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Lectures 10-11

Lectures 10-11. Particle interactions with matter. 9.0 Overview. 9.3 Photons in matter Overview Photoelectric effect Raleigh scattering Compton scattering Pair production Comparison of cross-sections Z & A dependence of cross-sections

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Lectures 10-11

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  1. Lectures 10-11 Particle interactions with matter Nuclear Physics Lectures, Dr. Armin Reichold

  2. 9.0 Overview • 9.3 Photons in matter • Overview • Photoelectric effect • Raleigh scattering • Compton scattering • Pair production • Comparison of cross-sections • Z & A dependence of cross-sections • Similarities between pair production and Bremsstrahlung • 9.1 Introduction • 9.2 Charged particles in matter • Classification of interactions • Non-radiating interactions (ionisation) • Radiating interactions • Ionisation and the Bethe-Bloch formula (BBF) End of lecture 9 • Radiating interactions • Cherenkov-radiation • Bremsstrahlung • Synchrotron-radiation • The em-shower Nuclear Physics Lectures, Dr. Armin Reichold

  3. 12.1 Introduction(why do we need to know this) • Measure properties of nuclei through decay products • Measure energy, momentum, mass & charge of particles with • M  [0 (g) ; few 100 GeV (fission fragment)] • Ekin [keV (Radioactivity) ; few GeV (accelerator experiments)] • Q/e  [0 (g,n); O(100) (fission fragments)] • Need to translate microscopic particle properties into quantitatively measurable macroscopic signals • Do this by interactions between particles and matter • Which interactions would be useful? • Weak?  Too weak at low (nuclear) interaction energies • Strong?  Some times useful but often noisy (strong fluctuations, few interactions per distance) • EM?  Underlies most nuclear and particle physics detectors (L9&10) • Energies released ≤ Ekin(particle) often too small for direct detection  need amplification of signals (see detector section L11) Nuclear Physics Lectures, Dr. Armin Reichold

  4. a) b) c) 12.1 Introduction • Particle Ranges • If smooth energy loss via many steps (i.e. ionisation from light ions)  sharply defined range, useful for rough energy measurement • If a few or a single event can stop the particle (i.e. photo-effect)  exponential decay of particle beam intensity,  decay constant can have useful energy dependence  No range but mean free path defined • Sometimes several types of processes happen (i.e. high energy electrons)  mixed curves, extrapolated maximum range Nuclear Physics Lectures, Dr. Armin Reichold

  5. 12.1 Introduction • Particles we are interested in • photons • exponential attenuation at low E, often get absorbed in single events • detect secondary electrons and ions liberated in absorption process. • charged particles • sharper range (continuously loose energy via ionisation) • leave tracks of ionisation in matter  measure momentum in B • sometimes radiate photons  can be used to identify particle type • neutrons • electrically neutral  no first-order em-interaction  devils to detect • react only via strong force (at nuclear energies!) • long exponential range (lots of nuclear scattering events followed by absorption or decay) • need specific nuclear reactions to convert them into photons and/or charged particles when captured by a target nucleus • if stopped, measure decay products, e- + p + n Nuclear Physics Lectures, Dr. Armin Reichold

  6. 12.2 Charged particles in matter(classification of interactions) • If particle or medium emit photons, coherent with incoming particle  radiation process • Bremsstrahlung, Synchrotron-radiation: emitted by particle • Cherenkov-radiation: emittted by medium • If no coherent radiation  non-radiating process • Ionisation, scattering of nuclei or atoms • Note: Scintillation is a secondary process in which the light is emitted after ionisation or atomic excitation. It is NOT a radiation process Nuclear Physics Lectures, Dr. Armin Reichold

  7. 12.2 Charged particles in matter(non radiating interactions, what to collide with) • What could a charged particle collide with • Atomic electrons (“free”)  large energy loss DE≈q2/2me (small me, q=momentum transfer)  small scattering angle • Nuclei  small energy loss (DE=q2/2mnucleus)  large scattering angle • Unresolved atoms (predominant at low energies)  medium energy loss DE<q2/2meeff because: meeff(bound)>me(free)  medium scattering angle  atoms get excited and will later emit photons (scintillation) Nuclear Physics Lectures, Dr. Armin Reichold

  8. 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula) • Deal with collisions with electrons first since these give biggest energy loss. • Task: compute rate of energy loss per path length, dE/dx due to scattering of a charged particle from electrons in matter. • Remember a similar problem? • Scatter alpha particles of nuclei = Rutherford scattering Nuclear Physics Lectures, Dr. Armin Reichold

  9. Rutherford Scattering any charged particle X (original used a’s) scatters of nucleus Charge(X)=Ze Charge(nucleus)=Z’e Mnucl >> MX no nuclear-recoil first order perturbation theory (Z*Z’*aem<<1) point  point scattering  no form-factors Bethe-Bloch situation any charged particle X scatters of electron (in matter) Charge(X)=Ze Charge(electron)=1e MX >> Me no X-recoil (not true for X=e-) first order perturbation theory (Z*1*aem<<1) point  point scattering  no form-factors commonalities differences 12.2 Charged particles in matter(Comparison between Rutherford Scattering and EM-scattering of free electrons) • spin-0scatters of spin-½ • could be relativistic • electron is often bound • spin-0 scatters ofspin-0 • non-relativistic • nucleus assumed unbound Nuclear Physics Lectures, Dr. Armin Reichold

  10. 12.2 Charged particles in matter(Comparison between Rutherford Scattering and EM-scattering of free electrons) • Will initially ignore the spin and relativistic effects when deriving first parts of Bethe Bloch formula • Start with Rutherford like scattering using electron as projectile • Later introduce more realistic scattering crossection (Mott) to get full Bethe Bloch formula • Add effects for bound electrons at the end Nuclear Physics Lectures, Dr. Armin Reichold

  11. 12.2 Charged particles in matter(From Rutherford Scattering to the Bethe-Bloch Formula) • Differential Rutherford-scattering crossection for electrons as projectiles P,V = momentum and relative velocity of electron wrt. nucleus Z = charge of nucleus q = scattering angle of the electron wrt. incoming electron direction W= stereo angle • If we want to turn this process around to describe energy loss of a particle X scattering of electrons in a solid we need to initially assume: • X scatters of free electrons i.e. Ekin,projectile >> Ebin,electronor Vprojectile>>Vbound-e (deal with bound electrons later) • MX>>me so that reduced Mreduced(X) ≈ Mrest(X)  will need recoil corrections to apply results to dE/dx of electrons passing through matter Nuclear Physics Lectures, Dr. Armin Reichold

  12. P’electron,out q q Pelectron,in 12.2 Charged particles in matter(normal Rutherford Scattering: e- on nucleus, change of variables) • Change variables from W to q2 (q = momentum transfer to electron) to get to frame independent form Nuclear Physics Lectures, Dr. Armin Reichold

  13. 12.2 Charged particles in matter(normal Rutherford Scattering: e- on nucleus, change of variables) Nuclear Physics Lectures, Dr. Armin Reichold

  14. 12.2 Charged particles in matter(Rutherford Scattering, change of frame to nucleus on e) • Change frame to: • electron stationary (in matter), nucleus moving with V towards electron • p in formula is still momentum of electron moving with relative V p =megV • q2is frame independent • non-relativistic this is obvious (do it at home) • Energy transfer to the electron is defined via: • DE=n=|q2|/2me dn/dq2=1/2me • relativistic need to define q as 4-momentum transfer, but we assume non relativistic for Rutherford anyway. Nuclear Physics Lectures, Dr. Armin Reichold

  15. |q2|=2nme number of collissions with electrons in length dx per unit crossection area crossection weighted avg. energy lost per collision 12.2 Charged particles in matter(From inverse Rutherford scattering to the Bethe-Bloch Formula) • Above is crossection for a non relativistic heavy particle of charge z to loose energy between n and n+dn in collision with a spin-less electron it approaches with velocity V • We want as a useful quantity: • kinetic energy lost by projectile = -dT • per path length dx • in material of atomic number density n • with Z’ electrons per atom Nuclear Physics Lectures, Dr. Armin Reichold

  16. 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, simple integral) • Two of our assumptions justifying the use of Rutherford scattering were: • Electrons in matter have no spin • Projectile travels at non relativistic speed • None of these are met in practise • We have to do all of the last 5 slides again starting from a relativistic crossection for spin ½ electrons. Nuclear Physics Lectures, Dr. Armin Reichold

  17. Rutherford term Mott term 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, Mott) • Differential Mott-scattering crossection for relativistic spin ½ electrons scattering off a finite mass nucleus (finite mass  e- could be target) • If we perform the same transformations (Wq2n) with this crossection and then perform the integral: • we get … Nuclear Physics Lectures, Dr. Armin Reichold

  18. Mott term A list of limits for nmax follows: 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, Mott integral) • Valid for all charged particles (not limited to heavy particles) • nmax can be computed via kinematics of “free” electron since Ebind <<Ekin (see Williams problem 11.1 on p.246) Nuclear Physics Lectures, Dr. Armin Reichold

  19. 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, nmin) • But what about nmin ? • can not assume that e is free for small energy transfers • n≠q2/2me because electron bound to atom • can get excited atoms in final state (not just ions)  our integral was wrong for the lower limit! (can’t get from first to second line on slide 15 any more) • For small n need 2-D integraldn dq depending on detailed atomic structure • We need to find some average description of the atomic structure depending only on Z and A if we want to find a universal formula • This gives sizable fraction of integral but is very hard to do • The result is the Bethe-Bloch Formula Nuclear Physics Lectures, Dr. Armin Reichold

  20. 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula = BBF) • Stopping power = mean energy lost by ionisation upon perpendicularly traversing a layer of unit mass per area. • Units: Mev g-1 cm2, Range: 4.1 in H to 1.1 in U • I=mean excitation energy; depends on atom type, I≈11*Z [eV] Nuclear Physics Lectures, Dr. Armin Reichold

  21. 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, Bethe-Bloch features) • d=density correction: dielectric properties of medium shield growing range of Lorenz-compacted E-field that would reach more atoms laterally. Without this the stopping power would logarithmically diverge at large projectile velocities. Only relevant at very large bg • BBF as a Function of bg is nearly independent of M of projectile except for nmax and very weak log dependence in d  if you know p and measure b  get M (particle ID via dE/dx): See slide 23 • Nearly independent of medium. Dominant dependence is Z’/A ≈½ for most elements. • Limitations: • totally wrong for very low V (ln goes negative  particle gains Energy = stupid) • correct but not useful for very large V (particle starts radiating, see next chapter) (off syllabus) Nuclear Physics Lectures, Dr. Armin Reichold

  22. m+ can capture e- Bethe Bloch 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, variation with bg) • Broad minimum @ bg≈3.0(3.5) for Z=100(7) • At minimum, stopping power is nearly independent of particle type and material Emc = critical energy defined via: dE/dxion.=dE/dxBrem. • Stopping Power at minimum varies from 1.1 to 1.8 MeV g-1 cm2) • Particle is called minimum ionising (MIP) when at minimum Nuclear Physics Lectures, Dr. Armin Reichold

  23. in drift chamber gas 12.2 Charged particles in matter(Ionisation and the Bethe-Bloch Formula, variation with particle type) • P=mgv=mgbc • variation in dE/dx is useful for particle ID • variation is most pronounced in low energy falling part of curve • if you measured P and dE/dx you can determine the particle mass and thus its “name” e Nuclear Physics Lectures, Dr. Armin Reichold

  24. 12.2 Charged particles in matter(Radiating Interactions) • Emission of scintillation light is secondary process occurring later in time. • Has no phase coherence with the incident charge and is isotropic and thus SCINTILLATION NOT A RADIATING INTERACTION in this sense. • Primary radiation processes which are coherent and not isotropic are: • Cherenkov radiation is emitted by the medium due to the passing charged particle. • Bremsstrahlung and Synchrotron Radiation are emitted by charged particle itself as result of its environment. Nuclear Physics Lectures, Dr. Armin Reichold

  25. A ct/n  ct particle trajectory O P 12.2 Charged particles in matter(Cherenkov Radiation) • Source of E-field (Q) passing through medium at a v > vphase(light in medium) creates conical shock wave. Like sonic boom or bow wave of a planing speed boat. • Not possible in vacuum since v<c. Possible in a medium when v>c/n. • The Cerencov threshold at  = 1/n can beused to measure b and thus do particle ID if you can measure the momentum as well. • Huygens secondary wavelet construction gives angle of shockwave as cos = 1/n, This can be used to measure particle direction and b. • In time that the particle goes from O to P, light goes from O to A. • Cherenkov radiation first used in discovery of antiproton (1954). • Now often used in large water-filled neutrino detectors and for other particle physics detectors (see Biller). • Total energy emitted as Cherenkov Radiation is ~0.1% of other dE/dx. Nuclear Physics Lectures, Dr. Armin Reichold

  26. 12.2 Charged particles in matter(Cherenkov Radiation) • Picture of Cherenkov light emitted by beta decay electrons in a working water cooled nuclear reactor. Nuclear Physics Lectures, Dr. Armin Reichold

  27. g e-* e- e- Ze 12.2 Charged particles in matter(Bremsstrahlung = BS = Brake-ing Radiation) • Due to acceleration of incident charged particle in nuclear Coulomb field • Radiative correction to Rutherford Scattering. • Continuum part of x-ray emission spectra. • Electrons “Brem” most of all particles because • radiation ~ (acceleration)2 ~ mass-2. • Lorentz transformation of dipole radiation from incident particle frame to laboratory frame gives “narrow” (not sharp) cone of blue-shifted radiation centred around cone angle of =1/. • Radiation spectrum falls like 1/E (E=photon Energy) because particles loose many low-E photons and few high-E photons. I.e. It is rare to hit nuclei with small impact parameter because most of matter is “empty” • Photon energy limits: • low energy (large impact parameter) limited through shielding of nuclear charge by atomic electrons. • high energy limited by maximum incident particle energy. Nuclear Physics Lectures, Dr. Armin Reichold

  28. 12.2 Charged particles in matter(Bremsstrahlung  EM-showers, Radiation length) • dT/dx|Brem~T (see Williams p.247, similar to our deriv. of BBF and plot on slide 22) dominates over dT/dx|ionise ~ln(T) at high T. • Ecrit = Energy at which BR-losses exceed ionisation losses (see slide 22) • For electrons Bremsstrahlung dominates in nearly all materials above few 10 MeV. Ecrit(e-) ≈ 600 MeV/Z • If dT/dx|Brem~T  T(x)=T0 exp(-x/X0) • Radiation Length X0 of a medium is defined as: • distance over which electron energy reduced to 1/e via many small BS-losses • X0 ~Z 2 approximately as it is the charge that particles interact with • Bremsstrahlung photon can produce e+e--pair (see later) and start an em-shower (also called cascade, next slide)  The development of em-showers, whether started by primary e or  is measured in X0. Nuclear Physics Lectures, Dr. Armin Reichold

  29. 12.2 Charged particles in matter(simple EM-shower model) • Simple shower model assumes: • e≈2 • E0 >> Ecrit • only single Brem-g or pair production per X0 • The model predicts: • after 1 X0, ½ of E0 lost by primary via Bremsstrahlung • after next X0 both primary and photon loose ½ E again • until E of generation drops below Ecrit • At this stage remaining Energy lost via ionisation (for e+-) or compton scattering, photo-effect (for g) etc. • Abrupt end of shower happens at t=tmax = ln(E0/Ecrit)/ln2 • Indeed observe logarithmic dependence of shower depth on E0 Nuclear Physics Lectures, Dr. Armin Reichold

  30. 12.2 Charged particles in matter(Synchroton Radiation) • Appears mainly in circular accelerators (mainly to electrons) and limits max. energy achievable. • Similar to Bremsstrahlung • Replace microscopic force from E-field in Bremsstrahlung with macroscopic force from vxB to keep electron on circular orbit • Electrons radiate only to the outside of circle because they are accelerated inward • Angle of maximum intensity of synchrotron radiation with tangent of ring =1/ • Synchrotron radiation = very bright source of broad range of photon energies up to few 10 keV used in many areas of science • Many astrophysical objects emit synchrotron radiation from relativistic electrons in strong magnetic fields Nuclear Physics Lectures, Dr. Armin Reichold

  31. 13.1 Photons in matter(Overview-I) • Rayleigh scattering • Coherent, elastic scattering on the entire atom (the blue sky) • g + atom  g + atom • dominant at lg>size of atoms • Compton scattering • Incoherent scattering on electron from atom • g + e-bound g + e-free • possible at all Eg > min(Ebind) • to properly call it Compton requires Eg>>Ebind(e-) to approximate free e- • Photoelectric effect • absorption of photon and ejection of single atomic electron • g + atom  g + e-free + ion • possible for Eg < max(Ebind) + dE(Eatomic-recoil, line width) (just above k-edge) Nuclear Physics Lectures, Dr. Armin Reichold

  32. 13.1 Photons in matter(Overview-II) • Pair production • absorption of g in atom and emission of e+e-pair • Two varieties: • a) dominant: g+ nucleus e+ + e- + nucleusrecoil • b) weak: g + Z*atomic e- e+ + e- + Z *atomic e-recoil • Both variants need: Eg>2mec2 + Erecoil • bigger Mrecoil gives lower threshold because Erecoil = Precoil2/2Mrecoil • type a) has lower threshold then type b) because Mnucl>>Meeff • Nucleus/atom has to recoil to conserve momentum  coupling to nucleus/atom needed  strongly charge-dependent crossection (i.e. growing with Z) • type a) has aproximately Z times larger coupling  dominant Nuclear Physics Lectures, Dr. Armin Reichold

  33. 13.1 Photons in matter (Crossections) Lead R  Rayleigh PE  Photoeffect C  Compton PP  Pair Production on nucleus PPE  Pair Production on atomic electrons PN  Giant Photo-Nuclear dipole resonance Carbon • As Z increases • PE extends to higher E due to stronger atomic e- binding • PP & PPE extend to lower E due to stronger coupling of projectile to target • Threshold for PPE decreases as nucleus contributes more to recoil via stronger atomic electron-nucleus bond • As A increases Erecoil (nucleus) decreases and threshold for PP gets closer to minimum of 2*mec2 Nuclear Physics Lectures, Dr. Armin Reichold

  34. Pair production Bremsstrahlung Typical Lenth = Pair Production Length L0 Typical Lenth = Radiation Length X0 e- e- g g e-* e-* e- e- Ze Ze 13.1 Photons in matter(Comparison of Bremsstrahlung and Pair Production) X0 : distance high E e- travels before it reduces its energy by 1/e or E(e-)=E0*exp(-x/X0) X0=attenuation length L0 : distance high E g travels before prob. for non interaction reduced to 1/e P(g)=1/L0*exp(-x/L0) L0=mean free path • Very similar Feynman Diagram • Just two arms swapped L0=9/7 X0 Nuclear Physics Lectures, Dr. Armin Reichold

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