Lecture 1
Momentum measurement Multile scattering Bethe-Bloch formula / Landau tails Ionization of gases Wire chambers Drift and diffusion in gases Drift chambers Micro gas detectors Silicon as a detection medium Silicon detectors stris/ixels
Momentum measurement Β Β
Momentum measurement x y B L s mv ρ = q( v B) = qbρ ( GeV c) = 0.3Bρ ( m) θ ρ sagitta L = sinθ θ ρ he sagitta s is determined by 3 measurements with error s(x): Error on the sagitta θ s = ρ( 1 cosθ ) ρ 8 0.3L B θ 0.3 8 L B 1 s = x ( x + ) 3 1 x ( ) σ meas. σ ( s) = = s 3 σ ( x) = s 3 σ ( x) 8 0.3 BL For N equidistant measurements (R.L. Gluckstern, NIM 4 (1963) 381) meas. σ ( ) σ ( x) (for N 10) = 70 /( N + 4) 0.3 BL Examle: = 1 GeV/c, L = 1 m, B = 1, s(x) = 00 mm, N = 10 ( ) σ meas. 0.5% (s 3.75 cm)
Momentum resolution Resolution is generally defined as 1 standard deviation - 1σ for a gaussian distribution dominated by Poisson fluctuations: σ = FWHM-full width at half-maximum /.36 E resolution = σ/e For box distribution (uniform within an interval ): σ = / 1
Can we distinguish curved track from the straight line? s = ρ - (ρ L /4) CLEO (electron-ositron collider): Maximum momentum = 5 GeV/c, B field = 1.5 àρ= /(0.3 B) = 11.11 m rack radius = 1.0 m à s = 0.011 m (1.1 mm) EASY!!! ALAS racking length L = 1.15 m B field = = 50 GeV/c ρ = 144.9 m s = 0.0011m (1.1 mm) = 1000 GeV/c ρ = 1666.7 m s = 0.0001m (0.1 mm)!!!! must consider measurement errors
Scattering An incoming article with charge z interacts with a target of nuclear charge Z via exchange of the virtual hoton. he cross-section for this e.m. rocess is dσ = dω 4zZr e mec β sin 1 4 θ Rutherford formula Deends on charge of the assing article and on the charge and density of the material dσ/dω Average scattering angle θ = 0 Cross-section for θ 0 is infinite! θ
Multile Scattering In sufficiently thick material layer the article will undergo multile scattering. Each individual scattering is indeendent of the revious one à statistical rocess in 3 dimensions L P Gaussian sin -4 (θ/) Aroximation: r lane θ lane θ RMS 0 = θ lane = θ lane = 1 θ0 L X 0 1 θ RMS sace 0 θ 0 θ lane X 0 is radiation length of the medium
Contribution of multile scattering to σ ( ) σ ( ) σ ( x) σ 1 ( x) MS θ 0 σ ( ) MS = constant indeendent of! More recisely: σ ( ) MS = 0.045 B 1 LX 0 σ()/ total error σ()/ meas. σ()/ MS examle: Argon gas (X 0 =110m), L=1m, B=1 σ ( ) MS 0.5%
he Helix Equation he helix is described in arametric form hs cos λ x ( s ) =x o o ( + R ) + R cos Φ cos Φ 0 R ( m ) = (GeV ) 0.3B ( ) P L y ( s ) = y o + R sin Φ hs cos λ ( o + ) R sin Φ0 λ z ( s ) = z o + s sinλ B P P λ is the di angle h =±1 is the sense of rotation on the helix he rojection on th x-y lane is a circle ( x x o + R cos Φ o ) + (y y o + R sin Φ o ) = R x o and y o the coordinates at s = 0 Φ o is also related to the sloe of the tangent to the circle at s = 0 ( x o, y o ) R sin Φ o Φ o R cos Φ o right-handed system 1
Gas detectors Fast charged articles ionize the atoms of a gas. Often the resulting rimary electron will have enough kinetic energy to ionize other atoms. Primary ionization 10-40 airs/cm ΔE/air ~ 0-40 ev n otal ionization 3 4 total n rimary Assume detector, 1 cm thick, filled with Ar gas: 1 cm ~ 100 e-ion air à 100 electron-ion airs are not easy to detect! (Noise of amlifier ~1000 e - ) We need to increase the number of e-ion airs.
Gas amlification Simlest case - cylindrical field geometry Proortional Counter E gas cathode b a anode E threshold 1/r E ( r) = V ( r) = CV0 πε 0 CV0 πε 0 1 r ln r a C = caacitance / unit length a r MECHANISM OF SIGNAL AMPLIFICAION u Electrons drift towards the anode wire. u Close to the anode wire the field is high ( ~kv/cm) u Electrons gain enough energy for further ionization à exonential increase of number of e - -ion airs.
Signal Formation electron (F. Sauli, CERN 77-09) wire Avalanche formation within a few wire radii and within t < 1 ns! Signal is induced by the moving charges both on anode and on cathode (electrons and ions). dv = Q lcv 0 dv dr Electrons collected by anode wire, i.e., dr is small (few µm) dr (F. Sauli, CERN 77-09) Ions have to drift back to cathode, dr is big. Signal duration limited by total ion drift time! Need electronic signal differentiation to limit dead time.
Cross section for ionization by collisions of electrons with atoms of noble gasses n α( E) x α( r)x = n0e or n = n0e α 1 λ M = λ: mean free ath r n = = C ex α n0 a ( r) dr α: First ownsend coefficient (e - -ion airs/cm) Gain M ke CV 0 otal signal (number of collected electrons) is roortional to the total ionization generated by the assage of charged article. -> roortional chambers E
Multi wire roortional chambers G. Charak et al. 1968, Nobel rize 199 field lines and equiotentials around anode wires Negative signals on all wires. yical arameters: L=5mm, d=1mm, a wire =0µm. Digital readout: satial resolution limited to x d 1 σ ( d=1mm, s x =300 µm )