Ionization Energy Loss of Charged Projectiles in Matter Steve Ahlen Boston University
Almost all particle detection and measurement techniques in high energy physics are based on the energy deposited by energetic charged particles in matter by means of atomic excitation and ionization.
Drift chambers for tracking Silicon detectors for vertex Electromagnetic calorimeters Hadronic calorimeters
Examples of the tracks in photographic emulsions of primary nuclei of the cosmic radiation moving at relativistic velocities. Taken from Powell s Nobel Prize lecture.
Projectile Atoms Projectile mass = M = Am u Electron mass = m << M (we assume) Electron charge = e Projectile charge = Z e Target atomic number = Z Target atomic mass = A Projectile velocity = v = βc Projectile Lorentz factor = γ N = number density of electrons
Electrons scatter off projectile in center of mass frame
Rutherford scattering cross section: dσ dω = 4m v q 4 q sin 4 ( Θ ) q = electron charge = e q = projectile charge = Z e m = electron mass v = relative speed Θ== center of mass scattering angle
Mott scattering cross section for electrons off nuclei (lowest order): 4 dσ Z e = m c + p cos dω 4p 4 c sin 4( Θ ) Θ 4 dσ Z e m γ = β sin dω 4p 4 sin 4( Θ ) Θ
Some kinematics Energy transfer to electron in lab frame: ε = mv Θ Θ γ sin = ε sin m Scattered electrons produced in thickness dx in lab: dn dσ dσ = N = N dεdx dε dω ε dσ dn = NAdx A Scattered electron production per unit energy transfer: m πsin Θ Θ sin cos = Θ 4πN ε m dσ dω
Θ β ε ε γ ε π = ε sin 4p m e N Z 4 dx d dn 4 m 4 m ε ε β ε π = ε m 4 mv e NZ dx d dn
Energy loss per path length for large energy transfers (> ε ): de dx >ε = ε ε m ε dn dε dεdx πnz mv e 4 n ε ε m β
de/dx = force exerted on projectile by medium = Z ee
de dx >ε = πnz mv e 4 n ε ε m β de dx <ε = πnz mv e 4 n mv (if below Cherenkov threshold, v < c/n) I γ ε β I = logarithmic average of atomic excitation energies
Bethe Formula: Ann. Physik 5, 35 (930); Z. Phys. 76, 93 (93) de dx = 4πNZ mv e 4 n ε m I β Half the energy is lost in close collisions (small number of events, large energy transfer - delta rays) Half lost in distant collisions (large number, energy lost per collision is about same as ionization energy)
Energy loss expressed per grammage X = ρx ρ = mass density de dx = mv 4πNZ ( m NA Z ) u e 4 n ε m I β Minimum ionization rate (γ= =3): de dx min g MeV cm
Material I (ev) Plastic scintillator 63 Aluminum 63 Iron 85 Lead 85
A nucleus of magnesium or aluminium moving with great velocity, collides with another nucleus in a photographic emulsion. The incident nucleus splits up into six alpha - particles of the same speed and the struck nucleus is shattered.
Delta Ray production: dn dεdx = πnz mv ε e 4 ε de β ρ ε dx 0 ε m dn dx >ε 0. 0 ε g MeV cm
Electron range for electron energy ε: R 0.537ε g cm MeV + 0.985 3.3MeV ε But lots of straggling for electrons
Impact parameter approach for distant collisions γzeb p y = ee y() t dt = e dt 3 ( b + γ v t ) Collision time: t b γv γvt = b tan θ π γzeb b py = e sec θdθ = π Z e ( b b tan ) 3 + θ γv bv
p y = Ze bv de dx dis tan t = b b max min ( p ) y m πnbdb = 4πNZ mv e 4 n b b max min b max γ = v ω Collision time equals oscillation time of electron b min = a 0 Impact parameter approach valid only for distances larger than size of atoms
de dx = 4πNZ mv e 4 n mv I γ β Relativistic rise One γ from increasing impact parameter; The other γ from increase in kinematic limit to energy transfer
Relativistic rise does not continue indefinitely: Density effect - polarization of material screens field of projectile at distant atoms. de 4πNZ e mv γ = n β dx mv I average 4 δ
Asymmetric energy loss fluctuations skew energy loss distribution (Landau distribution). At high energies (γ ~ 0 for solids, 000 for gases), the most probable energy loss in thickness x is: mp ( β = ) = πnz FWHM mp mc = e 4 x n.9z ( 4.0).9Z n a 0 a 0 x x.6 0% g MeV cm
If maximum energy delta ray is small compared to mean energy loss, ε m < 0 de dx x energy loss distribution is nearly gaussian and square of standard deviation of energy loss is: σ = 4πNZ e 4 x ( ) β ( ) β
Some observational examples
Large Hadron Collider (LHC) at Geneva, Switzerland proton-proton collisions with 4 trillion electron volts energy
ATLAS
Endcap Muon Chambers made at Harvard (4 so far) Mod-0 - May 7, 000
Mod-0 on Test Stand
Scintillators and Phototubes
BMC Test Stand scintillators Mod-0 XP-00 phototubes concrete
Layer Side View Layer 4 Layer 3 Layer
Shower event if trigger on coincidence of top two scintillators - these are electrons and positrons
Another shower, this one made locally
Single tracks for trigger that includes scintillators above AND below the concrete - these are muons
Scintillator signal vs position from timing mu0690a_f; y_=5 500 000 500 q 000 500 0-400 -300-00 -00 0 00 00 300 400 t - t ( channel = 35 ps)
Landau distribution for one scintillator q mu0690a_f; y_=5; deltb:(90,300) 80 70 60 50 40 30 0 0 0 00 400 600 800 000 00 400
Ionization Measurements and Particle Identification
Range is well defined for muons, pions, nuclei, if energy is not too high R ( ) β + β ( ) = R ( ) B ( ) β proton Z Z M m p nd term is range extension due to electron capture: this is small for small Z or large velocity σ R R % M m p
Range R(Z,A) vs Ionization Range R(Z,A) vs rigidity 0000 000 00 R(,) R(,4) R(,0.) R(,0.4) 0000 000 00 Range (g/cm) 0 0. 0.0 Range (g/cm) 0 0. 0.0 0.00 R(,) R(,4) R(,0.) 0.00 0 5 50 75 00 5 50 0.000 R(,0.4) 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (Z/beta)^ rigidity (A beta gamma/z) Ranges are for air
Discovery of positron in cloud chamber Lead plate thickness = 6 mm Magnetic field = 4000 gauss into picture Dimensions: 7 x 7 x 3 cm Change of curvature implies particle moving up, so charge is positive Track density => Z/β = Aβγ/Z = 0.07 below plate, 0.0 above plate: so A < 0.0 For very low energy particles like these, mass and charge can be measured, showing them to be positrons.
Pion decays in nuclear emulsion Photo-micrographs of four examples of the successive decay π - µ - e as recorded in photographic emulsions. Note that the muon ranges are nearly the same, so they must have same energy when pion decays at rest - a two body decay.
event in bubble chamber * the slow proton is identifed by its heavier ionisation, * the K0 subsequently decays into a pair of charged pions, * the antineutron annihilates with a proton a short distance downstream from the primary interaction, to produce three charged pions, * the neutral pion decays into two photons, which (unusually for a hydrogen chamber) both convert into e+e- pairs, * (external particle detectors were used to identify the charged kaon).
Velocity Measurements are the Most Effective Way to Identify Particles de dx K Z Z β if velocity known Rigidity = Mcβγ Z e M if velocity and charge are known
Scintillator timing can measure time-of-flight to 00 ps Position from scintillator timing compared to drift tube position
Charge Mass E ( ) ( ) MeV amu = 938 β
Cherenkov Radiation if v > c/n: pi = pf + Dk Ei = Ef + ω ω = ck n cosθ C = c n v
Cherenkov Yield: de dx Cherenkov = ω ω Z c e n β ωdω A few hundred photons per cm for n =.5 for typical photo-multiplier tubes
Cherenkov response for roughened.3 cm Pilot 45 radiator in white box for neon nuclei
Ring imaging Cherenkov detectors using small light sensors provide best velocity resolution
Use of relativistic Rise in Gas Ionization Counters From Detection of cosmic-ray antiprotons with the HEAT-pbar instrument (Proceedings of ICRC 00: 69)
At very high energies, transition radiation can determine Lorentz factor