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

Transcription

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

2 Optimal Operating Voltage: Procedure [This section requires the GM tube from the Tel-X-Ometer X-Ray apparatus. If another group is currently using this detector you will need to negotiate with them for its use. You only need it for 30 minutes or less and you can do this part of the experiment at any time (so go on to the next part if the X-ray GM tube is not currently available).] Using the Geiger tube from the Tel-X-Ometer, with the voltage set at about 400 V, find the most active source among those in the red box in the lead brick enclosure. Simply set the Geiger counter on the table and balance a source right up against the end of it. You will need to determine which side of the source is the most active. Once you ve identified the most active source, use that to measure counting rate versus voltage. Take readings across the full range of voltages available. The counting rates will vary significantly, so watch the number of counts for at least 30 seconds and record what seems to be the average. In particular, you will want to take careful reading around the "turn-on" voltage. Estimate an uncertainty for each reading. Graph counting rate vs. voltage. On your graph, identify the regions from those described in the Introduction. Are all regions present? If not, why do you think they might not be represented? Given your data, identify the optimum operating voltage range for this GM detector. Dead Time: Theory and Procedure As noted above, once a discharge has been initiated in the GM tube a new pulse will not be detected until the previous discharge has extinguished itself. Thus there is a dead time τ associated with each counting event. If we measure a counting rate r in a time interval Δt, the total detector dead time will be T dead = (rδt)τ. Thus the true counting rate R and the measured counting rate r are related by R(Δt T dead ) = r Δt so if we know τ we can correct our measured count rates as follows r R = 1 rτ. (1) We can determine τ by measuring the individual and combined counting rates from two high flux samples. If we call these rate r 1, r, and r c we have the following three equations: r 1 = R 1 (1 r 1 τ) (a) r = R (1 r τ) (b) r c = (R 1 +R )(1 r c τ) (c) with the three unknowns R 1, R, and τ. Solve this set of equations for τ and show that the dead time is approximately given by: τ r 1 + r r c r 1 r. (3) Here we will make such measurements using our two 5µC 137 Cs sources (located in the clear plastic box) and DataStudio to determine the dead time for the PASCO Geiger counter. (Note

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

5 FIGURE 1. Gaussian fit to counting frequency (using a 10 cps bin width) for 60 Co. A plot of experimental counting data, binned using ΔA = 10cps, and the fit ƒ(a) = M ΔA P Gaussian (A) are shown in Fig. 1. The data in Fig. 1 are compiled from M=1,04 consecutive measurements of the number of detected gammas per second emitted by a 60 Co source. The frequency of the measured counting rates is well represented by the Gaussian distribution curve of Eq (8) with A = 7,540. Note that only the sample average and the total number of samples are necessary to calculate the distribution curve. In principle, Eq. (4) could also be used to model the data of Fig. 1, however, application of Eq. (4) is extremely cumbersome when dealing with large numbers. In an actual experiment, there is always some background radiation present. This background is mostly due to cosmic radiation reaching the earth, but is also composed of radiation from very small amounts of radioactive material present in the walls, floor, and tables of the laboratory. If the intensity of radiation from the radioactive material being used is very large compared to the background, then the background may be ignored. If, however, a weak radiation source is being used, it is important to subtract the background in order to determine the decay rate of the radioactive material itself. In this experiment, correction for the background will not be necessary. However, the background radiation will be used as a very low intensity source of radiation, hence the Poisson distribution will best approximate the data. In this experiment on statistics of nuclear counting, the rate at which radiation reaches a detector from a long lived radioactive source is determined by measuring the number of events occurring in the detector during a specified time interval (we will use 10s). Many measurements are made and the average is calculated from the values obtained. Even though each value represents the measurement of the same quantity, the values will be different. The cause for the differences is the statistical fluctuation in the amount of radiation reaching the detector. A careful study is then 5

6 made of the fluctuations, and the precision of the measurement is determined. A similar procedure is followed for background radiation, and the results of the two studies are compared. Procedure: Collect count rate data for the 5µC 137 Cs and the 1µC Na sources using the PASCO Geiger tube and Data Studio. Acquire a series of at least 00 measurements (more is better) of the number of counts using a time interval of ten seconds. Place each of the sources back in the lead brick housing when you are done with them. With no sources out, take a series of at least 00 measurements (more is better) of the number of room background counts using the same time interval. In the same manner, take a "shielded background" count rate by using lead bricks to shield the top of the detector (but do not set the bricks on the detector!). Data Analysis: For each of the four data sets, use the DataStudio statistics functions to determine the sample average A and standard deviation σ. In each case compare the actual standard deviation with the approximation σ A. Do you find agreement? Do you expect to? Copy the Data Studio data to Excel or KaleidaGraph and construct a frequency distribution curve for each data set. You will need to bin your data to create these plots. You have to do this manually in Excel. Thus, it is recommended that you use KG where data can be automatically binned using the "bin data" function. For each data set, determine the fraction of data points that are within the range A ± σ. Make similar determinations for the fraction of measurements within the ranges A ± σ and A ± 3σ. Numerically integrate the Gaussian probability function (Eq. (6); let µ = 0) to determine what fraction of normally distributed data is expected to fall within each of the above ranges. Do your data sets appear to be normally distributed? Compare each experimental frequency graph with a plot of the theoretical Gaussian (for all data sets) and Poisson (for background data only) distributions by plotting these distributions on the graph along with your data (as done in Fig. 1). You can construct these functions in KG using the "Formula Entry" window. The Gaussian distribution should be symmetric about A while the Poisson distribution should be skewed slightly toward the lower counts. How well is your data described by these distribution functions? Questions: 1. Do any of the counts in this part (i.e., Part II) of the lab need to be corrected for coincidence (i.e., dead-time) losses? Why or why not?. Are the half-lives of 137 Cs and Na "long" compared to your counting times? Could you have carried out the same type of analysis using 116 In as the source? 6