Expermental Technques for Nuclear and Partcle Physcs Interactons of partcles n matter (1)
Incdent beam/radaton feld on materal ("fxed" target / detector) N partcles per sec Φ flux (partcles/sec/unt area) Cross secton, σ t σ [ ] -8 barn 10 m n t target nucle per unt volume n t N A ρ M [cm -3 ] σ number of reactons per unt tme beam partcles per unt tme scatterngcentres per unt area
Lumnosty dn! N! n σdx t : L Φ N N! N! (0) e mean freepath between nteractons Reacton yeld, Y : Y N! (0) N! ( t) N! Thn target ( t << Lumnosty and reacton yeld (1 e 1 ) : Y nσ N! nσt Φ ) t n σ x t n t t n v N t (fxed target) Beam s exp. attenuated by 1/e at σ A l 1 n σ ( n At) σ N! ( n At) N! A ntσ L σ N! σ Collder: Two partcle beams crculate n opposte drectons wth velocty v n a storage rng wth radus R and collde at a reacton pont. If there are j partcle packets wth N a partcles a and N b partcles b the lumnosty: L N N j v / π R a b A The reacton rate (yeld) s gven by L σ where A s the beam cross secton at the reacton pont. For a total ntegrated lumnosty L dt the total number of reactons n tme T s L dt σ. Example: If the cross secton σ 1nb and a 100 pb -1 ntegrated lumnosty then 10 5 reactons are expected. The frst long LHC run endng n December 01 reached an ntegrated lumnosty around 30 fb -1 T 0 N t ρ t M Hggs dscovery
dσ ( Ω) dω (Doubly) Dfferental cross secton, dσ/ (de)dω number of partcles per unt tme scatterednto dω ncdent beam partcles per unt tme scatterngcentres per unt area θ A d ΔΩ A d r dω N! ( E, θ, ΔΩ) L dσ ( E, θ ) dω ΔΩ
Interactons of charged partcles wth matter 1) Interactons wth atomc electrons (electronc stoppng). ) Elastc and nelastc scatterng on nucle (nuclear stoppng). For 1) and ) electronc stoppng domnates completely. 3) Emsson of Cherenkov radaton. 4) Nuclear reactons. 5) Bremsstrahlung. Stoppng power: de dx ( E, z, A,...) Range: de dx 1 de
In most detectors the mportant nteracton s the electromagnetc (but not n all applcatons) Bohr, Bethe & Bloch developed the theory Phenomena on a mcroscopc scale : Ionzaton (.e. n gases) and exctaton of atoms (and fluorescence) Creaton of electron-hole pars Creaton of phonons (heat) Channelng n crystallne structures Radaton damage of crystal structure Tme scale of stoppng Heavy ons v << c ~ ps Very hgh energy v c Tme x/c 1 m 3.33 ns, 1 cm 33.3 ps At the LHC experments at CERN (Geneva) new reactons are possble every 5th ns. A detector sze > 10 m means that a new event mght have started n the detector before the partcles created n the prevous event have left the detector.
M, z,t, v Electronc stoppng Protons and ons e - Elastc collsons: ΔT max m e (v) ΔT T max 4m M e ΔT max. kev (protons) ~ straght lne unform range Alpha partcles : T 8MeV > 15000 collsons
Bethe-Bloch formula for heavy charged partcles The approxmate Bethe-Bloch formula: ( ) de dx e 4π ε 0 4π z m c e N AZ β A abs abs ρ ln mec I β ln(1 β ) β where : velocty z, β are the charge and the velocty of lght ρ, Z I abs, A abs are the densty, atomc number and atomc weght of "exctaton potental" 7 Z(1 + ) ev Z Z(9.76+ 58.8Z ( β 1.19 Z < 13 ) ev v / c) of Z 13 the projectle the absorber A more complete formula can be found n e.g. Leo, chapter..
Bethe-Bloch formula energy and momentum dependence Fgure from Leo Fgure from Partcle data book
Stragglng Fgure from Leo
At very low energes, the Bethe-Bloch formula breaks down Energy dependence of de/dx for β<<1 not completely understood As the velocty ~ velocty of atomc electrons de/dx reaches a maxmum and then drops sharply Pck-up of electrons s one mportant effect
Statstcal smulatons - SRIM He on Au 10 MeV α partcles on gold
Bragg curves for carbon-1 ons n water
where Snce the knetc energy β g( T We can wrte the Bethe-Bloch formula n the form: f ( β ) s a functon of / M ) de dx de If we know ( T ) for a dx type n the same medum : Scalng law for de/dx T z de z f dx the partcle ( f '( T de dx 1- β / M ) partcle type 1n a ( T 1 z ) z ( β ) 1) Mc 1 velocty only. de dx 1 medum we can scale t to a ( T the velocty s a functon of M M 1 ) T M : partcle
Range Scalng laws for range Snce R T 0 n the same medum : 1 T 0 " The range R $ de % ' de (gnorng Coulomb scatterng) # dx & 0 1 z f (β ) de we have a smlar scalng law for R for dfferent partcles 0 R (T ) M z 1 M 1 z R M 1 1(T ) M For the same partcle n dfferent meda there s also an approxmate scalng law: R 1 R ρ ρ 1 A 1 A (Bragg-Kleeman rule)
The tme requred to stop a For nonrelatvstc partcles : v E m We assume that the average velocty as t slowsdown s where v c E mc s the ntal velocty.if the partcles were unformly decelerated then k 0.5. But snce they slow down faster towards the end of The stoppng tme, T T stop R v R kc mc E c stop partcle n an absorber can be calculated from the average velocty. E 931MeV / u A, s then : R kc Stoppng tme 931MeV / u A E proj proj the range, k s somewhat larger;k 0.6. 1. 10 v 7 kv, R A proj E typcally pcoseconds n solds and nanoseconds n gases
Mass stoppng power When de/dx s expressed n unts of mass thckness t vares lttle over a wde range of materals. de dε 1 de ρ dx z Z A f ( β, I) dε > The mass stoppng power scales as Z/A for the same partcle type n dfferent meda. Example: A 10 MeV proton loses about the same amount of energy n 1g/cm of copper as n 1g/cm of alumnum.
de/dx for mxtures and compounds 1 de ρ dx where : w If of w a a and 1 fracton by weght of s the the number of molecular compound then : A m w1 ρ a A A m, a de dx 1 + where : A w ρ A de dx atoms of stands +... element element n the molecules for the atomc weght of element
S charged partcle detector telescope 185 MeV 40 Ca + 58 N E ΔE ΔE E de dx ( T z ) z 1 de dx 1 ( T M M 1 )
Channelng Taken from Leo r n 4πε n! r 0 ; a 0 1 mee
Interactons of electrons and postrons wth matter Most often relatvstc Non-lnear (zg-zag) stoppng path Bethe- Bloch : where τ T / m e de dx c Two basc processes: 1) Collsons wth atomc electrons ) Energy loss by radaton; bremsstrahlung de de de coll and dx e 4π ε 0 tot dx coll + dx π N AZabsρ τ ( τ + ) ln mec β Aabs ( I / mec ) + F( τ ) τ e ( + 1)ln 8 4π ε 0mec F( τ ) 1 β + for ( τ + 1) β 14 10 4 F( τ ) ln 3 + + + 1 τ + ( τ + ) ( τ + ) rad 3 e for e +
de dx rad e 4π ε 0 de dx de dx N rad coll A Zabs( T + mec 4 137m c A e e T + mec m c abs Zabs 1600 ) ρ ( T + mec 4ln mec ) 4 3 de/dx n Cu Crtcal energy,above whch radaton losses domnate: E c 800 MeV Z + 1.
Cherenkov radaton Cherenkov radaton s emtted when charged partcles move faster than the velocty of lght n a certan medum. Ths speed s gven by: v partcle βc > c / n An electromagnetc shockwave s created whch has a concal shape emtted at an angle: 1 cosθcherenkov βn( ω) The energy radated per unt frequency nterval per sold angle n a slab of materal of thckness d s gven by: d E dωdω where : α n s the refractve ndex of ξ ( θ ) z e! c4πε 0 α! nβ sn c 1 137 ωd (1 βncosθ ) βc ωd θ πβc sn ξ ( θ ) ξ ( θ ) s the fnestructureconstant the medum Fgure from Leo
Transton radaton A transton radaton detector (TRD) s a partcle detector usng the veloctydependent threshold of transton radaton n a layered materal. Many layers of materals wth dfferent ndces of refracton are typcally used. At each nterface between layers, the probablty of transton radaton ncreases wth the relatvstc gamma factor. Thus partcles wth large γ produce many photons, and those wth lower velocty gve rse to less radaton. For a gven energy, ths allows a dscrmnaton between a lghter and heaver partcles. ALICE TRD
Example (Leo.1) In an experment nvolvng cosmc ray muons, a cm thch plastc scntllator (densty 1.03 g/cm 3 ) s used as detector. What s the mean energy deposted n the detector per muon? Cosmc ray muons are hgh-energy partcles and are assumed to be mnmum onzng. At these energes (> ~300 MeV, 1/ρdE/dx s ~.0 MeV/(g/cm ) and ~ constant) t de 1 de ΔE dx ( ρt) dx ρ dx 0.0 1.03 MeV 4.1MeV.
Problem (Knoll.) Wth the ad of Fg.7 (next) estmate the energy remanng n a beam of 5 MeV protons after passng through 100 µm of slcon. Soluton R 10 µm for 5 MeV protons n S. Therefore the range remanng after 100 µm of slcon s estmated as 110 µm. Ths corresponds to 3. MeV.