CHARGED PARTICLE INTERACTIONS

Background

Charged Particles Heavy charged particles Charged particles with Mass > m e α, proton, deuteron, heavy ion (e.g., C +, Fe + ), fission fragment, muon, etc. α is one of the most common heavy charged particles Electrons (light charged particles) e, β -, β + Originally electron or positron

Charged Particle Behavior in Medium Heavy charged particles Light charged particles

What Happens During Travel in Medium? Charged particle loses its energy by ionization or excitation Medium ++ + Loss of energy by ionization or excitation - Ionization Excitation

Ionization and Excitation Ionization Neutral Atom Ionization After Ionization Excitation Electron in ground state Excitation De-excitation

General Characteristics of Charged Particle Interactions Interact through Coulombic forces with one or more electrons or with the nucleus of practically every atom in its vicinity Most interactions results in the transfer of only a small fraction of the particle s kinetic energy (10 100 ev) Many collisions are required to stop a particle (~10 4 to 10 5 interactions for a 1 MeV particle) Each particle traces out a given path length which is approximately linear for heavy particles, but is very nonlinear for light particles such as β - or β +.

Types of Interactions

Interaction Types of Charged Particles 1. Soft collision (b>>a) 2. Hard collision (b ~ a) 3. Coulomb interactions with Nuclei (b << a) 4. Annihilation of β + 5. Nuclear interactions by heavy particles

(1) Soft Collision (b>>a) Interaction Charged particle interacts with entire atoms Interaction probability The most numerous type of interaction encountered Energy transfer Transfer of a small amount of energy (few ev) resulting in either (1) ionization or (2) excitation Accounts for ~1/2 of the total energy transferred to the medium Small part of the energy transferred in soft collision is in the form of Cerenkov photon radiation.

Cerenkov Radiation EM radiation is emitted when a charged particle (e.g., β-particle) travel in a medium with a speed > speed of light. (No matter can travel faster than light speed in vacuum. But, it is possible in a medium. When light passes through a transparent material (glass or water) the speed is less than its speed in vacuum (Speed = c/n where n = refractive index)) The fraction is very small: 0.1% of β energy The radiation energy is continuous (eg) The characteristic blue glow of nuclear reactors is due to Cerenkov radiation.

(2) Hard Collision (b ~ a) Interaction Interaction with primary a single atomic electron The collided electron is ejected from the atom and called secondary electron or delta ray (δ-ray). The δ-rays are energetic enough to undergo their own Coulomb-force interaction ( another charged particle radiation). Interaction probability Very small compared to soft collision Energy transfer Few in number of interactions, but these interactions account for about remaining 1/2 of the total energy transferred. Energy transfer/collision is much bigger than that of soft collision What happens if inner shell is ejected? Characteristic x-ray Auger electron

Maximum Energy Transfer (T max ) in a Single Collision (Now, we are interested how much of energy is transferred by a hard collision) Heavy particles (classic results)

Heavy particles (Relativistic results) (Example) T max value of proton 1 MeV proton (β 2 = 0.0002128) T max = 2.18 kev (0.2%) 50 kev proton (β 2 = 0.0001066) T max = 109 ev (0.2%) ( Only small fraction of energy is transferred for heavy charged particles)

Positron If annihilation does not occur: T max = T Electron The primary and struck electron are indistinguishable according to quantum mechanics By convention, the one with the greater energy is referred to as the primary electron: T max = ½ T

(3) Coulombic Interactions with Nuclei (b << a) 97-98% of encounters: elastic scattering Insignificant amount of energy loss Major mechanisms for deflection of both β - and β + σ es Z 2 of absorber (eg) Rutherford gold foil experiment 2-3% of encounters: inelastic radiative interactions Bremsstrahlung (continuous x-rays) σ brem Z 2 absorber 2 0 M α-particle and proton does not emit Bremsstrahlung because too heavy Results in deflection as well as up to 100% energy transfer to photons

- - - - - - - -

(4) Inflight Annihilation (β + ) Only β + particle Annihilation process Positron loses its kinetic energy by exciting or ionizing atoms through its path Finally it is annihilated by colliding with a free electron. Creates two (or more) gamma rays with energy of 0.511 MeV Two photons direction is exactly opposite (180 ) A positron is not necessarily annihilated when all the kinetic energy is lost. The remaining kinetic energy before annihilation is given to one or both of annihilation photons. So, photons energy may exceed the usual 0.511 MeV

Energy Loss Mechanism of Positron Energy loss mechanisms of positron Inelastic scattering (= Ionization and excitation) Bremsstrahlung

Sources of Positron Pair production Positron emitting nuclides F-18 P-30 (commonly used for PET)

(5) Nuclear Interactions by Heavy Particles Inelastic collision at T 100 MeV Two step process Individual nucleons driven out of nucleus Excited nucleus subsequently decays through the emission of elaboration particles (mostly nucleons of low energy) and γ-rays

Stopping Power

Stopping Power Stopping power Expectation value of the instantaneous rate of energy loss per unit path length by a charged particle of kinetic energy T in a given medium of atomic number Z. Three components Collision stopping power (electron, heavy ions) Radiative stopping power (electrons) Nuclear recoil stopping power (heavy ions @ low energies)

LET (Linear Energy Transfer) LET is closely related to stopping power. So, let s review LET here. LET is used to quantify the effects of ionizing radiation on biological specimens (eg) LET was used to determine radiation Quality Factor for radiation protection concept Linear energy transfer A measure of energy transferred to material per unit path length when an ionizing particle travels through it

(1) Collision Stopping Power of Heavy Charged Particles Derivation of stopping power (Stopping power was driven by Bethe and Bloch) (Thy drove the stopping power of (1) heavy charge particles and (2) electron separately because collision stopping power of electrons (or β) is different from that of heavy charged particle by two factors) Electron (or β) can lose a large fraction of its energy in a single collision Incident electron is identical to atomic electron (Now we will review stopping powers of the (1) heavy charge particles and (2) electron.)

Bethe-Bloch formula for heavy charged particles

10000 Collision Stopping Power (MeV cm 2 /g) 1000 Alpha particle Proton Electron 100 10 1 0.01 0.1 1 10 100 1000 10000 Energy (MeV)

(2) Collision Stopping Power of Electron Collision stopping power of electron (β) vs. heavy particle Collision stopping power of electrons (or β) is different from that of heavy charged particle by two factors: Electron can lose a large fraction of its energy in a single collision Incident electron is identical to atomic electron Bethe-Bloch formula for electrons Mass collision stopping power for electrons

(3) Radiative Stopping Power Radiative stopping power Stopping power resulting from bremsstrahlung Electrons can lose its energy by bremsstrahlung as well as collision (cf) Heavy charged particles does not emit bremsstrahlung radiation Radiative stopping power vs. Collision stopping power (electron)

Radiation Yields of Electron and Beta β-rays have continuous energy while electron is mono-energetic. Therefore, total fraction of electron and β kinetic energy T 0 lost to bremsstrahlung will be different Electron Beta

(4) Radiation Yield Radiation yield [ Y(T 0 ) ] Total fraction of initial kinetic energy T 0 of a charged particle lost to bremsstrahlung production (See Appendix E of Attix) Average fraction of electron energy lost to bremsstrahlung (g) g = Average value of Y for all electron of various initial energies T 0 µ en = µ tr ( 1 g)

(5) Restricted Collision Stopping Power Restricted collision stopping power Use of (dt/ρdx) c will overestimate absorbed dose to small volumes due to the escape of energetic delta rays (dt/ρ dx) = restricted collision mass stopping power Includes all soft collisions as well as hard collisions resulting in energy transfers less than the cutoff value. Restricted Linear Energy Transfer (LET or L )

Bragg Curve Bragg Curve Plot of stopping power (energy loss per path length) along the track of a charged particle Typical Bragg curve shape As energy falls, stopping power increase Stopping power rapidly increase near the end and reach its peak before particle stop according to Bethe-Bloch formula Bragg peak Bragg peak is observed only for heavy charged particles. For electrons, there is no increase in energy deposited near the end of the tract and the Bragg peak for electrons is never observed.

Bragg curve vs. Stopping power by energy

Energy Straggling Energy straggling A spread of energies (= energy straggling) always results when an initially mono-energetic beam of particles encounters an absorber Details of the microscopic interaction undergone by any specific particle vary somewhat randomly. Energy loss in a material is a statistical or stochastic process. Many particles lose the average energy, although some will lose not so much and some will lose more than the average This results in a finite width to the energy distribution curve known as energy straggling. Straggling peak is approximately Gaussian shaped

Range Straggling Range straggling Fluctuation in path length for individual particles. Details of microscopic interaction vary depending on particle It results in a spread of particle energy in medium energy straggling Like this, the stochastic (or statistical) factors of charged particle interaction results in fluctuation not only in energy but also in path length of each particle range straggling

Average Ionization Energy (W-value) Energy loss by charged-particles in medium Ionization: Each ionization results in an ion-pair (IP) consisting of a positive gas ion and a free electron. Excitation W-value Average energy needed to produce an ion pair W = T (initial kinetic energy of particle) N (total # of ion pairs formed) Generally W = 30 ~ 35 ev W for air = 34 ev (Dry air at STP) Why W > Ionization potential of gases (10-20 ev)? Charged particle loses energy by not only ionization but also excitation The excitation does not produce ion pair

Specific Ionization Specific ionization Number of ion pairs per path length

Range of Charge Particles

Two Definitions of Range Range (R project ) Average maximum depth of penetration CSDA range (R CSDA ) Average total path length in the medium

CSDA Range CSDA : Continuous Slowing-Down Approximation How to calculate a CSDA range (R CSDA )?

Range of Heavy Charged Particles Experiment for heavy charged particle range

3 different types of ranges in alpha particles Mean range: Absorber thickness that reduce the alpha particle counts to ½ of its values in the absence of the absorber Extrapolating range: Obtained by extrapolating the linear portion of the end of the transmission curve to zero Maximum range: Absorber thickness where count is zero

Range of heavy charged particles can be obtained by: (Graph or table of experimental or calculated range data) (Semiempirical formula) Semiempirical formula of alpha range in air at STP R R ( cm) = exp ( 0.161 T ) for 1 T 4 MeV 1.5 ( cm) = ( 0.005T + 0.285) T for 4 T 15 MeV T is the kinetic energy of the particle in (MeV) Generally, energy of alpha particles from radionuclides is 4 ~ 8 MeV Range of 4 MeV alpha is about 2.5 cm in air

Range (Alpha and Proton in Water)

Semiempirical Formula of Alpha Range in Air (Semiempirical formula approximates experimental data. Therefore, the formula can vary depending on approximation.) You can find different formulae for charged particle range In Attix R ( cm) = exp ( 0.161 T ) for 1 T 4 MeV R In Turner and Cember R R 1.5 ( cm) = ( 0.005T + 0.285) T for 4 T 15 MeV ( cm) = 0.56 T for T < 4 MeV ( cm) = 1.24 T 2.62 for 4 < T < 8 MeV Simple one R 1.5 ( cm) = 0.31 T for 4 T 8 MeV

Experimental data vs. Formula

Charged Particle Range from Radiation Protection Aspect Generally charged particle (especially heavy charged particles) range is very short You can protect alpha even with a piece of paper Generally, charged particles (α, p, β) are not important for external exposure because they can not penetrate skin and give dose (radiation energy) to organs and tissues in human body But, when they are in human body, they are very much concern because there is no protective layer (like skin) in human internal tissues all the energy from the particle will be absorbed by human tissue In addition, of α particle is 20. Radiation weighting factor = Relative biological effect by radiation type Radiation weighting factor of photon = 1 ( α particle is more dangerous about 20 times compared to photon)

Characteristics of transmission curve for heavy charged particle Flat at first and sharp decrease at the end Because increasing thickness of absorbers serves merely to reduce the energy; the number is not reduced until the approximate range is reached

Scaling Laws (It is not possible to make range experiments for all possible incident particles, all possible absorbing materials at all possible energies.) Range of a given particle in another medium (1 vs. 2) Bragg-Kleeman Rule For a compound or mixture, use the effective mass number:

Range of another particle (1 vs. 2) in the same medium Example (alpha vs. proton in the same medium))

Ranges of Electrons and Positrons Characteristics of projected ranges of electron (or positron) Number of electrons decreases with absorber thickness Long tails at the end Due to bremsstrahlung x-ray In attenuation experiments for mono-energetic fast electrons, even small values of absorber thickness lead to a loss of detected electrons because scattering within the absorber effectively removes an electron from the flux striking the detector (cf) Alpha particle goes straight after collision Collision does not result in loss of detected alpha particles

Different types of ranges in beta particles Extrapolating range: obtained by extrapolating the linear portion of the curve to zero Maximum range: Absorber thickness where count is zero

Semiempirical formula: Projected ranges of electrons in any media

You can find other equations in other texts Turner, Cember, and Lamarsh (electron or β) R R 2 1.27 0.0954 ln T ( g / cm ) = 0.412 T for 0.01 T < 2 ( g / cm ) = 0.530 T 0.106 for T > 2.5 MeV 2.5 MeV For low Z materials R CSDA R prjoect For example, R project /R CSDA of carbon is 0.95 (for 0.025 MeV ~ 10 MeV) For β, T is maximum energy of β

Characteristics of projected ranges of β particle The transmission curve for beta particles emitted by a radioisotope source differs significantly from that of monoenergetic electron sources because of the continuous distribution of energies The soft or low-energy particles are rapidly absorbed with increasing depth and thus the initial slope on the attenuation curve is much greater. For the majority of the spectrum, the curve happens to have a near exponential shape like photon (only an empirical approximation).

Range of Alpha vs. electron (β) Alpha Alpha range is very short You can shield alpha even with a piece of pater Range = ~ 1.7 cm in air (for E = 3 MeV) Electron (or β) Longer than alpha particle (but much shorter than photon or neutron) You can shield beta particle with thin aluminum Range 13 cm in the air for E max = 3MeV