Fig. m-13  Pulse sampled from the pulse ionization chamber in case of: Full charge collection (optimal time constant tin=RLCd~10Ti); Electron charge collection (optimal time constant tin=RLCd~10Te). Where: Ti is  full collection time of the positive charged ions (~10-3s); Te is full collection time of the negative charged electrons (~10-6s)

Impact of the input integration chain

It was already mentioned that for most scintillation spectrometer is acceptable t_in=R_LC_d~ t_i, whith some drop off pulse amplitude. For better quality comparing, concept of "ballistic error" D A=A_max-A₀ was established.

Fig. m-14

Light pulse and his current response i(t) from scintillation counter; Shape of the voltage pulse uin(t)on output of the integration chain. (In case of exponential shape of scintillation flash is maximal voltage pulse height Amax=ITS/Cd.) Dashed curve suits ideal time constant tin=RLCd~infinite.

 

Scintillation pulse current id=I*exp(-t/TS) in equivalent detector circuit is divided in RL and Cd. After Laplacce transform we obtain

From time-domain response Uin(t) in time tmax we obtain then pulse height A0 on the output of the scintillation detector whit real time constant tin=RLCd<>infinite.

 

Fig. m-15  llustration of the so called "ballistic error" D A=Amax-A0 vs. ratio TS/tin scintillation time constant TS and input ciruit time constant tin .

From this figure you can see that for scintillation spectrometer applications has no reason to make t_in= R_LC_d shorter than 10T_S, e.g. in scintillator NaI(Tl) (T_S=0,3ms) with C_d~5pF and R_L=0,5MW preserves time constant t_in=R_LC_d~3ms.

Nuclear detector analog signal processing

Fig. m-16  Pulses pile-up affect on baseline fluctuations and cause loss of resolution of pulse amplitude measurement: signal from output of a resistive-feedback preamplifier output Delay-line shaping amplifier connected to the preamplifier can stabile base line

Before amplification, the pulse-shaping amplifier must replace the long decay time of the preamplifier output pulse with a much shorter decay time (with preserving the energy information representing by pulse amplitude). Otherwise the acceptable counting rate would be severely restricted.

Possible ways to short pulse duration:

• CR – RC pulse shaping;
• Pole-zero cancellation shaping;
• Delay-line shaping .

CR - RC pulse shaping

Fig. m-17 Equivalent circuit of amplifier: CR differentiator => high-pass filter with attenuation for low frequency  RC integrator => low pass filter attenuates high frequency region

Fig. m-18 CR Differentiation - simple tool for pulse shorting.

Fig. m-19  RC Integration

Fig. m-20  CR-RC pulse shaping

Best pulse shape from point of noise/signal (S/N) consideration is theoretical calculated pulse form in next figure (a).

To improve S/N ratio in amplifiers typically is set shaping time constant:

This noise performance (case - e in next fig.) is only 36% worse than optimum filter with the same time constant. The simple arrangement is useful for estimates, since simple to evaluate.

Fig. m-21 Pulse shape for optimal S/N in amplifiers. Factor CF makes possible to compare the optimal theoretical calculated pulse (a) form with real filter pulse shaping. optimal theoretical calculated pulse triangle pulse form (ideal integrator+ delay line) Gaussian pulse form (CR+RCn) Semi-Gaussian pulse form (CR+RC5) Unipolar CR-RC shaping For bipolar pulses: (CR)2-RC shaping is CF=1,88 delay clip +(CR)2-RC shaping is CF=1,38

By replacing the simple RC integrator with more complicated active network => S/N improved by 17% to 19% at noise corner time constant.
 

Fig. m-22  Semi-Gaussian (CR+RCn) pulse shaping. Increasing number n of RC output pulse shape changes its form to symmetrical pulse.

 

Fig. m-23  Example of so called active filter, based on operation amplifier - modern method pulse shaping in the Semi-Gaussian Shaping Amplifier.

Delay-line shaping

Drawback of CR-RC shaping is much longer pulse duration compared with delay-line shaping.

Simple delay line clipping of scintillator pulses illustrates next figure. Goal is to short pulses for improved count rate capability.
 

	Fig. m-27a  Delay line clip as a shaping circuits.
	Fig. m-27b Coax line terminated with short circuits allow to use reflected and delayed waveform (inverted pulse) for shaping pulse of duration 2td .

Fig. m-28 Two stage shortening circuits: Differentiator CR on input amplifier helps in preliminary pulse shortening (In short duration pulse is pulse height drop negligible and wave 3 and 4 are practically identical and the pulse shape is improved.). Delay line clip in output amplifier 1 - originally wave; 2 – modified output; 3 – attenuted reflected wave; 4 – reflected wave.

 

Fig. m-29 Cable delay in one input path to differential amplifier. Distance unit of line clip makes delay T0=(LC)0,5 R0=(L/C) 0,5 is terminating resistance; (L,C – equivalent parameters of the transmissions line).

Pole-zero cancellation (PZC)

Bipolar output pulse is obtained when 2 differentiate chains are inserted. Double CR -differentiate shaping produces a bipolar pulse with equal area in its positive/ negative lobes. CR-RC-CR shaping case to undershoot and loss of resolution due to baseline fluctuations. At medium to high counting rates a substantial fraction of the amplitude pulses will ride on the undershoot from the previous pulse, pulse have longer duration and worse S/N ratio, what cause broadening of the energy peaks.
 

Fig. m-25 Doubly-Differentiated CR-RC-CR Shaping - thereason way can an undershoot be generated.

Feedback capacitance in charge sensitive preamp must be discharged too. Conventionally it is done with resistor. Output is no longer a step, but decays exponentially. Exponential decay superimposed on shaper output. Small amplitude overshoot decays back to baseline, with long time constant, provided by the preamplifier output pulse often occurs.
 

	Fig. m-26a
Issue of pulse undershoot on preamp output

Because conventional CR-RC shaping case to undershoot and loss of resolution due to baseline fluctuations in Pole-Zero Cancellation (PZC) the resistor R_pzis added in parallel with capacitor. Resistance R_pz must be precisely adjusted for good baseline recovery (=> simple exponential decay to baseline) especially at high counting rates.
 

The benefit of Pole-Zero cancellation

"Zero"compensates "pole"of the transfer function.

Fig. m-26b => no undershoot and loss of resolution due to baseline fluctuation.

Unipolar versus bipolar shaping

The bipolar shape is useful in minimizing baseline shift with varying counting rates when ac-coupling is used. Electronic resolution with bipolar shaping is typically 25-50% worse than for corresponding unipolar shaper. However:

• Bipolar shaping eliminates baseline shift (dc comp.=0), pole zero adjustment is less critical + added suppression of low frequency pick up.
• Not all measurements require optimum noise performance. Bipolar shaping is much more convenient for user (important in larger systems!) Often the method of choice.

Fig. m-30 Unipolar pulse after second differentiator changes to bipolar pulse.	Typical unipoloar and bipolar output pulse shapes from a semi-Gaussian shaping amplifier.

Fig. m-31 Creating bipolar pulse on base of double delay lines, terminated with short circuits (from input side is coax terminated with resistor R0~Z0 and no reflection occurs). Duration of bipolar pulse is 2td ( td - wave propagation time across line)

Base-line restoration (BLR)

Any series capacitor in system prevents transmission of DC component. A sequence of unipolar pulses has a DC base line component that depends on duty factor, i.e. event rate.
 

Fig. m-32 Baseline shift: The baseline shifts to make overall transmitted charge zero. Fluctuations in baseline shift due to random rate lead to spectral broadening.

Baseline restorer (BLR) reduces baseline shift.
 
 

Fig. m-33 Basic principle of baseline restorer: Connect signal line to ground in absence of signal to establish baseline just to arrival of pulse.

BLR are originally performed with diodes (passive restorer), today BLR circuits tend to include active loops with adjustable thresholds to sense presence of signal ("gated" restorer) + asymmetric charge and discharge time constants for the "memory" capacitor.

Pile-up rejection

Pile-up causes false amplitude measurement.
 

	Fig. m-34 Two cases of pile -up false amplitude measurement: DT < time to peak  --> both amplitudes affected by superposition. In the case both pulses are rejected. Dead time: DT+inspect time (~pulse width);
	DT < time to peak and DT < inspection time, i.e. time where amplitude of first pulse << resolution  --> peak amplitude of first pulse is unaffected and  second pulse is rejected only. No additional dead time is necessary if first pulse is accepted for conversion and dead time of ADC>( DT+inspect time)

 

Fig. m-35 Basic waveforms in the pile-up rejector. Input pulse with rise time TP and duration TW; TF - duration of the fast discriminator output pulse; TINS - inspection interval for single pulse ; TINH - inhibit pulse accompanies of pile-up.

When 2 gamma-rays (according to figure above) arrive at the detector within the width of the spectroscopy amplifier output pulse a pulse pile-up to form an output pulse of distorted amplitude can occur.

The pile-up rejector is implemented by adding a "fast" pulse-shaping amplifier with very short shaping time constant in parallel with the "slow" spectroscopy amplifier.

Fast discriminator is set above the much higher noise level => digital logic pulses.

The trailing edge of fast output triggers an inspection interval T_INS pulse , that covers the width T_W of the slow pulse.

If second fast pulse arrives during T_INS an inhibit pulse is generated (prevents A/D analysis of the pile-up event) contribute to dead time of the spectrometry system:
 

TD =	TP+TW
TW -	width of the pulse above noise level,
TP -	time from start of the pulse until ADC detects the peak and closed its linear gate.

 

Fig. m-36 Demonstrations of Effectiveness of the Pile-Up Rejector in Suppressing the Pile-Up Spectrum with a Germanium Detector and a 60Co Spectrum at 50 000 count/s.

Noise

In almost every area of measurement the ultimate limit of detectability of weak signals is set by noice - unwanted signals that obscure the desired signal. Even if the quantity being measured is not weak, the presence of noise degrades the accuracy of the measurement.

Noise- random variation in the output current of electronics circuits and amplifiers. These fluctuations in current are primarily due to the inherent particle nature of electrical current, and to variations in path, recombination rates, and diffusion caused by the randomness of charge movement. While the statistical average of diffusion current may be constant, on an instant-by-instant basis the diffusion rate may vary. At very low signal levels the signal current may be comparable in magnitude to the variation produced by such noise.

The noise has no specific frequency. Each impact of charges with atoms produces a very short energy pulse, which represents a very broad band of frequencies. Being random in nature, the distribution has a statistical effective power P_n .(Usefully interpreted signal has on time interval t_j - t_i its mean value, different from zero, but mean value of noise is zero!) Measurement of noise must be made over specified frequency band on power or voltage-square basis.

The noise power can be stated as P_n/D f - effective power per Hz at a center frequency, or V_rms/D f - effective voltage per Hz at a center frequency .

In a multistage amplifier, the noise power is attenuated by circuits or stages of amplifier. Because additive contribution of noise from the second stage to all noise is less than contribution of the first stage, it is conventional to refer noise rms voltage to the input an amplifier.

In this conclusions we suppose a proper grounding and shielding and the elimination of interface and pickup. Then the noise is applied to anything that obscures a desired signal, the term describing "random" noise of physical (often thermal) origin. Noise can be characterized by its:

• frequency spectrum;
• amplitude distribution;
• physical mechanism responsible for its generation.

Resistor generates a noise voltage across its terminals. The thermal noise is developed by the random thermal motion and impacts of the charges and atoms in all conductors. Since is due to thermal energies, this noise is temperature dependent. It is also found to be uniformly distributed over all frequencies presently in use. (Noise whit a flat spectrum is called "white noise") The thermal noise due to a resistance R is:
 

Vn(rms)= VnR=(4kTRD f)0.5

The term S_nR=4kTR is called spectral density of noise and is often used by describing properties of this type of noise too. (T - absolute temperature in degrees Kelvin, k - Boltzmann's constant, Df is the bandwidth in hertz). In graphical mapping of noise is often used density units like V_n/(Hz)^0.5 (- V_n per root Hz), or V_n²/Hz (- volts squared per Hz).
 

Spectral density of the termal noise

 

Fig. m-37a Real "noisy" resistor R and his equivalent circuit (spectral density 4kT) with "noiseless" R and Voltage noise generator Vn=4kTR Current noise generator in=4kT/R

For example 10kW resistor has on open-circuits rms voltage of 1.3mV, measured with bandwidth Df=10kHz. A real resistor may be considered as equivalent thermal noise generator of rms voltage V_n (or source I_n ) in series (parallel) with noiseless resistor R.

The significance of thermal noise is that it sets a lower limit on the noise voltage in any detector or amplifier, having resistance. The noise power is dependent on the frequency band Df =>wide Df gives more noise => The bandwidth should be made as small as possible to get reduced termal noise.
 

Shot noise

An electric current is the flow of discrete electric charges. The finiteness of the charge quantum result in statistical fluctuation of the current. The noise is uniformly distributed over frequency spectrum. (white noise as "the rain in a tin roof"). The noise energy increases with the current in device, so collector currents are maintained at a few hundred mA for low noise application (I_C min) :
 

Inoise(rms)= In=(2e Idc D f)0.5

 

Spectral density of the shot noise e - electron charge 1,6. 10-19C; IDC= direct current in device, e.g. detector leakage Ib .

Flicker noise - 1/f noise

Shot noise and Johnson noise are irreducible forms of noise generating according to physical principles. Real devices have in addition to thermal and shot noise variation, various sources of "excess noise". This noise depends on many factors having to do with the construction of the particular resistor, including the resistive material (carbon-composition, metal film, wire-wound) and especially the end-cap connections. This noise has approximately a 1/f spectrum and is sometimes called  "pink noise".   => Base current noise in transistor is reduced to low level by suitable fabrication techniques. Flick noise seems largely confined bellow 1000Hz and varies inversely with frequency.
 

Voltage and current noise transistor amplifier 

It is conventional to refer noise voltages to the input of an amplifier (although the measurements are usually made at the output), i.e., to describe source noise and amplifier noise in terms of microvolts at the input that would generate the observed output noise. This makes sense when you want to think of the relative noise added by the amplifier to a given signal, indipendent of amplifier gain; it's  also realistic, because most of the amplifier noise is usually contributed by the input stage.
 

Fig. m-37b The  noise model of an amplifier: VS-is the noiseless signal source  RS-the source resistance

The noise generated by an amplifier is easily described by a simple noise model (fig. above in which :

• e_n - represents noise voltage source in series with the input, and
• i_n - input noise current ((in bipolar trasistor).

The two terms are the amplifier input noise voltage and the noise voltage generated by the amplifier's input noise current passing through the source resistance R_S .Since the two terms are usually uncorrelated , their squared amplitudes add to produce the effective noise voltage seen by the amplifier. For low value of resistance R_S noise voltage e_n dominates, whereas for high R_S the noise current generally dominates. The fact that e_n drops and i_n rises with increasing collector current provides a simply way to optimize transistor operating current to give lowest noise whit given source.

JFET noise voltage e_n is essentially thermal noise of channel resistance given approximately by
 

	en2 = 4kT(0.7/gm)
where - 1/gm	- inverse transconductance takes the place of resistance. (gm- characterizes the change in current per volt change in gate input => slope of the transfer curve).

 

For the JFET, where is the current noise component in very low  depends en2 first of all on the inverse transconductance of the transistor gm:

Analysis of practical system

According the figure bellow noise current and voltage sources can be described by their spectral densities, which depends on:

1. Detector bias shot current I_b (part (a) on fig. below) ;
2. Shunt resistance thermal noise (load resistor R of the detector - part (b) on fig. below);
3. Series resistance thermal noise ( for JFET R_S=0,7/g_m - part (c) on fig. below);

Fig. m-38 The detector  current signal source is on fig. represented with capacitor C only. The noiseless detector signal source has added to it an irreducible noise voltage from the Johnson noise of its source resistance and from noise current shot noise.

Equivalent input noise circuit in fig. up depends on current:

Then noise on the output of the equivalent voltage circuits, based on the input impedance Z (R parallel with C) is:

Fig. m-39

Then the equivalent noise charge:
 

consist of parts:

• The "parallel" noise component, proportional to shaping time constant t_i =t_d =t and independent of capacitance of the detector C.

• The "series" noise component, proportional to capacitance of the detector C and calculated proportional ~1/t .

• Maximal resistance R_p and minimal bias current I_b (-> for extremely small bias currrent is recomending to work at low temperatures).
• Minimal resistance R_S,- can be accomplished by FET with high g_m, or by connecting 2, 3 FET in parallel (individual selection FET according to minimal flicker noise component 1/f too).

True in general must be experimentally checked, e.g. for system with Si - barrier detector can be optimal t~0,5 - 1ms , and for germanium or Si(Li) detector t~6 - 20ms.

Next figure illustrates dependence parallel and series component to total noise of the preamplifier. Short time constant t is dominant for series component of the noise and long time constant again for parallel component (which depends on thermal R_p~R, thus on detector leakage I_b or gate leakage current of FET).
 

Fig. m-40 The dependence of preamplifier noise contribution on the amplifier shaping time constant.

For comparing purpose the system noise can be expressed:

• as "noise" charge Q_n in Coulombs ;
• in noise electrons N_n= Q_n /e, where e is electron charge 1,6. 10^-19C;
• or in energy units (eV, keV), e.g. FWHM (usually requires specification of detector material too).

Fig. m-41 Noise (expressed in FWHM keV) vs. input capacitance C of detector dependence for 2 type of preamplifiers.

Preamplifier

Fig. m-42 Application of preamplifier in the timing (output 1) and energy (output 2) measurement.

Preamplifier is located close to the detector and accepting signal from detector preserves max S/N ratio whit minimum shaping, or sometimes a fast rise time output pulse for good time resolution measuremnt too.
 

	Fig. m-43 Functional schematic of preamplifier with fast/slow output.
	Fig. m-44 Pulse response on energy / timing output.

The optimum configuration of preamplifier in system depends on characteristics of the detector and type of signal processing following after.

There are generally used 3 type of preamplifiers:

1. Voltage preamplifiers;
2. Current preamplifiers;
3. Charge sensitive preamplifiers.

1. Voltage - sensitive preamplifier:

Fig. m-45 Typical charge - sensitive preamplifier consists of 3 parts: Active integrator; P/Z cancellation circuit; Buffer with gain ~ 10, which performs well terminating to coax cable characteristic impedance.

Charge sensitive preamplifier integrates the charge Q_D on its feedback capacitor C_f . Its gain A_q~V_o/Q_D~1/C_F is not sensitive to variations of detector capacitance. The rise time of its output pulse is equal to the detector current pulse width in ideal case.

Voltage from preamplifier output V_o~Q_D/C_F. The decay time constant t_F=R_FC_F (C_F~0.1 to 5pF). Feedback resistor R_F is a noise source and is made as large as possible with the signal energy rate product and the detector leakage current in the direct-coupled system.

Fig. m-46 Typical shape of output pulse from charge - sensitive preamplifier: Pulse rise time ~ charge collection time in detector; Pulse height e.g. 10-50 mV/MeV.	Fig. m-47 The same pulse output in other time scale (at middle counting rate).

Sensitivity (usually is expressed in mV/MeV ) V_o/E =e/(eC_F) to energy E deposited in given detector material. The charge released by the detector is a function of the gamma - ray or particle energy and the detector material Q_D=eE/e (e - the amount of energy required to produce e.g.  an electron-hole par in the detector.)

Preamplifier noise depends on FET, resistance to the input, input leakage current from detector and FET and total capacitance at the input (C_F, C_D).

The noise can be measured in the system shown on next figure.
 

Fig. m-48 System for Measurement Charge Sensitive Preamplifier Noise

• At first system gain V_p/E must be determined. Charge Q, equivalent to the known energy E, is injected into preamplifier by detector radiation source or step pulse generator connected through a capacitor (V_p - pulse amplitude from the charge Q).

• Then noise can be determined by measuring out the voltage V_rms at the output of the filter amplifier:

Linear pulse-shaping amplifier

Linear pulse-shaping amplifier is special type of wide-band amplifier, which is used with scintillation, silicon charge particle detectors for energy spectroscopy measurement.

• Primary purpose to magnify the amplitude of the preamplifier output pulse from the mV range into the 0.1 - 10V range (ADC level).
• Shapes the pulses to optimize the energy resolution.
• Minimize the risk of overlap between successive pulses.
• Baseline ensure that the baseline between pulses is held rigidly at ground potential in spite of changes in counting rates or temperature

For detector with very short charge collection times, the rise time of the preamplifier output pulse is controlled by the amplifier itself, and the rise time is usually in the range from 10ns to 100ns.

For detector with long charge collection times, such as NaI(Tl) detectors, proportional counters, coax Ge - detectors, the output rise time of the preamplifier is controlled by the detector charge collection time. The output rise time can range up to 700ns, for large Ge - detectors, and ~500ns for NaI(Tl) detectors.
 

Conflict with the need of handling high counting rates and optimum energy resolution

The fast amplifiers – tries for minimal degradation of rise time => good time resolution.

The slow or wide-band linear amplifiers, used with scintillation, silicon charge particle detectors for energy spectroscopy purpose apart from magnify the amplitude of the preamplifier pulses are assign to :

1. Shape the pulses for optimizing S/N ratio to optimize the energy resolution:

• CR - RC pulse shaping.
• Semi-Gaussian pulse shaping.

Fig. m-49 Typical CR - RC pulse shaping in linear amplifier: Preamplifier output pulse shape; After CR – shaping; After CR-RC – shaping (unipolar output); After RC – double CR-shaping (bipolar output).

1. Minimize the risk of overlap between successive pulses by:

• Pole-zero cancellation (has a sense in the first stages of amplification, e.g. in preamplifier).
• Unipolar versus bipolar shaping (by height counting rates is bipolar shaping performed)
• Base-line restoration (is performed in the last stages of amplification).
• Pile-up rejection.

Fig. m-50 Illustration of baseline shift vs. counting rates. Simple diodes BLR or active fixation BLR can be used to minimize fluctuations in baseline shift due to random rate.

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