From the Nucleus to the Muon. Karlsruhe, Fall 2002

Transcription

1 From the Nucleus to the Muon Paolo Franzini Università di Roma, La Sapienza Karlsruhe University Karlsruhe, Fall 2002

2 THE BEGINNING 1. Rutherford Scattering 2. Ionization 3. Neutral Particles 4. The Pion Is the Muon th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 2

3 While I certainly had nothing to do with the dcoveries of the first half of the 19 th centuries, I want to go over some of them, for many reasons. New ways of thinking and new instruments were developed in those years. To a very large extent the instruments are the same today. In a somewhat general sense so the way we understand new results. For example, often today we say Rutherford scattering when we talking about experiments probing the structure of some particles. th The understanding of the basic processes by which we see particles was developed very early in the last century and I wh to dcuss some of the underlying principles. Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 3

4 Radioactivity and cosmic rays were first observed as ionizing phenomena. Thanks to ionization we sense particles, and because of it we can even see their trajectories. Together with ionization go scintillation, conversion of Ag salts to metallic silver, condensation of droplets in the Wilson or cloud chamber, formation of bubbles in the bubble chamber and finally the production of tiny electric signals in gas or semiconductors by which the microscopic and gigantic modern detectors can track charged particles. The Geiger and Marsden experiment, often called the Rutherford experiment, used scintillation to detect the scattered α-particles. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 4

5 Rutherford scattering, classical 6 MeV -particles Thomson r0 10 cm V Z/r Rutherford Au nucleus r 0 10 cm V= Z/r The scattering angle a function of the impact parameter. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 5

6 Geiger and Marsden, idealized The experiment often sketched as without even mentioning that it all must be in vacuum. Marsden was the student and so did all the observation, verifying the 1/ sin 4 θ/2 dependence of the scattered α s. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 6

7 Geiger and Marsden, in reality The real set up. 8 MeV α-particles from a radioactive source R in a piece of brass D with a hole Gold foil F 400 atom thick A ZnS coated screen S A telescope M to observe light flashes All in vacuum B th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 7

8 Q 2, t The classical mechanics scattering. The QED scattering. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 8

9 2. Rutherford Scattering p 2b b O db The deflection angle of a particles travelling at a dtance b from an infinite mass charge Ze nucleus given by p/p where the momentum change. equals the impulse of the force F acting for the time interval required to travel the dtance 2b at a speed β The relation between the scattering angle and the impact parameter b : θ = p/p = F t/p = αz/b 2 2b β 1 p = 2αZ 2αZ or b = βbp θβp. The scattering cross section then: th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 9

10 dσ dθ = 2πb db dθ = 8πα2 Z 2 β 2 p 2 θ d dθ ( ) 1 = 8πα2 Z 2 θ β 2 p 2 θ 3 The Rutherford cross section : dσ dcosθ = πα2 Z 2 2β 2 p 2 θ 1 sin 4 θ/2. In the small angle limit, sin 4 θ/2 θ 4 /16, and with d cos θ θdθ : dσ dθ = 8πα2 Z 2 β 2 p 2 θ 3 The reason why we find the correct answer because the crude approximation used for p in fact exact as derived below. th Rutherford was very proud that the scattering formula that he derived in classical mechanics was the correct answer also in QM. Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 10

11 The sin 4 θ/2 term comes from dσ/dθ 1/( p) 4. p the momentum transfer p in p out or p 1 p 2. Then ( p) 2 = p p2 2 2p 1 p 2 cos θ. In the Rutherford limit, p 1 = p 2 = p, ( p) 2 =2p 2 sin 2 θ/2. In the same limit, it also t, the Lorenz-invariant 4-momentum transfer squared. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 11

12 Exact computation of p b Ze Referring to fig., consider an infinite cylinder, of radius b around an ax z through the position of the heavy particle, parallel to the trajectory of the charge e moving withe velocity β. To compute p = F dt we use Gauss theorem: E d σ = Ze = 2πbE dz or E dz = ez S cyl 2πb from which F dz = e2 Z 2πb = 2αZ b and p = F dt = F dz/β = 2αZ βb. e z th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 12

13 In QED, Rutherford scattering the scattering of a light, spin-zero particle (we call it a pion) from a heavy spin zero nucleus. The appropriate amplitude given in the Feynman graph. ( A, Z) p 1 p 2 k p p 1 2 ( A, Z) q 1 q 2 The upper ver given by J µ (π)a µ and the lower by J µ (nucleus)a µ. Both currents must be (p in + p out ) to satfy gauge invariance and the proportionality constant the particle charge. Including the photon propagator, 1/k 2 and summing over its polarization we find M = e 2 1 Z(p 1 + p 2 ) µ k 2(q 1 + q 2 ) µ th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 13

14 th Finally we compute: d 3 σ = p 2E 2 de 2 dω δ(e in E out ) β (2π) 2 M 2 4E 1 E 3 4E 2 E 4 dσ dcosθ =8πα2 Z 2 E 2 1 k 4 From p 1 p 2 = {0, p 1 p 2 }, k 2 = (p 1 p 2 ) 2 = 4p 2 sin 2 θ/2 and dσ dcosθ = πα2 Z Rutherford 2 β 2 p 2 sin 4 θ/2. (1) Eq. (1) acquires multiplicative factors when spin taken into account (Mott, Rosenbluth) as well as from recoil. The result however valid strictly only for point-like particles. Otherwe: dσ dcosθ = dσ dcosθ F (k 2 or t) 2 point where the form factor F (t) equals 1 for point like structures and F 2 < 1 signals the presence of structure. Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 14

15 3. Ionization When a charged particle crosses a medium it suffers repeated collions with electrons. It easy, for M > m e, to compute the average energy loss per unit path length, usually measured in g/cm 2. th By reasoning as in my derivation above of the Rutherford formula we get that the energy transferred to an electron at dtance b from the trajectory of a charge z particle : p 2 = 2z2 e 4 2m e m e β 2 b 2 If N the electron number density (N e /cm 3 ), the energy lost per cm of trajectory to the electrons between b and b +db de(b) = 4πz2 e 4 N db mβ 2 b Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 15

16 and the total energy loss per unit path lenght: de dx = 4πz2 e 4 N mβ 2 ln b max b min b max limited by the suppression of energy transfer if 1/ t < f, where f the average frequency of the bound electrons. b min obtained from the max KE T max that can be transferred to the electron in a collion. The result not too different from the Bethe-Block formula: de dx = Kz2Z A 1 β 2 [ ] 1 2 ln 2m eβ 2 γ 2 T max I 2 β 2 δ 2 The energy loss in MeV/(g/cm 2 ), with K = 4πN A re 2 m e c 2. There a region where de/dx 1/β 2. Th has been very useful for particle identification. Deviation from Bethe-Bloch becomes large for β<0.03 and βγ, > 600. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 16

17 Stopping power [MeV cm 2 /g] Lindhard- Scharff Nuclear losses µ Anderson- Ziegler Bethe-Bloch µ + on Cu Radiative effects reach 1% [GeV/c] Muon momentum Radiative Radiative losses Without δ βγ [MeV/c] Minimum ionization E µc [TeV/c] th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 17

18 de/dx (kev/cm) µ π K p e D de/dx (MeV g 1 cm 2 ) H 2 liquid He gas Fe Sn Pb Al C Momentum (GeV/c) Specific ionization vs momentum as measured in the Time Projection Chamber βγ = p/mc Muon momentum (GeV/c) Pion momentum (GeV/c) Proton momentum (GeV/c) th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 18

19 A convenient value to remember for the minimum of de/dx 2 MeV/g/cm 2, valid for He, D 2, n-(ch 2 ) (polyethylene, plastics). For H 2, A = Z, de/dx min =4. For U it becomes about 1. The simple picture fails at low energy and for electrons. At high energy radiation or bremsstrahlung dominates as we will dcuss. The energy lost in the medium by a charged particle goes initially into excitation and ionization, i.e. formation of electron-ion pairs. Ultimately the energy will thermalize. In gases especially, where atomic and molecular collion are relatively rarer, the ion pairs can survive a long time and are the bas for seeing the trajectory of a charged particles and measuring some of its properties. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 19

20 The initial ion-electron pairs can be drifted apart by an electric field. Drift velocities are vastly different for ions and electrons. Saturated drift velocities for electrons in gases are few cm/µs, 1000 time smaller for ions th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 20

21 Ion pairs are made dramatically evident in the Wilson chamber by liquid droplets condensing on the positive ions and the electrons, often a negative ion formed by the electron attached to electronegative atoms, oxygen foremost. An electric field has drifted apart positive and negative charges. Diffusion also evident. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 21

22 It an experimental fact that the number of pairs proportional to the energy deposited. (Until saturation at very high specific ionization.) Gas Pairs/cm ev/pair He Ne Ar Xe In noble gases and some materials, some fraction of the excitation energy emitted as vible light. Th process called scintillation. It was dcovered in the early days of dcharge tubes, used by Thomson and many others. It still very important today in particle detection. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 22

23 Drift velocities: 4 cm/µs, saturated, Ar, He cm/µs liq Ar. There are some complications. 1. Drift velocity, larger for electrons than ions, field dependent 2. Recombination 3. Formation of negative ions in the presence of oxygen, halogens and so on 4. The signal, 100 e C, below the thermal noe of most, not all, amplification mechanm th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 23

24 Charged particles leave behind a trail of ions which allows us to track them. They are also deflected in magnetic fields. p = 300 B r, B in Gauss, p in ev/c, r in cm Detection of photons Neutral particles do not ionize. Neutron can be detected easily since they interact strongly with matter. Photons interact with matter by several processes: Photoelectric effect Compton effect Electron-positron pair production. At high energy, E m, orγ 1, the last process important. The others are still relevant but we will not dcuss them. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 24

25 Pair Production Pair production, γ e + e can only take place in an external field, in the real world that provided by a nucleus of charge Ze. The nucleus absorbs the recoil momentum necessary not to violate Lorentz-invariance as illustrated. e e e Z Z e th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 25

26 Before dealing with pair production, we consider the two photon process γγ e + e. By counting powers of e and dimensional arguments we get a familiar result σ(γ + γ e + + e )= α2 s for high enough energies so that the electron mass can be neglected. The connection between th last process and the one we are interested in, clear: the Coulomb field of the nucleus, which we take to be of infinite mass, provides an almost real photon which, together with the initial photon, produces the e + e pair. The electrostatic Coulomb potential Ze/r whose Fourier transform, 1/q 2, the spectrum of the virtual photons corresponding to the potential. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 26

27 The 4-momentum of the virtual photon p = (0, p), i.e. the nucleus absorbs momentum but not energy because its mass infinite. Because of th exchanged momentum the momenta of the two final electrons become independent in pair creation, contrary to the case of γγ e + e. Therefore, the final state of the pair production process contains an extra factor d 3 p/e for the two electrons. To get the total cross section, we must integrate. Dimensionally th means that an extra factor p 2 s must appear in the cross section, cancelling the 1/s factor in the γγ result. Counting powers of e and Z, and taking into account the above, we finally write σ = Z 2 α 3 f(m e ) and therefore: th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 27

28 σ Pair Z2 α 3. The complete result the Bethe-Heitler formula: σ Pair = 28 Z 2 α 3 9 m 2 ln E γ e m e which valid for a point nucleus, without screening. The nucleus Coulomb field does not extend to infinity because of the atomic electron screening. Taking th into account, the log term becomes a constant, ln(183 Z 1/3 ). m 2 e th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 28

29 Th a very important result. At high energy, photons have a finite, non-zero, mean free path in matter, almost energy independent. A naive expectation could have been that absorption dappears at high energy. Bremsstrahlung The process by which a high energy electron, or positron, radiates crossing matter identical to that of pair creation. The bremsstrahlung amplitude the same as that of pair creation. A) B) Z Z e e e Z e Z th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 29

30 High energy electrons (positrons) also have a finite, non-zero, mean free path in matter. The radiated photon spectrum 1/E. Integration gives the electron energy loss which proportional to the energy or 1 de E dx const The radiation length X 0 defined as the mean dtance over which a high energy electron looses in average a fraction 1-e of its energy. (7/9)X 0 also the mean free path for pair creation of high energy electrons. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 30

31 X 0 for various material Material Z X 0 g/cm 2 cm H 2 liq He m C Fe Pb U BGO NaI th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 31

32 1 Mb σ p.e. (b) Lead (Z = 82) experimental σ tot Cross section (barns/ atom) 1 kb σ Rayleigh κ nuc 1 b σ Compton κ e 10 mb 10 ev 1 kev 1 MeV 1 GeV 100 GeV Photon Energy Total cross section for γ in lead th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 32

33 Positrons Lead (Z = 82) de (X 1 E dx 0 ) Electrons Møller (e ) Ionization Bremsstrahlung (cm 2 g 1 ) Bhabha (e + ) Positron annihilation E (MeV) Fractional energy loss per radiation length, for electrons (positrons) in lead th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 33

34 Electromagnetic cascade At high energy both radiation and pair-production will happen repeatedly in cascade. It makes little difference whether it all started with an electron or a photon. As the cascade develops the number of particles and photons increases. Charged particles loose energy to the medium, the beginning a signal of the shower. At some point the energy of photon and electrons becomes low enough that radiation and pair production stop. EM cascade th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 34

35 (1/E 0 ) de/dt Energy Photons 1/6.8 Electrons 30 GeV electron incident on iron Number crossing plane t = depth in radiation lengths Shower profiles for fractional energy loss and number of particles. Detection of the number of particles or the energy measures the photon (or electron or positron) energy. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 35

36 EM shower developing in the plates in a Wilson Chamber. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 36

37 4. The muon and the pion In 1937 Anderson and Neddermeyer proved the extence of a particle of mass m e, it did not radiate in a 1 cm platinum plate. Street and Stevenson measured 130 m e. It could have been Yukawa s mesotron, expected mass 1/r p 200 MeV (m µ =105 MeV). The mass now MeV. In 1947 Conversi, Pancini and Piccioni observed that the lifetime of negative muons stopped in carbon equals that of positive muons. A definitive proof that the muon not Yukawa s mesotron. During the last war years the Rossi group had developed a method to select positive and negative muons in the cosmic radiation at sea level. Th was done with the so-called magnetic lenses. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 37

38 The magnetic lens Two Fe blocks, magnetized with a current in opposite directions, have a focussing effect on particles of one sign and defocussing on the opposite sign. N S th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 38

39 The Conversi Pancini Piccioni experiment A late signal from the D counters in coincidence with the A and B counters delayed, signals the arrival of a positive (negative) muon which stops in the absorber and then decays into an electron. Magnetized Fe Pb D D A B D C C C C C Absorber A B D D D D D C Pb N N' th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 39

40 So what? The muon lifetime in vacuum 2.2 µs, measured since the early days. Th immediately tells us that the decay due to the weak interaction. The Yukawa mesotrons have however strong interaction with nucleons. Th means that a mesotron interacts with a nucleon with a reaction rate of per second (s 1 ). When muons come at rest in matter, (<10 ps), positive muons can only decay as in vacuum while negative muons will bind in hydrogen-like structures. In condensed matter, thermal muons bind into S-wave orbits also in very short times. Mostly by Stark effect. th For muons in a 1S states we estimate the overlap integral as just Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 40

41 the ratio of the nucleus radius to the Bohr radius, cubed. If you do not remember the Bohr radius, begin writing the energy: E = V + T = α r + p2 /2m e. Quantize the system by the uncertainty principle, p = 1/r ( h=c=1) and find the minimum energy: d dp ( αp + p2 /2m e )= α + p/m e =0 giving p = m e α from which the Bohr radius a =1/(m e α), in natural units. Reintroducing h and c we find a (maybe??) more familiar expression a = 4πɛ 0 h 2 /m e e 2 in SI units, a = h 2 /m e e 2 in cgs units. a cm. In natural units the Rydberg constant α 2 m e /2orR = ev/( )= eV. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 41

42 The Bohr radius R Bhor =1/(m r Zα)= /Z cm and the overlap integral A( ) 3 Z 3 ( ) 3 = Z 3 A. Even for hydrogen the absorption rate for Yukawa mesons Γ abs., Yukawa = s 1 or 10 8 times larger then the decay rate. Very grossly, taking the reaction rate for µ + N N + ν e as 1/τ µ 10 6 /s, the weak µ absorbtion rate Γ weak 0.12 Z 3 A 0.24 Z 4 s 1. For carbon, Z = 6 and Γ weak 300, which much smaller than the decay rate of the muon. Positive and negative muons equally decay when stopped in carbon. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 42

43 However, for Z =(10 6 /.24) 1/4 45, the weak capture rate becomes significant. Conversi et al. did observe that for negative muons stopping in Fe, Z=26, the free decay almost not present. The estimates above are of course very crude. The Z 3 A dependence of the capture rate has been however verified (later) by measuring τ µ =1/(Γ(dec)+Γ(cap)) in different materials. From the observation of Conversi et al. the muon was born. The names π, µ are due to the Brtol nuclear emulsion group of Lattes, Occhialini and Powell. The muon a basic constituent of the so called standard model and as best as we can tell an elementary particle. The π-meson the Yukawa mesotron. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 43

44 The pion Yukawa s mesotrons was found soon after, observing the full decay chain π µ e. In fact the cosmic ray muons that reach the earth surface all come from the reaction steps: p +(A, Z) π ± +..., π µν. Pions and muons are not present in the primary radiation because of their short lifetimes. Pions hardly reach sea level because of their short lifetime and large absorption. They are found at high altitude or in interactions. Negative pions stopped in H 2 NEVER decay, as expected. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 44

45 Reflections Establhing the extence of the muon, as was done in the 47 experiment, one of the milestones of today s physics. Leptons, the electron dcovered in 1897, the muon and a few (too many) other particles are what we like today to call elementary particles. The pion today has lost its status. It just another of the few hundred states made up of a quark and an antiquark. It not an elementary particle. Conversi was my teacher in the early fifties and I have always admired h passion for physics. th Karlsruhe, Fall 2002 Paolo Franzini -... to the muon 45