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PHYS 5012 Radiation Physics and Dosimetry

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Radiation Physics Lecture 4 Interactions of Charged Particles with Matter General Aspects Nuclear Reactions Elastic Scattering PHYS 5012 Inelastic Scattering Stopping Power Collision Stopping Power Radiation Physics and Dosimetry (Heavy Particles) Collision Stopping Power (Light Particles) Lecture 4 Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power Tuesday 27 March 2012 Radiation Physics Lecture 4 Interactions of Charged Particles with Matter Interactions of Charged Particles with Matter General Aspects Nuclear Reactions Elastic Scattering Inelastic Scattering Stopping Power Collision Stopping Power (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power Radiation Physics Lecture 4 General Aspects of two-particle Collisions Interactions of Charged Particles Collisions between two particles involve a projectile and a with Matter General Aspects target. Nuclear Reactions Elastic Scattering Inelastic Scattering Types of targets: whole atoms, atomic nuclei, atomic Stopping Power Collision Stopping Power orbital electrons, free electrons. (Heavy Particles) Collision Stopping Power (Light Particles) Types of projectiles: Radiative Stopping Power Total Mass Stopping Power I heavy charged particles (protons, α-particles, heavy Range Mean Stopping Power ions) I light charged particles (electrons, positrons) I photons (considered previously) I neutrons (not considered here) Henceforth, we will consider only charged particle projectiles. Two-particle collisions are then Coulomb collisions. Radiation Physics Lecture 4 Two-Particle Collisions Interactions of Charged Particles with Matter General Aspects 3 categories: Nuclear Reactions Elastic Scattering Inelastic Scattering 1. Nuclear reactions – final reaction products differ from Stopping Power Collision Stopping Power initial particles; charge, momentum and mass-energy (Heavy Particles) Collision Stopping Power conserved; e.g. deuteron bombarding nitrogen-14: (Light Particles) 14 15 Radiative Stopping Power 7 N(d, p)7 N Total Mass Stopping Power Range 2. Elastic collisions – final products identical to initial Mean Stopping Power particles; kinetic energy and momentum conserved; e.g. Rutherford scattering of α particle on gold 197 197 nucleus: 79 Au(α, α)79 Au 3. Inelastic collisions – final products identical to initial particles; kinetic energy not conserved Radiation Physics Lecture 4 Interactions of Charged Particles with Matter In inelastic collisions, some kinetic energy is converted to General Aspects excitation energy in the form of: Nuclear Reactions Elastic Scattering Inelastic Scattering I nuclear excitation of target resulting from heavy Stopping Power Collision Stopping Power charged particle striking target nucleus; e.g. (Heavy Particles) A A ∗ Collision Stopping Power X(α, α) X (Light Particles) Z Z Radiative Stopping Power Total Mass Stopping I atomic excitation or ionisation of target resulting from Power Range heavy or light charged particle colliding with target Mean Stopping Power orbital electron I bremsstrahlung emission by light charged particle projectile resulting from Coulomb interaction with target nucleus Radiation Physics Lecture 4 Nuclear Reactions Interactions of Charged Particles with Matter General Aspects Nuclear Reactions Elastic Scattering Inelastic Scattering Stopping Power Collision Stopping Power (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power Schematic illustration of a general nuclear reaction. (Fig. 5.1 in Podgoršak.) I intermediate compound produced temporarily; spontaneously decays into reaction products P P I conservation of atomic number: Zbefore = Zafter P P I conservation of atomic mass: Abefore = Aafter Radiation Physics Conservation of momentum Lecture 4 p1 = p3 + p4 (1) Interactions of Charged Particles with Matter p1 = p3 cos θ + p4 cos φ k to p1 General Aspects =⇒ Nuclear Reactions 0 = p3 sin θ + p4 sin φ ⊥ to p1 Elastic Scattering Inelastic Scattering Stopping Power Conservation of mass-energy Collision Stopping Power (Heavy Particles) Collision Stopping Power 2 2 2 2 (Light Particles) m1c + EK,1 + m2c = m3c + EK,3 + m4c + EK,4 (2) Radiative Stopping Power Total Mass Stopping Power = = (γ − ) 2 Range where EK particle kinetic energy 1 mc Mean Stopping Power 2 2 2 2 Q = m1c + m2c − m3c + m4c Q value (3) Also, Q = EK,final − EK,initial I Q > 0 ⇒ exothermic collision I Q = 0 ⇒ elastic collision I Q < 0 ⇒ endothermic collision Radiation Physics Lecture 4 Threshold Energy Interactions of Charged Particles A minimum projectile energy Ethr is required for an with Matter General Aspects endothermic reaction to proceed. Nuclear Reactions Conservation of 4-momentum, = ( / , ): Elastic Scattering p E c p Inelastic Scattering Stopping Power 2 2 Collision Stopping Power p + p = p + p ⇒ (p + p ) = (p + p ) (Heavy Particles) 1 2 3 4 1 2 3 4 Collision Stopping Power (Light Particles) 2 2 2 2 2 Radiative Stopping Power and using p = (E1/c) − |p1| = m c and Total Mass Stopping 1 1 Power p2 = (E /c)2 = m2c2, gives Range 2 2 2 Mean Stopping Power 2 2 2 4 2 4 2E1E2 = (p3 + p4) c − (m1c + m2c ) Note that p3 + p4 is the centre-of-mass 4-momentum, pcm, 2 2 2 2 2 2 2 and so (p3 + p4) = pcm = (Ecm/c) = (m3c + m4c ) /c since the modulus of a 4-vector is invariant and has the same value in any frame of reference. So the threshold energy E1 for the projectile is: Radiation Physics Lecture 4 2 2 2 2 4 2 4 (m3c + m4c ) − (m1c + m2c ) Ethr = 2 (4) Interactions of 2m2c Charged Particles with Matter General Aspects corresponding to a threshold kinetic energy: Nuclear Reactions Elastic Scattering 2 2 2 2 2 2 Inelastic Scattering (m3c + m4c ) − (m1c + m2c ) Stopping Power EK,thr = (5) Collision Stopping Power 2 (Heavy Particles) 2m2c Collision Stopping Power (Light Particles) Radiative Stopping Power in terms of the Q value: Total Mass Stopping Power Range 2 2 Mean Stopping Power m1c + m2c Q EK,thr = −Q 2 − 2 (6) m2c 2m2c 2 If Q m2c (as is often the case), then m1 EK,thr ≈ −Q 1 + (7) m2 The Q value is defined for general two-particle collisions. Radiation Physics Lecture 4 Interactions of Charged Particles with Matter General Aspects Nuclear Reactions Example: pair production and triplet production. Elastic Scattering Inelastic Scattering For pair production, m1 = 0, m2 = m3 me and Q = Stopping Power 2 Collision Stopping Power −2mec , so (Heavy Particles) pp 2 Collision Stopping Power E = 2mec (Light Particles) γ thr Radiative Stopping Power Total Mass Stopping 2 Power For triplet production, Q = −2mec but m2 = me, so Range Mean Stopping Power Etp = 4m c2 γ thr e Radiation Physics Lecture 4 Elastic Scattering Interactions of Charged Particles I initial and final particles remain the same (i.e. with Matter General Aspects m3 = m1 and m4 = m2), so Q = 0 Nuclear Reactions Elastic Scattering I kinetic energy transfer ∆EK from m1 to m2 Inelastic Scattering Stopping Power Collision Stopping Power I total kinetic energy conserved (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power Schematic illustration of elastic scattering. θ is the scattering angle, φ is the recoil angle and b is the impact parameter. (Fig. 5.2 in Podgoršak.) Radiation Physics Lecture 4 Classical kinematics Interactions of Charged Particles with Matter General Aspects Kinetic energy transfer determined from conservation of Nuclear Reactions Elastic Scattering momentum and energy: Inelastic Scattering Stopping Power Collision Stopping Power 1 2 4m1m2 2 (Heavy Particles) ∆EK = m2u2 = EK1 2 cos φ (8) Collision Stopping Power 2 (m + m ) (Light Particles) 1 2 Radiative Stopping Power Total Mass Stopping Power Head-on collisions: Range Mean Stopping Power I b = 0 and φ = 0 I maximum energy and momentum transfer I θ = 0 (forward scattering) when m1 > m2 I θ = π (back-scattering) when m1 < m2 I projectile stops when m1 = m2 Radiation Physics Lecture 4 Example: proton colliding with orbital electron. Interactions of Charged Particles Maximum energy transfer (for a head-on collision), noting with Matter General Aspects that mp me: Nuclear Reactions Elastic Scattering Inelastic Scattering me Stopping Power −3 ∆Emax ≈ 4EK ≈ 2 × 10 EK Collision Stopping Power p p (Heavy Particles) mp Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Collisions between particles of the same mass (m = m ): Power 1 2 Range Mean Stopping Power I distinguishable particles (e.g. electron colliding with positron): ∆Emax = EK1 ⇒ head-on collision transfers all projectile’s kinetic energy to target I indistinguishable particles (e.g. free electron colliding 1 with bound electron): ∆Emax = 2 EK1 Radiation Physics Lecture 4 Interactions of Charged Particles with Matter General Aspects Relativistic formula for energy transfer in a head-on Nuclear Reactions Elastic Scattering collision: conservation of mass–energy ⇒ Inelastic Scattering Stopping Power Collision Stopping Power (Heavy Particles) Collision Stopping Power (Light Particles) 2 2(γ + 1)m1m2 Radiative Stopping Power ∆Emax = (γ2 − 1)m2c = 2 2 EK1 (9) Total Mass Stopping m + m + 2γm1m2 Power 1 2 Range Mean Stopping Power 2 where γm1c = initial energy of incident projectile, 2 γ1m1c = final energy of incident projectile and 2 γ2m2c = final energy of target particle. Radiation Physics Lecture 4 Classical Rutherford scattering Rutherford scattering is the elastic
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