PHYS 5012 Radiation Physics and Dosimetry

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

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 scattering of a point Interactions of Charged Particles charge by a stationary fixed point charge. The original with Matter General Aspects Geiger-Marsden experiment, which led Rutherford to Nuclear Reactions Elastic Scattering propose the currently accepted Rutherford–Bohr atomic Inelastic Scattering Stopping Power model, was conducted with α particles scattering off gold Collision Stopping Power (Heavy Particles) foil. The classical derivation of the differential Collision Stopping Power (Light Particles) cross-section is cumbersome and requires knowledge of Radiative Stopping Power Total Mass Stopping Power the trajectory of the α as it is deflected by the Coulomb Range Mean Stopping Power field of the gold nuclei. This depends on knowing the position and momentum of the charge at all times, which is forbidden in quantum mechanics. The classical derivation gives

dσ zZ c2 α 2 θ = ~ sin−4 (10) dΩ 4 EK 2 where z is the charge of the incident projectile (2 for an alpha particle) and α is the ﬁne structure constant. Radiation Physics Lecture 4 Quantum derivation

Interactions of Recall from Lec. 2 that general problem of scattering is to Charged Particles with Matter ﬁnd, for a given initial state i, the probabilities of various General Aspects Nuclear Reactions ﬁnal states f . 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

Fermi’s Golden Rule is equivalent to the ﬁrst order Born approximation:

dσ 1 = |M |2 (11) dΩ 64π2 ﬁ for elastic scattering. Radiation Physics Lecture 4

Interactions of Charged Particles with Matter The transition matrix element amplitude Mﬁ is calculated General Aspects Nuclear Reactions using spherical plane waves of the form Elastic Scattering Inelastic Scattering Stopping Power Collision Stopping Power Ψ(r) ∝ exp(ip · r/~) (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power and a screened Coulomb potential known as the Yukawa Total Mass Stopping Power potential: Range 2 Mean Stopping Power zZe V(r) = exp(−ηr) 4π0r This gives the same result for dσ/dΩ as the classical derivation, eqn. (11), but is much less lengthy to derive. Radiation Physics Lecture 4 Inelastic collisions

Interactions of I hard collisions: Coulomb interactions with orbital Charged Particles with Matter electron for b ≈ a General Aspects Nuclear Reactions I soft collisions: Coulomb interaction with orbital Elastic Scattering Inelastic Scattering electron for b a Stopping Power Collision Stopping Power (Heavy Particles) I radiative collisions: Coulomb interactions with Collision Stopping Power (Light Particles) nuclear ﬁeld for b a Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power

The three different types of collisions depend on the classical impact paramater b and atomic radius a. (Fig. 6.1 in Podgoršak.) Radiation Physics Lecture 4 Stopping Power

I Stopping power measures how readily charged Interactions of Charged Particles particles come to rest in matter with Matter General Aspects I incident charged particle loses all kinetic energy via Nuclear Reactions Elastic Scattering multiple Coulomb interactions (mostly elastic, but Inelastic Scattering Stopping Power sometimes inelastic) Collision Stopping Power (Heavy Particles) I gradual loss of kinetic energy called continuous Collision Stopping Power (Light Particles) slowing down approximation (CSDA) Radiative Stopping Power Total Mass Stopping I e.g. 1 MeV charged particle typically undergoes Power 5 Range ∼ 10 interactions before losing all its kinetic energy Mean Stopping Power Radiation Physics Lecture 4 Linear stopping power, dE/dx = rate of energy loss per unit path length of charged particle Interactions of Charged Particles −1 with Matter Mass stopping power, S = −ρ dE/dx, is the commonly General Aspects 2 −1 Nuclear Reactions used measure of stopping power (in units MeV m kg ) Elastic Scattering Inelastic Scattering Stopping Power 2 types of stopping powers: Collision Stopping Power (Heavy Particles) Collision Stopping Power 1. Collision stopping power, Scol – for hard and soft (Light Particles) Radiative Stopping Power collisions involving both light and heavy charged Total Mass Stopping Power particles; can result in atomic excitation and Range Mean Stopping Power ionisation

2. Radiative stopping power, Srad – for radiative collisions; only light charged particles (i.e. electrons and positrons) experience appreciable energy losses; can result in bremsstrahlung emission

Stot = Scol + Srad total stopping power Radiation Physics Lecture 4 Collision Stopping Power for Heavy Charged Particles Interactions of Charged Particles < with Matter I for Ei ∼ 10 MeV, heavy charged particles undergo soft General Aspects Nuclear Reactions and hard collisions Elastic Scattering Inelastic Scattering small angle scattering (θ ' 0) Stopping Power I Collision Stopping Power (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power

Schematic diagram of a heavy charged particle collision with an orbital electron. The scattering angle θ is exaggerated for clarity. (Fig. 6.3 in Podgoršak.) Radiation Physics Lecture 4 Classical Derivation Momentum transfer: Interactions of Charged Particles Z Z ∞ with Matter General Aspects ∆p = F∆pdt = Fcoul cos φ dt Nuclear Reactions −∞ Elastic Scattering 2 −2 Inelastic Scattering where Fcoul = (ze /4πε0)r , giving Stopping Power Collision Stopping Power 2 +(π−θ)/2 (Heavy Particles) ze Z cos φ dt Collision Stopping Power ∆p = dφ (Light Particles) 2 Radiative Stopping Power 4πε0 −(π−θ)/2 r dφ Total Mass Stopping Power Range Hyperbolic particle trajectory ⇒ angular displacement Mean Stopping Power varies with time ⇒ dφ/dt = ω and conservation of angular 2 momentum requires L = Mv∞b = Mωr ⇒ ze2 1 Z +(π−θ)/2 ∆p = cos φ dφ 4πε0 v∞b −(π−θ)/2 ze2 1 θ = 2 cos 4πε0 v∞b 2 ze2 1 ≈ 2 (12) 4πε0 v∞b Radiation Physics Lecture 4

Interactions of Charged Particles with Matter General Aspects Nuclear Reactions Energy transferred to electron in a single collision with Elastic Scattering Inelastic Scattering impact parameter b: Stopping Power Collision Stopping Power (Heavy Particles) 2 2 2 2 Collision Stopping Power (∆p) e z (Light Particles) ∆E(b) = = 2 (13) Radiative Stopping Power πε 2 2 Total Mass Stopping 2me 4 0 mev∞b Power Range Mean Stopping Power Total energy loss obtained by integrating ∆E(b) over all possible b and accounting for all electrons available for interactions. Radiation Physics Lecture 4

Interactions of Charged Particles no. electrons in volume annulus between b and b + db with Matter General Aspects Nuclear Reactions = no. electrons per unit mass × mass in annulus Elastic Scattering Inelastic Scattering Stopping Power ZNA Collision Stopping Power (Heavy Particles) ⇒ ∆n = dm Collision Stopping Power A (Light Particles) Radiative Stopping Power Total Mass Stopping where Power Range Mean Stopping Power dm = ρ dV = ρ[π(b + db)2∆x − πb2∆x] ≈ 2π ρ b db ∆x

⇒ ∆n ≈ 2π ρ (ZNA/A) b db ∆x Multiply ∆E(b) by this and integrate over b to get the total energy transfer to electrons. Radiation Physics Lecture 4

Interactions of Charged Particles Mass collision stopping power with Matter General Aspects Nuclear Reactions Elastic Scattering 2 2 2 Z bmax Inelastic Scattering 1 dE ZNA e z db Stopping Power Scol = − = 4π 2 Collision Stopping Power ρ dx A 4πε0 mev b (Heavy Particles) ∞ bmin Collision Stopping Power 2 4 2 (Light Particles) ZNA re mec z bmax Radiative Stopping Power = 4π ln (14) Total Mass Stopping 2 Power A v∞ bmin Range Mean Stopping Power 2 I Scol ∝ z , where z = atomic number of heavy charged particle (e.g. z = 2 for an α particle) −2 I Scol ∝ v∞ , where v∞ = initial velocity of heavy charged particle Radiation Physics Lecture 4

Interactions of I bmax ⇔ ∆Emin = minimum energy transfer Charged Particles with Matter corresponding to minimum excitation or ionisation General Aspects Nuclear Reactions potential of orbital electron from (13) Elastic Scattering Inelastic Scattering 2 4 2 re mec z Stopping Power ∆E = 2 = I (15) Collision Stopping Power min 2 2 (Heavy Particles) v∞bmax Collision Stopping Power (Light Particles) Radiative Stopping Power I = mean ionisation-excitation potential of medium Total Mass Stopping Power Range −2/3 Mean Stopping Power I ≈ 9.1Z(1 + 1.9Z ) eV (16) e.g. I ≈ 78 eV for carbon. But (16) is poor approximation for compounds because chemical bonds are neglected (e.g. I ≈ 75 eV for water). Typical I values for various compounds of interest are listed in Table 6.4 in the textbook. Radiation Physics Lecture 4 I bmin ⇔ ∆Emax = maximum energy transfer corresponding to head-on collisions: Interactions of Charged Particles me 2 ∆Emax ≈ 4 EK,i = 2mev (for M me), so with Matter M ∞ General Aspects Nuclear Reactions Elastic Scattering 2 2 2 Inelastic Scattering e z 2 Stopping Power ∆E = 2 = 2m v (17) max 2 2 e ∞ Collision Stopping Power 4πε0 m v b (Heavy Particles) e ∞ min Collision Stopping Power (Light Particles) Putting together (15) and (17) gives Radiative Stopping Power Total Mass Stopping Power 1/2 Range 1/2 2 bmax ∆Emax 2mev Mean Stopping Power = = ∞ (18) bmin ∆Emin I =⇒ classical collision stopping power for heavy charged particles:

2 2 2 2 ZNA re mec z 1 2mev∞ Scol = 4π 2 2 ln (19) A v∞/c 2 I Radiation Physics Lecture 4

Interactions of Charged Particles with Matter Generalised solution for the collision stopping power for General Aspects Nuclear Reactions heavy charged particles: Elastic Scattering Inelastic Scattering Stopping Power 2 2 2 Collision Stopping Power NA re mec z (Heavy Particles) Scol = 4π 2 Bcol (20) Collision Stopping Power A (v∞/c) (Light Particles) Radiative Stopping Power Total Mass Stopping Power z2 Range ≈ 3.070 × 10−5 B MeV m2 kg−1 Mean Stopping Power Aβ2 col

with A in units of kg and where β = v∞/c and Bcol = atomic stopping number includes relativistic and quantum-mechanical corrections and is ∝ Z Radiation Physics Lecture 4

Interactions of Charged Particles Bcol with Matter 2 1/2 General Aspects 2mev Nuclear Reactions classical Z ln I Elastic Scattering Inelastic Scattering (Bohr) Stopping Power 2 Collision Stopping Power 2mev (Heavy Particles) non-rel, qm Z ln Collision Stopping Power I (Light Particles) Radiative Stopping Power (Bethe-Bloch) Total Mass Stopping h 2 β2 i Power rel, qm Z ln 2mec + ln − β2 Range I 1−β2 Mean Stopping Power (Bethe) rel, qm, shell h 2 2 i 2mec β 2 CK polarisation, Z ln I + ln 1−β2 − β − Z − δ (Fano) Radiation Physics Lecture 4

Interactions of Corrections to Bethe formula: Charged Particles with Matter C /Z = shell correction accounting for General Aspects I K Nuclear Reactions non-participation of K-shell electrons at low Elastic Scattering Inelastic Scattering energies; negligible energy transfer when velocity of Stopping Power Collision Stopping Power (Heavy Particles) incident particle is comparable to that of orbital Collision Stopping Power (Light Particles) electrons (K-shell electrons are fastest). Radiative Stopping Power Total Mass Stopping Power I δ = polarisation (density effect) correction in Range Mean Stopping Power condensed media; accounts for reduced participation by distant atoms resulting from effective Coulomb ﬁeld being reduced by dipole of nearby atoms; important for heavy charged particles at relativistic energies (but important for light charged particles at all energies). Radiation Physics Lecture 4 Example: The stopping power of water for protons. Interactions of Charged Particles Using the Bethe formula (relativistic and quantum- with Matter mechanical derivation, but without shell and polarisation General Aspects Nuclear Reactions corrections), with z = 1 for protons and for H O, A = Elastic Scattering 2 Inelastic Scattering 18.0 g = 0.0180 kg, Z = 10, and I = 75 eV giving Stopping Power Collision Stopping Power (Heavy Particles) 2 Collision Stopping Power −2 −2 β 2 (Light Particles) Scol = 1.71 × 10 β 9.520 + ln − β Radiative Stopping Power 2 Total Mass Stopping 1 − β Power Range 2 −1 Mean Stopping Power in units of MeV m kg . For 1 MeV protons, for instance, β2 = 0.00213, giving

2 −1 Scol = 26.97 MeV m kg

which compares well with the exact value obtained from 2 −1 the NIST/pstar database: Scol = 26.06 MeV m kg . Radiation Physics Lecture 4 I Scol ∝ Z/A, but Z/A does not vary appreciably Interactions of between different materials (Z/A ≈ 0.4 − 0.5 typically) Charged Particles 2 with Matter I Scol ∝ z ⇒ an α-particle of a given β has 4 times the General Aspects Nuclear Reactions collision stopping power of a proton Elastic Scattering Inelastic Scattering Stopping Power I Z dependence of Scol mostly through I, which Collision Stopping Power (Heavy Particles) increases with Z; Bcol has term − ln I, so stopping Collision Stopping Power (Light Particles) power gradually decreases with higher Z Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power

Stopping powers of protons in aluminium (Z=13) and lead (Z=82) (data from NIST/pstar). Radiation Physics Lecture 4 I dependence of Scol on particle kinetic energy EK varies strongly from non-relativistic to relativistic Interactions of Charged Particles regimes with Matter General Aspects I peak in Scol occurs at non-relativistic energies and is Nuclear Reactions Elastic Scattering responsible for the Bragg peak in depth dose curves Inelastic Scattering Stopping Power for heavy charged particles Collision Stopping Power (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power

Schematic plot of the mass collision stopping power for a heavy charged particle as a function of kinetic energy; M0 is rest mass of the charged particle, I is mean excitation/ionisation energy of the target medium (Fig. 6.7 in Podgoršak.) Radiation Physics Lecture 4 Collision Stopping Power for Light Charged Particles Interactions of Charged Particles with Matter General Aspects Nuclear Reactions Elastic Scattering Inelastic Scattering Stopping Power Collision Stopping Power 3 differences from heavy particle collisions: (Heavy Particles) Collision Stopping Power (Light Particles) 1. relativistic effects important at lower energies Radiative Stopping Power Total Mass Stopping 2. larger fractional energy losses Power Range Mean Stopping Power 3. radiative losses can occur Hard and soft collisions combined using Møller and Bhabba cross sections for electrons and positrons, respectively. Radiation Physics Lecture 4

Interactions of 2 Charged Particles 2 ZNA mec EK(1 + τ/2) ± with Matter Scol = 2πre 2 ln + F (τ) − δ General Aspects A β I Nuclear Reactions (21) Elastic Scattering Inelastic Scattering where Stopping Power Collision Stopping Power (Heavy Particles) − 2 2 Collision Stopping Power (τ) = ( − β )[ + τ / − ( τ + ) ] for electrons (Light Particles) F 1 1 8 2 1 ln 2 Radiative Stopping Power Total Mass Stopping Power and Range Mean Stopping Power β2 14 10 4 F+(τ) = 2 ln 2 − 23 + + + 12 τ + 2 (τ + 2)2 (τ + 2)3

for positrons and where

EK τ = 2 mec Radiation Physics Lecture 4

For light charged particles, Scol dependence on Z is Interactions of Charged Particles similar to that for heavy charged particles, but with Matter dependence on EK differs: 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

Mass collision stopping power (solid curves) and radiative stopping power (dashed curves) for electrons. (Fig. 6.10 in Podgoršak.) Radiation Physics Lecture 4 Radiative Stopping Power Electrons and positrons can undergo radiative losses as Interactions of a result of Coulomb interactions with atomic nuclei. Charged Particles with Matter Larmor formula predicts radiative power P ∝ a2 ∝ Z2/m2. General Aspects Nuclear Reactions Bethe and Heitler derived the cross-section for Elastic Scattering Inelastic Scattering bremsstrahlung radiation: Stopping Power Collision Stopping Power 2 2 (Heavy Particles) σ ∝ αr Z Collision Stopping Power rad e (Light Particles) Radiative Stopping Power Total Mass Stopping which contributes to mass stopping power: Power Range Mean Stopping Power N S = A σ E (22) rad A rad i 2 Ei = EK,i + mec = initial total energy

EK,i = initial kinetic energy

Srad can be written in terms of a weakly varying function Brad of Z and Ei (see Table 6.1 in in Podgoršak): N S = αr2Z2 A B E (23) rad e A rad i Radiation Physics Lecture 4

Radiative stopping powers for electrons in different material (solid curves) and collision stopping powers (dashed curves) for the same material. (Fig. 6.2 in Podgoršak.) Radiation Physics Lecture 4 Total Mass Stopping Power

Interactions of S = S + S (24) Charged Particles tot col rad with Matter General Aspects I for heavy charged particles, Srad ≈ 0 Nuclear Reactions < Elastic Scattering I for light charged particles, Scol > Srad for EK ∼ 10 MeV Inelastic Scattering Stopping Power typically Collision Stopping Power (Heavy Particles) I critical kinetic energy, (EK)crit, where Ecol = Erad Collision Stopping Power (Light Particles) Radiative Stopping Power 800 MeV Total Mass Stopping (EK)crit ≈ (25) Power Z Range Mean Stopping Power

Total mass stopping power (solid curves) and radiative and collision stopping power (dashed curves) for electrons. (Fig. 6.11 in Podgoršak.) Radiation Physics Lecture 4 Range

I heavy charged particles experience small fractional Interactions of Charged Particles energy losses and small angle deflections in elastic with Matter General Aspects collisions Nuclear Reactions Elastic Scattering I light charged particles experience larger fractional Inelastic Scattering Stopping Power energy losses and large angle deflections per elastic Collision Stopping Power (Heavy Particles) or inelastic collision Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power Radiation Physics Lecture 4 I range, R, of a particular charged particle in a particular medium measures the expected linear Interactions of distance the particle will reach in that medium before Charged Particles with Matter coming to rest (i.e. cannot penetrate beyond R) General Aspects Nuclear Reactions I depends on particle charge and kinetic energy, as Elastic Scattering Inelastic Scattering well as absorber composition Stopping Power Collision Stopping Power (Heavy Particles) I CSDA range, RCSDA, measures average geometric Collision Stopping Power (Light Particles) path length traversed by charged particles of a Radiative Stopping Power −2 Total Mass Stopping specific type in a given medium (in units kg m ) in Power Range the continuous slowing down approximation Mean Stopping Power I RCSDA > R always

Z EK,i Z EK,i dEK dEK RCSDA = = −ρ (26) 0 Stot(EK) 0 dEK/dx

I RCSDA difﬁcult to solve using analytic Stot(EK) solutions, (20) and (21) for heavy and light charged particles, respectively Radiation Physics Lecture 4 CSDA range for heavy charged particles

Interactions of Charged Particles 2 2 with Matter I for heavy particles, Stot(EK) = Scol(EK) ∝ z Bcol(β)/β , General Aspects where β is related to via = (γ − ) 2, where Nuclear Reactions EK EK 1 Mc − / Elastic Scattering γ = (1 − β2) 1 2, so E = E (β) and Inelastic Scattering K K Stopping Power Collision Stopping Power (Heavy Particles) Z 2 β dEK(β) Collision Stopping Power R ∝ (Light Particles) CSDA 2 Radiative Stopping Power z Bcol(β) Total Mass Stopping Power Range use = (β) β and let (β) = (β)/β2: Mean Stopping Power I dEK Mg d G Bcol

M Z β g(β) M RCSDA ∝ 2 dβ = 2 f (β) z 0 G(β) z

I f (β) independent of heavy particle type (only depends on β) ⇒ can calculate values of RCSDA for heavy particles relative to protons Radiation Physics Lecture 4 M p RCSDA(β) = 2 RCSDA(β) (27) mpz Interactions of Charged Particles p with Matter RCSDA(β) = proton range (obtain from NIST), General Aspects Nuclear Reactions M/mp = heavy charged particle mass / proton mass, Elastic Scattering Inelastic Scattering z = heavy particle charge Stopping Power Collision Stopping Power (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power Total Mass Stopping Power Range Mean Stopping Power

CSDA Range of protons in water (ρ = 1 g cm−3 so depth in cm has same value as R). From the NIST/pstar database. Radiation Physics Lecture 4 Example 1: Range of an 80 MeV 3He2+ ion in soft tissue. Interactions of 3 p Charged Particles We have z = 2 and M = 3mp, so R(β) = 4 R (β). Now with Matter General Aspects we need to ﬁnd the energy of a proton having the same β Nuclear Reactions 3 2+ p Elastic Scattering as the He ion. For a ﬁxed β, EK/M = const, so EK = Inelastic Scattering Stopping Power (mp/M)E = 80/3 MeV = 26.7 MeV. Using the NIST/pstar Collision Stopping Power (Heavy Particles) database, and using water as a soft tissue equivalent, Collision Stopping Power (Light Particles) Radiative Stopping Power p −2 −2 Total Mass Stopping RCSDA = 0.7173 g cm = 7.173 kg m Power Range Mean Stopping Power −2 −2 =⇒ RCSDA = 0.5380 g cm = 5.380 kg m Since water has ρ = 1 g cm−3, the average distance a 3He2+ ion can penetrate into soft tissue is ≈ 0.5 cm. Note: this exceeds the minimum thickness of outer layer of dead skin cells (epidermis, ∼ 0.007 cm), so 3He2+ ions can reach living cells from outside the human body. Radiation Physics Lecture 4

Interactions of Example 2: Range of a 7.69 MeV α particle in soft tissue. Charged Particles α p Using z = 2 and M = 4mp gives R (β) = R (β). For with Matter p General Aspects the same β, the proton energy is EK = (7.69/4) MeV ≈ Nuclear Reactions Elastic Scattering 1.923 MeV. For this proton energy, the NIST/pstar Inelastic Scattering p −3 −2 Stopping Power database gives R = 7.077 × 10 g cm . So the aver- Collision Stopping Power (Heavy Particles) Collision Stopping Power age depth to which 7.69 MeV α particles can penetrate into (Light Particles) Radiative Stopping Power soft tissue is close to the thickness of the epidermis. This Total Mass Stopping Power means that external sources of these particles are less Range 3 2+ Mean Stopping Power of a health hazard than He ions. However, 7.69 MeV α 214 particles are emitted by the radon daughter 84 Po, which is present in the atmosphere of uranium mines. These α’s pose a serious radiological hazard when ingested through the lungs. This has been linked to the higher incidence of lung cancer among uranium miners. Radiation Physics Lecture 4 CSDA range for light charged particles

Interactions of Charged Particles I for light particles, need to also take into account with Matter General Aspects radiative losses 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

CSDA Range of electrons in water (ρ = 1 g cm−3 so depth in cm has same value as R). From the NIST/pstar database. Radiation Physics Lecture 4 Mean Stopping Power

I in practice, charged particle beams are generally not Interactions of Charged Particles monoenergetic with Matter General Aspects I electrons in an initially monoenergetic beam will lose Nuclear Reactions Elastic Scattering different amounts of energy through a medium Inelastic Scattering Stopping Power I produces an energy spectrum: Collision Stopping Power (Heavy Particles) Collision Stopping Power (Light Particles) Radiative Stopping Power dφ(E) N Total Mass Stopping = (28) Power dE Stot(E) Range Mean Stopping Power I N = no. of monoenergetic electrons of initial kinetic

energy EK,0 per unit mass in medium

I collision stopping power for a single energy EK,0 should be deﬁned as an average over energy spectrum produced as a result of all collisions:

R EK,0 dφ 0 dE Scol(E) dE Scol(EK,0) = (29) R EK,0 dφ 0 dE dE Radiation Physics Lecture 4

Using (28) and the deﬁnition (26) for RCSDA, the Interactions of Charged Particles denominator is: with Matter General Aspects Z EK,0 φ Z EK,0 Nuclear Reactions d dE Elastic Scattering dE = N = NRCSDA Inelastic Scattering 0 dE 0 Stot(E) Stopping Power Collision Stopping Power (Heavy Particles) Similarly, the numerator is Collision Stopping Power (Light Particles) Radiative Stopping Power Z EK,0 Z EK,0 Total Mass Stopping dφ Scol(E) Power Scol(E) dE = N dE Range dE S (E) Mean Stopping Power 0 0 tot

and Scol = Stot − Srad implies

Z EK,0 Z EK,0 dφ Srad(E) Scol(E) dE = N 1 − dE 0 dE 0 Stot(E)

= NEK,0 [1 − Y(EK,0)] (30) Radiation Physics Lecture 4

Interactions of Charged Particles with Matter where General Aspects Nuclear Reactions Elastic Scattering Z EK,0 1 Srad(E) Inelastic Scattering ( ) = radiation yield Stopping Power Y EK,0 dE (31) Collision Stopping Power EK,0 0 Stot(E) (Heavy Particles) Collision Stopping Power (Light Particles) Putting together gives the mean collision stopping power: Radiative Stopping Power Total Mass Stopping Power Range 1 − Y(EK,0) Mean Stopping Power Scol(EK,0) = EK,0 (32) RCSDA

For heavy charged particles, Y(EK,0) = 0, so

Scol(EK,0) = EK,0/RCSDA. Radiation Physics Lecture 4 Example 1: Mean stopping power for a 5 MeV α in lead.

Since Y(EK,0) = 0, then Scol(EK,0) = EK,0/RCSDA. From the Interactions of −2 −2 Charged Particles NIST/astar database, we ﬁnd RCSDA = 1.70 × 10 g cm , with Matter General Aspects so Nuclear Reactions Elastic Scattering 2 2 −1 2 −1 Inelastic Scattering Scol(EK,0) = 2.94 × 10 MeV cm g = 29.4 MeV m kg Stopping Power Collision Stopping Power (Heavy Particles) 2 −1 Collision Stopping Power c.f. the collision stopping power is Scol = 23.3 MeV m kg . (Light Particles) Radiative Stopping Power Why is Scol > Scol? Total Mass Stopping Power Range Example 2: Mean stopping power for a 5 MeV electron in Mean Stopping Power lead. From the NIST/estar database, we ﬁnd −2 RCSDA = 3.67g cm and Y(EK,0) = 0.205, so

2 −1 2 −1 Scol(EK,0) = 1.08 MeV cm g = 0.108 MeV m kg

2 −1 c.f. Scol = 0.112 MeV m kg . Why is Scol ≈ Scol?