PHYS 5012 Radiation Physics and Dosimetry

Radiation Physics Lecture 5

Interactions of Charged Particles with Matter Radiation Yield Bremsstrahlung Targets Thin Targets PHYS 5012 Thick Targets Restricted Stopping Power Straggling and Scattering Radiation Physics and Dosimetry Electron Range Dose Linear Energy Transfer Lecture 5

Tuesday 3 April 2012 Radiation Physics Lecture 5 Charged Particle Interactions (cont.)

Stopping power of water for 10 MeV electrons: 2 −1 2 −1 Scol = 1.968 MeV cm g , Srad = 0.1814 MeV cm g , 2 −1 Stot = Scol + Srad = 2.149 MeV cm g −2 Range: RCSDA = 4.975 g cm ⇒ average path length of 10 MeV electrons in water is ≈ 5 cm. Radiation yield: −2 2 −1 Y = 4.072 × 10 ⇒ S¯col = 1.928 MeV cm g Radiation Physics Lecture 5 Bremsstrahlung (Radiation) Yield Recall the mean collision stopping power (c.f. eqn. 32, Interactions of Lec. 4): Charged Particles with Matter 1 − Y(EK,0) Radiation Yield Scol(EK,0) = EK,0 Bremsstrahlung Targets RCSDA Thin Targets Thick Targets which takes into account the continuous degradation in Restricted Stopping Power Straggling and Scattering energy of an initially monoenergetic beam of radiation. Electron Range Dose Linear Energy Transfer I Y(EK,0) = radiation yield = fraction of initial kinetic

energy EK,0 emitted as bremsstrahlung radiation through continuous slowing down of charged particle in a medium

I Y(EK,0) ≈ 0 for heavy charged particles I for light charged particles (electrons and positrons),

Z EK,0 1 Srad(E) Y(EK,0) = dE (1) EK,0 0 Stot(E)

I electron-positron annilhilation radiation is generally negligible compared to bremsstrahlung emission Radiation Physics Lecture 5

Interactions of Charged Particles with Matter Radiation Yield Bremsstrahlung Targets Thin Targets Thick Targets Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer Bremsstrahlung yield for electrons in different media plotted against initial electron kinetic energy. (Fig. 6.12 in Podgoršak.) Recall from Lec. 4 (c.f. eqn. 22), the formula for radiative stopping power: N S (E) = A σ E rad A rad i

2 where σrad ∝ Z Brad(Z, Ei) is the Bethe-Heitler total radiative cross section derived from the quantum mechanical theory for bremsstrahlung radiation, and 2 where Ei = EK,0 + mec . Radiation Physics Lecture 5

Interactions of Charged Particles with Matter Radiation Yield Bremsstrahlung Targets I energy radiated per charged particle: Thin Targets Thick Targets Restricted Stopping Power Z EK,0 Straggling and Scattering Srad(E) Electron Range Erad = EK,0Y(EK,0) = dE (2) Dose 0 Stot(E) Linear Energy Transfer

I energy lost to ionisation per charged particle:

Z EK,0 Scol(E) Ecol = EK,0 − Erad = EK,0 [1 − Y(EK,0)] = dE 0 Stot(E) (3) Radiation Physics Lecture 5 Example: Radiation yield for 10 MeV electrons. 1. In water (Z = 10): Y ≈ 4.1% (from NIST/estar) =⇒ Interactions of Charged Particles I energy radiated per charged particle: with Matter Radiation Yield Bremsstrahlung Targets Erad = EK,0Y(EK,0) ≈ 0.41 MeV Thin Targets Thick Targets Restricted Stopping Power Straggling and Scattering I energy lost to ionisation per charged particle: Electron Range Dose Linear Energy Transfer Ecol = EK,0 [1 − Y(EK,0)] ≈ 9.59 MeV

1. In tungsten (Z = 74): Y ≈ 30% (from NIST/estar) =⇒

I energy radiated per charged particle:

Erad = EK,0Y(EK,0) ≈ 3.0 MeV

I energy lost to ionisation per charged particle:

Ecol = EK,0 [1 − Y(EK,0)] ≈ 7.0 MeV Radiation Physics Lecture 5 Radiation Yield and Radiative Fraction

Interactions of Charged Particles with Matter I Y(EK,0) = radiation yield = fraction of initial kinetic Radiation Yield energy EK,0 of an electron (or positron) lost to Bremsstrahlung Targets Thin Targets radiative (bremsstrahlung) losses Thick Targets Restricted Stopping Power Straggling and Scattering I g¯ = radiative fraction = average fraction of energy Electron Range Dose transferred to light charged particles by photons that Linear Energy Transfer is subsequently lost to radiation (predominantly bremsstrahlung); c.f. µ µ ab = tr (1 − g¯) ρ ρ

I photon interactions produce electrons (and

positrons) with a spectrum of kinetic energies EK,i ⇒

g¯ = average value of Y(EK,i) for all light charged particles produced by photon interactions Radiation Physics Lecture 5 Bremsstrahlung Targets

Interactions of Charged Particles with Matter I monoenergetic electron beams incident on solid Radiation Yield Bremsstrahlung Targets target material commonly used to generate Thin Targets Thick Targets (bremsstrahlung) radiation beams Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer Radiation Physics Lecture 5 Thin Targets

Interactions of Charged Particles with Matter Radiation Yield Bremsstrahlung Targets thin X-ray targets – thickness ∆x R sufﬁciently Thin Targets I Thick Targets small such that incident electrons experience Restricted Stopping Power Straggling and Scattering I no ionisation losses Electron Range Dose I no elastic collisions Linear Energy Transfer I only one bremsstrahlung interaction

I total energy radiated:

dE (E ) = ρN ∆x S = N ∆x − rad rad tot e rad e dx

where Ne = no. of incident electrons Radiation Physics Lecture 5 I intensity spectrum, IE (where E = hν = ~ω), obtained Interactions of by taking Fourier transform of classical Larmor Charged Particles with Matter formula (c.f. Lec. 1) for electromagnetic power Radiation Yield −2 Bremsstrahlung Targets I → dI(b)/dω ∝ b ⇒ more photons emitted for Thin Targets small-b interactions of monoenergetic electrons Thick Targets Restricted Stopping Power I integrate over all impact parameters: Straggling and Scattering R Electron Range IE ∝ (dI(b)/dω) bdb ∝ ln(bmax/bmin) → approx. Dose Linear Energy Transfer independent of E = hν ⇒ thin-target bremsstrahlung spectrum is approximately

ﬂat up to a cut-off hνmax = EK,0

I also consistent with Bethe-Heitler quantum mechanical differential cross-section, which predicts

Iω ∝ hω dσrad ∝ Brad(ω)

where Brad(ω) is almost independent of ω R 1 n.b. Brad = 0 Brad d(hω/EK,0) Radiation Physics Lecture 5 Thick Targets

Interactions of Charged Particles with Matter Radiation Yield ∆x ∼ R such that Bremsstrahlung Targets I Thin Targets I no incident electrons can traverse the medium Thick Targets Restricted Stopping Power I attenuation of bremsstrahlung photons is minimised Straggling and Scattering Electron Range Dose I intensity spectrum is a superposition of multiple Linear Energy Transfer thin-target spectra for different EK,i, since incident monoenergetic electrons gradually lose energy through multiple collisions

IE = CZ (EK,0 − hν) Kramer’s spectrum (4)

where C = constant and EK,0 = initial kinetic energy of electrons incident on thick target Radiation Physics Lecture 5 I ﬁltration of resulting X-ray beam through attenuators

Interactions of preferentially removes low enegy photons and Charged Particles with Matter hardens the spectrum; used for clinical applications Radiation Yield Bremsstrahlung Targets Thin Targets Thick Targets Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer

Typical bremsstrahlung spectra produced by 100 keV electrons striking a target: 1. thin target; 2. thick target (unfiltered) showing superposition of multiple thin target spectra; 3. thick target, with resulting beam filtered by an X-ray tube window; 4. thick target with additional filtration. In general, spectra should also include characteristic emission from the target and filters. (Fig. 6.19 Podgorsak) Radiation Physics Lecture 5 2 I for EK,0 mec , Interactions of Charged Particles with Matter 2 2 NA 2 2 NA Radiation Yield Srad = αre Z Brad(EK,0 + mec ) ∝ Z Brad Bremsstrahlung Targets A A Thin Targets Thick Targets Restricted Stopping Power ⇒ energy radiated: Straggling and Scattering Electron Range Dose Z EK,0 Z EK,0 Linear Energy Transfer Srad(E) dE Erad = dE ≈ Srad 0 Stot(E) 0 Stot(E)

and Stot ≈ Scol ∝ ZNA/A ⇒

Erad ∝ Z

⇒ production of kilovoltage X-ray beams determined by high-Z targets, but independent of initial electron energy Radiation Physics Lecture 5

Interactions of Charged Particles with Matter Radiation Yield 2 Bremsstrahlung Targets I for EK,0 mec , Thin Targets Thick Targets Restricted Stopping Power Srad Srad EK,0 Straggling and Scattering = = Electron Range Dose Stot (Scol + Srad) (n/Z + EK,0) Linear Energy Transfer

where n = n(Z, EK,0) is a weak function of Z and EK,0 ⇒ production of megavoltage X-ray beams largely independent of target Z, but more efﬁcient for higher initial electron energies Radiation Physics Lecture 5 radiation pattern

2 ◦ I for EK mec , most radiation emerges at θmax ≈ 90 (dipole pattern) 2 ◦ I for EK mec , most radiation emerges at θmax ' 0 (forward beaming) Radiation Physics Lecture 5

Interactions of Charged Particles This is why X-ray tubes, which produce kilovoltage with Matter Radiation Yield X-rays, have an exit window positioned at an angle to the Bremsstrahlung Targets Thin Targets target, while medical linacs, which produce megavoltage Thick Targets Restricted Stopping Power X-rays, have an X-ray window placed on the other side of Straggling and Scattering Electron Range the target: Dose Linear Energy Transfer Radiation Physics Lecture 5 Restricted Stopping Power

Interactions of Charged Particles with Matter I Scol includes both hard and soft collisions; hard Radiation Yield Bremsstrahlung Targets collisions can produce secondary electrons (δ rays) Thin Targets Thick Targets carrying signiﬁcant kinetic energy away from primary Restricted Stopping Power Straggling and Scattering particle track ⇒ non-localised energy deposition Electron Range Dose I localised energy transfer measured by excluding δ Linear Energy Transfer rays above a threshold energy ∆

I ∆ < ∆Emax = maximum energy transfer to δ rays

I L∆ = restricted mass collision stopping power < Scol rate of energy loss due only to collisions in which energy transfer does not exceed threshold ∆

I L∆ = Scol when ∆ = ∆Emax

I for nonrelativistic heavy charged particles, 2 2 ∆Emax ≈ 2meβ c (c.f. Lec. 4 eqn. 8) Radiation Physics Lecture 5

L∆ associates energy loss with energy absorbed in a Interactions of Charged Particles target (especially if small, e.g. biological scales) with Matter Radiation Yield Bremsstrahlung Targets Example: What is the energy of a proton than can pro- Thin Targets Thick Targets duce a δ-ray with enough energy to traverse a cell with a Restricted Stopping Power Straggling and Scattering diameter 2.5 µm? From the NIST/estar database, the en- Electron Range −4 Dose ergy of an electron with RCSDA/ρ = 2.5 × 10 cm is 10 keV Linear Energy Transfer (assuming the cell tissue is water-equivalent). Equating 2 2 2 this to ∆Emax = 2meβ c gives β = 0.01 and hence, 2 (γ−1)mpc = 4.7 MeV. A proton with more energy than this would produce more energetic δ-rays that could deposit their energy on scales larger than the cell size, thus over- shooting the irradiation target. In this case, a restricted stopping power of L10 keV for protons in water would be ap- propriate. Radiation Physics Lecture 5 1 I for incident electrons, ∆Emax = 2 EK, so L∆ = Scol when 2∆ = EK (see ﬁgure below) Interactions of Charged Particles with Matter Radiation Yield Bremsstrahlung Targets Thin Targets Thick Targets Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer

Unrestricted (solid curves) and restricted collision stopping powers, with ∆ = 10 keV and ∆ = 100 keV (dashed curves) for electrons in carbon. (Fig. 6.16 in Podgoršak.) Radiation Physics Lecture 5 Straggling and Scattering energy straggling is the formation of a distribution of Interactions of I Charged Particles particle energies resulting from the stochastic nature with Matter Radiation Yield of energy loss interactions in a medium Bremsstrahlung Targets Thin Targets Thick Targets Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer Radiation Physics Lecture 5

Interactions of Charged Particles I range straggling is the formation of a distribution of with Matter Radiation Yield pathlengths traversed by particles in a medium Bremsstrahlung Targets Thin Targets before they stop; it also results from stochastic Thick Targets Restricted Stopping Power changes in rate of energy loss Straggling and Scattering Electron Range Dose Linear Energy Transfer Radiation Physics Lecture 5 I multiple Coulomb scattering results in an angular spread and dispersion of an initially parallel beam Interactions of Charged Particles ("pencil beam") of charged particle into a diverging with Matter Radiation Yield 3D conical beam Bremsstrahlung Targets Thin Targets Thick Targets Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer Radiation Physics Lecture 5

Interactions of Charged Particles with Matter Radiation Yield Bremsstrahlung Targets I many collisions with large b, but small θ Thin Targets Thick Targets I fewer collisions with small b, but large θ Restricted Stopping Power Straggling and Scattering symmetric (gaussian) angular dispersion Electron Range I Dose perpendicular to beam Linear Energy Transfer

1 Z dσ θ2 = θ2Nl dΩ rms angular spread (5) rms 2 dΩ where N = number density of target material, l = thickness of material. Radiation Physics Lecture 5 Electron Range

I electrons suffer many deviations in their trajectories Interactions of Charged Particles as a result of elastic and inelastic scattering → with Matter Radiation Yield haphazard electron tracks, with secondary δ rays Bremsstrahlung Targets Thin Targets Thick Targets Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer

I lower energy electrons suffer more scatterings and deviations

I scattering increases with Z (RCSDA ∝ Z at low EK) Radiation Physics −2 Lecture 5 I maximum penetration depth, Rmax (in units kg m ),

can be RCSDA, especially for low EK electrons Interactions of Charged Particles I Rmax/RCSDA ' 0.5 for high-Z; e.g. for 1 MeV electrons: with Matter Radiation Yield in Pb, Rmax/RCSDA ' 0.57; in C, Rmax/RCSDA ' 0.95 Bremsstrahlung Targets Thin Targets I Rmax measured from percentage depth-dose curve: Thick Targets Restricted Stopping Power Straggling and Scattering Electron Range Dose Linear Energy Transfer

Typical Percentage Depth Dose (PDD) curve for an arbitrary electron beam in water. R50 is the mean range, Rp is the projected range. (Fig. 6.15 in Pod- goršak.) Radiation Physics Lecture 5 Dose Deposition

I absorbed dose is the energy deposited by charged Interactions of Charged Particles particles in a medium per unit mass of the medium: with Matter Radiation Yield Bremsstrahlung Targets Thin Targets ∆Eab −1 dE Thick Targets D = = Φ = Φ Scol (6) Restricted Stopping Power ∆m ρ dx Straggling and Scattering Electron Range Dose Linear Energy Transfer where Φ = ﬂuence (no. particles per unit area).

Recall from Lec. 4, Scol vs. EK for electrons:

Mass collision stopping power (solid curves) and radiative stopping power (dashed curves) for electrons. (Fig. 6.10 in Podgoršak.) Radiation Physics S for electrons is smoothly varying w.r.t. E Lecture 5 I col K 2 ⇒ as incident electrons with EK mec lose energy, D

Interactions of remains relatively steady, with a gradual build up due Charged Particles with Matter to increase in number of secondary low-EK electrons Radiation Yield 2 Bremsstrahlung Targets and increase in Scol when EK < mec , followed by a Thin Targets Thick Targets gradual drop in D as electron energy is depleted Restricted Stopping Power decreases as increases, so dose deposited Straggling and Scattering I Rmax Scol Electron Range over shorter depth for lower E Dose K,0 Linear Energy Transfer Radiation Physics Lecture 5

I compare electron Rmax, Scol and depth dose curves to Interactions of Charged Particles those for heavy charged particles: with Matter Radiation Yield Bremsstrahlung Targets Thin Targets The localised deposition of energy Thick Targets Restricted Stopping Power (’Bragg peak" in depth-dose curve) Straggling and Scattering for heavy charged particles results Electron Range from the sharp increase in Scol as Dose the particles become nonrelativis- Linear Energy Transfer tic. For the same Rmax, electrons deposit their energy through- out most of the depth traversed. Radiation Physics Lecture 5 Linear Energy Transfer

I Bethe formula for Scol for heavy charged particles (c.f. Interactions of Charged Particles Lec. 4) ⇒ with Matter Radiation Yield 2 2 2 2 Bremsstrahlung Targets dE z 2mec γ β 2 Thin Targets ∝ ρ 2 ln − β (7) Thick Targets dx β I Restricted Stopping Power Straggling and Scattering Electron Range I lim β 1 (Bethe-Bloch formula): Dose Linear Energy Transfer dE z2 2m v2 ∝ ρ ln e dx β2 I

−2 I dE/dx for light charged particles has similar β dependence and somewhat weaker logarithmic

dependence; but, β is higher for a given EK I heavy charged particles have a higher linear energy transfer (LET) – energy transfer is more localised

I dose from high-LET radiation (e.g. p, α) causes more biological damage because it is localised, thus reducing chances for DNA repair Radiation Physics Lecture 5