Experiment 1. Statistics in counting experiments

Experiment 1. STATISTICS IN COUNTING EXPERIMENTS

Emissions from a long-lived radioactive source are thought to occur randomly in time. The distribution of time intervals between these emissions is then described by the Poisson distribution. In this experiment, you will gain a thorough understanding of the statistics in counting experiments in which events occur randomly in time. This knowledge is basic to interpretation of many of the experiments you will carry out later in this course.

Equipment: Geiger detector, scintillation detector, time-interval measuremenr (period measurement and scaling)..

Readings: Radiation and detectors (general), sect. 5.1; Interaction of charged particles with matter, 5.2.2; Geiger detectors, sect. 5.3.1 and 5.3.4; Counting statistics, sect. 10.5 and and 5.3.5. [Unless noted otherwise, all readings are from A. Melissinos, Experiments in Modern Physics,(Academic Press, New York 1966)]; Data Analysis for Physical Scientists, Louis Lyons (Cambridge, 1991), chapter 1.

Key concepts: Random processes, empirical frequency distribution functions, Poisson and Gaussian distributions; Geiger tube operation (plateau, continuous discharge region, operating point, dead time).

1.1 Counts detected in a fixed time interval

Instrumentation here includes a scintillation detector (Ludlum) and frequency counter. The scintillation detector generates an output pulse for each pulse that is detected above a certain threshold voltage, with no information about the energy of the pulse. The active element of the counter is a 1-inch by 1-inch NaI crystal. The frequency counter acts as a pulse counter. For each of parts 1 and 2 below, measure in 100 trials the number of radiations detected in a given time interval by the scintillation counter. You may either count cosmic and background radiation without a radioacive source nearby or also radiation from a beta source placed near the counter.

1.1.1 Small average number of counts

Select a time interval so that the mean number of counts is very close to 2, using lead shielding if necessary. Make a frequency histogram for 100 trials, and compare your normalized histogram against the hypothesis that the cosmic-ray flux is time-independent, on average, with the same average counting rate, using Poisson statistics. Compare the standard deviations of experimental and theoretical distributions having the same mean.

1.1.2 Large average number of counts

Select a time interval and arrange shielding so that the averate number of counts is close to 100. Make a frequency histogram for 100 trials and compare your normalized distribution to the Gaussian approximation of the Poisson distribution having the same mean counting rate. This approximation is excellent when the mean number of counts in the interval is large (greater than about 5-10). Compare standard deviations of experimental and theoretical distributions having the same means.

1.2 Distributions of time intervals between counts

Instrumentation here includes a Geiger detector and time interval counter run in 'period' mode. The Geiger detector has an active volume of about 30 cm³and is most sensitive to impinging charged particles. About 1 cosmic ray (mostly muons) will be detected per second for that volume. In period mode, a counter starts a clock when a first count is detected and counts periods of an oscillator until a second pulse is detected. The displayed count is a measure of the time interval between successive pulses.

1.2.1 Between successive counts

Make 100 measurements of time intervals between successive pulses using the Geiger detector. Histogram the distribution of time intervals using an appropriate bin-width, and compare your distribution with the prediction for a random series of pulses that are Poisson distributed and having the same mean counting rate.

1.2.2 Between every other count

A divide-by-n circuit may be available that can be connected to the output of the Geiger counter. Depending on the switch setting, only the 2nd, 4th or 8th pulse following an initial pulse will generate an output. One can thus measure the distribution of time-intervals between a pulse and the 2nd, 4th or 8th subsequent pulse from the circuit output. Make 100 measurements as in part 1.2.1 for time intervals to the 2nd pulse. Compare experimental and theoretical distribution functions for the same average counting rate.

If you have time and interest, you may do the same for the distributions of intervals to the 4th or 8th pulses. In the absence of a divide-by-n circuit, you may instead sum pairs of intervals between subsequent pulses to get a representation of the distribution of time intervals until a second pulse is detected. Compare experimental and theoretical frequency histograms.

Questions and Considerations

a. Know how a Geiger detector works. If it is possible to vary the operating voltage of your tector, determine a safe operating point by measuring the counting rate as a function of the applied high voltage. Never allow the tube to operate in the continuous discharge region, which will destroy the tube!

b. For what kinds of radiation is the Geiger detector most sensitive? Why?

c. Can a Geiger detector be used to measure the energy of incident radiation? Explain.

d. The Geiger tube has a dead time of ~300 microseconds following each pulse. Can you think of a way to measure the dead time? Do you need to take the dead time into account in or any of the measurements in this experiment?

Diagram of Apparatus