Excitation / ionization. Interaction theory Electrons. Lesson FYSKJM4710. Eirik Malinen. Incoming charged particle interacts with atom / molecule:

Transcription

1 Interaction theory Electrons Lesson FYSKJM4710 Eirik Malinen Source: F. H. Attix: Introduction to radiological physics and radiation dosimetry (ISBN ) Excitation / ionization Incoming charged particle interacts with atom / molecule: Excitation An ion pair is created Ionisation

2 Elastic collision 1 Interaction between two particles where kinetic energy is preserved: m 1, v m m, v m 1, v 1 χ θ Classical mechanics: T = m v = m v + m v mv= mv cosθ+ mv cosχ = m v sinθ m v sinχ 1 1 Elastic collision m vcosχ 4m m cos v =, v = v 1 tan θ= m1+ m (m1+ m ) sin χ m1 cos m χ Equations give, among others, maximum energy transferred: 1 mm E = m v = 4 T 1 max,max 0 (m1+ m ) χ

3 Elastic collision 3 a) m 1 >>m b) m 1 =m c) m 1 <<m 0 χ π/ 0 χ π/ 0 χ π/ 1 m 0 θ tan ( sin χ) 0 θ π/ 0 θ π m1 m m 1 Emax = 4 T0 Emax = T E = 4 T 0 m m 1 max 0 Proton-electron collision: θ max = 0.03º, E max = 0. % Electron-electron (or e.g. neutron-neutron) coll.: θ max = 90º, E max = 100 % Elastic collision cross section Rutherford showed that the cross section is: dσ 1 Ω θ 4 d sin ( /) small scattering angles are most probable With respect to energy: dσ 1 de E small energy transfers most probable

4 Stopping power S=dT/dx; expected energy loss per unit lenght T 0 dx T 0 -dt n V targets per volume Emax Emax dσ NAZ dσ dt = EnVdxσ = nvdx EdE =ρ dx EdE de A de E E min N Z σ = = ρdx ρ A de max dt S A d EdE E E min min Impact parameter Charged particles: Coulomb interactions Most impprtant: interactions with electrons Impact parameter b: a: classical atomic radius

5 Soft collisions 1 b >> a : incoming particle passes atom at long distance Weak forces, small energy transfers to the atom Inelastic collisions: Predominantly excitations, some ionizations Energy transfer range from Emin to H Hans Bethe. Quantum mechanical considerations In the following, theory for heavy charged particles Soft collisions Sc,soft dtsoft NAZ πrmcz 0 e mc e β H = = ln β ρ ρdx A β I (1 β ) c r 0 : classical electron radius = e /4πε 0 m e c I: mean excitation potential β = v/c z: charge of incoming particle ρ: density of medium N A Z/A : numbers of electron per gram H: maksimum energy transferred by soft collisions

6 Soft collisions Quantum mechanics (atomic structure) is reflected in the mean excitation potential Hard collisions 1 b << a : charged pass particle pass through atom Large (but few) energy transfers Energy transfers from H to E max May be considered as an elastic collision between free particles (binding energy is negligible) S dt N Z π r m c z E ρ ρ β c,hard hard A 0 e max = = ln β dx A H c

7 Collision stopping power For inelastic collisions, the total cross section is thus: S S c c,soft S = + ρ ρ ρ c,hard NAZ z mec β 0 e = 4πr m c ln β A β (1 β )I Important: Increases with z, decreases with v, not dependent on particle mass S c /ρ, different substances Singly charged heavy particles I and electron density (ZN A /A) give differences

8 S c /ρ, electrons and positrons Electron-electron scattering is more complicated; scattering between two identical particles S c, hard /ρ (el-el) is described by the Möller cross section S c, hard /ρ (pos-el) is described by the Bhabha c.s. S c, soft /ρ was given by Bethe, as for heavy particles Characteristics similar to that for heavy charged particles Shell correction Derivation of S c assumes v >> v atomic electrons When v ~ v atomic electrons, no ionizations Most important for K-shell electrons Shell correction C/Z takes this into accout, and thus reduces S c /ρ C/Z depends on particle energy and medium

9 Density correction Charged particles polarizes medium which is being traversed E ur ur pol E E eff = E part + E pol, E eff < E part part Charged (+z) particle ur ur ur ur ur + Weaker interactions with remote atoms due to reduction in electromagnetic field strenght Polarization increases with energy and density Most important for electrons and positrons Density correction Density correction δ reduces S c /ρ for liquids and solids S c /ρ (water vapor) > S c /ρ (water) Dashed line: S c /ρ without δ

10 Linear Energy Transfer 1 LET Δ is denoted the restricted stopping power dt/dx: mean energy loss per unit lenght but how much is deposited locally? Electron released by ionization Track from charged particle Electrons leaving local volume energy transfer > Δ S c : energy transfers from E min to E max How much energy is deposited within the range of an electron given energy Δ? Linear Energy Transfer Energy loss (soft + hard) per unit lenght for E min < E < Δ: L Δ dt NAZ dσ = =ρ EdE dx A de Δ For Δ=E max, we have L =S c ; unrestricted LET LET Δ is often given in [kev/μm] 30 MeV protons in water: LET 100 ev / L = 0.53 E Δ min NAZ z mec β Δ 0 e =ρπr m c ln β A β (1 β )I

11 Brehmsstrahlung 1 Photon may be emitted from charged particle accelerated in the field from an electron or nucleus e.g. electron Larmor s formula (classical electromagnetism) for radiated effect from accelerated charged particle: (ze) a P = 6 πε c e.g. nucleus 0 3 Brehmsstrahlung For particle accelerated in nuclear field: zze zze F= ma = a = 4πε r 4πε mr Z P m Comparison of protons and electrons: P p m e 1 = P e m p Brehmsstrahlung not important for heavy charged particles

12 Brehmsstrahlung 3 Energy loss by brehmsstrahlung is called radiative loss Maksimum energy loss is the totale kinetic energy T Radiative loss per unit lenght: radiative stopping power: S dt NAZ = α r 0 (T+ mec )B r(t,z) ρ ρdx A r r B(T,Z) r weakly dependent on T and Z Brehmsstrahlung increases with energy and atomic number Total stopping power, electrons Total mass stopping power: dt dt dt = + ρdx ρdx ρdx tot c r

13 Radiation yield Y(T) (dt / ρdx) Sr (dt / ρ dx) + (dt / ρdx) S r = = c r TZ n n=750 MeV Y(T) Water vann Tungsten wolfram T [MeV] S c /ρ, protons and electrons S c /ρ [MeV/cm g] Electrons, total Electrons, collisions Electrons, radiative Protons, collision T [MeV]

14 Cerenkov effect High energy electrons (v > c/n) polarizes medium (e.g. water) and blueish light (+ UV) is emitted Low energy loss Other interactions Nuclear interactions: Inelastic process where charged particle (e.g. proton) excites nucleus Scattering of charged particle 4 Emission of neutron, photon, or α-particle ( He ) Not important below ~10 MeV (protons) Positron annihilation: Positron interacts with electron a pair of photons with energy x MeV is created. Photons are emitted in opposite directions. Probability decreases as ~ 1/v

15 Range 1 The range R of a charged particle in matter is the (expectation value) of it s total pathlenght p The projected range <t> er is the (expectation value) of the largest depth t f a charged particle can reach along it s incident direction Electrons: <t> < R Heavy charged particles: <t> R CSDA-range The range may be approximated by R CSDA (continuous slowing down approximation) Energy loss per unit lenght dt/dx gives implicitly a measure of the range: T 0 Δx dt T0 Δ T= T0 Δx dx n n dx dx Δ x = ΔT, R= Δ x = ΔT dt i i= 1 i= 1 dt i T 0 1 dt R CSDA = dx 0 dt

16 Range 3 The range is often given multiplied by the density: T 1 0 dt R CSDA = ρdx 0 dt Unit thus becomes [cm] [g/cm 3 ] = [g/cm ] Range of charged particle depends on: Charge and kinetic energy Density, electron density and mean excitation potential of absorber Range 4

17 Multiple scattering and straggling In a beam of charged particles, one has: Variations in energy deposition (straggling) Variations in angular scattering The beam, where all particles originally had the same velocity, will be smeared out as the particles traverses matter v v 1 v 4 v v 3 Straggling Number of electrons Primary beam Beam at shallow depths Beam at large depths Energy [MeV]

18 Projected range Energy deposition, protons 187 MeV

19 Energy deposition, electrons Monte Carlo simulations Monte Carlo simulations of the track of an electron (0.5 kev) and an α-particle (4 MeV) in water Note: e - is most scattered α has the highest dt/dx e - nm Excitation Ionisation α

20 Hadron therapy Heavy charged particles may be used for radiation therapy conforms better to the target than photons or electrons Web pages For stopping powers: For attenuation coefficients: