Interaction theory Photons Eirik Malinen
Introduction Interaction theory Dosimetry Radiation source Ionizing radiation Atoms Ionizing radiation Matter - Photons - Charged particles - Neutrons Ionizing radiation Effects Molecules Cells Humans H O DNA
Objectives To understand primary and secondary effects of ionizing radiation How radiation doses are calculated and measured To understand the principles of radiation protection, their origin and applications
Contents FYSKJM4710 Interactions between ionizing radiation and matter Radioactive and non-radioactive sources Calculations and measurement of absorbed doses (dosimetry) Radiation chemistry Biological effects of ionizing radiation Principles of radiation protection
Relevant issues X-ray and CT investigations Radiotherapy Positron emission tomography Radiation protection Radiation Biology
X-ray contrast: only a matter of differences in density?
Cross section 1 Cross section s: target area, effective target covering a certain area Proportional to the interaction strength between an incoming particle and the target particle Consider two discs, one target and one incoming: electron, nucleus... radius r incoming particle radius r 1 (direction into the paper) s is the total area: π (r + r ) 1
Cross section N particles move towards an area S with n atoms Atomic cross section, σ Incoming particle S interaction no interaction Probability of interaction: p = nσ/s Number of interacting particles: Np= Nnσ/S
Cross section 3 Separate between electronic and atomic cross section The cross section depends on: - Type of target (nucleus, electron,..) - Type of and energy of incoming particle (photon, electron ) Cross section calculated with quantum mechanics - here visualized in a classical window
Cross section 4 Differential cross section with respect to scattering angle dσ number of particles scattered into dω 1 = dω number of particles per unit area dω
Photon interactions Photon represented by a plane wave in quantum mechanical calculations A in (r,t) ~ e i( p in r ω t ) in In principle, two different processes: Absorption r A inn ( Ain( r,t ) r,t ) ψ 0 ψ k * θ r A ut ( r,t ) Scattering r A inn ( r,t ) Ain( r,t ) Scattering: coherent (elastic) og incoherent (inelastic) ψ 0 ψ k * θ r A ut ( r,t ) Aout ( r, t)
Coherent (Rayleigh) scattering Scattering without loss of energy: hν=hν Photon is absorbed by atom, thereby emitted at a small deflection angle Depends on atomic structure and photon energy Atomic cross section: σ R Z hν
Incoherent (Compton) scattering Scattering with loss of energy: hν <hν Photon-electron scattering; electron may be assumed free (i.e. unbound) Thomson scattering: low energy limit, hν 0
Compton scattering kinematics Conservation of energy and momentum: hν = h ν ' + T hν h ν ' h ν ' = cosθ + p cos ϕ, sinθ = p sinϕ c c c ( pc ) = T + Tm c e hν hν θ h ν ' =, cotϕ = 1+ tan 1 + (1 cos ) m c hν θ mec e
Compton scattering kinematics
Compton scattering example An X-ray unit is to be installed, with the beam direction towards the ground. Employees in the floor above the unit are worried. Maximum X-ray energy is 50 kev. What is the maximum energy of the backscattered photons? hν hν θ = 180 h ν ' = = hν hν 1 + (1 cos θ ) 1+ m c m c e e 50 h ν = 50 kev h ν ' = = 16 kev 50 1+ 511
Compton scattering cross section 1 Klein and Nishina derived the cross section for Compton scattering, assuming free electron Differential cross section: r 0 d σ ν ' ν ' ν = + sin θ dω ν ν ν ' dω = sinθ dθ dφ r 0 : classical electron radius z θ dω incoming photon along z-axis φ y x
Compton scattering cross section Cylinder symmetry results in: 0 d σ ν ' ν ' ν = π r + sin θ sinθ d θ ν ν ν ' ~ probability of finding a scattered photon in the interval [θ, θ+dθ] Total electronic cross section: π ν ' ν ' ν e = 0 + ν ν ν ' 0 σ π r sin θ sinθdθ Atomic cross section: a σ=z e σ
Compton scattering cross section 3 Scattered photons are more forwardly directed with increasing photon energy: 10 kev 00 kev MeV
Compton scattering cross section 3 Cross section may be modified with respect to energy: dσ dσ dω dσ dθ = = π sinθ d( h ν ') dω d( h ν ') dω d( h ν ') h ν ' = hν hν 1 + (1 cos θ ) m c e dσ π r m c h ν ' hν hν m c = + + d( h ν ') ( h ν ) hν h ν ' h ν ' hν 0 e e 1 1 1
Compton scattering cross section 4 Backscatter edge Compton edge
Compton scattering cross section 5 Correct atomic cross section: Effect of electron binding energy Klein-Nishina Carbon (Z=6) Copper (Z=9) Lead (Z=8)
Compton scattering transferred energy 1 The energy transferred to an electron in a Compton process: T = hν h ν ' The cross section for energy transfer: dσ tr dσ T dσ hν h ν ' = = dω dω hν dω hν Mean energy transferred: T dσ hν h ν ' dσ T dω dω dω hν dω σ tr dσ dω σ σ dω = = = hν
Compton scattering transferred energy The fraction of incident energy transferred:
Photoelectric effect 1 Photon is absorbered by atom/molecule; the result is an excitation or ionization θ γ e - X X + T a Atom may deexcite and emit characteristic radiation:
Photoelectric effect In the kinematics, the binding energy of the ejected electron should be taken into account: T = hν E T hν E b a b Assuming E b =0, the atomic cross section is: 7 / 4 5 mec h e 0 ν e dτ ν = r α Z sin θ 1+ 4 cosθ d Ω h m c e W α: The fine-structure constant Solid angle e Ω gives the direction of the ejected electron
Photoelectric effect Photoelectric cross section (dσ/dθ)/s
Characteristic radiation Energy of characteristic radiation depends on elektronic structure and transition probabilities K- and L-shell vacancies hν K and hν L Isotropic emission Fraction of photoelectric interactions: P K [hν>(e b ) K ] and P L [(E b ) L <hν<(e b ) K ] Probability for emission: Y K og Y L (flourescence yield) Energy emitted from the atom: P K Y K hν K +(1-P K )P L Y L hν L
Auger effect Energy release by ejection of losely bound electron Energy of emitted electron equal to deexcitation energy Low Z: Auger dominates High Z: characteristic radiation dominates
Photoelectric cross section General formula: τ n Z ( h ν ) m, 4<n<5, 1<m<3 Fraction of energy transferred to photoelectron: T = hν hν E b hν However: don t forget Auger electron(s) Cross section for energy transfer to photoelectron: τ = τ tr ( hν P Y hν -(1-P )P Y hν ) K K K K L L L hν
Pair production 1 Photon absorption in the nuclear electromagnetic field where an electron-positron pair is created Triplet production: in the electromagnetic field of an electron
Pair production Conservation of energy: + hν = mec + T + T Average kinetic energy after absorption: T = hν mec Estimated electron/positron scattering angle: θ m c Total cross section: e T κ αr0 Z P
Discovery of pair production e + e - Magnetic field
Triplet production In the electromagnetic field from an electron, an electron-positron pair is created Energy conservation: hν = m c + T + T + T + e 1 Average kinetic energy: T hν mec = 3 Primary electron is also given energy Threshold: 4m 0 c
Pair- and triplet production Pair production dominates: k pp /Z or k tp /Z [cm ] Pair
Photonuclear reactions Photon (energy above a few MeV) excites a nucleus Proton or neutron is emitted (γ, n) interactions may have consequences for radiation protection Example: Tungsten W (γ, n)
Summary, interactions 1 Photonnuclear reactions Processes Compton Rayleigh Photoel. eff. Pair/triplet Kinematics hν hν' = 1 + ( hν / m c )( 1 cosθ ) T = hν hν' cotϕ = e [ 1 + ( hν / m c )] tan( θ / ) e T=0 T = hn - E b - T a P: hn = m 0 c + T = hn - E b T: hn = m 0 c + 3T Atomic cross section d σ ν' ν' ν Z + sin θ dω ν ν ν' Z σ R hv Z n τ ( hv ) m κ Z P( hν ) κ Z P( hν ) t p Mean energy transferred σ T = hν tr σ T = hν P Y hν ( 1 P K K K K )P Y hν L L L T T p t hν m0c = hν m0c = 3
Attenuation coefficients 1 n V atoms per volume = ρ(n A /A) Number of atoms: n=n V V=n V Sdx Interaction probability p=nσ/s=n V σdx Probability per unit length: µ=p/dx=n V σ=ρ(n A /A)σ µ: linear attenuation coefficient σ S dx
Attenuation coefficients N A : Avogadro s constant; 6.0 10 3 mole -1 A: number of grams per mole N A /A: number of atoms per gram N A Z/A: number of electrons per gram Number of atoms per volume: ρ(n A /A) Etc.
Attenuation coefficients 3 Total mass attenuation coeffecient: µ τ σ κ σ = + + + R ρ ρ ρ ρ ρ Coefficient for energy transfer: µ tr µ T = ρ ρ hν Braggs rule for mixture of atoms: µ µ n = f i, fi = n ρ mix i= 1 ρ i m i= 1 i m i
Attenuation coefficients 4 Mass attenuation coefficient [cm /g] hn [MeV] Oxygen (Z=8) hn [MeV] Uranium (Z=9) Rayleigh Compton Photoelectric Pair/triplet Total
Attenuation 1 Beam with N photons impinge absorber with thickness dx: dx N N-dN Probability for interaction: µdx Number of photons interacting: Nµdx dn dn = Nµ dx = µ dx N = x N N0e µ
Attenuation Note that µ is the interaction probability per unit lenght not the absorption probability e -µx corresponds to a narrow beam measurement geometry: Shield Scattered photons detector Attenuator, thickness dx
Attenuation 3 Probability for photon not interacting: e -µx Normalized probability! µ x µ x iv ni iv iv µ ni ni 0 p = Ce, p dx = 1, p = e Mean free path: = = µ x iv µ = ni 0 0 x xp dx x e dx 1 µ
Attenuation 4 Thickness of absorber (cm)
Attenuation - example MeV photons Pb: µ = 0,516 cm -1 H O: µ = 0,049 cm -1 e µ HO x x x H Pb HO O = e µ = µ µ 10 times as much water necessary H Pb Pb O x Pb
Scattered photons e -µx : number of primary photons at a given depth What about the scattered photons? Monte Carlo simulations
Primary and scattered photons, 100 kev e -µx
Primary and scattered photons, 1 MeV e -µx