. m m A pair of nucleons with the same l have combined parity of +1 therefore parity of a nucleus depends on the parity of the last unpaired proton and/or neutron. 48

49 Spin and Parity Individual and/or collective motions of nucleons (determined by nuclear structure and dynamics) determine symmetry and other properties of the nuclear wave function. A nuclear state is labelled with both its spin and parity in the form J π e.g ,1,0,... Some observations for ground states: Even A nuclei have J = 0,1,2,... Odd A nuclei have J = , 2, 2,... All even-even nuclei have J π =

50 Recall the magnetic moment of the electron Classically for an electron of charge e and mass m orbiting with angular momentum L associated orbital magnetic moment is e μl = L 2m And similarly define an intrinsic (spin) magnetic moment e μ= g s 2 m where g-factor = (or from Dirac) Actual value of electron intrinsic magnetic moment is an eigenvalue of when electron is in m = + substate. 1 µ z s 2 µ e e = 2 g = 2g = 2g B 2m 2m µ where the Bohr magneton µ B = e 2m = J T -1 50

51 Nuclear magnetic dipole moments Both the proton and the neutron have an intrinsic magnetic moment and since the proton is charged, it can also produce a magnetic moment when in orbital motion. Write the magnetic moments for spin up protons and neutrons as or Measure Then e 1 1 µ p = 2 g p µ n = 2 gn 2M p 2 µ = g µ 1 p 2 p N e M µ = g µ p 1 n 2 n N where the nuclear magneton µ = e 2 M = J T -1 g = and g = p µ = µ and µ = µ p Note that N µ µ = 32 p n N n n p (predicted by quark model) N 51

52 Magnetic moments of nuclei involve spin and orbital component, and are indictors of structure. For a nucleus the intrinsic magnetic moments of the constituent protons and neutrons will contribute to the total magnetic moment with further contributions from any orbital motion of the protons. Then µ = Jg µ J J N g J where is the nuclear g-factor the ratio of the magnetic moment in nuclear magnetons to the total angular momentum quantum number for the nuclear (or particle) state. Find µ = 0 for all J = 0 ground states (why?) 52

53 Nuclear electric quadrupole moments Define quadrupole moment as ρ ( 2 2 r ) Q = ()3z r dv 0 ch with z-axis along symmetry axis defined by nuclear spin. Then can write ( ) 2 2 Q0 = Z 3 z r and has units of barns z where r = x + y + z For a spherical nucleus: z = r and thus Q 0 = For a non-spherical nucleus: z r z z > r Prolate ellipsoid Q 0 > z z < r Oblate ellipsoid Q 0 < Find Q 0 = 0 1 for all J = 0 and J = 2 ground states (why?) 53

54 A nucleus may be deformed away from spherical into a (a) prolate shape (rugby ball) or (b) (b) oblate (flying saucer) 54

55 8 May 2013 A representation of the radium-224 nucleus. The atomic nucleus is a many-body quantum system with a shape determined by the number of nucleons that it contains and the interactions between them. Most of the several thousand known stable and radioactive atomic nuclei, with differing numbers of protons and neutrons, are spherical or rugby-ball shaped. But there is circumstantial evidence that some heavy, unstable nuclides are distorted into a pear shape through the phenomenon of octupole deformation. Samples of these rare atomic species can be accelerated to 8% of the speed of light in the REX-ISOLDE facility at CERN, and now Coulomb excitation experiments on beams of the short-lived isotopes radium-224 and radon-220 have demonstrated clear octupole deformation in the former. The results make it possible to discriminate between the various theoretical models of octupole-deformed nuclei, and are also relevant to the pursuit of physics beyond the standard model. 55

56 The nucleon-nucleon potential Martin 7.1 We infer that the N-N force must be 1. Short range (from Rutherford scattering) 2. Attractive at short range (to bind nucleons in nuclei) 3. Strong relative to Coulomb (to bind protons) 4. Repulsive at very short range (i.e. must saturate (to prevent collapse of all nuclei to the dimension of the force range) F V() r = r 56

57 Electrostatic and gravitational potential is long range V 1 r. Near constancy of nuclear binding energy per nucleon B/A means that each nucleon feels only the effect of a few neighbours. This is called saturation implies the strong inter-nucleon potential is short range. Range is of order of the 1.8 fm inter-nucleon separation. Since volume A, nuclei do not collapse, there is a very short range repulsive component. Reminiscent of interatomic potential in molecules. Is nuclear physics just quark chemistry? Depth of potential is of order of binding energy, perhaps tens of MeV. 57

58 The deuteron The deuteron 2 H is the bound state of a neutron and a proton. Experimental properties: Binding energy B = ± MeV Angular momentum and parity J π = 1 + Magnetic dipole moment μ D = nuclear magnetons Electric quadrupole moment Q D = m 2 Deuteron has very small binding energy, and no excited states. Also µ = µ + µ = ( ) µ = 0.88µ n p N N And with very small Q D, infer that deuteron is in L = 0 state where L= lp + ln 58

59 Deuteron ground state L = 0 Then since J = L+ S J = 1 (measured) implies proton and neutron spins are aligned (S = 1) Thus using the standard spectroscopic notation 2S + 1 L J where L = S, P, D, F, G, H for L = 0, 1, 2, 3, 4 the deuteron has a triplet ground state 3 S1 59

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61 The radioactive decay law Radioactive decay is a statistical (random) process. Consider a set of identical unstable nuclei. The set will obtain N 0 = N(0) of these nuclei at t =0, and N(t) radioactive nuclei at time t. i.e. N(t) < N(0) because of the decay. The rate at which the nuclei decay is proportional to N(t): dn() t i.e. =λnt ( ) dt λ is called the decay constant and depends on the nuclide. λ represents the probability that any one nucleus will decay during the next second (or any other time unit). 61

62 For a particular set of nuclides, λ is the same for each nucleus, at all times. dn() t dn = λn() t = λdt dt N N() t N 0 dn() t Nt () = λ t 0 dt [ log ( ) () ] N t [ ] t e Nt = λ t ( ) N (0) 0 log = ln Nt () N(0) = e λt e Radioactive decay law: Nt ( ) = N(0) e λt 62

63 Mean life of decay λ: decay constant Nt () N(0) The mean life, or lifetime, is defined to be τ = 1 λ N(0) e N(0) When t = τ, Nt ( ) =, e τ Nt () = Ne λt 0 t so the lifetime τ is the time taken for N(0) to drop to 1 N(0) e 63

64 Half life of decay, T12 T 12 is the time taken for N(t) to drop to 1 2 N(0) So at t = T : 12 1 Nt ( ) = NT ( 12) = N(0) 2 λt i.e. N(0) = N(0) e ln 2 = e λt = λt ln T = τ λ = λ = T 12 is convenient, since when t = m e.g. after 3 half-lives, N(3 T 12) = 12 N N(0) T 12 N(0) , N(t) is reduced to N(0) 2 m 64 m

65 Activity dn() t The decay rate Rt () = or activity of the sample is dt often more interesting than Nt () itself. Write dn() t Rt ( ) = = λn(0) e dt λt or Rt ( ) = R(0) e λt where R(0) = λn(0) The number of disintegrations (decays) per second also reduces exponentially with decay constant. λ SI units of activity: 1 bequerel (Bq) = 1 decay per second Older unit: 1 curie (Ci) = decays per second 65

66 For a decay chain A B C A N ( ) (0) t A t = NA e λ λa λat λb For a two stage decay sequence NB( t) = NA(0) e e λb λ A For a three stage decay sequence λ t λ t e e e NC( t) = λλ A BNA(0) + + A B λct ( λ λ )( λ λ ) ( λ λ )( λ λ ) ( λ λ )( λ λ ) B A C A A B C B A C B C t

67 Interaction of radiation with matter 67

68 Electromagnetic radiation Three primary processes of interaction with matter: Photoelectric effect, Compton scattering, Pair production 68

69 7.6 cm x 7.6 cm NaI 1.9 cm x 0.5 cm HPGe 69

70 Probability per unit length for removal of a photon Linear absorption coefficient μ Fractional loss in intensity: I( x) = Ie µx 0 Mass attenuation coefficient: µ α = ρ 70

71 Discovery of the neutron James Chadwick, 1932 Outside the nucleus, free neutrons are unstable and have a half life of ± 1.0 s. n p+ e + ν e 71

72 Neutrons Uncharged, do not interact via Coulomb forces Interact (via collisions or reactions) with the nuclei of absorbing material Secondary radiation is almost always heavy charged particles Probability of interaction given by a cross section σ (see later) Fractional loss in intensity of a neutron beam irradiating a material of N nuclei per unit volume, and thickness x: N I( x) = Ie σ 0 x 72

73 Pulse height spectrum n Organic scintillator p PMT n Energy spectrum of recoil protons 73

74 Frank Brooks ( ) UCT Professor of Nuclear Physics developed in the early 1960s the first practical systems of pulse shape discrimination (PSD) which allows the identification of different types of charged particles in certain scintillator detectors by means of the characteristics of the scintillation decay. gammas PSD in a 5 cm x 5 cm organic liquid scintillator neutrons AmBe source Beam of 66 MeV neutrons

75 Charged particles The linear stopping power (or specific energy loss) for charged particles in a given absorber de/dx can be calculated using the Bethe-Bloch formula better via Monte Carlo method see 75

76 Specific energy loss in air 76

77 Bragg peak 77

78 Radiation units Activity: the rate of spontaneous decay transitions in a sample SI unit: 1 bequerel (Bq) = 1 decay per second Traditional unit: 1 curie (Ci) = Bq Exposure: quantity of radiation to which an object is exposed which is the ionization that the radiation would produce in dry air SI unit: 1 roentgen (R) = C/kg in dry air amount of radiation which delivers 8.78 mj of energy to 1 kg of dry air Exposure (and roentgens) seldom used nowadays 78

79 Absorbed dose: a measure of the energy deposited in a medium by ionizing radiation per unit mass SI unit: 1 gray (Gy) = 1 J kg -1 Traditional unit: 1 rad = 0.01 gray The absorbed dose depends not only on the incident radiation but also on the absorbing material: a soft X-ray beam may deposit four times more dose in bone than in air, or none at all in a vacuum. Kerma: kinetic energy released in matter the sum of the initial kinetic energies of all the charged particles liberated by uncharged radiation (i.e., indirectly ionizing radiation such as photons and neutrons) in a sample of matter, per unit mass of the sample kerma always greater than absorbed dose since some of the energy escapes from the absorbing volume in the form of bremsstrahlung x-rays or fast moving electrons. SI unit: grey (G) 79

80 Equivalent dose: a computed average measure of the radiation absorbed by a fixed mass of biological tissue, that attempts to account for the different biological damage potential of different types of ionizing radiation. It is therefore a less fundamental quantity than the total radiation energy absorbed per mass (the absorbed dose), but is a more significant quantity for assessing the health risk of radiation exposure. SI unit: 1 sievert (Sv) = 1 J kg -1 Traditional unit: 1 rem = 0.01 Sv equivalent dose (H) = absorbed dose (D) radiation weighting factor (W R ) 80

81 equivalent dose (H) = absorbed dose (D) radiation weighting factor (W R ) Radiation weighting factors (legislated): Radiation W R X-rays, gamma rays, electrons, muons 1 Neutrons: thermal 5 Neutrons: < 0.1 MeV 10 Neutrons: < 2 MeV 20 Neutrons: > 2 MeV 10 High energy protons 5 Alpha particles, fission fragments, heavy nuclei 20 One sievert of two different radiations produces the same biological effect (more or less). Dose rate also useful equivalent dose per unit time. 81

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