Pair (and Triplet) Production Effect:

Transcription

1 Pair (and riplt Production Effct: In both Pair and riplt production, a positron (anti-lctron and an lctron (or ngatron ar producd spontanously as a photon intracts with a strong lctric fild from ithr a nuclus ( production or an lctron (triplt production. Pair Production hν initial positron Annihilation, if in-flight thn s nuclus initial Elctron: oftn, { triplt Production if lctron instad of nuclus } θ θ hs intractions ar dominant at high incidnt photon nrgy: h ν >> m c Nobl Pri in physics, 948, wnt to P. Blacktt s bubbl! It works by condnsation in a supr-hatd liquid (cloud chambr, or boiling in a supr-coold liquid (bubbl chambr. h particls crat local trail of bubbls lik airplans mak condnsat tracks. his ffct also invokd to xplain Hawking radiation. Lctur 7 MP 5 Kissick 3

2 Aftr som math that you should do, th thrshold nrgy for ths ffcts to tak plac is: ( hν min m c Mc mc Whr, if: M M M m nuclus >> m thn it's production thn it's triplt production Assuming that th rcoil of th nuclus is small, th availabl kintic nrgy is simply avail avail hν ( hν hν m c min for and triplt production (m c.mv Focusing now on Pair Production: h man kintic nrgy givn to ach of th two particls is half of th availabl kintic nrgy [actually, th positron gts a bit mor nrgy bcaus of th push from th positivly chargd nuclus]. avail Lctur 7 MP 5 Kissick 3

3 3 h man angl givn to ach of th two particls is with rspct to th incidnt photon dirction is m c θ with units: [θ ] radians h / dpndnc is similar to brmsstrahlung! {and that s not all as w will s}. Highr nrgy particls gt mor forward dirctd! Cross-sction for Pair production: Bth and Hitlr (934 drivd th atomic diffrntial cross-sction as follows: {pr atom} d a κ d avail P And all of th complications ar in P. -- h quantity,, is dfind as follows: r x 8 cm / atom h quantity, r, is th classical lctron radius, and also rprsnts th rang of th strong nuclar forc: r.879x m c 5 m Not that /37 α, th fin structur constant. On of th most important numbrs in th univrs!! S Appndix for this lctur for mor information about th fin structur constant. Lctur 7 MP 5 Kissick 3

4 4 P is a function of photon nrgy and almost indpndnt of atomic numbr,, as shown in th following figur from Attix, pag 49: Notic th symmtry in th abov figur: Enrgy not givn to th positron is givn to th lctron and th othr way around h cross-sction is again obtaind by intgrating th diffrntial cross-sction: a κ avail P d( Pd( / avail avail With: d ( / avail P thn, Pd( / avail d( / avail Lctur 7 MP 5 Kissick 3

5 5 Using th abov to dfin an avrag P P : a κ P And this P also has littl dpndnc. If th intraction is too far from th nuclus, thn many orbital lctrons will scrn th nuclar lctric fild. Whn scrning can b nglctd, thr is no dpndnc and just a wak logarithmic dpndnc on nrgy: 8 hν 8 P ln 9 m c 7 (m c << hν << 37m c / 3 Whn scrning is maximid, at high nrgy, thr is a wak logarithmic dpndnc and basically no nrgy dpndnc: 8 P ln(83 9 ( hν >> 37m c /3 /3 7 (If 6, this is hν>>35mv At nrgis around m c, no analytical form is possibl. W can us th approximation that aκ is proportional to for all photon nrgis. hr is a grat similarity btwn production and brmsstrahlung!! Lctur 7 MP 5 Kissick 3

6 6 Asid: th similarity btwn production and brmsstrahlung: hr is a concpt about anti-particls, proposd by P. Dirac in 93, that th ngativ nrgy root from E th Schodingr Equation with som rlativity. ( pc ( mc is an anti-particl. his cam out of A photon with nough nrgy, and an lctric fild to xchang momntum with, can librat somthing out of th infinit sa of ngativ nrgy (th Dirac sa, compltly filld and occupid stats vacuum!, and th hol lft bhind is th anti-mattr: nrgy ngatron m c positron Dirac sa Anothr mathmatical oddity is that on can prhaps viw th positron as moving backwards in tim. his is usd in Fynman Diagrams whr th similarity btwn brmsstrahlung and production is profoundly obvious! tim Pair production tim Brmsstrahlung positron lctron lctron nuclus nuclus position position Lctur 7 MP 5 Kissick 3

7 7 h cross-sctions (cm /atom ar VERY similar: Pair production at vry high nrgis: aκ r 37 8 ln(83 9 / 3 7 Brmsstrahlung (radiativ losss: r ln(83 9 /3 4 8 hrfor, a κ 7 9 h Pair Production Mass Attnuation cofficint is as follows thn: N A κ aκ with units of ρ A aκ N A A cm ( atoms / mol atom g / mol cm g Rcall for Compton: ρ N A A N A A a if a Lctur 7 MP 5 Kissick 3

8 8 Focusing now on riplt Production: h lctric fild is now from an lctron, a vry light particl which bcoms indistinguishabl from th cratd particl! riplt Production hν initial And anothr! initial initial positron θ θ Annihilation, if in-flight thn s Elctron: oftn, Impossibl to tll which is which!!! h man kintic nrgy givn to ach of th thr particls is a third of th availabl kintic nrgy [actually, th positron still gts a bit mor nrgy bcaus of th push from th positivly chargd nuclus that most availabl lctrons ar finding thmslvs nar]. avail 3 Of cours, sinc M m in ( hν It is all du to momntum consrvation. m c Mc min mc, th thrshold is now m! 4 c Asid: Bcaus of atomic xcitations, th thrshold is actually small m c, but vry vry Lctur 7 MP 5 Kissick 3

9 9 h riplt production Cross-sction It would b th sam but w actually us a factor, C, to rlat th triplt crosssction to th production cross-sction bcaus of lctron xchang ffcts: Whr, aκ κ triplt C a C.6. C.. -- And C has a ngligibl dpndnc on. P C (5MV < hν < MV (MV < hν < MV Combind riplt and Attnuation Cofficints: κ aκ aκtriplt P( / C a Combind riplt and Mass Attnuation Cofficints: κ N A P( / C ρ A Combind riplt and Mass Enrgy ransfr Cross-sctions: h fraction of photon nrgy transfrrd to chargd particls is thrfor, avail / hν, κ κ ρ ρ κ hν mc hν ρ hν tr avail Not a typo! h m c is for BOH! (riplt has an xtra momntum issu, but still just crating particls. his is shard btwn ithr or 3 particls. Right nar th thrshold, th amount of nrgy transfrrd is small, but this crosssction approachs th attnuation cross-sction at larg nrgis! Lctur 7 MP 5 Kissick 3

10 Positron annihilation in flight : Whn th positron mts an lctron, two gamma photons ar rlasd in opposit dirctions in th cntr-of-mass (or cntr-of-momntum fram. hrfor, an isotropic angular distribution in this fram. Both photons hav th sam circular polariation: both RHC or both LHC. cntr-of-mass fram laboratory fram hν CofM hν - hν CofM - hν h annihilation gammas in th lab fram will hav a sum of nrgis: hν hν mc W will nd to includ th lost chargd particl (positron kintic nrgy in th calculation of µ n. h mass annihilation cofficint was drivd by Hitlr: annihil ρ N A A πr 4 ln ( ( 3 Whr: m c ( v / c {S Fynman, Chaptr 8, ~.% of th tim, 3 photons com off, not two, if positronium had l (momntum not qual to ro.} Lctur 7 MP 5 Kissick 3

11 Not: annihil ( / A for >> mc ρ So that annihilation is much mor likly at low nrgis! positronium is nam givn to th tmporary positron-lctron thing that xists for a short tim whn both hav basically no kintic nrgy thn thy giv off two idntical, opposit gammas of.5 gammas in ithr fram! hν. 5MV /- hν. 5MV Nxt lctur: som xtras lik photonuclar ractions and Raligh scattring. hn, w start to form a cohsiv pictur of photon intractions. Lctur 7 MP 5 Kissick 3