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24. PARTICLE DETECTORS - Particle Data Group

24. PARTICLE DETECTORS - Particle Data Group

22 24.

22 24. Particle detectors A pulse shaper formed by a single differentiator and integrator with equal time constants has F i = F v =0.9andF fv = 4, independent of the shaping time constant. The overall noise bandwidth, however, depends on the time constant, i.e. the characteristic time T s . The contribution from noise currents increases with shaping time, i.e., pulse duration, whereas the voltage noise decreases with increasing shaping time. Noise with a 1/f spectrum depends only on the ratio of upper to lower cutoff frequencies (integrator to differentiator time constants), so for a given shaper topology the 1/f contribution to Q n is independent of T s . Furthermore, the contribution of noise voltage sources to Q n increases with detector capacitance. Pulse shapers can be designed to reduce the effect of current noise, e.g., mitigate radiation damage. Increasing pulse symmetry tends to decrease F i and increase F v (e.g., to 0.45 and 1.0 for a shaper with one CR differentiator and four cascaded integrators). ( ) Q 2 n = 2eI d +4kT/R b + i 2 na F i T S + ( 4kTR s + e 2 ) (24.25) na Fv Cd 2/T S + F vf A f Cd 2 . As the characteristic time T S is changed, the total noise goes through a minimum, where the current and voltage contributions are equal. Fig. 24.9 shows a typical example. At short shaping times the voltage noise dominates, whereas at long shaping times the current noise takes over. The noise minimum is flattened by the presence of 1/f noise. Increasing the detector capacitance will increase the voltage noise and shift the noise minimum to longer shaping times. 10000 Equivalent noise charge (e's) 5000 2000 1000 1/f noise 500 200 Current noise Voltage noise 100 0.01 0.1 1 10 100 Shaping time (µs) Figure 24.9: Equivalent noise charge vs shaping time. For quick estimates, one can use the following equation, which assumes an FET amplifier (negligible i na ) and a simple CR–RC shaper with time constants τ (equal to November 26, 2001 09:18

the peaking time): [ (Q n /e) 2 =12 1 nA · ns ] [ ] I d τ +6×10 5 kΩ τ ns R b [ ] +3.6×10 4 ns (pF) 2 (nV) 2 /Hz 24. Particle detectors 23 e 2 C 2 n τ . (24.26) Noise is improved by reducing the detector capacitance and leakage current, judiciously selecting all resistances in the input circuit, and choosing the optimum shaping time constant. The noise parameters of the amplifier depend primarily on the input device. In field effect transistors, the noise current contribution is very small, so reducing the detector leakage current and increasing the bias resistance will allow long shaping times with correspondingly lower noise. In bipolar transistors, the base current sets a lower bound on the noise current, so these devices are best at short shaping times. In special cases where the noise of a transistor scales with geometry, i.e., decreasing noise voltage with increasing input capacitance, the lowest noise is obtained when the input capacitance of the transistor is equal to the detector capacitance, albeit at the expense of power dissipation. Capacitive matching is useful with field-effect transistors, but not bipolar transistors. In bipolar transistors, the minimum obtainable noise is independent of shaping time, but only at the optimum collector current I C , which does depend on shaping time. Q 2 n,min =4kT C √ βDC √ Fi F v at I c = kT e C√ β DC √ F v F i 1 T S , (24.27) where β DC is the DC current gain. For a CR–RC shaper, [ 1 ] √ C Q n,min /e ≈ 800 √ × . (24.28) pF (β DC ) 1/4 Practical noise levels range from ∼ 1e for CCDs at long shaping times to ∼ 10 4 e in high-capacitance liquid argon calorimeters. Silicon strip detectors typically operate at ∼ 10 3 e electrons, whereas pixel detectors with fast readout provide noise of several hundred electrons. In timing measurements, the slope-to-noise ratio must be optimized, rather than the signal-to-noise ratio alone, so the rise time t r of the pulse is important. The “jitter” σ t of the timing distribution is σ n σ t = ≈ t r (dS/dt) ST S/N , (24.29) where σ n is the rms noise and the derivative of the signal dS/dt is evaluated at the trigger level S T . To increase dS/dt without incurring excessive noise, the amplifier bandwidth should match the rise-time of the detector signal. Increasing signal-to-noise ratio also improves time resolution, so minimizing the total capacitance at the input is also important. At high signal-to-noise ratios, the time jitter can be much smaller than the rise time. The timing distribution may shift with signal level (“walk”), but this can be corrected by various means, either in hardware or software [69]. November 26, 2001 09:18

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