High Energy Physics. Lecture 5 The Passage of Particles through Matter

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Hilary Lamb
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1 High Enrgy Physics Lctur 5 Th Passag of Particls through Mattr 1

2 Introduction In prvious lcturs w hav sn xampls of tracks lft by chargd particls in passing through mattr. Such tracks provid som of th most important information on particls Th main mchanism by which chargd particls bcom visibl is ionization In this lctur w shall larn mor about th passag of particls through mattr W bgin by discussing th passag of chargd particls through mattr

3 Th Passag of Chargd Particls through Mattr Considr a bam of chargd particls travlling along paralll paths in vacuum and ntring a rgion of spac filld with mattr Most particls will continu travlling along roughly straight paths and los nrgy through ionization of th atoms along thir paths Som of th particls ar scattrd through larg angls: this rsults in loss of bam intnsity (attnuation) Chargd particls ar ithr lctrons or havy chargd particls: th lightst particl that is havir than th lctron is th muon whos mass is about 00 tims that of th lctron 3

4 Artist s imprssion of havy particls travlling through mattr bam (μ, π, p, α, ) wid angl scattr: bam attnuation (of cours if thr is also a magntic fild, thn th tracks ar bnt. But for now w work in a magntic fild fr mdium) 4

5 Th main mchanism of nrgy loss is ionization. Ignoring th atomic Binding Enrgy (BE), th maximum nrgy loss pr collision rsulting in ionization is from nrgy and momntum consrvation in nonrlativistic approximation ΔT T max m 4 whr m m m hr T is th initial KE and ΔT max th nrgy impartd to an lctron and hnc lost by th particl m is th particl mass and m is th lctron mass 5

6 y atomic lctron passing particl m θ vt b (impact paramtr) x t=0 From classical mchanics: momntum = impuls p x Fx dt 0 py = y = = F dt 6

7 Th forc F is th Coulomb forc: F y = From our figur w hav: hnc and z 4πε r 0 sinθ vt = b cot θ, r = b / sinθ 1 dt = b d cotθ = b dθ = r dθ θ F dt y v v sin vb z = 4πε 0 1 sin θ dθ vb 7

8 at t = w hav θ = 0 and at t = + θ = π and hnc th y componnt of momntum acquird by th lctron is p y π z z = sinθ dθ = 4πε vb 4πε vb and th KE transfrrd from th particl to th lctron is (in nonrlativistic approximation!) T = p m 8

9 whr 3-1 is Avogadro s numbr N = mol A Lt th particl pass through a foil of thicknss dx, dnsity ρ atomic numbr Z mass numbr A thn th numbr of lctrons pr cm 3 is N = N Zρ A A 9

10 a cylindrical ring, concntric with th flight path of th particl and of radius b and width db has a volum givn by and contains dv NdV = π bdbdx lctrons thrfor th nrgy loss to all lctrons lying btwn b and b+db is = TNdV and hnc th rat of nrgy loss of th particl is bmax dt = π N dx bmin T bdb 10

11 and aftr intgration w gt 4 dt z Z ρ 1 max N A ln dx 4πε 0 A mv bmin = Considr th limits of intgration: naivly w would lik to put b max = but that would imply a minimum nrgy transfr to th lctron of T min =0, and that mans that no ionization can tak plac; so w put th minimum nrgy transfr qual to an avrag ionization potntial I, hnc b b z 1 1 I 4 max = 8πε0 mv 11

12 b min : naivly w would put b min = 0 but that corrsponds to a had-on collision in which F y = 0 and th impuls in x dirction is nonzro. A simpl stimat of b min is to st sinc v = thrfor 4 v b 1 T = m v = m v max max in a had-on collision = z 1 min 4 πε0 mv 8 1

13 and hnc b b max min = mv I A mor carful analysis that taks account of QM and rlativistic ffcts yilds th following formula: 4 dt z Z ρ 1 mv = N ln ln(1 ) A β β dx 4πε 0 A mv I whr β = v/c This is calld th Bth-Bloch formula. 13

14 Commnts: (i) For practical calculations it is convnint to rplac th lmntary charg by th fin structur constant α: α = 4πε 0 1 c α is a dimnsionlss numbr and is qual to 1/137 in vry good approximation (bttr than 1%): α =1/ c is Planck s constant dividd by π and c is th spd of light 00MV fm and hnc α c =.13MV fm ( ) 14

15 (ii) In nuclar physics on liks to masur th thicknss of matrial through which particls pass as ρx (in mg/cm ) (but nuclar physicists still writ x for this quantity!) (iii) Th quantity dt/dx is calld th stopping powr of th matrial; w can rwrit it in th following form: dt = dx A 4π α c ( ) whr B N z m A v mc β = Z ln ln(1 β ) β I is calld th stopping numbr. B 15

16 (iv) At low spds vc 1 th stopping powr is roughly invrsly proportional to th squar of th particl vlocity: dt dx (v) At vlocitis clos to th spd of light th rlativistic corrction givs ris to an incras of th stopping powr (vi) Btwn th 1/v drop and th rlativistic ris th stopping powr gos through a minimum: this is known as minimum ionization. Th muon has a vry wid, flat minimum, thrfor a minimum ionizing particl of larg rang is a muon candidat. 1 v 16

17 (vii) Avrag ionization potntial: This dpnds on th matrial; it is a function of th atomic numbr Z. On thortical grounds Bloch has shown that it is to a rasonabl approximation proportional to Z: I = kz whr k 11.5V Exprimntal valus ar shown in th tabl blow: Z 4 B 6 C 13 Al 9 Cu 8 Pb I (V) k(v)

18 (from PDT 004) 18

19 Rang Th rang R is dfind as th distanc travlld bfor th particl stops: R T 1 4πε 0 Am 4 ρ 0 T A v R= dx= de = de de dx z N B 0 For rough stimats put and for nonrlativistic particls thn 1 B const Z hnc E = mv de = mvdv A m R= const ρz z v

20 Discussion of th Rang Formula (i) Compar th rangs of two distinct particls of qual initial vlocity travlling through th sam mdium: R m z R m z = for instanc proton and α particl: R R = m m 4 1 p α p ( ) (ii) Compar th rangs of a particl travlling in diffrnt mdia X and Y approximation: ignor th Z dpndnc in th log trm, thn RX AX ρ XZX = RY AY ρyzy and if w masur th rang in g/cm, thn α 0

21 w hav stting ( ρr) ( ρr) 1 X AZ Y Th lattr approximation is good nough for rough work; for light lmnts, up to iron, it is practically xact but not so good for havy lmnts: ( AZ ) = 56 6 =. F ( AZ ) = 07 8 =.5 Pb 1

22 In HE physics xprimnts on is intrstd in highly rlativistic particls. Thrfor th intgration of th (invrs) stopping powr must b don numrically. Th rsult is displayd in th following figur.

23 3

de/dx Effectively all charged particles except electrons

de/dx Effectively all charged particles except electrons de/dx Lt s nxt turn our attntion to how chargd particls los nrgy in mattr To start with w ll considr only havy chargd particls lik muons, pions, protons, alphas, havy ions, Effctivly all chargd particls