Th Unitd Stats Nuclar Rgulatory Commission and Duk Univrsity Prsnt: Rgulatory and Radiation Protction Issus in Radionuclid Thrapy Copyright 008 Duk Radiation Safty and Duk Univrsity. All Rights Rsrvd.
Wlcom! This is th svnth of a sris of training moduls in Radiation Physics. Ths moduls provid a basic introduction to radiation physics and th intraction of radiation with mattr. Sponsord by th Unitd Stats Nuclar Rgulatory Commission and Duk Univrsity Author: Dr. Rathnayaka Gunasingha, PhD
Your Instructor Dr. RathnayakaGunasinghais an Acclrator Physicist with background in High Enrgy physics. Dr. Gunasinghais a physicist in th Radiation Safty division and mmbr of th Faculty of th Duk Mdical Physics Graduat Program. Contact: rathnayaka.gunasingha@duk.du
Goals of th Cours Upon complting ths instructional moduls, you should b abl to: undrstand th Basic Intraction of Radiation with Mattr apply th knowldg in various calculations usd in Mdical and Halth Physics undrstand th basic principls bhind various instrumntation usd in Mdical and Halth Physics
This Modul Will Covr Soft Collision s or Inlastic Collisions Hard Collisions Inlastic Collisions with nuclus Elastic Collisions with nuclus Bth-Bloch Formula Stopping powr, Rang, Stopping Tim
Intraction of Chargd Particls with Mattr Whn a chargd particl ntrs into a mdium, it intracts with lctrons and nucli in th mdium Ths intractions ar calld "collisions" btwn chargd particls and th atomic lctrons and nucli. Th collisions ar lad to th (a) Ionizations - productions of ion-lctron pairs, and (b) Excitations - raising th nrgy of th oribital lctrons into highr stats in th atom
Intraction of Chargd Particls with Mattr A chargd particl is calld havy, if its mass is gratr than th rst mass of th lctron xampls: alpha, muon, proton and som fission products A havy chargd particl losss a ngligibl amount of nrgy in a collision with nuclus In this lsson, w nglct th nuclar forcs, and considr only lctromagntic forc ( Coulomb forc) btwn th chrgd particl and th lctrons
Intraction of Chargd Particls with Mattr Dpnding on th rlativ siz of th impact paramtr b (i.. shortst distanc btwn th cntr of th atom and th path of th chargd particl), and th radius of th atom a, w can considr four typs of intractions. Q chargd particl b impact paramtr a
Intraction of Chargd Particls with Mattr Four typs of intractions: (a). Inlastic or soft collisions ( b >>> a) (b). Hard collsions ( b a) (c). Inlastic collsions with a nuclus ( b << a) (d). Elastic collisions with a nuclus ( b << a)
Inlastic or soft collisions ( b>>>a) In this typ, th Coulomb forc du to moving particl affcts th atom as a whol, lading to xcitation of th atomic lctrons or ionization of th atom Largr valus of b ar mor probabl than th nar hits for a chargd particl and thrfor many soft collisions occur This is th main procss of intractions with mattr nrgy transfr for chargd particl
Inlastic or soft collisions ( b>>>a) In th procss of soft collisions, whn th vlocity of v p c n and ( βc), xcds th spd of chargd particl light in th mdium with rfractiv indx n,, a small amount of absorbd nrgy is rlasd as photons. this radiation is calld"crnkov ~ Radiation".
Crnkov Radiation At tim t 0, particl is at O at t t, particl is at O distanc travlld by th wav front, distanc travlld by th particl, ct OM n OO βct Crnkov ~ photons form a particl. cosθ ct n βct wav front of βn half angl ( 90-θ), bhind th
Hard Collisions( b ~ a) Whn b a, chargd particl can xrt an impuls which is nough to jct an lctron from th atom. This jctd lctron is calld a " δ - ray"and it carris kintic nrgy lost by th chargd particl. In th collision, atomic lctron is tratd as "fr" δ - ray has nough nrgy to undrgo its its nrgy along a sprat track own intractions, and losss
Inlastic collisions with a nuclus (b<<a) In this typ, Coulomb forc intraction is th nuclus mainly with If th particl is an lctron, Brmstrahlung can occur If b < nuclar radius and chargd particl has nough nrgy, inlastic intraction with th nuclus can lav th nuclus in th xcitd stat. xcitd nuclus thn dcays by mission of nuclons or γ rays.
Elastic collisions with a nuclus This typ of intractions ar known as Ruthrford scattring. Thr is no xcitation or radiation. Particl losss nrgy through rcoil of th nuclus
Enrgy transfr in a singl collision Considr a havy particl of with an lctron ( mass m ) mass M, vlocity V, collids Assumptions : a. havy particl movs rapidly compard to lctrons orbiting spd ( lctron is considrd fr) b. Enrgy transfr is largr than th binding nrgy of th lctron ( collsion is lastic)
Enrgy transfr in a singl collision M, V m M, V m, v BEFORE Collision AFTER Collision Consrvation of MV MV MV MV m v Elastic Collision, consrvation of + Solving (.) and (.) linar momntum + V m v - - - - (.) nrgy - - - - (.) ( M m ) ( M + m ) V
Enrgy transfr in a singl collision Max. Enrgy transfr to th lctron max Q max MV MV whr E 4mM Q. ( M + m ) ( M + m ) ( M + m ) MV kintic nrgy of MV m 4m M ME V havy particl
Enrgy transfr in a singl collision If incidnt particl, Q Q max m 500 4m E V If incidnt particl, proton max lctron 3 m V M M E m 836 m
Enrgy transfr in a singl collision Rlativistic cas Q m max Q m γ V Q max m M << M m γ V m γ + + M γ β γ β β m c whr, max V c
Intractions of Chargd Particls with Mattr Considr a particl of mass M, charg straight lin with vlocity v. z moving along a M Q z x O v b impact paramtr Elctron is at O r y θ b q Q z q x O
Intractions of Chargd Particls with Mattr kqq Coulomb forc F r Whn a particl passs O', F and no nt motion of kqq nt forc on lctron, F cosθ r momntum impartd to th lctron dp F dt p cosθ b r ; F dt dx dt x x υ ; kz r tanθ x lctron along x - dirction cosθ dt x b componnt of ; sc θ dθ forc rvrs dx b vdt b ; dt bsc θ dθ υ
Intractions of Chargd Particls with Mattr ( ) 4 4 4 Th nrgy transfrd to th lctron, cos sc. cos sc b m V k z E m b V k z m p E Vb zk p d b kz d b b kz p π π π π θ θ υ θ υ θ θ θ
Intractions of Chargd Particls with Mattr x m c b k z m p E c b zk p p E dy c V 4 Enrgy transfr to th lctron, Kintic unchangd is incrasd by factor dcrasd by a Rlativistic Cas β β γ γ β
Intractions of Chargd Particls with Mattr This nrgy is dtrmind by b db b p dx Probability that nrgy lost btwn E, E + de is givn by probability that b b, b + db b E b + db E + de thn P( E) de p '( b) db ( b incras rsults dcras in E)
Intractions of Chargd Particls with Mattr ZN A p ( E) de πbdb ρdx, A ZN A whr ρdx is th numbr of lctrons pr unit ara A b p( E) de 4 z k β m c E p( b) db Man nrgy loss dt πzn Aρdxk z β m c A dt 4 bdb E E max min E [ ln E ] max 4 z k β m c 4 πzn Aρdx z k A β m c E P( E) de E min de E de E
Intractions of Chargd Particls with Mattr Dfin constant C πn dt m c C β Whn E to priod of lctron in atomic orbit τ < f rot kzz bv min from ρ ( b F Zz A max max y kzz b A k m c Emax ( ρdx)[ ln E] C ( ρdx)[ lnb] ) collision is soft and collision tim shortr compard but τ τ γ τ τ b γβc 4 b γβc 4 E min m c β Zz A b b max min
Intractions of Chargd Particls with Mattr b γβc b γβc b max < < f h I rot γβch I hf rot I h plank constant man ionization potntial
Intractions of Chargd Particls with Mattr b min : to M At th rst fram, lctron movs with βc with rspct Minimum valu of b, (uncrtainity principl dos not allow b Brogli's chargd particl) is quivalnt wav lngth in th rlativ coordinat systm of b min h p h ( m C) cγβ to th d Brogli wav lngth to b mor prcis than d and
Intractions of Chargd Particls with Mattr Substituting, mc dt C β According to this quation : (a.) (b.) dt dt kintic nrgy of b dt mc C β Zz A Zz A ( ρdx) ( ρdx) γ ln havy particl γβch mγβc ln. I h β m C I c
Stopping Powr Considr any chargd particl of typ Y and kintic nrgy T, in a mdium of atomic numbr Z Th xpctation valu of th rat of nrgy loss pr unit lngth is calld th stopping powr dt is th nrgy loss in stopping powr dx, thn dt dx Y, T, Z Units: MV. cm or J. m
Mass Stopping Powr Dividing th stopping powr, by th dnsity of th mdium is calld Mass Stopping Powr mass stopping powr dt ρdx units: Mv. cm g
Stopping Powr Dpnding on th nrgy lost by chargd particls, stopping powr is subdividd into two:. Collision stopping powr: rat of nrgy loss from th sum of soft and hard collisions ( collision intractions). Radiativ stopping powr: rat of nrgy loss from radiativ intractions. i.. mainly from brmstrahlung productions
Bth Formula for dt/dx Using rlativistic quantum mchanics (assuming havy particl vlocity [ v βc]is much gratr than th Bohr - orbit vlocity of atomic lctron) Bth drivd th following hard collisions) whr dt ρdx K m C K ln I Cm c β z formula (combining soft and ( β ) β β C N π A A k 4 ( z) ( ) m c
Bth Formula for dt/dx Substituting valus for N, π, K, m c K 0.535 Zz Aβ a MV cm g Th trms in th brackts ar unitlss; I and m c sould b in V dt Zz β 0.3070 ln ( m c ) + ln ln ( I ) ρ dx Aβ β β dt Zz β 0.3070 3.8373 ln ln + ( I ) β ρ dx Aβ β z atomic numbr of th particl Z atomic numbr of th mdium A mass numbr of mdium
Important faturs of this formula Man ionization potntial I ( ) V (xitation nrgy) dfind as th gomtric man valu of all ionizations and xitation potntials of an atom in th absorbing mdium I is calculatd using th quation for dt. Also it can b ρ dx dt masurd if all othr quantitis in ar known ρ dx
Important faturs of this formula Logarithm of I is ntrd in abov formula. Thrfor following approximat mpirical valus can b usd to stimat I in V. I 9.0 ( V ) Z 0.0 +.7 Z (V) Z 3 5.8 + 8.7 Z ( V ) Z > 3
Important faturs of this formula I for a compound or mixtur considr th individual contribution from with lmnt thn, nln N ( I ) Z N ln( I ) i n atoms cm N i Z i i i -3 i for i th lmnt with and and ach total numbr of lctrons in th mixtur Z i Z i I i For a pur compund n, n i z i rplacd by lctron numbrs
Important faturs of this formula Exampl : ( ) Calculat I for watr H O. I H 9 V I. +.7 8 05 V n 0 N N n ln I N Z ln I i i 0ln I ln 9 + 8 ln05 74.6 V H ( ( )) ( )
Important faturs of this formula Exampl: Solution : dt Zz β 0.3070 3.8373 + ln dx A β ρ β β For proton z watr Z 0 A 8 ρ at ρ dt dx dt dx Calculat th mass collision stopping powr of for MV MV 0.3070 70.93 protons. 0 8β β β 3.8373 + ln β β 0.003 MV cm g ln I ln watr ( I ) 4.3 4.3 V
Dpndnc on th mdium dt Zz β + ρ dx Aβ β 0.307 3.8373 ln β ln ( I ) Z Whn Z (mdium) is incrasd, numbr of lctrons pr unit mass A dcrasd ( I ) Whn Z incrasd ln incrasd ; dt Thrfor dcrasd. ρdx
Dpndnc on particl vlocity Whn v incrass, β incrass. Bcaus of β brackt dt ρdx dcrass outsid of th Whn particl charg z incrass dt ρdx incrass
Dpndnc on particl mass Particl mass dos not appar in dt ρdx formula. So, dt ρdx dos not dpnd on particl mass.
Rlation btwn kintic nrgy T and β (Rlativistic scaling) ( ) -. is connctd to th paramtr mass rst particl of a of how th kintic nrgy Following shows 0 0 0 0 0 0 0 + + + c M T c M T c M T c M T c M c M T M T β β β β β γ β
Rlativistic Scaling Particl Rst mass for som havy particls Mc ( MV) Z muon pion proton nutron dutron α 05.66 39.60 938.8 939.57 875.63 377.4 0
Rlation btwn kintic nrgy T and β (Rlativistic scaling) For any two particls A and nrgis, T T A B M M ( γ ) ( γ ) which givs th rlation, A B c c B, for a givn valu β, T T A B M M A B kintic i.. If two particls travl at th sam spd, thir ratio of kintic nrgis ar proportional to th ratios of thir rst masss.
Minimum ionizing particls Variation figur. of dt ρdx with T, for som particls is shown in
Minimum ionizing particls dt Th valu of for many diffrnt particls approachs a constant ρdx broad minimum valu at highr nrgis. dt ρdx For light matrials thus valu corrsponds to MV. cm g. Bcaus of thir similar nrgy loss bhavior, ths rlativ particls ar calld "Minimum Ionizing Particls"
Shll Corrction trm In Born approximation, vlocity of passing gratr than that of atomic lctrons. particl is much K - shll lctron of atom has th highst vlocity. th highst nrgy and hnc Whn passing particl vlocity falls blow, K - shll vlocity, thn K - shll lctrons do not contribut to dt ρdx.
Shll Corrction trm Corrctd for combind ffct of all i shlls into a singl trm C Z β + ρdx Aβ β Z dt Zz C 0.307 3.8373 ln ln β ( I ) Anothr corrction is th δ ( β ), which is a function of β and dilctric constant of mdium, is known as "dnsity ffct" or "polarization ffct" and important for dns mdia such as solid. It is ngligibl for havy chargd particls.
Rang Considr a charg particl ntring into a mdium with kintic nrgy T0. Thn th avrag valu of th distancs ( l) coming to rst, is calld th "Rang". that a particl travls bfor A B T 0 l t f
Projctd Rang Avrag valu of th farthst dpth of pntration of th particl in its initial dirction is calld th "Projctd Rang". units g.cm t f Th rciprocal of th mass stopping powr is usd to calculat th rang. Thn, T 0 0 dt Rang R( T ) dt, T0 strating nrgy of th particl ρdx units: g.cm
Projctd Rang This quantity is somtims mntiond as Continuous Slowing Down Approximation (CSDA) rang. dt Using th valu for in th dfinition of ρdx a rlation for R in th following way, rang R, w can driv dt ρdx Zz 0.307 Aβ 3.8373 + β ln β β ln I 0 R T dt 0 dt ρdx z dt ' G( β )
Projctd Rang ( ) ( ) th havy particl. th rang of givs and, function of a is ) ( ) ( ) ( Thn,. of ar functions ) and g( ) ( whr ) ( Sinc, 0 R f f z M d G g z M R G d Mg dt Mc Mc T β β β β β β β β β β β β γ β
Projctd Rang ( ) For two havy paricls A M, z and B( M, z ) with th A A B B sam vlocity, th ratio of thir rangs R and R is givn by, A B R A M A z B RB M B za If w know th rang of particl A, thn w can find th rang partilc B using this formula.
Projctd Rang 3 + Exampl: Find rang for 40 MV H ions in watr H Z M 3 T 40 MV 3 + p Z M T? T T A B M M A B 3 + 40 MV H corrsponds to 3.3 MV proton which has a rang R p ( β ) 0.93 g.cm 3 40 T 3 RA M A Z B RH 3 RB M B Z A Rp 4 R H T T H 0.45 g.cm p R H p 0.93 ( 3 ) 4
Projctd Rang What w hav larnd All particls with sam vlocity ( β ) hav kintic nrgis in proportion to thir rst mass T T A B M M A B All singlly chargd havy particls with sam β, hav sam stopping powr Rang of singlly chargd particls of sam β, ar proportional to thir rst masss RA M A z B RB M B za
Stopping Tim Stopping tim ( Slowing down tim): Tim takn by a havy chargd particl to stop in th mattr Slowing down for a chargd particl in a mdium, can b calculatd using th rat of nrgy loss. By using th chain rul of diffrntiation, an xprssion for rat of nrgy loss can b drivd as follows, dt dt ρdx dt dt ( ρv) ρv dt ρdx dt ρ dx ρdx
Stopping Tim Thn, assuming th rat of nrgy loss is constant, slowing down tim is givn by, T T t dt dt ρv dt ρdx If kintic nrgy T is givn, valu for v can b found using th rlation btwn T and β.
Stopping Tim Exampl: Calculat th slowing down tim for 0 MV protons in watr. dt dt ρ gcm ; T 0 MV ; 45.9 MVcm g vρ dt ρdx -3 at 0 MV, β 0.0099 v 0.0099c dt 0.0099c 45.9.99 0 MV s dt stimat stopping tim t is 0 MV 0 0 t 5 0 dt dt.99 0 ρv ρdx dt - s
Bragg Curv A plot of rat of nrgy loss along a track of a chargd particl is known as "Bragg Curv". Figur shows a bragg curv for a particl. /ρ(dt/dt) or (de/dx) Distanc of pntration
Bragg Curv Bragg curv is a consqunc of β dpndnc of stopping powr. A pak occurs at th nd of th track, bcaus th intraction cross sction incrass as th th particl vlocity dcrass. At th nd of and th curv falls off th track, charg is rducd through lctron pickup, Maximum nrgy loss occurs at th nd of th track
Enrgy Straggling Enrgy loss by a havy chargd particl collision in a statistical or stochastic procss. mdium is a Whn mononrgtic bam of particls passs through a mdium, a sprad of nrgis rsults around th avrags as it passs a givn dpth. This unqual nrgy losss for th bam of calld "Enrgy Straggling". similar condition is
Enrgy Straggling Figur shows th nrgy distribution of mononrgtic chargd particls at various points along its path. First portion, nrgy distribution bcoms widr with distanc. At th nd of th track distribution narrows, sinc th particl has lss nrgy.
Rang Straggling Fluctuation of path lngth of individual particls of th sam nrgy is calld "Rang Straggling". Following stup, a dtctor and an absorbr of changabl thicknss can b usd to dtrmin th "Rang Straggling". Sourc I/I 0 Dt 0.5 t R m R t
Rang Straggling In figur, w dfin two rangs: Man Rang and th Extrapolat Rang. Man Rang ( ): Th valu of th absorbr thicknss at which rlativ count I rat 0.5 is dfind as man rang. I 0 R m R Extrapolat Rang( ) : Th valu of thicknss obtaind by xtrapolating th linar portion of th nd of th curv, is calld th xtrapolat rang.
Scaling Laws ( Bragg-Klman s Rul) dt Som tims Rang (R) or stopping powr ar not availabl ρdx for mixturs. Assuming stopping powr pr atom of compound is additiv, w can dfin th stopping powr for a compound, by dt dti Wi N dx i N dx c c i i dt stopping powr N atomic dnsity Wi wight fraction dx
Scaling Laws ( Bragg-Klman s Rul) Rang for a compound or a mixtur is givn by : M c Rc A i ni R i R Rang of lmnts, n numbr of atoms of lmnt i in A i Atomic wight M c th molcul molcular wight
Scaling Laws ( Bragg-Klman s Rul) Whn Rang data is not availabl for on lmnt, w can us th valu of a known rang. if R is known and R is unknown, R R ρ ρ A A ρ dnsity, i A i atomic wight of absorbing matrial
Crdits and Rfrncs Attix, F.H: Introduction to th Radiological Physics and Radiation Dosimtry, Wily-VCH