Charged particle tracking and momentum resolution Why do we need charged particle tracking in an experiment? FDetermine the number of charged particle produced in a reaction. FDetermine the identity of a charged particle (e.g. p, K, p ID using dE/dx). FDetermine the momentum of a charged particle. We measure the momentum of a charged particle by determining its trajectory in a known magnetic field. Simplest case: constant magnetic field and p^B trajectory is a circle with p=0.3Br We measure the trajectory of the charged particle by measuring its coordinates (x, y, z or r, z, f, or r, q, f) at several points in space. We measure coordinates in space using one or more of the following devices: Multiwire Proportional Chamber low spatial resolution (1-2 mm) Drift Chamber moderate spatial resolution (50-250mm) Silicon detector high spatial resolution (5-20 mm) Most common Simplest case: determine radius of circle with 3 points Better momentum resolution better mass resolution Many particles of interest are observed through their decay products: D0K-p+,D+K-p+ p+,pp-, K0p+ p+ By measuring the momentum of the decay products we measure the mass of the parent. 1 m m1+m2 m2=(E1+E2)2-(p1+p2)2 = m12+ m22 +2[(m12+ p12 )1/2 (m22+ p22 )1/2 - p1p2 cosa] a For fixed a: m2 / m2 p / p 2 Richard Kass 880.A20 Winter 2002 Momentum and Position Measurement (L/2, y ) y 2 Trajectory of charged particle s=sagitta x (0, y1) (L, y3) z Assume: we measure y at 3 equi-spaced measurements in (x, y) plane (z=0) each y measurement has precision y have a constant B field in z direction so p^=0.3Br Note: The exact The sagitta is given by: expression for s is: y1 y3 L2 L2 0.3BL2 s y2 2 8r 8 p^ /(0.3B) 8 p^ L2 sr r 4 2 The error on the sagitta, s, due to measurement error is found using propagation of errors to be: s 3 / 2 y Thus the momentum (^to B) resolution due to position measurement error is: p^ p^ s s 880.A20 Winter 2002 3 / 2 y 2 (0.3L B) /(8 p^ ) 8 p^ 3 / 2 y 2 0.3L B 32.6 p^ y 2 LB (m, GeV/c, T) Richard Kass Momentum and Position Measurement From previous page the momentum resolution due to measurement error is: p^ p^ s s 3 / 2 y 2 (0.3L B) /(8 p^ ) 8 p^ 3 / 2 y 2 0.3L B 32.6 p^ y 2 LB (m, GeV/c, T) Typical numbers for CLEO (or BELLE or BABAR) are: B=1.5T, L=0.8m, y=150mm p ^ p^ 32.6 1.5 10 4 p ^ 0.82 (1.5) 5.1 103 p ^ Thus for a particle with transverse momentum (p^) = 1GeV/c: p ^ 0.5% The above momentum resolution expression can be generalized for the case of n position measurements, each with a different y. The expressions are worked out in Gluckstern’s classic NIM article, NIM, 24, P381, 1963. A popular formula is for the case where we have n>>3 equally spaced points all with same y resolution: p^ p^ 720 y p^ (m, GeV/c, T) 2 n 4 (0.3BL ) Note: The above expression tells us that the best way to improve this component of momentum resolution is to increase the path length (L). 880.A20 Winter 2002 Richard Kass More on Momentum Resolution On the previous page we calculated the position measurement contribution to the momentum resolution. This is only part of the story. We also have contributions from multiple scattering (MS) and angular resolution. Previously we saw that the momentum resolution contribution to MS was given by: p p s rms plane sB L 13.6 10 3 z L / Lr p 4 3 with L L / sin q, p^ p sin q 0.3BL2 z /(8 p^ ) p 52.3 10 3 p B LLr sin q MS depends on the total path length (L) and momentum (p). Bending in the magnetic field depends on p^(=psinq) and projected path length (L). The MS contribution is independent of the position resolution contribution so the combined resolution is the two added in quadrature: p^ p^ 2 52.3 10 3 720 y p^ 2 B LL sin q r n 4 (0.3BL ) 2 2 ( m, GeV/c, T) Technically speaking, the above is only the transverse momentum (p^) resolution. We want an expression for the total momentum resolution! 880.A20 Winter 2002 Richard Kass Even More on Momentum Resolution We can get an expression for the total momentum (p) resolution using: p p^ / sin q p^ 1 cot 2 q and treating qand p^ as independent variables. Using propagation of errors (L) we find: 2 2 Often detectors measure the r-f p p^ cotqq 2 p p^ coordinate independently of the z coordinate. In these cases p^ and q are independent. Putting it all together we have for the total momentum resolution: p p GeV/c, T, m, radians 2 52.3 10 720 y p sin q cot q q 2 2 B LL sin q n 4 ( 0 . 3 BL ) r 2 2 Position resolution 3 Multiple scattering Angular resolution While the above expression is only approximate it illustrates many important features: a) p^ resolution improves as B-1 and depends on p as L-2 or L-1/2. b) For low momentum (0), MS will dominate the momentum resolution. c) Improving the spatial resolution (y) only improves momentum resolution if the first term is dominate. d) Angular resolution is not usually the most important term since qmin30-45o and q10-3 rad. For more detailed information must do a Monte Carlo simulation (GEANT+detector). Include: hit efficiencies, discrete scattering off of wires, non-gaussian tails, etc, etc…. 880.A20 Winter 2002 Richard Kass Still More on Momentum Resolution Let’s examine the momentum resolution equation for a CLEO-like system: B=1.5T y=1.5x10-4 m N=50 q=10-3 radians L=0.8 m Lr=166.7 m (gas+wires) p 2.85 103 p sin q p 2 2 2 3.0 103 3.0 103 cotq sin q momentum resolution at 900 2 momentum resolution at 450 0.012 0.012 spatial resolution MS combined 0.01 angular resolution spatial resolution MS combined 0.01 0.008 0.008 p/p p/p 0.006 0.006 0.004 0.004 0.002 0.002 0 0 0 0.5 1 1.5 2 p(GeV/c) 880.A20 Winter 2002 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 p (GeV/c) Richard Kass 3.5 4 Mass Resolution and Physics Discovery of the Upsilon at Fermilab in 1977 using a double arm spectrometer. Had to do an elaborate fit to find 3 resonances: U(1S), U(2S), U(3S) pBemmX 1977 PRL 39, 252 (1977) PRL 39, 1240 (1977) Double arm spectrometer (E288) mm 1986 Upgraded double arm spectrometer (E605) clearly separates the 3 states: improved mass resolution and particle ID (RICH) 880.A20 Winter 2002 Better fit Richard Kass Wire Chamber Operation A wire chamber is just a gas tight container with a wire inside. The gas is the medium that gets ionized by a passing charged particle. The wire helps define an electric field and “collects” ionization, part of signal path. A typical cylindrical wire chamber has: a wire (anode) held at +V outside of cylinder (cathode) held at ground Charged particle passing through cylinder creates ions movement of ions creates a voltage or current pulse signal pulse travels down wire to “outside world” usually to preamplifier Location of charged particle is measured relative to wire Operating characteristics depend on the applied voltage (E-field) recombination: no signals ionization: signals, but no gas gain proportional: “big” signals due to gas gain Geiger-Muller: gas gain so large it produces sparks (discharge) Have to run in proportional or geiger mode to detect single particles like e’s, p’s, K’s, p’s. 880.A20 Winter 2002 Richard Kass Signal Production First consider a very simple case: A capacitor in a gas tight box. +Vo anode R d --gas +++ cathode c The electric field inside the chamber is E=Vo/d. signal The parallel plates of the chamber have capacitance C with stored charge Qo=CVo. If N ions are produced by r a charged particle passing through the gas then the electrons will drift to the anode and the positive ions will drift to the cathode. Assuming that the e-’s and positive ions make it to the plates long before the power supply can recharge the plates back to Vo (RC very large) the charge on each plate will be diminished by N|q|, with |q| the charge of an electron. Therefore the voltage across the plates will drop by DV=N|q|/C and we would see this voltage drop as our signal. C =Chamber capacitance c =pulse shaping (and HV decoupling) capacitor R =power supply series resistor (large) r =pulse shaping resistor 880.A20 Winter 2002 Richard Kass Pulse formation in a cylindrical wire chamber In a cylindrical chamber the electric potential, E-field and capacitance are given by: (r ) CV0 ln( r / a) 2pL wire radius= a, tube radius=b, length of tube= L E (r ) CV0 1 2pL r C 2pL ln(b / a) Note: I put the L dependence in , E, and C. Leo’s C is my C/L. The potential energy stored in the electric field is W=1/2CV02. Assume a charged particle goes through the cylinder and ionizes the gas. As a charge, q, moves a distance dr there is a change in the potential energy (dW): d (r ) q d (r ) dW q dr and dW CV0 dV dV dr dr CV0 dr The total induced voltage from electrons produced at r is: q a d (r ) q a (CV0 ) dr (q) a r V dr ln CV0 a r dr CV0 a r 2pL r 2pL a The total induced voltage from positive ions produced at r is: q b d (r ) q b (CV0 ) dr q b V d r ln CV0 a r dr CV0 a r 2pL r 2pL a r Note: the total induced voltage is: DV=V++V- = -q/C 880.A20 Winter 2002 Richard Kass Pulse formation in a cylindrical wire chamber Note: the positive ions and electrons do not contribute equally to the DV if there is multiplication in the gas. Since the avalanche takes place near the wire (r=1-2mm) and the electrons are attracted to the wire the positive ions travel a much greater distance. For typical values of a (10mm) and b (1cm) we find: b 3 V ln 10 ar 75 a r ln( 11 / 10 ) V ln a We can find the voltage vs time by looking at V(t) for the positive ions: r (t ) dV (r ) q r (t ) V (t ) V (t ) dr ln 2pL a r ( 0 ) a dr The problem now is to find r(t). ln By definition, the mobility, m, of a gas is the ratio of its drift velocity to electric field. 1 dr E (r ) dt For cylindrical geometry we have: dr CV0 1 CV0 mE (t ) m rdr m dt dt 2pL r 2pL m v / E (r ) 880.A20 Winter 2002 Richard Kass Pulse formation in a cylindrical wire chamber From previous page we had: rdr m CV0 dt 2pL CV0 t 2 mCV0 t rdr m dt r (t ) a 2 p L p L r (0) a 0 1/ 2 r (t ) q r (t ) q mCV0 q t V (t ) ln ln(1 t ) ln( 1 ) 2 2pL a 4pL 4pL t0 pLa a 2 ln(b / a) With: t0 2mV0 t0 2 b2 2 The total drift time is: T 2 (b a ) 2 t0 a a Typical gas mobilities are m=1-2 cm2s-1V-1. Example: Let m=1.5 cm2s-1V-1, V=1500V, a=10mm, b=1cm then: t0=1.5x10-9 s and T=1.5x10-3 s. t 0 t0 10t0 102t0 103t0 T ln[1+t/t0] 0 0.69 2.4 4.6 6.9 13.8 Time development of voltage pulse 880.A20 Winter 2002 Richard Kass Gas Gain In many gases (e.g. argon) ion multiplication occurs as the “original” electrons get close to the wire. If the electric field is high enough the electrons will be accelerated to the point where they have enough kinetic energy to liberate electrons in collisions with other atoms/molecules. A cartoon of the multiplication process. Over a limited range of electric field the final amount of ionization is proportional to 1st Townsend coefficient for argon+organic vapors the amount of primary ionization. The total amount ionization, n, is related to the primary ionization by, np,: r2 n=Mnp where: M exp[ a ( x)dx] a is the first Townsend coefficient. r1 Gas gains up to M=108 are possible. Above this limit breakdown (sparking) occurs. This limit (ax<20) is the Raether limit. 880.A20 Winter 2002 Richard Kass Multiwire Proportional Chamber (MWPC) In late 1960’s early 1970’s techniques were developed that allowed many sense wires (anodes) to be put in the same gas volume. The MWPC was born! The spatial resolution () of an MWPC is determined by the sense wire spacing (Dx): Dx 12 Typical wire spacings are several mm, but MWPC with 1mm spacing have been built. cathode . . . . . Dx anodes sense wires Gas volume Charpak wins 1992 Nobel Prize for developing MWPCs Advantages of MWPC: Disadvantages of MWPC can cover large area poor spatial resolution elaborate electronics need low noise premps systems with thousands of wires planar or cylindrical geometry can get pulse height info dE/dx position info along wire using charge division easy to get a position measurement (digital) can handle high rates works in magnetic field ease of construction 880.A20 Winter 2002 miniaturization of electronics elaborate gas system must understand electrostatics forces on wires Richard Kass Drift Chambers Drift Chambers are MWPCs where the time it takes for the ions to reach the sense wire is recorded. This time info gives position info: ts x v(t )dt t0= start time, ts=stop time=time electrons reach sense wire t0 For some gases the drift velocity is constant (independent of E-field): x=v(ts-t0) A gas with almost constant drift velocity is 50-50 Argon-Ethane, drift velocity 50mm/nsec By using the drift time information we can improve our spatial resolution by a factor of 10 over MWPCs (1mm 100 mm). Hex-cell drift chamber 880.A20 Winter 2002 drift times are circles around the sense wires Richard Kass Drift Chambers Resolution The spatial resolution of a drift chamber is limited by three effects: Statistics of primary ionization location of the primary ionizations (a few 100mm apart) Diffusion of the electrons as they drift to the wire 1 2 Dx n mE N=# of primary ions D=gas diffusion constant m=mobility x=drift distance E=electric field magnetic field changes alters drift path: drift path depends on “lorentz angle”, ExB How well the electronics measures time must measure time to < 1nsec, must know start time (t0) Contributions to spatial resolution 880.A20 Winter 2002 Richard Kass Drift Chambers Drift chambers come in all sizes, shapes and geometries: planar fixed target cylindrical colliding beam Time information gives a “circle” of constant distance around the sense wire (more complicated in B field) In almost all cases, wires in different layers are staggered to resolve the left-right ambiguity Typical cylindrical DC: Many wires in same gas volume. Use small angle stereo for z. Usually use single hit electronics. Sense (anode) and field wires. CLEO, CDF, BELLE, BABAR Tube Chamber: Single sense wire in a cylinder Can make out of very thin wall tubes. very little material Small drift cell single hit electronics Good cell isolation broken wire only affects one tube CLEO’s PTL detector 880.A20 Winter 2002 Jet chamber: optimized to resolve two tracks in a “jet”. Drift direction roughly perpendicular to wire plane. Single track gives multiple hits on several wires. Use multi-hit electronics so two tracks on a wire can be resolved. Lorentz angle must taken into account wires are “slanted” Richard Kass Time Projection Chamber TPC measures all 3 space coordinates Many hits per track (>100) good dE/dx measurement Used at LEP, RHIC Very complicated electric field shaping: E||B to reduce effects of diffusion Long drift times complicated gas system Lots of electronic channels complicated electronics 880.A20 Winter 2002 Richard Kass Silicon Strip Detectors SSD’s are solid state proportional chambers Approximately 1000X more ionization in silicon compared to a gas. Not necessary to have charge multiplication to get useable signals. • silicon strip detector measures position to ~10mm. • silicon detector has many thin metal strips on top and (sometimes) bottom surface of silicon wafer • charged particle ionizes the silicon as it passes through • electric field in silicon guides ions to top/bottom • ions are collected on one or or two (or 3) strips • knowing which strip has signal gives position of charged track relative to silicon detector 880.A20 Winter 2002 Richard Kass Silicon Strip Detectors Dx Resolution is mainly determined by strip pitch: 12 Dx=3.5 need strips every 50mm to get 15 mm resolution 00strips per cm Strips can only be 5 cm long (technological limit) Modern silicon strip detectors have 105-106 strips! CLEO III hybrid (one of 122) Require custom electronics electronics must be small electronics must be radiation hard low power dissipation wire bond connections (105-106) Mechanical Structure must be rigid/strong must be low mass to minimize MS mechanical tolerances ~mm preamps Much more engineering involved with silicon detectors compared to drift chambers! 880.A20 Winter 2002 Digital ADC capacitors Richard Kass Advanced Silicon Detectors Double sided silicon detector (CLEO) Put orthogonal (x,y) strips on top and bottom surface. Allows 2 coordinate measurements per silicon wafer minimizes amount of material less MS Problems in high rate environments poor two track separation Pixel detector (BTEV, LHC) Get position location (x,y) from hit pad (50mm x 50mm) minimizes amount of material less MS Radiation hard(er) Quick response time Small detector capacitance good s/n with thin detector less MS Good two track resolution 880.A20 Winter 2002 Richard Kass CLEO III Silicon Detector Installation of CLEO III silicon detector 1.25x105 strips Each strip has its own: RC, preamp, ADC Readout cables 880.A20 Winter 2002 hybrids Silicon wafers (layer 4) Drift chamber Richard Kass CLEO II.V charged particle tracking CLEO II.V had: 3 layer silicon detector 10 layer drift chamber (VD) 51 layer drift chamber (DR) All in a 1.5T B field DR Si VD 880.A20 Winter 2002 Richard Kass