Interaction of particles with matter

Transcription

1 Introduction to Elmntary Particl Physics. Not 1 Pag 1 of 15 Intraction of particls with mattr 1. Particls and intractions. Wak intractions (nutrinos) 3. Elctromagntic intractions (chargd particls) 3.1. Ionization nrgy losss - Bth-Bloch formula for de/dx - Landau Fluctuations - Multipl scattring - Scintillation (spcial cas of d-xcitation) 3.. Brmsstrahlung radiation 3.3. Chrnkov radiation 3.4. Transition radiation 4. Elctromagntic intractions (photons) 4.1. Photo-ffct 4.. Compton 4.3. Pair production 5. Elctromagntic showr (photons and lctrons) 6. Strong intractions: Hadronic showr (hadrons)

2 Introduction to Elmntary Particl Physics. Not 1 Pag of Particls and intractions Four typs of intractions: gravitational, wak, lctro-magntic, strong. Th typical rlativ magnituds of ths forcs: 1-39 : 1-7 : 1 - : 1. Th gravitational forc plays no rol in th high nrgy physics. Particls. Considr only rlativly long-livd particls so that thy could travl som distanc and had a chanc to intract with mattr bfor dcaying. Not that du to γ-factor, th distanc particls can travl bfor dcaying will dpnd on thir nrgy. W will us 1 GV as a bnchmark nrgy (particls of highr nrgy hardly vr producd vn at Tvatron, th most powrful collidr as of today). Lptons: nutrinos (stabl) lctron (stabl), muon (τ= µs, cτ=5 m, γcτ=5 km), τ-lpton (τ=.3 ps, cτ=9 µm, γcτ=5 mm) Hadrons: considr only ground stats of quarks (xcitd stats liv ~ s) Msons quark-antiquark pairs lightst msons mad of u,d-quarks, M~14 MV o π ± (τ=3 ns, cτ=1 m, γcτ=6 km) o π (τ=1-16 s, cτ=3 nm, γcτ= µm) lightst msons mad of s- and u,d-quarks, M~5 MV o K ± (τ=1 ns, cτ=4 m, γcτ=8 m) o K L (τ=5 ns, cτ= m, γcτ= 4 km), K S (τ=.1 ns, cτ=3 cm, γcτ=6 m) o η (τ=1-18 s, ) lightst msons mad of c- and u,d,s-quarks, M~ GV o D ± (τ=1 ps, cτ=3 µm, γcτ=15 mm) o D (τ=.4 ps, cτ=1 µm, γcτ= 6 mm) o D ± S (τ=.5 ns, cτ=15 µm, γcτ=8 mm) o J/ψ (τ=1 - s,.) lightst msons mad of b- and u,d,s,c-quarks, M~5 GV o B ±, B, B s (τ= ps, cτ=5 µm, γcτ=1 mm) o B ± c (τ=.5 ps, cτ=15 µm, γcτ= 3 mm) o Υ (.) no msons ar vr formd with t-quarks (thy liv much lss than 1-3 s, hadron formation tim) Baryons quark-quark-quark particls lightst baryons mad of u,d-quarks, M~1 GV o p ± (stabl) o n (τ=15 min, cτ=3 1 8 km, γcτ=3 1 1 km) lightst baryons mad of s- and u,d-quarks, M~1 GV o Λ (τ=.3 ns, cτ=8 cm, γcτ=8 m) o Carrirs of th forcs: photons (stabl) gluons (do not occur as fr particls) W/Z (liv lss than 1-5 s) Wak Elctromagntic Strong rlativ strngth nutrinos Ys - -, µ, τ Ys Ys - hadrons: chargd nutral Ys Ys Ys - Ys Ys photons - Ys -

3 Introduction to Elmntary Particl Physics. Not 1 Pag 3 of 15. Wak intractions (nutrinos) Th only particls for which this intraction is of any significanc ar nutrinos, as thy do not participat in th othr intractions. Th forc is xtrmly wak: th man fr pass of 1 MV anti-nutrino in watr is about 1 cm or 5 light yars! So dtction of nutrino is actually dtction of missing nrgy-momntum in th balanc of th outgoing particls. W nvr obsrv multipl nutrino intractions along its passag through mattr. Whnvr an xtrmly rar + singl intraction happns, othr particls ar producd (for xampl, ν + p + n ) or knockd off from th mdia (for xampl, ν + ( atomic) ν + ( scattrd ) ) it is ths particls that w will dtct by making us of thir intractions with mattr via lctromagntic or strong forc.

4 Introduction to Elmntary Particl Physics. Not 1 Pag 4 of Elctromagntic intractions (chargd particls) 3.1. Ionization nrgy losss for a chargd particl passing through mdia Bth-Bloch formula for nrgy losss de/dx (193s) de z Z m β γ T = 4π ln dx Am m I α ρ 1 max δ β β N whr: m, m N, α univrsal constants: lctron and nuclon masss; fin structur constant; z, β, γ incoming particl paramtrs (charg in units of, vlocity β=v/c, gamma factor); Z, A, ρ, I mdia proprtis: charg and atomic numbr of mdia atoms, dnsity, avrag ionization potntial for th mdia (I ~ 16*Z.9 V for Z>1) δ small corrction du to mdia polarization (for gasss, it is ngligibly small). To undrstand th origin of th Bth-Bloch formula, w will considr th following simplifid modl. Momntum transfr from an incoming particl of charg z to an atomic lctron is:, dx z dx a z = z = z = dq F dt F, v 4 π ( x + a ) v x + a whr a th shortst distanc btwn particl's trajctory and an atomic lctron; v particl vlocity, z particl charg, x chargd particl's coordinat along its trajctory. Aftr intgration from to + in x w will hav th total momntum transfrrd to an lctron q z α =. av Th nrgy transfrrd to lctron (and, consquntly, lost by th incoming particl) is thn: q z α 1 T = =. m m v a Summing up ovr all lctrons in th mdia, th avrag nrgy losss xprincd by th incoming particl aftr passing distanc dx of th mdia will b: amax z α a max de = T π andadx = 4π n ln dx m amin v a, min Not th two cutoffs in th intgration a min and a max. At larg distancs, for which th calculatd T bcoms smallr than th ionization/xcitation potntial I, no nrgy transfr bcoms possibl this imposs a cutoff a max : z α 1 I = mv amax Th maximum nrgy an incoming particl can transfr to an lctron is limitd by th nrgy-momntum consrvation laws. For an incoming particl of mass M, vlocity v (and gamma-factor γ): m β γ Tmax = m m 1+ γ + M M z α 1 This can b accountd for introducing a cutoff a min : Tmax =. m v a min

5 Introduction to Elmntary Particl Physics. Not 1 Pag 5 of 15 Thrfor, th formula for nrgy losss that w ar driving can b r-writtn as follows (now th ngativ sign is also includd to rflct that th nrgy is bing lost): de z α 1 1 Tmax = 4π n ln dx β m I Th numbr of lctrons in th mdia of dnsity ρ and mad of lmnts (Z,A) is ρ n = Z, from whr: Am de z α Z ρ 1 Tmax 4π ln dx β AmN m I = Dpndnc on th charg z of th incoming particl is as z. Th nrgy looss dcras as 1/v, whr v is th vlocity of th incoming particl. Slow moving particls would los mor nrgy and as thir momnta incrass (and th vlocity saturats at th spd of light), on xpcts flattning out of de/dx. If de/dx is normalizd on ρ, th dpndnc on mdia bcoms vry wak as Z/A~1/ for most lmnts and =1 for hydrogn. On can s that in comparison to th formula w hav drivd, th Bth-Bloch formula has an xtra factor of m β γ I undr th log, which actually lads to a slow ris of ionization losss with th particl momntum. This is du to accounting for th rlativistic flattning of th lctric fild of th incoming particl. As th rsult, th particl can ioniz atoms at fathr and farthr distanc as its fild bcoms mor and mor squashd at largr momnta. This ris vntually flattns out du to polarization ffcts in th mdia. Th plots blow show th actual dnsity-normalizd de/dx curvs for a fw matrials. Th typical valu of nrgy losss at th minimum is about MV/(g/cm ). On can s that th ris aftr th minimum is vry slow and hardly xcds 5% vn at p~1 GV. Th particls with vlocitis corrsponding to βγ>3 ar usually calld minimum-ionizing particls (mip). N

6 Introduction to Elmntary Particl Physics. Not 1 Pag 6 of 15 Landau fluctuations Th Bth-Bloch formula givs th avrag nrgy losss for ionization and xcitation. Th fluctuations around th most probabl valu can b paramtrizd by th Landau distribution (ths fluctuations ar spcially larg for thin layrs and gass): 1 1 L( ) xp ( λ λ = λ + ) π, E E λ ξ whr λ is th dviation from most probabl nrgy losss: W = ( E nrgy losss in a layr with thicknss qual to x, E W most probabl nrgy losss, ξ - is a paramtr charactrizing th width of th distribution ). Not that th disprsion of this distribution quals to infinity, indpndntly of how small ξ is. In gass, typical numbrs of primary clustrs ~ 3/cm typical total numbr of rlasd lctrons ~1/cm In solids, th numbrs ar ~1 tims largr (plainly du to thir highr dnsity). Landau fluctuations (Landau tail) corrspond to rar larg nrgy transfrs from th incidnt particl to atomic lctrons. Such lctrons ar calld δ-lctrons. Thy typically can caus additional ionization, lading to a clustr of a fw lctrons. If w know th avrag nrgy losss E for a particl in mdia, th avrag numbr of rlasd lctrons can b stimatd as E/W, whr W is th avrag nrgy spnt pr on rlasd lctron. W is somwhat largr than th ionization potntial (I ) bcaus of som nrgy going into xcitation of atoms and braking of molculs. Th numbr lctrons appard bcaus of th intractions of th ionization particl with mattr is n primary, whras n total is th numbr of lctrons apparing aftr intractions of initial lctrons with mattr. Gas dnsity, ρ [g/cm 3 ] I [V] W [V] n initial [cm -1 ] n total [cm -1 ] H 8.99* O 1.43* Ar 1.78* CO 1.98* CH * C 4 H 1.67*

7 Introduction to Elmntary Particl Physics. Not 1 Pag 7 of 15 Multipl scattring of chargd particls An intraction of a particl of charg z with a nuclus of charg Z charg is dscribd by Ruthrford formula (spins and not point-lik structur of nucli ar ignord): dσ 1 zzα 1 = dω 4 4 β p sin ( θ ) Aftr passing through distanc L and as a rsult of multipl scattrings on nucli, th incidnt particl will xprinc som typical displacmnts and dflctions. Ths will hav approximatly Gaussians distributions, whos avrags ar zro and sigmas ar givn as follows: θ 14MV z β p L X = and r = Lθ Th Gaussian shap dscribs wll th bulk of scattrings (98%), whil th tails xhibit ~1/sin 4 (θ/) dpndnc coming from th Ruthrford formula. Dpndnc of θ on particl's charg, vlocity and momntum can b asily tracd to th Ruthrford formula as wll. Also, it is clar that th dpndnc on th matrial thicknss should b lik squar root sinc th ovrall scattring is a accumulation of indpndnt scattrings. X is th charactristic of mdia and is calld th radiation lngth. Th dpndnc of X on matrial paramtrs is intuitivly clar: it must b invrsly proportional to n A =ρ/a (dnsity of nucli, scattring cntrs, in mdia) and invrsly proportional to Z +Z=Z(Z+1) (scattring on nucli ~Z plus scattring on lctrons ~, but whos numbr Z tims largr): X m 1 1 = ( 1) ln(183/ ) 3 1/3 Z Z + α na Z Th following formula is good for quick stimats (log trm is diffrnt, which ffctivly accounts for small xtra factor omittd in th formula abov): X ( 716 g / cm ) A Z( Z + 1) ln(87 / Z ) ρ Typical valus of X ar.6 cm (lad), 1.8 cm (iron), 36 cm (watr), 3 m (air). Aftr travrsing 1 m of iron, a muon of 1-GV nrgy will b - typically dflctd by ~.6 and - typically displacd by ~6 mm Sam numbrs for air will b about sqrt(3/1.8)=4 tims smallr. 1 3

8 Introduction to Elmntary Particl Physics. Not 1 Pag 8 of 15 Scintillation A chargd particl travrsing mattr lavs bhind it a wak of xcitd molculs. Crtain typs of molculs, will rlas a small fraction of this nrgy in th form of optical or clos UV photons, for which th mdia may b quit transparnt with attnuation lngths raching as much as a fw mtrs. Amount of nrgy carrid away by scintillation light is typically 1% or lss of de/dx. This light may b usd for dtcting th fact of a particl travrsing th mdia. Scintillating matrials ar classifid in inorganic and organic. Inorganic high-z, vry high dnsity (good for γ and dtction and nrgy masurmnt) high light yild rlativly slow Classical Exampls dcay tim (ns) λ (nm) γ pr MV de/dx X (cm) NaI (vry hygroscopic!) CsI Bi 4 G 3 O 1 (BGO) PbWO Organic (plastics, liquid) low cost and as of fabricating various shaps fast rlativly smallr light yild Hydrogn rach (good nutron dtction) Classical Exampls dcay tim (ns) λ (nm) γ pr MV de/dx X (cm) NE BC Blow is an xampl of scintillation mchanism in nobl gass (not that th original xcitations hav much largr nrgy than th nrgy of th scintillation light).

9 Introduction to Elmntary Particl Physics. Not 1 Pag 9 of Brmsstrahlung (radiation losss or braking radiation ) Bth and Hitlr, 1934 Rlativistic chargd particls, as thy propagat through mattr and wiggl du to multipl scattring on nucli, xprinc acclrations and, thrfor, must b radiating lctromagntic wavs mission of such photons is calld "braking" radiation, or brmsstrahlung. Th nrgy losss turn out to b proportional to th incoming particl nrgy, whil th proportionality cofficint 1/X dpnds on th composition of th matrial as wll as th incoming particl mass and charg. de E dx = X, whr X is th radiation lngth (discussd in sction on multipl scattring): Z( Z + 1) α 183 = n ln A 1/3 X m Z Th nrgy at which brmsstrahlung radiation bcoms qual to ionization losss is calld critical nrgy E c. Blow E c, th ionization losss dominat; abov E c, th main sourc of nrgy losss is brmsstrahlung. For mdia atoms with Z 13 th critical nrgy valus for incidnt lctrons can b stimatd as follows: 55MV EC =. Z Th critical nrgy for lctrons in iron (Z=6) is ~ MV, 7 MV in lad (Z=8). Th amount of multipl scattring dpnds on particl's momntum and dos not dpnd on particl's mass. Th amount of radiation, howvr, dpnds quadraticly on th acclration and, thrfor, is proportional to ~1/m. For xampl, braking radiation for muons, th lightst chargd particl aftr an lctron, will b 4, tims smallr than for lctrons of th sam momntum. Th critical nrgy for muons will b ~1 TV.

10 Introduction to Elmntary Particl Physics. Not 1 Pag 1 of Chrnkov radiation As a chargd particl travrs mattr, it producs a wak of polarizd molculs along its path. As th molculs gt polarizd and thn dpolarizd, thy mit radiation in all dirctions. If th particl movs with a spd v xcding spd of light in th mdia c/n (n is th rfraction indx for th mdia), in a crtain dirction th radiation bcoms cohrnt and rsults in significant amount of light producd. Th dirction in which wavs would add up cohrntly can b asily constructd using Huygns tchniqu of circular fronts. This radiation is calld Chrnkov (oftn splld Črnkov) aftr Chrnkov who has studid it in mid-193s and showd that it was not du to luminicns or any othr known radiation mchanism. Th first obsrvation is actually attributd to Mallt in 196. Th thory bhind th Chrnkov radiation was put togthr by Frank and Tamm in Th main formulas rlatd to Chrnkov radiation ar th ons for its dirction and intnsity: cosθ = 1/ nv - angl btwn a particl dirction and photon mition dirction dn C = πα z numbr of phtons mmitd pr unit of nrgy (flat spctrum) dxdε n v On can s that thr is a thrshold minimul vlocity v=1/n, blow which thr is no radiation. As th particl surpasss this thrshold vlocity, it starts mitting small amount of light in th forward dirction (θ~). As its vlocity approachs spd of light, th intnsity rachs its maximum and opns up to th maximum angl θ max =acos(1/n). Amount of nrgy mittd is about 1-4 of de/dx losss. A fw rfrnc numbrs for som common matrrials: Mdia n θ max dn/dxdε (cm -1 V -1 ) visibl dn/dx (cm -1 ) γ thrshold Air Isobutan Watr Quartz Chrnkov, Frank, Tamm awardd Nobl Priz for discovry and xplanation of th ffct.

11 Introduction to Elmntary Particl Physics. Not 1 Pag 11 of Transition radiation This radiation was prdictd by Ginsburg and Frank in Th corrct rlativistic tratmnt is du to Garibian, Th fild of chargd particl in vicinity of a boundary with dilctric matrial can b calculatd using charg imag tchniqu. Th particl and its imag form a dipol. If th particl movs towards or away from th boundary, th dipol momnt d will b changing with tam and, thrfor, on should xpct th charactristic dipol radiation to occur. Avrag radiatd nrgy pr boundary: whr ω N W = p ε m (~ V for plastics). 1 αω pγ 3 =, + _ Th typical nrgy of photons is ε = ( ω / 4) γ (thrfor, fast moving particls produc X-rays) Th avrag numbr of mittd X-ray photons pr boundary N=α=1/137. p Not that th intnsity of transition radiation (nrgy of X-rays) incrass linraly with particls nrgy, or mor accuratly with γ. Th angular distribution of transition radiation is pakd forward with a sharp maximum at θ=1/γ. Radiation from vacuum-mdia and mdia-vacuum boundaris has diffrnt phass and thrfor a minimum thicknss of th film is rquird to prvnt cohrnt conclation. Th minimum thicknss of matrials is ~(c/ω p )γ : ~ µm for CH plastic polimrs and ~1 mm for air gaps.

12 Introduction to Elmntary Particl Physics. Not 1 Pag 1 of Elctromagntic intractions (photons) Thr ar thr vry distinct procsss of photon intracting with mdia: + γ + atom atom + (photolctric ffct; dominant for ε γ <.1 MV) γ + γ + (Compton ffct; dominant for.1 MV < ε γ < 1 MV) + γ + nuclar + + nuclar (Pair production; dominant for ε γ > 1 MV) Th photon bam intnsity in a mdia falls xponntially: whr λ is a fr man path, 1 1 λ = = n σ n σ. A total A i i I x / λ = I, 4.1 Photolctric procss, or photo-ffct This is th procss of photo absorption lading to ionization of an atom. If photon nrgy is sufficint, an lctron from th most innr atomic shlls (K-shll) will b prdominantly knockd off. In this cas, an lctron from a highr nrgy shll can fall and mit a charactristic frquncy light. (Augr-lctrons can b discussd hr as wll). Approximatly, th photo-ffct cross sction away from th charactristic nrgy paks dpnds on th lmnt proprtis and photon nrgy as σ ph 5 4 Z α. 3 ε 4. Compton scattring This is th procss of scattring of photons on atomic lctrons. Approximatly, th Compton cross sction has th following dpndnc on mdia proprtis and photon nrgy as: ε ln σ C Zα ε Th t-channl diagram is shown on th right. Thr is also a similar s-channl diagram. On can s from ths diagrams that th matrix lmnt will b proportional to. Thrfor, th atomic cross sction will b proportional to Z ( ), whr th xtra factor Z accounts for th numbr of lctrons in an atom. Th nrgy spctrum of scattrd photons and knockd off lctron is approximatly flat from ε= to ε=ε γ. 4.3 Pair Production + -pair production by photons in th nuclar fild. This procss has an nrgy thrshold and only possibl for ε γ > m = 1 MV. For photon nrgis ε γ > (7 MV) / Z 1/3, th cross sction rachs th platau lvl and bcom practically nrgy indpndnt (th dpndnc on α and Z is obvious): This is th procss σ pair 3 7 4α Z 183 = ln 1/3 9 m Z Not th intrsting rlationship that follows from this quation: λ pair ~ (9/7) X. From th obvious lctron-positron symmtry in this procss, th avrag nrgis of lctrons and positrons must b qual to ε γ /.

13 Introduction to Elmntary Particl Physics. Not 1 Pag 13 of 15 Total photon cross sction σ tot in lad, as a function of nrgy: σ p.., atomic photo-ffct (lctron jction, photon absorption); σ Rayligh, cohrnt scattring (Rayligh scattring atom nithr ionizd nor xcitd); σ Compton, incohrnt scattring (Compton scattring off an lctron); κ nuc, pair production, nuclar fild; κ, pair production, lctron fild.

14 Introduction to Elmntary Particl Physics. Not 1 Pag 14 of Elctromagntic showrs (lctrons and photons) Th pictur abov illustrat a simplifid modl of an lctromagntic showr. Only brmsstrahlung and pair λ = X. production procsss ar considrd. For simplicity, w will assum pair On can s that th numbr of particls grows with th distanc as: N( t ) = t stp as E( t) = E t. Procss continus until E( t) Ec and nrgy pr particl falls at ach < --aftr this point lctrons will b losing thir nrgy prdominantly via ionization losss and photons will b r-scattrd a fw tims (Compton) and finally absorbd (photo-ffct). Th procss of particl multiplication stops whn E tmax Not th showr siz in lngth grows with nrgy only logarithmically. ln E / E = Ec, from whr t max = c. ln Total numbr of particls producd in th showr is proportional to th incidnt particl nrgy: N total tmax t anq a 1 E = = = = q 1 1 E t= ( tmax + 1) tmax. 95% of th showr cor is containd in a cylindr of radius R 95% = R M and lngth L 95% ~t max +1, whr 1MV R X g cm is Molir radius. M = / E c For xampl, for lad and 1-GV lctrons, R 95% ~4 cm and L 95% ~16 cm. Pictur blow shows an lctromagntic showr rcordd by th ICARUS xprimnt in thir Liquid Argon Tim Projction Chambr. c

15 Introduction to Elmntary Particl Physics. Not 1 Pag 15 of Strong intractions: Hadronic showr (hadrons) High nrgy collisions of hadrons typically rsult in prolific production of pions, th lightst particls mad of quarks. Strong intractions of a kind pion + p k pions + p will, thrfor, naturally lad to a hadronic cascad similar in many rspcts to an lctromagntic showr producd by lctrons and photons. Th main diffrncs will b in a) man fr path btwn collisions; b) numbr of particls producd pr collision; c) minimum nrgy aftr which th showr dvlopmnt trminats. From point of viw of strong intractions, protons and nutrons approximatly can b viwd as non-transparnt sphrs of radius r~1 fm. Thrfor, π+p cross sction is σ ~πr ~3 mb. A nuclus of A protons and nutrons, in its turn, can b viwd as a bag of tightly packd nuclons and its cross sction, thrfor, can b stimatd as σ=σ A /3. Man fr pass for a hadron in mdia of atoms (Z,A) is thn Thus, on can stimat man fr pass btwn intractions 1 λ nσ 1/3 1 AmN A mn λ = = =. nσ ρσ ρσ =, whr n is atomic dnsity: n=ρ/(am N ). For iron, this givs λ~19 cm. Th actual numbr is clos and somwhat smallr (17 cm) this is bcaus w ignord intractions that would lad to nuclus xcitations. Numbr of particls producd pr on collision k typically will b somwhat largr than as in th cas lctromagntic showr ( +γ or γ ). Th hadron cascad will stop whn th nrgy of products rachs ~ MV, aftr which no mor pions can b mad (pion mass ~14 MV). Th total numbr of particls in th showr and th point at which it rachs its maximum can b stimatd th sam way as was don for m-showr (albit small modification: k): L ln E / E ln k min max = λ, which for 1 GV hadrons givs L max ~6λ. Th transvrs showr siz ~λ. Thr ar two mor important diffrncs from th m showrs that wr not mntiond abov: d) Th numbr of particl spcis that can in principal b producd is larg ) Th most prolifically producd particls ar pions: π +, π -, π. Howvr, π is a short livd particl that dcays into two photons. Thrfor, all π s, onc producd, dcay to photons, which start dns and rlativly short lctromagntic showrs. Thrfor, although th numbr of chargd tracks producd in a hadronic cascad is proportional to th nrgy of an incoming particl, th fluctuations ar typically much largr than thos in th lctromagntic showr initiatd by an lctron or photon.