Pulse count distribution

In Fig. 2 we show a quasi-sinusoidal LED output as measured from the unattenuated signal on the oscilloscope, and the pulse count distributions obtained with ND1 and ND3 filters (average counts per 400 ysec channel of 100 and 1, respectively). Also shown in this figure are two typical PDF fits to the latter data (average count per cycle = 1), compared to the PDF calculated from the LED signal variation displayed on the oscilloscope. 

Distribution Alternate Protocol I Electrical Pulse Counting Determination of Droplet D3.3.1... 

ELECTRICAL PULSE COUNTING DETERMINATION OF DROPLET SIZE DISTRIBUTION... 

Figure D3.3.2 Schematic diagram of electrical pulse counting instrument for determining droplet size distribution.

The major disadvantage of the laser diffraction and electrical pulse counting techniques is that they are only directly applicable to dilute emulsions or emulsions that can be diluted without disturbing the particle size distribution. However, many food emulsions are not dilute and cannot be diluted, either because dilution alters the particle size distribution or because the original sample is partially solid. For concentrated systems it is belter to use particle-sizing instruments based on alternative technologies, such as ultrasonic spectrometry or NMR (Dickinson and McClements, 1996). 

Table 1. Basic Quantities in Analyses of CW Laser Scattering for Probability Density Function. In Eq. 1 within the table, F(J) is the photon count distribution obtained over a large number of consecutive short periods. For example, F(3) expresses the fraction of periods during which three photons are detected. The PDF, P(x), characterizes the statistical behavior of a fluctuating concentration. Eq. 1 describes the relationship between Fj and P(x) provided that the effects of dead time and detector imperfections such as multiple pulsing can be neglected. In order to simplify notation, the concentration is expressed in terms of the equivalent average number of counts per period, x. The normalized factorial moments and zero moments of the PDF can be shown to be equal by substitution of Eq.l into Eq.2. The relationship between central and zero moments is established by expansion of (x-a)m in Eq.(4). The trial PDF [Eq.(5)] is composed of a sum of k discrete concentration components of amplitude Ak at density xk. [The functions 5 (x-xk) are delta functions.]...

Because of the high speed of the biased amplifier and the ADC in a TCSPC board, the photon pulses delivered to the ADC need not be broader than 50 to 100 ns. Therefore an extremely high preamplifier gain is not required, and AC coupling can be avoided. The setup can therefore be used up to a count rate of several 10 pulses per second. Examples for pulse height distributions recorded this way are shown in Fig. 6.13, page 227. 

The pulse height distribution of the dark pulses is almost the same as for the photon pulses. Increasing the threshold reduces the dark counts and the signal counts by the same ratio. Please note that this applies only to dark pulses originating from the detector itself. Counts caused by electrical noise pickup can, of course, be suppressed. However, the better solution is correct shielding of the detector. 

This statement can be reversed, i.e., if the waiting time between pulses has a y) , v) distribution, then the number X(f) of the pulses counted over the period tis a random variable with a il(vt) Poisson distribution. Since the time is an explicit parameter here, the word process appears to be an appropriate expression. As a matter of fact, the function X t) is referred to as Poisson process in the theory of stochastic processes. 

The photomultiplier, as shown in Fig. 6, is almost universally used as a photon counter, that is, the internal electron multiplication produces an output electrical pulse whose voltage is large compared with the output electric circuit noise. Each pulse in turn is the result of an individual photoexcited electron. The numbered electrodes, 1-8, called dynodes, are each successively biased about 100 V positive with respect to the preceding electrode, and an accelerated electron typically produces about 5 secondary electrons as it impacts the dynode. The final current pulse collected at the output electrode, the anode, would in this case contain 5 400,000 electrons. The secondary emission multiplication process is random, the value of the dynode multiplication factor is close to Poisson distributed from electron to electron. The output pulse amplitude thus fluctuates. For a secondary emission ratio of = 5, the rms fractional pulse height fluctuation is 1 /V<5 — 1 = 0.5. Since the mean pulse height can be well above the output circuit noise, the threshold for a pulse count may be... 

Note that, since the pulse height Vjn is proportional to the detector gain (photomultiplier and channeltron alike), the setting of G will determine the fraction of the pulse height distribution that will exceed the threshold voltage, and hence the measured count rate. Thus, for example, for Vin to surpass typical discriminator thresholds in practice either the gain needs to be increased, or a xlO preamplifier is required (for the cited numerical example of a 2-20 mV signal). 

EM Noise Although Electron Multipliers are a type of pulse counting device, the signal measured from many pulses (that resulting from many secondary ion impacts per second) displays a distribution in current (the number of electrons produced per ion impact). This distribution arises from the statistical nature of the ion to electron conversion process, as well as the processes responsible for the additional electrons formed in subsequent electron-surface collisions. 

Figure 4.18 Schematic example of Pulse Height Distribution analysis carried out for 4500 eV secondary ions impinging on an ETP Discrete Dynode Electron Multiplier operated at the listed voltages. Pulse counting was carried out using custom built ECL logic pre-amplifier/discriminator units. Discriminator voltage in this case should be set at 5 mV. Reproduced with permission from van der Heide and Fichter (1998) Copyright 1998 John Wiley and Sons.

Electron speetro.scopic data acquired using pulse counting should have a count rate which shows a Poi.ssonian distribution. For the count rates typically-found in practical electron spectroscopy, this approximates well to a Gaussian distribution, The following discussion assumes a Gaussian distribution of random noise in the spectrum. 

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