Radiation Dosimetry Attix 8 Charged-Particle Interactions in Matter Ho Kyung Kim hokyung@pusan.ac.kr Pusan National University
References F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry, John Wiley and Sons, Inc., 1986 2
Uncharged radiation May not interact with matter at all and consequently not lose energy; or may interact and thus lose energy in one or a few "catastrophic" events Impossible to predict how far an individual photon (or neutron) will penetrate through matter Described by the mean-free path instead of the range Charged particle Surrounded by its Coulomb electric force field Interacts with electron(s) or with nucleus of every atom it passes; and these interactions transfer only minute fractions of KE (as if losing KE gradually in a frictionlike process) often referred to as the "continuous slowing-down approximation" (CSDA) The probability of a charged particle passing through a layer of matter without any interactions is nil (e.g., A 1-MeV charged particle undergoes ~10 5 interactions before losing all of KE) Characterized by a pathlength Range = the expectation value of pathlengths for a very large population of charged particles 3
Types of Coulomb-force interactions Charged particle Coulomb-force interactions can by characterized in terms of the relative size of the classical impact parameter bb vs. the atomic radius aa A. "Soft" collisions (bb aa) B. "Hard" (or "knock-on") collisions (bb~aa) C. Interactions with the external nuclear field (bb aa) D. Nuclear interactions by heavy charged particles Attix Fig. 8.1 4
Soft collisions When a charged particle passes an atom at a considerable distance (bb aa) Particle's Coulomb-force field excites the atom to a higher energy level, and sometimes ionizes it by ejecting valence-shell electron Transfer a very small amount of energy (a few ev) to an atom Most numerous (or probable) interactions Some (< 0.1%) energy spent by the soft collision can be emitted by the absorbing medium as coherent bluish-white light called Čerenkov radiation at an angle ξξ relative to the particle direction When vv = ββcc > cc/nn cc/nn = the velocity of light in the medium with refractive index nn ξξ = cos 1 1 ββnn Conical wavefront of half angle 90 ξξ behind the charge particle like the shock wave trailing 5
Hard (or knock-on) collisions When bb~aa, charged particles more likely interact with a single atomic electron, which is then ejected from the atom with considerable KE, called a delta (δδ) ray δδ-ray undergoes additional Coulomb-force interactions and dissipates its KE along a separate track (called a "spur") from that of the primary charged particle Can accompany characteristic x rays and/or Auger electrons Although hard collisions are few in number compared to soft collisions, the fractions of the primary particle's energy that are spent by these two processes are generally comparable (half & half???) 6
Interactions with the external nuclear field When bb aa, the Coulomb-force interaction takes place mainly with the nucleus Elastic scattering (2-3%) Not important to HCP Deflects incident electrons Not an energy-transfer mechanism Diff'l atomic x-sec. ZZ 2 Main reason why electrons follow very tortuous paths, especially in high-zz media, and why electron backscattering increases with ZZ Inelasticradiative interaction (2-3%) Electron deflects and emits bremsstrahlung by transferring KE Diff'l atomic x-sec. ZZ2 MM 0 2 Not important to HCP & low-zz material Positron annihilation Remaining KE of the positron is given to in-flight annihilation photon(s) Monte Carlo calculations of electron transport often assume for simplicity that the energy-loss interactions may be treated separately from the scattering (i.e., change-of-direction) interactions 7
Nuclear interactions by heavy charged particles When HCPs with KE > ~100 MeV & bb < aa Usually ignored in the context of radiological physics & dosimetry 8
STOPPING POWER The expectation value of the rate of energy loss per unit of path length xx by a charged particle of type YY & kinetic energy TT in a medium of atomic number ZZ dtt dxx YY,TT,ZZ In units of (MeV/cm) Mass stopping power in (MeV cm 2 /g): 1 ρρ Collisional stopping power + Radiative stopping power = dtt dxx 1 dtt Collisional stopping power due to soft & hard collision interactions = Radiative stopping power due to bremsstrahlung production Separately account for the in-flight annihilation dtt + 1 ρρ dxx cc ρρ dxx rr 1 ρρ dtt ss dxx cc + 1 ρρ dtt h dxx cc 9
1 dtt = ρρ dxx cc HH TT mmmmmm TT QQ cc ss dtt + HH TT mmaaaa TT QQ cc h dtt TT = the energy transferred to the atom or electron in the interaction HH = an arbitrary energy boundary between soft & hard collisions in term of TT TT mmmmmm = the max. energy transferred at a head-on collision TT mmmmmm 2mm 0 cc 2 ββ 2 MeV for a HCP with TT < MM 1 ββ 2 0 cc 2 20 kev & 0.2 MeV for protons with TT = 10 MeV & 100 MeV, respectively TT mmmmmm = TT for positrons if annihilation does not occur TT mmmmmm = TT for electrons 2 2 TT mmmmmm 2mm 0cc 2 ββ 2 TT mmmmmm II II = the mean excitation potential of the struck atom QQ cc ss & QQ cc h are the respective differential mass collision coefficients for soft & hard collisions In units of (cm 2 /g MeV) 10
Soft-collision term The Born approximation The particle velocity (vv = ββcc) is much greater than the max. Bohr-orbit velocity (uu) of the atomic electrons The fractional error ~ uu vv 2 The Bethe soft-collision formula 1 ρρ dtt ss = 2CCmm 0cc 2 zz 2 dxx cc ββ 2 Dimensionless ln 2mm 0cc 2 ββ 2 HH II 2 (1 ββ 2 ) ββ 2 Valid for uu vv CC = ππ NN AAZZ AA kk 2CCmm 0cc 2 zz 2 ββ 2 2 ~ ZZ 137ββ 2 1 rr 0 2 = 0.150 ZZ AA = 0.1535 ZZzz2 AAββ 2 cm 2 g MeV g/cm 2 II = the geometric-mean value of all the ionization & excitation potentials of an atom of the absorbing medium Generally derived from stopping-power or range measurements 11
Hard-collision term Dependent upon the particle types (electron, positron, or HCP) For HCPs and HH TT mmmmmm 1 ρρ dtt h dxx = kk ln TT mmmmmm cc HH ββ 2 Then we have: 1 dtt = ρρ dxx cc 1 ρρ dtt ss + 1 dtt h dxx cc ρρ dxx = kk ln 2mm 0cc 2 ββ 2 TT mmmmmm cc II 2 (1 ββ 2 ) = 0.371 ZZzz2 AAββ 2 13.8373 + ln ββ 2 1 ββ 2 ββ2 ln II 2ββ 2 = 2kk ln 2mm 0cc 2 ββ 2 II(1 ββ 2 ) ββ2 12
Dependence on the stopping medium 1 dtt ρρ dxx cc 1 dtt as ZZ ~ ZZ ρρ dxx cc AA 1 ρρ dtt dxx cc ~ ln II = the number of electrons per unit mass of the medium Dependence on particle velocity 1 dtt as ββ ; ρρ dxx cc slowly rises 1 dtt ρρ ln 1 ρρ dxx cc ~ 1 ββ 2 ~ 1 TT ββ 2 dtt TT gradually flattens (to a min. at 3) as ββ 1; and then 1 dtt dxx cc MM 0 cc 2 ρρ dxx cc 1 ββ 2 ββ 2 increases as ββ 1 13
Attix Fig. 8.2 14
Dependence on particle charge 1 ρρ dtt dxx cc ~zz 2 InDependence on particle mass Relativistic scaling considerations TT = MM 0 cc 2 1 1 & ββ = 1 1 1 ββ 2 TT MM0cc 2 +1 2 1/2 The KE required by any particle to reach a given velocity is proportional to its rest energy, MM 0 cc 2 The heavier the particle, the slower it will be going to at a given TT e.g., A 10-MV accelerator accelerates a proton to TT = 10 MeV (ββ = 0.1448); a deuteron to TT = 10 MeV (ββ = 0.1029); and an alpha to TT = 20 MeV (ββ = 0.1032) Attix Table 8.1 15
Shell correction Intended to account for the error when ββ uu Where shell (orbital) electrons gradually decrease their participation in the collision process; then, 1 dtt the Bethe underestimates the stopping power ρρ dxx cc Apply the shell correction term, CC ZZ 1 dtt = 2kk ln 2mm 0cc 2 ββ 2 ρρ dxx cc II(1 ββ 2 ) ββ2 CC ZZ = 0.371 ZZzz2 AAββ 2 13.8373 + ln ββ 2 1 ββ 2 ββ2 ln II CC ZZ Attix Fig. 8.3 16
Mass collision stopping power for electrons/positrons ττ = TT mm 0 cc 2 1 dtt = kk ln ρρ dxx cc ττ 2 (ττ + 2) 2(II/mm 0 cc 2 ) 2 + FF± ττ δδ 2CC ZZ FF ττ = 1 ββ 2 + ττ 2 8 (2ττ+1) ln 2 (ττ+1) 2 for electrons FF + ττ = 2 ln 2 ββ2 14 23 + + 10 + 4 for positrons 12 ττ+2 (ττ+2) 2 (ττ+2) 3 δδ = the correction term for the polarization or density effect 17
Polarization or density-effect correction Negligible for all HCPs but important for electrons Stopping power is decreased in condensed media due to the polarization or density effect Influences the soft collision process in condensed media (liquids or solids) Higher density ( ~10 3 10 4 ) & less atomic spacing ( 0.1) compared to those of a gas Decreases in energy loss because the dipole distortion of the atoms near the track of the passing particle weakens the Coulomb-force field experienced by distant atoms δδ is a function of the composition & density of the medium, and the parameter χχ χχ = log 10 mmmm mm 0 vv = log 10 ββ 1 ββ 2 18
Attix Fig. 8.4 19
Attix Fig. 8.5 20
Mass radiative stopping power In units of (MeV cm 2 /g) 1 dtt NN AA ZZ 2 = σσ ρρ dxx 0 rr AA (TT + mm 0cc 2 ) BB rr σσ 0 = rr 0 2 137 = 1 137 ee 2 mm 0 cc 2 TT = the particle KE in MeV 2 = 5.80 10 28 cm 2 /atom BB rr = a slowly varying function of ZZ & TT 16/3 for TT << 0.5 MeV; ~6 for TT = 1 MeV; 12 for TT = 10 MeV; and 15 for TT = 100 MeV BB rr ZZ 2 is dimensionless 1 ρρ dtt ~ NN AAZZ 2 while 1 dtt ~ NN AAZZ dxx rr AA ρρ dxx cc 1dTT ρρdxx rr 1dTT ρρ TTTT nn dxx cc nn = 700~800 AA 1 dtt ρρdxx rr 1dTT ρρ dxx cc ZZ 21
Attix Fig. 8.6 22
Radiation yield YY TT 0 = yy TT 0 = TT 0 yy TT dtt 0 TT 0 0 dtt Radiation yield of a charged particle of initial E of TT 0 YY TT 0 0 for HCPs = 1 TT 0yy TT dtt TT 0 0 Customarily neglecting in-flight annihilation in calculating YY TT 0 yy TT 1dTT ρρdxx rr 1dTT ρρdxx The amount of energy radiated per electron = YY TT 0 TT 0 Refer to Ch. 2 & assuming that only Compton interactions occurs: gg = YY TT 0 = 0 TT mmmmmm YY TT 0 0 TT mmmmmm dσσ dtt 0 EE γγ dtt 0 dσσ dtt 0 EE γγ dtt 0 23
Stopping power in compounds Bragg's rule Atoms contribute nearly independently to the stopping power, and hence the effects are additive 1 ρρ dtt 1 dtt 1 dtt = ff dxx ZZ1 + ff mmmmmm ρρ dxx ZZ2 + ZZ1 ρρ dxx ZZ2 ff ZZii = weight fraction II values for compounds ZZ ln II = ZZ AA ii ff ZZii ln II AA ii ii Polarization correction factor for compounds δδ = ii ff ZZ ii ZZ AA δδ ii ii ZZ AA 24
Restricted stopping power If one is calculating the dose in a small object or thin foil, the use of mass collision stopping power will overestimate the dose unless the escaping δδ-rays are replaced (i.e., unless δδ-ray CPE exists) The δδ-rays resulting from hard collisions may be energetic enough to carry KE a significant distance away from the track of the primary particle Restricted stopping power, 1 dtt ρρ dxx Fraction of the collision stopping power that includes all the soft collisions plus hard collisions resulting in δδ-rays with energy less than a cutoff value Alternative form: the linear energy transfer (kev/µm) LL kkkkkk μμmm = ρρ 10 If = TT mmmmmm, 1 ρρ 1 ρρ dtt dxx = dtt dxx (MeV cm 2 /g) 1 dtt ρρ dxx cc Then, LL = LL, called the unrestricted LET 25
1 dtt = kk ln 2mm 0cc 2 ββ 2 ρρ dxx II 2 (1 ββ 2 ) is in units of (ev) 2ββ 2 2CC ZZ For electrons & positrons 1 ρρ dtt dxx = kk ln ττ 2 (ττ+2) 2(II/mm 0 cc 2 ) 2 + GG ± ττ, ηη δδ 2CC ZZ GG ττ, ηη = 1 ββ 2 + ln 4 1 ηη ηη + 1 ηη 1 + (1 ββ 2 ) ττ2 ηη 2 + (2ττ + 1) ln(1 ηη) for electrons GG + ττ, ηη = ln 4ηη ββ 2 1 + 2 ξξ 2 ηη 3 + ξξ 2 ξξξξ 2 ηη2 + 1 + ξξξξ ξξ2 ττ 2 3 ηη3 ξξ3 ττ 3 4 ηη4 for positrons ξξ ττ + 2 1 Note that GG ττ, 1 2 = FF ττ and GG + ττ, 1 = FF + ττ 2 26
RANGE The expectation value of the pathlength pp that a charge particle follows until it comes to rest (R R CCCCCCCC ) The projected range tt is the expectation value of the farthest depth of penetration tt ff of a charged particle in its initial direction In units of (g/cm 2 ) Attix Fig. 8.7 27
CSDA range Range in the continuous slowing down approximation TT 0 dtt R CCCCCCCC dtt 0 ρρdxx R CCCCCCCC is greater for higher ZZ because the stopping power decreases with increasing ZZ 1 R CCCCCCCC TT 0 1.77 415 + 1 670 in C Attix Fig. 8.8 28
The range of other heavy particles can be estimated from that: a. All particles with the same velocity have KEs in proportional to their rest masses b. All singly charged heavy particles with the same velocity have the same stopping power c. Consequently the range of singly charged heavy particles of the same velocity is proportional to their rest mass For example: R of the deuteron with a KE of 20 MeV = 2 R of the proton with a KE of 10 MeV The same velocity & stopping power as a 10-MeV proton R of the alpha with a KE of 40 MeV = R of the proton with a KE of 10 MeV The same velocity as a 10-MeV proton 4 larger stopping power of the alpha because of zz 2 dependence A rule of thumb R CCCCCCCC = 1 MM 0 PP zz 2 MMPP R CCCCCCCC 1 MM 0 0 zz 2 MMPP 0 TT 0 1.77 415 + 1 670 29
Projected range tt 0 tt ttff tt dtt = dnn tt 0 tt dtt dtt = 1 dnn tt 0 ttff tt dtt 0 dtt dtt NN 0 0 tt ttff tt dtt Attix Fig. 8.9 30
Straggling & multiple scattering There will be a distribution of tt ff (refer to Attix Figs. 8.9a & b), giving rise to an S-shaped descending curve, due to: Multiple scattering Predominant effect Range straggling Stochastic variations in rates of energy loss cf., Energy straggling: the spread in energies observed in a population of initially identical charged particles after they have traversed equal path lengths 31
Electron range See Attix Fig. 8.9c 32
33
Attix Fig. 8.10 34
Photon "projected" range See Attix Fig. 8.9d, where scattered photons are ignored tt mmmmmm at tt = tt = 1/μμ employing NN tt = NN 0 ee μμtt 35
DOSE IN THIN FOILS Consider a parallel beam of charged particles with TT 0 perpendicularly incident on a thin (ρρtt < R) foil of ZZ with the assumptions: a. Constant collision stopping power b. Passing straight through the foil without scattering c. Thick compared to the average δδ-ray range or CPE for δδ rays The energy lost (in units of MeV cm -2 ) by a fluence Φ passing through a foil of ρρtt EE = Φ dtt ρρdxx cc ρρtt ρρtt = the particle pathlength through the foil The absorbed dose in the foil = Φ dtt MeV ρρtt ρρdxx cc g The dose in the foil is independent of its thickness DD = Φ dtt ρρdxx cc ρρtt = 1.602 10 10 Φ dtt ρρdxx cc (Gy) 36
Estimating δδ-ray energy losses When the CPE violates for δδ rays, replace dtt ρρdxx cc by dtt ρρdxx dtt : portion of the collision stopping power that includes only the interactions transferring ρρdxx less than the energy Choose to be the energy of δδ rays having tt = ρρtt 37
Estimating path lengthening due to scattering A correction of path lengthening is not necessary for heavy particles A correction is required for electrons due to significant path lengthening due to multiple scattering Correct with tt (> 1) tt Use Attix Fig. 8.11 Ordinate: 100(tt tt)/tt, the mean % path increase of electrons traversing a foil of ρρtt Abscissa: ξξ = ρρtt, the normalized (dimensionless) foil thickness XX 0 XX 0 = the radiation length = the mass thickness in which electron KE would be diminished to 1/ee of its original value due to radiative interaction only For compounds,» 1 = ff ZZ1 + ff ZZ2 + XX 0 (XX 0 ) ZZ 1 (XX 0 ) ZZ 2 38
e.g., Consider a fluence of 10 10 cm -2 1-MeV electrons on a Cu foil with 0.01 g cm -2 XX 0 = 13.04 g cm -2 ξξ = 7.7 10 4 tt tt tt = 2.4%, which implies that the mean electron pathlength is 2.4% greater than the Cu thickness, increasing the absorbed dose by the same amount Attix Fig. 8.11 39
DOSE IN THICKER FOILS The assumption a) will not be valid Foils may be thick enough to change the stopping power significantly but not to stop the incident particles Use the CSDA range tables instead of stopping-power tables The assumption b) will be not valid Straight tracks through the foil will not be satisfied, but the path-lengthening error is small (~1%) for heavy particles The assumption c) may be valid δδ-ray effects can be neglected 40
Dose from heavy particles Determine the residual CSDA range from the heavy-particle range tables R eexx = R CCCCCCCC (ρρtt) ffffffff (g/cm 2 ) Find the corresponding residual KE from the tables, then the energy spent in the foil by each particle will be TT = TT 0 TT eeee (MeV) The energy imparted per unit cross-sectional area of particle beam: EE = Φ TT (MeV/cm 2 ) The average absorbed dose considering the incident angle θθ of beam: DD = 1.602 10 10 Φ TT ρρtt/ cos θθ (Gy) 41
Dose from electrons Correction for the path lengthening is required Bremsstrahlung production will further affect the range Estimate the true mean pathlength ρρtt for the electrons using Attix Fig. 8.11 Determine R CCCCCCCC for electrons with TT 0 from the electron range tables R eexx = R CCCCCCCC ρρtt TT = TT 0 TT eeee Some E will be carried away by bremsstrahlung x-rays (usually negligible to the dose) Assume all the x-rays escape Determine the radiation yield from the tables The energy spent in collision interactions in the foil: TT cc = (TT 0 TT eeee ) cc = TT 0 1 YY(TT 0 ) TT eeee 1 YY(TT eeee ) cc DD = 1.602 10 10 Φ TT cc ρρtt 42
Example) Consider a fluence of 10 10 cm -2 10-MeV electrons on a Pb foil with 1.13 g cm -2 (or 1 mm) XX 0 = 6.496 g cm -2 ξξ = 0.174 tt tt = 8.5% from Attix Fig. 8.11 tt ρρtt = 1.23 g cm -2 R eexx = R CCCCCCCC ρρtt = 6.133 1.23 = 4.90 TT eeee = 7.29 MeV TT cc = TT 0 1 YY(TT 0 ) TT eeee 1 YY(TT eeee ) cc = 10 1 0.3162 7.29 1 0.2607 = 1.449 MeV DD = 1.602 10 10 Φ TT cc = 1.602 10 10 1010 1.449 = 2.05 Gy ρρtt 1.13 Compare to DD = 1.92 Gy if both path lengthening and the change in the stopping power are ignored 43
Mean dose in foils thicker than the max. projected range EE = ΦTT 0 1 YY(TT 0 ) MeV/cm 2 DD = 1.602 10 10 ΦTT 0 1 YY(TT 0 ) ρρtt Gy If the radiative losses are considerable and the foil thickness is great enough to reabsorb, a very crude estimate of the reabsorbed fraction of the energy due to x-rays can be given by Then, we have DD = 1.602 10 10 ΦTT 0 1 YY(TT 0 )ee μμeeee ρρ ρρtt 2 μμ eeee ρρ ρρtt ee μμ eeee ρρ ρρtt 2 can be evaluated at some mean x-ray energy, e.g., 0.4TT 0 for thick-target bremsstrahlung 44
Electron backscattering Backscattering due to nuclear elastic interactions can be an important cause of dose reduction, especially for high ZZ, low TT 0, & thick target layers For thin foils; Backscattering events are equally likely to occur in the first & last infinitesimal layer of the foil do not require correction to the absorbed dose On average, backscattering can be assumed to occur in the midplane of the foil For thicker foils; Electron energy backscattering coefficient, ηη ee (TT 0, ZZ, ) Fraction of incident energy fluence that is redirected into the reverse hemisphere implies an infinitely thick (> tt mmmmmm /2) layer Attix Fig. 8.12 45
Otherwise, backscattered-electron numbers can be used as an upper limit on the backscattered energy ηη TT 0, ZZ, = 1.28exp 11.9ZZ 0.65 1 + 0.103ZZ 0.37 TT 0 0.65 Fractional number of perpendicularly incident electrons that are backscattered from an infinitely thick layer of ZZ Applicable for TT 0 at least up to 22 MeV ηη increases with ZZ & decreases with TT 0 ηη ee TT 0, ZZ, < ηη TT 0, ZZ, 46
Bragg curve Unique the dose vs. depth distribution of heavy charged particles due to the TT 0 2 dependence of the range at low energies (resulting from the ββ 2 dependence of the stopping power) Spending the first half of initial KE along a pathlength xx, and the remaining half of the KE in distance xx/3 Attix Fig. 8.13a 47
Dose vs. depth for electron beams A diffuse maximum at roughly half of the maximum penetration depth due to multiple scattering of (small mass) electrons Attix Fig. 8.14a 48
Broad beam when the beam radius rr = R CCCCCCCC Attix Fig. 8.14b 49
Absorbed dose at depth At any point PP at depth xx in a medium ww where the charged-particle fluence spectrum Φ xx (TT) (excluding δδ-rays) is known: DD ww = 1.602 10 10 0 TT mmmmmm Φxx (TT) dtt ρρdxx cc,ww dtt CPE at PP for the δδ-rays Note that the range of most δδ-rays 1 mm in condensed media How to determine Φ xx (TT)? Solve the radiation transport equations Simply, use the range tables Remaining range R rr = R CCCCCCCC TT 0 xx Then, find the remaining KE TT rr from R rr Replace Φ 0 (TT 0 ) by Φ xx (TT rr ) 50