PHYS 5012 Radiation Physics and Dosimetry

Radiative PHYS 5012 Radiation Physics and Dosimetry Mean Tuesday 24 March 2009

Radiative Mean

Radiative Mean Collisions between two particles involve a projectile and a target. Types of targets: whole atoms, atomic nuclei, atomic orbital electrons. Types of projectiles: heavy charged particles (protons, α-particles, heavy ions) light charged particles (electrons, positrons) neutrons (not considered here)

Two-Particle Collisions Radiative Mean 3 categories: 1. Nuclear reactions final reaction products differ from initial particles; charge, momentum and mass-energy conserved; e.g. deuteron bombarding nitrogen-14: 14 7 N(d, p)15 7 N 2. Elastic collisions final products identical to initial particles; kinetic energy and momentum conserved; e.g. Rutherford scattering of α particle on gold nucleus: 197 79 Au(α, α)197 79 Au 3. Inelastic collisions final products identical to initial particles; kinetic energy not conserved

Radiative Mean In inelastic collisions, some kinetic energy is converted to excitation energy in the form of: nuclear excitation of target resulting from heavy charged particle striking target nucleus; e.g. A Z X(α, α)a Z X atomic excitation or ionisation of target resulting from heavy or light charged particle colliding with target orbital electron bremsstrahlung emission by light charged particle projectile resulting from Coulomb interaction with target nucleus

Radiative Mean Schematic illustration of a general nuclear reaction. (Fig. 4.1 in Podgoršak.) intermediate compound produced temporarily; spontaneously decays into reaction products conservation of atomic number: Z before = Z after conservation of atomic mass: A before = A after

Conservation of momentum Radiative p 1 = p 3 + p 4 (1) = p 1 = p 3 cos θ + p 4 cos φ to p 1 0 = p 3 sin θ + p 4 sin φ to p 1 Conservation of mass-energy ( m1 c 2 + E K,1 ) + m2 c 2 = ( m 3 c 2 + E K,3 ) + ( m4 c 2 + E K,4 ) (2) Mean where E K = particle kinetic energy = (γ 1)mc 2 Q = ( m 1 c 2 + m 2 c 2) ( m 3 c 2 + m 4 c 2) Q value (3) Q > 0 exothermic collision Q = 0 elastic collision Q < 0 endothermic collision

Threshold Energy Radiative Mean minimum projectile energy E thr required for endothermic reaction to proceed Conservation of 4-momentum, p = (E/c, p): p 1 + p 2 = p 3 + p 4 (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 and using p 2 1 = (E 1/c) 2 p 1 2 = m 2 1 c2 and p 2 2 = (E 2/c) 2 = m 2 2 c2, gives 2E 1 E 2 = (p 3 + p 4 ) 2 c 2 (m 2 1c 4 + m 2 2c 4 ) Note that p 3 + p 4 is the centre-of-mass 4-momentum, p cm, and so (p 3 + p 4 ) 2 = p 2 cm = (E cm /c) 2 = (m 3 c 2 + m 4 c 2 ) 2 /c 2 since the modulus of a 4-vector is invariant and has the same value in any frame of reference. So the threshold energy E 1 for the projectile is:

Radiative Mean E thr = (m 3c 2 + m 4 c 2 ) 2 (m 2 1 c4 + m 2 2 c4 ) 2m 2 c 2 (4) corresponding threshold kinetic energy: E K,thr = (m 3c 2 + m 4 c 2 ) 2 (m 1 c 2 + m 2 c 2 ) 2 2m 2 c 2 (5) in terms of the Q value: [ m1 c 2 + m 2 c 2 E K,thr = Q m 2 c 2 Q ] 2m 2 c ( 2 Q 1 + m ) 1 m 2 (6) if Q m 2 c 2 (usually the case).

Radiative Mean The Q value is defined for general two-particle collisions. For pair production, for example, m 1 = 0, m 2 = m 3 m e and Q = 2m e c 2, so ( E pp γ )thr = 2m ec 2 while for triplet production, Q = 2m e c 2 but m 2 = m e, so ( ) E tp γ thr = 4m ec 2

Radiative initial and final particles remain the same (i.e. m 3 = m 1 and m 4 = m 2 ), so Q = 0 kinetic energy transfer E K from m 1 to m 2 Mean Schematic illustration of elastic scattering. θ is the scattering angle, φ is the recoil angle and b is the impact parameter. (Fig. 4.2 in Podgoršak.)

Classical derivation of kinetic energy transfer: conservation of momentum and energy Radiative Mean Head-on collisions: E K = 1 2 m 2u 2 2 = E K1 4m 1 m 2 (m 1 + m 2 ) 2 cos2 φ (7) b = 0 and φ = 0 maximum energy and momentum transfer θ = 0 (forward scattering) when m 1 > m 2 θ = π (back-scattering) when m 1 < m 2 projectile stops when m 1 = m 2

Radiative Example: proton colliding with orbital electron Maximum energy transfer (for a head-on collision), noting that m p m e : E max 4E Kp m e m p 2 10 3 E Kp Collisions between particles of the same mass (m 1 = m 2 ): Mean distinguishable particles (e.g. electron colliding with positron): E max = E K1 head-on collision transfers all projectile s kinetic energy to target indistinguishable particles (e.g. free electron colliding with bound electron): E max = 1 2 E K1

Radiative Mean Relativistic formula for energy transfer in a head-on collision: E max = (γ 2 1)m 2 c 2 = 2(γ + 1)m 1m 2 m 2 1 + m2 2 + 2γm 1m 2 E K1 (8) where γm 1 c 2 = energy of incident projectile

Radiative Mean Stopping power measures ability of matter to stop charged particles incident charged particle loses all kinetic energy via multiple Coulomb interactions (mostly elastic, but sometimes inelastic) gradual loss of kinetic energy called continuous slowing down approximation (CSDA) e.g. 1 MeV charged particle typically undergoes 10 5 interactions before losing all its kinetic energy stochastic process need to use probabilities and average quantities

Radiative hard collisions: Coulomb interactions with orbital electron for b a soft collisions: Coulomb interaction with orbital electron for b a radiative collisions: Coulomb interactions with nuclear field for b a Mean The three different types of collisions depend on the classical impact paramater b and atomic radius a. (Fig. 5.1 in Podgoršak.)

Radiative Mean Linear stopping power, de/dx = rate of energy loss per unit path length of charged particle Mass stopping power, S = ρ 1 de/dx, is the commonly used measure of stopping power (in units MeV m 2 kg 1 ) 2 types of stopping powers: 1. Radiative stopping power, S rad for radiative collisions; only light charged particles (i.e. electrons and positrons) experience appreciable energy losses; can result in bremsstrahlung emission 2. Collision stopping power, S col for hard and soft collisions involving both light and heavy charged particles; can result in atomic excitation and ionisation S tot = S rad + S col total stopping power

Radiative For electrons and positrons: Radiative Mean S rad = N A A σ rade i (9) E i = E K,i + m e c 2 = initial total energy E K,i = initial kinetic energy σ rad = total cross section for bremmstrahlung production S rad can be written in terms of a weakly varying function B rad of Z and E i (see Table 5.1 in in Podgoršak): S rad = αr 2 ez 2 N A A B rade i (10) derived theoretically by Bethe and Heitler.

Radiative Mean Radiative stopping powers for electrons in different material (solid curves) and collision stopping powers (dashed curves) for the same material. (Fig. 5.2 in Podgoršak.)

Radiative for Heavy Charged Particles for E i < 10 MeV, heavy charged particles undergo soft and hard collisions small angle scattering (θ 0) Mean Schematic diagram of a heavy charged particle collision with an orbital electron. The scattering angle θ is exaggerated for clarity. (Fig. 5.3 in Podgoršak.)

Radiative Mean Classical Derivation Momentum transfer: p = F p dt = where F coul = (ze 2 /4πε 0 )r 2, giving p = ze2 4πε 0 +(π θ)/2 (π θ)/2 F coul cos φ dt cos φ r 2 dt dφ dφ Hyperbolic particle trajectory angular displacement varies with time dφ/dt = ω and conservation of angular momentum requires L = Mv b = Mωr 2 p = ze2 4πε 0 1 v b +(π θ)/2 (π θ)/2 = 2 ze2 1 4πε 0 v b cos θ 2 2 ze2 1 4πε 0 v b cos φ dφ (11)

Radiative Mean Energy transferred to electron in a single collision with impact parameter b: ( ) E(b) = ( p)2 e 2 2 z 2 = 2 2m e 4πε 0 m e v 2 b 2 (12) Total energy loss obtained by integrating E(b) over all possible b and accounting for all electrons available for interactions.

Radiative Mean no. electrons in volume annulus between b and b + db = no. electrons per unit mass mass in annulus ( ) ZNA n = dm A where dm = ρ dv = ρ[π(b + db) 2 x πb 2 x] 2π ρ b db x n 2π ρ (ZN A /A) b db x Multiply E(b) by this and integrate over b to get the total energy transfer to electrons.

Radiative Mean Mass collision stopping power S col = 1 de ρ dx = 4π ZN A A = 4π ZN A A ( e 2 ) 2 z 2 4πε 0 m e v 2 ( ) e 2 2 z 2 4πε 0 m e v 2 bmax b min db b ln b max b min (13) S col z 2, where z = atomic number of heavy charged particle (e.g. z = 4 for an α particle) S col v 2, where v = initial velocity of heavy charged particle

Radiative Mean b max E min = minimum energy transfer corresponding to minimum excitation or ionisation potential of orbital electron from (12) ( ) e 2 2 z 2 E min = 2 4πε 0 m e v 2 b 2 = I (14) max I = mean ionisation-excitation potential of medium I 9.1Z(1 + 1.9Z 2/3 ) ev (15) e.g. I 78 ev for carbon. But (15) is poor approximation for compounds (e.g. I 75 ev for water).

Radiative b min E max = maximum energy transfer corresponding to head-on collisions: E max 4 me M E K,i = 2m e v 2 (for M m e ), so ( ) e 2 2 z 2 E max = 2 4πε 0 m e v 2 b 2 min Putting together (14) and (16) gives = 2m e v 2 (16) Mean b max b min = ( Emax E min ) 1/2 = ( 2me v 2 ) 1/2 (17) I = classical collision stopping power for heavy charged particles: S col = 4π ZN A A ( e 2 4πε 0 ) 2 z 2 m e v 2 1 2 ln 2m ev 2 I (18)

Radiative Mean Generalised solution for the collision stopping power for heavy charged particles: S col = 4π N A A ( e 2 4πε 0 ) 2 z 2 m e c 2 (v /c) 2 B col 5 z2 3.070 10 Aβ 2 B col MeV m 2 kg 1 (19) with A in units of kg and where β = v /c and B col = atomic stopping number includes relativistic and quantum-mechanical corrections and is Z

Radiative Mean classical (Bohr) non-rel, qm (Bethe-Bloch) rel, qm (Bethe) rel, qm, shell, polarisation B col ( ) 1/2 2mev 2 Z ln Z ln [ Z ln [ Z ln I ( 2mev 2 I ) ( 2mec 2 I ( 2mec 2 I ) ( ) ] + ln β 2 β 2 1 β 2 ) ( ) ] + ln β 2 β 2 C 1 β 2 K Z δ C K /Z = correction accounting for non-participation of K-shell electrons; important for low-e K,i δ = polarisation (density effect) correction; accounts for reduced participation by distant atoms resulting from effective Coulomb field being reduced by dipole of nearby atoms; important for light charged particles

Radiative Mean Example: The stopping power of water for protons. Using the Bethe formula (relativistic and quantummechanical derivation, but without shell and polarisation corrections), with z = 1 for protons and for H 2 O, A = 18.0 g = 0.0180 kg, Z = 10, and I = 75 ev giving ( ) ] β S col = 1.71 10 2 β [9.520 2 2 + ln 1 β 2 β 2 in units of MeV m 2 kg 1. For 1 MeV protons, for instance, β 2 = 0.00213, giving S col = 26.97 MeV m 2 kg 1 which compares well with the exact value obtained from the NIST/pstar database: S col = 26.06 MeV m 2 kg 1.

Radiative S col Z/A, but Z/A does not vary appreciably between different materials (Z/A 0.4 0.5 typically) Z dependence of S col mostly through I, which increases with Z; B col has term ln I, so stopping power decreases with higher Z Mean Stopping powers of protons in aluminium (Z=13) and lead (Z=82) (data from NIST/pstar).

dependence of S col on particle kinetic energy E K varies from non-relativistic to relativistic regimes Radiative Mean Schematic plot of the mass collision stopping power for a heavy charged particle as a function of kinetic energy. (Fig. 5.4 in Podgoršak.)

for Light Charged Particles Radiative Mean 3 differences from heavy particle collisions: 1. relativistic effects important at lower energies 2. larger fractional energy losses 3. radiative losses can occur Hard and soft collisions combined using Møller and Bhabba cross sections for electrons and positrons, respectively.

Radiative Mean S col = 2πre 2 ZN A m e c 2 [ A β 2 ln where ( EK (1 + τ/2) I ) ] + F ± (τ) δ F (τ) = (1 β 2 )[1 + τ 2 /8 (2τ + 1) ln 2] for electrons and F + (τ) = 2 ln 2 β2 12 (20) [ 23 + 14 ] τ + 2 + 10 (τ + 2) 2 + 4 (τ + 2) 3 for positrons and where τ = E K m e c 2

For light charged particles, S col dependence on Z is similar to that for heavy charged particles, but dependence on E K differs: Radiative Mean Mass collision stopping power (solid curves) and radiative stopping power (dashed curves) for electrons. (Fig. 5.5 in Podgoršak.)

Radiative Mean Total Mass S tot = S rad + S col (21) for heavy charged particles, S rad 0 for light charged particles, S col > S rad for E K < 10 MeV typically critical kinetic energy, (E K ) crit, where E col = E rad 800 MeV (E K ) crit Z (22) Total mass stopping power (solid curves) and radiative and collision stopping power (dashed curves) for electrons. (Fig. 5.6 in Podgoršak.)

Radiative heavy charged particles experience small fractional energy losses and small angle deflections in elastic collisions light charged particles experience larger fractional energy losses and large angle deflections per elastic or inelastic collision Mean

Radiative Mean range, R, of a particular charged particle in a particular medium measures the expected linear distance the particle will reach in that medium before coming to rest (i.e. cannot penetrate beyond R) depends on particle charge and kinetic energy, as well as absorber composition CSDA range, R CSDA, measures average path length traversed by charged particles of a specific type in a given medium (in units kg m 2 ) in the continuous slowing down approximation R CSDA > R always R CSDA = EK,i 0 de EK,i K S tot (E K ) = ρ 0 de K de K /dx (23)

Radiative Mean R CSDA difficult to solve using analytic S tot (E K ) solutions, (19) and (20) for light particles, need to also take into account radiative losses for heavy particles, S tot (E K ) = S col (E K ) z 2 B col (β)/β 2, where β is related to E K via E K = (γ 1)Mc 2, where γ = (1 β 2 ) 1/2, so E K = E K (β) and R CSDA β 2 de K (β) z 2 B col (β) use de K = Mg(β)dβ and let G(β) = B col (β)/β 2 : R CSDA M z 2 β 0 g(β) G(β) dβ = M z 2 f (β) f (β) independent of heavy particle type (only depends on β) can calculate values of R CSDA for heavy particles relative to protons

R CSDA (β) = M M p z 2 Rp CSDA (β) (24) R p CSDA (β) = proton range M/M p = heavy charged particle mass / proton mass z = atomic number of heavy charged particle Radiative Mean of protons in water (ρ = 1 g cm 3 so depth in cm has same value as R). From the NIST/pstar database.

Radiative Mean Example 1: of an 80 MeV 3 He 2+ ion in soft tissue. We have z = 2 and M = 3M p, so R(β) = 3 4 Rp (β). Now we need to find the energy of a proton having the same β as the 3 He 2+ ion. For a fixed β, E K /M = const, so E p K = (M p /M)E = 80/3 MeV = 26.7 MeV. Using the NIST/pstar database, and using water as a soft tissue equivalent, R p CSDA = 0.7173 g cm 2 = 7.173 kg m 2 = R CSDA = 0.5380 g cm 2 = 5.380 kg m 2 Since water has ρ = 1 g cm 3, the average distance a 3 He 2+ 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 3 He 2+ ions can reach living cells from outside the human body.

Radiative Mean Example 2: of a 7.69 MeV α particle in soft tissue. Using z = 2 and M = 4M p gives R α (β) = R p (β). For the same β, the proton energy is E p K = (7.69/4) MeV 1.923 MeV. For this proton energy, the NIST/pstar database gives R p = 7.077 10 3 g cm 2. So the average depth to which 7.69 MeV α particles can penetrate into soft tissue is close to the thickness of the epidermis. This means that external sources of these particles are less of a health hazard than 3 He 2+ ions. However, 7.69 MeV α particles are emitted by the radon daughter 214 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.

Radiative Mean Mean in practice, charged particle beams are generally not monoenergetic electrons in an initially monoenergetic beam will lose different amounts of energy through a medium produces an energy spectrum: dφ(e) de = N S tot (E) N = no. of monoenergetic electrons of initial kinetic energy E K,0 per unit mass in medium (25) collision stopping power for a single energy E K,0 should be defined as an average over energy spectrum produced as a result of all collisions: S col (E K,0 ) = EK,0 dφ 0 EK,0 0 de S col(e) de dφ de de (26)

Radiative Mean Using the definition for R CSDA : Similarly, EK,0 0 EK,0 0 dφ EK,0 de de = N 0 and S col = S tot S rad implies EK,0 0 de S tot (E) = NR CSDA dφ EK,0 de S S col (E) col(e) de = N 0 S tot (E) de dφ EK,0 [ de S col(e) de = N 1 S ] rad(e) de 0 S tot (E) = NE K,0 [1 B(E K,0 )] (27)

Radiative Mean where B(E K,0 ) = 1 E K,0 EK,0 0 S rad (E) de radiation yield (28) S tot (E) Putting together gives the mean collision stopping power: S col (E K,0 ) = E K,0 1 B(E K,0 ) R CSDA (29) For heavy charged particles, B(E K,0 ) = 0, so S col (E K,0 ) = E K,0 /R CSDA.

Radiative Mean Example: Mean stopping power for a 5 MeV α in lead. Since B(E K,0 ) = 0, then S col (E K,0 ) = E K,0 /R CSDA. From the NIST/astar database, we find R CSDA = 1.702 10 2 g cm 2, so S col (E K,0 ) = 2.94 10 2 MeV cm 2 g 1 = 29.4 MeV m 2 kg 1 c.f. the collision stopping power is S col = 23.3 MeV m 2 kg 1.