University of Oslo. Department of Physics. Interaction Between Ionizing Radiation And Matter, Part 2 Charged-Particles.

Transcription

1 Interaction Between Ionizing Radiation And Matter, Part Charged-Particles Audun Sanderud

2 Excitation / ionization Incoming charged particle interact with atom/molecule: Ionization Excitation Ion pair created from ionization

3 Elastic collision Interaction between two particles with conservation of kinetic energy ( and momentum): m, v m 1, v m m 1, v 1 Classic mechanics give: T = m v = m v + m v mv= mv cosθ+ mv cosχ = m v sinθ+ m v sinχ 1 1 χ θ

4 Elastic collision() m vcosχ 4m m cos v =, v = v 1 tan θ = m1+ m ( m1+ m) m1 cos m sin χ χ These equations gives the maximum transferred energy: χ 1 mm E = m v = 4 T m m 1 max,max 0 ( + ) 1

5 Elastic collisions(3) a) m 1 >>m a) m 1 =m a) m 1 <<m 0 χ π m χ m 1 0 θ tan sin m E 4 T 1 max = 0 max 0 m1 0 χ π 0 χ π 0 θ π 0 θ π E = T m E = 4 T 1 max 0 m Proton(#1)-electron(#): θ max =0.03 o, E max =0. % T 0 Electron(#1)-electron(#): θ max =90 o, E max =100 % T 0

6 Elastic collisions-cross section Rutherford proved that the cross section of elastic scattering is: dσ 1 dω 4 sin θ ( ) Small scattering angels most probable Differentiated by the energy dσ 1 de E Small energy transferred most probable

7 Stopping power Stopping power, (dt/dx): the expectation value of the rate of energy loss per unit of pathlength. Dependent on: -type of charged particle dx -its kinetic energy -the atomic number of the medium T 0 T 0 -dt n v targets per volume unit Emax Emax dσ NAZ dσ dt = EnVdxσ = n vdx EdT = ρ dx EdT min E N Z = = ρdx A dt max S dt A dσ EdT ρ E E min dt A dt E min

8 Impact parameter The charged particle collision is a Coulomb-force interaction Most important: the interaction with electrons The impact parameter b useful versus the classic atomic radius a

9 Soft collisions b>>a: particle passes an atom in a large distance Small energy transitions to the atom The result is excitations (dominant) and ionization; amount energy transferred range from E min to a certain energy H Hans Bethe did quantum mechanical calculations on the stopping power of soft collision in the 1930 We shall look at the results from particles with much larger mass then the electron

10 Soft collisions() S c,soft dt soft NAZ πr0 mec z mec β H = = ln β ρ ρdx A β c I ( 1 β ) r 0 : classic electron radius = e /4πε 0 m e c I: mean excitation potential β:v/c z: charge of the incoming particle ρ: Density of the medium N A Z/A: Number of electrons per gram in medium H: Maximum transferred energy at soft collision

11 Soft collisions(3) The quantum mechanic effects are specially seen in the excitation potential I Mean excitation potential, I/Z [ev] Atomic number, Z High Z small transferred energy less likely

12 Hard collisions b<<a: particle passes trough the atom Large (but few) energy transactions to single electron Amount energy transferred range from H to E max Can be seen as an elastic collision between free particles (bonding energy nelectable) S c,hard dt hard NAZ πr0 mec z E max = = ln β ρ ρdx A β H c

13 Collisions stopping power The total collision stopping power is then (soft + hard): S S S N Z 4πr m c z m c β ρ ρ ρ A β ( 1 β ) I c c,soft c,hard A 0 e e = + = ln β Important: increase with z, decrease with v, not dependent on particle mass

14 S c /ρ in different media I and electron density (ZN A /A) gives the variation Department of Physics

15 S C for electrons/positrons Electron-electron scattering more complicated; interaction between identical particles S c,soft /ρ: Bethe s soft coll. formula S c,hard /ρ: electron-elektron; Møller cross section positron-electron; Bhabha cross section S ρ N Z πr m c z τ ( τ+ ) ± C = ln + F ( τ) δ, τ T / mec A β I/mc ( e ) Z c A 0 e ( ) ( ) ( τ + 1) /8 1 ln τ τ+ F τ = 1 β + + β F ( τ ) = ln τ + τ + τ + ( ) ( ) 3 The characteristics similar to that of heavy particles

16 Shell correction The approximation used in the calculations of S C assume v>>v atomic electron When v~v atomic electron no ionizations will occur Occur first in the K-shell - highest atomic electron speed Shell correction C/Z handles this, and reduce S C /ρ C/Z depend on particle velocity and medium

17 Density-effect correction Charged particles polarizes the medium Charged (+z) particle E = E + E Eeff < Epol eff eff pol Weaker interaction with distant atoms because of the reduction of the Coulomb force field Polarization increase with (relativistic) speed But: polarization not important at low ρ Most important for electrons / positrons

18 Density-effect correction() Density-effect correction δ reduces S c /ρ in solid and liquid elements S c /ρ (water vapor) > S c /ρ (water) Dashed curves: S c without δ

19 Radiative stopping power When charged particles are accelerated by the Coulomb force from atomic electrons or nucleus photons can be emitted; Bremsstrahlung Charged particle atomic electron The Lamor equation (classic el.mag.) denote the radiation power from an acceleration, a, of a charged particle: (ze) a P = 6πε c 0 3 ε 0 : Permittivity of a vacuum

20 Radiative stopping power() The case of a particle accelerated in nucleus field: zze zze Zz F= ma= a= P a z 4πε r 4πε mr m 0 0 Comparison of proton and electron as incoming: P proton m electron 1 = P electron mproton 1836 Bremsstrahlung not important for heavy charged particles

21 Radiative stopping power(3) The maximum energy loss to bremsstrahlung is the total kinetic energy of the electron Energy transferred to radiation per pathlength unit: radiative stopping power: S dt NAZ = = α r0 ( T+ mec ) B r(t,z) ρ ρdx A r r B r (T,Z) weak dependence of T and Z Radiative energy loss increase with T and Z

22 Total stopping power, electrons Total stopping power, electrons: dt dt dt = + ρdx ρdx ρdx tot c r Comparison: S S r c TZ n n = 750MeV

23 Radiation yield Estimated fraction of the electron energy that is emitted as bremsstrahlung: ( dt / ρdx) r Sr YT ( ) = = ( dt / ρ dx) + ( dt / ρdx) S c r Radiation yield, Y(T) Water Tungsten Kinetic energy, T (MeV)

24 Comparison of S c Electrons, total Electrons, collision Electrons, radiative Protons, total Kinetic energy, T [MeV]

25 Other interactions Cerenkov effect: very high energetic electrons (v>c/n) polarize a medium (water) of refractive index n and bluish light is emitted (+UV) Little energy is emitted

26 Other interactions() Nuclear interactions: Inelastic process in which the charged particle cause an excitation of the nucleus. Result: - Scattering of charged particle - Emission of neutron, γ-quant, α-particle Not important below ~10 MeV (proton) Positron annihilation: Positron interact with atomic electron, and a photon pair of energy x0.511mev is created. The two photons are emitted 180 o apart. Probability decrease by ~1/v

27 Braggs rule Braggs rule for mixtures of n-atoms/elements: n S c S c = f, f = i i ρ ρ mix Z = 1 i n Z Z n i f ( Z ) ( ) ( ) Z ln I i i f Z Z δ A i i i A i Zi= 1 Zi= 1 ln ( Imix ) =, δ n mix = n f ( Z ) f ( Z ) A Zi A i Z Z = 1 Zi i Z = 1 Z = 1 i m Z i m Z i n i i

28 Linear Energy Transfer LET Δ ; also known as restricted stopping power Δ, cutoff value; LET Δ includes all the soft and the fraction of the hard collision δ-rays with energy<δ δ-electron as a result of ionization Trace of charged particle δ-electrons living the volume energy transferred > Δ S c includes energy transitions from E min to E max LET Δ the amount of energy disposed in a volume defined by the range of an electron with energy Δ

29 Linear Energy Transfer() The energy loss per length unit by transitions of energy between E min < E < Δ: L dσ EdE dx A de Δ dt NAZ Δ = = ρ Δ E If Δ = E max then L =S c ; unrestricted LET min NAZ z mec β = ρπr0mec ln Δ β A β (1 β )I LET Δ given in kev/μm 30 MeV protons in water: LET 100eV /L = 0.53

30 Range The range R of a charge particle in a medium is the expectation value of the pathlength p The projected range <t> is the expectation value of the farthest depth of penetration t f in its initial direction Electrons: <t> < R Heavy particles: <t> R

31 Range() Range can by approximated by the Continuous Slowing Down Approximation, R CSDA Energy loss per unit length is given by dt/dx gives an indirect 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 T0 dt R CSDA = ρdx 0 1 dt

32 Range(3) Range is often given multiplied by density T0 dt R CSDA = ρ dx 0 1 dt Unit is then [cm][g/cm 3 ]=[g/cm ] Range of a charge particle depend on: - Charge and kinetic energy - Density, electron density and average excitation potential of absorbent

33 Range(4) Department of Physics

34 Straggling and multiple scattering In a radiation field of charged particles there is: - variations in rate of energy loss - variations in scattering The initial beam of particle at same speed and direction, are spread as they penetrate a medium v v v v 1 3 v 4

35 Multiple scattering Electrons experience most scattering characteristic of initially close to monoenergetic beam: Initial beam Beam at small depth in absorbent Beam at large depth in absorbent Number Energy [MeV]

36 Projected range <t> Characteristic of different type of particles penetrating a medium:

37 Energy disposal Protons energy disposal at a given depth: Department of Physics

38 Energy disposal() Electrons energy disposal at a given depth; multiple scattering decrease with kinetic energy:

39 Monte Carlo simulations Monte Carlo simulations of the trace after an electron (0.5 MeV) and an α-particle (4 MeV) in water Notice: e - most scattered α has highest S

40 Hadron therapy Heavy charged particles can be used in radiation therapy gives better dose distribution to tumor than photons/electrons

41 Tables on the web Stopping power Attenuation coefficients

42 Summary