de/dx Effectively all charged particles except electrons

Transcription

1 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 xcpt lctrons Th man nrgy loss of a chargd particl through mattr is dscribd by th Bth-Bloch quation 1

2 de/dx You ll s de dx de ρdx de dx whr MV with units of cm MVcm with units of g with units x is of th mass MVcm g thicknss ρdx

3 de/dx In particl physics, w call de/dx th nrgy loss In radiation and othr branchs of physics, de/dx is calld th stopping powr or linar nrgy transfr (LET) and de/ρdx is calld th mass stopping powr 3

4 de/dx Assum Elctrons ar fr and initially at rst Δp is small so th trajctory of th havy particl is unaffctd Rcoiling lctron dos not mov apprciably W ll calculat th impuls (chang in momntum) of th lctron and us this to giv th nrgy lost by th havy chargd particl 4

5 Gauss's law says z ET dx = b z thn I = bv I ΔE m z m v ( b) = = 4 de/dx dx I = Fdt = ET dt = ET v by symmtry th nt forc is in th transvrs dirction E T b πbdx = 4πz 5

6 Now in a thicknss thr ar πbdbdxnz N is de - dx de - dx - de dx πnz 4 4π Nz m v de/dx th # atoms/v and = = = 4 4πz NZ m v ΔE Z ( ) b bdb ln dx db b b b btwn lctrons Z max min is b and b + db th atomic numbr 6

7 Not de/dx W only considr collisions with atomic lctrons and can nglct collisions with atomic nucli bcaus Z m de dx Z Am lctrons p >> de dx nucli Excpt for ions on high Z targts at low nrgy 7

8 de/dx b min (short distanc collisions) In an lastic collision btwn a havy particl and an lctron Δp ΔT ΔT ΔE b min max max max ( b ) = = = min m v m v z γm v = m γ v = m γ v = z m v b 4 min 8

9 Classical de/dx b max (long distanc collisions) W can invok th adiabatic principl of QM Thr will b no chang if th intraction tim is longr than th orbital priod b 1 τ v(vlocity) ν (frquncy) γv bmax = v v is a man frquncy avragd ovr all stats 9

10 de dx 4πz m v de/dx Substituting w hav 4 de 4πz = NZ ln dx m v = 4 NZ This is vry clos to Bohr s 1915 rsult Actually Bohr calculatd th nrgy transfr to a harmonically bound lctron and found b b max min γ mv ln z v 3 de dx = 4πz m v 4 NZ 1.13γ mv ln z v 3 v c 10

11 de/dx Our approximation is not too bad 11

12 de/dx Nots on th ssntial ingrdints de = dx c z Z 1 ln c γ β Enrgy loss dpnds only on th vlocity of th particl, not its mass At low vlocity, de/dx dcrass as 1/β Rachs a minimum at β=0.96 or βγ=3 At high vlocitis, de/dx incrass as lnγ Calld th rlativistic ris Enrgy loss dpnds on th squar of th charg of th incidnt particl Enrgy loss dpnds on Z of th matrial 1

13 Bth-Bloch de/dx β=p/e γ=e/m 13

14 Quantum Effcts Ral nrgy transfrs ar discrt QM nrgy > classical nrgy but th transfr occurs in a fw collisions Bth calculatd probabilitis that th nrgy transfrrd would caus xcitation or ionization b min must b consistnt with th uncrtainty principl On nds to us th largr of Z γm v and h γm v Bth also includd spin ffcts 14

15 Dnsity Effct So far w calculatd th nrgy loss to on lctron of on atom and thn prformd an incohrnt sum For larg γ, b max > atomic dimnsions Th atoms in btwn will b affctd by th filds and ths atoms thmslvs can produc prturbing filds at b max Atoms along th fild will bcom polarizd thus shilding lctrons at b max from th full lctric fild of th incidnt particl Dnsity ffct = inducd polarization will b gratr in dnsr mdiums 15

16 Calculation Dnsity Effct Frmi (1940) was first Strnhimr Phys Rv 88 (195) 851 givs additional gory dtails Jackson contains a calculation as wll Th nt ffct is to rduc th logarithm by a factor of γ Instad of a rlativistic ris w obsrv a lss rapid ris calld th Frmi platau And th rmaining slow ris is du to larg nrgy transfrs to a fw lctrons 16

17 Dnsity Effct 17

18 Dnsity Effct Th dnsity ffct is usually stimatd using Strnhimr s paramtrization S tabls on nxt slid 18

19 19

20 Bth-Bloch Equation K= MVcm /g I = man xcitation nrgy T max is th maximum kintic nrgy that can b impartd to a fr lctron Accurat to about 1% for pion momnta btwn 40 MV/c and 6 GV/c At lowr nrgis additional corrctions such as th shll corrction must b mad 0

21 Bth-Bloch de/dx β=p/e γ=e/m 1

22 T max T max is th maximum nrgy that can b impartd to lctrons Not it is in th logarithm and is also rsponsibl for part of th de/dx incras as th nrgy incrass T max is givn by mc β γ Tmax = 1+ γm / M + m ( / M ) Somtims a low nrgy approximation is usd T max = mc β γ

23 Alpha particls from 5 Cf fission T max 3

24 Man Excitation Potntial I Approximatly I/Z = 1 V for Z < 13 I/Z = 10 V for Z > 13 Constants xist for most lmnts and should b usd if mor accuracy is ndd 4

25 Shll ffct Othr Effcts At low nrgis (whn v~orbital vlocity of bound lctrons), th atomic binding nrgy must b accountd for At vlocitis comparabl to shll vlocitis, th de/dx loss is rducd Shll corrctions go as -C/Z whr C=f(β) Rlativly small ffct (1%) at βγ=0.3 but it can b as larg as 10% in th rang MV for protons Brmsstrahlung For havy chargd particls, this is important only at high nrgis (svral hundrd GV muons in iron) 5

26 Low Enrgy On larg ffct at low nrgis is that th incidnt particl will captur an lctron for som of th tim thus nutralizing itslf Thus th ionization losss will dcras Enrgy losss from lastic scattring with nucli also bcom important (and may dominat for havy ions) 6

27 Low Enrgy de/ρdx~(z/β) Z 7

28 de/dx Valus For low nrgis (< 1000 MV) tabls of stopping powr ar availabl from NIST For high nrgis, on can us de/dx min as a good stimat 8

29 de/dx Valus How much nrgy dos a cosmic ray muon (E>1 GV) dposit in a plastic scintillator 1 cm thick? de ρdx ρ Δx = MV 9