# Nuclear Physics Lab I: Geiger-Müller Counter and Nuclear Counting Statistics

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1 Nuclear Physics Lab I: Geiger-Müller Counter and Nuclear Counting Statistics PART I Geiger Tube: Optimal Operating Voltage and Resolving Time Objective: To become acquainted with the operation and characteristics of the Geiger-Müller (GM) counter. To determine the best operating voltage and the resolving time of a Geiger counter. The resolving or dead time is used to correct for coincidence losses in the counter. Experimental Apparatus: A typical Geiger-Müller counter consists of a cylindrical gas filled tube, a high voltage supply, a counter and timer. A large potential difference is applied between the tube body which acts as a cathode (negative potential) and a wire down the tube axis which acts as an anode (positive potential). The sensitivity of the instrument is such that any particle capable of ionizing a single gas molecule in the GM tube (thus producing an electron-ion pair) will initiate a discharge in the tube. What happens next depends on the voltage across the gas-filled tube. For the lowest applied voltages, only the ions created by direct interaction with the incoming radiation are collected. In this mode, the detector is called an ion chamber. For higher voltages, the ions created are accelerated by the potential difference gaining sufficient energy to create more ion pairs. This results in a localized avalanche of ions reaching the wire. This is the proportional region. The pulse height (or voltage of the signal) is proportional to the number of initial ion pairs created by the incoming radiation. This in turn is proportional to the energy of the incoming radiation. For even higher voltages, the new ions can create additional photons which move out of the local region and further down the tube; essentially the discharge propagates an avalanche of ionization throughout the entire tube, which results in a voltage pulse--typically a volt in amplitude. Since the discharge is an avalanche and not a pulse proportional to the energy deposited, the output pulse amplitude is independent of the energy of the initiating particle and, therefore, gives no information as to the nature of the particle. This is the Geiger-Müller region. In spite of the fact that the GM counter is not a proportional device, it is an extremely versatile instrument in that it may be used for counting alpha particles, beta particles, and gamma rays. Such a large output signal obviates the need for more than a single stage of amplification in the associated electronic counter. Geiger-Mueller tubes exhibit dead time effects due to the recombination time of the internal gas ions after the occurrence of an ionizing event. The actual dead time depends on several factors including the active volume and shape of the detector and can range from a few microseconds for miniature tubes, to over 1000 microseconds for large volume devices. When making absolute measurements it is important to compensate for dead time losses at higher counting rates. Please keep all sources in the lead brick house. Take out only the one you need, and return it as soon as you are done taking a measurement. 1

3 that the operating voltage for this detector cannot be adjusted). Set up the DataStudio for a 30s counting interval. Devise a way to place the 137 Cs sources so that when both are in place they touch one another, are positioned midway between the ends of the tube, and so that each source can be removed then replaced in exactly the same position. With only one cesium radioactive source in place, take a five-minute count (10 measurements). The count rate should be in the range 10,000-0,000 counts per 30s. Record the count as r 1. Place the second source beside the first (being careful not to disturb the first) and take a fiveminute count of the combined sources. Record this count as r c. Now remove the first source and take a third five-minute count. Record this as r. Repeat with the source positions reversed because these sources are not of equal strength. (Note: If the count rate exceeds 65,000 DataStudio will "reset" its counter). Calculate the dead time τ of the PASCO GM detector for both arrangements using Eq. (3). Now that we know τ for the PASCO GM tube we can use Eq. (1) to correct any counting rates measured with this detector. Apply such a correction (if necessary) for data taken in the next section. PART II Statistics of Nuclear Counting* *[Portions of the Theoretical Background are taken from Experimental γ -Ray Spectroscopy and Investigations of Environmental Radioactivity Experiment 9 by Randolph S. Peterson, Spectrum Techniques, Inc.] Objective: To study the statistical fluctuations which occur in the disintegration rate of an essentially constant radioactive source (one whose half-life is very long compared to the time duration of the experiment). Theoretical Background: We can never know the true value of something through measurement. If we make a large number of measurements under (nearly) identical conditions, then we believe this sample s average to be near the true value. Sometimes the underlying statistics of the randomness in the measurements allows us to express how far our sample average is likely to be from the real value. Such is the situation with radioactive decay, with its probability for decay, λ, that is the same for identical atoms. Radioactive materials disintegrate in a completely random manner. There exists for any radioactive substance a certain probability that any particular nucleus will emit radiation within a given time interval. This probability is the same for all nuclei of the same type and is characteristic of that type of nucleus. There is no way to predict the time at which an individual nucleus will decay. However, when a large number of disintegrations take place, there is a definite average decay rate 3

4 which is characteristic of the particular nuclear type. Measurements of the decay rate taken over small time intervals will yield values which fluctuate randomly about the average value and consequently which follow the laws of statistics. Hence in dealing with data from measurements of radioactivity, the results of the laws of statistics must be applied. Given that λt is the probability of decay for a single nucleus in time interval t (and thus 1 λt is the probability for non-decay), the probability P(n,t) of n nuclei decaying in time t from a sample of N identical atoms is given exactly by the binomial distribution P(n,t) = N! n!(n n)! (λt)n (1 λt) N n. (4) The mean and variance of this distribution are µ = Np and σ = Np(1 p), respectively, where p = λt. If λt is small and N is large such that µ = λtn remains small, this binomial distribution can be approximated by the Poisson distribution P(n,t) = µn n! e µ where µ = λnt is the average number of decays in time interval t. (5) If λt is small and N is large such that µ = λtn is not small (perhaps greater than 100), the binomial distribution can be approximated by the normal (Gaussian) distribution function, P(n,t) = (n µ) 1 πσ e σ. (6) where σ µ is the square of the standard deviation, and gives a measure of the width of the distribution. Experimentally we measure a sources activity or count rate. We expect a large number of independent measurements to be described by the above probability functions where we approximate µ with our sample average A and the standard deviation σ with the square root of A. Thus, given a large number of measurements M of a source's activity, A, the frequency ƒ(a) = MP(A)ΔA with which we measure A (per interval ΔA) is expected to follow Poisson statistics if A is small: P Poisson (A) = A A (7) A! e A and Gaussian statistics if A is large: P Gaussian (A) = 1 ) A. π A (8) Note that ƒ(a) is the number of times our measurement falls in the range A A+ΔA. 4

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