The beam delivered by by the accelerator comes in "bunches" which are spaced according to the RF frequency, 53 MHz. Thus each bunch is said to occupy an RF "bucket" of about 19 nanosecond duration. Ideally the beam to E831 is extracted uniformly from the accelerator throughout the (approximately) 20 sec duration of the slow spill. Spill structure (i.e. deviations from uniformity) are important to E831 because RF buckets which contain too many beam particles have higher probabilities of producing "fake" triggers which can dominate our data taking at high beam intensity. They also cause problems for detectors such as the PWC's and Straws by causing large instantaneous current draws.

- Trigger Dependence on Bucket Multiplicity
- Calculating the Hadronic Fraction
- Estimating Bucket Occupancy
- Accelerator Duty Factor
- Effect of Super Buckets
- Monitoring the Spill

By comparing the beam particle multiplicity distribution obtained from RF buckets which generated an event trigger with that from an unbiased sample of RF buckets from the same spill, one can determine the triggering probabilty's dependence on beam particle multiplicity. For good triggers one expects a linear dependence. In practice one requires an extra quadratic term to decribe the observed distributions. This can be expressed algebraically as follows :-

Pt = K*P(n)*[n + 0.3*n*(n-1)] where: Pt = Probabilty that an RF bucket forms a trigger P(n) = Probability that the RF bucket contains n beam particles K = Normalization constantThe coefficient, 0.3, was determined for the standard E831 trigger set using the tagging multiplicity distributions as described above. It will of course vary with the choice of trigger set.

One can use the above equation to determine the expected Hadronic Fraction under various conditions. To start with we will assume that all RF buckets in a spill are populated according to a Poisson distribution and that all good (hadronic) triggers are generated by a single beam particle. The hadronic fraction, HF, becomes:-

HF = [Hadronic triggers]/[All triggers] = F*avg{P(n)*n}/[ avg{P(n)*n} + 0.3*avg{P(n)*n*(n-1)}]where F is the fraction of single particle triggers that produce good hadronic events. If P(n) is a poisson distribution with mean, m, then we obtain :-

HF = F*m/[m + 0.3*m*m] = F/[1 + 0.3*m]

The expected value for m for a given spill can be determined by dividing the total number of beam particles by the total number of RF buckets in a spill. For E831 the secondary beam yield is roughly 1.5E-4 times the value of PB3SEM. If all available RF buckets were uniformly filled then a 23 sec spill would contain 1.22E9 RF buckets. Thus a reasonable approximation for m is:-

m =~ PB3SEM*1.2E-13

In practice however, not all the available RF buckets are occupied in a given spill. The fraction of buckets occupied is known as the accelerator Duty Factor, DF, and can be measured using a reference beam counter as follows:-

DF = R*R/(N*C) where: R = No. of counts recorded by the reference counter in the spill N = No. of RF buckets in the spill C = No. of coincidences of the reference counter with itself after some fixed delay (typically 10 RF buckets).If all N buckets have the same occupancy probabilty then C/R should equal R/N and DF would be 1.0. However in practice a 100% duty factor is never achieved. For a start the RF monitor is active between T5 and T6 which is

To first order the effect of a low duty factor is to increase the average bucket occupancy to m/DF. Thus the Hadronic Fraction becomes :-

HF = F/[1 + 0.3*m/DF]

Typically, the Tevatron has a duty factor of about 65% which means that for
a spill with PB3SEM = 4.0E12, and assuming F is about 75% (a wild guess), we
discover that one should expect a hadronic fraction of about 60%. This is
somewhat higher than what we normaly see. The reason for the discrepancy is
that the micro-structure of the spill is not very uniform and hence the bucket
occupancy is not well described by a Poisson distribution. The only place that
this really affects our calculations is in the estimation of the expectation
value of P(n)*n*(n-1). For a poisson distribution this is simply m*m, however,
for spill containing a disproportionately large number of high multiplicity
buckets (**super buckets**) it can be much larger.

The loss of events due to non-Poisson fluctuations in bucket multiplicity can be expressed in terms of the spill deadtime, the measured hadronic fraction, and the optimal hadronic fraction one would obtain if the spill were purely Poisson. If the reduction factor, R, is defined as the number of good triggers obtained in a spill divided by the number that could have been obtained if the spill had been Poisson, and we assume that the deadtime per event is the same for both good and bad triggers then it can be shown that:-

R = 1 - (HF'-HF)*DT/HF' where: HF'= Expected Hadronic Fraction for a "good" spill. HF = Measured Hadronic Fraction for the "bad" spill. DT = Fractional deadtime for the "bad" spill.Since the number of good events in a spill is given by the product of the number of ungated triggers (UT), the fractional livetime (LT), and the hadronic fraction (HF), then R can also be written as:-

R = (UT*LT*HF)/(UT'*LT'*HF') = LT/LT' where: LT = Livetime for the "bad" spill. LT'= Livetime for a "good" spill with the same beam intensity.

In order to maximize our data taking rate we need to keep the hadronic fraction as high as possible so as to reduce the deadtime generated while reading out uniteresting events. One way of doing this is to monitor the spill structure. We do this in a number of ways:-

- Macro Spill Structure: The large time scale structure of the spill is monitired by displaying the current draw in the PWC's and Straw Chambers on an oscilliscope. The current draw is proportional to the rate at which beam is extracted. Spikes in these plots translate into a corresponding increase in m for that part of the spill and hence a decrease in hadronic fraction.
- Micro Spill Structure: The micro structure of the beam is determined by the distribution of the beam within the Tevatron and how evenly it is extracted. It take a proton roughly 21 microseconds to complete a circuit in the Tevatron. This means that if the Tevatron is not evenly filled or is not being evenly extracted then the resulting spill structure will tend to have a frequency of about 50kHz. This is characteristic of the so called "super buckets".
- Duty Factor: The beam duty factor is monitored by a special CAMAC module built by the RD/EED department. This module takes the RF signal, T5, T6, and the signal from any beam counter to calculate and display the beam duty factor as a percentage.